Does a potential infinity actually exist?
david petry
It seems that there is a lot of confusion about the notion of a
"potential" infinity. Now I had always assumed that the people who
were confused by this concept didn't know how to use Google, but then
I tried myself to find a really clear explanation of the concept, and
I was unable to. So I guess it's up to me to explain it.
The idea of a potential infinity may be equated with the view of
infinity held by Gauss, Poincare, and Weyl. The following quotes
characterize this view:
Gauss (paraphrased): "The notion of a completed infinity doesn't
belong in mathematics; infinity is merely a figure of speech which
helps us talk about limits"
Poincare (quoted from Morris Klein): "Actual infinity does not exist.
What we call infinite is only the endless possibility of creating new
objects no matter how many exist already"
Hermann Weyl: "... classical logic was abstracted from the mathematics
of finite sets and their subsets .... Forgetful of this limited origin,
one afterwards mistook that logic for something above and prior to all
mathematics, and finally applied it, without justification, to the
mathematics of infinite sets. This is the Fall and original sin of
[Cantor's] set theory ...."
So the basic idea, as so eloquently stated by Weyl, is that infinite
sets do not exist in exactly the same sense as the sense in which
finite sets exist, and that mathematics should distinguish between the
two senses. The language of classical mathematics (in any sense the
reader may chose to interpret that phrase) does not make any
distinction whatsoever between different senses of existence, and
hence it impossible, or nearly so, to even talk about the potential
existence of infinity within the language of classical mathematics.
That seems to explain why there is so much confusion about the notion
of potential infinity.
The mathematical entities that have an "actual" existence are the
things that we can actually compute. Thus, for example, we can
actually compute an approximation of sqrt(2) accurate to seven digits,
and hence such an approximation actually exists. Likewise, with a
computer, we can compute a hundred digit approximation of sqrt(2), or
even a million digit approximation, and so such approximations
actually exist. In general, we can compute arbitrarily many digits of
sqrt(2), but we cannot actually compute infinitely many such digits.
That is, we cannot complete that task, and so the infinite string of
digits representing sqrt(2) exactly does not actually exist. Instead,
we say that it has only a potential existence, by which we mean that
we can only actually compute arbitrarily accurate approximations to
it.
What Gauss, Poincare and Weyl realized, but many mathematicians today
fail to realize, is that we don't actually need infinity to do
mathematics. That is certainly true about the mathematics of the real
world (i.e. the mathematics used in physics and computer science,
etc.) What we are actually interested in is the things that have an
actual existence, and the things with a potential existence should be
thought of as useful fictions or figures of speech which help us
reason about the things that actually exist. In other words, the
notion of infinity introduces no new theorems into mathematics;
everything that can be said using the notion of infinity could also be
said without it. (Or equivalently, those assertions that absolutely
require the axiom of infinity or the axiom of choice have nothing to
tell us about the things that actually exist)
To illustrate these points, consider the following two sentences:
1) sum (k=1..oo) 1/2^k = 1
2) Ap En Am (m>n) -> |sum (k=1..m) 1/2^k - 1| < 1/p
Here, 'A' and 'E' represent the universal and existential quantifiers,
and the variables p,n,m are natural numbers.
The two sentences say exactly the same thing, but the first sentence
invokes infinity, while the second doesn't. Most people would agree
that the first sentence is far more readily comprehended, and so we
see that we don't actually need infinity, but infinity is indeed a
very useful figure of speech.
So how would we develop a set theory if we insist that infinity has
only a potential existence? For starters, finite sets of integers are
no problem. They have an exact representation as data structures in a
computer, for example. The set of all integers is also not much of a
problem. We can define some sense in which the set of integers {1..N}
is an approximation to the set of all integers {1..oo}, and then we
can say that the set of all integers has a potential existence (i.e.
arbitrarily accurate approximations to it actually exist).
But what about the real numbers? The basic idea is that we have to
find a way to approximate the set of real numbers by entities that
actually exist. Let me outline one way in which the theory of real
numbers can be developed within the constraints imposed by maintaining
that infinity only has a potential existence.
Recall that with the usual topology, the set of real numbers has a
countable neighborhood base. The elements in that neighborhood base
could be taken to be the pairs of rational numbers, and we can
certainly say that that neighborhood base has a potential existence in
the same sense as the set of all integers does. Then, instead of
classical real analysis, we would develop a theory of interval
analysis (such a theory has indeed been developed). And it's a good
bet that interval analysis is capable of providing us with all the
tools we need for real world mathematics--after all, in the real
world, our measurements only yield real numbers within some interval;
they never give us infinite precision.
So in conclusion, when we say that infinity has only a potential
existence, we are making a distinction between different senses of the
word "existence". Classical mathematics fails to make the
distinction, and that failure pushes mathematics in the direction of
make-believe, and that's a deficiency of classical mathematics.
MoeBlee
wrote:
> It seems that there is a lot of confusion about the notion of a
> "potential" infinity. Now I had always assumed that the people who
> were confused by this concept didn't know how to use Google, but then
> I tried myself to find a really clear explanation of the concept, and
> I was unable to. So I guess it's up to me to explain it.
>
> The idea of a potential infinity may be equated with the view of
> infinity held by Gauss, Poincare, and Weyl. The following quotes
> characterize this view:
>
> Gauss (paraphrased): "The notion of a completed infinity doesn't
> belong in mathematics; infinity is merely a figure of speech which
> helps us talk about limits"
>
> Poincare (quoted from Morris Klein): "Actual infinity does not exist.
> What we call infinite is only the endless possibility of creating new
> objects no matter how many exist already"
>
> Hermann Weyl: "... classical logic was abstracted from the mathematics
> of finite sets and their subsets .... Forgetful of this limited origin,
> one afterwards mistook that logic for something above and prior to all
> mathematics, and finally applied it, without justification, to the
> mathematics of infinite sets. This is the Fall and original sin of
> [Cantor's] set theory ...."
In post after post after post, you depend too much on quotes - pretty
much argument by weight of authority.
> So the basic idea, as so eloquently stated by Weyl, is that infinite
> sets do not exist in exactly the same sense as the sense in which
> finite sets exist, and that mathematics should distinguish between the
> two senses. The language of classical mathematics (in any sense the
> reader may chose to interpret that phrase) does not make any
> distinction whatsoever between different senses of existence, and
> hence it impossible, or nearly so, to even talk about the potential
> existence of infinity within the language of classical mathematics.
> That seems to explain why there is so much confusion about the notion
> of potential infinity.
No, the problem is that so many people, such as you, intone over and
over about a difference between potential and actual infinity, but you
give only informal descriptions of certain intuitions and not formal,
mathematical definitions or axioms regarding potential infinity.
If you would either give 'potential infinity' as a primitive and
axioms with it or 'potential infinity' defined from primitives, then
there might be something of specific mathematical interest there.
> The mathematical entities that have an "actual" existence are the
> things that we can actually compute. Thus, for example, we can
> actually compute an approximation of sqrt(2) accurate to seven digits,
> and hence such an approximation actually exists. Likewise, with a
> computer, we can compute a hundred digit approximation of sqrt(2), or
> even a million digit approximation, and so such approximations
> actually exist. In general, we can compute arbitrarily many digits of
> sqrt(2), but we cannot actually compute infinitely many such digits.
> That is, we cannot complete that task, and so the infinite string of
> digits representing sqrt(2) exactly does not actually exist. Instead,
> we say that it has only a potential existence, by which we mean that
> we can only actually compute arbitrarily accurate approximations to
> it.
Hilbert long ago distinquished between the contentual and the ideal,
and since then mathematics has developed the notion of primitive
recursion and classes of certain kinds of mathematical sentences such
as to distinguish "degrees" of "distance" from primitive recursion.
But you don't offer a mathematical theory to make rigorous your
informal notion of potential infinity as approximation.
> What Gauss, Poincare and Weyl realized, but many mathematicians today
> fail to realize, is that we don't actually need infinity to do
> mathematics.
Depends on what "do mathematics" means. For certain people,
mathematics is "done" axiomatically. And the axioms of set theory
(which include the axiom of infinity) are efficient, intuitive, and
easy to work with as an axiomatization of ordinary working
mathematics. Perhaps there are finitistic axiomatizations of ordinary
working mathematics, but we can't compare them with the axioms of set
theory unless those finitistic axioms are presented to us. Of course,
you don't present anything like that, as you are too busy repeating
over and over your polemics and from your cache of quotations.
> That is certainly true about the mathematics of the real
> world (i.e. the mathematics used in physics and computer science,
> etc.) What we are actually interested in is the things that have an
> actual existence, and the things with a potential existence should be
> thought of as useful fictions or figures of speech which help us
> reason about the things that actually exist.
That is indeed the view of certain people who work in set theory and
set theoretic axiomatized mathematics.
> In other words, the
> notion of infinity introduces no new theorems into mathematics;
> everything that can be said using the notion of infinity could also be
> said without it.
Could be SAID...perhaps. But can be PROVEN is another matter. When you
provide some axioms and primitives that PROVE things about
approximations and the like, THEN you'll have gotten past your
currently over TWENTY YEAR rut of primarily intoning polemics over and
over again.
> (Or equivalently, those assertions that absolutely
> require the axiom of infinity or the axiom of choice have nothing to
> tell us about the things that actually exist)
Even in the very sense of a "useful fiction" that you mentioned, the
axiom of infinity DOES have a LOT to do with PROVING various theorems
of ordinary working mathematics.
