Impossibility of physical contact
Michael Hand
but owes much to the ancients. (I hope I'm not repeating something
that's already been here. :-] )
To show: Physical contact (without overlap) is impossible.
(i) If a physical body maximally occupies a closed space, then it occupies
its boundary. Contact of two bodies cannot consist of the two boundaries
occupying the same point(s), for then the bodies OVERLAP at the point(s).
But points cannot be *adjacent*, so if the boundaries do not share point(s),
then there is a distance between them and contact is absent.
(ii) But if a body maximally occupies an open space, it doesn't occupy
its boundary, so even if the boundaries share points, the bodies don't --
the boundaries separate them!
Q.E.D.
Whoa!
B-o} Michael
(F-ups to sci.philosophy.tech is prolly best.)
Russell Turpin
Does anyone in this day and age really place any importance on
this argument?
The underlying assumption of the argument is that physical bodies
are well modeled by either (1) characteristic functions on open
sets, or (2) characteristic functions on closed sets, all in
Euclidean space. Modern physicists have done work on modeling
friction, transmission of force, diffusion, and other surface
interactions. The underlying models are all more complex than
the above assumption allows. Indeed, basic quantum mechanics
excludes the kind of characteristic function assumed by the
argument. It seems to me the clear response to the argument is a
rejection of its assumed model of physical bodies.
Russell
Paul Callahan
>To show: Physical contact (without overlap) is impossible.
> (i) If a physical body maximally occupies a closed space, then it occupies
>its boundary. Contact of two bodies cannot consist of the two boundaries
>occupying the same point(s), for then the bodies OVERLAP at the point(s).
>But points cannot be *adjacent*, so if the boundaries do not share point(s),
>then there is a distance between them and contact is absent.
> (ii) But if a body maximally occupies an open space, it doesn't occupy
>its boundary, so even if the boundaries share points, the bodies don't --
>the boundaries separate them!
> Q.E.D.
It seems to me that all this is saying is that a contiguous interval on the
real number line (or any higher dimensional space) cannot be a disjoint union
of either (i) two closed intervals or (ii) two open intervals. On the other
hand, it can be a disjoint union of an open interval and a closed interval.
E.g. [1,2) and [2,3] do not overlap, and their union is [1,3]. It seems to me
that this example fits the criteria for two "bodies" that "touch" without
overlapping. In any case, I hope the cited paragraph isn't considered a recent
breakthrough in philosophy, because it's fairly old mathematics.
Applying the given argument to physical bodies makes the assumption that
for any two points in space, there is a point in between. I consider this
to be a mathematical idealization that makes it easier to think about space,
rather than a true description of space itself. If space were quantized into
a discrete array of cells, for example, the argument would not hold up. I have
no idea whether such a notion has any physical validity, but in any case, the
issue cannot be resolved without reference to a specific set of assumptions
about the nature of physical space.
--
Paul Callahan
call...@cs.jhu.edu
Neil Rickert
>In article <16...@helios.TAMU.EDU> e34...@tamuts.tamu.edu (Michael Hand) writes:
>>To show: Physical contact (without overlap) is impossible.
>It seems to me that all this is saying is that a contiguous interval on the
>real number line (or any higher dimensional space) cannot be a disjoint union
>of either (i) two closed intervals or (ii) two open intervals. On the other
Nah. This was merely a silly statement based in different possible meanings
of 'physical contact'. Mere semantic disputes like this have no relevance to
mathematics. Whether they have any relevance to philosophy I will leave to
the philosophers.
--
=*=*=*=*=*=*=*=*=*=*=*=*=*=*=*=*=*=*=*=*=*=*=*=*=*=*=*=*=*=*=*=*=*=*=*=*=*=*=
Neil W. Rickert, Computer Science <ric...@cs.niu.edu>
Northern Illinois Univ.
DeKalb, IL 60115 +1-815-753-6940
David M Tate
>>To show: Physical contact (without overlap) is impossible.
[assumptions and argument deleted]
>It seems to me that all this is saying is that a contiguous interval on the
>real number line (or any higher dimensional space) cannot be a disjoint union
>of either (i) two closed intervals or (ii) two open intervals. On the other
>hand, it can be a disjoint union of an open interval and a closed interval.
>E.g. [1,2) and [2,3] do not overlap, and their union is [1,3]. It seems to me
>that this example fits the criteria for two "bodies" that "touch" without
>overlapping.
Yes, but the first assumption in the original statement was an axiom which
was basically equivalent to "every physical body is a closed, connected set
of points". Thus, as you say, there can be no union of two bodies which is
also closed and connected.
>Applying the given argument to physical bodies makes the assumption that
>for any two points in space, there is a point in between. I consider this
>to be a mathematical idealization that makes it easier to think about space,
>rather than a true description of space itself.
You mean, the real number system (and mathematics in general) might be just
a *model* of reality, which is possibly inaccurate in nontrivial ways?!?!?
:-)
(When I make statements like that in this group, people usually tell me I'm
no longer talking about mathematics, and that I should go elsewhere...)
--
David M. Tate | "What is this person's name?"
dt...@unix.cis.pitt.edu | "This person is dead. I do not know whether
Motto: | dead people make use of names."
Gramen artificiosum odi | --Jack Vance, _Araminta_Station_
David M Tate
>-----
>Does anyone in this day and age really place any importance on
>this argument?
I'm not sure this is the right question. If you ask instead "Is there
anyone in this day and age who would benefit from considering this argument",
then I think the answer is "yes".
>The underlying assumption of the argument is that physical bodies
>are well modeled by either (1) characteristic functions on open
>sets, or (2) characteristic functions on closed sets, all in
>Euclidean space. Modern physicists have done work on modeling
>friction, transmission of force, diffusion, and other surface
>interactions. The underlying models are all more complex than
>the above assumption allows.
Exactly. Unfortunately, the Exclusive and Illuminated Guild of Pure
Mathematicians haven't caught on yet. They are still (as a group; I don't
mean to overgeneralize here) either ignorant or apathetic about this failure
of the traditional mathematical models. Or, to put it another way, they are
more interested in pursuing the structural quirks of the old, inaccurate
models than they are in developing new, accurate models and studying *them*.
That's why the work you mention above has been done by physicists and other
"applied" scientists.
>It seems to me the clear response to the argument is a
>rejection of its assumed model of physical bodies.
Yep.
Tom Burke
>To show: Physical contact (without overlap) is impossible.
> (i) If a physical body maximally occupies a closed space, then it occupies
>its boundary. Contact of two bodies cannot consist of the two boundaries
>occupying the same point(s), for then the bodies OVERLAP at the point(s).
>But points cannot be *adjacent*, so if the boundaries do not share point(s),
>then there is a distance between them and contact is absent.
> (ii) But if a body maximally occupies an open space, it doesn't occupy
>its boundary, so even if the boundaries share points, the bodies don't --
>the boundaries separate them!
physical bodies: If one physical body (call it Bob) maximally occupies
a closed space (so that it occupies its boundary) and a second (call
it Mary) maximally occupies an open space (so that it doesn't occupy
its boundary), then contact without overlap will occur if Bob and Mary
share parts of their boundaries.
Russell Turpin
I wrote:
>> The underlying assumption of the argument is that physical bodies
>> are well modeled by either (1) characteristic functions on open
>> sets, or (2) characteristic functions on closed sets, all in
>> Euclidean space. Modern physicists have done work on modeling
>> friction, transmission of force, diffusion, and other surface
>> interactions. The underlying models are all more complex than
>> the above assumption allows.
In article <132...@unix.cis.pitt.edu> dt...@unix.cis.pitt.edu (David M Tate) writes:
> Exactly. Unfortunately, the Exclusive and Illuminated Guild of Pure
> Mathematicians haven't caught on yet. They are still (as a group;
> I don't mean to overgeneralize here) either ignorant or apathetic
> about this failure of the traditional mathematical models. ...
I don't know any mathematicians who claim that the above model of
physical bodies is accurate for any except limited purpose.
Indeed, pure mathematicians, in general, make few claims about
the applicability of physical models. (And, one might ask, why
should they?) Do mathematicians as a group deserve Mr Tate's
criticism? Perhaps he can tell us of which mathematicians he
complains?
> ... That's why the work you mention above has been done by physicists
> and other "applied" scientists.
Well, the study of friction, surface interaction, etc, *is*
part of physics, not mathematics. (I don't mean this as any
prescriptive division of the fields, but as a practical
observation of what is done in each discipline.)
It is worth noting that there is lots of fruitful collaboration
between physicists and mathematicians in these areas. Physicists
use Hilbert spaces, Sobolev spaces, partial differential
equations, and many other parts of mathematics in quantum
field theory, modeling solid-fluid interaction, etc.
I am not sure I understand Mr Tate's complaint.
Russell
David M Tate
[Why it isn't reasonable to model physical objects as closed, connected
sets of points in Euclidean space]
>In article <132...@unix.cis.pitt.edu> dt...@unix.cis.pitt.edu (David M Tate) writes:
>> Exactly. Unfortunately, the Exclusive and Illuminated Guild of Pure
>> Mathematicians haven't caught on yet. They are still (as a group;
>> I don't mean to overgeneralize here) either ignorant or apathetic
>> about this failure of the traditional mathematical models. ...
>
>I don't know any mathematicians who claim that the above model of
>physical bodies is accurate for any except limited purpose.
>Indeed, pure mathematicians, in general, make few claims about
>the applicability of physical models.
OK, perhaps that's a better way of putting it. I intended the word
"apathetic" above to convey that particular interpretation, but I can
see that I should have been more explicit.
> (And, one might ask, why
>should they?)
I'll get back to that in a minute.
>Do mathematicians as a group deserve Mr Tate's
>criticism? Perhaps he can tell us of which mathematicians he
>complains?
Well, at the risk of sounding like an Objectivist (i.e. sweeping complaints
about behaviour that is characteristic of no individual), it's the current
culture of pure mathematical inquiry that I'm bothered by. I'm not so much
trying to censure the behaviour of individual mathematicians; after all, they
have to make a living, too, and the way to do that is to do well what the
mathematics powers-that-be expect you to do. I'm more concerned with what
those collective goals are.
>> ... That's why the work you mention above has been done by physicists
>> and other "applied" scientists.
>
>Well, the study of friction, surface interaction, etc, *is*
>part of physics, not mathematics. (I don't mean this as any
>prescriptive division of the fields, but as a practical
>observation of what is done in each discipline.)
Of course. I never implied otherwise. What I said, and you deleted, was
that the most recent new *mathematical* tools used in advanced physics tend
to be developed by physicists (i.e. non-mathematicians).
>It is worth noting that there is lots of fruitful collaboration
>between physicists and mathematicians in these areas. Physicists
>use Hilbert spaces, Sobolev spaces, partial differential
>equations, and many other parts of mathematics in quantum
>field theory, modeling solid-fluid interaction, etc.
You have a strange notion of "collaboration". This looks to me like a
completely one-sided exchange. Mathematicians work on whatever they feel
like working on, and physicists (and engineers, and chemists, and ...) use
whatever techniques turn out to be applicable to their own problems. It's
a great arrangement for the mathematicians, but a pain in the ass to the
physicists/chemists/etc. who have to sit around waiting on the whim of the
mathematical community, or ignore their own chosen fields long enough to
become mathematicians, in order to do their own theoretical/analytical work.
Of course, back when the great mathematicians were all also natural scientists,
this wasn't a problem.
>I am not sure I understand Mr Tate's complaint.
I can see that. I hope this has helped.
The whole question "what should mathematicians study?" is, I think, a
complex one. Mathematicians, not surprisingly, argue in favor of the
answer "Whatever they like." I'd be the last person to deny them that
option, but not if I (or my tax dollars, or society as a whole, etc.) am
footing the bill.
Now, I understand very well that today's pure theory often leads to tomorrow's
applications. However, the original posting in this thread was talking about
areas of physics where we already *know* that certain mathematical approaches
are dead ends, and that no amount of further extension of those particular
frontiers will be relevant to the problem.
Consider two extreme views:
1. Mathematics is purely a formal pursuit, which is not (and should
not be) concerned with modelling or interpretation. It is its
own justification; an art form, if you will.
2. Mathematics is a subfield of the physical sciences whose purpose
is to provide modelling tools. While some blue-sky work is
necessary to keep the creativity of the field alive, most work
should be at least partially motivated by physical issues and
concerns.
I've known people who ascribe to one or the other of these extremes. Many
people who have corresponded with me in the sci.math newsgroup have expressed
sentiments equivalent to #1. Some posters in sci.physics have expressed
ideas essentially equivalent to #2. My own view lies somewhere in the middle.
Jeff Dalton
[Some uncharacteristically wrongheaded stuff.]
>Of course. I never implied otherwise. What I said, and you deleted, was
>that the most recent new *mathematical* tools used in advanced physics tend
>to be developed by physicists (i.e. non-mathematicians).
So? It's hardly suprising that such should usually be the case.
Mathematicians who do only mathematics that's relevant to physics
might as well be considered physicists.
New mathematics can turn out to be useful in variety of other
fields; it's not just mathematics and physics. Think of category
theory and computer science, for example.
Some mathematics isn't yet useful anywhere outside of methematics,
and that some never will be. I don't see this as a greater problem
than that some physics will never be applied, that much of science
does not address the current needs of industry, or that literary
criticism isn't going to advance quantum mechanics.
I am entirely happy to have tax money go to such things nonetheless.
I prefer a society in which people can investigate things that
interest them without always having to produce an immediate financial
or physical science return to justify it.
>You have a strange notion of "collaboration". This looks to me like a
>completely one-sided exchange. Mathematicians work on whatever they feel
>like working on, and physicists (and engineers, and chemists, and ...) use
>whatever techniques turn out to be applicable to their own problems.
What a prejudicial way of saying it: work on whatever they like. Are
the physicists working on what they like? No, they're phisics slaves,
and so the mathematicians should be maths slaves. Give me a break.
Remember that mathematicians are seldon employed to do whatever they
like, and when they are it's usually because of something they've
already done that makes it look like a good idea to give them free
reign. At the very least, they must at least start by doing things
that are of interest to other methematicians. Else no PhD and no
mathematics career.
>It's
>a great arrangement for the mathematicians, but a pain in the ass to the
>physicists/chemists/etc. who have to sit around waiting on the whim of the
>mathematical community, or ignore their own chosen fields long enough to
>become mathematicians, in order to do their own theoretical/analytical work.
They don't wait around, they use existing mathematics. If they
develop something new, then they are themselves the useful
mathematicians. Is there supposed to be something wrong with
this?
>The whole question "what should mathematicians study?" is, I think, a
>complex one. Mathematicians, not surprisingly, argue in favor of the
>answer "Whatever they like." I'd be the last person to deny them that
>option, but not if I (or my tax dollars, or society as a whole, etc.) am
>footing the bill.
See above.
> 2. Mathematics is a subfield of the physical sciences whose purpose
> is to provide modelling tools.
I see. Only the physical sciences. Not economics, computer science,
etc, etc. Perhaps they aren't important enough to get us to think
pure mathematicians are Wasting Our Tax Money.
>I've known people who ascribe to one or the other of these extremes. Many
>people who have corresponded with me in the sci.math newsgroup have expressed
>sentiments equivalent to #1. Some posters in sci.physics have expressed
>ideas essentially equivalent to #2. My own view lies somewhere in the middle.
But rather close to 2, it seems.
-- jd
Ian Sutherland
>>The underlying assumption of the argument is that physical bodies
>>are well modeled by either (1) characteristic functions on open
>>sets, or (2) characteristic functions on closed sets, all in
>>Euclidean space. Modern physicists have done work on modeling
>>friction, transmission of force, diffusion, and other surface
>>interactions. The underlying models are all more complex than
>>the above assumption allows.
>
>Exactly. Unfortunately, the Exclusive and Illuminated Guild of Pure
>Mathematicians haven't caught on yet. They are still (as a group; I don't
>mean to overgeneralize here) either ignorant or apathetic about this failure
>of the traditional mathematical models. Or, to put it another way, they are
>more interested in pursuing the structural quirks of the old, inaccurate
>models than they are in developing new, accurate models and studying *them*.
>That's why the work you mention above has been done by physicists and other
>"applied" scientists.
It's not a bit clear to me that Mr. Turpin is referring to the kind of
"new models" that you are fond of bringing up Mr. Tate. The notion of
modeling physical objects by (completely classically acceptable) wave
functions constitutes a rejection of (1) and (2) which I think many
modern mathematicians would accept. Rejecting (1) and (2) does not
require the rejection of things like countable additivity of measures.
P.S. Before you flame me like you did the last time, note that nothing
in the above paragraph implies that countable additivity of measures
is good, or an adequate model of probability, or that the classical
construction of the reals is the best way to model physical
quantities or objects, etc. It means exactly what it says.
--
Ian Sutherland i...@cambridge.oracorp.com
Sans peur
David M Tate
>In article <132...@unix.cis.pitt.edu> dt...@unix.cis.pitt.edu (David M
>Tate) writes:
>
>[Some uncharacteristically wrongheaded stuff.]
I'm sorry you think so. On the other hand, it seems to me that you are
trying very hard to read things between the lines that I don't intend, so
maybe I can still change your mind a bit.
(Of course, it's also possible that I'm just putting things very badly.
This has been known to happen... :-) ).
>Mathematicians who do only mathematics that's relevant to physics
>might as well be considered physicists.
The word "only" here is your addition. To anticipate a later objection in
your posting: no, I do not grant physics any special stature above other
natural and social sciences. It merely made a convenient example. I don't
intend to exclude mathematics relevant to economics, computer science, game
theory, chemistry, geology, engineering of various sorts, etc.
>New mathematics can turn out to be useful in variety of other
>fields; it's not just mathematics and physics. Think of category
>theory and computer science, for example.
See above.
>Some mathematics isn't yet useful anywhere outside of methematics,
>and that some never will be. I don't see this as a greater problem
>than that some physics will never be applied, that much of science
>does not address the current needs of industry, or that literary
>criticism isn't going to advance quantum mechanics.
What if we reach a point where almost *no* mathematics is applied outside of
mathematics? Would that be a problem? Consider it as a hypothetical, if you
will; I think we're moving in that direction, but that's a separate argument.
>I am entirely happy to have tax money go to such things nonetheless.
>I prefer a society in which people can investigate things that
>interest them without always having to produce an immediate financial
>or physical science return to justify it.
That's a lovely sentiment, Jeff, and I of course agree with it wholeheartedly
as long as there's infinite bucks to go around. However, when the money gets
tight, I'd just as soon *not* have my bucks going to support somebody's study
of flower symbolism in the works of Baudelaire, or the latest innovations of
a Pulitzer-prize-winning thrash metal composer. Not that I deny these people
the right to pursue those things, or even the cultural worth of those
activities, but they're not real high on my list of priorities.
>>You have a strange notion of "collaboration". This looks to me like a
>>completely one-sided exchange. Mathematicians work on whatever they feel
>>like working on, and physicists (and engineers, and chemists, and ...) use
>>whatever techniques turn out to be applicable to their own problems.
>
>What a prejudicial way of saying it: work on whatever they like. Are
>the physicists working on what they like? No, they're phisics slaves,
>and so the mathematicians should be maths slaves. Give me a break.
If the physicists want to be doing physics, but can't because they have to
do theoretical math instead, then they're not working on what they want to
be working on. Mathematicians are free to disdain physics (or any other
area of application) and follow their whims, but physicists are constrained
by their subject matter. Why is this controversial?
>Remember that mathematicians are seldon employed to do whatever they
>like, and when they are it's usually because of something they've
>already done that makes it look like a good idea to give them free
>reign. At the very least, they must at least start by doing things
>that are of interest to other methematicians. Else no PhD and no
>mathematics career.
Didn't I say this?
Look, let's start over without prejudice. I'm interested in the following
questions:
1. What is the goal of mathematical investigation?
2. What are the criteria by which mathematicians evaluate the
"worth" of particular work?
3. What *should* be the goal(s) of mathematical investigation?
4. Do the criteria in (2) accurately reflect (1) or (3)? If not,
what would be a better alternative?
As for the rest, forget I said anything, and I'll try to clarify my beliefs
(including those adopted merely for Devilish Advocacy) in the ensuing
discussion. Clearly, I have said things so far which begged to be mis-
understood, and were.
Shane D. Deichman
invoke the inverse square law of electrostatic repulsion.
As r --> 0, the repulsive force between like charges (in
this case, the valence electrons of the objects in ques-
tion) approaches infinity. Hence, one can say that no two
objects ever share the same point in space -- i.e. that the
magnitude of the separation vector is nil.
However, what happens when, due to chemical reactions occurring
on the boundaries of the objects, electrons are transferred from
one object to the other? One can always go to smaller and
smaller scales where SOME spatial seaparation exists -- even
in the midst of exchange reactions.
Of course, I'm sure quantum physics will have some stochastic
explanation of the whole thing, relating the separation to
Heisenberg's Uncertainty Principle. If delta-x is sufficiently
large that the electron orbitals are small in comparison, then
the whole argument is moot. No? :-)
-shane
--
^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^
| Shane D. Deichman deic...@cod.nosc.mil |
| "There's no heavier burden than a |
| <affix favorite disclaimer here> great potential!" -Linus Van Pelt |
Keith Ramsay
In article <132...@unix.cis.pitt.edu>, David M Tate writes:
> You mean, the real number system (and mathematics in general) might be just
> a *model* of reality, which is possibly inaccurate in nontrivial ways?!?!?
> :-)
Mathematics can be inaccurate in nontrivial ways, yes; mathematicians
are only human, after all. I am uneasy, however, with the idea of the
real number system being inaccurate "as a model". If the real number
system were to turns out to be "inaccurate", it would not be because
it fails to model any particular thing, but because it would fail in
some more subtle way to be a coherent structure. I see NO sign of this
being the case.
When a mathematical structure turns out not to be identical in form
with a physical object, because of certain idealizations, it still
often remains a crucial tool for constructing and examining the more
complex and accurate models. For instance, you may model a computer as
an idealized Turing machine with unlimited storage, or with registers
which can hold integers. This is inaccurate, of course, but the new
analysis does not start by discarding the integers in favor of new
number systems which only go out to 2^64.
The integers are the natural structure; the word-size is arbitrary. So
we evaluate the functioning of our machine relative to the natural
mathematical structure, rather than trying to hack the mathematics to
fit the machine.
--
Keith Ramsay
ram...@zariski.harvard.edu <--Only through early June
Keith Ramsay
> that the most recent new *mathematical* tools used in advanced physics tend
> to be developed by physicists (i.e. non-mathematicians).
I am not so sure of this claim. Examples might help illuminate what
you mean. The line between physicists and mathematicians is not always
a sharp one. I also find that physicists and other non-mathematicians
often see greater novelty in reformulations, so that for all I know
what you see as "new tools" are in fact applications of "old tools".
> The whole question "what should mathematicians study?" is, I think, a
> complex one.
Certain structures appear repeatedly in scientific investigations. In
some cases we can even give an explanation for why these structures
should be regarded as natural, and the questions about them as natural
and basic. We should study them.
A thorough answer to the question would require examining specific
structures which are studied, how they arose, how they are connected,
and so on. A major problem for us is that it is often very difficult
for us to describe the connections without giving extensive technical
background, which most people are unwilling to absorb.
> [...] However, the original posting in this thread was talking about
> areas of physics where we already *know* that certain mathematical approaches
> are dead ends, and that no amount of further extension of those particular
> frontiers will be relevant to the problem.
In what areas do we "know" that certain mathematical approaches are
"dead ends"? I suspect you may be taking too narrow a view of what
constitutes an "approach". There are generally many forms of what is
essentially the same line of attack on a problem. Much of the
information which has been applied to one approach will apply to
others as well.
Mathematicians are, in part, the guardians of the "big picture".
Physicists may need to go chasing after one field theory after
another, but we try to study the common structures involved in all of
them, so as to (if we are fortunate) gain insights which will remain
stable in time.
David M Tate
>
>In article <132...@unix.cis.pitt.edu>, David M Tate writes:
>> You mean, the real number system (and mathematics in general) might be just
>> a *model* of reality, which is possibly inaccurate in nontrivial ways?!?!?
>> :-)
>
>Mathematics can be inaccurate in nontrivial ways, yes; mathematicians
>are only human, after all. I am uneasy, however, with the idea of the
>real number system being inaccurate "as a model". If the real number
>system were to turns out to be "inaccurate", it would not be because
>it fails to model any particular thing, but because it would fail in
>some more subtle way to be a coherent structure. I see NO sign of this
>being the case.
