FOM: defining ``mathematics''

Matt Insall montez at
Sat Jan 22 01:06:00 EST 2000

This is the (belated) fourth part of my reply to professor Sazonov.

Matt Insall

> "Matt Insall" <montez at>
>     Date: Thu, 30 Dec 1999 18:44:01 -0800
> wrote in reply to my posting on the definition of mathematics
> given by Buss:
> Sam Buss wrote:
> > "Mathematics is the study of objects and constructions, or of aspects
> of
> > objects and constructions, which are capable of being fully and
> completely
> > defined.  A defining characteristic of mathematics is that once
> mathematical
> > objects are sufficiently well-specified then mathematical reasoning
> can be
> > carried out with a robust and objective standard of rigor."
> I am afraid that these "fully and completely defined objects" would lead
> us imperceptible again and again back to Platonism. Also what does it
> mean "capable of being fully and completely defined"? I think that this
> or other way it is inevitable to explain all of these in terms of
> appropriate
> formal systems.
> ++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++
> ++++++++++
> ++++++++++
> I'm not sure, professor Sazonov, why you should be ``afraid'' of being
> lead
> back to Platonism.  Are you convinced that our mathematics has no
> reality
> beyond the marks on the page, as I understand the pure formalist
> position to
> be?
> As I wrote in the cited posting, I understand mathematics as
> a kind of formal engineering and of course as a science investigating
> formal systems. Formal systems are very useful things. Their goal
> *in general* is not finding a "mathematical" truth (what does it
> mean?),

Just ``the truth''.  For example, one problem of mathematics is that of
deciding whether PA is consistent or inconsistent.  The current type of
activity seems to have no hope of showing that PA is consistent, and seems
unlikely to show that it is inconsistent.  The theorems that have been
proved in this regard are ``true'' about PA.  They are mathematical, having
been proved by mathematical logicians.

>but rather to serve as levers, accelerators for human thought.
> Formalisms organize and govern our intuition and thought. E.g., how
> would we efficiently multiply natural numbers without formal rules of
> multiplication of decimal numbers learned at school? Of course
> formalisms we are considering in mathematics usually have
> some relation to the real world (or to some other formal systems
> having some relation to the real world, etc). When we write a formal
> symbol "5" we think, say, on a set of five pebbles. When we multiply
> numbers according to the mentioned formal rules we expect that the
> input and output of this process will correspond to such and such real
> experiment with pebbles. Otherwise we hardly would be interested in
> these formal rules of multiplication. Mathematical formalism should be
> helpful for human thought.

I completely agree with this discussion of formalisms.  However, I still
object to the reduction of mathematics to (the study of) formalisms.

> "Reality beyond the marks on the page" lies in pragmatics, not in the
> subject matter of mathematics.

Why not?  The early geometers presumably carried out actual geometric
constructions physically as well as mentally.  I expect that, to them, the
reality of the geometric objects they studied was quite important to their
discoveries and understandings in geometry.  Also, I feel certain that a
formal system can be, and probably has been, developed to deal with (at
least certain aspects of) pragmatics, and so the study of pragmatics becomes
a part of mathematics, according to your definition in terms of formal

> The question was ``What is
> mathematics?'', not ``What is your philosophical belief about what parts
> of
> mathematics possess some metaphysical property such as `existence beyond
> marks on the page'.''
> I think, without a correct philosophy of mathematics it is impossible
> to define what it is, and vice versa.

You may be right, but what do you mean by a ``correct philosophy'' of
mathematics?  Then, even once you have the notion of ``correct philosophy''
down, what you have said here (vice versa) makes it impossible, in practical
terms, to obtain either a definition of mathematics or a ``correct
philosophy'' of mathematics, since your claim is that each requires the
other.  (BTW, if your claim is correct, then this claim itself is part of
the ``correct philosophy'' of mathematics, is it not?)

> MATT INSALL [Sun, 2 Jan 2000 07:10:11 -0800 ] citing my posting:
> <snip>
>      Mathematics is a kind of *formal engineering*, that is engineering
>      of (or by means of, or in terms of) formal systems serving as
>      "mechanical devices" accelerating and making powerful the human
>      thought and intuition (about anything - abstract or real objects or
>      whatever we could imagine and discuss).
> ++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++
> ++++++++++
> ++++++++++++
> I like this ``definition'' also, as far as it goes.  The problem I see
> in
> all this is that the ``definitions'' all seem to appeal to terms not
> previously defined.  Thus I would not really call this a definition,
> because
> too much of the description is undefined.  In particular, how does one
> define ``engineering'', or ``human thought and intuition''?  I guess
> this
> might make a good Webster's type of definition, but it is hardly
> mathematical itself.
> ++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++
> ++++++++++
> ++++++++++++
> Why a definition of mathematics should be "mathematical itself"?