Granted, it would be welcome to have an axiomatization that proves
ONLY what you take to be computational and scientific mathematical
theorems and doesn't prove also all the stuff about infinities. I'd be
very interested in studying such a thing (and I know there are
proposed theories along such lines; so I'm not YET well informed about
them only because my study time is limited and it has seemed to me
that the finitistic proposals I've perused briefly are a lot more
complicated - even in the syntax of the language - than set theory).
> To illustrate these points, consider the following two sentences:
>
> 1) sum (k=1..oo) 1/2^k = 1
> 2) Ap En Am (m>n) -> |sum (k=1..m) 1/2^k - 1| < 1/p
>
> Here, 'A' and 'E' represent the universal and existential quantifiers,
> and the variables p,n,m are natural numbers.
>
> The two sentences say exactly the same thing, but the first sentence
> invokes infinity, while the second doesn't. Most people would agree
> that the first sentence is far more readily comprehended, and so we
> see that we don't actually need infinity, but infinity is indeed a
> very useful figure of speech.
Again, you're going right past the difference between EXPRESSING in a
language and PROVING in a theory. To PROVE things about limits of
functions and such, we use an axiomatization that also happens to
prove things about the fictional (or 'ideal' or whatever) infinities.
So, again, let us know when you have axioms and primitives to make the
proofs but without infinities.
> So how would we develop a set theory if we insist that infinity has
> only a potential existence? For starters, finite sets of integers are
> no problem. They have an exact representation as data structures in a
> computer, for example. The set of all integers is also not much of a
> problem. We can define some sense in which the set of integers {1..N}
> is an approximation to the set of all integers {1..oo}, and then we
> can say that the set of all integers has a potential existence (i.e.
> arbitrarily accurate approximations to it actually exist).
When you get to "the set of all integers" you resort to vagueness and
posturing. I'd just like to know what is the syntax of your formal
language, what are your primitives, and what are your axioms.
> But what about the real numbers? The basic idea is that we have to
> find a way to approximate the set of real numbers by entities that
> actually exist. Let me outline one way in which the theory of real
> numbers can be developed within the constraints imposed by maintaining
> that infinity only has a potential existence.
>
> Recall that with the usual topology, the set of real numbers has a
> countable neighborhood base. The elements in that neighborhood base
> could be taken to be the pairs of rational numbers, and we can
> certainly say that that neighborhood base has a potential existence in
> the same sense as the set of all integers does. Then, instead of
> classical real analysis, we would develop a theory of interval
> analysis (such a theory has indeed been developed). And it's a good
> bet that interval analysis is capable of providing us with all the
> tools we need for real world mathematics--after all, in the real
> world, our measurements only yield real numbers within some interval;
> they never give us infinite precision.
As soon as you got the word "could", what followed was not a proposal
but some ruminations about what might be a proposal.
I'm not against informal brainstorming. It's fine that one has
informal notions and intutitions to propose, but it would help, at
least me, to sustain any interest if I had some assurance that the
ultimate objective is a rigorous, formal mathematical theory and not
just layer after layer after layer of informal notions.
> So in conclusion, when we say that infinity has only a potential
> existence, we are making a distinction between different senses of the
> word "existence". Classical mathematics fails to make the
> distinction, and that failure pushes mathematics in the direction of
> make-believe, and that's a deficiency of classical mathematics.
First, even finitistic mathematical objects are abstractions. And
aside from my point of view about that, Hilbert's notions about the
contentual and the ideal are a vastly richer and vastly superior point
of departure for explanation than your polemics. Second, when you come
up with a theory (not just a bunch of polemical objections and a
smattering of ruminations), then there might be something to talk
about in regards to a challenge to set theory.
MoeBlee
Calvin
collection is the collection of natural
numbers, whether you call it a set or not.
If this is not a finished infinity, as distinct
from a potential infinity, then there must be
at least one natural number n that doesn't yet
exist. But since n is a natural number, it exists.
Reductio ad absurdum.
aatu.kos...@xortec.fi
> If you would either give 'potential infinity' as a primitive and
> axioms with it or 'potential infinity' defined from primitives, then
> there might be something of specific mathematical interest there.
You'll find a systematic treatment of "potential infinity" in
intuitionistic mathematics, especially the theory of choice sequences.
Of course, as with any piece of mathematics, the principles that
concern choice sequences must be seen justified on the "informal"
intuitionistic understanding if we are to adopt them as axioms (of
intuitionistic analysis, say).
Intuitionistic analysis and intuitionistic mathematics in general will
probably not be to Petry's liking, though, highly abstract and
infinitary as they are.
--
Aatu Koskensilta (aatu.kos...@xortec.fi)
"Wovon man nicht sprechen kann, daruber muss man schweigen"
- Ludwig Wittgenstein, Tractatus Logico-Philosophicus
MoeBlee
> MoeBlee wrote:
> > If you would either give 'potential infinity' as a primitive and
> > axioms with it or 'potential infinity' defined from primitives, then
> > there might be something of specific mathematical interest there.
>
> You'll find a systematic treatment of "potential infinity" in
> intuitionistic mathematics, especially the theory of choice sequences.
> Of course, as with any piece of mathematics, the principles that
> concern choice sequences must be seen justified on the "informal"
> intuitionistic understanding if we are to adopt them as axioms (of
> intuitionistic analysis, say).
>
> Intuitionistic analysis and intuitionistic mathematics in general will
> probably not be to Petry's liking, though, highly abstract and
> infinitary as they are.
Here's where I'm somewhat in the fog: As far as formal theories go, I
know that Heyting arithmetic is the intuitionistic answer to PA, but
that doesn't give us an intuitionistic analysis. So, okay, I know that
there are various proposed constructivist set theories (I think
specifically intuitionistic ones too?) (e.g., in Troelstra & van
Dalen), but I don't know where there is a treatment that derives
analysis from such a set theory (hmm, I seem to have recalled looking
at Kleene & Vesley's 'The Foundations Of Intuitionistic Analysis' and
not finding a formal axiomatization of analysis, though maybe I
misremember that). And though I am familiar with the notion of a
choice sequence and with various intutionistic ideas about analysis
(mainly, I've read, but not throroughly studied, Heyting's book
'Intuitionisim'), I've not seen a book that clearly derives
intuitionisitc analsysis from axioms, while, on the other hand, an
axiomatic set theoretic basis for classical analysis is found in
numerous textbooks.
I do know the intuitionistic predicate calculus, so I'd like to see
some exact set of axioms (whether set theoretic or otherwise) then to
derive intutitionistic analysis. And then in that language and with
those axioms, if there is a formal definition of 'potential infinty'
then I'd like to see that too. Also, as I understand, Bishop's school
is constructivist but not necessarily intuitionistic, and I don't find
an axiomatization in Bishop & Bridges nor in Beeson (maybe this school
even eschews axiomatization?) (I haven't seen Bridges & Richman). Then
there's Martin-Lof, who I just don't understand from the beginning; I
think I need more study to catch up to approaching his work.
MoeBlee
david petry
> > If you would either give 'potential infinity' as a primitive and
> > axioms with it or 'potential infinity' defined from primitives, then
> > there might be something of specific mathematical interest there.
It would appear that Moeblee didn't understand anything I wrote. I'm
arguing that when we acknowledge that infinity only has a potential
existence, then we see that infinity is merely a figure of speech, and
anything that can be said using the word "infinity" could also be said
without it. The best I could do is present rules for transforming
sentences using "infinity" into sentences not using it, but the
example I gave in the article gives the main idea.
> Intuitionistic analysis and intuitionistic mathematics in general will
> probably not be to Petry's liking, though, highly abstract and
> infinitary as they are.
That is true. I argue that the ultimate purpose of mathematics is to
provide tools that help us understand the world we live in (i.e. it
should be of practical use to physicists, engineers, computer
scientists, etc.), and also that we should make liberal use of
Ockham's razor and KISS (keep it simple). It baffles me that
mathematicians, including most intuitionists, won't grudgingly
acknowledge that those are worthy ideals.
MoeBlee
wrote:
> On Aug 23, 9:39 am, aatu.koskensi...@xortec.fi wrote:
>
> > MoeBlee wrote:
> > > If you would either give 'potential infinity' as a primitive and
> > > axioms with it or 'potential infinity' defined from primitives, then
> > > there might be something of specific mathematical interest there.
>
> It would appear that Moeblee didn't understand anything I wrote. I'm
> arguing that when we acknowledge that infinity only has a potential
> existence, then we see that infinity is merely a figure of speech, and
> anything that can be said using the word "infinity" could also be said
> without it. The best I could do is present rules for transforming
> sentences using "infinity" into sentences not using it, but the
> example I gave in the article gives the main idea.
I addressed what you said specifically and in fair detail. Your not
responding to what I said but instead just saying that I don't
understand what you said is purely a rhetorical ploy by you.
MoeBlee
Virgil
david petry <david_lawr...@yahoo.com> wrote:
They may be worthy enough for those only interested in the mundane, but
not for those who are interested in the ideal.
toni.l...@gmail.com
wrote:
Unfortunately for David Petry and his agenda, reality is often far
more complex then the mathematical model that attempts to mimic it.
Furthermore, the mathematical analysis relying on infinite structures
that David Petry and other cranks like him attack is a million times
more useful in solving real-life problems than any of the proposed
"solutions" the cranks come up with. The mathematics championed by
David Petry that is used to tote up grocery bills is not suitable for
cracking, say, Navier-Stokes.
Occam's Razor thus says it is the cranks who we should expunge, not
completed infinities.
david petry
> Occam's Razor thus says it is the cranks who we should expunge, ...
