FOM: Goedel: truth and misinterpretations

V. Sazonov V.Sazonov at
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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