Let me revise my comment to the following:  I guess this might make a good
Webster's type of definition, but it is hardly rigourous.

My reason for desiring a rigourous definition is that others (non-rigourous
ones) are not definitions at all.

> As to definition of ``engineering'' and ``human thought and
> intuition'' I do not think that I could or need do anything
> better than Webster. Any of us knows sufficiently well what
> it is. (Say, it is enough to know that engineering is creating
> any useful devices.) I do not understand why it bothers you.

Here is a ``Webster-type'' definition of engineering:  1.  The application
of scientific principles to practical ends as the design, construction, and
operation of efficient and economical structures, equipment, and systems.
2. The profession of or work performed by an engineer.

Now, to study engineering formally with this definition, one must find the
definitions of the other words used in the definition, etc.  Of course, in a
Webster's dictionary, there will eventually be a definition cycle, in which
one must know the meaning of some term t in order to understand the
definition given for t.  These terms are essentially ``undefined'', even
though the dictionary gives a definition for them.  Of course, one may take
all such terms as undefined in the formalist sense, so that they may be
applied to anything one wishes to apply them to.  My experience with natural
language dictionaries is that almost every term I find in such a dictionary
is  a member of such a definition cycle, and so cannot be made rigourous.
As such, it might be understandable in natural language, but cannot be
effectively studied formally.  (It can certainly be studied informally,
however, making the study of natural language a viable subject.)  If it is
your purpose to only give an ``informal definition'' of mathematics, you may
or may not have succeeded, but what have you gained over Webster, and

Mathematics: ``The study of number, form, arrangement, and associated
relationships, using rigourously defined literal, numerical, and operational

> Mathematics deals with formal systems ("devices" of a special kind).
> This is the main point.

By ``devices'', do you mean certain idealized, or actual physical objects?
If you accept the ``existence'' of ideal ``devices'', then you are a
Platonist.  If you do not, then you must produce, physically, every such
device (formal system) that you claim exists.  Thus you are a
constructivist, finitist, etc.  How do you even use FOL at all, in this
case?  Previously, you appeared to question the existence of the number
10^10^10, presumably because it is so enormous as to be physically
unattainable.  Now you wish to allow the formal system of FOL to exist,
although it has formulas of length 10^10^10^10 in it?  This makes no sense.
If the number 10^10^10 does not exist, then neither does a formula of that
length, for how can an existent formula have a nonexistent length?  This
destroys the closure property for your formulas under the conjunction,
disjunction, negation, etc, operations.  Suppose you attempt to get around
this problem by developing a new and improved FOL in which abbreviations are
allowed.  When a formula gets to be a certain size, so that its negation,
for instance, would have a nonexistent length, the formula is
``abbreviated'' by a ``new'' symbol.  However, this cannot be iterated for
as many as 10^10^10 times, so one must stop before this.  Thus all the
formulas, including those which are abbreviations, can be constructed in
less than 10^10^10 time periods of at most 1 in 10^10^10 parts of a second,
but many cannot be used, because when that much time has passed, there is no
more time left, since such times are described using numbers larger than
10^10^10.  Thus, in one second (10^10^10 iterations of these very short time
periods), all of useful FOL has been constructed, but now cannot be used,
because there are no other seconds left in which to apply them.  This is
paradoxical, to say the least.  You may object that ``time'' should be
quantized differently.  Please tell me how.  What is the shortest unit of
time?  Is it not 1 part in n seconds, where n is the ``largest integer'' in
your formal system which identifies 10^10^10 with some infinite number?  If
not, why?  This consequence of the formalist position brings us to the
question of ``truth in the real world'', even when we wish to know what our
formal system actually is, which we must know, or else, the formal system
does not exist.

>Are these formal systems meaningless or
> useless? In principle they can, but mathematicians, like engineers,
> prefer to do something reasonable, rational. This is the only
> (pragmatic) restriction on the class of formal systems considered.
> Platonists pretend to know on existence of an ABSOLUTE TRUTH (but I
> do not believe them!) and restrict the formalisms of mathematics to
> those which are true (what does it mean?).

