# Does science need infinity?

7 views

### Nancy McGough

Feb 17, 1993, 1:39:22 AM2/17/93
to
Hellow net scientists -

First, let me ward off flames about the massive crossposting
I've done. I would really like input from all branches of
science. I've directed followups to sci.math so this should
be the one and only cross-posted article in this thread.

I'm writing a paper with the working title of "Does Science
Need Infinity?" My background is math, especially set theory,
logic, and philosophy of math. In graduate school I avoided
applied math classes so I'd really like some input from
people who are applied mathematicians. YEARS ago as an
undergraduate I took quantum mechanics, physical chemistry,
and some applied math classes like PDEs, but if you asked
me to do an applied math problem it would take me weeks
(maybe months or years!) to get up to speed.

So now, with that background, here's what I'm interested
in. Infinity is used throughout applied math, both implicitly
and explicitly. For example:
* Analysis assumes the space and time are a continuum (i.e.,
infinitely many points, dense, complete, connected, etc.)
* Infinite dimensional spaces (like Hilbert space)
* Dirac delta function has value infinity at 0 but the
integral from -inf to +inf is 1.
* Fractal boundaries are infinitely long.
* Infinite series.
* Computer scientists have theorems about computability,
etc., where they assume that there is infinite storage
and/or time.
* Penrose tiles - do these have applications like in
crystallogrophy?
* Game of Life - does this have applications like in
Biology?
* Large Cardinals - do these have any application in
real life?
* <lots of things I haven't thought of...>

My questions:
I'd like to get some concrete examples of infinity being
used in science - including areas I've listed above
and areas I've missed. I'd also like your thoughts
about whether it's used in a "potential" or "actual
(completed)" way. And is this use of infinity
necessary for solving the problem, or is there another
(possibly more cumbersome) way to solve the problem.

Here's an example of what I'm looking for:
Trigonometric functions are used to solve problems
in most of the sciences. For example, when bodies
collide the mass, velocity, and angles of recoil
are related by formulas that use trig functions.
Values of trigonometric functions are equal to the
sum of an infinite Taylor Series, for example:

Sin(x) = x - x^3/3! + x^5/5! - x^7/7! +...

In practice this is only used in a "potentially
infinite" way. I.e., computers or calculators
only sum up a finite number of terms. Higher
degrees of accuracy can be gotten by summing
up more (but still finitely many) terms. (Note
that computers these days don't use Taylor series
but they do use other (potentially) infinite series
for calculations.)

[???Does anyone know what infinite series are used
these days by computers???]

This is not a necessary use of infinity. Trig functions
were evaluated for thousands of years without infinite
series. For example, Ptolemy (85 - 165 AD) described
his trigometric methods and provided trig tables in his
"Mathematikes Suntaxis," which Greek scholars accepted
as the greatest book on astronomy.

I'm looking forward to posts about lots of different
areas of science and lots of different uses of infinity.
I'd like examples of things I've listed, like the Dirac
delta function (I don't really know how this is used)
and things I haven't listed (like maybe Lie algebras,
which I know essentially nothing about, except I think
they have something to do with infinity).

Thanks much,
Nancy
nan...@u.washington.edu

### Marc GIRONDOT p75783

Feb 17, 1993, 3:06:32 AM2/17/93
to
In article <1lsmiq...@shelley.u.washington.edu> nan...@u.washington.edu writes:
>Hellow net scientists -

>
>I'm writing a paper with the working title of "Does Science
>Need Infinity?" My background is math, especially set theory,
>logic, and philosophy of math. In graduate school I avoided
cut....
>
>Thanks much,
>Nancy
>nan...@u.washington.edu

In population biology/ evolution we use the infinity when searching
of equilibria. For example when having all the parameters
to described one particular point concerning a population, we make
a simple (or less simple!) model and we are looking for all the
equilibrium state. Then we compare the observed population with all
the equilibirum state, and if there is one we cannot reject, generally
we stop. The system is described.
For example, one of the most classical point:
Imagine a population with two alleles (a A) at one locus
a is a p frequence in population and A is at q frequence (=1-p)
Then the frequency of genotype at equilibrium are:
AA p^2
Aa 2pq Called Hardy-Weinberg equilibrium
aa q^2
In this case the equilibrium is evident but when you observed a
population without such proportion you need to introduce others
paramaters (migration, mutation, small population, selection, non-random
mating, etc...) and then to calculate the equilibrium states we need to
evaluate your system at the infinity.

Population biologists like equilibrium, and then the infinity !

Bye.
Marc Girondot
Dpt Dynamique du genome et evolution

I dont known how to cross-posted... sorry
Also, sorry for the faults.

### joa...@vaxsar.vassar.edu

Feb 17, 1993, 9:59:01 AM2/17/93
to
In article <1lsmiq...@shelley.u.washington.edu>, nan...@hardy.u.washington.edu (Nancy McGough) writes:

> My questions:
> I'd like to get some concrete examples of infinity being
> used in science - including areas I've listed above
> and areas I've missed. I'd also like your thoughts
> about whether it's used in a "potential" or "actual
> (completed)" way. And is this use of infinity
> necessary for solving the problem, or is there another
> (possibly more cumbersome) way to solve the problem.>

I can think of a couple of examples that you might be interested in which have
to do with physics (with applications in astronomy, of course) off the top of
my head. The first is the relativistic mass equation which tells you the mass
of an object as a function of its speed: M = Mo / sqr(1-sq(v/c))
This shows that when the velovity of the object (v) gets close to the speed of
light (c) the mass of the object will go to infinity. The second one has to do
with optics. To find whether an angle of incidence of a parallel polarized
light wave is completely internally reflected, you rely on the outcome of the
equation having infinite results:
Rp=(tan(AngleIncidence-AngleTransmitted)/
(tan(AngleIncidence+AngleTransmitted))
Take for instance: AngleIncidence = (1/2 pi - AngleTransmitted) will yield an
infinite result and show that the AngleIncidence is the angle of total internal
reflection.

You can take these results for what they are worth. The first is a pretty
real result that has been (somewhat)proven on a very small scale, and the
second can be found other ways using the Fresnel Equations, but as you say, in
a very cumbersome manner.

In regards to:

> * Computer scientists have theorems about computability,

> etc., where the assume that there is infinite soorage
> and/or time.

I assume that you are familiar with what is called Big Oh notation. This is
the idea that you can find the relative speeds of algorithims by taking their
limits as they go to infinity. (Examples of Big Oh notation would be O(Logn) is
faster than O(n^2).) This is, of course acadmeic at a certain point due to the
fact that O(Logn^Lgn) seems to be a much faster algorithm than O(n^n) simply
because of the fact that the limit as n->infinity of n^n/Logn^Logn is zero.
Even though it starts off being very fast, the function slowly grows to be much
slower then almost any other algorithm (except ones on the order of O(n!)).
You can see that for almost all practical purposes, an algorithm with an order
of magnitude (Big Oh) of O(Logn^Logn) will still be better, but when seen at
infinitey, it is O(n^n) that is the winner.

I hope this meager sample helps.

+-------------------------------+---------------------------------------------+
| | |
| J. Trent Adams | "People can increase their reading speed by |
| Department: Physics/Astronomy | skipping words. Poople who can't read are |
| EMS Crew Chief | skipping everything. Does this mean /|||\ |
| Vassar College | they have an infinite reading speed?" }> <{ |
| joa...@vaxsar.vassar.edu | - Gregory Rawlins | v | |
| | \_/ |
+-------------------------------+---------------------------------------------+

### Mark

Feb 18, 1993, 2:55:37 AM2/18/93
to
>My questions:
>I'd like to get some concrete examples of infinity being
>used in science - including areas I've listed above
>and areas I've missed. I'd also like your thoughts
>about whether it's used in a "potential" or "actual
>(completed)" way. And is this use of infinity
>necessary for solving the problem, or is there another
>(possibly more cumbersome) way to solve the problem.

Abraham Robinson answered the question about the utility of idealizations
in Mathematics as part of his long-term (successful) venture in fleshing
out the larger part of Leibnitz's philosophy on a strikingly solid
foundation.

Reading all of what he wrote from the 1950's on will almost certainly resolve
the issues you're bringing up to your satisfaction,

Then look for some material written by Rudy Rucker...

### Renato Ghica

Feb 18, 1993, 12:21:49 PM2/18/93
to

If you didn't have infinity then you could never evaluater any integral for
any number of point except zero.

This is because when ine says "sum of (blah) from n=0 to n=infinity",
one is assuming time is infinite. Otherwise the universe might
cease to exist around n=13 or so. :-)

So if time is not infinite, you could only evaluate zero number
of points with any accuracy.

You don't want scientist to say "n=0 to n=end_of_the_universe", do you?

makes sense?
--

"This will just take a minute."
"I'm 90% done."
"It worked on my machine."

### G.J. McCaughan

Feb 18, 1993, 6:19:51 PM2/18/93
to

>I assume that you are familiar with what is called Big Oh notation. This is
>the idea that you can find the relative speeds of algorithims by taking their
>limits as they go to infinity. (Examples of Big Oh notation would be O(Logn) is
>faster than O(n^2).) This is, of course acadmeic at a certain point due to the
>fact that O(Logn^Lgn) seems to be a much faster algorithm than O(n^n) simply
>because of the fact that the limit as n->infinity of n^n/Logn^Logn is zero.
>Even though it starts off being very fast, the function slowly grows to be much
>slower then almost any other algorithm (except ones on the order of O(n!)).
>You can see that for almost all practical purposes, an algorithm with an order
>of magnitude (Big Oh) of O(Logn^Logn) will still be better, but when seen at
>infinitey, it is O(n^n) that is the winner.

WHAT?????

I presume you are confusing the speed at which a function grows with the speed
at which an algorithm with that running time runs, or something, but what you
have written makes no sense at all.

--
Gareth McCaughan Dept. of Pure Mathematics & Mathematical Statistics,
gj...@cus.cam.ac.uk Cambridge University, England. [Research student]

### joa...@vaxsar.vassar.edu

Feb 19, 1993, 6:09:33 AM2/19/93
to
In article <1993Feb18.2...@infodev.cam.ac.uk>, gj...@cus.cam.ac.uk (G.J. McCaughan) writes:
> In article <1993Feb18....@vaxsar.vassar.edu> joa...@vaxsar.vassar.edu writes:
>
>>I assume that you are familiar with what is called Big Oh notation. This is
>>the idea that you can find the relative speeds of algorithims by taking their
>>limits as they go to infinity. (Examples of Big Oh notation would be O(Logn) is
>>faster than O(n^2).) This is, of course acadmeic at a certain point due to the
>>fact that O(Logn^Lgn) seems to be a much faster algorithm than O(n^n) simply
>>because of the fact that the limit as n->infinity of n^n/Logn^Logn is zero.
>>Even though it starts off being very fast, the function slowly grows to be much
>>slower then almost any other algorithm (except ones on the order of O(n!)).
>>You can see that for almost all practical purposes, an algorithm with an order
>>of magnitude (Big Oh) of O(Logn^Logn) will still be better, but when seen at
>>infinitey, it is O(n^n) that is the winner.
>
> WHAT?????
>
> I presume you are confusing the speed at which a function grows with the speed
> at which an algorithm with that running time runs, or something, but what you
> have written makes no sense at all.

> Gareth McCaughan Dept. of Pure Mathematics & Mathematical Statistics,
> gj...@cus.cam.ac.uk Cambridge University, England. [research student]

I certainly hope that I am not confusing the two, yet since I have only a
minor in computer science (and a major in astronomy with another minor in
physics) I might be mistaken. If I remember correctly (and you can look in
Gregory Rawlins' "Compared to What?" p. 44) when you say that an algorithm has
an order of magnitude of O(n) you assume that the speed (meaning the time that
the program takes to run) of the algorithm with input n items will be
proportional to the speed with which the function grows. If you have to compute
the nth Fibonacci number, you would rather use an algorithm with an order of
magnitude of O(Logn) than O(3n) since the one with O(Logn) will be completed
much sooner.

Does that make it clearer?