I'm not sure I understand this. Suppose, for example, that physical space
turns out to be discrete. Just for kicks, suppose that time is also discrete.
Now, any model of spacetime, be it Euclidean or not, which uses continua as
models of intervals is going to be inaccurate (though not necessarily
incoherent or inconsistent). The formal definition (via Dedekind cuts or
whatever) of "real numbers" may not correspond to any physical aspect of the
thing(s) we model with them. That's a contingent thing, though--in some
universes, we get it right; in others, we don't.
Have I misunderstood what you were saying?
>When a mathematical structure turns out not to be identical in form
>with a physical object, because of certain idealizations, it still
>often remains a crucial tool for constructing and examining the more
>complex and accurate models.
Of course; I never meant to imply otherwise. On the other hand, sometimes
the model is sufficiently divergent from reality in some way that we have to
find a better model for at least some purposes. Thus, Euclidean models of
spacetime seem to work perfectly well for "most" of space, away from large
masses. Near large masses, though, the model is sufficiently inaccurate that
we need to find a better one. There is, a priori, no way we can know that
there might not arise an example for which the "real numbers" are sufficiently
inaccurate that we need something else. The point was merely that we tend to
get mentally lazy, and think/talk about the reals as if they were "out there"
somewhere in nature, and not merely a conceptual model akin to the planetary
model of the atom.
>The integers are the natural structure; the word-size is arbitrary. So
>we evaluate the functioning of our machine relative to the natural
>mathematical structure, rather than trying to hack the mathematics to
>fit the machine.
Well, yes and no. The integers are a model, too; they aren't any more "real"
than the reals, if you can follow such an ugly sentence. Some other race of
beings might come along and decide that the ordinals are the "natural"
structure, or something even more bizarre.
David M Tate
>
>It's not a bit clear to me that Mr. Turpin is referring to the kind of
>"new models" that you are fond of bringing up Mr. Tate. The notion of
>modeling physical objects by (completely classically acceptable) wave
>functions constitutes a rejection of (1) and (2) which I think many
>modern mathematicians would accept. Rejecting (1) and (2) does not
>require the rejection of things like countable additivity of measures.
Who said anything about countably additive measures?
>P.S. Before you flame me like you did the last time, note that nothing
>in the above paragraph implies that countable additivity of measures
>is good, or an adequate model of probability, or that the classical
>construction of the reals is the best way to model physical
>quantities or objects, etc. It means exactly what it says.
Fine. Relax. I had no intention of flaming you. (Besides, I thought you
were "sans peur" :-)...). As I've already said, in response to Jeff Dalton's
comments, I fear I phrased my comments rather badly, and I have in fact more
or less retracted them. If I have flamed you in the past, please accept my
public apology.
(Appearances to the contrary notwithstanding, I have neither agenda nor
manifesto with regard to promulgating some particular flavor of Truth in
mathematics and mathematical modelling. I just have some questions about
what math is for, and some concerns about where it's going...)
David M Tate
>> The whole question "what should mathematicians study?" is, I think, a
>> complex one.
>
>Certain structures appear repeatedly in scientific investigations. In
>some cases we can even give an explanation for why these structures
>should be regarded as natural, and the questions about them as natural
>and basic. We should study them.
OK, this is the view I'm questioning. You say "certain structures appear
repeatedly; we should study them". It sounds as if you are equating these
"structures" with particular mathematical constructs. I would argue that
these constructs *do not* appear in scientific investigations; they are
_models_ of things which appear in scientific investigations. (I assume we
are both using the phrase 'scientific investigations' in the broadest sense
imaginable).
Example: the idea of "quantity" occurs over and over in every conceivable
practical application. However, it is an enormous leap from that primitive
notion of "quantity" to the Dedekind reals. When you talk about the reals,
you are no longer talking about the "certain structure", but about one
particular attempt to model that structure.
Second Example: the idea of "grouping" occurs over and over (etc.). However,
Zermelo-Fraenkel set theory with the countable axiom of choice is not the only
way of formalizing this notion of "grouping".
Third Example: the idea of "likelihood" or "chance" occurs (...). However,
the Kolmogorov axioms of probability theory are not the only option in how to
model this notion.
(My apologies to those who are getting sick of these same examples over and
over; I'd be delighted to take suggestions for some new ones.)
I'm not saying there's necessarily anything wrong with the way we model these
things at the moment; only that it's a conceivable possibility. Every time we
formulate a mathematical object, we are imposing structure on nature that may
not really be there. There are no sets, no integers, no parabolae, no vector
spaces in "real life": these are mental conveniences of ours, which help us
think about the way things are/behave.
Jeff Dalton
>In article <48...@skye.ed.ac.uk> je...@aiai.UUCP (Jeff Dalton) writes:
>>Mathematicians who do only mathematics that's relevant to physics
>>might as well be considered physicists.
>
>The word "only" here is your addition. To anticipate a later objection in
>your posting: no, I do not grant physics any special stature above other
>natural and social sciences. It merely made a convenient example. I don't
>intend to exclude mathematics relevant to economics, computer science, game
>theory, chemistry, geology, engineering of various sorts, etc.
There are many people who are doing mathematics. Some of them do
mainly mathematics and are called Mathematicians (this connection is
usual, not necessary). Others do mainly something else, or maths
that is so close to something else that they're not classed as
mathematicians. Some theoretical physicists might be examples.
Some computer scientists certainly are.
Of the rest, many do applied maths or, say, statistical theory (which
is usually fairly close to applications).
Let's suppose there is a core or "pure" mathematicians who just do
mathematics that has no applications and that these mathematicians
don't care about it ever having applications.
Some of this mathematics will turn out to have application in
unexpected areas anyway. Category theory might be an example.
I think what's left is only a small part of the mathematics that's
done. So I think you're worried about something that isn't a very
big problem. At least, I don't think it's any bigger than the
"problem" of philosophy that no one but other philosophers in
the same area ever reads, for example.
>>Some mathematics isn't yet useful anywhere outside of mathematics,
>>and that some never will be. I don't see this as a greater problem
>>than that some physics will never be applied, that much of science
>>does not address the current needs of industry, or that literary
>>criticism isn't going to advance quantum mechanics.
>
>What if we reach a point where almost *no* mathematics is applied outside of
>mathematics? Would that be a problem? Consider it as a hypothetical, if you
>will; I think we're moving in that direction, but that's a separate argument.
Literary criticism is almost never applied in that way, and I don't
think that's a problem
>>I am entirely happy to have tax money go to such things nonetheless.
>>I prefer a society in which people can investigate things that
>>interest them without always having to produce an immediate financial
>>or physical science return to justify it.
>
>That's a lovely sentiment, Jeff,
It's not just a sentiment; I vote accordingly. I also pay taxes.
>and I of course agree with it wholeheartedly as long as there's
>infinite bucks to go around.
Infinite bucks are never going to be required.
>However, when the money gets tight, I'd just as soon *not* have my bucks
>going to support somebody's study of flower symbolism in the works of
>Baudelaire, or the latest innovations of a Pulitzer-prize-winning thrash
>metal composer. Not that I deny these people the right to pursue those
>things, or even the cultural worth of those activities, but they're not
>real high on my list of priorities.
Do you think money is so tight right now?
In any case, when you think it is you can vote for those parties who
will restructure education to eliminate such things from Universities.
(As well as other such funding from taxes, of course). I'd prefer to
keep the English Lit departments, etc, myself.
>If the physicists want to be doing physics, but can't because they have to
>do theoretical math instead, then they're not working on what they want to
>be working on.
The maths they need is also part of physics. If they don't like
what their work in physics entails, they can try something else,
but I don't see that they have any right to have other people do
it for them.
>Mathematicians are free to disdain physics (or any other area of
>application) and follow their whims, but physicists are constrained
>by their subject matter. Why is this controversial?
Mathematicians have to do mathematics, just as physicists have to
do physics.
>>Remember that mathematicians are seldon employed to do whatever they
>>like, and when they are it's usually because of something they've
>>already done that makes it look like a good idea to give them free
>>reign. At the very least, they must at least start by doing things
>>that are of interest to other methematicians. Else no PhD and no
>>mathematics career.
>
>Didn't I say this?
You seemed to think tax money might be funding mathematicians to
do whatever they liked.
>Look, let's start over without prejudice. I'm interested in the following
>questions:
>
> 1. What is the goal of mathematical investigation?
To extend mathematics.
> 2. What are the criteria by which mathematicians evaluate the
> "worth" of particular work?
Beauty, power, importance to something else of interest, etc.
I don't think they all use the same criteria, by the way.
> 3. What *should* be the goal(s) of mathematical investigation?
To extend mathematics.
> 4. Do the criteria in (2) accurately reflect (1) or (3)? If not,
> what would be a better alternative?
Yes.
-- jd
Russell Turpin
>> What if we reach a point where almost *no* mathematics is
> Literary criticism is almost never applied in that way, and
> I don't think that's a problem
This may have been true at one time, but these days, theoretical
literary criticism usually has political or philosophical
implications, and sometimes even aspires to have scientific
implications, and its practitioners do it conscious of this.
Russell
David M Tate
>In article <133...@unix.cis.pitt.edu> dt...@unix.cis.pitt.edu (David M Tate) writes:
>
>There are many people who are doing mathematics. Some of them do
>mainly mathematics and are called Mathematicians (this connection is
>usual, not necessary). Others do mainly something else, or maths
>that is so close to something else that they're not classed as
>mathematicians. Some theoretical physicists might be examples.
>Some computer scientists certainly are.
This is the traditional view, but I'm beginning to doubt it (as you can
certainly tell by now :-) ). I think that "pure" mathematics and "applied"
mathematics are completely different fields of endeavor, in the same way that
composing poetry and writing UNIX manuals are. Both use language, but their
goals (and aesthetics) are sufficiently different that they aren't directly
comparable. Similarly, the goals (and aesthetics) of pure vs. applied math
are sufficiently different that I don't see them as instances of the same
activity.
>>What if we reach a point where almost *no* mathematics is applied outside of
>>mathematics? Would that be a problem? Consider it as a hypothetical, if you
>>will; I think we're moving in that direction, but that's a separate argument.
>
>Literary criticism is almost never applied in that way, and I don't
>think that's a problem
It's not a problem because our ability to build bridges, cure cancer, feed
people, etc. doesn't depend on the quality of our literary criticism. It
does, however, depend on the quality of our applied sciences, which in turn
depend on the available mathematical tools.
I'm not saying we're anywhere near the hypothetical state-of-mathematics
that I described above. However, I think it's clear that we're moving in
that direction, and I don't see any limits (be they social, cultural, fiscal,
whatever) on how far in that direction the field of math may eventually move.
>>However, when the money gets tight, I'd just as soon *not* have my bucks
>>going to support somebody's study of flower symbolism in the works of
>>Baudelaire, or the latest innovations of a Pulitzer-prize-winning thrash
>>metal composer. Not that I deny these people the right to pursue those
>>things, or even the cultural worth of those activities, but they're not
>>real high on my list of priorities.
>
>Do you think money is so tight right now?
Not in the US. Probably won't be in our lifetimes, either, but I worry about
it all the same.
>In any case, when you think it is you can vote for those parties who
>will restructure education to eliminate such things from Universities.
Heavens no! There's no reason to eliminate them; just move them over to
Fine Arts where they will by then belong...
>(As well as other such funding from taxes, of course). I'd prefer to
>keep the English Lit departments, etc, myself.
This is a straw man, Jeff. I haven't said anything which can remotely be
construed as advocating the elimination of arts departments in universities.
What I *have* advocated is that we not fund arts posing as sciences with money
targeted for science research, if we get to that point.
>You seemed to think tax money might be funding mathematicians to
>do whatever they liked.
No, merely to do what the mathematical community approves of. What they
collectively "like", if you will.
>>Look, let's start over without prejudice. I'm interested in the following
>>questions:
>>
>> 1. What is the goal of mathematical investigation?
>
>To extend mathematics.
Simpliciter? Regardless of relevance, application, etc.? If so, then we're
talking about an art form.
>> 2. What are the criteria by which mathematicians evaluate the
>> "worth" of particular work?
>
>Beauty, power, importance to something else of interest, etc.
>I don't think they all use the same criteria, by the way.
In other words, aesthetic criteria? (And I agree, different mathematicians
have different aesthetics.)
>> 3. What *should* be the goal(s) of mathematical investigation?
>
>To extend mathematics.
>
>> 4. Do the criteria in (2) accurately reflect (1) or (3)? If not,
>> what would be a better alternative?
>
>Yes.
So it doesn't matter in what direction we extend mathematics; they're all
equally fine, so long as they meet the aesthetic needs of mathematicians?
And if we wander off into mathematical never-never land, in which the
mathematical community has decided that the only *really* interesting work
involves the properties of lattices in 73-dimensional vector spaces, and that
nothing else is worthwhile, that's still OK? Even when that's all they'll
teach?
I'm not trying to sound snide here; I'm really interested in whether you would
answer "yes" to all of these questions. I would not, and I'm trying to
formulate for myself just what it is that I'd object to...
Chris Holt
[after a flurry with Jeff Dalton]
>Look, let's start over without prejudice. I'm interested in the following
>questions:
> 1. What is the goal of mathematical investigation?
the study of relationships among abstract structures, themselves
expressed as abstract structures (IMHO, of course)
> 2. What are the criteria by which mathematicians evaluate the
> "worth" of particular work?
elegance
> 3. What *should* be the goal(s) of mathematical investigation?
Idealistically, what mathematicians want to do. Pragmatically,
developing structures that are useful in modelling aspects of
our environment. However, since the latter cannot be known
ahead of time ("George, go and develop a model for economics
like a good mathematician, would you?") we treat it like any
field of research, in which people are given a fairly free
rein (else it's not research).
> 4. Do the criteria in (2) accurately reflect (1) or (3)? If not,
> what would be a better alternative?
An interesting question that ought to be researched; but the
skills involved in applied math are not the same as those of
pure math, and demanding that pure mathematicians concentrate
on areas of obvious application sounds like a recipe for
generating a lot of second-rate applied mathematicians at
the expense of first-rate pure mathematicians, to the benefit
of neither.
[Note: I of course am an applied mathematician; at least, in
my more arrogant moments I have delusions and pretensions
along those lines.]
-----------------------------------------------------------------------------
Chris...@newcastle.ac.uk Computing Lab, U of Newcastle upon Tyne, UK
-----------------------------------------------------------------------------
"They have been at a great feast of languages, and stolen the scraps." - WS
Christopher Menzel
>You say "certain structures appear
>repeatedly; we should study them". It sounds as if you are equating these
>"structures" with particular mathematical constructs. I would argue that
>these constructs *do not* appear in scientific investigations; they are
>_models_ of things which appear in scientific investigations.
But doesn't the fact that they are *models* mean that there is some
similarity of structure between the models and what they're models
*of*, a similarity of structure *in virtue of which* they are
models? If so, it *does* sound like these structures appear
in scientific investigation.
>There are no sets, no integers, no parabolae, no vector
>spaces in "real life": these are mental conveniences of ours, which help us
>think about the way things are/behave.
When you say "mental convenience," do you mean that these are mental
entities? So that sets and numbers are in our heads? Frege had
some pretty strong words against this sort of view, e.g., it seems
to entail that each of us has his or her own number 2. If this
isn't your view, could you say more?
Chris
Keith Ramsay
In article <133...@unix.cis.pitt.edu>, David M Tate writes:
> Mathematicians are free to disdain physics (or any other
> area of application) and follow their whims, but physicists are constrained
> by their subject matter. Why is this controversial?
We believe ourselves to be "constrained by our subject matter" as well
as the physicists are. You make it sound as though "subject matter"
means "area of application", and hence that mathematics is not a
subject matter in its own right. I don't buy that.
David M Tate
>dt...@unix.cis.pitt.edu (David M Tate) writes:
>
>> 3. What *should* be the goal(s) of mathematical investigation?
>
>Idealistically, what mathematicians want to do. Pragmatically,
>developing structures that are useful in modelling aspects of
>our environment.
Which is it? I'm asking for a normative here. From what you've said, it
looks like your possible replies are (a) what they want to do, (b) developing
structures that are useful in modelling, or (c) there is no normative goal of
mathematics. You can't pick more than one of the above, though, because they
conflict directly. (I suppose you could claim that the ideal is some sort of
balance between (a) and (b)... is that what you were saying?)
>An interesting question that ought to be researched; but the
>skills involved in applied math are not the same as those of
>pure math, and demanding that pure mathematicians concentrate
>on areas of obvious application sounds like a recipe for
>generating a lot of second-rate applied mathematicians at
>the expense of first-rate pure mathematicians, to the benefit
>of neither.
For the record, I have nowhere advocated restricting mathematicians to work
in "areas of obvious application". I have mentioned it as one extreme end of
a range of possible views toward mathematical investigation. Sorry to sound
paranoid, but's it's going to be hard enough to come out of this discussion
without the label "that guy who thinks mathematicians should all do physics",
which couldn't be farther from the truth.
David M Tate
>In article <133...@unix.cis.pitt.edu>, dt...@unix.cis.pitt.edu (David M Tate) writes...
>>You say "certain structures appear
>>repeatedly; we should study them". It sounds as if you are equating these
>>"structures" with particular mathematical constructs. I would argue that
>>these constructs *do not* appear in scientific investigations; they are
>>_models_ of things which appear in scientific investigations.
>
>But doesn't the fact that they are *models* mean that there is some
>similarity of structure between the models and what they're models
>*of*, a similarity of structure *in virtue of which* they are
>models?
No, I wouldn't say that. They are models by virtue of being attempts to
capture certain essential characteristics of the system/thing being modelled.
Just how similar they really are to that system/thing depends on how well we
do the modelling (or, perhaps, on how much accuracy we feel we need for that
particular application). The modelling process always involves some degree
of abstraction from the actual; how much "similarity" remains is variable,
and somewhat in the eye of the modeller.
>If so, it *does* sound like these structures appear
>in scientific investigation.
Only because those are the aspects *we* decide are interesting. There is no
abstraction in nature; only *things* and *behaviour*. Our mental models of
what is going on automatically attempt to impose generalizations on these
things and behaviours, so that we can predict and plan and *understand* what's
going on around us. All thought is modelling; some is just more explicit than
others.
>>There are no sets, no integers, no parabolae, no vector
>>spaces in "real life": these are mental conveniences of ours, which help us
>>think about the way things are/behave.
>
>When you say "mental convenience," do you mean that these are mental
>entities? So that sets and numbers are in our heads? Frege had
>some pretty strong words against this sort of view, e.g., it seems
>to entail that each of us has his or her own number 2. If this
>isn't your view, could you say more?
Sure. I've read Frege, and I was unconvinced by much of his attempt to
explicate, say, the integers. As I see it, we do all have our own number 2.
By talking with other people, refining the use of the language, etc. we
achieve some degree of convergence in our notions, but it's still basically
a personal model.
I think this really makes the most sense when you ask yourself "What is the
motivation for formalism in mathematics?". The answer is, it seems to me,
to make sure that we're all talking about the same "2". To this end, we
formulate axioms which attempt to capture what is common among our various
personal notions of "2" in a systematic way. Probability theory is perhaps
the best example I know of this. Everyone has a different idea (or "mental
model", if you prefer) of what randomness/chance/probability/likelihood/etc.
is. Rather than attempt to work in that quagmire of differing interpretation
and intuition, we instead try to capture the essentials in a few axioms, and
explore the consequences of those axioms. [NB. For those who have heard me
rant a bit on this topic, I don't object to this approach... I just don't
think the question of "which axioms are best" is closed.]
(I should also add that, in anticipation of an almost-sure objection, I have
read Wittgenstein's "Of Pain and Private Language", and he think he's very
wrong...)
David M Tate
>as the physicists are. You make it sound as though "subject matter"
>means "area of application", and hence that mathematics is not a
>subject matter in its own right. I don't buy that.
OK, fair enough. I didn't mean it to sound that way. I suppose the
distinction I was trying to draw was between internal constraints (e.g.
mathematics is defined to be that which mathematicians are doing at the
moment, as a group) and external constraints (e.g. physics is the attempt
to model/understand/predict the behaviour of matter and energy in this
universe). It is conceivable that the "subject matter" of mathematics
could shift almost arbitrarily far from its present scope, given enough
time, but physics is anchored by the necessity of dealing with The Way
Things Are (tm).
So, yes, any one particular mathematician is constrained to do work which is
related in some definite way to the current canon of math. However, the
field as a whole has no such constraint. Think of a school of fish (no
disparagement intended here; it's just a visual image that came to me): each
fish must remain more or less within the school, but the school as a whole
can move great distances.
Stephen Carrier
>In article <48...@skye.ed.ac.uk> je...@aiai.UUCP (Jeff Dalton) writes:
>>In any case, before it looked like you objected to mathematicians
>>doing certain things (or to too many mathematicians doing them)
>No, it was more like not enough mathematicians doing certain other things on
>which many other fields of great practical importance depend.
Is this really a problem? Engineers crying for a solution to a mathematical
problem and no one working on it because all qualified persons are staring
in their navel?
[...]
>>Not all of them. For example, a "something else of interest" might
>>be physics.
>
>But even then the mathematician ties his work to physics because he finds the
>link aesthetically pleasing, not because the link is inherently valuable to him
>as a criterion in evaluating the worth of mathematical research.
This is caricature of the "Pure Mathematician". Mathematics relates
not just to physics, of course, but computer science and society via
statistics and economics. "Pure Mathematician" is an abstract type.
Many, perhaps most, real mathematicians have some interest in those
real-world applications. If their particular work is not obviously
related to those real-world applications, their "pure" work can support
the work of mathematicians and non-mathematicians who do "real"
things. For the purpose of justifying public funds for pure math, you can
think of pure math as a layer on top of applied math, which itself is
grounded in the real world. If pure math helps applied math, and applied
math helps the world, isn't pure math helping the world, indirectly.
Even if 99% of mathematicians were working on what you consider pure
rather than applied mathematics, mathematics as a discipline is still
touching society and its real world. The idea of a monolithic
"mathematics" becoming 100% inner-directed and disconnected from the
practical world is absurd. The issue is applied research and pure
research. The difference between pure and applied is this: The less
idea you have whether something you work on will be practially useful,
or what part of it might be practically useful, or in how many decades
or centuries it would be practically useful, if ever, then the more
pure it is. Whether or not the scientific worker cares or not
shouldn't be important, because even if he does care, he couldn't
tell. To never spend money on such research is to never risk, and how
much progress of any kind can come from that?
>The contrast I'm trying to get at here is that between evaluation mechanisms
>in mathematics and evaluation mechanisms in other research areas that affect
>quality-of-life in concrete ways. I know very few people who would be willing
>to allow the field of medecine, for example, to rely on aesthetics-based
>evaluation mechanisms. I'm wondering if this alleged qualitative difference
>between medecine and math isn't really more a matter of degree, in such a way
>that there are extreme cases wherein we would want to exercise the same kind
>of normative control over math that we do over medecine...
We already do: the NSF hardly funds mathematical research
at random, and anyways some mathematicians are inclined to
be bored with math that is too much "abstract nonsense", and criticise those
that are.
Stephen
Jeff Dalton
>In article <48...@skye.ed.ac.uk> je...@aiai.UUCP (Jeff Dalton) writes:
>>In article <133...@unix.cis.pitt.edu> dt...@unix.cis.pitt.edu (David M Tate) writes:
>>>What if we reach a point where almost *no* mathematics is applied outside
>>>of mathematics?
>>Literary criticism is almost never applied in that way, and I don't
>>think that's a problem
>It's not a problem because our ability to build bridges, cure cancer, feed
>people, etc. doesn't depend on the quality of our literary criticism. It
>does, however, depend on the quality of our applied sciences, which in turn
>depend on the available mathematical tools.
So if those mathematicians were doing lit crit instead, the problem
would vanish?
>I'm not saying we're anywhere near the hypothetical state-of-mathematics
>that I described above. However, I think it's clear that we're moving in
>that direction, and I don't see any limits (be they social, cultural, fiscal,
>whatever) on how far in that direction the field of math may eventually move.
I take it that the hypothetical state is:
What if we reach a point where almost *no* mathematics is applied
outside of mathematics?
So far as I can tell, there are social, cultural, and fiscal limits.
What we don't have is something that clearly and explicitly prevents
it.
>>In any case, when you think it is you can vote for those parties who
>>will restructure education to eliminate such things from Universities.