Knowing the existence of absolute truth is not the same as knowing what that
absolute truth is.  Thus claiming to know there is absolute truth, does not
constitute a claim that one knows what is absolutely true.  However, the
claim that there is no absolute truth is formalizable as a refutable
formula, and as such, is untenable.  For an informal argument, suppose
person A claims that there is no absolute truth.  Is the claim of person A
true or false in the real world?  If it is true, then it is absolutely true,
being true in the real world, and this is contrary to the claim of person A,
so this claim is false in the real world.  One may claim that the denial of
absolute truth is here restricted to mathematics, and hence not applicable
to ``truth in the real world''.  However, the notion of ``truth in the real
world'', while perhaps not definable, is formalizable, via Tarski's
definition of satisfaction, and, as such, is a part of mathematics,
according to your definition.

>Formalist (or rationalist,
> as Prof. Mycielski probably would say) position consists in considering
> ANY REASONABLE formal system. There is no pretension here on knowing
> what is reasonable or not. Only our experience could help to judge
> this in each concrete situation.

Ah but the claim that the Platonist position is unreasonable amounts to a
pretention to know what is reasonable or not, or at least to know (some of)
what is not reasonable.  It seems to me we do not (completely) know either
way at this point in history.  The only currently clear criterion for
``reasonableness'' is the criterion of consistency.  Others, such as
``effectiveness'', ``recursiveness'', ``feasibility'', etc., place
mechanical bounds on the apparently nonmechanical human mind, and so are
themselves unreasonable limits on human endeavours such as mathematics.

> MATT INSALL continues citing my posting:
> <snip>
> Finally, I would like to stress that mathematics actually deals
> nothing with truth. (Truth about what? Again Platonism?) Of course
> we use the words "true", "false" in mathematics very often.
> But this is only related with some specific technical features of
> FOL. This technical using of "truth" may be *somewhat* related
> with the truth in real world. Say, we can imitate or approximate
> the real truth. This relation is extremely important for possible
> applications. But we cannot say that we discover a proper
> "mathematical truth", unlike provability.

Would you say then that it is possible that Cohen's theorem, namely that CH
is independent from ZFC, is not ``absolutely true''?  If that is the case,
what kind of truth value does it have?  Do you have in mind a model of
reality in which this theorem is formalizable, but fails to be true?  I know
of no way to come up with such a model.  I believe no such model exists,
either in physical reality or abstractly.  Now, this particular theorem is a
theorem of mathematics, being a theorem of mathematical logic, and is
absolutely true.

>This formalist point of
> view is not related with rejection of intuition behind formal
> systems. But the intuition in general is extremely intimate thing
> and cannot pretend to be objective. Also intuition is *changing*
> simultaneously with its formalization. (Say, recall continuous
> and nowhere differentiable functions.) Instead of saying that
> a formal system is true it is much more faithful to say that it is
> useful or applicable, etc. Some other formalism may be more
> useful. There is nothing here on absolute truth.
> ++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++
> ++++++++++
> ++++++++++++
> Okay, so if we do not deal with truth, then what would you say is the
> ``truth in the real world'' of the following statement:  ``If f is a
> continuous function defined on the real numbers, then f has the
> intermediate
> value property.''  I submit that as mathematicians, we do, and should,
> care
> about the ``truth in the real world'' of such a statement.
> Yes, of course! But this is ``truth in the real world'' (which is
> related rather with pragmatic and possible applications of
> corresponding mathematical theory), not the ``absolute mathematical
> Platonist truth''.
> When I say that "mathematics actually deals nothing with truth"
> I mean that *ideally* its goal is not finding a truth, whether it
> is
> (i) mathematical truth (let me ask again, what does this truth
> mean?), except, of course, the facts of provability in a formal
> system or the like, or
> (ii) a truth on the real world (which I respect very much).
> The latter truth is the goal of *other* sciences like physics,
> biology, etc. mathematics can/should only supply them with some
> machinery (formalisms) which will help them. Occasionally,
> during working on a formal system, mathematicians can find some
> real truth in a "real" science for which this formalism was
> especially created or in which it is applied. But this is not the
> goal of mathematics *in general*. This is only a (very important)
> witness that the formalism considered is good.  Of course, the real
> situation with concrete mathematical practice is not so pure. (Say,
> recall Newton!) But it seems it is unreasonable to heap up everything,
> especially in a short definition. However, recall this definition:
>      Mathematics deals with formal systems making powerful the human
>      thought and intuition (about anything - abstract or real objects or
>      whatever we could imagine and discuss).
> "Thought and intuition about anything"! Which else truth and meaning
> do you need? However they are moved from the subject matter of
> mathematics, they are here, very close. Nothing is lost, except of
> the nonsense Platonism as a philosophy. Even the naive Platonism of
> working mathematicians can be used freely.