+-------------------------------+---------------------------------------------+
| | |
| J. Trent Adams | "People can increase their reading speed by |

| Department: Physics/Astronomy | skipping words. People who can't read are |

### Scott Brown

Feb 19, 1993, 5:55:41 PM2/19/93
to

> My questions:
> I'd like to get some concrete examples of infinity being
> used in science - including areas I've listed above
> and areas I've missed. I'd also like your thoughts
> about whether it's used in a "potential" or "actual
> (completed)" way. And is this use of infinity
> necessary for solving the problem, or is there another
> (possibly more cumbersome) way to solve the problem.>

I think the question is not stated very precisely yet. There
are two similar questions, "does science need the idea of infinity"
and "does science assume the universe is infinite, or simply use
the idea of infinity as a (to date, tremendously effective) model
of a 'finite-but-very-large' universe". "Concrete" examples of
inifinity...something strange about the wording there. There
are lots of examples of mathematical calculations, such as limits,
and hence integrals, derivatives, and differential equations, in
which sense is made of 'infinite' processes; but to actually
evaluate, for example, a Riemann sum, one uses techniques that
are themselves necessarily finite. In teaching elementary calculus
to college undergraduates, I find that absorbing the techniques
and ideas involved in dealing with limiting processes is often very
challenging. The epsilon-delta proof is feared, and students
seek refuge in mere symbolic manipulation. Ahh, but I digress.

Another example:

infinity
--- 1 1 1 1
Claim: \ ----- = 1 ; i.e., - + - + - + ... = 1.
/ k 2 4 8
--- 2
k=1

Don't be fooled by the infinity in the notation; any elementary
calculus book will explain that this just means a certain limit is
equal to 1. To whit:

n
--- 1
lim \ ----- = 1
n->oo / k
--- 2
k=1

And what to make of the n->infinity here? Well, flip back a few
pages and look at the definition of the limit of a sequence. It will
say something like

lim a = L if (defn) for any epsilon > 0, one can find
n->oo n

an integer M such that if n is greater than M, a is no
n
further than epsilon from L.

That is, the definition is entirely in terms of finite numbers.
Here there is an assumption that the set of integers is infinite,
which is not exactly a 'concrete' example of infinity.

Does science "need infinity"? Or merely find it useful?

Other opinions?

Scott Brown
sbr...@symcom.math.uiuc.edu

oooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooo

"The aim of proof is, in fact, not merely to place the truth of a
proposition beyond all doubt, but also to afford us insight into the
dependence of truths upon one another."
-Gottlob Frege

"Jane, how do I stop this crazy thing!?!?!"
-George Jetson

oooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooo

### John Donald Collier

Feb 20, 1993, 7:19:30 AM2/20/93
to
Hartrey Field argues eloquently that science doesn't need numbers, in
_Science Without Numbers_. What is meant by "need" here? If it just
means "useful", then the answer is one thing; "require logically", and
the answer is another.

--
John Collier Email: jcol...@ariel.ucs.unimelb.edu.au
HPS -- U. of Melbourne Fax: +61 3 344 7959
Parkville, Victoria, AUSTRALIA 3052

### Noam Shomron

Feb 21, 1993, 3:11:23 AM2/21/93
to
In article <1m57ki...@ariel.ucs.unimelb.EDU.AU> jcol...@ariel.ucs.unimelb.E

DU.AU (John Donald Collier) writes:
>Hartrey Field argues eloquently that science doesn't need numbers, in
>_Science Without Numbers_. What is meant by "need" here? If it just
>means "useful", then the answer is one thing; "require logically", and
>the answer is another.
>

In my opinion (and there are many others who share it), most people who claim
that science would be better off without mathematics simply don't *grasp* what
science _is_. How can one test a theory without an objective way to make
predictions? Anyone who knows anything of the fundamentals of ANY field of
modern science can tell you that in terms of "usefulness", well, mathematics is
crucial to science in the sense that it's necessary, i.e., we'd be in the Dark
Ages without it, and in terms of "require logically", it is possible to make
qualitative predictions to a certain extent, but I don't see how you'd describe
something such as, say, quantum vaccuum polarization without mathematical
terms. Science doesn't "require" anything, but by definition it deals with
_physical_ things in the same way mathematics deals with _mathematical_ things.
Care to describe "Mathematics without Numbers"? If I say that when I throw a
ball up into the air it will dome down as one, not two, balls than I've
introduced numbers right there. If you accept that, *ANY* mathematical object
in a scientific context is a step away. If not, well.... In conclusion, I
ask: when you say science doesn't need numbers, what kind of science do you
mean?

--- Noam
--

### John Donald Collier

Feb 21, 1993, 9:52:34 PM2/21/93
to
nsho...@magnus.acs.ohio-state.edu (Noam Shomron) writes:

Nice example of a flame by someone who is completely ignorant of the
subject. Field's approach a nominalistic approach to science. Your
case would be stated in terms of identity. There is no difference
between ball A and ball B. All scientific expressions are out in terms
of similarities and dissimilarities on the nominalistic approach. Of
course these have a cardinality, but their cardinality need not be
used to do science. Other numbers are reduced out the same way.

Field is a brilliant philosophical logician. His book isn't just
fooling around. It would take one or two courses in logic, and a
course or two in analytic philosophy just to understand the concepts
in it (and, no, they can't be paraphrased in simpler language -- I
am talking about the concepts, not the way they are expressed).

On Field's approach, numbers are not required to make precise
predictions. ("The pointer is closer to line A than line B,"
might be an example.) Numbers are convenient ways to express
such claims, but it is unclear they are logically necessary.

### Mikhail Zeleny

Feb 23, 1993, 12:00:12 AM2/23/93
to
In article <1m9f5i...@ariel.ucs.unimelb.EDU.AU>
jcol...@ariel.ucs.unimelb.EDU.AU (John Donald Collier) writes:

>nsho...@magnus.acs.ohio-state.edu (Noam Shomron) writes:

>>In article <1m57ki...@ariel.ucs.unimelb.EDU.AU>
>>jcol...@ariel.ucs.unimelb.EDU.AU (John Donald Collier) writes:

JDC:

>>>Hartrey Field argues eloquently that science doesn't need numbers, in
>>>_Science Without Numbers_. What is meant by "need" here? If it just
>>>means "useful", then the answer is one thing; "require logically", and
>>>the answer is another.

NS:

>>In my opinion (and there are many others who share it), most people who claim
>>that science would be better off without mathematics simply don't *grasp* what
>>science _is_. How can one test a theory without an objective way to make
>>predictions? Anyone who knows anything of the fundamentals of ANY field of
>>modern science can tell you that in terms of "usefulness", well, mathematics is
>>crucial to science in the sense that it's necessary, i.e., we'd be in the Dark
>>Ages without it, and in terms of "require logically", it is possible to make
>>qualitative predictions to a certain extent, but I don't see how you'd describe
>>something such as, say, quantum vaccuum polarization without mathematical
>>terms. Science doesn't "require" anything, but by definition it deals with
>>_physical_ things in the same way mathematics deals with _mathematical_ things.
>> Care to describe "Mathematics without Numbers"? If I say that when I throw a
>>ball up into the air it will dome down as one, not two, balls than I've
>>introduced numbers right there. If you accept that, *ANY* mathematical object
>>in a scientific context is a step away. If not, well.... In conclusion, I
>>ask: when you say science doesn't need numbers, what kind of science do you
>>mean?

JDC:

>Nice example of a flame by someone who is completely ignorant of the
>subject. Field's approach a nominalistic approach to science. Your
>case would be stated in terms of identity. There is no difference
>between ball A and ball B. All scientific expressions are out in terms
>of similarities and dissimilarities on the nominalistic approach. Of
>course these have a cardinality, but their cardinality need not be
>used to do science. Other numbers are reduced out the same way.

I am always happy to see someone display manners inferior to my own.
Please apply your criteria to yourself, John: the semantics of
identity involves either an explicit second-order extensional
characterization, or a first-order introduction as a primitive binary
predicate with appropriate axioms, in which case the second-order
quantifiers will be called for in giving the semantics of the language
in which these axioms are stated. In either case, the ontological
commitment is far in excess of that required for defining either von
Neumann or Frege integers.

Field tries to avoid this predicament by claiming that his logic is
special, in that it does not depend on a metatheoretic interpretation.
Yeah, right. In any case, the sort of nominalism advocated by your
pal Hartry is not nearly sufficiently eliminative to warrant any real
philosophical interest. For one thing, Field implicitly opts for
countenancing type-identities, insofar as he recognizes the need for
proof theory; so there you already have one species of abstract
objects, which, by dint of arbitrary pronouncement, are ruled not to
be mathematical objects. For another, in taking as the basic building
block of his physicalism a theory of concrete aggregates of space-time
points, Field commits himself to the ontology of such choice abstract
universals, as upper semi-lattices. Wasn't it Karl Marx who observed
that one cannot be just a little bit pregnant?

JDC:

>Field is a brilliant philosophical logician. His book isn't just
>fooling around. It would take one or two courses in logic, and a
>course or two in analytic philosophy just to understand the concepts
>in it (and, no, they can't be paraphrased in simpler language -- I
>am talking about the concepts, not the way they are expressed).

Harumph! Field is a brilliant "philosophical" logician, as opposed to
some of his accomplished contemporaries, brilliant *mathemathical*
logicians Kripke, Kreisel, Scott, Kaplan, Church, or Martin? Methinks
this is just a question of competence. Methinks Field is just fooling
around, cause it's the best he can do. As for you, John, your snubs
seem utterly groundless. Granted that you can fart, but do you have
the shit to back it up?

JDC:

>On Field's approach, numbers are not required to make precise
>predictions. ("The pointer is closer to line A than line B,"
>might be an example.) Numbers are convenient ways to express
>such claims, but it is unclear they are logically necessary.

No, John, of course numbers are not required to make precise
predictions! After all, you only need the concept of a metric to
verify the truth of "The pointer is closer to line A than line B."
And every schoolboy knows that metric does *not* depend on the icky
concept of number.

>--
>John Collier Email: jcol...@ariel.ucs.unimelb.edu.au
>HPS -- U. of Melbourne Fax: +61 3 344 7959
>Parkville, Victoria, AUSTRALIA 3052

cordially,
mikhail zel...@husc.harvard.edu
"Le cul des femmes est monotone comme l'esprit des hommes."

### Mikhail Zeleny

Feb 23, 1993, 11:30:18 AM2/23/93
to
In article <1mcko6...@ariel.ucs.unimelb.EDU.AU>

jcol...@ariel.ucs.unimelb.EDU.AU (John Donald Collier) writes:

>zel...@husc10.harvard.edu (Mikhail Zeleny) writes:

>>In article <1m9f5i...@ariel.ucs.unimelb.EDU.AU>
>>jcol...@ariel.ucs.unimelb.EDU.AU (John Donald Collier) writes:

MZ:

>>I am always happy to see someone display manners inferior to my own.
>>Please apply your criteria to yourself, John: the semantics of
>>identity involves either an explicit second-order extensional
>>characterization, or a first-order introduction as a primitive binary
>>predicate with appropriate axioms, in which case the second-order
>>quantifiers will be called for in giving the semantics of the language
>>in which these axioms are stated. In either case, the ontological
>>commitment is far in excess of that required for defining either von
>>Neumann or Frege integers.

JDC:
>I didn't mention identity. What the hell do you think you are

It's a very bad idea to deny your words on a network equipped with a

JDC:
>>>Nice example of a flame by someone who is completely ignorant of the
>>>subject. Field's approach a nominalistic approach to science. Your
>>>case would be stated in terms of identity. There is no difference
>>>between ball A and ball B. All scientific expressions are out in terms
>>>of similarities and dissimilarities on the nominalistic approach. Of
>>>course these have a cardinality, but their cardinality need not be
>>>used to do science. Other numbers are reduced out the same way.

Now that your lie has been dealt with, what about that identity?

MZ:

>>Field tries to avoid this predicament by claiming that his logic is
>>special, in that it does not depend on a metatheoretic interpretation.
>>Yeah, right. In any case, the sort of nominalism advocated by your
>>pal Hartry is not nearly sufficiently eliminative to warrant any real

JDC:
>Don't presume Field is a friend of mine. Secondly, the question
>concerned the need of infinity for physics, not your philosophical
>interest, which we all know to be warped.

Sorry, John, I didn't mean to cast aspersions. But the question, as
you restated it by appealing to Field's thesis, concerns the need of
mathematical entities for physics, rather than the truth of any
proposition about infinity. So kindly set aside the issue of y'all's
attitude towards my phililosophical interest, and try to deal with the
mess you've dragged yourself into, by dint of your blithe assertions.

MZ:

>>philosophical interest. For one thing, Field implicitly opts for
>>countenancing type-identities, insofar as he recognizes the need for
>>proof theory; so there you already have one species of abstract
>>objects, which, by dint of arbitrary pronouncement, are ruled not to
>>be mathematical objects. For another, in taking as the basic building
>>block of his physicalism a theory of concrete aggregates of space-time
>>points, Field commits himself to the ontology of such choice abstract
>>universals, as upper semi-lattices. Wasn't it Karl Marx who observed
>>that one cannot be just a little bit pregnant?