>>(As well as other such funding from taxes, of course). I'd prefer to
>>keep the English Lit departments, etc, myself.
>
>This is a straw man, Jeff. I haven't said anything which can remotely be
>construed as advocating the elimination of arts departments in universities.
Think back. You wrote:
However, when the money gets tight, I'd just as soon *not* have my
bucks going to support somebody's study of flower symbolism in the
works of Baudelaire, or the latest innovations of a Pulitzer-prize-
winning thrash metal composer. Not that I deny these people the
right to pursue those things, or even the cultural worth of those
activities, but they're not real high on my list of priorities.
How do you propose to keep your bucks from going to support them
if some of them are in arts departments, if you pay taxes, and if
the government continues to support universities? There are options
other than the voting strategy I suggested above, but you may want to
take back the "remotely be construed" and perhaps the "straw man" too.
>What I *have* advocated is that we not fund arts posing as sciences with
>money targeted for science research, if we get to that point.
Is that what we're doing? Giving science money to these mathematicians?
In any case, before it looked like you objected to mathematicians
doing certain things (or to too many mathematicians doing them), and
to having any of your money support it. Now you seem to be saying
it's ok to have money support it so long as it isn't _science_ money.
I'm finding it difficult to determine exactly what your objection is.
>>Beauty, power, importance to something else of interest, etc.
>>I don't think they all use the same criteria, by the way.
>
>In other words, aesthetic criteria? (And I agree, different mathematicians
>have different aesthetics.)
Not all of them. For example, a "something else of interest" might
be physics.
-- jeff
Keith Ramsay
> related in some definite way to the current canon of math. However, the
> field as a whole has no such constraint. Think of a school of fish (no
> disparagement intended here; it's just a visual image that came to me): each
> fish must remain more or less within the school, but the school as a whole
> can move great distances.
Physics is not "anchored" by physical reality alone (nature is not
forcing us to be physicists), but by physical reality in combination
with an intuitive sense of what constitutes a proper scientific study
of it. The connection between nature and physics is sufficiently
direct and uncontroversial that we don't worry about slippage. The
fact that physicists are interested in superconductivity seems
completely natural, so much so that we aren't worried that physicists
will choose instead to become fascinated by the sounds that children's
toys make when banged together. Nevertheless it is only this intuitive
sense of what is important in physics which keeps physics from
wandering off in such directions.
My position is, first, that mathematics is in fact anchored to certain
concerns which are stable with time, but, second, that one ought not
to try to bind the range of concern of mathematics to any particulars
of nature.
For example, in my field (number theory) there has been for thousands
of years an interest in determining the rational and integer solutions
to equations. Some of the simplest equations are polynomial equations
with two unknowns. We have managed to classify these equations by a
property called the "genus". We understand the genus 0 case well.
There is a remarkable theorem to the effect that when the genus is 1,
there are only finitely many integer solutions. In a brilliant piece
of work (early 1980's, I think), Gerd Faltings succeeded in showing
that when the genus is >2, there are only finitely many rational
solutions.
I do not see a field which has pursued essentially the same aims for
thousands of years as wandering around very much.
This is not to say that there is no danger of mathematicians
"wandering off into space", but I think the danger is less than it
appears. A field becomes suspect when it loses touch with the abiding
concerns of mathematics. Some people are suspicious of logic for this
reason, although I think this suspicion partly the result of
ignorance.
On the other hand, I think mathematics need room to rove around a good
bit. If the physicists think that a certain group (of symmetries) is
the group of a field occurring in nature, they will naturally study
that particular group intensively. For mathematicians, though, it is
appropriate to study the whole range of groups, finding properties in
common to all of them, and finding those groups which have special
properties unlike the others. The two roles are complementary.
David Petry
>> Mathematicians are free to disdain physics (or any other
>> area of application) and follow their whims, but physicists are constrained
>> by their subject matter. Why is this controversial?
>
>We believe ourselves to be "constrained by our subject matter" as well
>as the physicists are. You make it sound as though "subject matter"
>means "area of application", and hence that mathematics is not a
>subject matter in its own right.
Physicists are constrained by an objective, observable reality. Mathematicians
could be so constrained, if they chose to be. They could limit their inquiry
to phenomena observable through computation. But they have chosen not to.
Pure mathematicians (analysts, anyways) are studying what they think they would
see if they could perform infinitely many computations, each with infinite
precision. The world of the pure mathematician exists in the imagination only.
It is not an objective, observable reality.
I think it's reasonable to say that pure mathematicians feel free to disdain
reality and follow their whims. The only constraints on mathematicians are
social constraints - what other mathematicians won't ridicule.
Atleast, that's the way I see it.
David Petry
Ignore me and I'll go away. Flame me and maybe I'll stick around.
David M Tate
>In article <133...@unix.cis.pitt.edu> dt...@unix.cis.pitt.edu (David M Tate) writes:
>
>>It's not a problem because our ability to build bridges, cure cancer, feed
>>people, etc. doesn't depend on the quality of our literary criticism. It
>>does, however, depend on the quality of our applied sciences, which in turn
>>depend on the available mathematical tools.
>
>So if those mathematicians were doing lit crit instead, the problem
>would vanish?
What?
I don't follow this at all. By "the problem", do you mean cancer/hunger/etc.,
or what? I can't map this occurrence of the word back onto the "problem"
that I was talking about in the quoted passage...
>>I'm not saying we're anywhere near the hypothetical state-of-mathematics
>>that I described above. However, I think it's clear that we're moving in
>>that direction, and I don't see any limits (be they social, cultural, fiscal,
>>whatever) on how far in that direction the field of math may eventually move.
>
>I take it that the hypothetical state is:
>
> What if we reach a point where almost *no* mathematics is applied
> outside of mathematics?
>
>So far as I can tell, there are social, cultural, and fiscal limits.
Really? I don't see any. It's hard for there to be social limits on a
field that 99% of society don't know anything about. Ditto cultural limits,
unless you mean "within the culture of mathematics", in which case I'd be
interested to hear what you think those might be. Fiscal limits come when
somebody turns off the cash flow; given that the government/society/etc. are
already well beyond their ability to judge whether math is doing "what they
want it to" (whatever that might be), I don't see any reason why they should
stop funding math at least at the levels they are now...
Or were you thinking along different lines?
>>This is a straw man, Jeff. I haven't said anything which can remotely be
>>construed as advocating the elimination of arts departments in universities.
>
>Think back. You wrote:
>
> However, when the money gets tight, I'd just as soon *not* have my
> bucks going to support somebody's study of flower symbolism in the
> works of Baudelaire, or the latest innovations of a Pulitzer-prize-
> winning thrash metal composer. Not that I deny these people the
> right to pursue those things, or even the cultural worth of those
> activities, but they're not real high on my list of priorities.
>
>How do you propose to keep your bucks from going to support them
>if some of them are in arts departments, if you pay taxes, and if
>the government continues to support universities?
OK, I see the problem. We think about "where the money is going" differently,
it would seem. If the government wants to use some of my money to support
education in general, then that's fine with me, and always will be. No strings
attached; LitCrit is just as valid as Fluid Dynamics, microeconomics, music,
or whatever. I think of education (and the educational system) as separate
from public funding of research, despite the overlap of personnel. I can see,
though, how this might have looked contradictory to you.
>There are options
>other than the voting strategy I suggested above, but you may want to
>take back the "remotely be construed" and perhaps the "straw man" too.
To the extent that you weren't deliberately twisting my words, I retract my
tone. However, you were still responding to a position which is not mine,
and which I (still) don't think my words imply. But this is silly; I'm not
mad at you, and I didn't intend my objection to be provocative. (Do the words
"straw man" have strong pejorative connotations to you? I suppose I'll have
to be more careful where I use them; I simply meant a shorthand for "That
position isn't the one I was proposing"...)
>>What I *have* advocated is that we not fund arts posing as sciences with
>>money targeted for science research, if we get to that point.
>
>Is that what we're doing? Giving science money to these mathematicians?
Jeff, I'm _not_ talking about what we are or aren't doing _now_. This is a
hypothetical future, remember?
>In any case, before it looked like you objected to mathematicians
>doing certain things (or to too many mathematicians doing them)
No, it was more like not enough mathematicians doing certain other things on
which many other fields of great practical importance depend.
>Now you seem to be saying
>it's ok to have money support it so long as it isn't _science_ money.
In the sense outlined above. If the people think they're buying something
useful (in whatever the public's sense of "useful" is at the time), they
shouldn't be misled. They aren't (and probably never again will be) competent
to judge for themselves.
>>>Beauty, power, importance to something else of interest, etc.
>>>I don't think they all use the same criteria, by the way.
>>
>>In other words, aesthetic criteria? (And I agree, different mathematicians
>>have different aesthetics.)
>
>Not all of them. For example, a "something else of interest" might
>be physics.
But even then the mathematician ties his work to physics because he finds the
link aesthetically pleasing, not because the link is inherently valuable to him
as a criterion in evaluating the worth of mathematical research.
The contrast I'm trying to get at here is that between evaluation mechanisms
in mathematics and evaluation mechanisms in other research areas that affect
quality-of-life in concrete ways. I know very few people who would be willing
to allow the field of medecine, for example, to rely on aesthetics-based
evaluation mechanisms. I'm wondering if this alleged qualitative difference
between medecine and math isn't really more a matter of degree, in such a way
that there are extreme cases wherein we would want to exercise the same kind
of normative control over math that we do over medecine...
>
>-- jeff
--
David M. Tate |
dt...@unix.cis.pitt.edu | The guilty flee when no man pursueth; the
Motto: | innocent get hit in the head with a rock.
Gramen artificiosum odi |
greg Nowak
}This is the traditional view, but I'm beginning to doubt it (as you can
}certainly tell by now :-) ). I think that "pure" mathematics and "applied"
}mathematics are completely different fields of endeavor, in the same way that
Do you know much about the history of the fields? Or how they work in
practice? Career demographics of both?
}It's not a problem because our ability to build bridges, cure cancer, feed
}people, etc. doesn't depend on the quality of our literary criticism. It
Not necessarily. Are you familiar with Susan Sonntag's book "Illness
as Metaphor" ?
}>> 1. What is the goal of mathematical investigation?
}>To extend mathematics.
}Simpliciter? Regardless of relevance, application, etc.? If so, then we're
}talking about an art form.
Whoops. So any mathematics that isn't applied mathematics is an art
form. Are you familiar with what Hardy said about his own work? If
you were making funding decisions and had an interview with Hardy,
would you fund his work?
}So it doesn't matter in what direction we extend mathematics; they're all
}equally fine, so long as they meet the aesthetic needs of mathematicians?
}And if we wander off into mathematical never-never land, in which the
}mathematical community has decided that the only *really* interesting work
}involves the properties of lattices in 73-dimensional vector spaces, and that
}nothing else is worthwhile, that's still OK? Even when that's all they'll
}teach?
Mathematical fields tend to get played out. Plane geometry was real hot for
a few centuries, but there's not much left to do. If mere human
beings want to spend a few centuries studying lattices in
73-dimensional vector spaces, so what? The rest of mathematics can
wait; it's not like it will evaporate if it's not discovered before
tuesday. There's an infinite amount of mathematics to be done; it is
unlikely that there's an optimal path for us to take through the space
of provable mathematical truths.
rutgers!phoenix.princeton.edu!greg Gregory A Nowak/Phoenix Gang/Princeton NJ
"Take 2*3*5*7*11*13. It's divisible by 59." --Matt Crawford
Joseph O'Rourke
>[T]he NSF hardly funds mathematical research at random...
In article <134...@unix.cis.pitt.edu> David M Tate writes:
>It's hard for there to be social limits on a
>field that 99% of society don't know anything about. Ditto cultural limits,
>unless you mean "within the culture of mathematics", in which case I'd be
>interested to hear what you think those might be.
I believe the society of mathematicians exerts very strong control
over *funded* research. My most recent NSF proposal was reviewed
by eight(!) of my peers, each of whom wrote a review of the proposal
and an estimation of worth of my entire research career. They
then summarized their opinion in one of five categories: E, VG, G, F, P.
The current statistics are that essentially *all* evaluations must
be Excellent or Very Good to receive funding. One can hardly imagine
tighter control over funded research.
I would be interested to learn if the U.S. is unusual in
the degree of peer consensus required for funded research. For
example, my understanding is that the Canadian equivalent, NSERC,
is more lenient in its control of research.
Jeff Dalton
> It is conceivable that the "subject matter" of mathematics
>could shift almost arbitrarily far from its present scope, given enough
>time, but physics is anchored by the necessity of dealing with The Way
>Things Are (tm).
If it shifts far enough, we wouldn't call it mathematics any more.
That some other people might use the same word, mathematics, for it
is a separate issue.
In short, our mathematics can't shift arbitrarily. If something
else is called mathematics, we could easily have a different opinion
about that than we do about mathematics.
-- jd
David M Tate
>
>Is this really a problem? Engineers crying for a solution to a mathematical
>problem and no one working on it because all qualified persons are staring
>in their navel?
Not at the moment (unless Han de Bruijn is right), but I wasn't necessarily
talking about the status quo. I'm wondering if it's a *potential* problem.
>In article <134...@unix.cis.pitt.edu> dt...@unix.cis.pitt.edu (David M Tate) writes:
>>But even then the mathematician ties his work to physics because he finds the
>>link aesthetically pleasing, not because the link is inherently valuable to him
>>as a criterion in evaluating the worth of mathematical research.
>
>This is caricature of the "Pure Mathematician". Mathematics relates
>not just to physics, of course, but computer science and society via
>statistics and economics. "Pure Mathematician" is an abstract type.
Well, I thought so too once, but 30 or 40 pointed Email messages to the
contrary from the readers of sci.math got me to wondering. Many of those
people were downright irate at the idea that "mathematics relates to physics",
or to any other applied discipline. They all saw math as being its own
justification, on aesthetic grounds, with the "applications" parasitic. I'm
not just making these opinions up; I've seen them stated over and over.
>If pure math helps applied math, and applied
>math helps the world, isn't pure math helping the world, indirectly.
Of course. That's the whole point here; what happens if that first link
gets broken, and pure math no longer helps applied math? You seem to be
saying this is impossible, but I'm not sure what your argument is.
>To never spend money on such research is to never risk, and how
>much progress of any kind can come from that?
I should never have mentioned money; this isn't about money. The question is,
is there a point beyond which "blue-sky" math research becomes irrelevant to
applications, and is the field of mathematics moving corporately in that
direction? If so, is there anything that might stop it/them from going that
far? My claim has been that there doesn't seem to be any sort of control
mechanism that would stop such a shift, were it to occur.
David M Tate
>In article <133...@unix.cis.pitt.edu>, dtate@unix (David M Tate) writes:
>}>> 1. What is the goal of mathematical investigation?
> Jeff Dalton replied:
>}>To extend mathematics.
>
>}Simpliciter? Regardless of relevance, application, etc.? If so, then we're
>}talking about an art form.
>
>Whoops. So any mathematics that isn't applied mathematics is an art
>form. Are you familiar with what Hardy said about his own work? If
>you were making funding decisions and had an interview with Hardy,
>would you fund his work?
For the nth time, I'm not talking about funding issues here.
Whether mathematics is an art form depends on exactly one thing: the
criteria by which mathematicians distinguish good math from bad math.
If the criteria are fundamentally internal to mathematics, then it is
being treated as an art form; art is the only activity we do which is
considered to be its own justification. If Jeff has correctly characterized
the way mathematicians evaluate each others' work, then math has become an
art form, at least for those mathematicians who employ those criteria. This
is not an attempt to denigrate art or math; it's just an observation.
So what did Hardy say about his work?
David M Tate
>> It is conceivable that the "subject matter" of mathematics
>>could shift almost arbitrarily far from its present scope, given enough
>>time, but physics is anchored by the necessity of dealing with The Way
>>Things Are (tm).
>
>If it shifts far enough, we wouldn't call it mathematics any more.
I'm not so sure. What is math, if not that activity pursued by people who
call themselves mathematicians? It's not clear to me that Diophantus would
recognize category theory as "mathematics", but that hasn't caused us to
stop and invent a new name.
As long as the journey to new and different interests is made in small steps,
I see no reason why we should assume that we wouldn't call it mathematics
any more...
Chris Holt
About 15 years ago I heard Dirac give a talk (note the name-dropping :-)
and he made a special point that the motivation behind almost all
of his contributions was aesthetic. Would you therefore call him
a mathematician?
>I'm wondering if this alleged qualitative difference
>between medecine and math isn't really more a matter of degree, in such a way
>that there are extreme cases wherein we would want to exercise the same kind
>of normative control over math that we do over medecine...
What kind of normative control ought we have over medecine? Who
(besides medical researchers) can make accurate decisions as to
the relative value of the genome project vs. other uses of funds?
Chris Holt
>In article <1991May29.1...@newcastle.ac.uk> Chris...@newcastle.ac.uk (Chris Holt) writes:
>>dt...@unix.cis.pitt.edu (David M Tate) writes:
>>
>>> 3. What *should* be the goal(s) of mathematical investigation?
>>
>>Idealistically, what mathematicians want to do. Pragmatically,
>>developing structures that are useful in modelling aspects of
>>our environment.
>Which is it? I'm asking for a normative here. From what you've said, it
>looks like your possible replies are (a) what they want to do, (b) developing
>structures that are useful in modelling, or (c) there is no normative goal of
>mathematics. You can't pick more than one of the above, though, because they
>conflict directly. (I suppose you could claim that the ideal is some sort of
>balance between (a) and (b)... is that what you were saying?)
I'm saying that *I* think the goal should be to allow mathematicians
to study what they want, because chaining people's ideas of what is
interesting in terms of other people's value systems is dangerous.
However, as we are part of a political process, there must be some
unfortunate but inevitable compromises.
Keith Ramsay
. repeatedly; we should study them". It sounds as if you are equating these
. "structures" with particular mathematical constructs. I would argue that
. these constructs *do not* appear in scientific investigations; they are
. _models_ of things which appear in scientific investigations. (I assume we
. are both using the phrase 'scientific investigations' in the broadest sense
. imaginable).
Models do *not* appear and, in particular, are *not* constructed in
scientific investigations? Even in the broadest sense of the word? :-)
I was not using "scientific investigations" to mean the *objects* of
the investigations.
Keith Ramsay
G.H.Hardy eventually came to feel he had lost his powers of doing
create mathematical research, and in this sad state wrote a book
called "A Mathematician's Apology". "Apology" means here an attempt to
justify one's actions. It is a somewhat interesting book, although
somewhat depressing for a mathematician; he paints a pessimistic
picture.
His main arguments as I recall were that pure mathematics, though
useless, at least did no harm, and that mathematicians were unlikely
to be nearly as good at doing anything else. You would enjoy it I'm
sure. :-) People have noted however that even number theory, which he
cites as being particularly "pure", has been applied since he wrote.
--
If you want to write to me after June 6,
Keith Ramsay write c/o Harvard math dept., 1 Oxford St.,
ram...@zariski.harvard.edu Cambridge MA, 02138.
David M Tate
>
>Models do *not* appear and, in particular, are *not* constructed in
>scientific investigations? Even in the broadest sense of the word? :-)
No, this is just the opposite of what I wanted to say. Looking back at the
original, I see that I typed "constructs" when I meant "structures", thus
creating the absurd claim you noticed :-).
What I was saying is that the mathematical and logical models are purely
human things, built by us to reflect/represent/approximate "reality". We
*invent* mathematics; we don't discover it.
Keith Ramsay
In article <133...@unix.cis.pitt.edu>, David M Tate writes:
> [...] The formal definition (via Dedekind cuts or
> whatever) of "real numbers" may not correspond to any physical aspect of the
> thing(s) we model with them. That's a contingent thing, though--in some
> universes, we get it right; in others, we don't.
The points are that the validity and the applicability (which may not
be exactly the same thing) of the real numbers are distinct questions
from whether any individual physical phenomenon is accurately modeled
by them. Saying that the reals are a model, and that as a model they
may fail, leaves the impression that their success is due to their
success in directly modeling some particular thing or things.
Much of the power of mathematics comes from its being able to
translate questions about one structure into questions about another
one. (I mean "structure" here in the sense that it is used in
mathematics.) We study the reals not simply because various physical
phenomena can be modeled using them, but because the questions which
are asked, both within and from outside mathematics, may often be
answered using them.
This power of application is relatively robust and independent on
particulars of the world. An analogy may be useful. Suppose that you
want to examine the exports produced by some remote country. Some of
the cities of that country are more or less directly connected by
ports to the outside world. They are located where they are because
they are near corresponding places in other countries. Other cities
farther in the interior of the country, however, are placed where they
are because they are convenient means of transporting goods within the
country. Particularly well-placed cities in the interior may remain in
business independent of whether particular goods and services are
selling well, or whether particular ports are open.
Such is the case in mathematics: the place of such a construction as
the real line is secure, even if many applications which use them turn
out only to be approximations.
--
If you want to contact me after this week,
Steve Stevenson
>In article <1991May29.1...@newcastle.ac.uk> Chris...@newcastle.ac.uk (Chris Holt) writes:
>>dt...@unix.cis.pitt.edu (David M Tate) writes:
>>
>>> 3. What *should* be the goal(s) of mathematical investigation?
>>
Look in a good book on the philosophy of mathematics which should discuss
the typical requirements. I've also seen some good stuff out of the
philosophy of science when written by a scientist and not just a philosopher.
As a good beginning, try Suppes {\em Introduction to Logic}, the last
chapter.
--
===============================================================================
Steve (really "D. E.") Stevenson st...@hubcap.clemson.edu
Department of Computer Science, (803)656-5880.mabell
Clemson University, Clemson, SC 29634-1906
David M Tate
>translate questions about one structure into questions about another
>one. (I mean "structure" here in the sense that it is used in
>mathematics.) We study the reals not simply because various physical
>phenomena can be modeled using them, but because the questions which
>are asked, both within and from outside mathematics, may often be
>answered using them.
OK, this is a very good point. Part of the reason we love the real numbers
is that they're so damn flexible. This robustness of application provides a
unifying affect over many, many different areas of application, in the same
way that fields like Graph Theory or Algebra do at a higher level; we learn
more when we can use the same methods in widely different areas.
However, I don't think this changes the problem of potentially coming up
against an application area where the Dedekind reals are not a sufficiently
accurate/valid/whatever model. Just as not all relational systems can be
handled by standard graph theory or standard algebra, even the most robust
models have their limitations. This doesn't mean we need to chuck the reals;
it just means we may need something else as well.
>This power of application is relatively robust and independent on
>particulars of the world.
I'm not sure I see this, even given your example. In fact, I'm not sure
what it is you mean by "independent on (of?) particulars of the world".
Are you saying that the integrative power of the real numbers to bring
together diverse areas of math which use them does not depend on what those
particular areas and applications are? If so, I agree; but see above.
Han de Bruijn
> (i) if a physical body maximally occupies a closed space ...
> (ii) But if a body maximally occupies an open space ...
> [ then paradox arises. ]
Physically speaking, there is NO empirical evidence for any possible difference
between "closed" and "open" spaces. For this would require an infinite accuracy
which is not present in the real world. The paradox is resolved by INACCURACY.
I'm going to repeat this again and again, for it is the _key_ to understanding.
Without such inaccuracy, a shaft would stick in his bearing, you wouldn't be
able to drive a car, and ... you couldn't make love (: like Bob and Mary :-).
So, the whole mathematics machinery of Nature is SWINGING a bit ... Tada tada.
As a matter of fact, most of Point Set Topology, where the closed and open sets
come from, is _questionable_ from a physics or engineers or any-other-activity-
-than-pure-math point of view. But not only from any point of view _outside_
mathematics.
Quoting without permission from "Constructive Analysis" (Chapter 3. Set Theory)
by Errett Bishop, Douglas Bridges:
> Very little is left of general topology after that vehicle of classical
> mathematics has been taken apart and reassembled constructively.
> With some regret, plus a large measure of relief, we see this flamboyant
> engine collapse to constructive size.
Let's get physical (: Olivia Newton John) Yeah, swing it out babe ...
-
* Han de Bruijn; Applications&Graphics | "A little bit of Physics * No
* TUD Computing Centre; P.O. Box 354 | would be NO idleness in * Oil
* 2600 AJ Delft; The Netherlands. | Mathematics" (HdB). * for
* E-mail: Han.de...@RC.TUDelft.NL --| Fax: +31 15 78 37 87 ----* Blood
Han de Bruijn
> invoke the inverse square law of electrostatic repulsion.
> Hence, one can say ...