I guess I no longer understand your notion of Platonism.  ``Naive
Platonism'' refers to what, in particular, other than the human intuition?
``Platonism'', as I understand it, contends that there are facts that can be
discovered and reasoned about using mathematics and its formalisms, and that
some of these facts are purely mathematical in nature.  It does not pretend
to know which purported facts are actual facts, i.e., absolutely true (true
in the real world), except for a small fraction of them, such as the
tautologies, and results of FOL and other demonstrably sound systems of
reasoning.  Some Platonists may claim to know what is absolutely true beyond
this, but not all.  Remember that the Greek geometers, Platonists
extraordinaire, considered Euclidean geometry to embody absolute truth about
the real world, although they considered the real world to be merely an
approximation to the truths of geometry.  However, they realized that there
are undefined terms in any such theoretical development of abstract
geometry, and so the theory could have other applications than the one
originally intended.  This does not mean that the theory embodies no
absolute truth, for there are conclusions it draws which we observe as true
in the real world.  That is, it makes predictions, as any scientific theory,
which are then observed in the real world.  This does not mean that
Euclidean geometry is the best (or only) theory for describing the universe,
although at the time it was the best available.  However, the fact that
Bolyai and Lobachevsky, and Gauss and Riemann were able to obtain a
consistent geometry by denying the fifth postulate of Euclid is observable,
or, if you prefer, demonstrable, and so is true in the real world.  This
corresponds to a theorem of mathematics, namely that there are consistent
geometries which deny the fifth postulate, which is therefore absolutely

> The problem I
> see is that it is either true or false, but not both, but the formalist
> approach would have us believe that no one even knows what the statement
> means.
> Where you discovered this?

Where have I discovered what?  Your comments have frequently indicated that
mathematics not only does not know what is absolutely true, but that the
statement that there is absolute mathematical truth does not even have any

>Look more carefully on the definition
> above.
> As to "either true or false, but not both", I recall my
> favorite example which I already presented in FOM. Look on
> a nice picture in a computer display.
> Is it continuous OR discrete,
> or both continuous AND discrete?

The question is not well-posed.  If you tell me what topology you want me to
use, then I can, in theory, determine whether the picture represents a
continuous function, by checking finitely many sets of pixels, a yes-no
condition that tells me the answer to my question.  Perhaps you do not mean
``continuous'' in the same sense I mean?  Well, aside from saying you are
abusing the well-defined topological term ``continuous'', I would say what
you are asking may be ``Do the pixels occupied by the nice picture form a
connected set or a discrete set?''  Again, I say you have not presented me
with a well-posed problem.  The topology must be given for you to ask me
this.  In some topologies, the picture may be connected, and in others, it
may be discrete, and in still others, it may be neither, but in no topology
can it be both unless the computer screen contains only one pixel.  (In any
topology, a connected discrete set is a singleton.)  However, once the
question is well-posed, in a true-or-false format, rather than as a query,
it is either absolutely true or absolutely false, although we may be unable
to determine which.

> Why should we take the formal dogmas of FOL with its contradiction
> law (A & ~A => anything) as a rule governing the real world?

Because it is observed to be so.  Where, in the real world, is an actual
contradiction not abhorred?  When any physical theory has been appropriately
formalized, so that it is considered correct by scientists, contradictions
are not accepted, because they are observed to not exist in the real world.
Consider, for example, the fact that Physicists generally agree, currently
that a ``Theory of Everything'' has not been found.  Why are they not happy
with the current situation which describes certain phenomena using
relativistic mechanics, and explains other phenomena using quantum
mechanics?  As I understand it, the reason the ``relativity-quantum'' theory
pair is not considered sufficient is that there are overlaps in the domains
of the two theories at which the two theories make contradictory
predictions.  Since the Physicists agree that a contradictory prediction
cannot be true, they do not consider this theory-pair to be a sufficient
description of reality.

