[FOM] Points of View in Philosophy of Mathematics

[Dmytro Taranovsky]

This posting surveys the points of view in philosophy of mathematics,
especially the disagreements between Platonists and formalists, and
outlines the main arguments.  The purpose is to clarify rather than to
persuade the readers of the correct philosophy.

1.  The nature of mathematical objects.
Generally agreed:  It appears that we can refer to mathematical objects.
Points of view:
A.  Referring to mathematical objects is an error; the appearance of
possibility of reference is an illusion.
B.  There is a vast repository of objects, called human imagination,
that includes mathematical objects.  Objects in human imagination exist
in the sense that one can refer to them, but formal statements about
them are not guaranteed to have truth values.
C.  Apparent references to mathematical objects are likely to be, absent
evidence to the contrary, references to the existing mathematical
objects.

I believe in C; Sazonov believes in B; and Harvey Friedman claims that
his unidentified correspondent and Fields medalist believes in A.  A
single person can have different positions with respect to different
mathematical objects.

2.  The nature of mathematical truth
A.  Mathematical statements, except ultrafinitistic ones, have no
semantics.
B.  Mathematical statements are statements about objects in human
imagination, and as such they have some semantics, but are inherently
vague.
C.  Mathematical truth exists, but it exists independently of whether
mathematical sets themselves exist.
D.  Every possible mathematical structure metaphysically exists, but
mathematical statements cannot refer to ALL sets, only to all sets in a
particular structure, and as such are inherently vague because they mean
different things as applied to different structures.
E.  A formal statement about the set theoretical universe which consists
of all sets that exist is either true about the universe, or not true in
the universe.

My position is E.  Sazonov's position is B.  From behavioral point of
view, position D is essentially formalism and is similar to Sazonov's
views, and position C is essentially Platonism.

3.  Human ability to gain mathematical knowledge.
A.  Mathematical knowledge solely consists of knowledge of which
theorems are effectively provable in which formal systems.
B.  Because humans do not interact with mathematical objects, all that
we believe we know about mathematical objects may be wrong.
C.  Humans are not finite automata and have special (non-recursive)
powers to learn mathematical truth.
D.  Humans know the mathematical axioms through spiritual interaction
with mathematical objects.

My position is C.  I view position D as mysticism, and reject it as
such.  
Question:  To what extent can human civilization learn mathematical
truth?  Is there a fundamental limitation on human mathematical
knowledge?

4.  Physical realization of mathematical objects.
Generally agreed:  Mathematics is extremely important in making
empirical predictions.  
Points of view:
A.  Physical world is bounded and discrete, so only small integers are
physically realized.
B.  Real numbers are realized as physical quantities, but higher
mathematical objects are not.
C.  Every mathematical object, including the least inaccessible
cardinal, is realized in some physical universe.

I find choice B plausible, but I am not sure.  Sazonov is inclined to
believe A.

----------------------------------------------------
Arguments to Support Points of View
5.  Platonism
A.  It appears that we can refer to mathematical objects, in particular
the least inaccessible cardinal.
There is no evidence for such a reference to be an error.
If it appears that we can refer to something and there is no evidence of
error, then highly likely
the apparent reference is semantically valid.
A reference cannot be semantically valid unless it refers to something.
To exist means to be something rather than nothing.
Conclusion:  The least inaccessible cardinal is highly likely to exist.

I accept argument A.


B.   Real numbers are almost universally used in current physical
theories.
An object that is used throughout physical theories, such as an
electron, is likely to exist.
Conclusion:  Real numbers are likely to exist.
Objection:  The invocations of real numbers are different from
invocations of ordinary physical objects in that physical objects cause
observable events, while real numbers, such as the location of the
center of mass, are merely used as properties of physical objects.  An
observation gives only an approximation to the real number, and it is
possible to rewrite physical theories to use only finite structures.

I believe that argument B, despite the objection, provides substantial
evidence for existence of real numbers.


C.  If real numbers are just imagined, then their theory, like the
theory of a fictional world, should be dominated by incompletenesses
without a natural solution and require many complex and arbitrary rules.
The theory of real numbers requires only very basic axioms along with
projective determinacy and is so coherent, beautiful, and precise.  All
natural incompletenesses appear to have unique natural solutions.
Conclusion:  Real numbers are real rather than just imagined.

Technical Note:  Statements about real numbers at a high quantifier
level are best visualized as statements about projective sets.
Note:  At present, the argument would be much less persuasive for
uncountable sets, but I believe this will change in the future.
-------------------------
6.  Arguments for formalism.
A. There is no evidence that one can observe or interact with
mathematical objects, so minimization of unexplainable requires that
mathematical objects do not exist.
Objection:  The argument shows that mathematical objects are not
observable, but that does not imply their non-existence.
Note:  The extent to which A is persuasive is a complex philosophical
issue.

B.  Higher set theory contains plenty of incompletenesses that set
theorists do not know how to solve, but if the sets are real then the
incompletenesses must have a solution.
Objection:  Given that humans do not interact with the mathematical
objects themselves, incompletenesses of our knowledge are to be
expected, even if the objects exist.


Did I miss any major arguments?  Which points of view do you find most
persuasive?

Dmytro Taranosky
http://web.mit.edu/dmytro/www/main.htm