Yes, of course, that would be the final solution, wouldn't it.
Jesse F. Hughes
That's pretty cute, coming from someone that advocates exterminating
"humanists".
--
Jesse F. Hughes
"Usenet is demonstrably dangerous. It needs to be regulated."
--James S. Harris, voice of reason and moderation
T.H. Ray
of finite sets and their subsets .... Forgetful of this limited origin,
one afterwards mistook that logic for something above and prior to all
mathematics, and finally applied it, without justification, to the
mathematics of infinite sets. This is the Fall and original sin of
[Cantor's] set theory ...."
So the basic idea, as so eloquently stated by Weyl, is that infinite
sets do not exist in exactly the same sense as the sense in which
finite sets exist, and that mathematics should distinguish between the
two senses. The language of classical mathematics (in any sense the
reader may chose to interpret that phrase) does not make any
distinction whatsoever between different senses of existence, and
hence it impossible, or nearly so, to even talk about the potential
existence of infinity within the language of classical mathematics.
That seems to explain why there is so much confusion about the notion
of potential infinity.<<<
"This is the essence of set theory: It considers not
only the sequence of numbers but also the totality of
its subsets as a closed aggregate of objects existing
in themselves. In this sense it is based on the
actually infinite. But once this has been accepted,
the vast structure of analysis has an unshakeable
firmness ..."
Weyl, Hermann [1949] Philosophy of Mathematics & Natural
Science, Princeton, p.46
"As I see it, mathematics owes its greatness precisely
to the fact that in nearly all its theorems what is
essentially *infinite* is given a finite resolution.
But this "infinitude" of the mathematical problems
springs from the very foundations of mathematics--
namely, *the infinite sequence of the natural numbers
and the concept of existence relevant to it.*
(* ...* = italics in the original)
Weyl, H. [1918] The Continuum: A critical examination
of the foundation of analysis, translated by
Stephen Pollard and Thomas Bole, Dover Books 1994
republication originally published by Thomas
Jefferson University Press,1987. p. 49
Stop abusing Weyl with your tendentious cherrypicked
quotes.
Tom
David R Tribble
>> If you would either give 'potential infinity' as a primitive and
>> axioms with it or 'potential infinity' defined from primitives, then
>> there might be something of specific mathematical interest there.
>
david petry wrote:
> It would appear that Moeblee didn't understand anything I wrote. I'm
> arguing that when we acknowledge that infinity only has a potential
> existence, then we see that infinity is merely a figure of speech, and
> anything that can be said using the word "infinity" could also be said
> without it. The best I could do is present rules for transforming
> sentences using "infinity" into sentences not using it, but the
> example I gave in the article gives the main idea.
Well, this is partially true, in that the limits in calculus and
analysis can be replaced with epsilon/delta statements.
MoeBlee wrote:
>> Intuitionistic analysis and intuitionistic mathematics in general will
>> probably not be to Petry's liking, though, highly abstract and
>> infinitary as they are.
>
david petry wrote:
> That is true. I argue that the ultimate purpose of mathematics is to
> provide tools that help us understand the world we live in (i.e. it
> should be of practical use to physicists, engineers, computer
> scientists, etc.), and also that we should make liberal use of
> Ockham's razor and KISS (keep it simple). It baffles me that
> mathematicians, including most intuitionists, won't grudgingly
> acknowledge that those are worthy ideals.
They are, but why should mathematics be limited to
only practical applications?
While it''s true that the early days of mathematics were
devoted to deriving useful applications to model reality,
that hasn't been the case for the last 300 years or so.
Virgil
The ultimate purpose of mathematics, at least to many mathematicians, is
to make beauty.
That it is also useful is merely a lagniappe.
T.H. Ray
Virgil wrote:
> The ultimate purpose of mathematics, at least to many
> mathematicians, is
> to make beauty.
> That it is also useful is merely a lagniappe.
> >
What a lovely new word I learned: lagniappe. Even
being from the deep South, I had not heard it before.
Tom
david petry
> Stop abusing Weyl with your tendentious cherrypicked
> quotes.
Weyl seemed to be something of a flip-flopper on the foundations of
mathematics. I don't think I'm abusing him.
MoeBlee
> MoeBlee wrote:
> >> Intuitionistic analysis and intuitionistic mathematics in general will
> >> probably not be to Petry's liking, though, highly abstract and
> >> infinitary as they are.
Aatu wrote that, not me.
MoeBlee
MoeBlee
wrote:
Probably about half (maybe more) of your thread introduction essays
that I've seen in the last couple of years begin with a series of
quotes you take to be favorable to your views. Rarely (if ever) do you
delve into the context of those quotes to discuss their author's views
in any depth or detail, but rather you let them ride for all they're
worth as snippets chosen by you for maximum polemical effect and from
which you then embark to give your usual polemical whining. Your
approach to presenting quotes from the literature is markedly immature
in that way. It's akin to people who write editorials on political
matters by first establishing a wall of authority by quoting the likes
of Jefferson, Franklin, or Lincoln. Continual use of that kind of
argument by impression of authority gets cheap fast.
MoeBlee
Virgil
david petry <david_lawr...@yahoo.com> wrote:
The early developers of anything new in mathematics (as in many other
fields) sometimes have troubles finding their way, and the true path
only becomes clear to those following later when most, if not all, of
the dead ends and false trails have been eliminated.
Han de Bruijn
Sure. And that true path is yours, of course.
Han de Bruijn
Tonico
> Virgil wrote:
> > In article <1188234041.674074.164...@x35g2000prf.googlegroups.com>,
> > david petry <david_lawrence_pe...@yahoo.com> wrote:
>
> >>On Aug 25, 11:06 am, "T.H. Ray" <thray...@aol.com> wrote:
>
> >>>Stop abusing Weyl with your tendentious cherrypicked
> >>>quotes.
>
> >>Weyl seemed to be something of a flip-flopper on the foundations of
> >>mathematics. I don't think I'm abusing him.
>
> > The early developers of anything new in mathematics (as in many other
> > fields) sometimes have troubles finding their way, and the true path
> > only becomes clear to those following later when most, if not all, of
> > the dead ends and false trails have been eliminated.
>
> Sure. And that true path is yours, of course.
>
*************************************************
It may be, of course, that he meant his path, but I think Virgil was
talking about the path science in general, and maths in particular,
follow.
Was this SO hard to grasp?
Regards
Tonio
Han de Bruijn
Ah, ah, ah. But, according to Virgil, mathematics is NOT science!
Han de Bruijn
Virgil
Kind of you to suggest it, but no, it is not mine, though I seem better
able to follow it than you do. Perhaps it is because I do not try to
force it to be what it is not.
>
> Han de Bruijn
Virgil
Han de Bruijn <Han.de...@DTO.TUDelft.NL> wrote:
According to a lot of people whose opinions I value and have to a large
degree adopted.
But HdB is not among them.
T.H. Ray
It isn't. Though we speak colloquially of the
"mathematical sciences," mathematics is a liberal
art.
Tom
david petry
> Though we speak colloquially of the
> "mathematical sciences," mathematics is a liberal
> art.
Only for elite pure mathematicians. For applied mathematicians,
including physicists, engineers, computer scientists, and economists,
mathematics is a science.
"Some scientists and mathematicians have suggested that mathematics,
no longer tied to its origins in physics, is developing into a baroque
art form, a thing of great embellishments and few uses; that
mathematics has been reduced to a mere game of meaningless marks on
paper." (Eric Schechter)
T.H. Ray
Nope. Mathematics is a liberal art by strictly
objective standards, and not simply a figment of
your coarse and tendentious philosophy.
The sciences are driven by a process of
falsification. That is, new results that some theory
cannot incorporate, guarantee a continual pruning of
the scientific canon of knowledge. For example, the
phlogiston theory of combustion was discarded when
results showed that combustion is merely rapid oxidation.
Another example is the abandonment of the ether theory of
wave propagation, in favor of general relativity.
The mathematical canon of knowledge, on the other hand,
never discards a theorem. The canon only grows. Some
theorems become less useful or less convenient for
proving other theorems, but none cease being
theorems. Mathematics studies propositions of the
form, A==>B, and progresses in linguistic facility.
NEITHER science nor mathematics, however--contrary to
your personal belief--claims the anti-intellectual
standard of utility as a necessary condition. Both
are based on knowledge for the sake of knowledge alone.
Technology is no substitute for science. Computation
is no substitute for mathematics.
Tom
G. A. Edgar
> > "mathematical sciences," mathematics is a liberal
> > art.
In fact, two of the seven liberal arts.
--
G. A. Edgar http://www.math.ohio-state.edu/~edgar/
lui...@yahoo.com
wrote:
Many,many of us do not accept that argument.
For me and many of us, the ultimate purpose of mathematics is to
satiate the man desire of harmony by the creation of beautiful
structures through logic, sets ,numbers, spaces, functions, groups,
cellular automata etc.
Ludovicus.
Herman Rubin
david petry <david_lawr...@yahoo.com> wrote:
>On Aug 28, 3:21 pm, "T.H. Ray" <thray...@aol.com> wrote:
>> Though we speak colloquially of the
>> "mathematical sciences," mathematics is a liberal
>> art.
>Only for elite pure mathematicians. For applied mathematicians,
>including physicists, engineers, computer scientists, and economists,
>mathematics is a science.
No, it is still an art.