JDC:
>I thought the topic was numbers. Why are you talking about something
>else? Also, I wasn't aware that commitment to concrete aggregates of
>space-time points committed one to anything abstract, except perhaps
>one who was already ideologically committed.

You can get numbers out of sets, if not out of lattices. The point is
that there is little to choose between positing the need for axioms of
upper semi-lattices, or axioms for sets, as the issue of concreteness
is orthogonal to that of choosing the power of the resulting theory.
In short, sets may be interpreted as concrete manifolds, just as upper
semi-lattices may be interpreted as concrete aggregates. But
statements about sets or aggregates *must* be interpreted as abstract
syntactical types, for otherwise your proof theory never gets off the
ground. So ideological commitment is inescapable even for your side.

JDC:
>>>Field is a brilliant philosophical logician. His book isn't just
>>>fooling around. It would take one or two courses in logic, and a
>>>course or two in analytic philosophy just to understand the concepts
>>>in it (and, no, they can't be paraphrased in simpler language -- I
>>>am talking about the concepts, not the way they are expressed).

MZ:

>>Harumph! Field is a brilliant "philosophical" logician, as opposed to
>>some of his accomplished contemporaries, brilliant *mathemathical*
>>logicians Kripke, Kreisel, Scott, Kaplan, Church, or Martin? Methinks
>>this is just a question of competence. Methinks Field is just fooling
>>around, cause it's the best he can do. As for you, John, your snubs
>>seem utterly groundless. Granted that you can fart, but do you have
>>the shit to back it up?

JDC:
>I just did, you little twerp. All I had to do was quote you.
>Mathematical logic has little to do with the issue. Now try pulling
>your foot out of your mouth. Your irrelevancies don't do much to
>impress me.

Obviously not: you would first have to learn to understand them. And
do learn to curb your epithets, -- I happen to have a few choice ones,
just raring to stick to your person, -- and then where would we be?

MZ:

>>No, John, of course numbers are not required to make precise
>>predictions! After all, you only need the concept of a metric to
>>verify the truth of "The pointer is closer to line A than line B."
>>And every schoolboy knows that metric does *not* depend on the icky
>>concept of number.

JDC:
>Thanks for confirming what I was saying. Next time please do it
>more clearly. One gets the impression that you have trouble
>with English.

Very good! I see you have your work cut out for you; I shall be
eagerly awaiting the appearance of your sequel to Field's crackpot
classic. What are you going to call it, "Metric Without Numbers"?

I do so hate to intrude on your smug ignorance, but were you *really*
able to obtain a doctorate in philosophy, and to specialize in the
philosophy of science, without cracking a book of real analysis?

### Michael Tobis

Feb 23, 1993, 2:11:28 PM2/23/93
to
People take for gospel
Things that are imposs'ble.
-Piet Hein

Mr. Zeleny, I suggest you refrain from using sophisticated arguments your
opponent cannot understand in this debate, as he will certainly reply
with meaningless blather you cannot understand, all the while contending
that his arguments are meaningful and yours are not.

e.g.

In article <1993Feb23.1...@husc3.harvard.edu>, zel...@husc10.harvard.edu (Mikhail Zeleny) writes:
|> In article <1mcko6...@ariel.ucs.unimelb.EDU.AU>
|> jcol...@ariel.ucs.unimelb.EDU.AU (John Donald Collier) writes:
|>
|> >zel...@husc10.harvard.edu (Mikhail Zeleny) writes:
|>
|> >>In article <1m9f5i...@ariel.ucs.unimelb.EDU.AU>
|> >>jcol...@ariel.ucs.unimelb.EDU.AU (John Donald Collier) writes:
|>
|> MZ:
|> >>I am always happy to see someone display manners inferior to my own.
|> >>Please apply your criteria to yourself, John: the semantics of
|> >>identity involves either an explicit second-order extensional
|> >>characterization, or a first-order introduction as a primitive binary
|> >>predicate with appropriate axioms, in which case the second-order
|> >>quantifiers will be called for in giving the semantics of the language
|> >>in which these axioms are stated. In either case, the ontological
|> >>commitment is far in excess of that required for defining either von
|> >>Neumann or Frege integers.

...

A much better effort is this:

|> MZ:
|> >>No, John, of course numbers are not required to make precise
|> >>predictions! After all, you only need the concept of a metric to
|> >>verify the truth of "The pointer is closer to line A than line B."
|> >>And every schoolboy knows that metric does *not* depend on the icky
|> >>concept of number.
|>
|> JDC:
|> >Thanks for confirming what I was saying. Next time please do it
|> >more clearly. One gets the impression that you have trouble
|> >with English.
|>
|> Very good! I see you have your work cut out for you; I shall be
|> eagerly awaiting the appearance of your sequel to Field's crackpot
|> classic. What are you going to call it, "Metric Without Numbers"?
|>
|> I do so hate to intrude on your smug ignorance, but were you *really*
|> able to obtain a doctorate in philosophy, and to specialize in the
|> philosophy of science, without cracking a book of real analysis?

Of course. This sort of Philosophy of Science is not practiced by anyone
who has even cracked a book of introductory physics, as far as I can tell.

In that exchange, the ignorance of your opponent is clearly exposed
even for those of us who are really just mathematical dabblers. The
more accessible your criticisms, the more likely you will have an impact
on people who are in danger of catching this apalling 'all is text'
disease.

BTW I am especially taken by the following section:

JDC:
|> >>>Nice example of a flame by someone who is completely ignorant of the
|> >>>subject. Field's approach a nominalistic approach to science. Your
|> >>>case would be stated in terms of identity. There is no difference
|> >>>between ball A and ball B. All scientific expressions are out in terms
|> >>>of similarities and dissimilarities on the nominalistic approach. Of
|> >>>course these have a cardinality, but their cardinality need not be
|> >>>used to do science. Other numbers are reduced out the same way.

|> >>>Field is a brilliant philosophical logician. His book isn't just

|> >>>fooling around. It would take one or two courses in logic, and a
|> >>>course or two in analytic philosophy just to understand the concepts
|> >>>in it (and, no, they can't be paraphrased in simpler language -- I
|> >>>am talking about the concepts, not the way they are expressed).

Many of the most amazing symptoms appear here. Propaganda masquerading
as reason, elitism in the cause of anti-elitism, impenetrability posing
as profundity, rudeness taken for argument, nihilism taken for progress,
and hero worship taken for skepticism.

The most basic argument is this one: "I'm rubber and you're glue, whatever
you say goes back to you." It isn't very deep, but it is well-understood
by the average six-year-old, and long years of deep study allow the
practitioners to hone it to a fine weapon. Here elitism is used in an
attack on elitism, and a declaration of brilliance is used in place
of any semblance of reasoned argument.

Welcome to the topsy-turvy world of postmodern criticism. Prepare to
be deconstructed. Your elitist theorems will soon fall to the levelling
sword of sour grapes nihilism! etc. etc.

Again, Mr. Zeleny, I understand the temptation to be rude to these people,
but it only plays into their hands. Scrupulous politesse is something they
find much more irritating.

mt

### john baez

Feb 23, 1993, 5:29:07 PM2/23/93
to

Heh. CLEARLY science can do without numbers and make precise
predictions. For example, instead of referring to temperatures as
96, 97, 98, and 99 degrees (and so on) we can call them "pretty hot,"
"quite hot," "pretty darn hot," and "darn hot." Then, just as we can
say "the pointer is closer to line A than line B," we can say "it's
pretty darn hot." And if we like, we can, instead of saying "1 degree
hotter," say "a little bit hotter." Then we have such facts as "darn
hot is a little bit hotter than pretty darn hot." Keep this up and, lo,
we have eliminated numbers from the sciences!!

Perhaps, in fact, we should appoint Fields to head a committee in charge
of this conversion process, since he was the one who first came up with
this brilliant notion. Since most Americans are innumerate it
would simplify life immensely. And for once, the philosophy of science

### john baez

Feb 23, 1993, 5:45:35 PM2/23/93
to
In article <26...@galaxy.ucr.edu> ba...@guitar.ucr.edu (john baez) writes:
>And for once, the philosophy of science

My newsposter ate the rest of this snide remark, which might be just
as well.

### John Donald Collier

Feb 24, 1993, 12:46:14 AM2/24/93
to
zel...@husc10.harvard.edu (Mikhail Zeleny) writes:

>In article <1m9f5i...@ariel.ucs.unimelb.EDU.AU>
>jcol...@ariel.ucs.unimelb.EDU.AU (John Donald Collier) writes:

}I am always happy to see someone display manners inferior to my own.
}Please apply your criteria to yourself, John: the semantics of
}identity involves either an explicit second-order extensional
}characterization, or a first-order introduction as a primitive binary
}predicate with appropriate axioms, in which case the second-order
}quantifiers will be called for in giving the semantics of the language
}in which these axioms are stated. In either case, the ontological
}commitment is far in excess of that required for defining either von
}Neumann or Frege integers.

I didn't mention identity. What the hell do you think you are

%OAField tries to avoid this predicament by claiming that his logic is
%special, in that it does not depend on a metatheoretic interpretation.
%Yeah, right. In any case, the sort of nominalism advocated by your
%pal Hartry is not nearly sufficiently eliminative to warrant any real

Don't presume Field is a friend of mine. Secondly, the question
concerned the need of infinity for physics, not your philosophical
interest, which we all know to be warped.

%philosophical interest. For one thing, Field implicitly opts for
%countenancing type-identities, insofar as he recognizes the need for
%proof theory; so there you already have one species of abstract
%objects, which, by dint of arbitrary pronouncement, are ruled not to
%be mathematical objects. For another, in taking as the basic building
%block of his physicalism a theory of concrete aggregates of space-time
%points, Field commits himself to the ontology of such choice abstract
%universals, as upper semi-lattices. Wasn't it Karl Marx who observed
%that one cannot be just a little bit pregnant?

I thought the topic was numbers. Why are you talking about something
else? Also, I wasn't aware that commitment to concrete aggregates of
space-time points committed one to anything abstract, except perhaps
one who was already ideologically committed.

%JDC:
%%Field is a brilliant philosophical logician. His book isn't just
%%fooling around. It would take one or two courses in logic, and a
%%course or two in analytic philosophy just to understand the concepts
%%in it (and, no, they can't be paraphrased in simpler language -- I
%%am talking about the concepts, not the way they are expressed).

%Harumph! Field is a brilliant "philosophical" logician, as opposed to
%some of his accomplished contemporaries, brilliant *mathemathical*
%logicians Kripke, Kreisel, Scott, Kaplan, Church, or Martin? Methinks
%this is just a question of competence. Methinks Field is just fooling
%around, cause it's the best he can do. As for you, John, your snubs
%seem utterly groundless. Granted that you can fart, but do you have
%the shit to back it up?

I just did, you little twerp. All I had to do was quote you.
Mathematical logic has little to do with the issue. Now try pulling
your foot out of your mouth. Your irrelevancies don't do much to
impress me.

%No, John, of course numbers are not required to make precise
%predictions! After all, you only need the concept of a metric to
%verify the truth of "The pointer is closer to line A than line B."
%And every schoolboy knows that metric does *not* depend on the icky
%concept of number.

Thanks for confirming what I was saying. Next time please do it
more clearly. One gets the impression that you have trouble
with English.

### Mikhail Zeleny

Feb 24, 1993, 1:58:53 AM2/24/93
to
In article <1993Feb23.1...@daffy.cs.wisc.edu>
to...@skool.ssec.wisc.edu (Michael Tobis) writes:

MT:

>People take for gospel
>Things that are imposs'ble.
> -Piet Hein
>
>Mr. Zeleny, I suggest you refrain from using sophisticated arguments your
>opponent cannot understand in this debate, as he will certainly reply
>with meaningless blather you cannot understand, all the while contending
>that his arguments are meaningful and yours are not.

I believe that your suggestion is meant well, and on a certain level,
it makes sense. However, I cannot accept it as stated. Important
issues call for sophisticated arguments; anything short of that is a
mere palliative, and would never suffice to resolve the problem. And
even if my interlocutor is unlikely to understand the logical
consequences of his own position, this exchange, as you yourself
convincingly demonstrate, is by no means limited to interlocution.

>e.g.

>In article <1993Feb23.1...@husc3.harvard.edu>,
>zel...@husc10.harvard.edu (Mikhail Zeleny) writes:

>>In article <1mcko6...@ariel.ucs.unimelb.EDU.AU>
>>jcol...@ariel.ucs.unimelb.EDU.AU (John Donald Collier) writes:

>>>zel...@husc10.harvard.edu (Mikhail Zeleny) writes:

>>>>jcol...@ariel.ucs.unimelb.EDU.AU (John Donald Collier) writes:

MZ:
>>>>I am always happy to see someone display manners inferior to my own.
>>>>Please apply your criteria to yourself, John: the semantics of
>>>>identity involves either an explicit second-order extensional
>>>>characterization, or a first-order introduction as a primitive binary
>>>>predicate with appropriate axioms, in which case the second-order
>>>>quantifiers will be called for in giving the semantics of the language
>>>>in which these axioms are stated. In either case, the ontological
>>>>commitment is far in excess of that required for defining either von
>>>>Neumann or Frege integers.

