FOM: defining ``mathematics''

[Matt Insall]

This is my first reply to Professor Sazonov's reponses to challenges to his
advocacy of Formalism over Platonism.

Matt Insall



>
> ===================================
> Subject:
>          Re: FOM: defining ``mathematics''
>      Date:
>          Sat, 08 Jan 2000 22:22:42 +0100
>      From:
>          Vladimir Sazonov <sazonov at informatik.uni-siegen.de>
>        To:
>          FOM <fom at math.psu.edu>
>       CC:
>          Vladimir Sazonov <sazonov at informatik.uni-siegen.de>
>  References:
>          1
>
> The following are my comments on postings of Samuel Buss,
> Stephen G Simpson, Karlis Podnieks, Harvey Friedman,
> Jan Mycielski, Matt Insall, Vaughan Pratt and Joe Shipman
> related with formalism and defining what is mathematics.
> Cf. also my previous posting.
>
>
> Unfortunately, it is not easy to explain the formalist
> point of view as I understand it. (However, it seems to me
> so simple and clear!) Professor Mycielski is right that
> this term has a bad history. Therefore a lot of
> misunderstanding arise. But I like this term as short and
> linguistically most appropriate one and still hope it can
> be introduced again in a positive context to denote a
> "sober" and philosophically correct view on mathematics.
>

Please explain what the criteria are that you use to determine what
constitutes a ``philosophically correct'' view of mathematics.
It seems to me that the natural extension of formalism would be to
accept as ``sober'' any consistent view of mathematics, and to reject
any reference to ``correct'', since that term generally connotes a notion
of ``true'' vs ``false'', excluding intermediate possibilities.  Even
if one allows non-uniqueness of correct views of mathematics, so that
more than one view is judged correct, then the determination of what
constitutes a ``consistent'' view of mathematics needs to be made, in which
case, the formal definition of ``consistent'' comes into play, leading to
another ``true'' vs. ``false'' conundrum.  This may be carried on ad
infinitum, and one can formalize the process, so that it is part of
mathematics, and so the very statement that formalism is correct can be
formalized, and so must be either correct or incorrect, but in a formalistic
context, can be neither, since these terms have no meaning beyond the
context given.  This is what seems dangerous to me, rather than Platonism,
which
asserts that there is ``truth'' to be discovered, and that the formalisms
and rigour of mathematics can discover some of these truths.  Some of these
truths are statements about the formalisms of mathemtics themselves.  In
fact, some absolute truths are absolutely trivial.  For instance, it is
absolutely true that every abelian group is abelian.  The reason that this
is absolutely true is the following:  The statement itself is formulated in
a specific language which we have agreed to use (English), and can be
formalized in FOL (First Order Logic), as a part of ZFC.  In each case, the
reason this statement is true is that it means something that we know is
true, without formalisms.  The formalisms only encode this truth so that it
may be manipulated, as if by a machine.  If one asserts that the specific
symbols used to write the statement have no absolute meaning, I would
heartily agree, for that is one of the powers of language:  individual
meaningless-in-themselves symbols can be concatenated to express an
intuitive concept which has meaning.  The same fact can be expressed in
other informal languages, and in any of these languages, the truth of the
given statement will not be contested.  One may object that someone else may
interpret differently from me the words I used in the statement ``every
abelian group is abelian''.  Aside from the fact that most such
reinterpretations of my statement would lead to a neverteless true statement
(since most would be formalizable in FOL as
(forall x)[(A(x)-->A(x)])), the only thing I could say to that is that for
such persons, or reasoning systems, I have not made my meaning clear enough
to the reasoner.  For instance, if I am dealing with a reasoner who assigns
a different meaning to words when they are used a second time, I would have
to take that into account in expressing the truth I intend with the
statement that every abelian group is abelian.  This is merely a question
about communication, rather than a doubt whether the mathematical notion
``abelian group'' has any absolute meaning.  Since we can physically model
the axioms for a group, and in particular, an abelian group, there is no
question whether abelian groups exist:  they do.  ``The question may be:  Do
we know a significant amount about groups?'', or ``Can we know a significant
amount about groups?''  In the case of ZFC, there may be a (physical) model
of ZFC, but it may be impossible for us to find it.  Now, consider a
provably consistent fragment of ZFC.  Call it T.  If T has been proven
consistent by construction of a model, then the model exists, whether or not
one believes that ``actual sets'' exist.  Then, it may be true that the
intuitive notion I intend to capture with the term ``set'' is captured by
the fragment T.  Thus, the notion of ``set'' has meaning outside the
formalist/Platonist debate.  Now, if by ``absolute'', you mean that every
mathematician must agree on the meaning I assign to the term set, I agree
this is not guaranteed, but when I am referring to sets, I am referring to
some things that exist, but my knowledge about the properties of those
objects is (most definitely) incomplete; that is, it is absolutely true that
I do not know exactly which statements about those objects are true.  In
some cases, the absolute truth of this statement of incompleteness is a
direct consequence of Gödel's theorem:  We have only a (demonstrably)
recursive set of axioms for the theory of formal arithmetic (PA), but the
theory of the natural numbers, although complete, is (demonstrably) not
recursive.  This does not tell me that the natural numbers do not exist.  It
only means I have not determined enough facts about the natural numbers to
list in an axiomatic theory that will completely determine them.  I may
never know enough such facts.  It may be demonstrable that I can never know
enough such facts (especially if I grant you Church's Thesis as a bound on
the abilities of the human mind, which I do not), but this does not
guarantee that the natural numbers do not exist.  It only demonstrates a
limit on my quantity and quality of knowledge about the natural number
system.


