FOM: Goedel: truth and misinterpretations

Matt Insall montez at
Tue Oct 31 18:30:09 EST 2000

Professor Kanovei stated:
Thus you say "Goldbach's conjecture is true" meaning that
any physical amount of "pebbles" or whatever counting units
is taken it never yields a counterexample to the conjecture.
Note that, in this argument, the "pebbles" must be real
physical ones, not philosophical, imaginary counting units,
to retain the empirically valid character of (b).

My comments are as follows:
Firstly, using this (and the other two) criteria for the meaning of the 
concept of ``truth'' seems
to me to invalidate the most fundamental concepts in mathematics and science.
For example, if one requires that the only way nature can indicate
to us the truth of a postulate is to provide us with the physical materials
from which to construct models, then the following statement is indeterminate
in this sense:

``Every natural number has a successor.''

In particular, if there are only n pebbles in the universe, and I am 
required to
model natural numbers with pebbles, and not sets of pebbles, then I cannot
even assert the truth of the statement that the number n has a successor.
Moreover, in such a restricted sense, I can even ``prove'' that n does *not*
have a successor, since a successor of n must be a number which is in
one-to-one correspondence with a collection of pebbles, and there can
be no such collection of pebbles, under these circumstances.  Of course, all
that this means to mathematics is that the assertion

``Every natural number has a successor.''

is not provable in FOL (or in SOL, for that matter), augmented only with 
axioms corresponding to our
empirical observations about finitely many physical objects.  Of course, an 
branch of logic can be, and has been, developed from presumptions of
finiteness, and many other very interesting subjects work only with finite
structures.  (I am referring to Finite Model Theory, Finite Group Theory,
Finite Ring Theory, significant portions of Lattice Theory, Finite 
Finite Graph Theory, etc.)  However, the claim that there are only finitely
many objects worthy of study is not provable, or empirially observable, even
if there are only finitely many pebbles in the universe.  Similarly, the claim
that because some of the objects we study in mathematics have no physical
counterpart, we are working only with ``fictitious objects'' is no more 
than is the claim that those objects actually exist.

Secondly, let us consider all of Professor Kanovei's three criteria
for claiming the ``truth'' of some statement:

(a) it has been correctly proved mathematically
(b) it is true as a fact of the nature
(c) that it is true is given in a sacred scripts

Now, I want to consider the statement ``There are only finitely many 
Does criterion (a) apply to this statement?  No.  If you claim it has, then 
I will ask
``Where is the proof, and how many objects exist?''.  For if you have in 
hand a proof
that there are only finitely many objects, I expect you must know the exact 
of objects that exist, as a consequence of your proof, but I am fairly 
certain that
this is an impossible task for a human being.  Now, what about criterion (b)?
In order to demonstrate a statement to be ``true as a fact of nature'', 
there are various
accepted approaches, all of which I would question except, in this case, a 
enumeration of all of the finitely many objects in nature.  I doubt that 
any human being
can carry out this program.  This leaves criterion (c), which Professor 
Kanovei seems to
have denied acceptability in this forum. (He said it ``leads us out of the 
I conclude that the only way anyone can know that the statement ``There are 
finitely many objects.'' is a true statement according to these criteria is 
to go ``outside science''.
Then what is the ontological status in FOM of claims of the truth of the 
statement that there
are only finitely many objects?  Are we committed to the claim that the 
universe includes
only finitely many objects only because some modern scientific principle 
dictates this
(and, I believe, not all do)?  How is this different from accepting the 
word of some ``sacred script''?

Matt Insall

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