This is an important and subtle point, for which I claim no original
credit. Similar ideas are to be found more or less explicitly in such
standard texts on the virtues ob higher-order logic as Church, Kreisel
& Krivine, Barwise & Feferman, or Shapiro. In each instance, it is
all too easy to ignore its scope, or fail to understand its
implications.

MT:

>...
>
>A much better effort is this:

MZ:
>>>>No, John, of course numbers are not required to make precise
>>>>predictions! After all, you only need the concept of a metric to
>>>>verify the truth of "The pointer is closer to line A than line B."
>>>>And every schoolboy knows that metric does *not* depend on the icky
>>>>concept of number.

JDC:
>>>Thanks for confirming what I was saying. Next time please do it
>>>more clearly. One gets the impression that you have trouble
>>>with English.

MZ:

>>Very good! I see you have your work cut out for you; I shall be
>>eagerly awaiting the appearance of your sequel to Field's crackpot
>>classic. What are you going to call it, "Metric Without Numbers"?
>>
>>I do so hate to intrude on your smug ignorance, but were you *really*
>>able to obtain a doctorate in philosophy, and to specialize in the
>>philosophy of science, without cracking a book of real analysis?

MT:

>Of course. This sort of Philosophy of Science is not practiced by anyone
>who has even cracked a book of introductory physics, as far as I can tell.

The truly shameful aspect of this situation is that the above nonsense
gets passed for being representative of the field of philosophy graced
by the presences of Karl Popper, Willard Quine, David Armstrong,
Hilary Putnam, and Bas van Fraassen.

MT:

>In that exchange, the ignorance of your opponent is clearly exposed
>even for those of us who are really just mathematical dabblers. The
>more accessible your criticisms, the more likely you will have an impact
>on people who are in danger of catching this apalling 'all is text'
>disease.

Well, I am a mere dabbler myself; but the cause of my animus is not so
much the contemptuous dismissal of the value of mathematics, as the
discrediting of my own field, philosophy, that occurs when this sort
of garbage gets passed for its finest flower.

MT:

>BTW I am especially taken by the following section:

JDC:
>>>>>Nice example of a flame by someone who is completely ignorant of the
>>>>>subject. Field's approach a nominalistic approach to science. Your
>>>>>case would be stated in terms of identity. There is no difference
>>>>>between ball A and ball B. All scientific expressions are out in terms
>>>>>of similarities and dissimilarities on the nominalistic approach. Of
>>>>>course these have a cardinality, but their cardinality need not be
>>>>>used to do science. Other numbers are reduced out the same way.
>>>>>
>>>>>Field is a brilliant philosophical logician. His book isn't just
>>>>>fooling around. It would take one or two courses in logic, and a
>>>>>course or two in analytic philosophy just to understand the concepts
>>>>>in it (and, no, they can't be paraphrased in simpler language -- I
>>>>>am talking about the concepts, not the way they are expressed).

MT:

>Many of the most amazing symptoms appear here. Propaganda masquerading
>as reason, elitism in the cause of anti-elitism, impenetrability posing
>as profundity, rudeness taken for argument, nihilism taken for progress,
>and hero worship taken for skepticism.
>
>The most basic argument is this one: "I'm rubber and you're glue, whatever
>you say goes back to you." It isn't very deep, but it is well-understood
>by the average six-year-old, and long years of deep study allow the
>practitioners to hone it to a fine weapon. Here elitism is used in an
>attack on elitism, and a declaration of brilliance is used in place
>of any semblance of reasoned argument.
>
>Welcome to the topsy-turvy world of postmodern criticism. Prepare to
>be deconstructed. Your elitist theorems will soon fall to the levelling
>sword of sour grapes nihilism! etc. etc.

If you feel this way about the damage done to mathematics, you should
see what this crowd does to the foundations of idiographic disciplines
in the humanitities, like history and philology. The truly annoying
part of it all is the collateral trashing of rhetoric, and other areas
of linguistics, coopted into the service of ideology.

MT:

>Again, Mr. Zeleny, I understand the temptation to be rude to these people,
>but it only plays into their hands. Scrupulous politesse is something they
>find much more irritating.

I agree with you in principle, but alas, my daimon does not seem to
feel the same way.

>mt

### Tal Kubo

Feb 24, 1993, 6:21:55 AM2/24/93
to
In article <1m9f5i...@ariel.ucs.unimelb.EDU.AU>

jcol...@ariel.ucs.unimelb.EDU.AU (John Donald Collier) writes:
>
>Nice example of a flame by someone who is completely ignorant of the
>subject. [...]

>
>Field is a brilliant philosophical logician. His book isn't just
>fooling around. It would take one or two courses in logic, and a
>course or two in analytic philosophy just to understand the concepts
>in it (and, no, they can't be paraphrased in simpler language -- I
>am talking about the concepts, not the way they are expressed).

The high priest speaks!! I'm glad someone can grasp
the boggling complexity of Field's numberless thoughts,
since we plebes could never manage it ourselves.

### Gene W. Smith

Feb 24, 1993, 10:02:22 AM2/24/93
to
In article <1993Feb23.0...@husc3.harvard.edu>
zel...@husc10.harvard.edu (Mikhail Zeleny) writes:

>I am always happy to see someone display manners inferior to my own.

If you improved your manners you would greatly increase these happy
occasions.

--
Gene Ward Smith/Brahms Gang/IWR/Ruprecht-Karls University
gsm...@kalliope.iwr.uni-heidelberg.de

### Michael Zeleny

Feb 24, 1993, 6:15:59 PM2/24/93
to
In article <1993Feb24.1...@sun0.urz.uni-heidelberg.de>

gsm...@lauren.iwr.uni-heidelberg.de (Gene W. Smith) writes:

>In article <1993Feb23.0...@husc3.harvard.edu>
>zel...@husc10.harvard.edu (Mikhail Zeleny) writes:

>>I am always happy to see someone display manners inferior to my own.

>If you improved your manners you would greatly increase these happy
>occasions.

As much as I appreciate your ladylike advice, Gene, I find it far more
rewarding to apply my effort to the inherently fruitful and gratifying
task of degrading your manners, than to the futile and thankless one,
of improving mine.

>--
> Gene Ward Smith/Brahms Gang/IWR/Ruprecht-Karls University
> gsm...@kalliope.iwr.uni-heidelberg.de

cordially,

### Michael Tobis

Feb 25, 1993, 1:15:26 PM2/25/93
to
In article <1993Feb24.0...@husc3.harvard.edu>, zel...@husc10.harvard.edu (Mikhail Zeleny) writes:
|> In article <1993Feb23.1...@daffy.cs.wisc.edu>
|> to...@skool.ssec.wisc.edu (Michael Tobis) writes:

|> >Mr. Zeleny, I suggest you refrain from using sophisticated arguments your
|> >opponent cannot understand in this debate, as he will certainly reply
|> >with meaningless blather you cannot understand, all the while contending
|> >that his arguments are meaningful and yours are not.

|> I believe that your suggestion is meant well, and on a certain level,
|> it makes sense. However, I cannot accept it as stated. Important
|> issues call for sophisticated arguments; anything short of that is a
|> mere palliative, and would never suffice to resolve the problem.

I disagree. The importance of an issue is not obviously closely related to
the sophistication of the argument required to settle it. Why should it be?

|> And
|> even if my interlocutor is unlikely to understand the logical
|> consequences of his own position, this exchange, as you yourself
|> convincingly demonstrate, is by no means limited to interlocution.

Well, the subset of your audience that also fails to follow these particular
consequences is certainly nonempty.

|> >>>zel...@husc10.harvard.edu (Mikhail Zeleny) writes:

|> MZ:

|> >>>>the semantics of
|> >>>>identity involves either an explicit second-order extensional
|> >>>>characterization, or a first-order introduction as a primitive binary
|> >>>>predicate with appropriate axioms, in which case the second-order
|> >>>>quantifiers will be called for in giving the semantics of the language
|> >>>>in which these axioms are stated. In either case, the ontological
|> >>>>commitment is far in excess of that required for defining either von
|> >>>>Neumann or Frege integers.

|> This is an important and subtle point, for which I claim no original
|> credit. Similar ideas are to be found more or less explicitly in such
|> standard texts on the virtues ob higher-order logic as Church, Kreisel
|> & Krivine, Barwise & Feferman, or Shapiro. In each instance, it is
|> all too easy to ignore its scope, or fail to understand its
|> implications.

I haven't the least idea what you are saying. While this is not an unusual
state of affairs for me reading sci.math, I find it more than usually
frustrating because its subject is one that I suspect ought to be
accessible. And I believe that in resorting to such impenetrable arguments
to counter what appears to me to be nonsense, you are inadvertently
supporting the opponent's case by implying that such difficult arguments are
needed to refute him.

Is it possible to restate your point in a manner accessible to someone who
does have a smattering of real analysis and a fair amount of applied maths?
Otherwise I will indeed remain ignorant of the scope and implications of your
argument.

For now, I continue to suspect that the difficulty of much philosophy is
more a result of obfuscation rather than of sophistication. Even the rare
piece of philosophy that I read and appreciate (Dennett most recently) tends
to irritate me with what I take to be unnecessary and precious verbosity.

sincerely, but obviously out of his depth,
mt

### Mark Underwood

Feb 25, 1993, 7:13:41 PM2/25/93
to
In article <1993Feb21.0...@magnus.acs.ohio-state.edu>

wrote a good discussion of the question of the necessity of infinity and
the relationship between science and mathematics . . .

In my first Circuits class (Circuits I: Analysis of Linear Circuits) my
professor told the class that our entire job as Electrical Engineers would
be to create mathematical models of circuit designs, make sure that
*mathematically* they did what they were supposed to do, and then finally
try to come up with a *real* approximation of our *mathematical* circuit.
I'd say, then, that assuming the other sciences (or at least engineering
fields) follow a similar relationship, to say mathematics is crucial to
sicence is quite an understatement, wouldn't you???

_
Just my \$.019

Mark S. Underwood
EE Student, University of Kentucky
Lab Assistant, Boyd Hall Microlab
(a tiny little division of UK Library Microlabs)
E-Mail: msun...@mik.uky.edu

-- Men occasionally stumble over the truth, but most of them pick
themselves
-- up and hurry off as if nothing had happened.
-- Winston Churchill (1874-1965)

### John Donald Collier

Feb 26, 1993, 4:10:04 AM2/26/93
to
zel...@husc10.harvard.edu (Mikhail Zeleny) writes:

>>>>Nice example of a flame by someone who is completely ignorant of the
>>>>subject. Field's approach a nominalistic approach to science. Your
>>>>case would be stated in terms of identity. There is no difference
>>>>between ball A and ball B. All scientific expressions are out in terms
>>>>of similarities and dissimilarities on the nominalistic approach. Of
>>>>course these have a cardinality, but their cardinality need not be
>>>>used to do science. Other numbers are reduced out the same way.

>Now that your lie has been dealt with, what about that identity?

There is no mention of identity.

>You can get numbers out of sets, if not out of lattices. The point is
>that there is little to choose between positing the need for axioms of
>upper semi-lattices, or axioms for sets, as the issue of concreteness
>is orthogonal to that of choosing the power of the resulting theory.
>In short, sets may be interpreted as concrete manifolds, just as upper
>semi-lattices may be interpreted as concrete aggregates. But
>statements about sets or aggregates *must* be interpreted as abstract
>syntactical types, for otherwise your proof theory never gets off the
>ground. So ideological commitment is inescapable even for your side.

Field thinks there is something to choose. I am not sure I agree with
him, but your stating otherwise doesn't resolve the issue. I never
claimed that ideological commitment was not required. HF is a dogmatic
nominalist, who believes it is better to expand logic rather than to
accept abstract entities. My line of argument was that, given doubts
about the need for numbers in physics, the door is open to questioning
the need for infinity (otherwise there would be the reals to deal
with). Then I argued that decisions to throw away terms at infinity
in field theories were made on the basis of assumptions that what goes
on at infinity does not matter. Thus it is unclear that infinity is
required for science. Now you purport to have questioned the basis
for the doubts about the need for numbers, but you keep raising issues
which I don't seem to have relied on in my reasoning, as if they
were telling.