... that the repulsive force between an electron and itself
is infinite. So electrons immediately explode, huh? :-)
Han de Bruijn
> Not at the moment (unless Han de Bruijn is right), ...
Okay Dave. I hear every word. :-)
Jeff Dalton
>In article <48...@skye.ed.ac.uk> je...@aiai.UUCP (Jeff Dalton) writes:
>>In article <133...@unix.cis.pitt.edu> dt...@unix.cis.pitt.edu (David M Tate) writes:
>>> It is conceivable that the "subject matter" of mathematics
>>>could shift almost arbitrarily far from its present scope, given enough
>>>time, but physics is anchored by the necessity of dealing with The Way
>>>Things Are (tm).
>>
>>If it shifts far enough, we wouldn't call it mathematics any more.
>
>I'm not so sure. What is math, if not that activity pursued by people who
>call themselves mathematicians?
Well, it is that, but that isn't what determines what mathematics is.
If mathematicians all started to do something else tomorrow, that
wouldn't make that something else mathematics.
>It's not clear to me that Diophantus would recognize category theory
>as "mathematics", but that hasn't caused us to stop and invent a new name.
Yes, but by our rules it is mathematics and so is (most of?) what
Diophantus thought was mathematics. (Most of -- well maybe (I don't
know) he was one who included some mysticism in his mathematics.)
And why shouldn't we play by our rules (ie, use our language)?
>As long as the journey to new and different interests is made in small steps,
>I see no reason why we should assume that we wouldn't call it mathematics
>any more...
As I think I mentioned, it is not a question of whether we would be
using the word "mathematics" for it but rather the question of whether,
given that we mean something by "mathematics", a certain activity
would still count as mathematics in that sense. That is, we are
talking about mathematics (the subject magtter) and not whatever
is called "mathematics" (which might turn out to be almost anything.)
If I say that I think it's ok for mathematicians to do what interests
them, that is not saying that it's ok for anyone who is doing anything
that is called "mathematics" to do what interests them.
-- jd
David M Tate
>>>In article <133...@unix.cis.pitt.edu> dt...@unix.cis.pitt.edu (David M Tate) writes:
>>>> It is conceivable that the "subject matter" of mathematics
>>>>could shift almost arbitrarily far from its present scope, given enough
>>>>time, but physics is anchored by the necessity of dealing with The Way
>>>>Things Are (tm).
>>>
>>>If it shifts far enough, we wouldn't call it mathematics any more.
>>
>>I'm not so sure. What is math, if not that activity pursued by people who
>>call themselves mathematicians?
>
>Well, it is that, but that isn't what determines what mathematics is.
Isn't it?
Or, are you saying "What we mean by "mathematics", *today*, is not affected
by what people may or may not call themselves in the future." ? If so, I
agree, but you've changed the question. You feel that mathematicians (as we
use the word today) should be allowed to work on anything they like. Does
this include the possibility that they choose to cease being mathematicians?
Presumably so, whether they decide to stop *calling* themselves 'mathematician'
or not.
>If mathematicians all started to do something else tomorrow, that
>wouldn't make that something else mathematics.
Under the interpretation above, no. But what does that have to do with the
original worry? I suppose in the current terminology, I'm saying that it just
might happen that all the mathematicians would (gradually, corporately) decide
to stop doing math. Put this way, it sounds a bit odd, but I think that's
what follows from this particular terminology, and I still don't see what's
going to prevent it from happening...
>>It's not clear to me that Diophantus would recognize category theory
>>as "mathematics", but that hasn't caused us to stop and invent a new name.
>
>Yes, but by our rules it is mathematics and so is (most of?) what
>Diophantus thought was mathematics. (Most of -- well maybe (I don't
>know) he was one who included some mysticism in his mathematics.)
>And why shouldn't we play by our rules (ie, use our language)?
Wait a minute. I thought you just got through saying we were going to fix
our current meaning of 'mathematics' and *not* play language games. What are
you getting at here? I'm saying that, just as Diophantus (or Pythagorus, or
even Archimedes) might not wish to speak of the activities of some modern
mathematicians as "mathematics", there might come a day when most or all of
the relevant people (whatever they call themselves) are not doing anything we
would call 'mathematics', given our current sense of what math is. As for
the *activities* engaged in by these men still being considered math, that's
a fuzzy question. Much of what Archimedes did would be considered physics,
for example. Mathematicians recognize that these men pursued mathematical
goals, but mathematicians today do not do the same kinds of things that these
men did, except in a very broad sense. They don't have to; those fields have
been plowed already.
>>As long as the journey to new and different interests is made in small steps,
>>I see no reason why we should assume that we wouldn't call it mathematics
>>any more...
>
>As I think I mentioned, it is not a question of whether we would be
>using the word "mathematics" for it but rather the question of whether,
>given that we mean something by "mathematics", a certain activity
>would still count as mathematics in that sense.
I don't think I've denied this. You're the one who is worrying about whose
definition of "math" we use; I'm perfectly content to use ours. I still see
no reason to suppose that investigators in the future will continue to do that
which we call "pure mathematics", as opposed to engaging in some different
sort of formal games. I was merely suggesting that the persistence of the
name "mathematician" across such paradigm shifts could camouflage the fact
that a shift has in fact occurred.
>If I say that I think it's ok for mathematicians to do what interests
>them, that is not saying that it's ok for anyone who is doing anything
>that is called "mathematics" to do what interests them.
Is it saying that it's OK for no one at all to do mathematics?
Jeff Dalton
>In article <49...@skye.ed.ac.uk> je...@aiai.UUCP (Jeff Dalton) writes:
>>>>In article <133...@unix.cis.pitt.edu> dt...@unix.cis.pitt.edu (David M Tate) writes:
>>>>> It is conceivable that the "subject matter" of mathematics
>>>>>could shift almost arbitrarily far from its present scope, given enough
>>>>>time, but physics is anchored by the necessity of dealing with The Way
>>>>>Things Are (tm).
>>>>
>>>>If it shifts far enough, we wouldn't call it mathematics any more.
>>>
>>>I'm not so sure. What is math, if not that activity pursued by people who
>>>call themselves mathematicians?
>>
>>Well, it is that, but that isn't what determines what mathematics is.
>
>Isn't it?
No. And, BTW, if you start to say mathematics is whatever people who
call themselves mathematicians do, you ought to do the same for
physics and people who call themselves physicists. So much for the
anchor.
>Or, are you saying "What we mean by "mathematics", *today*, is not affected
>by what people may or may not call themselves in the future." ?
Well, it would be pretty bizarre if it _were_ so affected, wouldn't it?
What I am saying, however, is simply that the meaning of "mathematics",
the subject matter if you want, is not determined by whatever people
who call themselves mathematicians happen to be doing. It is entirely
possible for the people who call themselves mathematicians to start
doing something else, and we wouldn't automatically have to say that
new subject was mathamatics. _We_ wouldn't even have to call those
people mathematicians.
>If so, I agree, but you've changed the question.
I don't think so. The question (see quote above) is whether the
subject matter of mathematics could shift arbitrarily far from its
present scope. My answer is "no". It couldn't, for example, shift
into hairdressing.
Remember that when I said that it couldn't shift too far because we
wouldn't then regard it as mathematics I was responding to a part of
your argument, not to the whole thing. The particular worry that
mathematics could shift arbitrarily far is one we can dismiss, in my
opinion.
>You feel that mathematicians (as we use the word today) should be
>allowed to work on anything they like.
Within mathematics, yes; and they should be able to change to some
other subject too, if they want. Why not? Just what (if anything)
is your gripe about this?
>Does this include the possibility that they choose to cease being
>mathematicians?
Yes. See above.
>Presumably so, whether they decide to stop *calling* themselves
>'mathematician' or not.
That's right.
>>If mathematicians all started to do something else tomorrow, that
>>wouldn't make that something else mathematics.
>
>Under the interpretation above, no.
The interpretation under which it does happen must be pretty bizarre,
because it could make mathematics be hairdressing.
Note that this is quite different from the question of whether we, or
our descendents, or some other people, might, at some point, end up
using the word "mathematics" (that sound, those letters) for something
else.
>But what does that have to do with the original worry?
I said more about that later in my message, so I'll see what you say
about it below. However, I should warn you that I'm not entirely
sure what you mean by "the original worry" at this point.
>I suppose in the current terminology, I'm saying that it just might
>happen that all the mathematicians would (gradually, corporately)
>decide to stop doing math.
But of course they might. Maybe it gets harder to get funding for
maths, for example.
>Put this way, it sounds a bit odd,
Why?
>but I think that's what follows from this particular terminology, and
>I still don't see what's going to prevent it from happening...
Well, I happen to feel that if they all started doing pure mathamatics
that had no relevance to physics (or other approved subjects) they
would still be doing mathematics. You think that would be bad and may
even think something ought to be done to prevent it. I don't think it
necessarily would be bad. After all, some of that maths might turn
out to have relevance to physics (or to something else) later on
or might be sufficiently interesting in its own right.
But whatever we think about that possibility, it is very different
from the possibility that mathematicians will start doing something
that isn't mathematics at all.
I just want to make it clear that (1) mathematics can't make an
arbitrary shift even though the people that do it might, and (2)
if I say mathematicians ought to be able to develop mathematics
as they want I am not saying they ought to be able to turn it
into something else and still have us regard it in the same way
we do mathematics.
>>>It's not clear to me that Diophantus would recognize category theory
>>>as "mathematics", but that hasn't caused us to stop and invent a new name.
>>
>>Yes, but by our rules it is mathematics and so is (most of?) what
>>Diophantus thought was mathematics. (Most of -- well maybe (I don't
>>know) he was one who included some mysticism in his mathematics.)
>>And why shouldn't we play by our rules (ie, use our language)?
>
>Wait a minute. I thought you just got through saying we were going to
>fix our current meaning of 'mathematics' and *not* play language games.
>What are you getting at here?
I think there are some problems with the idea of fixing a current
meaning, but let's suppose that's more or less what I said further
back. Hence, when we use "mathematics" it means what it means in
the language we are using now, and not what it might mean in some
other language we might be using later on or might have been using
if we were ancient Greeks. Then I said it again: we should play
by our rules (ie, use our language) -- and our notion of mathematics.
>I'm saying that, just as Diophantus (or Pythagorus, or even
>Archimedes) might not wish to speak of the activities of some modern
>mathematicians as "mathematics", there might come a day when most or
>all of the relevant people (whatever they call themselves) are not
>doing anything we would call 'mathematics', given our current sense of
>what math is.
Yes, that might happen. And what I think of it would depend on
what they were doing.
>>>As long as the journey to new and different interests is made in small
>>>steps, I see no reason why we should assume that we wouldn't call it
>>>mathematics any more...
>>
>>As I think I mentioned, it is not a question of whether we would be
>>using the word "mathematics" for it but rather the question of whether,
>>given that we mean something by "mathematics", a certain activity
>>would still count as mathematics in that sense.
>
>I don't think I've denied this. You're the one who is worrying about whose
>definition of "math" we use; I'm perfectly content to use ours.
No you're not. You want to use some bogus definition in which
mathematics is defined by what people who call themselves
mathematicians are doing.
>I still see no reason to suppose that investigators in the future will
>continue to do that which we call "pure mathematics", as opposed to
>engaging in some different sort of formal games. I was merely
>suggesting that the persistence of the name "mathematician" across
>such paradigm shifts could camouflage the fact that a shift has in
>fact occurred.
If that's what you meant to say, I think you could have said it more
clearly.
In any case, if that it your worry, the same could happen for physics.
The reality anchor wouldn't rule out such camouflage. If we looked
more closely, we could detect that the physicists weren't doing
physics any more. We could do the same to the mathematicians.
-- jd
David M Tate
>>I don't think I've denied this. You're the one who is worrying about whose
>>definition of "math" we use; I'm perfectly content to use ours.
>
>No you're not. You want to use some bogus definition in which
>mathematics is defined by what people who call themselves
>mathematicians are doing.
Bogus? Jeff, I can't imagine a coherent notion of what mathematics is that
doesn't depend, eventually, on the self-definition of some group of people
who call themselves "mathematicians". If you want to pick mathematicians
today as the reference group, or the Ancient Greeks, or people living near
Tau Ceti in AD 3781, you're still choosing one of many possible 'definitions'
arbitrarily.
Let's put it this way: you seem to be using "mathematics" (unquoted in your
text) as a rigid designator of something. I'm trying to figure out what it's
supposed to be a rigid designator of, and all I can come up with is that you
want a word to rigidly designate those fields of enquiry that people who call
themselves "mathematicians", today, consider to be "mathematics". Fine. Let's
be explicit, and use capital-M Mathematics as this rigid designator.
Now, back to our original questions:
How far can mathematics shift? If you mean how far can Mathematics shift,
then the question is ill-formed. It can't shift; it's the same (arbitrarily
chosen) set of subjects/areas/endeavors it always was, by definition. If you
don't mean the rigid-designator Mathematics, but don't mean "whatever happens
to be called 'mathematics' in the language of whatever other place and time
you're looking at", then I don't know what you could be referring to. Could
you help me out here?
>The reality anchor wouldn't rule out such camouflage. If we looked
>more closely, we could detect that the physicists weren't doing
>physics any more. We could do the same to the mathematicians.
I don't see it. We could do this for physics because we can give a definition
of what 'physics' is that is indpendent of what physicists have done in the
past, or of what has been called 'physics'. It would be something along the
lines of "The systematic attempt to model physical systems and phenomena, and
to characterize and predict their behavior". We can know that this is what
physics is *supposed* to be, and detect deviation from it, even in the absence
of any actual physics ever having been done before.
What corresponding reference point are you proposing for mathematics, which
does not refer to what "mathematicians" have done in the past, yet could not,
as you say, be the same as hairdressing? What reason do you have to prefer
our 1991 idea of what mathematics is to the 500 BC or 3917 idea?
[I wrote and then deleted a long, point-by-point reply to your entire article.
At the end, I felt that we had gotten sufficiently muddled in trivia, mis-
understandings, and personality conflict that it wasn't worth posting. Had
I thought of it, I might have sent it by Email, but it may be better off dead.]
--
David M. Tate | "The greatest danger in scholarship [is not] that
dt...@unix.cis.pitt.edu | the individual should fail to master the thought
Motto: | of a school, but that a school may succeed in mas-
Gramen artificiosum odi | tering the thought of the individual." G. Sampson
David Petry
Jeff Dalton wrote:
>> ... You want to use some bogus definition in which
>>mathematics is defined by what people who call themselves
>>mathematicians are doing.
>
>Bogus? Jeff, I can't imagine a coherent notion of what mathematics is that
>doesn't depend, eventually, on the self-definition of some group of people
>who call themselves "mathematicians".
How about "Mathematics is the science of phenomena observable through
computation" ?
>How far can mathematics shift?
Before the time of Cantor, I think many or most mathematicians would have
accepted the definition I have given above. Since Cantor, mathematicians
have included the study of infinity as part of mathematics. Mathematics
has shifted pretty far.
> ... It can't shift; it's the same (arbitrarily
>chosen) set of subjects/areas/endeavors it always was, by definition.
Hmmm.
David Petry
David M Tate
>
>How about "Mathematics is the science of phenomena observable through
>computation" ?
I'm skeptical as to the possibility of defining "computation" in a way which
does not refer to mathematics, at least indirectly.
Jeff Dalton
>Odd # of >'s is Jeff Dalton; even # of >'s is David Tate
Now reverse that.
>>>I don't think I've denied this. You're the one who is worrying about whose
>>>definition of "math" we use; I'm perfectly content to use ours.
>>
>>No you're not. You want to use some bogus definition in which
>>mathematics is defined by what people who call themselves
>>mathematicians are doing.
>
>Bogus?
Yes, bogus. The definition of "mathematics" is not "the activity
performed by people who call themselves `mathematicians'" (or various
minor variations).
It should be trivial to see this. For instance, people could call
themselves "mathematicians" even if they did not do mathematics at
all, and the people who are doing mathematics might call themselves
something else. This is a _possibility_, but it would not be possible
if "mathematics" we defined as you seem to be suggesting.
However, there are two ways to read the "bogus" definition:
(1) To find out what "mathematics" means, find the people who call
themselves "mathematicians" and see whatever it is that they do.
(2) Treat "people who call themselves mathematicians" rigidly.
That is, it identifies some group of people, and whatever
those people are doing is mathematics.
It should be trivial to see that (1) is bogus, as indicated above.
For various reasons, (2) does not work as-is. We need to allow new
people to become mathematicians, for example. However, we could make
something like it work if we had a more flexible way of identifying
the people who are properly considered mathematicians. Perhaps we
start with a canonical set and then include people who are doing
sufficiently similar things. But how do we know what counts as
sufficiently similar? That is one way to see why we need some
independent notion of what counts as mathematics.
Even if the way we develop that notion is to look at what certain
people are, or were, doing, it can be independent after that. And
indeed it must be, for otherwise it would be impossible for
mathematicians to stop doing mathematics, something that they
clearly can do. They might, for example, all start doing physics.
Then they would be physicists rather than mathematicians, but not
because they've just called themselves physicists.
>Jeff, I can't imagine a coherent notion of what mathematics is that
>doesn't depend, eventually, on the self-definition of some group of people
>who call themselves "mathematicians".
It happens that mathematicians have called themselves that. But it
is not a necessary connection. Perhaps instead the mathematicians
never made much of a distinction between themselves and, say,
philosophers, and someone else invented the term "mathematics".
The term might catch on among mathematicians only after that.
>If you want to pick mathematicians today as the reference group, or
>the Ancient Greeks, or people living near Tau Ceti in AD 3781, you're
>still choosing one of many possible 'definitions' arbitrarily.
Start with the discussion of (2), above.
In any case, the reason for saying certain Ancient Greeks were doing
mathematics is that they are doing a certain thing, not that they
called themselves by a certain name. This would be so even if those
Greek were among the examples used to teach what mathematics is.
Looking at what matmematicians do is a way to see what mathematics
is, not a definion of it.
>Let's put it this way: you seem to be using "mathematics" (unquoted in your
>text) as a rigid designator of something.
I'm not sure I've used quotation consistently. But usually I quote
a word to show I'm talking about the word and not what it means or
refers to.
Whether that makes the word "mathematics" rigid I'm not sure. In any
case, I'm not trying to be Humpty Dumpty ("when I use a word, it means
what I say it means" or something to that effect). I think we have to
consider whether "mathematics" is a regid designator or not.
My knowledge of rigid designators comes from Kripke's _Naming and
Necessity_. A useful distinction from that book is between how we fix
the reference of a word and what the reference is. We might fix the
reference of "mathamatics" by seeing what mathematicians do, but the
reference doesn't therefore change if mathenaticians do something else.
>I'm trying to figure out what it's supposed to be a rigid designator of,
>and all I can come up with is that you want a word to rigidly designate
>those fields of enquiry that people who call themselves "mathematicians",
>today, consider to be "mathematics".
I don't quite agree with that. For instance, I don't think it's
very important that they call themselves "mathematicians". However,
it may be close enough for the present, depending on exactly what
you feel is implied by "rigid". I certainly think "mathematics"
refers to subject matter, and not just in my usage.
>Fine. Let's be explicit, and use capital-M Mathematics as this
>rigid designator.
>
>Now, back to our original questions:
>
>How far can mathematics shift? If you mean how far can Mathematics shift,
>then the question is ill-formed. It can't shift; it's the same (arbitrarily
>chosen) set of subjects/areas/endeavors it always was, by definition.
I don't know about shift, because it may be that things can't stop
being mathamatics; but mathematics can certainly change. Otherwise
no one could extend it to new areas, and mathematics is something
that has been and can be extended.
If Mathematics can't change in that way, then "Mathematics" is not
a synonym for "mathematics".
>If you don't mean the rigid-designator Mathematics, but don't mean
>"whatever happens to be called 'mathematics' in the language of whatever
>other place and time you're looking at", then I don't know what you
>could be referring to. Could you help me out here?
Does this mean that by "mathematics" _you_ mean whatever happens to be
called "mathematics" in the language of whatver time and place you're
looking at? If so, then the question of how far mathematics can shift
is a question about language change and not about the future
development of mathematics (the subject matter) at all.
>>The reality anchor wouldn't rule out such camouflage. If we looked
>>more closely, we could detect that the physicists weren't doing
>>physics any more. We could do the same to the mathematicians.
>
>I don't see it. We could do this for physics because we can give a
>definition of what 'physics' is that is indpendent of what physicists
>have done in the past, or of what has been called 'physics'. Along
>the lines of "The systematic attempt to model physical systems and
>phenomena, and to characterize and predict their behavior". We can
>know that this is what physics is *supposed* to be, and detect
>deviation from it, even in the absence of any actual physics ever
>having been done before.
Some words have neater definitions than others. That hardly means
the others can shift arbitrarily. I think Wittgenstein used "game"
as an example. Let's suppose that's right and that we can't give
a neat, necessary and sufficient conditions, definition for "game".
That does not mean we can't tell games from non-games, even if there
are some cases where we can't quite decide one way or the other.
I think we have a resonably good idea of what mathematics is,
sufficiently good so that arbitrary shifts are not included, even if
we can't produce a one-sentence definition. And for all I know, we
_can_ have a neat, one-sentence definitions. What I do know is that
it is not "what people who call themselves mathematicians do".
>What corresponding reference point are you proposing for mathematics, which
>does not refer to what "mathematicians" have done in the past, yet could not,
>as you say, be the same as hairdressing? What reason do you have to prefer
>our 1991 idea of what mathematics is to the 500 BC or 3917 idea?
Well, there are several things to say here.
Since mathematicians _have_ done mathematics in the past, that part
of what they've done may well be involved. But I would be using past
mathematicians only to help fix the reference (ie show what it is I
am talking about).
That doesn't mean that "mathematics" is _defined_ as what they did.
If I found out that I had been misinformed and that they had really
done something else, my concept of mathematics would not have to shift
accordingly. It _might_ shift if what they had actually been doing
led to a better notion of or understanding of what mathematics is.
Perhaps there are some times when it is difficult to distinguish
between those two cases, but that does not mean such a distinction
is impossible. For example, if it turned out that what they actually
did was hairdressing, that wouldn't mean that mathematics was
hairdressing. If it turned that they weren't formalists (and I'd
thought they were), I might decide that mathematics wasn't confined
to formalism.
This is always going to be a somewhat imperfect process but,
as someone said, that day fades gradually into night does not
mean we can't tell light from dark.
Now, why do I prefer our 1991 idea of what mathematics is to some
other one. It's because I think "maythematics" is a word in the
language I am using now, and if I mean something comletely different
by it then I'm using a different language (at least at that point).
I think something like that needs a pretty good reason.
>[I wrote and then deleted a long, point-by-point reply to your entire
>article. At the end, I felt that we had gotten sufficiently muddled
>in trivia, mis- understandings, and personality conflict that it
>wasn't worth posting. Had I thought of it, I might have sent it by
>Email, but it may be better off dead.]
I don't know about you, but I've enjoyed this discussion more than
any in months. Really. And the misunderstandings and personality
conflicts are nothing compared to uk.politics, for example.
-- jd
David M Tate
>>Odd # of >'s is Jeff Dalton; even # of >'s is David Tate
>
>>>mathematics is defined by what people who call themselves
>>>mathematicians are doing.
and, later,
>I don't quite agree with that. For instance, I don't think it's
>very important that they call themselves "mathematicians". However,
>it may be close enough for the present, depending on exactly what
>you feel is implied by "rigid". I certainly think "mathematics"
>refers to subject matter, and not just in my usage.
and, again, later:
>Does this mean that by "mathematics" _you_ mean whatever happens to be
>called "mathematics" in the language of whatver time and place you're
>looking at?
# # ### ##
## # # # ##
# # # # # ##
# # # # # ##
# ## # #
# # ### ##
I don't know how many times I have to deny this, Jeff, for you to believe
me. My point has absolutely nothing to do with linguistic variance, or what
name we choose to apply to anything.
>I think we have a resonably good idea of what mathematics is,
>sufficiently good so that arbitrary shifts are not included, even if
>we can't produce a one-sentence definition. And for all I know, we
>_can_ have a neat, one-sentence definitions. What I do know is that
>it is not "what people who call themselves mathematicians do".
What I'm trying to get at is that I think the definition of mathematics that
you are using, whatever it may be, is ill-formed. It either fails to refer
to something that is inherently interesting (which I assume you want to say
mathematics is), or fails to refer to anything that can't shift arbitrarily.
Let's see if I can express myself a little better about why I think this...
