FOM: Goedel: truth and misinterpretations
V. Sazonov
V.Sazonov at doc.mmu.ac.uk
Tue Nov 7 15:46:37 EST 2000
Matt Insall wrote:
> Now, the fact is that it is possible that unicorns exist and no one has
> recorded seeing one. Similarly, it is possible that electrons exist.
> Have you seen an electron?
> Has anyone? How do you know they exist?
As I can guess or partly know (I am not a specialist in Physics),
the concept of electron is included in some way in a system of
other concepts of physics some (or most) of which have clear
interpretations in terms of measurements and quite real, tangible
objects. Moreover, there is some intuitively plausible unified
picture including all these real objects and "objects" like
electrons. All of this is supported by a lot of concrete experiments.
This gives us right to say that electrons also exists. To my
opinion their existence is, nevertheless, not the same as existence
of the ordinary real objects. Moreover, the "unified picture" I
mentioned is probably not so perfect as it is desirable. But the
relations with the "normal" reality are so numerous and all-embracing
that nothing analogous can be said about, say, large cardinals in
mathematics and reality. In this sense (and with taking into account
some doubts I mentioned) electrons exists, unlike large cardinals
and many other mathematical "objects". Of course, there are some
mathematical objects ("denoted" by arithmetical numerals or epsilon
terms, according to Prof. Mycielski) which can be easily related with
the real world - sufficiently small numbers, finite graphs, etc.
here mathematics is occasionally somewhat analogous to Physics.
Mathematical theorems concerning these objects (i.e. deducible
formulas from feasibly consistent arbitrary theory, be this ZFC
or ANY theory else) are evidently TRUE in the sense of the real
world. But the given mathematical theory may contain much much more
of objects which have NO evident relation to the real world.
They should not, because this is mathematics, not physics.
Are so called unicorns even in a least degree analogous to electrons
or to large cardinals, where the latter are at least included in a
very elegant and formal (unlike physics, I think) mathematical theory?
I think that mathematical notions, in contrast to physical, have
absolutely different role. In general they are not intended to
immediately describe a reality (or any truth in reality; some
important exceptions are described above). They are only a very
specific kind of instruments (for thought) which can be used, say,
by physicists to make descriptions of the real world. Some "parts"
of these instruments (large cardinals, non-feasible natural numbers
like 2^1000, etc.) may have some important, but auxiliary role.
Instruments cannot be true or false. However, they may be suitable,
convenient or effective. It follows that philosophy of mathematics
also should not operate with the concept of truth (as with the main
concept of mathematics) as it is dealing with instruments.
Vladimir Sazonov
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