> Relations of mathematical formalisms with the truth in real world
> may be more complicated.
> Another (related) example: It is formally provable (and Platonists
> will say - mathematically true) the fact
>         limit of the sequence 10/(log log n) = 0
> with log base 2 logarithm. But it is definitely false in our world!
> Calculate this sequence by a real computer, step-by-step.
> The practical limit (if it makes a sense at all) should be > 1.

I presume that by ``practical limit'', you mean ``after the computer has
performed as many calculations as is reasonable''.  I do not know what
``reasonable'' would mean here.  Consider the recent advances of computation
devices:  Thousands and millions of iterations are possible now, where only
tens and hundreds were feasible only twenty years ago.  Would you then say
that ``the (practical) limit'' of the sequence you gave is dependent then on
the number of computations one can feasibly perform, and on how long one is
willing to wait for the computer to decide it cannot improve upon its level
of accuracy, and upon which computer one uses?  Your parenthetical comment,
then, is quite important here, for I contend that the notion of ``practical
limit'' that you seem to have in mind does indeed make no sense at all.
Moreover, because the technical definition of limit of such a sequence in
Mathematics involves the equivalent of the idealized situation of ``letting
an ideal computer (with no lower bound on its level of accuracy) never stop
calculating'', and observing how this ideal computer is guaranteed to behave
(in the particular case of calculating 10/(log log n) for ever larger values
of n), the only sensible way to interpret the statement that the given
sequence has a limit, which is some unique number x, is an interpretation
that yields (always) x=0.

> If this were correct about such statements as this, then do we, as
> human beings (not, per se, as mathematicians in particular) know what
> anything means?  In fact, would you say, professor Sazonov, that there
> is no
> such thing as ``truth in the real world''?
> I never said this.

I did not claim that you said this.  I only asked if it is a consequence of
your philosophy about (the lack of) truth in Mathematics.

> For if it is because we
> formalize Mathematics that we lose meaning, is it not the case that even
> the
> very statements we make about the ``real world'' are formalizations, of
> a
> sort,
> Not every statements can serve as *mathematical* formalization.
> We should have formal *rules* (not only symbols), like
> (x+y)z=xz+yz or (uv)'=u'v+uv', A=>B,A/B, etc. (of course, assuming
> that they have such and such intuitive meaning) which allow to deduce or
> calculate *mechanically* and therefore make stronger human thought.

I agree that such rules *help* make human thought stronger.  But how are
these rules *not* serving as mathematical formalization?  You even call them
``formal'' rules.

> and so can be interpreted any way one may choose.
> As to mathematical formalisms, it depends on our intentions.
> Arithmetical variables are usually interpreted, say, as finite
> sets of pebbles. Variables of the first order logic may have
> any imaginary interpretation.
> I am not against a meaning of symbols and intuition.
> I only cannot consider seriously "the" absolute Platonistic meaning
> because, by my opinion, it is nonsense.

What then do you consider (unseriously, as it were) to be ``the'' absolute
Platonistic meaning which you claim is nonsense?  If you claim only that
such an absolute meaning does not exist, then why?  Is it because the
undefined terms of a formal system can be assigned any meaning whatsoever,
and thereby yield a consistent system using that atomic assignment?  This is
not convincing to me, because this is true of any consistent informal system
as well, and yet there is ``truth in the real world'' which you claimed
previously to ``respect very much''.  My only (Platonic?) claim is that
there is an absolute (Mathematical) truth, and that our systems of classical
reasoning are capable of determining (some parts of) that absolute truth.

> After all, whether
> we are doing mathematics or not, we are only putting marks on the page.
> Thus, even the statement that  `` mathematics actually deals nothing
> with
> truth'' has no meaning outside the virtual marks on the virtual page on
> my
> computer monitor.  When you restrict mathematics to the tenets of pure
> formalism, everything must be so restricted.
> Formalism, as I understand it, (unlike Platonism) can only extend
> mathematics. Formalism does not reject meaning. It allows ANY kind
> of meaning. It only gives it somewhat different role. Mathematics
> considers meaningful formalisms. But the meaning of these formalisms
> is in a sense outside of mathematics, even if concrete mathematicians
> pay a lot of attention to it.

Would you say then that the meaning of the (formalizable) statement that
``CH is independent of ZF'' is outside Mathematics?  I would then disagree.
Its meaning is very much a part of Mathematics, for the meanings of similar
statements have been a part of Mathematics for a very long time.  For
example, the (equivalent of the) statement that ``Euclid's Parallel
Postulate is independent of the other axioms of euclidean geometry.'' has
been part of Mathematics since Riemann, Bolyai, Gauss and Lobachevski.  Its
meaning is clear to the educated reader, and represents absolute
mathematical truth.