>>What corresponding reference point are you proposing for mathematics, which
>>does not refer to what "mathematicians" have done in the past, yet could not,
>>as you say, be the same as hairdressing? What reason do you have to prefer
>>our 1991 idea of what mathematics is to the 500 BC or 3917 idea?
>
>Since mathematicians _have_ done mathematics in the past, that part
>of what they've done may well be involved. But I would be using past
>mathematicians only to help fix the reference (ie show what it is I
>am talking about).
>
>That doesn't mean that "mathematics" is _defined_ as what they did.
Yes, it does, in the crucial sense that if that's not what people (whatever
you call them) had been interested in, then it's not what *we* would be
interested in now, enough to give the subject a name. It is certainly
possible to rigidly designate our 1991 idea of what mathematics is (say by
"Mathematics"), but it's hopelessly parochial to suppose that this particular
referent has some priveleged ontological status among all the possible
approaches to quantitative modelling that we might have taken.
>If I found out that I had been misinformed and that they had really
>done something else, my concept of mathematics would not have to shift
>accordingly. It _might_ shift if what they had actually been doing
>led to a better notion of or understanding of what mathematics is.
It's this kind of use (in the last clause above) that confuses me. I can't
for the life of me figure out what 'mathematics' in that last clause is meant
to refer to. It isn't what people have done, nor even what you thought they
had done; what is it? You wouldn't *have* any idea of mathematics independent
of that history; as far as I can see, you *couldn't* have any such idea. So
just what is it that you think you might get a better notion of?
>I don't know about you, but I've enjoyed this discussion more than
>any in months. Really. And the misunderstandings and personality
>conflicts are nothing compared to uk.politics, for example.
Neat! I'm enjoying it, too, (though I get frustrated sometimes at my own
inability to convey the sense of my thoughts). I worry, though, about
sounding like I'm attacking people personally. I certainly have no such
intent. I think we're onto an interesting divergence in thought, here, and
I want to figure out exactly what it is...
--
David M. Tate |
dt...@unix.cis.pitt.edu | A journey of 1000 miles begins with
Motto: | locking your car keys in the trunk...
Gramen artificiosum odi |
Joseph O'Rourke
of the nature of mathematics, if David Tate and Jeff Dalton
could clarify what they mean by "mathematics." Here are two
dictionary definitions. Are either of you departing significantly
from the common definitions?
OED:
...[T]he abstract science which investigates deductively
the conclusions implicit in the elementary conceptions of
spatial and numerical relations, and which includes as its main
divisions geometry, arithmetic, and algebra.
Random House:
The systematic treatment of magnitude, relationships between
figures and forms, and relations between quantities expressed
symbolically.
Han de Bruijn
Several people attempted, but did not succeed in changing the subject line.
In article <135...@unix.cis.pitt.edu> David M. Tate in response to Keith Ramsay:
> However, I don't think this changes the problem of potentially coming up
> against an application area where the Dedekind reals are not a sufficiently
> accurate/valid/whatever model. [ ... ] it just means we may need something
> else as well.
Most of the time I agree with you, Dave. But here you are wrong. Reading the
last sentence, I must conclude that the real numbers, according to you, may
turn out to be a model which is _too limited_ for an accurate description of
real world phenomena. How sure I am that such a thing will never be the case!
The problem with the reals is _not_ that they are not general enough. Did it
ever occur to mathematicians, however, that the reverse also might represent a
problem? Yes, it did. Quoting without permission prof. Preston C. Hammer, from
an article called "Standards and Mathematical Terminology" (1972):
> It is my opinion, based on some years of consideration, that the inadequate
> generality of mathematics is one of its most serious drawbacks.
Therefore, how about a mathematical theory which is TOO GENERAL. Talking about
real numbers, there is evidence that a restricted setup would be sufficient for
all kind of applications. For example the Intuitionists have constructed a kind
of continuum which, in some respects, is more close to empirical evidence than
the classical model: Brouwer's theorem that every function which is _defined_
upon a closed interval of real numers is also _continuous_ at that interval.
Within the Intuitionistic approach, many pathological functions (like the one
that is 1 for all rational and 0 for all irrational numbers) drop out of the
system from the very start. You _never_ need them anyway.
I initiated a "Brouwer's theorem" thread, a year ago. Since everything can be
embedded in ZFC (:-) a person named Olson was provoked to prove the following
theorem: every real-valued function is continuous in a certain "sense". Where
sense is the integral of the function over (infinitesimally) small intervals.
A couple of weeks ago, Allan Adler posted a similar problem: given a suitable
dense subset of the reals, prove (or disprove) that every function which is
defined upon such a subset is also continuous.
So, a more restricted setup could rule out some of the fictious "possibilities"
which are "offered" by mathematics to Elementary Particle Physics, for example.
I have repeatedly pointed out that "additional", pure mathematical properties
of real numbers cannot be expected to be harmless, if they become applied in
such weird and uncomfortable areas of research like EPP or GR.
What Applied really needs is a "terse" continuum, with no more properties than
can be justified from an engineering point of view. Yes, Mathematics should be
based upon Engineering practice. (Jumping on my "hobby" horse, as allways :-)
Han de Bruijn
that mathematics could shift in a direction that has nothing to do anymore with
uhmm ... mathematics:
> In any case, if that is your worry, the same could happen for physics.
> The reality anchor wouldn't rule out such camouflage. If we looked more
> closely, we could detect that the physicists weren't doing physics anymore.
I couldn't agree with you more, Jeff! In My Not So Humble Opinion, Physics has
shifted already *a lot* towards a suspicious kind of pseudo Religion. Look at
area's like Elementary Particle Physics and Cosmology, where sound empirical
evidence has been overruled by the need for Nobel Prizes, where knowledge is
*owned* by a conspiracy of Big Institutes, funded by Military, uncontrollable
by the rest of the scientific community. I've been quite flamy about this in
the past (: perhaps someone remembers my "The Ultimate Arrogance" article in
sci.physics). Now taking my asbestos gloves ...
Han de Bruijn
> How about "Mathematics is the science of phenomena observable through
> computation" ?
Nice try, David.
I'm affraid, however, that for example _geometry_ drops out of your picture.
Quoting with his permission ;-), Sir Isaac Newton:
> Indeed however, geometry is just a part of mechanics. Since the _act_ of
> drawing a line or a circle belongs to mechanics; however, upon the drawing
> of those lines geometry is founded.
[Disclaimer: translated from a note in Dutch, back (?) into English. Don't know
what the precise reference is: "Principia" (latin)?]
What Newton says here, and what essentially is *my* viewpoint too: mathematics
is (= should be) based upon engineering practice, in a broad sense.
And vice versa, but that's another problem ...
David M Tate
>It would help at least this reader follow this interesting discussion
>of the nature of mathematics, if David Tate and Jeff Dalton
>could clarify what they mean by "mathematics."
But Joe, that's the goal here. Once we succeed, we're done :-).
>Here are two
>dictionary definitions. Are either of you departing significantly
>from the common definitions?
I certainly am. I think they're circular.
>OED:
> ...[T]he abstract science which investigates deductively
> the conclusions implicit in the elementary conceptions of
> spatial and numerical relations, and which includes as its main
> divisions geometry, arithmetic, and algebra.
Whose elementary conceptions? Mine? Yours? Archimedes'? In the days before
mathematics was studied, did this definition have a referent?
(I'm only worried about the first half of this one. The division into algebra,
geometry, etc. is clearly contingent, and not part of any definition of math
that tries to get beyond what we happen to have studied.)
>Random House:
> The systematic treatment of magnitude, relationships between
> figures and forms, and relations between quantities expressed
> symbolically.
A nice definition, but it could apply equally well to Abstract Impressionism
as far as I can tell. Necessary, but not sufficient.
Part of what I've been claiming is that there isn't really any closed form
definition like this that you could give for Jeff's notion of mathematics,
whatever that may be. Since I think his notion is probably pretty close to
a standard well-informed mathematical philosopher's definition, this makes me
wonder what math *really* is...
David M Tate
>
>In article <135...@unix.cis.pitt.edu> David M. Tate in response to Keith Ramsay:
>> However, I don't think this changes the problem of potentially coming up
>> against an application area where the Dedekind reals are not a sufficiently
>> accurate/valid/whatever model. [ ... ] it just means we may need something
>> else as well.
>
>Most of the time I agree with you, Dave. But here you are wrong. Reading the
>last sentence, I must conclude that the real numbers, according to you, may
>turn out to be a model which is _too limited_ for an accurate description of
>real world phenomena. How sure I am that such a thing will never be the case!
Oh, I don't think you *must* conclude that, especially since you'd be wrong.
I believe, as I think you do, that if we run into such a case, it will be one
in which the real numbers are too strong; that is, that the funny measure-
theoretic properties of the continuum which so endears it to analysts will
imply physical behaviors which are counterfactual. So far, with a few *very*
controversial possible exceptions, there's nothing about the reals that
causes us to make wildly inaccurate predictions about the processes we model
with them. It seems reasonable, for example, to use the Dedekind reals as
a model of 'quantity' in such everyday applications as measuring lengths and
temperatures, even though we can't measure with irrational accuracy. The
price of being wrong is smaller than our measurement error anyway. However,
that may not always be true for every conceivable application; when it isn't,
we will need a better (i.e. more accurate, less powerful) model of 'quantity'.
Since this new model will almost certainly be much harder to work with, though,
it behooves us to keep the computationally-pleasant Dedekind reals around for
ordinary applications, just as we keep Euclidean geometry and Newtonian
mechanics.
Matt Crawford
) Look at area's like Elementary Particle Physics and Cosmology, where sound
) empirical evidence has been overruled by the need for Nobel Prizes,
Give an example ... any example!
) where knowledge is *owned* by a conspiracy of Big Institutes, funded by
) Military, uncontrollable by the rest of the scientific community.
Every piece of physics produced by the big conspiracy universities gets
printed in a journal you can look at for free in any major science library.
You do not need to pay royalties to the scientists in order to read, build
on, or criticize what they publish.
) I've been quite flamy about this in the past
Cranks always are.
Jeff Dalton
>In article <49...@skye.ed.ac.uk> je...@aiai.UUCP (Jeff Dalton) writes:
>>In article <137...@unix.cis.pitt.edu> dt...@unix.cis.pitt.edu (David M Tate) writes:
Even # of >'s is Jeff Dalton; odd # of >'s is David Tate
>>>>No you're not. You want to use some bogus definition in which
>>>>mathematics is defined by what people who call themselves
>>>>mathematicians are doing.
> ["NO" in a very large font, perhaps aiming to be a picture and
> hence worth 1000 words.]
>I don't know how many times I have to deny this, Jeff, for you to believe
>me. My point has absolutely nothing to do with linguistic variance, or what
>name we choose to apply to anything.
I'm glad to hear it, but you keep saying things that I find hard to
interpret any other way. In particular, you introduced the idea that
mathematics is "that activity pursued by people who call themselves
mathematicians"; and you seem to think that mathematics can make
arbitrary shifts in its subject matter.
My point is simply that the word "mathematics" has some particular
meaning in our language; it is a particular subject matter or, if
you'd prefer, a particular (kind of) activity. Moreover, I am taking
"mathematics" as the more basic term, so that it isn't defined in
terms of mathematicians or whatever they happen to be doing. So far
as I an tell, the only way the word "mathematics" can refer to some
different subject matter (or activity) is if it changes its meaning.
Now, there clearly can be some change in mathematics. For example,
a new area can be opened up. Category theory and topology were new
areas at relatively recent points, for example. Then we can ask: does
this new area count as mathematics? If we say "yes", then this is,
in a sense, a change in the meaning of the word; but perhaps it's
better to think of it as a change in our knowledge of the referent.
(Consider an analogy with natural kind terms. If we learn more about
tigers, is that a change in the meaning of the word "tiger"? In any
case, it still refers to tigers.)
However, I think it must also be the case that for some things we
would say "no". So mathematics can't make an arbitray shift without
some more radical language change.
>>I think we have a resonably good idea of what mathematics is,
>>sufficiently good so that arbitrary shifts are not included, even if
>>we can't produce a one-sentence definition. And for all I know, we
>>_can_ have a neat, one-sentence definitions. What I do know is that
>>it is not "what people who call themselves mathematicians do".
>
>What I'm trying to get at is that I think the definition of mathematics that
>you are using, whatever it may be, is ill-formed. It either fails to refer
>to something that is inherently interesting (which I assume you want to say
>mathematics is), or fails to refer to anything that can't shift arbitrarily.
>Let's see if I can express myself a little better about why I think this...
I'm not even sure what it _means_ for something to be "inherently
interesting". I also have difficulty understanding what you mean
by "ill-formed". Perhaps a definition is ill-formed when it fails
to refer? But I can't see how failure to refer to something
inherently interwsting (or that can't shift arbitrarily) shows
a definition is ill-formed.
In another article, Joseph O'Rourke asks us to clarify what we mean by
"mathematics" and whether we are departing significantly from the
common definitions (as in the OED and the Random House dictionaries).
The two definitions look fairly reasonable to me, but they may well
have problems, and I don't want to get into the game of devising
perfect definitions.
I don't think it matters all that much if we don't come up with a
neat, one- or two-sentence definition of "mathematics". Indeed, it
may not be possible, just as (if Wittgenstein is right) it isn't for
"game".
>>>What corresponding reference point are you proposing for mathematics,
>>>which does not refer to what "mathematicians" have done in the past,
>>>yet could not, as you say, be the same as hairdressing? What reason
>>>do you have to prefer our 1991 idea of what mathematics is to the 500
>>>BC or 3917 idea?
Well, I did answer this in my previous message. To put it simply, I
use the 1991 idea because the language I'm using is 1991 English. Is
this supposed to be mysterious? That mathematics certainly isn't the
same as hairdressing. And since the word refers to mathematics, and
since mathematics includes things that mathematicians have done in the
past, it does refer to things mathematicians have done in the past.
But it's defined a a particular subject matter, not as whatever A, B,
and C (various past mathematicians) did. if we found out we had been
mistake and they were all really hairdressers, that wouldn't mean that
mathematics was hairdressing.
>>Since mathematicians _have_ done mathematics in the past, that part
>>of what they've done may well be involved. But I would be using past
>>mathematicians only to help fix the reference (ie show what it is I
>>am talking about).
>>
>>That doesn't mean that "mathematics" is _defined_ as what they did.
>
>Yes, it does,
No it doesn't. Perhaps the problem is that we disagree on what it
means to be defined as something. To me, if "mathematics" is defined
as "what A, B, and C did", and we think they did X, but they really
did Z, then mathematics is Z -- because Z is what A, B, and C did.
However, if we use what we think they did just to identify a certain
subject matter, X, and then it turns out that they really did Z, then
that's ok, we already found X, and mathematics is still X. In this
case, the definition of "mathamatics" has to be something (I'm not
sure what) that gives us X. Since "what A, B, and C did" gives us Z,
it can't be the correct definition of the word "mathematics".
>Yes, it does, in the crucial sense that if that's not what people (whatever
>you call them) had been interested in, then it's not what *we* would be
>interested in now, enough to give the subject a name.
Of course, someone had to do it. But that hardly means it's defined
in terms of those people, much less that it's defined in terms of
people we _think_ did it.
>It is certainly possible to rigidly designate our 1991 idea of what
>mathematics is (say by "Mathematics"), but it's hopelessly parochial
>to suppose that this particular referent has some priveleged ontological
>status among all the possible approaches to quantitative modelling that
>we might have taken.
I wouldn't say mathematics is synonymous with quantitative modelling.
In any case, "mathematics" is a word in the language we're using,
so why shouldn't we use the meaning that language gives it? I think
that meaning includes the possibility of certain developments and
changes, but not of arbitrary changes.
>>If I found out that I had been misinformed and that they had really
>>done something else, my concept of mathematics would not have to shift
>>accordingly. It _might_ shift if what they had actually been doing
>>led to a better notion of or understanding of what mathematics is.
>
>It's this kind of use (in the last clause above) that confuses me. I can't
>for the life of me figure out what 'mathematics' in that last clause is meant
>to refer to.
To mathematics.
>It isn't what people have done, nor even what you thought they
>had done; what is it?
A particular subject matter (more or less). Some people, presumably,
have done it.
Perhaps a way to look at it is this: if completely different people
did the mathematics in the past (on a different planet, say), then
that mathematics can be the same as our mathematics. So once we've
decided what mathemartics is, it's independent of which particular
people did it.
Of course, we haven't figured out mathematics competely, so it
still makes sense to look more closely, or in different ways, at
the mathematics certain people have done. But we can drop people
from our list if it turns out that they weren't doing mathematics
after all.
Note that we have ways to decide whether or not something is
mathematics apart from who did it.
>You wouldn't *have* any idea of mathematics independent of that history;
>as far as I can see, you *couldn't* have any such idea. So just what is
>it that you think you might get a better notion of?
But of course our notion of mathematics depends on the particular
history that lead to it; our whole language depends on history.
But that doesn't mean we don't have a fairly definite idea of what
mathematics is or that we can't discover more about it.
>I think we're onto an interesting divergence in thought, here, and
>I want to figure out exactly what it is...
Me too.
-- jeff
Jon J Thaler
says:
> [...stuff deleted...] In My Not So Humble Opinion,
> Physics has shifted already *a lot* towards a suspicious kind of pseudo
> Religion. Look at area's like Elementary Particle Physics and Cosmology,
> where sound empirical evidence has been overruled by the need for Nobel
> Prizes, where knowledge is *owned* by a conspiracy of Big Institutes,
> funded by Military, uncontrollable by the rest of the scientific
> community. I've been quite flamy about this in the past (: perhaps
> someone remembers my "The Ultimate Arrogance" article in sci.physics).
> Now taking my asbestos gloves ...
This doesn't make any sense to me. But of course, I'm a particle physicist,
so what do I know. A few points to think about. I'm sure Mr de Bruijn hasn't:
* In the USA, particle physics used to get money from the Office of Naval
Research (ONR), but that ended about 30 years ago.
* I know of no examples where "sound empirical evidence has been
overruled by the need for Nobel Prizes." I'm not even sure what point
M de Bruijn is making here. Does he? Nobel Prizes aren't usually given
for flummery.
* Regarding the claim that "knowledge is owned by a conspiracy..." To the
extent that I can decipher this statement, it sounds like a "small
science" vs "big science" argument. If M de Bruijn can state his
argument in a meaningful way, I'll be happy to discuss this issue.
* Regarding "I've been quite flamy about this in the past..." I think
M de Bruijn has the wrong ajective. "Ignorant" fits much better.
Neil Rickert
>My point is simply that the word "mathematics" has some particular
>meaning in our language; it is a particular subject matter or, if
>you'd prefer, a particular (kind of) activity. Moreover, I am taking
Nonsense.
"Mathematics" has some particular meaning in YOUR language. It has some
particular meaning in MY language. Who can say whether those two meanings
are identical, or even similar? My meaning of "mathematics" is continually
evolving, and yours probably is too, but who is to say that they are evolving
in the same ways?
Since mathematicians deal in precision, and since there is no agreement on
a precise definition of mathematics, it is silly to waste time arguing that
there should be. Quite possibly Godel's results guarantee that mathematics
can never be precisely defined.
--
=*=*=*=*=*=*=*=*=*=*=*=*=*=*=*=*=*=*=*=*=*=*=*=*=*=*=*=*=*=*=*=*=*=*=*=*=*=*=
Neil W. Rickert, Computer Science <ric...@cs.niu.edu>
Northern Illinois Univ.
DeKalb, IL 60115 +1-815-753-6940
David M Tate
>My point is simply that the word "mathematics" has some particular
>meaning in our language; it is a particular subject matter or, if
>you'd prefer, a particular (kind of) activity.
So far, so good.
> Moreover, I am taking
>"mathematics" as the more basic term, so that it isn't defined in
>terms of mathematicians or whatever they happen to be doing.
Or have done? Here's where I start getting antsy...
>Now, there clearly can be some change in mathematics. For example,
>a new area can be opened up. Category theory and topology were new
>areas at relatively recent points, for example. Then we can ask: does
>this new area count as mathematics? If we say "yes", then this is,
>in a sense, a change in the meaning of the word; but perhaps it's
>better to think of it as a change in our knowledge of the referent.
Hmm. Platonist alert.
So, as you see it, Topology and Category Theory were "out there", part of the
reference of 'mathematics', even before anyone ever did any topology or
category theory?
As I see it, this view leads to such odd consequences as the possibility that
we are *wrong* about (say) graph theory being part of mathematics. We thought
it was, but we were mistaken. Do we really want to allow this possibility?
I think that part of our trouble is in talking about "the field" of math in a
circular fashion. More on this later...
>(Consider an analogy with natural kind terms. If we learn more about
>tigers, is that a change in the meaning of the word "tiger"? In any
>case, it still refers to tigers.)
And it's possible that we were wrong. We can discover that, for example,
lions are not a kind of tiger, even though we thought they were. Can we
make such discoveries about mathematics?
>However, I think it must also be the case that for some things we
>would say "no". So mathematics can't make an arbitray shift without
>some more radical language change.
I don't think language change is the issue. You want mathematics to be a
rigid designator of a "field", in all possible worlds. On a Kripkean view,
this doesn't hurt in worlds where there is no mathematics (although I think
your views also sound like you don't believe there are any such worlds), but
it forces you to say that mathematics is identically the same in all possible
worlds where there is math. In particular, the extension of mathematics is
not context dependent; the history of a given possible world does not affect
what constitutes mathematics in that world.
Consider a possible analogy. There is a 'school' of modern art called
Abstract Impressionism. It has certain distinguishing characteristics, both
static and behavioral, and various tenets. People who are knowledgable about
art can tell Abstract Impressionism from, say, Dadaism or Surrealism or
Pointillism. Many artists feel that Abstract Impressionism is an important
'field'.
Now, it is tempting to talk about Abstract Impressionism as if it were a
Thing, independent of how it came about and who is doing it. Conceptually,
I suppose this is possible; I can imagine other possible worlds in which
Abstract Impressionism exists, with different origins and adherents. But
I think it's nonsensical to say that there are aspects of AbImp out there
waiting to be discovered, or that we could (collectively) be wrong about
what constitutes AbImp. We can make a rigid designator to refer to such a
(larger than we have now) concept of AbImp, but the extension we give it will
be arbitrary. We have no way of knowing how the concept will develop (just
as Archimedes had no way of anticipating Category Theory).
I see mathematics the same way. Yes, we can rigidly designate what we mean
today by mathematics, but we cannot meaningfully rigidly designate any larger
(or smaller, or different) body of thought as being True Mathematics, the
same in all worlds. Any such choice is arbitrary; we can pick one, but we
have no *reason* to.
>I don't think it matters all that much if we don't come up with a
>neat, one- or two-sentence definition of "mathematics". Indeed, it
>may not be possible, just as (if Wittgenstein is right) it isn't for
>"game".
For once, I agree with Ludwig :-).
>To put it simply, I
>use the 1991 idea because the language I'm using is 1991 English.
But you're attempting to refer to something which is not time or location
dependent. Are you asserting the identity of the two? True Mathematics
(which includes the Topologies and Category Theories and such that we haven't
yet discovered) is the same as the 1991 English referent of 'mathematics'?
>Is
>this supposed to be mysterious?
Yes!
>That mathematics certainly isn't the
>same as hairdressing. And since the word refers to mathematics, and
^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^
>since mathematics includes things that mathematicians have done in the
>past, it does refer to things mathematicians have done in the past.
Our 1991 English "mathematics" refers to True Mathematics? I doubt it. In
fact, since there is no True Mathematics, this would imply that 'mathematics'
fails to refer, which I also doubt.
>But it's defined a a particular subject matter, not as whatever A, B,
>and C (various past mathematicians) did. if we found out we had been
>mistake and they were all really hairdressers, that wouldn't mean that
>mathematics was hairdressing.
But it would mean that whatever mathematics is, it is independent of what
anyone thinks it is, including us 1991 English speakers. Do you still claim
that 1991E "mathematics" refers to mathematics? That we can be wrong about
whether something is math or not? That we can be doing math and not know it?
>No it doesn't. Perhaps the problem is that we disagree on what it
>means to be defined as something. To me, if "mathematics" is defined
>as "what A, B, and C did", and we think they did X, but they really
>did Z, then mathematics is Z -- because Z is what A, B, and C did.
I'm not saying that's the definition, simpliciter. I'm saying that the
definition cannot be made without reference to what A, B, and C did (or are
doing, or will do) for some A, B, and C.