>"Some scientists and mathematicians have suggested that mathematics,
>no longer tied to its origins in physics, is developing into a baroque
>art form, a thing of great embellishments and few uses; that
>mathematics has been reduced to a mere game of meaningless marks on
>paper." (Eric Schechter)
These "meaningless marks on paper" often turn out later
to be of great practical value. Often they show that the
assumptions made by problem posers are incompatible with
each other, or that the computational problem the scientist
cannot solve can be handled by a device obtained by pure
mathematicians and apparently of no interest. The use of
group representations in physics is one. Another is the
fast Fourier transform, which Gauss did not think worthwhile
to pose. Computational complexity is aided by results in
algebraic geometry.
It is often the case that engineers do what mathematicians
and theoretical statisticians have already done.
--
This address is for information only. I do not claim that these views
are those of the Statistics Department or of Purdue University.
Herman Rubin, Department of Statistics, Purdue University
hru...@stat.purdue.edu Phone: (765)494-6054 FAX: (765)494-0558
T.H. Ray
> > > "mathematical sciences," mathematics is a liberal
> > > art.
>
> In fact, two of the seven liberal arts.
>
> --
> G. A. Edgar
>
>
>
>
(The Art of Mathematics) observation that one would
hardly consider a person educated, who could not
recite a least a few lines of poetry from memory, but
in our society a person can be considered highly
educated without ever having studied and memorized one
theorem. One.
Tom
>
>
>
>
>
>
>
> http://www.math.ohio-state.edu/~edgar/
Ken Pledger
david petry <david_lawr...@yahoo.com> wrote:
> On Aug 28, 3:21 pm, "T.H. Ray" <thray...@aol.com> wrote:
>
> > Though we speak colloquially of the
> > "mathematical sciences," mathematics is a liberal
> > art.
>
> Only for elite pure mathematicians. For applied mathematicians,
> including physicists, engineers, computer scientists, and economists,
> mathematics is a science....
Well, IMHO they're all wrong! The very idea that all knowledge
can be classified as art or science reeks of administrators trying to
put everything into the little boxes which they've chosen.
Mathematics is neither. It's unique. We don't rely on experiment
or observation of nature, but OTOH we're not free to choose the outcome
of a calculation as one might choose the next line of a poem.
Mathematics has its own written language, in which we may express
things which are beautiful, or of practical value, or both, or neither.
Much of that symbolic language is untranslatable - otherwise we wouldn't
need to use it. We can admire the achievements of mathematical heroes
fluent in that language, and we can encourage the halting efforts of
beginners in it.
Mathematical thinking is a very special mental activity, quite
different from any of the sciences or liberal arts. Mathematics is
mathematics. Be proud of it!
Ken Pledger.
Han de Bruijn
> It is often the case that engineers do what mathematicians
> and theoretical statisticians have already done.
How often? Any research done, or just handwaving?
I could easily state the opposite as well:
It is often the case that mathematicians do what engineers have already
done. (An example is the Finite Element Method in Numerical Analysis)
Han de Bruijn
Han de Bruijn
Yes. That happens in a society whih is essentially primitive
(except for its technology).
Han de Bruijn
mike3
> > > axioms with it or 'potential infinity' defined from primitives, then
> > > there might be something of specific mathematical interest there.
>
> arguing that when we acknowledge that infinity only has a potential
> existence, then we see that infinity is merely a figure of speech, and
> anything that can be said using the word "infinity" could also be said
> without it. The best I could do is present rules for transforming
> sentences using "infinity" into sentences not using it, but the
> example I gave in the article gives the main idea.
>
You need an axiom to define it.
What _axioms_ does "Potential Infinity"(TM) satisfy? Or how
can you derive it from some accepted set of axioms?
How can you manipulate "potential infinity"?
You say you can get rid of infinity, but how? And why, more
importantly? How do you say something that requires one to
invoke infinity without invoking infinity?
> > Intuitionistic analysis and intuitionistic mathematics in general will
> > probably not be to Petry's liking, though, highly abstract and
> > infinitary as they are.
>
mike3
> On Aug 27, 10:00 am, david petry <david_lawrence_pe...@yahoo.com>
> wrote:
>
> > On Aug 25, 11:06 am, "T.H. Ray" <thray...@aol.com> wrote:
>
> > > Stop abusing Weyl with your tendentious cherrypicked
> > > quotes.
>
> > Weyl seemed to be something of a flip-flopper on the foundations of
> > mathematics. I don't think I'm abusing him.
>
> Probably about half (maybe more) of your thread introduction essays
> that I've seen in the last couple of years begin with a series of
> quotes you take to be favorable to your views. Rarely (if ever) do you
> delve into the context of those quotes to discuss their author's views
> in any depth or detail,
I'd like to hear the context. Not saying "david petry" is right,
but I'm curious as to the correct interpretation of these quotes.
Han de Bruijn
> On Aug 23, 5:04 pm, david petry <david_lawrence_pe...@yahoo.com>
> wrote:
>
>>On Aug 23, 9:39 am, aatu.koskensi...@xortec.fi wrote:
>>
>>>MoeBlee wrote:
>>>
>>>>If you would either give 'potential infinity' as a primitive and
>>>>axioms with it or 'potential infinity' defined from primitives, then
>>>>there might be something of specific mathematical interest there.
>>
>>It would appear that Moeblee didn't understand anything I wrote. I'm
>>arguing that when we acknowledge that infinity only has a potential
>>existence, then we see that infinity is merely a figure of speech, and
>>anything that can be said using the word "infinity" could also be said
>>without it. The best I could do is present rules for transforming
>>sentences using "infinity" into sentences not using it, but the
>>example I gave in the article gives the main idea.
>
> You need an axiom to define it.
>
> What _axioms_ does "Potential Infinity"(TM) satisfy? Or how
> can you derive it from some accepted set of axioms?
The "need" for axioms is a _peculiarity_ which originally stems from
Euclidian geometry. It's a great invention, but it's _not_ a panacea
for everything. There is a school in mathematics, called constructivism,
the founder whereof, L.E.J. Brouwer, has said that axiomatics and logic
are accompanying the _act_ (i.e. construction) of mathematical entities,
but are by no means the foundations they can be based upon.
> How can you manipulate "potential infinity"?
Constructively, that is: not by means of an axiom system.
> You say you can get rid of infinity, but how? And why, more
> importantly? How do you say something that requires one to
> invoke infinity without invoking infinity?
>
>>>Intuitionistic analysis and intuitionistic mathematics in general will
>>>probably not be to Petry's liking, though, highly abstract and
>>>infinitary as they are.
>>
>>That is true. I argue that the ultimate purpose of mathematics is to
>>provide tools that help us understand the world we live in (i.e. it
>>should be of practical use to physicists, engineers, computer
>>scientists, etc.), and also that we should make liberal use of
>>Ockham's razor and KISS (keep it simple). It baffles me that
>>mathematicians, including most intuitionists, won't grudgingly
>>acknowledge that those are worthy ideals.
True, but: intuitionism < constructivism. Constructivism is everywhere
in Computer Science.
Han de Bruijn
T.H. Ray
You might be correct if, as you claim, some mathematics
could not be expressed in literary language. However,
in principle--though it may be pointless, impractical,
tedious and extremely difficult--all mathematics may
be so expressed.
Tom
T.H. Ray
You are begging the question. That Finite Element
Analysis for industrial applications inspired
new research techniques in number theory, does not
obviate the primacy of mathematical theory to explain
results--both physical results, from FEA in this case,
and abstract results derived from the same model.
Another example of experiment and observation leading
mathematical theory is the derivation of quantum
mechanics from Thomas Young's 2-slit experiment.
In both examples above, it is the mathematics which
explains the phenomenon. It is never the other
way around--phenonema do not explain themselves.
Theory is primary, and it requires no demonstration of
phenomena to stand on its own as a true result.
Whether it can be shown to model phenomena in nature or
not--a judgment still to be made, e.g., on string theory--
a true result mathematically stands as a true result.
If it is shown to model phenomena in nature, it is said
to be physically applicable; however, that is neither
a criterion nor a necessary condition to the mathematics.
Some theories, such as special and general relativity,
are created mathematically complete. That is, they
predict phenomena which later experiments validate.
Tom
Jesse F. Hughes
> T.H. Ray wrote:
>
>> Yes. I was always impressed with Jerry P. King's
>> (The Art of Mathematics) observation that one would
>> hardly consider a person educated, who could not recite a least a
>> few lines of poetry from memory, but
>> in our society a person can be considered highly
>> educated without ever having studied and memorized one
>> theorem. One.
>> Tom
>
> Yes. That happens in a society whih is essentially primitive
> (except for its technology).
Yeah, good point. Inability to recite poetry is a clear symptom of
social immaturity.
(Actually, Usenet is the clearest symptom I can find.)
--
Jesse F. Hughes
"And I will dream that I live underground and people will say, 'How
did you get there?'
"'I just live there,' I will tell them." -- Quincy P. Hughes, Age 4
mike3
So do you think the original poster's "potential infinity"
thing is a good idea?
Pubkeybreaker
david petry wrote:
> On Aug 28, 3:21 pm, "T.H. Ray" <thray...@aol.com> wrote:
>
> > Though we speak colloquially of the
> > "mathematical sciences," mathematics is a liberal
> > art.
>
> Only for elite pure mathematicians. For applied mathematicians,
> including physicists, engineers, computer scientists, and economists,
> mathematics is a science.
I have seen a lot of silly arguments in this newsgroup. This ranks
among them.
What <blankety-blank> DIFFERENCE does it make whether math is called
an art or a science?? It isn't worth arguing about!!!!! It is an
argument about a
LABEL.
