[FOM] special status of mathematics?

[Harvey Friedman]

In illustrating the special status of mathematics, I looked for a  
theorem of a thematic nature, whose proof is very well known, and  
where the appropriate axiomatic treatment of the statement and of the  
proof is also very well known. Also, the proof has substantial depth,  
both in an absolute sense, and as compared to "proofs" in other fields.

THEOREM. Any polynomial in one real variable with real coefficients,  
of degree at most 5, with at least six zeros, is degenerate (all  
coefficients are zero).

THEOREM. Let a,b,c,d,e,f,x,y,z,u,v,w be real numbers. Suppose that x <  
y < z < u < v < w, and for all real numbers t, if t = x or t = y or t  
= z or t = u or t = v or t = w, then at^5 + bt^4 + ct^3 + dt^2 + et +  
f = 0. Then a = b = c = d = e = f = 0.

In what subjects outside mathematics do we have comparable  
achievements or phenomena?

In normal attempts to give an answer to this, one does confront the  
interesting issue of proofs in physical science, where there is  
careful reasoning, with a (often very) substantial mathematical  
kernel, and that kernel is of the above general kind or generally much  
more complicated and deeper. E.g., various kinds of ordinary and  
partial differential equations are used, or differential geometry,  
etcetera.

However, it would seem that this is a conjoining of a piece of  
mathematics and some perceptive reasoning, where the latter is never  
cast as a proof in any comparable sense to what we have in mathematics.

There is the rather crucial problem as to whether the "non  
mathematical component" can actually be turned into a "proof" in any  
reasonable sense. This would require major advances in the foundations  
of physical science.

Computer science is another important area to try to analyze in these  
terms. I.e., are there proofs in any clear sense in computer science  
that are not mathematics? If so, what kind of depth do they exhibit?

This reminds me of a phrase that G. Kreisel used in print and in  
conversation: informal rigor. (Perhaps this phrase goes back to  
Goedel?) But the examples I remember him citing are extremely  
mathematical. He emphasized the example of a rigorous proof that the  
informal concept of validity in mathematical structures coincides with  
formal provability in a usual setup for predicate calculus. You don't  
have to formalize what a mathematical structure is in any usual way -  
like tuples of sets of certain kinds. You can instead use evident  
properties of the notion.

Of course, this is so heavily mathematical, that it doesn't really say  
much about the present FOM interchanges denying and affirming the  
special status of mathematics.

THESIS: Any remotely convincing case that mathematics does not have  
special status with regard to proof, will involve or precipitate a  
major breakthrough in the foundations of subjects other than  
mathematics.

CHALLENGE: Show us by specific convincing examples here on the FOM,  
not involving pointers to unstructured prose appearing elsewhere.

TECHNICAL PROJECT: Make a detailed study of the structure of proofs of  
the above Theorem (also for higher degrees).

Harvey Friedman