>However, if we use what we think they did just to identify a certain
>subject matter, X, and then it turns out that they really did Z, then
>that's ok, we already found X, and mathematics is still X. In this
>case, the definition of "mathamatics" has to be something (I'm not
>sure what) that gives us X. Since "what A, B, and C did" gives us Z,
>it can't be the correct definition of the word "mathematics".
No, in this case it will be something involving what we *thought* A, B, and C
were doing. A sociological phenomenon nonetheless, though... I can see why
you're not sure what the 'something' might be. That's where my big hangup is
here. I can't see that there could be any such something.
>Of course, someone had to do it. But that hardly means it's defined
>in terms of those people, much less that it's defined in terms of
>people we _think_ did it.
Your personal definition of mathematics is some extrapolation from what you
believe certain past and present folk to have done. It is _not_, as I see it,
some incomplete glimpse of an Eternal Frame of True Mathematics, which exists
independently of any past endeavor. In possible worlds which have no history
of mathematics (as an activity), 'mathematics' doesn't refer any more than
Abstract Impressionism does.
>That meaning includes the possibility of certain developments and
>changes, but not of arbitrary changes.
But True Mathematics can't change; only our knowledge of it. But if you mean
our knowledge of it when you say 'mathematics', then you aren't really talking
about it the way you said you were...
>>>It _might_ shift if what they had actually been doing
>>>led to a better notion of or understanding of what mathematics is.
>>
>>It's this kind of use (in the last clause above) that confuses me. I can't
>>for the life of me figure out what 'mathematics' in that last clause is meant
>>to refer to.
>To mathematics.
Cute.
This is the inconsistency I see: you can't make assertions like the last one
unless you have some Platonic notion of True Mathematics. But, on that view,
you can't talk about the possibility of 'shifts' or 'changes' in math, because
True Math doesn't change. What's more, you'd have to admit (as I see it) the
possbility that everyone could stop doing True Math in favor of something else,
even while thinking they were still doing math. This contradicts the things
you were saying early on in this discussion, as far as I can tell.
>Perhaps a way to look at it is this: if completely different people
>did the mathematics in the past (on a different planet, say), then
>that mathematics can be the same as our mathematics. So once we've
>decided what mathemartics is, it's independent of which particular
>people did it.
But our decision concerning 'what mathematics is' is not independent. I'd
say it follows pretty easily from that that our referent when we use the word
'mathematics' so depends. But you say our referent when we do that is in fact
mathematics. Yes, once we pick a referent, we can (if it's coherent) stick
with it. But this is uninteresting; it's how we pick the referent that makes
the difference.
>Note that we have ways to decide whether or not something is
>mathematics apart from who did it.
We just can't be sure we're deciding correctly.
>But of course our notion of mathematics depends on the particular
>history that lead to it; our whole language depends on history.
Whoa! 5 yards for illegal offensive shift :-). The dependence of language
on history is, explicitly, *not* what we're talking about here. The choice
of words is irrelevant; the concept's the thing. Unless you're going to get
Whorfian on me...
Han de Bruijn
> Every piece of physics produced by the big conspiracy universities gets
> printed in a journal you can look at for free in any major science library.
> You do not need to pay royalties to the scientists in order to read, build
> on, or criticize what they publish.
Every piece of religion produced by the Pope can be looked up too.
The only problem is that *He* can talk with Gd, while *we* can't ;-)
In article <91168.180...@SLACVM.SLAC.STANFORD.EDU> Jon J Thaler:
> * Regarding the claim that "knowledge is owned by a conspiracy..." To the
> extent that I can decipher this statement, it sounds like a "small
> science" vs "big science" argument. If M de Bruijn can state his
> argument in a meaningful way, I'll be happy to discuss this issue.
Only a Happy Few have access to those Huge Test Facilities in the first place.
And you need a _tremendous_ amount of Knowledge to comprehend the details of
those Big Physics Experiments and associated Theories in the second place.
At least that's what They are telling us all the time.
Ordinary people also have NO free access to the kitchen of a restaurant.
But at least they are allowed to smell, taste and even consume the meals
that come out. I don't know (and I don't care) how VLSI chips are cooked,
but I can buy them, and put them to work. So I "believe" in them.
Not so with Big Science.
You just have to rely on what They write and what They say.
Ha! Some of the Big Experiments seem to have been conducted only _once_,
such as the one that "proved" the existence of the Omega Minus particle.
While *I* allways thought that physical experiments should be repeatable.
Nevertheless, a Holy Belief in Omega Minus particles has been established.
> * Regarding "I've been quite flamy about this in the past..." I think
> M de Bruijn has the wrong ajective. "Ignorant" fits much better.
Being ambitious, I got myself involved with Quantum Electro Dynamics,
to begin with, and COULD'NT manage to comprehend. Indeed!
^^^^^^^^
Ignorant, hmm ... Why then can I manage so many *other* things?
I've learned six languages, I've done theoretical physics, I've been
innovative in numerical methods, I'm building my own hard- and software,
I'm playing organ, I'm writing poems and music, have a nice job and
a good salary. Moreover, my social and family life is quite healthy.
Something must be wrong with my intelligence? I am a cranck, eh?
Yeah, sure, that's what They are yelling at Us all the time.
No! There is something wrong with my preparedness to ACCEPT.
Couldn't accept the Strangeness Quantum Numbers, simply because they
are just plain BS. Gd may be good, but He is not stupid.
To my great relief, I discovered that those magnificent theories of
Them cannot answer even the simplest questions, such as how big the
binding energy of a deuteron is. Whew, and those calculations are
REALLY difficult, folks! I have them here on one of my bookshelves.
> * I know of no examples where "sound empirical evidence has been
> overruled by the need for Nobel Prizes." I'm not even sure what point
> M de Bruijn is making here. Does he? Nobel Prizes aren't usually given
> for flummery.
You must be deaf and blind. I've spoken to people who left the Big Science
departments (: Astronomy) for exactly that reason. And Nobel Prizes *are*
given for flummery. Take the Nobel Prizes for Economy (of which Samuelson
is a good example). Or take the Nobel Prizes for Peace. Grrr ...
Back to Physics. After having read some popularizations, sometimes written
by the Geniouses themselves, books like "The Key of the Universe", I became
gradually convinced that those high Priests and their flunkeys were telling
me just ordinary, dammned ... * L I E S *.
Well, I'm not mad, actually, just *VERY* dissapointed ...
Chris Holt
In the discussion between Jeff Dalton and David Tate about what
the nature and meaning of mathematics is, I think they're not
actually arguing. On the other hand, I found myself lost in
an e-mail discussion with David, so perhaps I'm just misinterpreting
him...
dt...@unix.cis.pitt.edu (David M Tate) writes:
>Jeff Dalton:
>>However, I think it must also be the case that for some things we
>>would say "no". So mathematics can't make an arbitray shift without
>>some more radical language change.
I think Jeff is emphasizing the word "arbitrary" here. If definitions
of this kind are derived by taking examples (algebra, geometry, etc.),
finding common principles among these examples (heavens, I sound
like an objectivist! :-), and defining the words as referring to
those common principles, subject to the caveat that one's
knowledge is inherently incomplete so that certain kinds of
additions, shifts, and even perhaps deletions are allowed, then
it is not possible for *arbitrary* shifts to occur. Shifts may
only be of particular kinds, even if we may not be able to
fully specify the nature of those shifts.
>I don't think language change is the issue. You want mathematics to be a
>rigid designator of a "field", in all possible worlds.
I don't think he wants it to be "rigid"; but it is partially
constrained. For instance, would you accept a shift that
disallowed geometry?
[analogy with abstract impressionism (expressionism?)]
>I see mathematics the same way. Yes, we can rigidly designate what we mean
>today by mathematics, but we cannot meaningfully rigidly designate any larger
>(or smaller, or different) body of thought as being True Mathematics, the
>same in all worlds. Any such choice is arbitrary; we can pick one, but we
>have no *reason* to.
So you're saying the definition has fuzzy edges; and that's fine.
I don't think Jeff would disagree; of course it has fuzzy edges,
every definition does (unless it's defined purely in terms of
other words, and the question at hand has to do with that definition).
Of course we can't *rigidly* designate a body of thought; but
we can say that the choices are not arbitrary.
-----------------------------------------------------------------------------
Chris...@newcastle.ac.uk Computing Lab, U of Newcastle upon Tyne, UK
-----------------------------------------------------------------------------
"They have been at a great feast of languages, and stolen the scraps." - WS
Bradley K. Sherman
This therefore is Mathematics, she reminds you
of the invisible forms of the soul; she gives
life to her own discoveries; she awakens the
mind and purifies the intellect; she brings
light to our intrinsic ideas; she abolishes
oblivion and ignorance which are ours by birth.
--Proclus Diadochus
David M Tate
Of course! How could I forget. Geez, it was so obvious all along...
:-)
Thanks, Brad; that's exactly what this discussion needed.
--
David M. Tate | Not all the knives of the lampposts/ Nor the chisels
dt...@unix.cis.pitt.edu | of the long streets/ Nor the mallets of the domes/
Motto: | Can carve/ What one star can carve/ Shining through
Gramen artificiosum odi | the grape leaves. Wallace Stevens
Joseph O'Rourke
dt...@unix.cis.pitt.edu (David M Tate) writes:
>Part of what I've been claiming is that there isn't really any closed form
>definition like this [OED definition] that you could give for Jeff
>[Dalton]'s notion of mathematics...
Perhaps there is some middle ground between the belief that there could
only be such a definition for a rabid Platonist, and your concern that
mathematics might shift arbitrarily far from what we currently consider
mathematics. There are other human endeavors that both change greatly
over time, and yet remain recognizably the same endeavor: cooking,
dancing, drawing. One does not need to be a Platonist about dance
to believe that there is a constant across all cultures and all of
history, and to imagine this constancy extending to the future.
Mathematics is more problematical because it does not have the physical
grounding that cooking does, for instance. But in so far as you believe
abstract reasoning about quantity and form is inherent to humans,
you can believe in constancy in mathematics, despite the vast topical
changes over time. Platonism grounded in human nature.
I take our continued delight in the gems of Euclid and
Archimedes as evidence that despite the passage of 2,200 years,
a good portion of what they were doing is mathematics in the
twentieth-century sense.
David M Tate
>over time, and yet remain recognizably the same endeavor: cooking,
>dancing, drawing. One does not need to be a Platonist about dance
>to believe that there is a constant across all cultures and all of
>history, and to imagine this constancy extending to the future.
>Mathematics is more problematical because it does not have the physical
>grounding that cooking does, for instance. But in so far as you believe
>abstract reasoning about quantity and form is inherent to humans,
>you can believe in constancy in mathematics, despite the vast topical
>changes over time. Platonism grounded in human nature.
But not all abstract reasoning about quantity and form is mathematics. Or,
at least, I don't think Jeff would say that. Some of it is philosophy; some
of it is theology; some of it is art; some of it is engineering.
As you point out, the problem with the given examples (cooking, dance,
drawing) is that they have obvious *functional* definitions: cooking is
the preparation of food. Dance is stylized body movement for artistic
purposes. We can give a similar description of drawing, it seems to me. But
we can't characterize math in such a way, or at least I can't think of any
form such a characterization could take. The fact that the range of change
inherent in these activities is limited by the functional characterization
(which is independent of current fashions or customs) merely shows that math
is not like cooking, I would think.
Ian Sutherland
>As you point out, the problem with the given examples (cooking, dance,
>drawing) is that they have obvious *functional* definitions: cooking is
>the preparation of food. Dance is stylized body movement for artistic
>purposes.
Are you by any chance a dancer Mr. Tate? I'd be very surprised if you
were. I think I'd get to a satisfactory definition of mathematics LONG
before I'd get to a satisfactory definition of "dance". It certainly
wouldn't look like your definition, and it probably wouldn't be
"functional".
>The fact that the range of change
>inherent in these activities is limited by the functional characterization
>(which is independent of current fashions or customs) merely shows that math
>is not like cooking, I would think.
I think "dance" as a whole evolves a hell of a lot faster than
mathematics, the constancy of certain FORMS of dance notwithstanding.
--
Ian Sutherland i...@cambridge.oracorp.com
Sans peur
Joseph O'Rourke
essay "The Mathematician" [reprinted in "The World of Mathematics"].
His comments are relevant to the discussion in this newsgroup:
he expresses a concern similar to David Tate's, about mathematics
drifting, but he suggests that mathematics is grounded in experience
and therefore can always "return to the source." This suggests
a reason to believe in its constancy over time. Here is the
passage that most speaks to this issue:
I think that it is a relatively good approximation to truth ...
that mathematical ideas originate in empirics, although the
genealogy is sometimes long and obscure. But, once they are
so conceived, the subject begins to live a peculiar life of its
own, and is better compared to a creative one, governed by almost
entirely aesthetical motivations... There is, however, a further
point which, I believe, needs stressing...[T]here is a grave
danger that the subject will develop along the line of least
resistance, that the stream, so far from its source, will
separate into a multitude of insignificant branches, and that
the discipline will become a disorganized mass of details and
complexities.
...[W]henever this stage is reached, the only remedy
seems to me to be the rejunvenating return to the source: the
reinjection of more or less directly empirical ideas.
David M Tate
>essay "The Mathematician" [reprinted in "The World of Mathematics"].
>His comments are relevant to the discussion in this newsgroup:
[Interesting quotation deleted.]
Thanks, Joe. I was starting to feel like a crackpot. But if von Neumann
shared my concern, it's at least an illustrious circle of crackpots that I
share. :-)
--
David M. Tate | Soap and education are not so sudden as a massacre,
dt...@unix.cis.pitt.edu | but they are more deadly in the long run.
Dept. of Industrial Eng.|
Gramen artificiosum odi | Mark Twain, _Sketches_New_and_Old_
Jeff Dalton
>dt...@unix.cis.pitt.edu (David M Tate) writes:
>>Jeff Dalton:
>
>>>However, I think it must also be the case that for some things we
>>>would say "no". So mathematics can't make an arbitray shift without
>>>some more radical language change.
>
>I think Jeff is emphasizing the word "arbitrary" here.
Yes, exactly. It couldn't become hairdressing, for example,
or at least not "hairdressing as we know it".
Quite simply, we mean something by the word "mathematics", and
we can tell, fairly well, whether something is mathematics or not.
I am just suggesting that we can extend the same ability over
time.
[(We can also distinguish between something that is mathematics and
something that's merely _called_ mathematics. _Anything_ might be
called mathematics. (Think of the "call me a cab" joke.) And that
means that the mere fact that some people call something mathematics,
now or in the future, doesn't determine whether it _is_ mathematics
for us. And so we can distinguish between (1) mathematics becomming
"useless" because no one works on the parts that have applications and
(2) some other activity being called mathematics and treated as if it
were mathematics (by getting funding intended for science, for example).
But perhaps it's more confusing than helpful to point this out.]
>If definitions of this kind are derived by taking examples (algebra,
>geometry, etc.), finding common principles among these examples
>(heavens, I sound like an objectivist! :-), and defining the words as
>referring to those common principles, subject to the caveat that one's
>knowledge is inherently incomplete so that certain kinds of additions,
>shifts, and even perhaps deletions are allowed, then it is not
>possible for *arbitrary* shifts to occur. Shifts may only be of
>particular kinds, even if we may not be able to fully specify the
>nature of those shifts.
That sounds reasonable to me. What I had in mind more directly was
simply that shifts couldn't take us arbitrarily far, but I like the]
idea that only certain kinds of changes are allowed and that we may
not be able to pin down completely what they are. (That's one way
to distinguish your position from the Objectivists'.)
It's also worth noting that if we regard some change as a "legal
move" that doesn't necessarily mean we would have to regard
arbitrarily many repetitions of it as legal, no mater what result
they produced. It should be clear that we can impose such limits
in games (eg, the repreated move rule in Chess); we can also
use them in other cases, even if we cannot say precisely what
the limit is (just as we don't have to be able to say precisely
how many snowflakes are required before we have a snowstorm).
-- jeff
David M Tate
>we can tell, fairly well, whether something is mathematics or not.
>I am just suggesting that we can extend the same ability over
>time.
But not over possible worlds. What we mean by mathematics (not "mathematics";
I'm not talking about possible different languages, but rather about what we
mean in English today by mathematics) is contingent; it depends intricately on
our past history of science and philosophy. Other races, or humans in other
possible worlds, might very well not see any sense in using a designator with
that particular extension. The fact that we do does not give that particular
field/subject/whatever any priveleged status among possible fields.
If we are going to allow mathematics to shift at all, then we are implicitly
talking about a new group of people and a new designator, since what we mean
today by "mathematics" cannot change. But this new group need not share our
priorities and convictions about what constitutes mathematics. Given our past,
I agree that it's extremely unlikely that anyone will ever consider
hairdressing to be identical to mathematics, but there are other directions
we can shift in. You seem to think "shift" has magnitude but not direction,
or that all directions must be equally (un)likely; I don't know why you think
this.
How much does the extension of the designator have to change before you would
say they're no longer talking about mathematics? (I don't intend this as a
sorites; I don't care where the line is, if there is one. My point is that
*their* rejection of *your* views about what is mathematics, and what isn't,
is no less priveledged than yours or mine.)
Jeff Dalton
Jeff Dalton (even # of >'s) and Dave Tate (odd # of >'s) continue:
>>My point is simply that the word "mathematics" has some particular
>>meaning in our language; it is a particular subject matter or, if
>>you'd prefer, a particular (kind of) activity.
>
>So far, so good.
Well, good. Because that's really all there is to it. If at some
future time the word "mathematics" (those letters, that sound) is
used to refer to something else, it is entirely possible that we
would not say that other thing was mathematics.
In order to even talk about such cases, however, we have to make use
of the language we're using now, in which "mathematics" refers to
mathematics, just so we have some way to refer to mathematics
no matter how some other people use the word.
For example, it is possible that for some random reason people
end up using "mathematics" to mean what we would mean by the
phrase "French cooking"; and in such a case surely we can say
"it doesn't mean mathematics any more". That we can make sense
of that story shows what the mathematics isn't just whatever's
called "mathematics".
Even if the change in meaning happens gradually, it is possible that
we would say "mathematics" stopped referring to mathematics. The only
way that would be impossible would be if whatever was called
"mathematics" just automatically _was_ mathematics.
Consequently, we can distinguish between two cases:
1. Mathemetics evolves in some way we don't like, while
remaining mathematics.
2. Mathematics makes a much greater shift, to something we
wouldn't consider mathematics.
I would say (2) is better described in some other way. It's
"no one does mathematics any more but the people who used to
now do some other thing" or "people are not calling some non-
methematics `mathematics', and giving it science money as if
it were mathematics".
>> Moreover, I am taking
>>"mathematics" as the more basic term, so that it isn't defined in
>>terms of mathematicians or whatever they happen to be doing.
>
>Or have done? Here's where I start getting antsy...
I am just trying to avoid a possible source of confusion. In a
dictionary, a word is often defined in terms of some other form of the
same word. Eg, "conductor: the person or thing that conducts". It
may be that it often doesn't matter which way around it's done. So
maybe it doesn't usually matter whether "mathematics" is defined in
terms of "mathematician" or vice versa. One reason it usually doesn't
matter is that dictionaries show a language as it is at a certain time
and are simply trying to get you to a meaning, one way or another.
But if one starts saying things like "what if mathematicians start
doing something else, such as hairdressing, rather than mathematics",
it will be confusing if it's thought that "mathematics" and "what
mathematicians do" must be synonymous.
>>Now, there clearly can be some change in mathematics. For example,
>>a new area can be opened up. Category theory and topology were new
>>areas at relatively recent points, for example. Then we can ask: does
>>this new area count as mathematics? If we say "yes", then this is,
>>in a sense, a change in the meaning of the word; but perhaps it's
>>better to think of it as a change in our knowledge of the referent.
>
>Hmm. Platonist alert.
>
>So, as you see it, Topology and Category Theory were "out there", part of
>the reference of 'mathematics', even before anyone ever did any topology
>or category theory?
I don't even know what it would mean for them to be "out there",
so I guess not.
But I don't think it matters for our discussion whether Topology is
invented or discovered, because we can talk about it only afterwards.
And at that point we can certainly ask whether it's part of
mathematics or not. I don't see any particular reason why we should
worry about whether it was _always_ part of mathematics or not.
>As I see it, this view leads to such odd consequences as the
>possibility that we are *wrong* about (say) graph theory being part of
>mathematics. We thought it was, but we were mistaken. Do we really
>want to allow this possibility?
Since graph theory is part of mathematics, I don't think we're going
to find out that it really isn't. However, I think it might be the
case that _something_ we though was mathematics turns out not to be,
why not? Are you suggesting we can never be mistaken about whether
something is mathematics?
>>(Consider an analogy with natural kind terms. If we learn more about
>>tigers, is that a change in the meaning of the word "tiger"? In any
>>case, it still refers to tigers.)
>
>And it's possible that we were wrong. We can discover that, for example,
>lions are not a kind of tiger, even though we thought they were. Can we
>make such discoveries about mathematics?
Ah, but we discover that because lions _aren't_ a kind of tiger.
To worry that we might discover that something that isn't part
of mathematics isn't part of mathematics just seems bizarre.
Moreover, in your example we don't discover that "tiger" doesn't refer
to tigers; we discover that some things we thought were tigers were
not.
>>However, I think it must also be the case that for some things we
>>would say "no". So mathematics can't make an arbitray shift without
>>some more radical language change.
>
>I don't think language change is the issue.
If the word "mathematics" can end up referring to something we
wouldn't regard as mathematics, how can that not be a language
change?
>You want mathematics to be a rigid designator of a "field", in all
>possible worlds.
I want "mathematics" to refer to mathematics, yes. For some reason,
you then add all sorts of rigidity to mathematics and start talking
of it as True Mathematics. Mathematics is somethign that can change
as new mathematics is invented/discovered, and ouir idea of what
mathematics is can change as we find out more about it.
>On a Kripkean view, this doesn't hurt in worlds where there is no
>mathematics (although I think your views also sound like you don't
>believe there are any such worlds), but it forces you to say that
>mathematics is identically the same in all possible worlds where
>there is math. In particular, the extension of mathematics is
>not context dependent; the history of a given possible world does
>not affect what constitutes mathematics in that world.
But I'm not forced to say anything of the kind. For example,
on a world where no one invented/discovered calculus, it wouldn't
be part of the mathematics known in that world. But when we pick
out what's mathematics in some world, we have to use our knowledge
of mathematics to do it. In particular, we don't say "whatever
these people _call_ "mathematics" is mathematics", because they
might call hairdressing "mathematics". We have to use our word,
but that doesn't mean mathematics can't have different contents
in different possible worlds.
If their mathematics was so different from ours that we couldn't
recognize it as mathematics, then it would look to us like a world
in which they didn't do mathematics, not a world in which maths
was some different thing.
Perhaps my mistake was to attempt to explain at length. I thought
the rigid designator stuff might be helpful; but if it turns out not
to be it can be dropped. I was using "tiger" as an example that
might be helpful, not saying mathematics was a natural kind in
the same way tiger is. Mathematics is a different kind of thing;
in particular it's a thing whose content can evolve as we invent/
discover more mathematics.
In any case, I think you are making a mistake about Kripke's view
of possible worlds. As I said above, it's our word "mathematics"
we use when describing a possible world.
>Consider a possible analogy. There is a 'school' of modern art called
>Abstract Impressionism. It has certain distinguishing characteristics, both
>static and behavioral, and various tenets. People who are knowledgable about
>art can tell Abstract Impressionism from, say, Dadaism or Surrealism or
>Pointillism. Many artists feel that Abstract Impressionism is an important
>'field'.
>
>Now, it is tempting to talk about Abstract Impressionism as if it were a
>Thing, independent of how it came about and who is doing it. Conceptually,
>I suppose this is possible; I can imagine other possible worlds in which
>Abstract Impressionism exists, with different origins and adherents.
Or maybe not. Maybe it isn't Abstract Impressionism unless it
fits into a certain history and the same pointings in a different
context wouldn't count as Abstract Impressionism. It depends on
what we mean by Abstract Impressionism.
>But I think it's nonsensical to say that there are aspects of AbImp
>out there waiting to be discovered, or that we could (collectively) be
>wrong about what constitutes AbImp.
Well there could certainly be AbImp _paintings_ out there waiting to
be discovered, and perhaps artists as well. Would that mean we were
(now) wrong about what constituted AbImp? I don't know, but I don't
think the answer will be able to resolve our dispute about maths.