Worrying about classifying a subject as opposed to discussing its
CONTENT
is just plain silly.
Math shares features with BOTH a liberal art and a science, so what
difference
does it make what we call it????
Han de Bruijn
> Math shares features with BOTH a liberal art and a science, so what
> difference does it make what we call it????
_This_ makes me feel better, MUCH better ..
Han de Bruijn
WM
Mathematics is driven by falsification too - and by majority decision.
Kant's friend Johann Schultz, among others, developed and used the
theory of infinite angular areas which was fashionable in the first
half of the 19th century. This theory was also propagated by Sylvestre
Franc Lacroix. In 1833 Adrien Marie Legendre present six "strict"
proofs of the parallel postulate to the Académie des Sciences, among
them three proofs using infinite triangular areas.
Cantor proudly wrote to his friend Schwarz about a proof that it had
been accepted as being strict by Weierstrass. Obviously this seemed
important to him - and there were other opinions.
Cantor's theory itself has been accepted by majority decision. It is
easy to falsify it by observing that elements of sets must be
distinguishable but the majority of the set of real numbers cannot
even be used individually (because only a countable number of real
numbers can be distinguished). Hence it is nonsense in maximal degree
to speak about uncountably many real numbers. It will last some time,
however, until the majority of mathematicians will recognize this
(most of them do not even know about the problem that non-adressable,
non-distiguishable, and hence unusable and, therefore, absolutely
useless numbers have to be acknowledged in order to maintain Cantor's
elegant "proof").
>
> The mathematical canon of knowledge, on the other hand,
> never discards a theorem.
This theorem is the first to be discarded. (Such claims belong to
matheology at most, they are foreign to sciences and arts.)
> The canon only grows. Some
> theorems become less useful or less convenient for
> proving other theorems, but none cease being
> theorems. Mathematics studies propositions of the
> form, A==>B, and progresses in linguistic facility.
or fallacy.
>
> NEITHER science nor mathematics, however--contrary to
> your personal belief--claims the anti-intellectual
> standard of utility as a necessary condition. Both
> are based on knowledge for the sake of knowledge alone.
> Technology is no substitute for science. Computation
> is no substitute for mathematics.
Mathematics can be shared, understood and done by foreign
intelligencies and computers. What cannot be computed is matheology.
Regards, WM
WM
Mathematics is driven by falsification too - and by majority decision.
Kant's friend Johann Schultz, among others, developed and used the
theory of infinite angular areas which was fashionable in the first
half of the 19th century. This theory was also propagated by Sylvestre
Franc Lacroix. In 1833 Adrien Marie Legendre present six "strict"
proofs of the parallel postulate to the Académie des Sciences, among
them three proofs using infinite triangular areas.
Cantor proudly wrote to his friend Schwarz about a proof that it had
been accepted as being strict by Weierstrass. Obviously this seemed
important to him - and there were other opinions.
Cantor's theory itself has been accepted by majority decision. It is
easy to falsify it by observing that elements of sets must be
distinguishable but the majority of the set of real numbers cannot
even be used individually (because only a countable number of real
numbers can be distinguished). Hence it is nonsense in maximal degree
to speak about uncountably many real numbers. It will last some time,
however, until the majority of mathematicians will recognize this
(most of them do not even know about the problem that non-adressable,
non-distiguishable, and hence unusable and, therefore, absolutely
useless numbers have to be acknowledged in order to maintain Cantor's
elegant "proof").
>
> The mathematical canon of knowledge, on the other hand,
> never discards a theorem.
This theorem is the first to be discarded. (Such claims belong to
matheology at most, they are foreign to sciences and arts.)
> The canon only grows. Some
> theorems become less useful or less convenient for
> proving other theorems, but none cease being
> theorems. Mathematics studies propositions of the
> form, A==>B, and progresses in linguistic facility
or fallacy.
>
> NEITHER science nor mathematics, however--contrary to
> your personal belief--claims the anti-intellectual
> standard of utility as a necessary condition. Both
> are based on knowledge for the sake of knowledge alone.
> Technology is no substitute for science. Computation
> is no substitute for mathematics.
>
Mathematics can be shared, understood and done by foreign
intelligencies and computers.
Regards, WM
Herman Rubin
In article <3974352.11884222033...@nitrogen.mathforum.org>,
>> --
>> G. A. Edgar
>Tom
On this topic, here is one that I read long ago.
At a meeting of a faculty committee to review
students not doing well, if someone named
Cicero flunked Latin or Shakespeare flunked
Literature, everyone laughed. But if Faraday
flunked physics or Gauss flunked mathematics,
only the scientists laughed.
G. A. Edgar
> >> > > "mathematical sciences," mathematics is a liberal
> >> > > art.
>
> >> In fact, two of the seven liberal arts.
>
Here is the listing...
http://i16.tinypic.com/6cs2hih.jpg
Herman Rubin
>> It is often the case that engineers do what mathematicians
>> and theoretical statisticians have already done.
>How often? Any research done, or just handwaving?
I have seen it many times; they do not know what
has been done, partly because they only know formulas
and do not have the basic concepts for the theorems.
Consider algebra. The German physicists and engineers
who coined the terms "Eigenwert" and "Eigenvektor"
were unaware that British algebraists had done this
years before, with different terminology, and not
in obscure journals.
Why did the normal distribution get the name Gaussian?
Now "normal" is not a good name, coined by Quetelet on
supposedly science grounds. But if anyone should be
the source of the name, it should be de Moivre, as he
had this 47 years before Gauss was born. This naming
was by physicists and engineers.
>I could easily state the opposite as well:
>It is often the case that mathematicians do what engineers have already
>done. (An example is the Finite Element Method in Numerical Analysis)
>Han de Bruijn
One should be careful about such methods, especially if
assumptions are not made clear. An excellent example is
the Feynmen path integral, which at first glance looks
simple and extremely useful. But when one looks at it, it
is not as indicated; the two "integrals" given are not as
stated, and only one of them can be even revived on a
mathematical basis. There are cases where there is a
supposed evaluation of the integral, but it is based on
showing that a method which gets the solution obtained by
other methods can be extended to work for a few cases.
Unless someone can come up with a general way of making
sense out of it, it is at best an approach, not
mathematics, in which certain limits happen to exist from
other considerations, and can be justified on still others.
Not all assumptions are consistent. It surprises many
that a Gamma process only changes by jumps, as it is
clearly continuous with probability one at each point.
High-order differentiability of initial conditions does
not imply that differentiability of results. One can
go wrong by jumping in with such solutions.
WM
Mathematics is driven by falsification too - and by majority decision.
Kant's friend Johann Schultz, among others, developed and used the
theory of infinite angular areas which was fashionable in the first
half of the 19th century. This theory was also propagated by Sylvestre
Franc Lacroix. In 1833 Adrien Marie Legendre present six "strict"
proofs of the parallel postulate to the Académie des Sciences, among
them three proofs using infinite triangular areas.
Cantor proudly wrote to his friend Schwarz about a proof that it had
been accepted as being strict by Weierstrass. Obviously this seemed
important to him - and there were other opinions.
Cantor's theory itself has been accepted by majority decision. It is
easy to falsify it by observing that elements of sets must be
distinguishable but the majority of the set of real numbers cannot
even be used individually (because only a countable number of real
numbers can be distinguished). Hence it is nonsense in maximal degree
to speak about uncountably many real numbers. It will last some time,
however, until the majority of mathematicians will recognize this
(most of them do not even know about the problem that non-adressable,
non-distiguishable, and hence unusable and, therefore, absolutely
useless numbers have to be acknowledged in order to maintain Cantor's
elegant "proof").
>
> The mathematical canon of knowledge, on the other hand,
> never discards a theorem.
This theorem is the first to be discarded. (Such claims belong to
matheology at most, they are foreign to sciences and arts.)
> The canon only grows. Some
> theorems become less useful or less convenient for
> proving other theorems, but none cease being
> theorems. Mathematics studies propositions of the
> form, A==>B, and progresses in linguistic facility
or fallacy.
>
> NEITHER science nor mathematics, however--contrary to
> your personal belief--claims the anti-intellectual
> standard of utility as a necessary condition. Both
> are based on knowledge for the sake of knowledge alone.
> Technology is no substitute for science. Computation
> is no substitute for mathematics.
>
Mathematics can be shared, understood and done by foreign
intelligences and computers.
Regards, WM
T.H. Ray
> In article
> <3974352.11884222033...@nitrogen.mathf
Ha!
Tom
T.H. Ray
example.
Tom
david petry
> david petry wrote:
> > On Aug 28, 3:21 pm, "T.H. Ray" <thray...@aol.com> wrote:
>
> > > Though we speak colloquially of the
> > > "mathematical sciences," mathematics is a liberal
> > > art.
>
> > Only for elite pure mathematicians. For applied mathematicians,
> > including physicists, engineers, computer scientists, and economists,
> > mathematics is a science.
>
> I have seen a lot of silly arguments in this newsgroup. This ranks
> among them.
>
> What <blankety-blank> DIFFERENCE does it make whether math is called
> an art or a science??
The underlying question being discussed is whether mathematics
*should* include Cantor's mythology of the infinite. If we think of
mathematics as a science, then the answer would be "no, we should
limit mathematics to more concrete objects as the constructivists do",
but if it's a liberal art, then the answer might be "sure, why not,
mathematics should include any story we might desire as long as it's
consistent and it resembles other mathematics in form".
T.H. Ray
art even when one adopts a constructivist philosophy. Constructivism
does not describe "concrete objects." It refers to the group
of mathematical statements that can be constructed in a finite
number of steps.