>Nobody could know what kind of
> intuition can be formalized in principle. Any attempt of describing
> a Platonist world including ANY potential mathematical intuition
> seems to me extremely non-serious. *In general* we can only mention
> the fact that our formalisms are meaningful. And this seems to me
> quite enough in the definition of mathematics.

It may be that nobody could know *all* intuition which can be formalized in
principle.  However, certain very simple intuitive concepts, such as the
notion of a two-element group, have already been formalized satisfactorily.
It is also clear that certain quite large groups can, in principle, be
formalized similarly, and the mathematical intuition behind work in those
groups would then be (physically) realized in such a formalization.  This is
only one way our formalisms are meaningful, but another is that these
groups, which may in some cases be large enough so as to make physical
realization (in a computation system) impractical, are frequently useful in
the physical sciences, both theoretically and for more concrete
applications.  If the formalization of such a group had no meaning, that
situation would be completely incongrous with the fact that those groups are
concretely applicable.  The question of defining Mathematics does not enter
in when trying to determine the status of the Platonistic Universe, which
you deny exists.  For if such a Platonistic Universe exists, its existence
is independent of our terminology used to describe it, such as
``mathematical'' or ``non-mathematical''.  It just ``is'', and we could call
its elements anything we want to, and describe them using any language we
wish.  The *actual* relationships between the members of this universe
exist, and some can be known within the framework of the language we choose
to use, to a certain extent, but others may be unknown from that ``vantage
point''.  Moreover, the same relationships that are known from one ``vantage
point'', using a particular language, may be known or knowable via another
language, but they are still the *same* relationships, and there is a
(possibly not ``effective'') translation between the two languages that
explains how the two languages are actually ``talking about'' the same
*actual* objects, and the same *actual* relationships between the objects.

> MATT INSALL citing my posting:
> ++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++
> ++++++++++
> ++++++++++++
> By the way, as an example of useful and meaningful formal system
> I recall *contradictory* Cantorian set theory. (What if in ZFC or
> even in PA a contradiction also will be found? This seems
> would be a great discovery for the philosophy of mathematics!)
> ++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++
> ++++++++++
> ++++++++++++
> I think this would be a disaster.  It is bothersome enough that
> ``Cantorian'' set theory (I think you actually mean Fregean set theory.
> Cantor's approach was decidedly NOT formalistic.)
> Read "formal" as "sufficiently formal" or "sufficiently rigorous".
> It is completely applicable to Cantor. Even contemporary
> logicians not always pay enough attention to using
> abbreviation mechanisms when proving in FOL. They are
> also not absolutely rigorous in this sense. Usually this
> subtlety (like Choice Axiom for many of working mathematicians)
> makes no value in everyday mathematics. (But who knows?)

I agree that many people are not very concerned about having absolutely
rigourous abbreviation mechanisms in place for anything at all (including,
but not limited to, Mathematics, and Mathematical Logic).  However, I ask
you, how does this subtlety ``make no value in everyday mathematics''?  It
seems to me to make a significant difference, especially in our discussions,
where you are denying the existence of objects I am quite comfortable with,
merely because they are sufficiently complex as to require millions of years
for a digital computer to completely unravel their meaning.

> is considered to be
> contradictory.  Why should ``philosophers of mathematics'' be so
> biased?
> Really, why? Someone would like to have a self-contained
> Platonistic world. It does not exist anyway.

How do you know this?  Have you a proof that such an absolute Mathematical
reality does not exist?

>Other one
> (a formalist) would like to have a *sufficiently* reliable and
> reasonable formalism. It is happy that ZFC is so reliable (until now).
> But *let us only imagine* that suddenly a contradiction will be
> found in ZFC and even in PA or PRA or even in the exponential
> arithmetic. (As I know, Edward Nelson even tried to find a
> contradiction based on using exponentiation; I also believe
> that this operation is rather problematic from the point of
> view of f.o.m.) Formalist view on mathematics will still exist.
> (Cantorian or Fregean set theory can serve us even as
> contradictory one. It actually serves to the most of mathematicians
> who use set theory without knowing the precise formulation
> of ZFC.) What about Platonism?