>We can make a rigid designator to refer to such a (larger than we have
>now) concept of AbImp, but the extension we give it will be arbitrary.
>We have no way of knowing how the concept will develop (just
>as Archimedes had no way of anticipating Category Theory).
We know something about how the concept will develop because we
know something about what the concept is. For example, we know
it won't develop to include dog catching. I don't see how you
can disagree with this without saying that we don't have any
particular concept in mind at all and that AbImp might just
turn out to be anything whatsoever.
Please note that I do not think that whatever Archimedes thought was
mathematics _is_ mathematics. If his mathematics couldn't include
Category Theory, then he doesn't have the same notion of mathematics
we do. If it turns out that his notion is sufficiently different,
then we might have to start adding footnotes to the translation of
certain Greek words as "mathematics". But I think we know enough
about Archimedes so that we can regard such possibilities as
very unlikely.
>I see mathematics the same way. Yes, we can rigidly designate what we mean
>today by mathematics, but we cannot meaningfully rigidly designate any larger
>(or smaller, or different) body of thought as being True Mathematics, the
>same in all worlds. Any such choice is arbitrary; we can pick one, but we
>have no *reason* to.
Well, fine, let's not designate anything as True Mathematics, the same
in all worlds. But let's designate something, indeed mathematics, as
mathematics.
We cannot say how mathematics will develop in the future. But that
doesn't mean that anything whatsoever that comes up in the future
might count as mathematics. We can have good reasons for saying
that something is not mathematics; we're not stuck with arbitrary
choices based on reasons no better than, say, whether a coin comes
up heads or tails.
>>To put it simply, I
>>use the 1991 idea because the language I'm using is 1991 English.
>
>But you're attempting to refer to something which is not time or location
>dependent. Are you asserting the identity of the two? True Mathematics
>(which includes the Topologies and Category Theories and such that we haven't
>yet discovered) is the same as the 1991 English referent of 'mathematics'?
And later:
>Our 1991 English "mathematics" refers to True Mathematics? I doubt it. In
>fact, since there is no True Mathematics, this would imply that 'mathematics'
>fails to refer, which I also doubt.
I said "mathematics" referred to mathematics, not that it referred to
True Mathematics. And it really shouldn't be seen as mysterious that
our word "mathematics" refers to mathematics, because the 2nd instance
of the word in this sentence (the instance not quoted) is again our
1991 English word. Furthermore, our 1991 English word refers to
something that can develop over time, in ways we haven't anticipated.
But in order for us to regard these developments as mathematics, they
have to be developments we would regard as mathematics. It's only by
such use of our current ideas that we even know which developments to
conseider.
>>But it's defined a a particular subject matter, not as whatever A, B,
>>and C (various past mathematicians) did. If we found out we had been
>>mistake and they were all really hairdressers, that wouldn't mean that
>>mathematics was hairdressing.
>
>But it would mean that whatever mathematics is, it is independent of what
>anyone thinks it is, including us 1991 English speakers.
No, it wouldn't mean that.
>Do you still claim that 1991E "mathematics" refers to mathematics?
Of course! What do you think it refers to? Ice cream?
>That we can be wrong about whether something is math or not?
See above on lions, tigers, etc.
>That we can be doing math and not know it?
Many people have done math and not known it. I suspect it is
rather common.
>>No it doesn't. Perhaps the problem is that we disagree on what it
>>means to be defined as something. To me, if "mathematics" is defined
>>as "what A, B, and C did", and we think they did X, but they really
>>did Z, then mathematics is Z -- because Z is what A, B, and C did.
>
>I'm not saying that's the definition, simpliciter. I'm saying that the
>definition cannot be made without reference to what A, B, and C did (or are
>doing, or will do) for some A, B, and C.
Fine. Then once we've found what we're talking about, we don't have to
get there through A, B, and C every time.
>>However, if we use what we think they did just to identify a certain
>>subject matter, X, and then it turns out that they really did Z, then
>>that's ok, we already found X, and mathematics is still X. In this
>>case, the definition of "mathamatics" has to be something (I'm not
>>sure what) that gives us X. Since "what A, B, and C did" gives us Z,
>>it can't be the correct definition of the word "mathematics".
>
>No, in this case it will be something involving what we *thought* A, B, and C
>were doing. A sociological phenomenon nonetheless, though... I can see why
>you're not sure what the 'something' might be. That's where my big hangup is
>here. I can't see that there could be any such something.
We happened to use A, B, and C. But that's it. They're not an
essential part of the definition. As for what the "something"
might be, I don't think we need a nice, neat definition any more
than we do for "game". That hardly means we don't have any
definite ideas on the subject at all.
>>That meaning includes the possibility of certain developments and
>>changes, but not of arbitrary changes.
>
>But True Mathematics can't change; only our knowledge of it. But if you mean
>our knowledge of it when you say 'mathematics', then you aren't really talking
>about it the way you said you were...
Our mathematics include things not discovered/invented yet, provided
that they're things we'd regard as maths.
>True Math doesn't change. What's more, you'd have to admit (as I see it)
>the possbility that everyone could stop doing True Math in favor of
>something else, even while thinking they were still doing math. This
>contradicts the things you were saying early on in this discussion,
>as far as I can tell.
I think you have to tell me more precisely just what this example
is. But leave out the True Math. I have no way of knowing whether
math is True Math or not.
There are a number of ways in which we could stop doing math, while
thinking we were doing it still. For example, we could all forget
what the word means now and start thinking it meant painting fire
engines. So "mathematics" wouldn't mean mathematics after the change.
(Note that I'm still using our language, in which "mathematics" means
mathematics, to talk about a case in which the same letters/sound
mean something else. This is the same sort of thing I've talked
about before, so I don't think it amounts to self-contradiction.)
>But this is uninteresting; it's how we pick the referent that makes
>the difference.
It matters more what "red" means that which particular red things
I used when finding out. Of course, maybe the latter is more
interesting. It depends.
>>Note that we have ways to decide whether or not something is
>>mathematics apart from who did it.
>
>We just can't be sure we're deciding correctly.
So? Who says we need a foolproof, perfect way of deciding such things?
-- Jeff
David M Tate
>case that _something_ we though was mathematics turns out not to be,
>why not? Are you suggesting we can never be mistaken about whether
>something is mathematics?
Well, you seemed to be quite certain that you knew what the extension of
the term was. If it is possible to be mistaken, that means (if we admit that
you understand the word correctly) that things are/aren't examples of math
independently of whether people who use the word "mathematics" to refer to
mathematics *think* they are examples of math. This seems absurd to me.
>>I don't think language change is the issue.
>
>If the word "mathematics" can end up referring to something we
>wouldn't regard as mathematics, how can that not be a language
>change?
For the 27th time, I don't *care* what word we use.
>I want "mathematics" to refer to mathematics, yes. For some reason,
>you then add all sorts of rigidity to mathematics and start talking
>of it as True Mathematics. Mathematics is somethign that can change
>as new mathematics is invented/discovered, and ouir idea of what
>mathematics is can change as we find out more about it.
OK, to me this is just contradictory. If mathematics can change, then it
isn't mathematics (in your sense), since you have already stipulated that
mathematics is a time-independent concept with a fixed (by our common use
of the concept) extension. How could such a thing ever change? If the
concept changes (i.e. we replace the current concept with a new one), then
the new thing is no longer mathematics in the sense that you use the word
now. And yet you seem to think it *would* still be mathematics, even though
it wouldn't be the same thing you mean when you talk about mathematics in
this discussion, which you also insist doesn't depend on people's opinions
about what math is. Help me out here; isn't this contradictory?
>But I'm not forced to say anything of the kind. For example,
>on a world where no one invented/discovered calculus, it wouldn't
>be part of the mathematics known in that world. But when we pick
>out what's mathematics in some world, we have to use our knowledge
>of mathematics to do it. In particular, we don't say "whatever
>these people _call_ "mathematics" is mathematics", because they
>might call hairdressing "mathematics". We have to use our word,
>but that doesn't mean mathematics can't have different contents
>in different possible worlds.
But you've already claimed that the content of mathematics is invariant,
although we may not have discovered it all yet. More contradiction?
>If their mathematics was so different from ours that we couldn't
>recognize it as mathematics, then it would look to us like a world
>in which they didn't do mathematics, not a world in which maths
>was some different thing.
How can you call it "their mathematics" if, under your definition, it isn't
mathematics at all? What gives you the right to describe it thus? (*Don't*
talk about how people use the word *mathematics*; we agree that this is
irrelevant). This is important; I believe that it is possible for them to
have mathematics, which is different from our mathematics. I don't think
you are allowed to agree with me, though, given the other opinions you've
expressed. I see that as a weakness in your account; perhaps the key weakness.
>Perhaps my mistake was to attempt to explain at length.
:-) I know the feeling...
>We know something about how the concept will develop because we
>know something about what the concept is. For example, we know
>it won't develop to include dog catching. I don't see how you
>can disagree with this without saying that we don't have any
>particular concept in mind at all and that [mathematics] might just
>turn out to be anything whatsoever.
But I don't disagree. The fact that math can't include dog-catching doesn't
mean that math can't shift arbitrarily far from what we think of it as; it
just means it can't shift arbitrarily far in any direction.
>We cannot say how mathematics will develop in the future. But that
>doesn't mean that anything whatsoever that comes up in the future
>might count as mathematics. We can have good reasons for saying
>that something is not mathematics; we're not stuck with arbitrary
>choices based on reasons no better than, say, whether a coin comes
>up heads or tails.
We're stuck with arbitrary choices based on the history of science. Is that
significantly better? Even from the point of view of Joe in the year 9363?
>I said "mathematics" referred to mathematics, not that it referred to
>True Mathematics.
Your other assertions implied the equivalence of mathematics and True
Mathematics.
>Furthermore, our 1991 English word refers to
>something that can develop over time, in ways we haven't anticipated.
As I've said, I think this contradicts other things you've said about what
mathematics is. In particular, it is incompatible with the claim that math
is independent of what people think it is.
>But in order for us to regard these developments as mathematics, they
>have to be developments we would regard as mathematics. It's only by
>such use of our current ideas that we even know which developments to
>conseider.
If they are developments we would regard as mathematics, then the idea hasn't
changed. If I discover a new genus of frog, that hasn't changed my notion of
frog (except connotatively); it was my notion of frog that led me to consider
the new discovery a frog.
>>I'm not saying that's the definition, simpliciter. I'm saying that the
>>definition cannot be made without reference to what A, B, and C did (or are
>>doing, or will do) for some A, B, and C.
>
>Fine. Then once we've found what we're talking about, we don't have to
>get there through A, B, and C every time.
But I think we do, because when we abstract the thing away from how we got
there, we're left with something arbitrary and accidental in extension. We
have to see it as the current scenery in a long journey, not the end of the
road.
Jeff Dalton
>In article <49...@skye.ed.ac.uk> je...@aiai.UUCP (Jeff Dalton) writes:
>>
>>Quite simply, we mean something by the word "mathematics", and
>>we can tell, fairly well, whether something is mathematics or not.
>>I am just suggesting that we can extend the same ability over
>>time.
>
>But not over possible worlds. What we mean by mathematics (not "mathematics";
>I'm not talking about possible different languages, but rather about what we
>mean in English today by mathematics) is contingent; it depends intricately on
>our past history of science and philosophy. Other races, or humans in other
>possible worlds, might very well not see any sense in using a designator with
>that particular extension. The fact that we do does not give that particular
>field/subject/whatever any priveleged status among possible fields.
You are making a mistake about the implications of possible worlds.
We have to use our idea of mathematics in order to identify the
mathematics in other possible worlds. If their mathematics was so
different from ours that we couldn't recognize it as mathematics, then
it would look to us like a world in which they didn't do mathematics,
not a world in which maths was some different thing.
The reason the word "mathematics" keeps coming in is that it is
essential to distinguish between what's mathematics in some world
and what's called "mathematics".
There's nothing in any of this that says the mathematics we refer
to cannot be extended to include other possible mathematics.
>If we are going to allow mathematics to shift at all, then we are implicitly
>talking about a new group of people and a new designator, since what we mean
>today by "mathematics" cannot change. But this new group need not share our
>priorities and convictions about what constitutes mathematics.
So? Perhaps they are just calling something else "mathematics".
The only way to tell is to see how much and in what ways it's
like _our_ idea of mathematics.
>How much does the extension of the designator have to change before you would
>say they're no longer talking about mathematics? (I don't intend this as a
>sorites; I don't care where the line is, if there is one. My point is that
>*their* rejection of *your* views about what is mathematics, and what isn't,
>is no less priveledged than yours or mine.)
Priveledge doesn't enter into it at all. You seem to think I
am claiming some universal validity to one meaning of the word.
I'm not. I'm just claiming it _has_ a meaning, and that that's
what we're talking about when we're talking about mathematics.
Quite simply, you must distinguish between a case where we
disagree on what mathematics is (eg, maybe you don't think
Intuitionism should count) and where we're just talking about
different things. My claim is simply that if you make a big
enough shift, it's talking about different things.
I'm going to be away for three weeks, so you may want to delay any
articles you would like me to answer. But I will resume the
discussion when I return.
-- Jeff
Jeff Dalton
>In article <49...@skye.ed.ac.uk> je...@aiai.UUCP (Jeff Dalton) writes:
>>
>>...I think it might be the
>>case that _something_ we though was mathematics turns out not to be,
>>why not? Are you suggesting we can never be mistaken about whether
>>something is mathematics?
>
>Well, you seemed to be quite certain that you knew what the extension of
>the term was. If it is possible to be mistaken, that means (if we admit that
>you understand the word correctly) that things are/aren't examples of math
>independently of whether people who use the word "mathematics" to refer to
>mathematics *think* they are examples of math. This seems absurd to me.
Don't be rediculous. _Of course_ some people can be wrong
about whether something's mathematics or not, just as they
can be wrong about whether something's science or not.
On the other hand, given that we speakers of English in 1991
agree more or less on what the word "mathematics" means (or
on what mathematics is), then we can't be mistaken that it
means that if we're to count as understanding the word at all.
This idea of mathematics doesn't exist independently of us,
because it's _our_ idea.
>>>I don't think language change is the issue.
>>
>>If the word "mathematics" can end up referring to something we
>>wouldn't regard as mathematics, how can that not be a language
>>change?
>
>For the 27th time, I don't *care* what word we use.
Nor do I. Everything I said would work just as well for
whatever word we happened to use.
>>I want "mathematics" to refer to mathematics, yes. For some reason,
>>you then add all sorts of rigidity to mathematics and start talking
>>of it as True Mathematics. Mathematics is somethign that can change
>>as new mathematics is invented/discovered, and ouir idea of what
>>mathematics is can change as we find out more about it.
>
>OK, to me this is just contradictory. If mathematics can change, then it
>isn't mathematics (in your sense), since you have already stipulated that
>mathematics is a time-independent concept with a fixed (by our common use
>of the concept) extension. How could such a thing ever change?
That's just silly. "Bill" is a rigid designator of my friend
Bill, but that doesn't mean he can't grow old, cut his hair,
etc. And mathematics is also something that can change.
>But you've already claimed that the content of mathematics is invariant,
>although we may not have discovered it all yet. More contradiction?
I have never claimed the content of mathematics is invariant.
Indeed, I have said again and again that we can invent/discover
new mathematics. You see a contradiction because I've said some
things that you, but not I, think imply that content can't change.
>>If their mathematics was so different from ours that we couldn't
>>recognize it as mathematics, then it would look to us like a world
>>in which they didn't do mathematics, not a world in which maths
>>was some different thing.
>
>How can you call it "their mathematics" if, under your definition, it isn't
>mathematics at all? What gives you the right to describe it thus?
1. Try to magine a world in which mathematics is so different from
ours that we can't recognize it as mathematics.
2. Can that really happen? Wouldn't it rather be that we don't
see any mathematics there at all?
That is, I am saying that worlds as in (1) don't exist; instead
there are worlds as in (2). So I'm not calling it "their
mathematics"; I'm saying that's what we wouldn't do.
>>We know something about how the concept will develop because we
>>know something about what the concept is. For example, we know
>>it won't develop to include dog catching. I don't see how you
>>can disagree with this without saying that we don't have any
>>particular concept in mind at all and that [mathematics] might just
>>turn out to be anything whatsoever.
>
>But I don't disagree. The fact that math can't include dog-catching doesn't
>mean that math can't shift arbitrarily far from what we think of it as; it
>just means it can't shift arbitrarily far in any direction.
Oh, give me a break! In what direction can maths shift arbitrarily
far without it shifting arbitrarily far from what we think of it as?
Only in a direction in which it's still mathematics. So why worry?
>We're stuck with arbitrary choices based on the history of science.
That we even care about science at all is an "arbitrary"
historical choice in that sens.
>Your other assertions implied the equivalence of mathematics and True
>Mathematics.
But they don't imply that. Mostly you've just said they imply it,
with very little in the way of explanation except for some
suppositions about regid designators and a confusion (perhaps my
fault) about the tiger analogy. I tried to explain that in my
previous message. I mean it when I say these things. I don't
think there's a contradiction.
>>Furthermore, our 1991 English word refers to
>>something that can develop over time, in ways we haven't anticipated.
>
>As I've said, I think this contradicts other things you've said about what
>mathematics is. In particular, it is incompatible with the claim that math
>is independent of what people think it is.
It should be clear by now that I do not think it's independent
of what we think it is, because it's our idea of mathamatics
that we're using.
>>But in order for us to regard these developments as mathematics, they
>>have to be developments we would regard as mathematics. It's only by
>>such use of our current ideas that we even know which developments to
>>conseider.
>
>If they are developments we would regard as mathematics, then the idea
>hasn't changed.
It depends on just what me mean by "idea". How abstract is it?
How much does it involved the actual content? Maybe there's some
level at which there's no change, and maybe that's what you mean
by "the idea". But that hardly means there's no change at all.
Entire new branches of matrhematics can be invented/discovered,
and as far as I'm concerned that can enlarge our idea of what
mathematics is. If you mean something else by "idea", then fine.
In any case, I cannot tell at all what issue you meant "the idea
hasn't changed" to resolve.
>If I discover a new genus of frog, that hasn't changed my notion of
>frog (except connotatively); it was my notion of frog that led me to
>consider the new discovery a frog.
Well, there's some music that changed idea of music. And yet I
think both ideas were ideas _of music_, so there's some continuity
as well.
>>>I'm not saying that's the definition, simpliciter. I'm saying that the
>>>definition cannot be made without reference to what A, B, and C did (or are
>>>doing, or will do) for some A, B, and C.
>>
>>Fine. Then once we've found what we're talking about, we don't have to
>>get there through A, B, and C every time.
>
>But I think we do, because when we abstract the thing away from how we got
>there, we're left with something arbitrary and accidental in extension. We
>have to see it as the current scenery in a long journey, not the end of the
>road.
Let's suppose the exact mathematics we've invented/discovered so
far is arbitrary and accidental. That doesn't mean we'd never
recognize anything else as mathematics. Our idea of mathematics
is at least a bit more abstract than that. Indeed, I too see
it as the current scenery in a long journey. Howver, I think
we can also do at leats a fairly good job of distinguishing
between continuing that journey and deciding to go somewhere
else.
As I said in a message earlier today, I'll be away for 3 weeks, and
articles probably expire in a shorter time. So if you want me to
respond to something, please post it in, say, two or three weeks.
I will try to think about this from time to time.
-- Jeff
Steven Daryl McCullough
(Note: By some strange quirk of the Net, our node doesn't receive News
articles until a week after they are posted. Therefore, there is a
possibility that the controversy between David Tate and Jeff Dalton
has already been resolved by now. Not likely, but possible.)
In this argument about what mathematics really is, I confess that I
really don't understand David Tate's point of view. I would better
understand this uncertainty about what mathematics really is if there
were any cases of substantial disagreement, of something that one
group calls mathematics, but that another group strongly denies is
mathematics. Where is there an example of such disagreement? David
seems to think that there is no more meaning to mathematics than "that
which mathematicians do". This definition is ultimately circular,
since the defining characteristic of a mathematician is that he or she
does mathematics. Mathematics is no more defined by what
mathematicians do than puns are defined by what punsters do. Once we
get the idea of a pun, or of mathematics, we can recognize new
instances.
I'm sure that there must have been some stabs at defining mathematics
early on in this discussion, but I must have missed it. How about the
following: Mathematics is the study of abstract structures. By
"abstract structure" I mean a collection of objects whose properties
are determined definitionally or axiomatically, without reference to
what those objects "really are". A good rule of thumb is that if you
have to appeal to an experiment to know whether a statement is true,
then the statement is not mathematics. Anything that can be learned
about mathematics can be learned using brain, pencil and paper (or
computers, which to me are essentially automated scratch pads).
If the above is still too fuzzy, I will stick my neck out further, and
say that a subject is a field of mathematics if and only if it can be
axiomatized so that its "truths" can be derived from logic alone. This
is not to say that mathematicians are simply logicians, deriving
theorems from axioms---most mathematicians do not use logic or even
axioms at all in their work. However, if what they are doing is
precise and unambiguous enough to count as mathematics, then it will
be possible to axiomatize what they do. Mathematics is thus contrasted
with physics, which must appeal to experiment, and with art, which
must appeal to taste (so far eluding axiomatization). Now, if instead
of mathematics you want to talk about *good*, *useful*, or
*interesting* mathematics, I would agree that such designations are
culturally and historically determined.
Daryl McCullough
Neil Rickert
>mathematics. Where is there an example of such disagreement? David
>seems to think that there is no more meaning to mathematics than "that
>which mathematicians do". This definition is ultimately circular,
But surely the contents of mathematics is a social convention. It
is not determined by a definition. Any definition has the effect of
excluding areas from mathematics, rather than including them.
Mathematics is no more defineable than is philosophy or physics or
history.
>what those objects "really are". A good rule of thumb is that if you
>have to appeal to an experiment to know whether a statement is true,
>then the statement is not mathematics. Anything that can be learned
>about mathematics can be learned using brain, pencil and paper (or
>computers, which to me are essentially automated scratch pads).
Defining mathematical truth is not the same as defining mathematics.
>If the above is still too fuzzy, I will stick my neck out further, and
>say that a subject is a field of mathematics if and only if it can be
>axiomatized so that its "truths" can be derived from logic alone. This
Some mathematicians may feel that this is a definition of the totality
of knowledge. That the only "truths" are those derived from logic alone.
A subject such as astrology might have no truths. Therefore it is part
of mathematics by this definition, since all of its truths (i.e. the
empty set) can be derived from logic alone.
David M Tate
Perhaps you can help Jeff and me stop talking in circles...
In article <23...@oravax.UUCP> da...@oravax.UUCP (Steven Daryl McCullough) writes:
>In this argument about what mathematics really is, I confess that I
>really don't understand David Tate's point of view.
Well, it's not so much a "point of view" as an objection to Jeff Dalton's
point of view. I'll try to be more explicit about what I believe instead.
>I would better
>understand this uncertainty about what mathematics really is if there
>were any cases of substantial disagreement, of something that one
>group calls mathematics, but that another group strongly denies is
>mathematics. Where is there an example of such disagreement?
In the discussion below your proposed definition of mathematics, about three
paragraphs from now :-).
But I really don't see how this is relevant to your objection. If there is
no disagreement over the extension of mathematics, surely this is evidence
that mathematics is a purely conventional notion. That seems a much more
likely explanation than that we all agree on what the defining characteristics
are... (see below).
>David
>seems to think that there is no more meaning to mathematics than "that
>which mathematicians do".
Mmm... not quite, but I can see how you might have gotten that impression.
Let me digress a moment to make explicit a distinction I should have been
making all along, between the extension of a concept and the concept/idea
itself. (I do *not* intend the word 'concept' in its Objectivist sense or
connotations). For example, the concept 'frog' determines which particular
things in our experience we will refer to as frogs, and which particular as-
yet-undiscovered things we would also consider to be frogs. The extension
of the concept, at a particular time and for a particular thinker, is that
set of things he/she would currently say were examples of the idea. So,
there may be things that are frogs that are not part of my extension of the
notion 'frog'.
But this notion 'frog' comes from somewhere. Without intending to sound like
an Objectivist, one could plausibly argue that the notion arose through a
process of generalization from observed creatures with similar skin, legs,
voices, behaviour, etc. More recently, the extension shifted for many people
to be based on genetic information, and not gross physical characteristics.