Tom
david petry
> That mathematics is a liberal art is not debatable.
And yet here we are debating it! Go figure.
G. A. Edgar
A. Edgar <ed...@math.ohio-state.edu.invalid> wrote:
> > >> > > Though we speak colloquially of the
> > >> > > "mathematical sciences," mathematics is a liberal
> > >> > > art.
> >
> > >> In fact, two of the seven liberal arts.
> >
>
> Here is the listing...
> http://i16.tinypic.com/6cs2hih.jpg
trying again
http://inst.santafe.cc.fl.us/~jbieber/HS/7lib-arts.htm
T.H. Ray
You may think you're debating. I am only reciting facts.
After all, to some, even the claim of creationism vs.
evolution is a "debate."
Tom
T.H. Ray
Thanks, G. I was about to ask. :-)
Tom
Herman Jurjus
[snip]
> The mathematical canon of knowledge, on the other hand,
> never discards a theorem. The canon only grows.
Hmm.
Once upon a time, Euclid's 5th postulate was considered irrefutably
true. Today, however...
--
Cheers,
Herman Jurjus
WM
Did I write too much in one go? Here is a shorter version: One counter-
example is the theorem for which Legendre presented six "strict"
proofs: "The parallel postulate is not an axiom but can be proven from
Euclid's other axioms."
Regards, WM
T.H. Ray
Which illustrates my point. It's still true. For
Euclidean geometry.
Tom
T.H. Ray
> > hand,
> > > never discards a theorem.
>
> > This theorem is the first to be discarded. (Such
> > claims belong to
> > matheology at most, they are foreign to sciences and
> > arts.)
>
> If you think you have a case, support it with an
> example.
>
Did I write too much in one go? Here is a shorter version: One counter-
example is the theorem for which Legendre presented six "strict"
proofs: "The parallel postulate is not an axiom but can be proven from
Euclid's other axioms."
Regards, WM
Actually, using Playfair's Axiom and not Euclid's 5th.
In any case, the parallel postulate is still true in
Euclidean geometry--no theorem of Euclid has been
rescinded. One can invent any manner of systems of
axioms; the self consistent results deduced therefrom
are added to the canon of mathematical knowledge. I
hope I was clear in identifying the term "knowledge"
with "theorem." Legendre's are as good as Euclid's.
Tom
Michael Press
<29848239.1188383860...@nitrogen.mathf
> The sciences are driven by a process of
> falsification. That is, new results that some theory
> cannot incorporate, guarantee a continual pruning of
> the scientific canon of knowledge. For example, the
> phlogiston theory of combustion was discarded when
> results showed that combustion is merely rapid oxidation.
> Another example is the abandonment of the ether theory of
> wave propagation, in favor of general relativity.
Phlogiston became an untenable theory when
it became evident that it cannot be formalized
as an exact differential.
--
Michael Press
Michael Press
<23698964.1188588087...@nitrogen.mathf
My first rule of debate is when there are two
well defined positions, they are both wrong.
--
Michael Press
David Bernier
This thread reminds me of the curious question:
Is the question as to whether some matter is a matter of fact (or else a
matter of opinion)
a matter of fact (or else a matter of opinion)? Does the answer depend
on the matter in question?
T.H. Ray
facts, characteristics, properties of the object are
in question--and that rhetoric will reveal a finer-
grained view of the object in order to--if not settle
the question--at least to focus it more clearly.
I am all in favor of Joseph Joubert's advice that "It
is better to debate a question without settling it,
than to settle a question without debating it."
There is no question, however, that mathematics is a
liberal art. This is not a position or an opinion of
mine. Mathematics shares all the properties and
characteristics of every other liberal art, and only
some of the properties and characteristics of physical
science. I exclude the "soft" social sciences
because the question of whether these belong to art
or science IS debatable. It may even be debatable into
which category computer science falls--and on that I
would be willing to take a position and participate in
debate.
By every objective standard, though, mathematics is a
liberal art.
Tom
Han.de...@dto.tudelft.nl
> > On Aug 31, 2:56 am, "T.H. Ray" <thray...@aol.com>
> > wrote:
>
> > > That mathematics is a liberal art is not debatable.
>
> > And yet here we are debating it! Go figure.
>
> You may think you're debating. I am only reciting facts.
These "facts" are only facts in the eye of their beholders.
> After all, to some, even the claim of creationism vs.
> evolution is a "debate."
Quite unfortunately for you, even evolution is not "settled".
You're just a dogmatic, T.H. Ray. Face it. It's hard to let
uncertainity and doubt enter into your life, isn't it?
Han de Bruijn
T.H. Ray
Actually, not at all. I am uncertain about a great
number of things. Of those things I have studied
and reached a rational conclusion, however, I am
certain. It would, after all, be irrational to say
that I am uncertain of my certainty.
I know for certain, for example, that the theory of
common ancestry that explains the fact of evolution
is a true scientific theory validated by overwhelmingly
compelling results. I know for certain that even should
the theory be falsified, the fact of evolution would
remain.
I know for certain that the art of mathematics shares
the properties and characteristics of every other
liberal art. If you think not--please provide a
counterexample.
You may be an irrational postmodernist who denies
the very existence of facts and truth. For that
discussion, I have no time at all.
Tom
T.H. Ray
> <29848239.1188383860...@nitrogen.math
Yes. Its most obvious flaw is empirical. Metallic
combustion (rusted iron, e.g.) was said to have
positive properties, because it appeared to add mass.
Negative phlogiston was said to inhabit fire, as it
appeared to subtract mass from combustibles such as
wood. That the positive and negative properties of
phlogiston theory are contradictory did not seem to
bother anyone until Lavoisier came along and set things
right.
You raise a critically important point. That is, the
closed judgment that characterizes a mathematically
complete theory is certain knowledge, when consistent
with experimental results. The art of mathematical
theory is primary to the scientifically objective
explanation of observed results.
Tom
WM
Legendre proved (or rather collected some proofs) of the theorem that
the fifth postulate can be proven from the rest of Euclid's axioms.
According to current mathematics this theorem is wrong. It has been
abolished. This makes your "theorem" that mathematics only adds
theorems also being wrong and to be abolished. It is simply a matter
of taste and fashion what one wants to understand by a "strict proof".
It is rather ridiculous to believe that the current "formalizations"
or the acceptance of actual infinity will be the end of the story.
Regards, WM
T.H. Ray
Your curiously nonstandard idea of what constitutes
a "theorem" renders your argument impotent in the
context of how mathematics is actually done. As I said,
and which is not controversial, any consistent
result from any self consistent system of axioms is true.
Legendre's failure simply supports the longstanding fact
that the fifth postulate is independent of the other
postulates of Euclidean geometry. Success would have
simply reduced the number of axioms, and not have
cancelled any theorem in that system. We know that
substitutions for the Fifth Postulate, beginning with
Playfair and extending to non-Euclidean geometry, leads
to new results, but does not cancel the old ones.
So far as "actual infinity" goes, it is neither a
theorem nor an axiom. So really, what is your point?
If it is merely that mathematicians argue over proof
theory from generation to generation, I only say, "So
what?" The objective standards of proof have not changed.
The judgment still has to be consistent with the
axioms chosen, without contradiction.
Tom
Aatu Koskensilta
> As I said, and which is not controversial, any consistent result
> from any self consistent system of axioms is true.
Really? So it is true that, say, Peano arithmetic is inconsistent?
--
Aatu Koskensilta (aatu.kos...@xortec.fi)
"Wovon man nicht sprechen kann, daruber muss man schweigen"
- Ludwig Wittgenstein, Tractatus Logico-Philosophicus
T.H. Ray
> s Logico-Philosophicus
Why would you say that? The axioms of Peano (I
prefer to say Dedekind-Peano) are certainly self-
consistent.
If I venture to try and interpret what you mean--that
(as Godel proved) no system of axioms is strong enough
to prove its own consistency--such does not obviate
anything that I said.
Tom
Aatu Koskensilta
> Why would you say that? The axioms of Peano (I
> prefer to say Dedekind-Peano) are certainly self-
> consistent.
You wrote that
As I said, and which is not controversial, any consistent result from
any self consistent system of axioms is true.
The theory PA + "PA is inconsistent" is consistent. Does it follow
that it is true that PA is inconsistent?
--
Aatu Koskensilta (aatu.kos...@xortec.fi)
"Wovon man nicht sprechen kann, daruber muss man schweigen"
- Ludwig Wittgenstein, Tractatus Logico-Philosophicus
T.H. Ray
You're not even reading my whole post. No point
in trying to have a dialogue on those terms.
Clever editing is no substitute for sound argument.
But just to be a good sport, and I promise you this
will be my last reply if you cut me short again:
That a system of axioms is not strong enough to prove
its own consistency (Godel) does not render untrue the
results deduced from that system. So, to your
question:
"The theory PA + 'PA is inconsistent' is consistent.
Does it follow that it is true that PA is inconsistent?"
It follows, metamathematically. Which is why Godel's
Theorem is sometimes characterized by "truth is stronger
than proof." Proofs of theorems, however, are not
affected thereby. We don't know of a stronger system
of axioms than Dedekind-Peano. Philosophy of
mathematics is to mathematics as book review is to
book.
Tom
Aatu Koskensilta
> That a system of axioms is not strong enough to prove
> its own consistency (Godel) does not render untrue the
> results deduced from that system.
That is indeed fortunately so. Do you think anyone has thought it did?
> So, to your question:
>
> "The theory PA + 'PA is inconsistent' is consistent.
