FOM: Ontology of Mathematics
Jan Mycielski
jmyciel at euclid.Colorado.EDU
Mon Jul 10 12:54:06 EDT 2000
(Reply to V. Sazonov and H. Friedman)
Commenting on my posting ONTOLOGY OF MATHEMATICS from July 24,
2000, V. Sazonov suggested (June 25) that I should write more clearly. And
his message was reinforced by H. Friedman (June 27) who tried to answer my
letter without (I believe) understanding well enough what I wrote.
Thus let me proceed with some clarifications. (I already tried to
make myself clear about these matters on FOM on Dec 10 and 29,
1999, and Jan 7 and Feb 15, 2000.)
1. The fourth sentence of my letter was intended to be:
A RATIONAL ONTOLOGY OF MATHEMATICS TELLS US THAT AT EVERY POINT IN
TIME THERE EXISTS ONLY A FINITE STRUCTURE OF MATHEMATICAL OBJECTS (which
we call sets, but they are more like containers which are intended to
contain other containers, such that not all of their intended content need
to be already constructed) WHICH ARE ACTUALLY IMAGINED (0R NAMED) BY
MATHEMATICIANS.
(Unfortunately, as pointed out by Friedman, this sentence was
somewhat garbled in my posting; but I believe not garbled to the point of
ruining its sense.)
However, it seems that Friedman did not take this sentence in, and
this caused him to write some irrelevant remarks.
The point is that this rational ontology tells us that there are
really such physical things in human brains which we call thought-objects
(sets), that there are only finitely many of them, and that those things
are all there is if we ask what is pure mathematics about. And that this
structure suffices to do pure mathematics. (See Hilbert 1904.)
On the other hand there are no such things as ideal or Platonic
objects which are not physical and exist independently of mankind.
Thus pure mathematics is pure art, in the sense that it talks
about an imagined universe and not the real one, however imagined means
really constructed in our brains and physical in there. Still pure
mathematics is not disconnected from experiment. Namely it is supported by
thought experiments dealing with those physical thought objets. If you
like the simile of a Turing machine (since we know what are the states of
such a machine), think of tha brain as a Turing machine which is still
very poorely known. I believe that a better model would require continuum
many states, but for our purpose here a Turing machine will do. You can
calculate mentally, and those calculations are those thought experiments.
Computers can also do mathematics (with some advantages and some
disadvanteges relative to us).
Now we ask for a mathematical description of this finite structure
in the brain. As I wrote Hilbert's epsilon extension of logic gives us
such a mathematical description. [Friedman complains that the epsilon
terms are too long. Why shold they be short? This is only an abstraction.
In reality we have short symbols and other abbreviations. (epsilon y fi)
stands for a single function symbol with as many argument places as there
are free variables in fi other than y. Then there are other important
abbreviations, namely quantifiers.]
2. I think that there is a more serious problem with Friedman's
philosophy. He writes that philosophy is a kind of loose talk, or a sand
castle, that it does not answer any questions in a definitive way. Many
philosophers (and philosophers of mathematics) believe that. But, as I
tried to demonstrate above, this is false. We have an absolutely clear and
compelling ontology of pure mathematics, a scientific answer to the
question what does mathematics really describe (namely that finite
structure of thought-objects).
What is unfortunate in the literature on philosophy of mathematics
is all the loose talk about the difference between abstract and concrete
objects. I believe that the only difference which makes real sense is the
difference between thoughts which are about something and thoghts without
any intended reference. For example, I do not know any reference of the
thought object 10^(10^10) or of a well ordering of the real line; But
they are just as concrete in my imagination as the thought object 3 or the
number pi. Still only the latter have a (many) references.
The concept of existence in common language is useful only because
it distinguishes thougts with a reference from thoughts without reference.
(Notice that nothing does not exist, so if the phrase "A does not exist"
is true, then A can be only a thought.
3. Later in his reply to my posting Friedman seems to complain
that what I wrote would invalidate the critique of classical mathematics
or the motivations expressed by the constructivists. Yes, I believe that
their philosophy is quite unclear and even misleading and antirational.
(I believe that it stems from a kind of sectarian ignorance of the
philosophy explained above, and expressed already in Hilbert 1904 and
1924, but perhaps not stressed enough in his papers. In Friedma's reply
the importance of those thoughts in those papers is negated.)
4. Finally, I do not understand why Friedman thinks that the
rational philosophy which I have expressed goes against the classification
of mathematics according to the strength of its axioms. I see no
connection, and am convinced that Reverse Mathematics and related work
initiated by him is interesting and beautiful mathematics.
As much as I understand, the only thing he might be unhappy about
(but his letter is not clear on this point) is that this philosophy makes
it visible that Arithmetic or Finite Combinatorics are no more justified
on ontological grounds than any other area of pure mathematics.
According to this ontology it is only applications which give an
additional ontological dimmention to mathematics. Indeed, in Friedman's
program of miniaturisation of the axioms of infinity one does not deal yet
with integers or sets small enough to be significant in science or in
engineering.
5. A thesis of rational philosophy which may be useful to a
mathematician-philosopher: The only ontological distinction between
thought objects is between those which have and those which have no
intended outer physical reference. All other such distinctions which were
attempted by many philosophers and philosophers-mathemnaticians are vague.
6. Of course I would be indebted for any criticism which could
improve or discard that rational philosophy.
Jan Mycielski
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