Platonism will still survive such a blow.  But, as with the ``crisis'' of
Russel's Paradox in Fregean set theory, the view of what actually exists
will change.  This does not mean that what actually exists will change, only
our understanding of what actually exists will change.  This is similar to
the Physical Sciences, in which there is an *actual* physical reality worthy
of our study (for practical as well as academic reasons), and there are
competing theories intending to explain that physical reality.  When the
Michelson-Morley experiment demonstrated that the predictions of the
aetehereal theory of light failed in the physical world, a new theory had to
be developed (because of the resulting contradictions).  But the various
theories did not determine what is *true* in the physical universe.  On the
contrary, what is true in the physical universe lead scientists to prefer
the new theory (relativity) over the old one.

> Vaughan Pratt <pratt at CS.Stanford.EDU>
>     Date: Thu, 30 Dec 1999 18:11:30 -0800
> wrote replying to the posting of Mycielski:
> >My prefered formalism (for ZFC) is not first-order logic, but
> >logic without quantifiers but with Hilbert's epsilon symbols. In this
> >formal language quantifiers can be defined as abbreviations. This has
> the
> >advantage that the statements in such a language do not refer to any
> >universes. So this does not suggest any existence of any Platonic (not
> >individually imagined) objects.
> When I do mathematics, regardless of what might be happening in my brain
> cells, I feel as though I am working in a world of mathematical objects.
> The perception of a Platonic universe is very strong for me,
> independently
> of its reality or lack thereof.   I'd find it hard if not impossible
> to prove things if I had to work in a framework expressly designed to
> eliminate that perception!
> I also cannot imagine a mathematician who is working in this strange
> manner, i.e. without any intuition behind the formalism considered.
> Karlis Podnieks has a Web page with his book where he argues
> that the *naive* Platonism of working mathematicians is normal
> an useful thing with which I completely agree. We can use *any* kind
> of intuition when working with a formal system if this intuition
> really helps. However, Platonism *as a philosophy* seems to me very
> dangerous and harmful for mathematics. The philosophy should not
> be based on a self-deception.

How is it dangerous?  How is it harmful?  How is it a self-deception?  (In
anticipation of your claim that it is self-deception because there is no
Platonic Universe, I wish to see a demonstration of this claim.  Your
continued repetition of the claim that there is no Platonic Universe does
not convince me.)


> JoeShipman at
>     Date: Fri, 31 Dec 1999 10:23:56 EST
> wrote:
> In a message dated 12/30/99 5:32:40 PM Eastern Standard Time,
> jmyciel at euclid.Colorado.EDU writes:
> <snip>
> <<Their definition of
>  mathematics (a description of a Platonic universe independent from
>  humanity) assumes more but it does not seem to explain more. Hence it
> is
>  inferior.>>
> On the contrary, it explains the unity (mutual consistency and
> interpretability) of almost all the mathematics developed by thousands
> of
> mathematicians over the centuries.
> It is only illusion of an explanation. It is a declaration
> but not explanation.

It is as much an ``explanation'' as the theory of relativity is an
explanation of certain observed phenomena in the physical universe.  What do
you require for an ``explanation''?  Traditionally, an hypothesis is said to
``explain'' an observed phenomenon if the given hypothesis predicts (i.e.
implies) the observed phenomenon.  Thus, supposing that an actual Platonic
Universe does exist, one would expect, on quite rational grounds, that the
Mathematics developed by many different Mathematicians would tend to
demonstrate the unity (i.e. mutual consistency and interpretability) to
which Professor Shipman refers.

>"Mutual consistency and interpretability"
> is very much alike to Church-Turing Thesis: any (reasonable)
> notion of computability may be reduced to Turing Machines. It
> is just a sufficient (sufficient for the present mathematics,
> probably not for the future mathematics) flexibility and
> expressibility of a language/theory considered. As for the
> case of Church-Turing Thesis no Platonism (as a philosophy)
> is needed to realize this fact.

And just how do you propose that one ``realize this fact''?  I do not
consider the Church-Turing Thesis as self-evident.  If it were self-evident,
then no philosophy (at all) would, IMHO, be needed to realize it.  That is,
its self-evidence would transcend the Formalist-Platonist debate, due to its
obvious nature.  (Of course, if it were self-evident, then it would be an
absolute Mathematical truth, which a Platonist would accept as fact, but the
Formalist would (presumably) reject as utter nonsense.)


Matt Insall

More information about the FOM mailing list