> Does it follow that it is true that PA is inconsistent?"
>
> It follows, metamathematically.
It follows "metamathematically" that it's true that PA is inconsistent
from the fact that the consistent theory PA + "PA is inconsistent"
proves "PA is inconsistent"? What does that mean?
> We don't know of a stronger system of axioms than Dedekind-Peano.
It's obscure what you have in mind. We do know all sorts of theories,
such as second-order arithmetic, Zermelo set theory, Zermelo-Fraenkel
set theory, and so on, that are stronger than Peano arithmetic, after
all.
> Philosophy of mathematics is to mathematics as book review is to
> book.
Luckily no philosophy of mathematics enters to the purely mathematical
observation that the consistency of a theory does not guarantee the
truth of its theorems.
--
Aatu Koskensilta (aatu.kos...@xortec.fi)
"Wovon man nicht sprechen kann, daruber muss man schweigen"
- Ludwig Wittgenstein, Tractatus Logico-Philosophicus
T.H. Ray
>
> > That a system of axioms is not strong enough to
> prove
> > its own consistency (Godel) does not render untrue
> the
> > results deduced from that system.
>
> That is indeed fortunately so. Do you think anyone
> has thought it did?
>
So then you make your irrelevant concluding statement
that consistency of a theory doesn't guarantee the truth
of its theorems. Problem is, you are the one
who has made reference to "theory"-- not I. A system
of axioms is not a theory. Theory embraces fundamental
theorems.
In fact, theory does guarantee the truth of theorems. That
it does not include all the truth that the theory
might embrace, does not obviate that guarantee.
Tom
> s Logico-Philosophicus
WM
> > Legendre proved (or rather collected some proofs) of
> > the theorem that
> > the fifth postulate can be proven from the rest of
> > Euclid's axioms.
> > According to current mathematics this theorem is
> > wrong. It has been
> > abolished. This makes your "theorem" that mathematics
> > only adds
> > theorems also being wrong and to be abolished. It is
> > simply a matter
> > of taste and fashion what one wants to understand by
> > a "strict proof".
> > It is rather ridiculous to believe that the current
> > "formalizations"
> > or the acceptance of actual infinity will be the end
> > of the story.
>
> > Regards, WM
>
> Your curiously nonstandard idea of what constitutes
> a "theorem" renders your argument impotent in the
> context of how mathematics is actually done.
It is not my idea but the opinion of Legendre who, as far as most
mathematicians believe, was among the greatest mathematicians and had
the ability to judge about proofs.
> As I said,
> and which is not controversial, any consistent
> result from any self consistent system of axioms is true.
The question is what is consistent and who determines that.
> Legendre's failure simply supports the longstanding fact
> that the fifth postulate is independent of the other
> postulates of Euclidean geometry. Success would have
> simply reduced the number of axioms, and not have
> cancelled any theorem in that system.
But the failure *has* cancelled a theorem.
> We know that
> substitutions for the Fifth Postulate, beginning with
> Playfair and extending to non-Euclidean geometry, leads
> to new results, but does not cancel the old ones.
This observation fails to discuss my argument.
>
> So far as "actual infinity" goes, it is neither a
> theorem nor an axiom.
It is an axiom in ZF. There *exists* a set ... (which is complete as
it has a cardinal number).
> So really, what is your point?
> If it is merely that mathematicians argue over proof
> theory from generation to generation, I only say, "So
> what?" The objective standards of proof have not changed.
You should rather say that you do not know the changes.
> The judgment still has to be consistent with the
> axioms chosen, without contradiction.
But the judgements what consistence is and what a contradiction is are
subject to human errors (even if done by machines programmed by
humans).
Regards, WM
Herman Rubin
>> As I said, and which is not controversial, any consistent result
>> from any self consistent system of axioms is true.
>Really? So it is true that, say, Peano arithmetic is inconsistent?
If it is, there is no proof of it, so the statement is
not violated. That we cannot prove that any system of
axioms containing Peano arithmetic is consistent if in
fact it is is no proof that it is not.
--
This address is for information only. I do not claim that these views
are those of the Statistics Department or of Purdue University.
Herman Rubin, Department of Statistics, Purdue University
hru...@stat.purdue.edu Phone: (765)494-6054 FAX: (765)494-0558
Herman Rubin
>> Why would you say that? The axioms of Peano (I
>> prefer to say Dedekind-Peano) are certainly self-
>> consistent.
>You wrote that
> As I said, and which is not controversial, any consistent result from
> any self consistent system of axioms is true.
>The theory PA + "PA is inconsistent" is consistent. Does it follow
>that it is true that PA is inconsistent?
No, because there is no proof in that theory that PA is
inconsistent. If PA is inconsistent, then of course
anything can be proved in PA.
Robert Israel
> In article <87wsvaz...@huxley.huxley.fi>,
> Aatu Koskensilta <aatu.kos...@xortec.fi> wrote:
> >"T.H. Ray" <thra...@aol.com> writes:
>
> >> Why would you say that? The axioms of Peano (I
> >> prefer to say Dedekind-Peano) are certainly self-
> >> consistent.
>
> >You wrote that
>
> > As I said, and which is not controversial, any consistent result from
> > any self consistent system of axioms is true.
>
> >The theory PA + "PA is inconsistent" is consistent. Does it follow
> >that it is true that PA is inconsistent?
>
> No, because there is no proof in that theory that PA is
> inconsistent.
Yes, there is: it's an axiom of that theory.
--
Robert Israel isr...@math.MyUniversitysInitials.ca
Department of Mathematics http://www.math.ubc.ca/~israel
University of British Columbia Vancouver, BC, Canada
Han de Bruijn
> In article <3a833$46d6753b$82a1e228$40...@news2.tudelft.nl>,
> Han de Bruijn <Han.de...@DTO.TUDelft.NL> wrote:
>
>>It is often the case that mathematicians do what engineers have already
>>done. (An example is the Finite Element Method in Numerical Analysis)
>
> One should be careful about such methods, [ ... ]
One should be more careful about what _mathematicians_ make out of such
engineering methods. They start formalizing, axiomatizing, theoretizing
without an even moderate understanding of the _real_ thing. Take a look
at the end result of such a process:
http://en.wikipedia.org/wiki/Finite_element_method
Especially read the section "A proof outline of existence and uniqueness
of the solution", where mathematicians have introduced - completely OT -
their Sobolev spaces, claiming that they can "prove" things with these.
Also read the section "Comparison to the finite difference method". And
beware! Because here comes the truth. The abovementioned section is pure
_nonsense_, because Finite Element Methods and Finite Difference Methods
are essentially equivalent:
http://hdebruijn.soo.dto.tudelft.nl/www/programs/suna01.htm
From:
http://hdebruijn.soo.dto.tudelft.nl/www/sunall.htm
Han de Bruijn
Han de Bruijn
Sure. Newtonian mechanics is still true. For Newtonian mechanics.
Han de Bruijn
Han de Bruijn
Or both are half wrong. Or both are half right. Depends upon whether you
are an optimist or a pessimist .. I am not a creationist. Yet I believe
that standard evolution theory is wrong. (Some people I know are experts
in the anatomy of the human body, that may be one reason)
Han de Bruijn
T.H. Ray
Your point is truly well taken. The truth of a theorem,
however, does not rest with the personal judgment
of any practitioner, however renowned.
And it is simply not the case that consistency and
contradiction are not objective standards. They are
objective standards because they are demonstrably so.
We change the domains to which the standards apply,
not the standards themselves. Fundamentally,
mathematics is the study of propositions of the form,
A==>B. Knowing what A is, objectively, is essential.
That is, necessary, though not sufficient.
Tom
T.H. Ray
Exactly. We use Newtonian mechanics alone to land
people on the moon. Einstein extended Newton; he did
not cancel him. Analogously, mathematical theories
grow.
Tom
T.H. Ray
One has to be careful of what one is talking about
here. Evolution is a fact in evidence, observable
and observed. The explanation for evolution is Darwin's
theory of common ancestry. Darwin's theory has been
much researched and refined in the last 150 years. One
of the more striking models of evolution in recent
history is the Gould-Eldredge model of Punctuated
Equilibrium. This is yet one more case where a
mathematically complete theory adds to our knowledge in
large leaps (pun intended).
Tom
Han de Bruijn
> And it is simply not the case that consistency and
> contradiction are not objective standards. They are
> objective standards because they are demonstrably so.
It has been pointed out to you by several debaters in this thread that
especially your concept of "consistency" is a bit, well: simple minded.
> We change the domains to which the standards apply,
> not the standards themselves. Fundamentally,
> mathematics is the study of propositions of the form,
> A==>B. Knowing what A is, objectively, is essential.
> That is, necessary, though not sufficient.
Is standard set theory consistent? If you think so, prove it! But ah,
since _everybody_ knows nowadays that the good old Hilbert programme
has been a pipe dream .. Am I wrong?
Han de Bruijn
Jesse F. Hughes
> I am not a creationist. Yet I believe that standard evolution theory
> is wrong. (Some people I know are experts in the anatomy of the
> human body, that may be one reason)
Wow. You know mathematics better than mathematicians and you know
biology better than biologists. You're a clever lad!
--
"Being who I am, I know that's a solution that will run in polynomial
time, but for the rest of you, it will take a while to figure that out
and know why [...But] it's the same principle that makes n! such a
rapidly growing number." James S. Harris solves Traveling Salesman
Han de Bruijn
> Han de Bruijn <Han.de...@DTO.TUDelft.NL> writes:
>
>>I am not a creationist. Yet I believe that standard evolution theory
>>is wrong. (Some people I know are experts in the anatomy of the
>>human body, that may be one reason)
>
> Wow. You know mathematics better than mathematicians and you know
> biology better than biologists. You're a clever lad!