> It could be objected that this term have been occupied and a
> new one should be used (say, rationalism - as Mycielski
> suggests; it seems to me that our points of view are
> very analogous, except the terminology). But I do not feel
> that "formalism"
>
> ***as it is usually treated***
>
> really denotes anything reasonable what deserves to have a
> name. What it usually denotes is rather some philosophical
> curiosity, nonsense. We should recover or reconstruct its
> *proper* meaning.
>
>
> I am sorry that this posting proves to be so long.
> Probably it would be better to present, instead,
> a concise description of this point of view. However,
> it is actually not the first my attempt to do this.
> Anyway, some objections would arise just because the
> term is "occupied". So, I am trying here to reply to
> these objections related to using this term or to clarify
> some points where there is no serious controversy but
> the danger of misunderstanding may arise.
>

I am not convinced that you have been misunderstood as badly as you may
think you have been.  I think I understand your position.  Let me see if I
can sum it up according to what I have read both above and below this point:
You consider mathematics to be the science of formalization, so that the
objects of study of this science are ``formal systems'' which do not, in and
of themselves, exist.  Please let me know if I have misunderstood.


>
>
> Stephen G Simpson <simpson at math.psu.edu>
>     Date: Tue, 21 Dec 1999 19:07:24 -0500 (EST)
> wrote:
>
> Buss hangs his definition of mathematics on a methodological
> issue: rigor.
>
> MY COMMENT:
>
> I think that rigor may be considered as methodological issue rather
> for other sciences. For mathematics rigor *in the form of various
> formalisms* constitutes its subject matter.
>

I would say that this type of rigour constitutes *part* of the subject
matter of mathematics.  It is a significant aspect of the study of
mathematics, undoubtedly, but is far from being all there is to mathematics.



> SIMPSON:
>
> But this doesn't seem to fit with the history of the
> subject.
>
> Our current standard of mathematical rigor evolved only in
> the 19th and 20th centuries.  Would Buss claim that there was no
> serious mathematics in the 17th and 18th centuries, prior to that
> evolution?
>
> MY COMMENT:
>
> As I understand, mathematicians always (at least beginning from
> Euclid) tried to be rigorous. Sometimes it was problematic. Before
> obtaining rigorous proof of a theorem we usually have some vague
> idea and then intermediate proofs. The process of finding a
> sufficiently satisfactory formal construction (including the very
> formalism where this construction will be done) may occupy even
> hundreds of years.


I agree with you on this score.  But does this not undermine your thesis
that mathematics is *merely* the study of these formal constructions?  The
fact that the ideas exist *prior* to the constructions, and are studied by
mathematicians, and are considered to be part of mathematics by those
mathematicians, means their existence is not contingent upon the
constructions themselves.  Thus the ideas are parts of mathematics that
have existence beyond the formal structures which represent them.