So, I feel that our concept of mathematics is so completely determined by our
past history and current culture that it is silly (although perfectly
consistent) to abstract out a concept of mathematics which is intended to be
timeless, in the sense that 'frog' is. It has no intrinsic interest, apart
from being the way we happen to think (today) about the past activities of
certain people. This is especially true if we wish to further divorce this
concept from any of the pragmatic goals/purposes of the people who did these
activities in the past. I think I could make a pretty good argument that
Archimedes, for example, would laugh out loud at the idea of abstracting away
the problem-solving environment from the particular modelling approach used...
>This definition is ultimately circular,
>since the defining characteristic of a mathematician is that he or she
>does mathematics. Mathematics is no more defined by what
>mathematicians do than puns are defined by what punsters do. Once we
>get the idea of a pun, or of mathematics, we can recognize new
>instances.
But the key question is "how do we get the idea"? Yes, we may have a
working definition of what mathematics is that doesn't depend on whether
anyone is actually doing math, or what they *are* in fact doing, but that
definition (or concept or idea or notion) is inevitably based on what
mathematicians _have_done_ in the past, many of them without any reference
to "our" definition of what math is. Because that's what it is: _our_
definition, based on _our_ history. We deliberately tailor this notion
so that (for example) its extension includes all historical activities (and
current activities) that we wish to think of as math. It's a very parochial
way of looking at things, as I see it. Other people with other histories
may formulate analogous ideas about some mathematics-analogue, but there's
no reason to suppose it will be the same as our notion, or have the same
extension. These other ideas will "not be mathematics" in the trivial sense
of not being *our* idea mathematics, but it's not clear at all that there is
some objective standard by which we can compare our idea with theirs.
>I'm sure that there must have been some stabs at defining mathematics
>early on in this discussion, but I must have missed it. How about the
>following: Mathematics is the study of abstract structures. By
>"abstract structure" I mean a collection of objects whose properties
>are determined definitionally or axiomatically, without reference to
>what those objects "really are".
>If the above is still too fuzzy, I will stick my neck out further, and
>say that a subject is a field of mathematics if and only if it can be
>axiomatized so that its "truths" can be derived from logic alone.
To me, this is far too broad. It includes a large variety of activities,
such as learning to play Go or to program in C, which are not instances of
mathematics.
>Mathematics is thus contrasted
>with physics, which must appeal to experiment, and with art, which
>must appeal to taste (so far eluding axiomatization). Now, if instead
>of mathematics you want to talk about *good*, *useful*, or
>*interesting* mathematics, I would agree that such designations are
>culturally and historically determined.
It's a start :-).
I would disagree that physics must (in the sense of logical necessity) appeal
to experiment. It is a convention of our time that *good* physics must appeal
to experimental verification (or lack of falsification), but this hasn't always
been true. I don't think we would have any trouble describing the activities
of certain ancient philosophers as "physics", despite the haphazard attempts
at justification of their theories... I'm much more inclined to say that
an activity qualifies as an instance of physics by virtue of its conceived
purpose. As you say, what then qualifies as good or useful or interesting
physics is a cultural phenomenon.
Scott J Brickner
we could separate mathematics into two areas: the study of the language
(mathology?) and the use of the language (mathics?)
The study of the language is what Newton did when inventing calculus.
He added a new body of notation to the language which allowed the
description of subjects which were previously inaccessible to the
language. I think this is what is normally called "pure mathematics".
The use of the language is, I think, normally called "applied
mathematics". This is where the physicist uses the language of
mathematics to describe some real-world phenomenon or to predict the
behavior of some real-world system.
So what mathematics "really" is, I think, is a language. Just like
English, only more concise. A statement in mathematics (proposition?)
provides a compact, concise, symbolic representation of an idea.
Scott
-----
'Becoming lightning, the cow throws back the veil.' - Rigveda
Steven Daryl McCullough
In article <144...@unix.cis.pitt.edu>, dt...@unix.cis.pitt.edu (David M Tate)
writes:
> So, I feel that our concept of mathematics is so completely determined by
> our past history and current culture that it is silly (although perfectly
> consistent) to abstract out a concept of mathematics which is intended to be
> timeless, in the sense that 'frog' is.
I don't get this at all. Are you saying that it is more meaningful to
come up with a timeless notion of "frog" than it is to come up with a
timeless notion of "mathematics"? To me, "mathematics" is much *more*
timeless than "frog". I can imagine creatures made of silicon or
ionized plasma doing mathematics, while the concept of "frog" is
completely bound to this Earth.
> It has no intrinsic interest, apart from being the way we happen to
> think (today) about the past activities of certain people.
That may be true, but why is specifically true of mathematics? The
case could be made that *nothing* has intrinsic interest, independent
of culture. Why do you think that mathematics is more bound to culture
than, say, biology or physics is?
> This is especially true if we wish to further divorce this
> concept from any of the pragmatic goals/purposes of the people
> who did these activities in the past.
Mathematics can be used to serve almost *any* goal. That is why my
definition of mathematics does not make reference to particular goals
and purposes: that would be unnecessarily limiting.
> I think I could make a pretty good argument that Archimedes,
> for example, would laugh out loud at the idea of abstracting
> away the problem-solving environment from the particular
> modelling approach used...
I confess to being ignorant about what would amuse Archimedes, but I
believe the Greeks (some of them) did have the notion that numbers and
geometric objects are abstractions. That was why such people as Euclid
considered it important to be able to prove facts about geometry using
axioms, instead of empirically determining what is true. It was not
enough to draw a bunch of right triangles and measure the sides to be
able to say that the square of the hypotenuse is equal to the sums of
the squares of the other two sides.
Archimedes was certainly more practical-minded than Euclid, but I
would bet that he understood that the field of mathematics was
distinct from the uses to which it could be put.
> ...the key question is "how do we get the idea"? Yes, we may have a
> working definition of what mathematics is that doesn't depend on whether
> anyone is actually doing math, or what they *are* in fact doing, but that
> definition (or concept or idea or notion) is inevitably based on what
> mathematicians _have_done_ in the past, many of them without any reference
> to "our" definition of what math is.
Yes, our culture is responsible for us having the concept of
mathematics. How is it any different for *any* concept?
> Because that's what it is: _our_ definition, based on _our_ history.
> We deliberately tailor this notion so that (for example) its extension
> includes all historical activities (and current activities) that we wish
> to think of as math.
David, I don't understand what you are getting at here. *Of course* we
came up with a definition of mathematics that covered the things that
we think of as mathematics. How could it be otherwise?
> It's a very parochial way of looking at things, as I see it.
I don't see what is parochial about it, any more than *any* definition
is parochial. Perhaps there is a culture that uses the same word to
mean both "frog" and "ginko tree". Does that mean that we are being
parochial in insisting that a ginko tree is not a frog?
> Other people with other histories may formulate analogous ideas
> about some mathematics-analogue, but there's no reason to suppose it will
> be the same as our notion, or have the same extension.
I agree with Jeff's comments on this matter: this possibility doesn't
make sense to me, either. What could you possibly mean by
"mathematics-analogue"? What would cause you to say that an activity
in another culture, while having nothing to do with our notion of
mathematics, was, in fact, a mathematics-analogue?
> These other ideas will "not be mathematics" in the trivial sense of
> not being *our* idea mathematics, but it's not clear at all that
> there is some objective standard by which we can compare our idea
> with theirs.
This still doesn't make sense to me. You seem to be thinking of the
following possibility: In some hypothetical land the natives have
exists a concept. Now, the natives don't use the word "mathematics"
for this concept (not being speakers of English); they use the word
"gug". Furthermore, this concept has nothing to do with what *we*
think of as mathematics---to us, it seems as if the concept means
"dancing". Yet, you think that we are being parochial in saying
"That's not mathematics, that's dancing!"
Of course, when people use the word "mathematics" they mean their own
idea of mathematics (or dancing, or frogs, or whatever). Does that
say anything about how mathematics differs from any other field?
> >I'm sure that there must have been some stabs at defining mathematics
> >early on in this discussion, but I must have missed it. How about the
> >following: Mathematics is the study of abstract structures. By
> >"abstract structure" I mean a collection of objects whose properties
> >are determined definitionally or axiomatically, without reference to
> >what those objects "really are".
> >If the above is still too fuzzy, I will stick my neck out further, and
> >say that a subject is a field of mathematics if and only if it can be
> >axiomatized so that its "truths" can be derived from logic alone.
>
> To me, this is far too broad. It includes a large variety of activities,
> such as learning to play Go or to program in C, which are not instances of
> mathematics.
I would disagree. I think that the rules for playing Go or for
programming in C are indeed mathematics. The first analysis of
computer programming (before there were any computers) was done by
mathematicicans; the fact that computer science is now considered a
field of its own is because good computer science involves, in
addition to mathematical notions, practical notions that stem from the
fact that the program is going to be run on a real machine and
interact with real people. Some aspects of computer science, such as
denotational semantics and asymptotic complexity of algorithms, are
pure mathematics (and are done by mathematicians, largely).
The study of abstract games are definitely a branch of mathematics, in
my opinion, although the actual playing of them is not.
Daryl McCullough
Jeff Dalton
>AHA! Some other interested parties (Daryl, Neil) are getting into it.
>Perhaps you can help Jeff and me stop talking in circles...
We'll see...
>In article <23...@oravax.UUCP> da...@oravax.UUCP (Steven Daryl McCullough) writes:
>>In this argument about what mathematics really is, I confess that I
>>really don't understand David Tate's point of view.
>
>Well, it's not so much a "point of view" as an objection to Jeff Dalton's
>point of view. I'll try to be more explicit about what I believe instead.
My point of view is simply that mathematics is something we humans
started doing at some point, and that by "mathematics" we mean that
thing and not something else. Moreover, that thing is not just an
enumeration of the particular mathematics we have discovered/invented
up to now; it is more abstract. That is why we can say (with a
reasonable degree of reliability) of some _new_ thing whether it is
part of mathematics or not. (If mathematics meant some particular
list, then new things wouldn't be in the list and so souldn't be
mathematics.)
Since we mean some particular thing by "mathematics", we don't have to
worry that it will turn into some arbitrary other thing. Cases that
look like they may be such a change will turn out to be something
else, such as the word "mathematics" being used for something else
(and not mathemtaics) or mathematicians starting to do something
else (and not mathematics).
_Of course_ our notion of mathematics depends on our history. After
all, we might have hunted and gathered all these years and not
bothered with mathematics at all. (But then we couldn't be having
this discussion.)
To worry that this notion of mathematics is too historical is to worry
that there's some True Mathematics out there and that we might have
missed it.
Mathematics is just something we've developed (or, if you want,
discovered). We might not have. We might have done something else
instead. We might have divided knowledge up in a different way so
that mathematics wasn't considered a separate subject.
But since we have identified this particular subject, we can talk
about it. That doesn't mean that it has some ideal existence in
the world for forms or that it's True Math. It's just something
we've developed and so can talk about.
>Let me digress a moment to make explicit a distinction I should have been
>making all along, between the extension of a concept and the concept/idea
>itself. (I do *not* intend the word 'concept' in its Objectivist sense or
>connotations). For example, the concept 'frog' determines which particular
>things in our experience we will refer to as frogs, and which particular as-
>yet-undiscovered things we would also consider to be frogs. The extension
>of the concept, at a particular time and for a particular thinker, is that
>set of things he/she would currently say were examples of the idea. So,
>there may be things that are frogs that are not part of my extension of the
>notion 'frog'.
This particular analogy may be more confusing than helpful. The
extension of "frog" is a collection of physical objects (the frogs).
If there are any objects in the extension of "mathematics" (assuming
that we can apply extension to mathematics at all), they will have
to be abstract objects of some sort. As soon as we go that way,
David tate will start to say "Platonism alert!"; which is one reason
I haven't tried to make my points in terms of extension.
>But this notion 'frog' comes from somewhere. Without intending to sound like
>an Objectivist, one could plausibly argue that the notion arose through a
>process of generalization from observed creatures with similar skin, legs,
>voices, behaviour, etc. More recently, the extension shifted for many people
>to be based on genetic information, and not gross physical characteristics.
You don't have to sound like an Objectivist; you can sound like Kripke
at al.
>So, I feel that our concept of mathematics is so completely determined by our
>past history and current culture that it is silly (although perfectly
>consistent) to abstract out a concept of mathematics which is intended to be
>timeless, in the sense that 'frog' is. It has no intrinsic interest, apart
>from being the way we happen to think (today) about the past activities of
>certain people.
But that's just false. It isn't just what he happened to think of
past activities. That we've sliced knowledge up in a particular way,
so that mathematics is seen as a distinct subject, doesn't mean
there's nothing in mathamatics. There may be no intrinsic interest in
the particular way we've sliced, but there can be lots of interest in
the stuff we've happened to place in mathatics. (Whether it's
intrinsic interest, I don't know. Does anything have intrinsic
interest? Worrying about this seems a lot like worrying about
whether out math is true math.)
>But the key question is "how do we get the idea"? Yes, we may have a
>working definition of what mathematics is that doesn't depend on whether
>anyone is actually doing math, or what they *are* in fact doing, but that
>definition (or concept or idea or notion) is inevitably based on what
>mathematicians _have_done_ in the past, many of them without any reference
>to "our" definition of what math is. Because that's what it is: _our_
>definition, based on _our_ history. We deliberately tailor this notion
>so that (for example) its extension includes all historical activities (and
>current activities) that we wish to think of as math.
You make it sound like we've fudged. Rather than talk about true
math, we deliberately tailor the notion to exactly fit what we
happened to do.
But we don't have to do any deliberate tailoring. Mathematics
included the things we think of as mathematics -- how could it be
otherwise? We don't have to say "I wish X were math too; let's
tailpor the definition".
>It's a very parochial way of looking at things, as I see it.
So is using English or some other natural language rather than
some universal, perfect language. But that's the way it has to
be (perfect, universal languages not being available).
> Other people with other histories
>may formulate analogous ideas about some mathematics-analogue, but there's
>no reason to suppose it will be the same as our notion, or have the same
>extension.
Exactly. But that isn't cause for concern. Different people with
different histories don't have to divide things into concepts in
exactly the same way. That we use a particular division doesn't
amount to a claim that that divisioin is the one true division.
>These other ideas will "not be mathematics" in the trivial sense
>of not being *our* idea mathematics, but it's not clear at all that there is
>some objective standard by which we can compare our idea with theirs.
Your supposition is that they have a mathematics-analogue. The
closest match for MA in English is mathematics, but there's not an
exact fit. So we say things like "we consider some parts of MA
mathematics, but not other parts." What's wrong with that? Do you
think we couldn't meaningfully say such things without some "objective
standard", whatever that is? If so, I'd say you're wrong.
-- jeff
David M Tate
helped me clarify my thinking.
>My point of view is simply that mathematics is something we humans
>started doing at some point, and that by "mathematics" we mean that
>thing and not something else.
Hard to argue with this, although I'd love to see you point to "that thing".
> Moreover, that thing is not just an
>enumeration of the particular mathematics we have discovered/invented
>up to now; it is more abstract.
So what's the abstraction? What is it that makes this mathematics, but not
that?
> That is why we can say (with a
>reasonable degree of reliability) of some _new_ thing whether it is
>part of mathematics or not. (If mathematics meant some particular
>list, then new things wouldn't be in the list and so souldn't be
>mathematics.)
Fine. So, what is this test that we apply? What is the intellectual litmus
that turns red when something is math, and doesn't when it isn't? Every time
I try to nail down that test, I get something that is either too restrictive
(e.g. a simple list, like you reject above), or too broad (e.g. includes
physics and chess-playing). This makes me wonder if maybe there really
isn't any underlying notion, other than arbitrary historical convention.
>>We deliberately tailor this notion [of mathematics]
>>so that (for example) its extension includes all historical activities (and
>>current activities) that we wish to think of as math.
>
>You make it sound like we've fudged. Rather than talk about true
>math, we deliberately tailor the notion to exactly fit what we
>happened to do.
Exactly. I'm not saying this is necessarily *conscious* fudging, but it
has the same effect. We're attempting to fit an ex-post-facto concept to
observed activities more than we are pursuing activities which fit some
a prioir concept.
>But we don't have to do any deliberate tailoring. Mathematics
>included the things we think of as mathematics -- how could it be
>otherwise?
How indeed, when we *designed* the notion of mathematics to account for
precisely those activities? Our (current) notion, that is; the people of
the time had their own notion, different from ours, designed to match their
own pasts.
--
David M. Tate | "There's *always* enough room to swing a cat;
dt...@unix.cis.pitt.edu | it's just that sometimes it hits things."
Dept. of Industrial Eng.|
Gramen artificiosum odi | --Karen Willmes
Steven Daryl McCullough
writes:
> > That is why we can say (with a
> >reasonable degree of reliability) of some _new_ thing whether it is
> >part of mathematics or not. (If mathematics meant some particular
> >list, then new things wouldn't be in the list and so souldn't be
> >mathematics.)
>
> Fine. So, what is this test that we apply? What is the intellectual litmus
> that turns red when something is math, and doesn't when it isn't? Every time
> I try to nail down that test, I get something that is either too restrictive
> (e.g. a simple list, like you reject above), or too broad (e.g. includes
> physics and chess-playing). This makes me wonder if maybe there really
> isn't any underlying notion, other than arbitrary historical convention.
David, I've said this before, but for the record, I believe there *is*
such a litmus test for what is mathematics. I think that, according to
this litmus test the theory of chess and the theories of physics are
indeed part of mathematics. You believe that any definition of
mathematics that includes (parts of) chess and physics can't be
correct, simply because you are committed to the position that
mathematics is defined to be what mathematicians do (which Jeff and I
do not believe).
Perhaps you would be happy with the following compromise: There is a
fairly rigid, precisely defined area which we can call "mathish
studies". This includes all sorts of things; all of what is called
"mathematics", plus parts of physics, economics, game theory, etc.
A subset of mathish studies is taken up by people who call themselves
"mathematicians"; this subset is called "mathematics". The definition
of "mathish studies" is precise and objective, while the definition of
that subset called "mathematics" is subjective and dependent on
history and on the tastes of mathematicians.
> >Mathematics included the things we think of as mathematics -- how
> >could it be otherwise?
>
> How indeed, when we *designed* the notion of mathematics to account
> for precisely those activities?
David, I consider this to be completely mistaken. Why would someone
want to lump algebra, geometry, and arithmetic together and call it a
single thing, "mathematics", rather than lumping algebra, caber
tossing, and poetry?
Daryl McCullough
David M Tate
>such a litmus test for what is mathematics. I think that, according to
>this litmus test the theory of chess and the theories of physics are
>indeed part of mathematics.
But I'm not talking about the *theory* of chess; I'm talking about *playing*
chess, or at least learning to play chess (which can reasonably be argued to
be a non-ending process which includes playing chess). I am unwilling to
allow that people who are playing chess for the standard reasons (i.e. for
the hell of it, in order to win, to pass the time, etc.) are engaged in
doing mathematics. That what they are doing is subject to mathematical
analysis is irrelevant, I would think, to this point. As I understand your
proposed definition of what is math, it includes recreational playing of
chess.
>You believe that any definition of
>mathematics that includes (parts of) chess and physics can't be
>correct, simply because you are committed to the position that
>mathematics is defined to be what mathematicians do (which Jeff and I
>do not believe).
I don't think I've said anything which implies this. Parts of physics are
indeed mathematics; *physics* simpliciter is not. Parts of chess can be
described mathematically; playing chess is not a form of doing math.
For the record, I also don't believe that math is defined to be what
mathematicians do--I believe that most mathematicians today act as if
math were defined to be what mathematicians do. Personally, I think the
definition of math is tied up in the goals of the individual practitioner
of math, which is what started this whole mess :^).
>Perhaps you would be happy with the following compromise: There is a
>fairly rigid, precisely defined area which we can call "mathish
>studies". This includes all sorts of things; all of what is called
>"mathematics", plus parts of physics, economics, game theory, etc.
>
>A subset of mathish studies is taken up by people who call themselves
>"mathematicians"; this subset is called "mathematics". The definition
>of "mathish studies" is precise and objective, while the definition of
>that subset called "mathematics" is subjective and dependent on
>history and on the tastes of mathematicians.
Hmm. I think you're on to something good here. I wonder if we'd be able
to agree on the exact definition of "mathish studies", but I think this
distinction you draw is fundamental, and describes what has happened
historically very well.
>> >Mathematics included the things we think of as mathematics -- how
>> >could it be otherwise?
>>
>> How indeed, when we *designed* the notion of mathematics to account
>> for precisely those activities?
>
>David, I consider this to be completely mistaken. Why would someone
>want to lump algebra, geometry, and arithmetic together and call it a
>single thing, "mathematics", rather than lumping algebra, caber
>tossing, and poetry?
Well, as a first cut, algebra, geometry, and arithmetic are traditionally
associated with people who called themselves "mathematicians", and that
nomenclature has cultural momentum. If all mathematicians had been Scot
athletes, then who knows what we might today consider part of the math bag?
:^)
Stranger things have happened. The partition between astronomy and
astrology is fairly recent, though it seems obvious to us. We draw a
strong line between physicians and barbers, too, but that hasn't been
true in all times and places. Check out the evolution of chiropraxy
for a really wild random walk.
Besides, this is a purely cultural thing. The mediaeval Japanese *did*
lump (their equivalents of) algebra, caber-tossing, and poetry under the
heading "bushido". Similarity is a *very* fuzzy philosophical concept.
>
>Daryl McCullough
--
David M. Tate | "Ford Prefect...was played by the estimable
dt...@unix.cis.pitt.edu | Geoffrey McGivern, star of several Footlights
Less hair than Gary | revues. He was virtually typecast as the
Huckabay has middle names | disreputable alien..." Douglas Adams.
Jeff Dalton
That is why we can say (with a reasonable degree of reliability) of
some _new_ thing whether it is part of mathematics or not. (If
mathematics meant some particular list, then new things wouldn't be
in the list and so souldn't be mathematics.)
dtate:
Fine. So, what is this test that we apply? What is the
intellectual litmus that turns red when something is math, and
doesn't when it isn't? Every time I try to nail down that test, I
get something that is either too restrictive (e.g. a simple list,
like you reject above), or too broad (e.g. includes physics and
chess-playing). This makes me wonder if maybe there really isn't
any underlying notion, other than arbitrary historical convention.
daryl:
David, I've said this before, but for the record, I believe there
*is* such a litmus test for what is mathematics. I think that,
according to this litmus test the theory of chess and the theories
of physics are indeed part of mathematics.
I think there's a (fairly reliable, though not perfect) test too,
because othwerwise we'd have a lot more trouble than we do when
deciding whether or not something is mathematics.
This is quite different, however, from saying we can state some
simple rules (necessary and sufficient conditions) for something
to count as mathematics. Wittgenstien said (more or less) that
we couldn't do it for "game", which seems pretty reasonable to me,
and I don't see why we should expect to do any better for maths.
dtate:
I'm not saying this is necessarily *conscious* fudging, but it has
the same effect. We're attempting to fit an ex-post-facto concept
to observed activities more than we are pursuing activities which
fit some a prioir concept.
jdalton:
But we don't have to do any deliberate tailoring. Mathematics
included the things we think of as mathematics -- how could it be
otherwise?
dtate:
How indeed, when we *designed* the notion of mathematics to account
for precisely those activities?
No, think about it. X includes the things we think of as X -- this
will (usually at least) be true regardless of whether we designed X
or not.
In any case, we didn't design mathematics that way. We didn't take
a bunch of activities that we didn't already think of as maths and
then decide to call them maths. Maybe that's how the word developed
historically, I don't know, but it isn't how it works now.
What happens now is that there are activities that we already consider
mathematics; new activities can be developed as mathematics or they
could develop independently and be recognized as mathematics later on.
What may be causing confusion is that if we want to state a definition
of mathematics that doesn't include or exclude too much then we may
have to design it precisely to fit. However, such definitions are
just attempts to express explicitly and precisely something that we
already understand fairly well (just as we understand what games are).
Such attempts may well fail precisely because they fit the current
content of mathematics while being too rigid about future content.
That hardly shows that our actual notion of mathematics has the same
defect or indeed that it is designed in the same way at all.
dtate:
Our (current) notion, that is; the people of the time had their own
notion, different from ours, designed to match their own pasts.
Different notion of what? Is there some common thing we have
different notions of? How would we know?
The most we can say, I think, is that certain notions are similar to
each other and possibly that there is an historical link between them.
When the notions are sufficiently similar, then we would probably say
they were different notions of the same thing (eg, of maths). Otherwise,
we probably wouldn't, even when there is an historical link.
-- jd