I am not especially sympathetic to Hartry's position myself, and
I have argued with him and others about it, but I don't see that
anything that you have raised really adds very much. If anything
you are merely muddying the waters. "Dirtying", perhaps, would
be more accurate.

>JDC:
>>>>Field is a brilliant philosophical logician. His book isn't just
>>>>fooling around. It would take one or two courses in logic, and a
>>>>course or two in analytic philosophy just to understand the concepts
>>>>in it (and, no, they can't be paraphrased in simpler language -- I
>>>>am talking about the concepts, not the way they are expressed).

>MZ:
>>>Harumph! Field is a brilliant "philosophical" logician, as opposed to
>>>some of his accomplished contemporaries, brilliant *mathemathical*
>>>logicians Kripke, Kreisel, Scott, Kaplan, Church, or Martin? Methinks
>>>this is just a question of competence. Methinks Field is just fooling
>>>around, cause it's the best he can do. As for you, John, your snubs
>>>seem utterly groundless. Granted that you can fart, but do you have
>>>the shit to back it up?

>JDC:
>>I just did, you little twerp. All I had to do was quote you.
>>Mathematical logic has little to do with the issue. Now try pulling
>>your foot out of your mouth. Your irrelevancies don't do much to
>>impress me.

>Obviously not: you would first have to learn to understand them. And
>do learn to curb your epithets, -- I happen to have a few choice ones,
>just raring to stick to your person, -- and then where would we be?

First learn that philosophical logic and mathematical logic are not on
a continuum, and that one can be very good in one without being very
good in the other. Then perhaps I would be willing to withdraw
my charge that you are a twerp.

>MZ:
>>>No, John, of course numbers are not required to make precise
>>>predictions! After all, you only need the concept of a metric to
>>>verify the truth of "The pointer is closer to line A than line B."
>>>And every schoolboy knows that metric does *not* depend on the icky
>>>concept of number.

>JDC:
>>Thanks for confirming what I was saying. Next time please do it
>>more clearly. One gets the impression that you have trouble
>>with English.

>Very good! I see you have your work cut out for you; I shall be
>eagerly awaiting the appearance of your sequel to Field's crackpot
>classic. What are you going to call it, "Metric Without Numbers"?

No, you misunderstood me. I was calling attention to your
invocation of uneccessary irrelevancies in your desparate
attempts to take issue with the point I raised.

>I do so hate to intrude on your smug ignorance, but were you *really*
>able to obtain a doctorate in philosophy, and to specialize in the
>philosophy of science, without cracking a book of real analysis?

No, obviously not. If you understood English better you would have
noticed that comparison of pointer positions does not depend on metric.
Incidentally, have you studied affine geometry?

Look, the confusions you have been making in this exchange are
so elementary that I really don't see much reason to continue.
You keep bringing in things that don't matter, like metrics
and identitiy, as if they do, and when I point out your
confusion you merely act like the things you want to talk
about are of GREAT IMPORTANCE, without explaining why they
have any relevance at all.

I have some idea why you might think the rather trivial relation of
identity is important, and why you might think that the comparison of
pointer positions requires a metric, but since you brought up the
issues, I really think it is your responsibility to explain their
relevance.

### Mikhail Zeleny

Feb 26, 1993, 8:02:39 AM2/26/93
to
In article <1mi9ec...@ariel.ucs.unimelb.EDU.AU>

jcol...@ariel.ucs.unimelb.EDU.AU (John Donald Collier) writes:

>zel...@husc10.harvard.edu (Mikhail Zeleny) writes:

JDC:

>>>>>Nice example of a flame by someone who is completely ignorant of the
>>>>>subject. Field's approach a nominalistic approach to science. Your
>>>>>case would be stated in terms of identity. There is no difference

^^^^^^^^

>>>>>between ball A and ball B. All scientific expressions are out in terms
>>>>>of similarities and dissimilarities on the nominalistic approach. Of
>>>>>course these have a cardinality, but their cardinality need not be
>>>>>used to do science. Other numbers are reduced out the same way.

MZ:

>>Now that your lie has been dealt with, what about that identity?

JDC:

>There is no mention of identity.

Sure thing: the term is used, rather than mentioned. Note that if you
try to cover your tracks by backpedaling to "terms of similarities and
dissimilarities", you would _ipso facto_ commit yourself to the first
horn of the dilemma, by opting for a second-order language, in which
the integers are definable. If you have a first-order explanation of
quantifiers over similarities and dissimilarities, please enlighten
me. Note that in that unlikely case, your ontological commitment to
second-order explanation will be merely pushed into the metalanguage,
which would have to be strong enough to derive elementary arithmetic.

MZ:

>>You can get numbers out of sets, if not out of lattices. The point is
>>that there is little to choose between positing the need for axioms of
>>upper semi-lattices, or axioms for sets, as the issue of concreteness
>>is orthogonal to that of choosing the power of the resulting theory.
>>In short, sets may be interpreted as concrete manifolds, just as upper
>>semi-lattices may be interpreted as concrete aggregates. But
>>statements about sets or aggregates *must* be interpreted as abstract
>>syntactical types, for otherwise your proof theory never gets off the
>>ground. So ideological commitment is inescapable even for your side.

JDC:

>Field thinks there is something to choose. I am not sure I agree with
>him, but your stating otherwise doesn't resolve the issue. I never
>claimed that ideological commitment was not required. HF is a dogmatic
>nominalist, who believes it is better to expand logic rather than to
>accept abstract entities. My line of argument was that, given doubts
>about the need for numbers in physics, the door is open to questioning
>the need for infinity (otherwise there would be the reals to deal
>with). Then I argued that decisions to throw away terms at infinity
>in field theories were made on the basis of assumptions that what goes
>on at infinity does not matter. Thus it is unclear that infinity is
>required for science. Now you purport to have questioned the basis
>for the doubts about the need for numbers, but you keep raising issues
>which I don't seem to have relied on in my reasoning, as if they
>were telling.

Field thinks many strange and wondrous things; among them is the gem
that completeness is an accidental property of first-order logic (he
was kind enough to repeat that chestnut for me during his 1989 UCLA
colloquium). Generally speaking, given a philosophical thesis of an
arbitrarily high degree of absurdity, there will be a professional
philosophaster capable of maintaining it with a straight face. In
this instance, notwithstanding the subject line, my challenge is
strictly limited to your slobbering encomium to Field, fully quoted in
my original article. In other words, the point is not the alleged
dispensability of infinity, but Field's putative ability to get along
without numbers and other mathematical objects.

JDC:

>I am not especially sympathetic to Hartry's position myself, and
>I have argued with him and others about it, but I don't see that
>anything that you have raised really adds very much. If anything
>you are merely muddying the waters. "Dirtying", perhaps, would
>be more accurate.

I said enough to make the matters clear to anyone who instructs others
to take a couple of courses in logic and analytic philosophy, so that
they may understand his sublime concepts. If you fail to understand
the implications of my point, it is incumbent upon you to stop telling
people who disagree with you, how stupid they must be, and get
cracking on that remedial education project yourself.

JDC:
>>>>>Field is a brilliant philosophical logician. His book isn't just
>>>>>fooling around. It would take one or two courses in logic, and a
>>>>>course or two in analytic philosophy just to understand the concepts
>>>>>in it (and, no, they can't be paraphrased in simpler language -- I
>>>>>am talking about the concepts, not the way they are expressed).

MZ:
>>>>Harumph! Field is a brilliant "philosophical" logician, as opposed to
>>>>some of his accomplished contemporaries, brilliant *mathemathical*
>>>>logicians Kripke, Kreisel, Scott, Kaplan, Church, or Martin? Methinks
>>>>this is just a question of competence. Methinks Field is just fooling
>>>>around, cause it's the best he can do. As for you, John, your snubs
>>>>seem utterly groundless. Granted that you can fart, but do you have
>>>>the shit to back it up?

JDC:
>>>I just did, you little twerp. All I had to do was quote you.
>>>Mathematical logic has little to do with the issue. Now try pulling
>>>your foot out of your mouth. Your irrelevancies don't do much to
>>>impress me.

MZ:

>>Obviously not: you would first have to learn to understand them. And
>>do learn to curb your epithets, -- I happen to have a few choice ones,
>>just raring to stick to your person, -- and then where would we be?

JDC:

>First learn that philosophical logic and mathematical logic are not on
>a continuum, and that one can be very good in one without being very
>good in the other. Then perhaps I would be willing to withdraw
>my charge that you are a twerp.

It seems to me that you are the one who must learn that dubious claims
do not get promoted to facts on your say-so. In this instance, you
come across as your nameless colleague, who pronounced before an IAS
audience including G\"odel, that no advances occurred in logic since
the time of Aristotle. But for the sake of charity, I will withdraw
my objection, provided that you find me three articles in the Handbook
of Philosophical Logic, that support your claim in being free of
pernicious mathematical methods. Otherwise, feel free to revel in
your Victorian philistine ignorance.

MZ:
>>>>No, John, of course numbers are not required to make precise
>>>>predictions! After all, you only need the concept of a metric to
>>>>verify the truth of "The pointer is closer to line A than line B."
>>>>And every schoolboy knows that metric does *not* depend on the icky
>>>>concept of number.

JDC:
>>>Thanks for confirming what I was saying. Next time please do it
>>>more clearly. One gets the impression that you have trouble
>>>with English.

MZ:

>>Very good! I see you have your work cut out for you; I shall be
>>eagerly awaiting the appearance of your sequel to Field's crackpot
>>classic. What are you going to call it, "Metric Without Numbers"?

JDC:

>No, you misunderstood me. I was calling attention to your
>invocation of uneccessary irrelevancies in your desparate
>attempts to take issue with the point I raised.

The article I responded to, contained no point, -- just blowhard
bluster and awe-struck name-dropping. Every word I said was utterly
relevant to the task of deflating your uppity presumption. If you
choose to dissemble further, help yourself to the last word.

MZ:

>>I do so hate to intrude on your smug ignorance, but were you *really*
>>able to obtain a doctorate in philosophy, and to specialize in the
>>philosophy of science, without cracking a book of real analysis?

JDC:

>No, obviously not. If you understood English better you would have
>noticed that comparison of pointer positions does not depend on metric.

Save your snide dismissals for your long-suffering students. Your
point had to do with verifying the truth of "The pointer is closer to
line A than line B," which clearly depends on metric. While the
property of the pointer appearing between line A and line B is clearly
invariant under a topological mapping, its being closer to line A than
line B depends on the metric properties of your space.

JDC:

>Incidentally, have you studied affine geometry?

Affine geometry will not get you out of your predicament, -- what you
need for calibrating your instruments, is a Euclidean affine space,
with its real-valued norm. Refer to Chapters 8 and 9 of Berger's
geometry textbook, to dispel your confusion over this point.

JDC:

>Look, the confusions you have been making in this exchange are
>so elementary that I really don't see much reason to continue.
>You keep bringing in things that don't matter, like metrics
>and identitiy, as if they do, and when I point out your
>confusion you merely act like the things you want to talk
>about are of GREAT IMPORTANCE, without explaining why they
>have any relevance at all.
>
>I have some idea why you might think the rather trivial relation of
>identity is important, and why you might think that the comparison of
>pointer positions requires a metric, but since you brought up the
>issues, I really think it is your responsibility to explain their
>relevance.

Yes, John, you are rubber and I am glue. Have it your way.

>--
>John Collier Email: jcol...@ariel.ucs.unimelb.edu.au
>HPS -- U. of Melbourne Fax: +61 3 344 7959
>Parkville, Victoria, AUSTRALIA 3052

cordially,
mikhail zel...@husc.harvard.edu
"Les beaulx bastisseurs nouveaulx de pierres mortes ne sont escriptz
en mon livre de vie. Je ne bastis que pierres vives: ce sont hommes."

### Gary Merrill

Feb 26, 1993, 10:42:27 AM2/26/93
to

I don't want what I say below to be taken as a defense of impoliteness,
but I found Mr. Collier's last remarks to be in some ways at least as
irritating as those about which he complains.

In article <1ml2b6...@ariel.ucs.unimelb.EDU.AU>, jcol...@ariel.ucs.unimelb.EDU.AU (John Donald Collier) writes:

|> Now, you may not like what I am saying, or what I think, but I
|> probably have better credentials and more experience than anyone else
|> who has been involved in this discussion. I have taught logic, set

This is quite presumptuous and quite likely false. (Also irrelevant,
but never mind that.)

|> It is unfortunate, but not surprising, that most of my colleagues
|> consider the Internet newsgroups a waste of time. If a book is mentioned

There are a number of reasons that academics (particularly in humanities
departments) might consider spending their time on newsgroups a waste.
Not all of them have to do with the level or type of discussion. Many
academics in humanities departments still lack either easy access to
or the requisite skills to take advantage of the net.

|> it was raised in the first place. It is especially dishonest to
|> attribute to someone things they never said or implied. It is
|> laughable when these attributions are supported by quotes that
|> clearly demonstrate that the attribution is false, but the humour
|> pales rather quickly, like most superficial things.