No. But some biologists have been more convincing to me than others,
especially those who have some _real_ expertise to share.
Han de Bruijn
Han de Bruijn
> One has to be careful of what one is talking about
> here. Evolution is a fact in evidence, observable
> and observed. The explanation for evolution is Darwin's
> theory of common ancestry.
What we actually _observe_ is kind of common "design".
> Darwin's theory has been
> much researched and refined in the last 150 years.
It has received so much criticism as well. Look, I can understand that
science should not allow God into its theories. But IF we are forced to
distort a theory, for the sole reason that kind of an intelligence would
be needed to proceed, are we still doing honest science then? (Mind the
"if" clause in the above)
> One of the more striking models of evolution in recent
> history is the Gould-Eldredge model of Punctuated
> Equilibrium. This is yet one more case where a
> mathematically complete theory adds to our knowledge in
> large leaps (pun intended).
I vaguely remember some contributions from statisticians. But don't know
how much they are worth. Personally, I don't trust reasonings like "give
me another million years to make the improbable to happen". Knowing what
a combinatorial explosion is and knowing that such explosions can happen
within very simple configurations .. I mean, folding a _protein_ is much
much much .. more complicated than this:
http://groups.google.nl/group/sci.math/msg/2ae54ca6c2614027
Han de Bruijn
Rotwang
> "Being who I am, I know that's a solution that will run in polynomial
> time, but for the rest of you, it will take a while to figure that out
> and know why [...But] it's the same principle that makes n! such a
> rapidly growing number." James S. Harris solves Traveling Salesman
I don't think I've seen that one. Could you tell me from what thread
it came?
T.H. Ray
>
> > And it is simply not the case that consistency and
> > contradiction are not objective standards. They
> are
> > objective standards because they are demonstrably
> so.
>
> It has been pointed out to you by several debaters in
> this thread that
> especially your concept of "consistency" is a bit,
> well: simple minded.
Naive would be a more polite word. I never claimed
rigor, though I am capable. For the purpose of these
discussions, however, my contexts for self-consistency
and consistency are entirely sufficient. And I wish
you would stop misusing the term, "debate." There
is no debate over the self consistency of axiom systems
and consistency of true results derived therefrom. It is
an objective standard--no individual arbitrarily decides
that a system of axioms is inconsistent.
>
> > We change the domains to which the standards apply,
>
> > not the standards themselves. Fundamentally,
> > mathematics is the study of propositions of the
> form,
> > A==>B. Knowing what A is, objectively, is
> essential.
> > That is, necessary, though not sufficient.
>
> Is standard set theory consistent? If you think so,
> prove it! But ah,
> since _everybody_ knows nowadays that the good old
> Hilbert programme
> has been a pipe dream .. Am I wrong?
>
> Han de Bruijn
>
Yes. You are. Hilbert's (and Russell's and Frege's)
desire to formalize all mathematical statements into
a final theory of truth failed only because truth is
too big for theory. That does not mean that true
statements derived from formal systems are therefore
not true. How this erroneous logic became popular
(especially among denizens of sci.math who seem to
think Cantor was some sort of anti-Christ)escapes me.
It is in fact absurd for you to demand that I (or anyone)
prove set theory consistent, when in fact it has already
been proved (Godel incompleteness) that no system can
prove its own consistency using its own self-consistent
axioms.
Going back to the example from science that you and I
have discussed before: that Newton failed to capture
theoretically all the principles of kinetics that nature
embodies does not render his theory wrong, or untrue.
Until Riemann and Einstein and Lorentz made new
language available to describe and predict unknown (and
counterintuitive) phenomena, Newton was assumed complete.
In mathematics, Euclid was considered complete until
Gauss, Lobachevsky and Bolyai invented new self-
consistent geometries.
Tom
T.H. Ray
>
> > One has to be careful of what one is talking about
> > here. Evolution is a fact in evidence, observable
> > and observed. The explanation for evolution is
> Darwin's
> > theory of common ancestry.
>
> What we actually _observe_ is kind of common
> "design".
>
the mutation of bacteria and virii that keep the drug
companies in business.
A model is not equivalent to a theory. However, one's
personal credulity or incredulity has nothing to do
with the science at all.
Tom
Jesse F. Hughes
If you don't know biology, how do you know which biologists have
"_real_ expertise" and which don't?
--
Jesse F. Hughes
"Why do the dirty villains always have to tie your hands *behind* ya?"
"That's what makes them villains."
--Adventures by Morse (old radio show)
Jesse F. Hughes
> T.H. Ray wrote:
>
>> One has to be careful of what one is talking about
>> here. Evolution is a fact in evidence, observable
>> and observed. The explanation for evolution is Darwin's
>> theory of common ancestry.
>
> What we actually _observe_ is kind of common "design".
>
>> Darwin's theory has been
>> much researched and refined in the last 150 years.
>
> It has received so much criticism as well. Look, I can understand that
> science should not allow God into its theories. But IF we are forced to
> distort a theory, for the sole reason that kind of an intelligence would
> be needed to proceed, are we still doing honest science then? (Mind the
> "if" clause in the above)
So you are sympathetic to Intelligent Design theory?
--
Jesse F. Hughes
"You shouldn't hate Mother Mathematics."
-- James S. Harris
shep
Although I have no trouble with the potential infinity of series, sums, limits, etc. as describing processes which continue indefinitely and which may approach but never reach some conclusion, I do have problems with the idea of a "completed infinity."
So, I find myself generally in agreement with David Petry's position. However, since I am not a mathematician I may boldly go where angels fear to tread and comment on the following statement:
"So how would we develop a set theory if we insist that infinity has
only a potential existence? For starters, finite sets of integers are
no problem. They have an exact representation as data structures in a
computer, for example. The set of all integers is also not much of a
problem. We can define some sense in which the set of integers {1..N}
is an approximation to the set of all integers {1..oo}, and then we
can say that the set of all integers has a potential existence (i.e.
arbitrarily accurate approximations to it actually exist)."
This sounds like a cop-out to me, an attempt to save the idea of infinite sets. However, if one thinks of sets as (at least theoretically) complete-able collections, then there are no infinite sets.
To put it another way. Sets are typically defined either extensively or intensively. I am proposing that any set must be theoretically capable of an extensive definition, that the intensive definition is a convenient shorthand, not an allowable substitute for an extensive definition.
In this much more restrictive use of the idea of a set, it is probably not true that "everything that can be said using the notion of infinity could also be said without it."
For example, there would be no question of comparing the cardinality of different infinite sets, since there are no infinite sets.
One may still speak of the real numbers, say, as an indefinitely long series or sequence with no largest or smallest element. One may still speak of series converging to some limit or diverging to infinity, but such series should not be treated in the same way as are finite sets.
There may already exist a class of mathematical objects with these characteristics (i.e those sets that are included in the definition of "set" which includes infinite sets but which are excluded by the definition which excludes infinite sets).
I'm feeling a bit like Leopold Kronecker here (who seems like not such a good guy) but I take this chance hoping for an interesting, not too vitriolic, response to my musings.
Thanks
david petry
I had written:
> "So how would we develop a set theory if we insist that infinity has
> only a potential existence? For starters, finite sets of integers are
> no problem. They have an exact representation as data structures in a
> computer, for example. The set of all integers is also not much of a
> problem. We can define some sense in which the set of integers {1..N}
> is an approximation to the set of all integers {1..oo}, and then we
> can say that the set of all integers has a potential existence (i.e.
> arbitrarily accurate approximations to it actually exist)."
Shep wrote:
> This sounds like a cop-out to me, an attempt to save the idea of
> infinite sets.
I certainly don't see it as a cop out in any way. Concepts like the
set of all integers and the set of all reals are very useful in
mathematics. There's no reason to deny that. But we can develop a
theory of such infinite sets within the constraints imposed by
treating infinity as having only a potential existence, and such a
theory will be sufficient for the mathematics that has applications in
the real world (i.e. in science and technology). That's my point.
shep
My real question here is something like "What are the consequences to mathematics if infinite sets are not allowed?"
I don't doubt the usefulness of concepts such as the set of all integers and the set of all reals, but I wonder where it would lead if, instead of answering your question ("So how would we develop a set theory if we insist that infinity has only a potential existence?") as you did, you said rather that only finite collections may be handled by set theory, and that other kinds of collections such as the integers, the reals, etc. require some other kinds of mathematical rules?
Forgive me if this is a stupid question but it is the one I was hoping to get an answer to.
Han de Bruijn
Uhm, I don't want to spoil the party, but I'm affraid that some such may
be not so easy to accomplish, when it comes to details.
I think I've found lately that the very notion of a set is incompatible
with potential infinity. This should hardly represent a problem if we
conceive the naturals or the reals as "proper classes". (This is done in
standard mathematics on a "higher level" for "all transfinite ordinals",
for the purpose of avoiding Russell's like paradoxes).
Meaning that you can say "x is a natural" or "x is a real", but without
having the naturals or the reals as completed, or "potential infinite",
sets; where the latter IMHO is impossible (or at last cannot be done in
a simple manner). We could also conclude that set theory should give up
its predominant position, one way or another. Or that it may be better
to give the name "set" to "proper class": a change of nomenclature only,
while leaving much of common mathematics (formally) the same.
Han de Bruijn