[FOM] Am I a Platonist?

[Martin Davis]

In a telephone conversation, Harvey Friedman once accused me of being an 
"extreme Platonist". I did not recognize myself in that description, and 
have thought for a long time about how to express my actual views in a 
coherent manner.

I take it for granted that our minds are the result of the activity of a 
brain that evolved to deal with the exigencies of life faced by our 
hunter-gatherer ancestors. I find it a source of wonder that as the result 
of a kind of logical "over-spill", our minds can obtain mathematical 
knowledge. I believe that mathematical knowledge is objective knowledge, 
that (as G\"odel has remarked), we have no ability to alter the properties 
of the natural numbers. We might decide that for every natural number to be 
the sum of three squares (rather than the four that Lagrange established to 
be the case) would be aesthetically pleasing, but we have no more ability 
to make this so than we have to alter the diameter of the earth. This 
contrasts with works of fiction where a writer can certainly produce a 
desired ending to order. I regard this objectivity as crucial, and regard 
questions concerning the actual "existence" of numbers or even measurable 
cardinals as having no clear content, and in any case, being beside the point.

I regard this objectivity as a hard-won result of mathematical practice and 
in no way given to some kind of a priori sense or intuition. It is a source 
of wonder and delight that with our finite brains, evolved for very 
different purposes, we are able to obtain objective knowledge about things 
that are infinite. Surveying the history of mathematics, we see 
mathematicians as almost reluctantly being driven to consider the infinite. 
Aristotle provided a safe mathematician's playground with his notion of 
"potential" infinity, suggesting avoidance of  any dealings with the 
"actual" infinite. Gauss explicitly warned mathematicians in a similar vein.

In the 17th century Torricelli stunned his contemporaries by flouting the 
Aristotelian doctrine by exhibiting a geometric body that was infinite in 
extent,  but had a finite volume. This discovery of what was OBJECTIVELY 
the case about the relation between the finite and the infinite was not the 
result of metaphysical speculation, but the direct result of the reach of 
our mathematical powers. When Cantor began his study of transfinite 
numbers, it was not at all clear that this could be carried out in a 
satisfactory manner, and of course many refused (and continue to refuse) to 
accept it. But the work of the set theorists continue to develop this 
domain, and even when assumptions that transcend ZFC are used, the 
objective nature of their results seems compelling. In a recent talk by 
John Steel, he spoke of the striking way in which bifurcations in the 
higher transfinite don't happen. As he put it, "There is only one road 
up."  To my mind this is clear evidence that one is dealing with objective 
knowledge.

In a recent post, Harvey Friedman has raised the question of the 
possibility of the coherence of a belief that every sentence in some 
language has a determinate truth value while accepting that this truth 
value will never be found by the human race. He mentions in particular, 
sentences of enormous length. But the fact is that there are many perfectly 
ordinary sentences of which it is perfectly coherent to believe they have a 
definite truth value which we are unable to determine. Does a planet 
recently discovered about some very distant star contain life? Or more 
mundanely, is the number of leaves on trees on my property at this moment even?

The name I suggest for my point of view is Empirical Objectivism: 
"empirical" because our understanding of the entities occurring in 
mathematics is not the result of any transcendent intuition, but rather of 
direct experience as practicing mathematicians, and "objectivism" to 
express the belief in the objective character of mathematical truth.

G\"odel was certainly not an empirical objectivist, but he approached that 
stance in the following statement from his 1951 Gibbs lecture: "If 
mathematics describes an objective world just like physics, there is no 
reason why inductive methods should not be applied in mathematics just the 
same as in physics. The fact is that in mathematics we still have the same 
attitude today that in former times one had toward all science, namely we 
try to derive everything by cogent proofs from the definitions (that is, 
 
from the essences of things). Perhaps this method, if it claims monopoly, 
is as wrong in mathematics as it was in physics."  Of course in mathematics 
we don't have the check that experiment provides in physics. But that 
comparison belongs in another post.

Martin



                           Martin Davis
                    Visiting Scholar UC Berkeley
                      Professor Emeritus, NYU
                          martin at eipye.com
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                        http://www.eipye.com