I presume you speak here, perhaps among other things, of Zeleny's remarks
concerning identity and his claim that you had raised this issue.

I've been pretty puzzled by this exchange. Consider the following, for
example:

You write ...

>>>Nice example of a flame by someone who is completely ignorant of the
>>>subject. Field's approach a nominalistic approach to science. Your
>>>case would be stated in terms of identity. There is no difference

^^^^^^^^

>>>between ball A and ball B. All scientific expressions are out in terms
>>>of similarities and dissimilarities on the nominalistic approach. Of
>>>course these have a cardinality, but their cardinality need not be
>>>used to do science. Other numbers are reduced out the same way.

After his claims about identity, Zeleny says ...

>>Now that your lie has been dealt with, what about that identity?

And you respond ...

>There is no mention of identity.

In fact there is a rather obvious mention of identity. Perhaps what
you *meant* to say was that *Zeleny* requires an appeal to identity
to state his case (but that Field does not). However, at best this
is rather unclearly expressed and your prose lends itself to alternative
interpretations. It is simply not a matter of debate that identity
was mentioned, but you persist in denying this. Perhaps this has
devolved to a trivial and childish point, but one would expect
more care and precision from a professional with the credentials
that have been cited.

--
Gary H. Merrill [Principal Systems Developer, C Compiler Development]
SAS Institute Inc. / SAS Campus Dr. / Cary, NC 27513 / (919) 677-8000
sas...@theseus.unx.sas.com ... !mcnc!sas!sasghm

### Jamie Andrews

Feb 26, 1993, 12:57:56 PM2/26/93
to
Er, could you guys take it to alt.flame or alt.homophobia
or something? I've got another .signature quote out of it, but
that's all I expect to get, given the way things have gone so far.

--Jamie.
ja...@cs.sfu.ca
"I am always happy to see someone display manners inferior to my own." -- MZ

### benjamin j elkins

Feb 26, 1993, 2:37:57 PM2/26/93
to
Hello out there in Net land !

I am looking for any e-mail address at ATT-Bell labs.

I am trying to locate some research material and
once I have a door in, then I think finding it will
be easy.

pax

### john baez

Feb 26, 1993, 3:03:30 PM2/26/93
to
In article <1ml2b6...@ariel.ucs.unimelb.EDU.AU>

jcol...@ariel.ucs.unimelb.EDU.AU (John Donald Collier) writes:
>There has been some question about my credentials in science
>and logic raised. I find this a bit annoying, because the information
>is available, but I summarise:

>Now, you may not like what I am saying, or what I think, but I
>probably have better credentials and more experience than anyone else
>who has been involved in this discussion.
>

>It is unfortunate, but not surprising, that most of my colleagues
>consider the Internet newsgroups a waste of time.

Why don't you answer the criticisms you so dislike with interesting and
informative facts rather than listing your degrees and claiming that you
have better qualifications than the rest of us? Or simply
ignore your detractors and post interesting stuff? Certainly it's not
surprising that folks would consider the above post a waste of time - we
aren't all *that* interested in your GRE scores etc. By the way, you
should generally expect a firestorm of criticism on the net when you say
"these ideas are so abstruse that I cannot possibly do them justice, but
don't criticize them until you've read X, Y, and Z." In the rare cases
where this is indeed true - if *any* such cases exist - it would be
wiser simply to let things drop; it's too much like telling someone "I
know a secret but I'm not going to tell you, so don't ask."

In order to follow my own advice let me change the subject.
Does anyone know a good treatment of the subject of fitting a curve to
data points? On the one hand this is the sort of philosophical problem
that people can wring their hands over endlessly: since there are
infinitely many curves through some finite set of points, any "best" fit
of a curve through those points implies an a priori notion of what
counts as the "best" fit - leading us into the problem of induction in
science, etc.. On the other hand, it's an eminently practical problem
that is treated in millions of particular cases in more or less well
thought out ways. Sometimes one has good reasons to seek a fit of a
certain form (a particular theory), sometimes ones only rationale is
convenience (linear fits or log-log fits are easy to do), and lots of
times it's some uncomfortable mix of the two. I'm wondering if there
are any good books that strike a balance between the two extremes
(philosophy of science on the one hand, nitty-gritty statistics on the
other). I know someone who's trying to fit some curves to some data,
and I think a good general discussion of the issues might help him out.

### Mikhail Zeleny

Feb 26, 1993, 3:09:18 PM2/26/93
to
In article <1ml2b6...@ariel.ucs.unimelb.EDU.AU>

jcol...@ariel.ucs.unimelb.EDU.AU (John Donald Collier) writes:

>There has been some question about my credentials in science
>and logic raised. I find this a bit annoying, because the information
>is available, but I summarise:

> MIT SB, Earth Physics, complete graduate theoretical physics sequence.
> MIT SB, Philosophy, courses with J. Thomson, G. Boolos on advanced
> logic and philosophy of mathematics, at graduate level.
> UCLA MA, courses on set theory and logic, Kaplan, Kalish
> philosophy of language with Burge, Donellan and Perry
> UWO PhD, courses on philosophy of math, advanced logic, philosophy
> of QM, space and time, and numerous others with Bub,
> Harper, Hooker, Demopoulos, van Fraassen.
> Earth Physics Branch, Govt of Canada, 2 years work as a research
> scientist.
> My grades were generally above average, and, in case for some reason
> it isn't evident, my verbal and math GRE scores were 790
> and 760 respectively.

I am quite impressed, especially by the "generally above average"
grades and the nearly perfect verbal and math GRE scores. While it
would be useful to see the analytical scores, and a proof of Mensa
membership, there seems to be no need to quibble about such minutiae.
As much as I am tempted to call the esteemed Professor's bluff, I
prefer to keep my own profile in this discussion depending solely on
what I say.

>Now, you may not like what I am saying, or what I think, but I
>probably have better credentials and more experience than anyone else

>who has been involved in this discussion. I have taught logic, set

>theory, philosophy of science and related subjects at Western, UCLA,
>Rice, U of British Columbia and U of Calgary, and most recently here.
>My current work is in the foundations of information theory and
>statistical mechanics, and in evolutionary theory and non- equilibrium
>approaches to biology.

So much the worse for your "better credentials and more experience",
which are utterly discredited by your thoughtless speech, if you seem
compelled to adduce them in support of indefensible nonsense. I am
still awaiting an answer on the semantics of identity and measurement
without a metric. All your huffing and puffing has the effect of
exposing you as a far bigger fool than you might appear solely on the
strength of your absurd pronouncements.

>My original remarks were in answer to a question of whether or not
>science _needed_ infinity. I offerred reasons to think it did not.
>(Though I don't accept those reasons myself, I am not satisfied that
>adequate answers have been found to refute them.) The consequent furor
>has been quite bizarre, given the context of the original question,
>and seems to me to reflect a lot of the thoughtless repartee that
>tends to substitute for reasoned and sympathetic consideration on many
>of these fora.

By the time I responded to your encomium of Field, the original
context had been elided, and the original message unavailable on my
machine. As is my custom, I replied to your message in its entirety,
as it was presented to me. Consequently, your invocation of context
is either a deliberate lie, or a careless oversight.

>It is unfortunate, but not surprising, that most of my colleagues

>consider the Internet newsgroups a waste of time. If a book is mentioned

>in a discussion, for example, it is a bit hasty to shoot from the
>hip that its conclusions must be wrong without at least skimming it.
>If an issue is raised in a given context, it is dishonest to pull
>it out of context and argue with it without consideration of why

>it was raised in the first place. It is especially dishonest to
>attribute to someone things they never said or implied. It is
>laughable when these attributions are supported by quotes that
>clearly demonstrate that the attribution is false, but the humour
>pales rather quickly, like most superficial things.

Now it appears that the lie is deliberate. You had the unmitigated
gall to question my attribution to you of claims about identity, even
after I demonstrated the passage in which you made the claims in
question. Now, instead of doing the honorable thing in admitting your
error, you are cravenly repeating your unsubstantiated accusation,
with the evidence of your blatant prevarication thoughtfully elided.
Consequently, I see no reasonable alternative to regarding you as a
liar and a coward.

>Yes, it really is most unfortunate, because computer networks
>have the potential to enhance communication and understanding.
>Instead, rather like television, it seems they seek the lowest
>common denominator.

Unfortunately for you, this will not be the case in the present matter.

### Mikhail Zeleny

Feb 26, 1993, 5:31:37 PM2/26/93
to
In article <1993Feb25.1...@daffy.cs.wisc.edu>
to...@skool.ssec.wisc.edu (Michael Tobis) writes:

>In article <1993Feb24.0...@husc3.harvard.edu>,
>zel...@husc10.harvard.edu (Mikhail Zeleny) writes:

>>In article <1993Feb23.1...@daffy.cs.wisc.edu>
>>to...@skool.ssec.wisc.edu (Michael Tobis) writes:

MT:

>>>Mr. Zeleny, I suggest you refrain from using sophisticated arguments your
>>>opponent cannot understand in this debate, as he will certainly reply
>>>with meaningless blather you cannot understand, all the while contending
>>>that his arguments are meaningful and yours are not.

MZ:

>>I believe that your suggestion is meant well, and on a certain level,
>>it makes sense. However, I cannot accept it as stated. Important
>>issues call for sophisticated arguments; anything short of that is a
>>mere palliative, and would never suffice to resolve the problem.

MT:

>I disagree. The importance of an issue is not obviously closely related to
>the sophistication of the argument required to settle it. Why should it be?

I think there is a good case to be made for the rule that issues not
calling for sophisticated argument tend to lose their importance in
virtue of receiving conclusive resolution.

MZ:

>>And
>>even if my interlocutor is unlikely to understand the logical
>>consequences of his own position, this exchange, as you yourself
>>convincingly demonstrate, is by no means limited to interlocution.

MT:

>Well, the subset of your audience that also fails to follow these particular
>consequences is certainly nonempty.

>>>>>zel...@husc10.harvard.edu (Mikhail Zeleny) writes:

MZ:
>>>>>>the semantics of
>>>>>>identity involves either an explicit second-order extensional
>>>>>>characterization, or a first-order introduction as a primitive binary
>>>>>>predicate with appropriate axioms, in which case the second-order
>>>>>>quantifiers will be called for in giving the semantics of the language
>>>>>>in which these axioms are stated. In either case, the ontological
>>>>>>commitment is far in excess of that required for defining either von
>>>>>>Neumann or Frege integers.

MZ:

>>This is an important and subtle point, for which I claim no original
>>credit. Similar ideas are to be found more or less explicitly in such

>>standard texts on the virtues of higher-order logic as Church, Kreisel

>>& Krivine, Barwise & Feferman, or Shapiro. In each instance, it is
>>all too easy to ignore its scope, or fail to understand its
>>implications.

MT:

>I haven't the least idea what you are saying. While this is not an unusual
>state of affairs for me reading sci.math, I find it more than usually
>frustrating because its subject is one that I suspect ought to be
>accessible. And I believe that in resorting to such impenetrable arguments
>to counter what appears to me to be nonsense, you are inadvertently
>supporting the opponent's case by implying that such difficult arguments are
>needed to refute him.

Alas, no simple argument known to me will do. Let me try again:
identity may be introduced as a defined notion in the language of
mathematics, in which case that very second-order language would
suffice for deriving elementary arithmetic, and, with the axioms of
choice and infinity, and appropriate impredicative devices required
for defining infima and suprema, -- even for deriving analysis;
otherwise it may be formulated in a first-order theory, in which case
the second-order formalism would be needed for formulating the
semantics of that theory, and the sam4e conclusion would follow on the
metalinguistic level. So either you get the means for defining
integers within your theory, or you get them within the theory fixing
the meaning of your theory.

MT:

>Is it possible to restate your point in a manner accessible to someone who
>does have a smattering of real analysis and a fair amount of applied maths?
>Otherwise I will indeed remain ignorant of the scope and implications of your
>argument.
>
>For now, I continue to suspect that the difficulty of much philosophy is
>more a result of obfuscation rather than of sophistication. Even the rare
>piece of philosophy that I read and appreciate (Dennett most recently) tends
>to irritate me with what I take to be unnecessary and precious verbosity.
>
>sincerely, but obviously out of his depth,
>mt

cordially,

### Tal Kubo

Feb 26, 1993, 6:00:58 PM2/26/93
to
In article <1993Feb26.1...@husc3.harvard.edu>
zel...@husc10.harvard.edu (Mikhail Zeleny) writes:

>I think there is a good case to be made for the rule that issues not
>calling for sophisticated argument tend to lose their importance in
>virtue of receiving conclusive resolution.