>
> As to another suggestion of Prof. Simpson to consider
> ``quantity'' as subject matter of mathematics, cf. below.
>
>
>
>        "Karlis Podnieks" <podnieks at cclu.lv>
> Fri, 24 Dec 1999 09:50:50 +0200
> wrote:
>
>
> Thus, Hegel defines quality as "first, direct definiteness", and
> quantity - as "definiteness that has become indifferent to the
> Existing" (sorry, I'm not very strong neither in German, nor in
> English). For me, "definiteness that has become indifferent to
> the Existing" is just the notion of a model that has become
> distracted from its "original", and hence, can be investigated
> independently of this "original". In other words, this is the
> notion of self-contained models (models that can be used by
> robots).
>
> I am fascinated by this equivalence of quantity and
> self-contained models since the beginning of 1970s when I read
> Hegel for the first time. I'm glad to see  outstanding people
> approaching this position from various angles...
>
> Could we agree on defining mathematics as the science of
> self-contained models? Let us call it Hegel's definition, not
> Podnieks's. Could this help?
>
>
> MY COMMENT:
>
> I like very much this clarification by Podnieks of what
> is quantity. But what are these "self-contained models
> (models that can be used by robots)"? I think, these are
> exactly formal systems. Thus, we have a chain of
> equalities
>
>             quantity = ... = formal systems
>
>
> with the most nontrivial philosophical step done by
> Hegel. It seems I also remember (rather vaguely) that step
> from a course of Marxist philosophy when I was a student.
> I am very grateful to Podnieks for recalling and for
> explaining it so clearly.

I am not directly familiar with Hegel, so I cannot comment on his work on
the philosophy here.  But I agree that the ``models that can be used by
robots'' should be formal systems.  On the other hand, I question the
identification of quantity with formal systems.  Some ``quantities'', I
contend, are ``happened upon'' by us human beings, and are ``unexplainable''
in the context of any formal system.  Yet these form a part of the study of
mathematics, especially of applied mathematics.  They may be the ``data''
with which one begins, and builds a formal system around, in an attempt to
describe physical (and intellectual) reality, of which mathematics is a
part.


>
> Thus, we can say that mathematics is the science on
> quantity or, essentially equivalently, the science on
> self-contained models (for "robots"!!!) or, equivalently,
> the science on formal systems.
>
> As I do not feel myself as a philosopher, I prefer
> the last more "operational" version in terms of formal
> systems. It seems to me simpler and sufficiently general
> and adequate. Also, how could we infer from the definition
> in terms of quantity the main feature of mathematics - its
> rigor? Only by the above chain reducing quantity to formalism.
> This seems somewhat complicated. How many people, even
> professional philosophers, will realize these equivalences
> without additional explanation? The first opinion would be
> that it is about numbers existing objectively. Again the
> danger of Platonism.
>
> However, I am also fascinated by this equivalence.
>

Would you really object to the statement that the number 1 exists
objectively?  When you question this, I question what you mean by
``objectively''.  If you reduce the question to one of language, again you
beg the question, because translations can be made, once one sufficiently
understands another's language.  Yes, there are theories T satisfied by any
model of PA in which the number 1 is not definable, but this is not the
question.  Questions such as that of whether or not the number 1 exists are
philosophical, and set the stage for the entire purpose of one's
mathematics.  However, if numbers do not exist objectively, then what do you
mean when you say ``number'', or more particularly, ``only one''?  I contend
that either one of us fails to reason consistently, or there is a way to
come to understand each other so that we both know what I mean when I say
the phrase ``the number 1''.  Thus at least the number 1 exists
``objectively''.  It may be debatable whether certain numbers exist
objectively.  For instance, ``the number of molecules in the universe'' may
have no clear meaning, if only because some molecules can be annihilated by
a nuclear reaction, making the entire notion of such a ``number'' not well
defined.  One may question whether infinite numbers, such as \aleph_0,
exist, on philosophical grounds, but one must accept the existence of
certain very simple numbers, especially certain finite ones.

Matt Insall
http://www.rollanet.org/~insall
montez at rollanet.org