There is as good a case to be made for the opposite rule: that
simplification increases importance. Certainly in math there are
situations when hard-to-prove results gained interest and importance
after new ideas led to simple proofs (sometimes trivial in hindsight).
Consider e.g. various theorems in algebra before and after the idea
of a vector space.

Tal

### SCOTT I CHASE

Feb 26, 1993, 2:55:00 PM2/26/93
to
In article <1ml2b6...@ariel.ucs.unimelb.EDU.AU>, jcol...@ariel.ucs.unimelb.EDU.AU (John Donald Collier) writes...

> My grades were generally above average, and, in case for some reason
> it isn't evident, my verbal and math GRE scores were 790
> and 760 respectively.

I have been reading posts on USENET for over a year now, almost every day.
I have never seen anyone, before now, defend themselves or their credibility
by posting their SAT scores. Regardless of what the argument was about,
you have lost it. If you don't see why, then maybe it just goes to show
how little SAT scores have to do with anything.

-Scott
--------------------
Scott I. Chase "It is not a simple life to be a single cell,
SIC...@CSA2.LBL.GOV although I have no right to say so, having
been a single cell so long ago myself that I
have no memory at all of that stage of my
life." - Lewis Thomas

### John Donald Collier

Feb 28, 1993, 12:49:04 AM2/28/93
to
Identity in the HF presentation is included as a primitive with no
need for explication. I my on remarks I focussed on diversity, or
diferentiation, which I believe can be used to avoid any problems
presented by those who feel identity must be formalised to make sense.

### John Donald Collier

Feb 28, 1993, 1:04:32 AM2/28/93
to
zel...@husc10.harvard.edu (Mikhail Zeleny) writes:

>In article <1mi9ec...@ariel.ucs.unimelb.EDU.AU>
>jcol...@ariel.ucs.unimelb.EDU.AU (John Donald Collier) writes:

>Sure thing: the term is used, rather than mentioned. Note that if you
>try to cover your tracks by backpedaling to "terms of similarities and
>dissimilarities", you would _ipso facto_ commit yourself to the first
>horn of the dilemma, by opting for a second-order language, in which
>the integers are definable. If you have a first-order explanation of
>quantifiers over similarities and dissimilarities, please enlighten
>me. Note that in that unlikely case, your ontological commitment to
>second-order explanation will be merely pushed into the metalanguage,
>which would have to be strong enough to derive elementary arithmetic.

It is entirely unclear to me how an explanation of human representations
of physical facts have any bearing on the ontology required for physics.
If you can explain why they do have such a bearing, perhaps there is
more to discuss.

>JDC:
>>First learn that philosophical logic and mathematical logic are not on
>>a continuum, and that one can be very good in one without being very
>>good in the other. Then perhaps I would be willing to withdraw
>>my charge that you are a twerp.

>It seems to me that you are the one who must learn that dubious claims
>do not get promoted to facts on your say-so. In this instance, you
>come across as your nameless colleague, who pronounced before an IAS
>audience including G\"odel, that no advances occurred in logic since
>the time of Aristotle. But for the sake of charity, I will withdraw
>my objection, provided that you find me three articles in the Handbook
>of Philosophical Logic, that support your claim in being free of
>pernicious mathematical methods. Otherwise, feel free to revel in
>your Victorian philistine ignorance.

I don't know what you are referring to here, but since you are at
Harvard, I might suggest that you enquire of Quine or Putnam whether
C.B. Martin is an accomplished philosophical logician. Nobody, including
Charlie himself, would claim that he is accomplished at mathematical
logic. I don't claim that philosophical logic has not been influenced
(to the better) by mathematical logic. I merely claim that they are
not to be compared directly with respect to their value in
discussing ontological issues.

### John Donald Collier

Feb 28, 1993, 5:09:17 AM2/28/93
to
zel...@husc10.harvard.edu (Mikhail Zeleny) writes:
>Save your snide dismissals for your long-suffering students. Your
>point had to do with verifying the truth of "The pointer is closer to
>line A than line B," which clearly depends on metric. While the
>property of the pointer appearing between line A and line B is clearly
>invariant under a topological mapping, its being closer to line A than
>line B depends on the metric properties of your space.

Sorry, you're right. What I should have said (I was assuming a
graduated set of markings when I said what I did) was that
one can test, for any two predictions, when there is a test
at all, whether or not one or the other conforms more closely,
according to previously agreed operations, according to
operations defined by the test. I was merely speaking in more
conventional terms that presumed these operations. In the
case I mentioned, the presumed operations are whatever is
used to compare closeness on the scale. This will be a set
of procedures, not a metric, though a metric is perhaps implied.
One doesn't have to assume its existence in order for the
procedures to exist. A nominalist will not assume the extravegance
of a metric where it is not required. It is convenient, but not
required.

### John Donald Collier

Feb 28, 1993, 5:11:58 AM2/28/93
to
sic...@csa1.lbl.gov (SCOTT I CHASE) writes:

>In article <1ml2b6...@ariel.ucs.unimelb.EDU.AU>, jcol...@ariel.ucs.unimelb.EDU.AU (John Donald Collier) writes...

>you have lost it. If you don't see why, then maybe it just goes to show
>how little SAT scores have to do with anything.

^^^
GRE.

### John Donald Collier

Feb 28, 1993, 5:21:49 AM2/28/93
to
sas...@theseus.unx.sas.com (Gary Merrill) writes:

}You write ...

}And you respond ...

I said that MZ's case would be stated in terms of identity. I
suggested another way to deal with the issue. If you thought
I was using the notion of identity, I apologise, but I thought
I had made it pretty clear that it was my opponent who would
consider the issue important, and that I was heading this off.
I meant that I had never mentioned identity in defense of the
Field position. Elsewhere, I remark that Field does use the
term as a primitive, but does not subject himself to the
requirements that MZ puts.

### Gene W. Smith

Feb 28, 1993, 9:27:10 AM2/28/93
to
In article <MARTINC.93...@hatteras.cs.unc.edu>
mar...@hatteras.cs.unc.edu (Charles R. Martin) writes:

>Oooh Oooh! A Zeleny/Smith flame war! Life is good.

I'm not sure. Here is a logical puzzle: If I were to ask, isn't it
neccesary to reply directly to Miss Zeleny in order to engage her in a
flame war, would that be a logical puzzle?

### Mikhail Zeleny

Feb 28, 1993, 12:59:30 PM2/28/93
to
In article <1mq30d...@ariel.ucs.unimelb.EDU.AU>

jcol...@ariel.ucs.unimelb.EDU.AU (John Donald Collier) writes:

>zel...@husc10.harvard.edu (Mikhail Zeleny) writes:

Given a graduated set of markings, you can indeed define a set of
procedures that would compare the outcome of measurements without
reference to numbers; for all I care you may call the designated
initial mark "Fred", and indicate succession of marks by the predicate
"son of ...". But to infer from this, rather far-fetched possibility,
the conclusion that mathematical objects are not presupposed, that is,
referred to in the underlying logical structure of your observation
language, is not unlike claiming that chemical compounds do not play a
part in trademarked over-the-counter cold remedies.

I take it that the right question to ask in this instance is
Russellian: what are the logical primitives of the language of
science? On such a view, I cannot conceive of a language of science,
wherein numbers would not be available, either through definition, or
by syntactical or semantical presupposition. In other words, either
you are using a language sufficiently powerful to derive arithmetic,
and, depending on your assumptions or inferences about infinity,
classical analysis, *or* arithmetic is presupposed in the explicit or
contextual definitions of your language, or in the interpretation of
your language. In short, mathematics is inevitable.

>--
>John Collier Email: jcol...@ariel.ucs.unimelb.edu.au
>HPS -- U. of Melbourne Fax: +61 3 344 7959
>Parkville, Victoria, AUSTRALIA 3052

### Mikhail Zeleny

Feb 28, 1993, 1:20:09 PM2/28/93
to
In article <1mpklg...@ariel.ucs.unimelb.EDU.AU>

jcol...@ariel.ucs.unimelb.EDU.AU (John Donald Collier) writes:

>zel...@husc10.harvard.edu (Mikhail Zeleny) writes:

>>In article <1mi9ec...@ariel.ucs.unimelb.EDU.AU>
>>jcol...@ariel.ucs.unimelb.EDU.AU (John Donald Collier) writes:

>>Sure thing: the term is used, rather than mentioned. Note that if you
>>try to cover your tracks by backpedaling to "terms of similarities and
>>dissimilarities", you would _ipso facto_ commit yourself to the first
>>horn of the dilemma, by opting for a second-order language, in which
>>the integers are definable. If you have a first-order explanation of
>>quantifiers over similarities and dissimilarities, please enlighten
>>me. Note that in that unlikely case, your ontological commitment to
>>second-order explanation will be merely pushed into the metalanguage,
>>which would have to be strong enough to derive elementary arithmetic.

>It is entirely unclear to me how an explanation of human representations
>of physical facts have any bearing on the ontology required for physics.
>If you can explain why they do have such a bearing, perhaps there is
>more to discuss.

If logic and semantics are concerned with explanation of human
representations of physical facts (a view that I would oppose), then
it is hard to see how physics, or indeed any other nomothetic
discipline, could be characterized as anything other than a human
representation of observable nomological regularities in nature.

Field's conception of "the ontology required for physics" is in no way
exempt from logical analysis. I have argued that such analysis
reveals an unavoidable dependence on an ontology of mathematical
objects. It is moreover my impression that in this instance, logical
dependence is a transitive relation. If Field's nominalist ontology
depends on a mathematical foundation (i.e. interpretation), then his
project fails.

JDC:
>>>First learn that philosophical logic and mathematical logic are not on
>>>a continuum, and that one can be very good in one without being very
>>>good in the other. Then perhaps I would be willing to withdraw
>>>my charge that you are a twerp.

MZ:

>>It seems to me that you are the one who must learn that dubious claims
>>do not get promoted to facts on your say-so. In this instance, you
>>come across as your nameless colleague, who pronounced before an IAS
>>audience including G\"odel, that no advances occurred in logic since
>>the time of Aristotle. But for the sake of charity, I will withdraw
>>my objection, provided that you find me three articles in the Handbook
>>of Philosophical Logic, that support your claim in being free of
>>pernicious mathematical methods. Otherwise, feel free to revel in
>>your Victorian philistine ignorance.

JDC:

>I don't know what you are referring to here, but since you are at
>Harvard, I might suggest that you enquire of Quine or Putnam whether
>C.B. Martin is an accomplished philosophical logician. Nobody, including
>Charlie himself, would claim that he is accomplished at mathematical
>logic. I don't claim that philosophical logic has not been influenced
>(to the better) by mathematical logic. I merely claim that they are
>not to be compared directly with respect to their value in
>discussing ontological issues.

OK, let us set aside the _ad hominem_ pleading, and consider the
intrinsic aspects of the issue. We are arguing whether it is
reasonable to claim that the language of science does not depend on
mathematics. In what sense, and to what extent do you admit the
language of science within the purview of scientific (that is,
logical) inquiry?

>--
>John Collier Email: jcol...@ariel.ucs.unimelb.edu.au
>HPS -- U. of Melbourne Fax: +61 3 344 7959
>Parkville, Victoria, AUSTRALIA 3052

cordially,

### Mikhail Zeleny

Feb 28, 1993, 1:35:22 PM2/28/93
to
In article <1mpjog...@ariel.ucs.unimelb.EDU.AU>

jcol...@ariel.ucs.unimelb.EDU.AU (John Donald Collier) writes:

>Identity in the HF presentation is included as a primitive with no
>need for explication. I my on remarks I focussed on diversity, or
>diferentiation, which I believe can be used to avoid any problems
>presented by those who feel identity must be formalised to make sense.

I think you are mistaken: the language of diversity or differentiation
is _prima facie_ second-order, and so is capable of defining extensional
identity as indiscernibility, or lack of diversity or differentiation,
along with the primitives of arithmetic. The fact that Field construes
identity as a primitive, does not exempt him from the need to give an
account of its interpretation. Surely it is arbitrary and capricious to
allow discussion of metaphysics, but prohibit all other metatheoretical
inquiries!

>--
>John Collier Email: jcol...@ariel.ucs.unimelb.edu.au
>HPS -- U. of Melbourne Fax: +61 3 344 7959
>Parkville, Victoria, AUSTRALIA 3052

cordially,

### hllh...@uctvax.uct.ac.za

Feb 28, 1993, 2:02:17 PM2/28/93
to

John Donald Collier writes:

> Now, you may not like what I am saying, or what I think, but I
> probably have better credentials and more experience than anyone else
> who has been involved in this discussion.

I'm afraid this kind of appeal to authority is quite useless as a debating
tool, and simply leaves one with a bad aftertaste. As for showering all of us
record supposed to enhance your position in this debate?

Hardy Hulley
Lab. for Formal Aspects of Computer Science
UCT

### Erin Zhu

Feb 28, 1993, 2:05:12 PM2/28/93
to
In article <1993Feb28.1...@sun0.urz.uni-heidelberg.de>

gsm...@clio.iwr.uni-heidelberg.de (Gene W. Smith) writes:

>In article <MARTINC.93...@hatteras.cs.unc.edu>
>mar...@hatteras.cs.unc.edu (Charles R. Martin) writes:

>>Oooh Oooh! A Zeleny/Smith flame war! Life is good.

At least it won't be bogged down in technicalities.

>I'm not sure. Here is a logical puzzle: If I were to ask, isn't it
>neccesary to reply directly to Miss Zeleny in order to engage her in a
>flame war, would that be a logical puzzle?

"Miss Zeleny"? "her"???

Lemme make something clear: Misha Zeleny, of decidedly unladylike
net.fame, have not and does not (and I suspect, will not) aspire to
the pinnacle of human achievement known as womanhood.

And I, for one, am content to live without dealing with MZ + PMS.

--Erin

### Michael Tobis

Feb 28, 1993, 4:02:53 PM2/28/93
to
Well, I understood the following, though I did not understand Mr. Zeleny's
first attempt at clarification. Why didn't he say this in the first place?

I continue to believe that making reason inaccessible is a significant
help to those who are propagating long-winded nonsense, to the extent
that it is problematic for a non-specialist to distinguish between them.

As for the importance of a question being proportional to the subtlety
of the argument required to settle it, it seems to me that Zeleny is
claiming that only unsettled questions are important. This is of course
true in a context where reason is actually respected. In the current
situation, however, where debating skills are more valued than solid results,
and where a reputation can be formed by the very absurdity of the position
one straight-facedly espouses, all questions become in some sense open
ones, e.g., whether mathematics is necessary for science.

I cannot accept that arguments accessible only to a specialist are required
to settle this issue. In the sense in which Zeleny argues for the correlation
between importance and subtlety, this is not an important question, as it
must be considered well and firmly settled. Indeed it was long settled
before anyone had the inspiration to ask the question. I wonder when they
will start arguing that pigs have wings.

In article <1993Feb28.1...@husc3.harvard.edu> Mikhail Zeleny writes:

|> Given a graduated set of markings, you can indeed define a set of
|> procedures that would compare the outcome of measurements without
|> reference to numbers; for all I care you may call the designated
|> initial mark "Fred", and indicate succession of marks by the predicate
|> "son of ...". But to infer from this, rather far-fetched possibility,
|> the conclusion that mathematical objects are not presupposed, that is,
|> referred to in the underlying logical structure of your observation
|> language, is not unlike claiming that chemical compounds do not play a
|> part in trademarked over-the-counter cold remedies.

|> I take it that the right question to ask in this instance is
|> Russellian: what are the logical primitives of the language of
|> science? On such a view, I cannot conceive of a language of science,
|> wherein numbers would not be available, either through definition, or
|> by syntactical or semantical presupposition. In other words, either
|> you are using a language sufficiently powerful to derive arithmetic,
|> and, depending on your assumptions or inferences about infinity,
|> classical analysis, *or* arithmetic is presupposed in the explicit or
|> contextual definitions of your language, or in the interpretation of
|> your language. In short, mathematics is inevitable.

mt

### Paul Lyon

Mar 1, 1993, 11:18:47 AM3/1/93
to
In article <1993Mar1.0...@husc3.harvard.edu> zel...@husc10.harvard.edu (Mikhail Zeleny) writes:

>Formalism is not an option for Field, if only because he wants his
>terms to refer to spatiotemporal aggregates. So semantics is
>unavoidable, and with it, mathematics. The salient point is that
>metaphysical objects arise as soon as you stipulate that your
>constants refer to, and your variables range over, concrete physical
>objects, merely by posing the question: "In virtue of what does the
>reference relation hold?" This, I believe, is one of the few points
>on which Church's intensional logic agrees with Putnam's critique of
>extensional Tarskian semantic theories: there is nothing in the
>latter, that would explain, or even fix, the referential connection
>between a name and the object it denotes.
>

Query: suppose it were possible to reconstruct Field's theory using
the calculus of individuals. Would the conclusion of the inevitability
of mathematics still hold? I have no idea whether such a reconstruction
is possible, but the point about terms referring to spatiotemporal
aggregates does suggest such a move, to me at least.

To be sure, the ``nominalism'' here would be that of Nelson Goodman:
abstract objects are okay, but sets are not. This may not be appropriate
in the present context...

Just a thought :-)

Paul Lyon

### Mikhail Zeleny

Mar 1, 1993, 1:21:33 AM3/1/93
to
In article <33...@optima.cs.arizona.edu>
gud...@cs.arizona.edu (David Gudeman) writes:

>In an effort to contain the article explosion, I'm answering several
>of Mikhail's articles at once. I don't think I have taken any of his
>remarks out of context to the extent that their meaning has changed.

>In article <1993Feb28.1...@husc3.harvard.edu> Mikhail Zeleny writes:

>>I take it that the right question to ask in this instance is
>>Russellian: what are the logical primitives of the language of
>>science? On such a view, I cannot conceive of a language of science,
>>wherein numbers would not be available, either through definition, or
>>by syntactical or semantical presupposition. In other words, either
>>you are using a language sufficiently powerful to derive arithmetic,
>>and, depending on your assumptions or inferences about infinity,
>>classical analysis, *or* arithmetic is presupposed in the explicit or
>>contextual definitions of your language, or in the interpretation of
>>your language. In short, mathematics is inevitable.

>You can leave out the *or* part. Any language adequate for science
>must be poweful enough to represent arithmetic. However, I don't see
>how this observation has any bearing on the ontological status of
>numbers. Any language powerful enough to represent arithmetic is
>powerful enough to represent FORTRAN also, but I don't think anyone is
>prepared to impute a special metaphysical significance to FORTRAN.

You are missing the implications of the Russellian project: once the
underlying logical structure of FORTRAN is uncovered, it will be found
to contain arithmetic. Or so we, mathematical chauvinists, would like
to think. On the other hand, it would take a very special kind of
loony to argue that FORTRAN is the logical foundation that underlies
the language of mathematics.

>In article <1993Feb28.1...@husc3.harvard.edu> Mikhail Zeleny writes:

>>If logic and semantics are concerned with explanation of human
>>representations of physical facts (a view that I would oppose), then
>>it is hard to see how physics, or indeed any other nomothetic
>>discipline, could be characterized as anything other than a human
>>representation of observable nomological regularities in nature.

>Do you expect nominalists to think this is a problem?

Yes, if they are honest. Representations are abstract objects; I hope
I have described the ensuing slippery slope in adequate detail.

>In article <1993Feb28.1...@husc3.harvard.edu> Mikhail Zeleny writes:

>>... the language of diversity or differentiation

>>is _prima facie_ second-order, and so is capable of defining extensional
>>identity as indiscernibility, or lack of diversity or differentiation,
>>along with the primitives of arithmetic. The fact that Field construes
>>identity as a primitive, does not exempt him from the need to give an
>>account of its interpretation.

>I don't see how this argues against Field's position either. All of
>the objects in his theories are physical objects (or so he claims), so
>identity is a primitive notion. Yes, if you want to investigate the
>_implications_ of identity, then you could use second order notions,
>but then it is you who is introducing second-order notions, not Field.
>Clearly, for each statement you can make about identity by refering to
>relations and sets, I can say something similar by refering to
>syntactical predicates. And if I am not willing to acknowledge the
>existence of metaphysical objects, then I don't think you can prove to
>me that your statement is different from mine.

Formalism is not an option for Field, if only because he wants his
terms to refer to spatiotemporal aggregates. So semantics is
unavoidable, and with it, mathematics. The salient point is that
metaphysical objects arise as soon as you stipulate that your
constants refer to, and your variables range over, concrete physical
objects, merely by posing the question: "In virtue of what does the
reference relation hold?" This, I believe, is one of the few points
on which Church's intensional logic agrees with Putnam's critique of
extensional Tarskian semantic theories: there is nothing in the
latter, that would explain, or even fix, the referential connection
between a name and the object it denotes.

>(So much for my promise not to defend the bad guys any more. But
>really Mikhail, your arguments against Field have not been nearly up
>to your usual standards...)

What usual standards?

>--
> David Gudeman
>gud...@cs.arizona.edu

### Michael Stemper

Mar 2, 1993, 1:51:50 PM3/2/93
to
The argument that "science doesn't need infinity" has been expressed (as
best I can interpret it) in this thread as:

"Terms at infinity are always dropped, because they have no effect.
Therefore, 'science' doesn't need infinity."

Now, I'm just a dumb-shit E.E., but when I went to school, that was called
circular reasoning.

Primus: You need the concept of infinity before you can say: "That thing
over there is at infinity". If you don't have infinity, how can you
identify something as being there?

Secundus: You need to run L'Hopital's rule right out to infinity before
you can say: "Because that thing over there is at infinity, I can ignore
it." If you don't know how to work with infinity, how can you justify
ignoring things with infinite separation, but not those with infinite
charge or mass?

Infinity - don't leave home without it!
--
#include <Standard_Disclaimer.h>
Michael F. Stemper
Power Systems Consultant
mste...@ems.cdc.com

### Mikhail Zeleny

Mar 2, 1993, 9:55:50 AM3/2/93
to
In article <1mun77...@emx.cc.utexas.edu>
pl...@emx.cc.utexas.edu (Paul Lyon) writes:

No reconstruction is needed, as Field's theory is already formulated
in mereological terms. To be sure, the theory of upper semi-lattices,
which is the mathematical backbone of mereology, is insufficient for
foundational purposes because of the homogeneity of its objects, as
opposed to the hierarchies arising in set or type theories. So on
this account, which in effect is endorsed by Field, you can indeed
avoid mathematics in the object language; however the nominalistic
gain achieved by such means is negated by the fact that, given the
need to explain the reference of your theoretical terms, you will be
confronted with the same need for *structured* abstract objects in
your semantic theory, inasmuch as lattice theory is insufficient for
that purpose.

>Just a thought :-)
>
>Paul Lyon

cordially,

### John Donald Collier

Mar 3, 1993, 5:41:34 AM3/3/93
to
zel...@husc10.harvard.edu (Mikhail Zeleny) writes:

>by syntactical or semantical presupposition. In other words, either
>you are using a language sufficiently powerful to derive arithmetic,
>and, depending on your assumptions or inferences about infinity,
>classical analysis, *or* arithmetic is presupposed in the explicit or
>contextual definitions of your language, or in the interpretation of
>your language. In short, mathematics is inevitable.

But, the nominalist would hold that the language does not imply
the independent existence of the entities postulated, beyond
concrete entities, and that the interpretation of, say the use
of arithmetic in physics is in terms of either some concrete
physical properties (Field), or in terms of operations (various
operationalists and instrumentalists, like, e.g., Mach). The
language istslef is just a way of conveniently tying together
individual instances of interest.

Now I think there are serious problems with this view (mostly
having to do with inner connections), but I don't think you
have raised any problems that people like Field have not
considered carefully.

### John Donald Collier

Mar 3, 1993, 6:12:22 AM3/3/93
to
to...@skool.ssec.wisc.edu (Michael Tobis) writes:

>Well, I understood the following, though I did not understand Mr. Zeleny's
>first attempt at clarification. Why didn't he say this in the first place?

>I continue to believe that making reason inaccessible is a significant
>help to those who are propagating long-winded nonsense, to the extent
>that it is problematic for a non-specialist to distinguish between them.

Well, I think MZ and I are engaging each other on the philosophical
issues now, but I really do doubt that these issues can be made
very comprehensible to someone who hasn't studied them for some time.
I give a semester course on realism/nominalism in science. Some
of the students catch on in five weeks or so, others take the whole
semester or longer. The ones who catch on generally recognise that
I have been making the same point in a variety of ways, and start
to anticipate me, but I doubt any of them could pick up the gist
in less than five weeks. Almost everyone tends to one side or
the other fairly strongly (someone I was on a PhD examining board with
once mentioned -- jokingly -- that she thought that realism or
anti-relaism was congenital). It takes some time to learn to jump
back and forth between views in order to compare them.