Hilbert's On the Infinite

He starts with some motivation:

    1. Because of Quantum Mechanics, there is no reason to believe that there is anything physical that is indefinitely small.
    2. Because of General Relativity, there is no reason to believe that there is anything physical that is indefinitely large.
    • That is, it is reasonable to suspect that there is both a smallest and a largest physical magnitude, and hence there is no physical realization of anything infinite.
  2. Because of Weierstrass, there is no reason to believe that there are infinitesimals. By eliminating the infinitesimal, Weierstrass produced a rigorous analysis, free of formal contradictions.
  3. Weierstrass eliminated the infinitesimal, but left the infinite in mathematics: the infinite series that have limits. This type of infinite is known as the actual infinite. There are various theories of the actual infinite, and they are incompatible and contentious. (Russell, Frege, Cantor) The Russell and Frege versions are inconsistent as is shown by the "Russell-Zermelo" paradox.
  4. Such problems led lots of people to follow Brouwer. Hilbert hates that: "No one shall throw us out of the paradise Cantor has created for us."
  5. In order to defeat Brouwer, he must be answered on his own grounds.
  6. To answer Brouwer on his own ground, one must find a common neutral territory that both Cantor and Brouwer can accept. Hilbert's "finitary mathematics" plays that role. In finitary mathematics, all claims are quasi-concrete, that is, "contentual" in the sense that all refer to quasiconcrete computations. Example: there is a prime number between $p$ and $p!+1$. This looks like an infinitary statement. But, according to Hilbert, the finitary version is not a statement at all, but a statement form.
  7. It follows that there is a prime number greater than $p$, and that is where the trouble begins.
  8. Everything that is consistent is allowed. For Hilbert, the complaint that there is something wrong with using complex numbers because "-1 doesn't really have a square root" is just silliness. If it is consistent to use complex numbers, there is no further mathematical question about whether we can. (Brouwer might disagree.) Hilbert identifies one clear sense in which this will be true: If the new theory is consistent, then anything it can prove that is finitary must be true: A finitary statement is genuinely true or false, and so if you can prove it in some nonfinitary way and it is actually false, you would have an inconsistency. So, whatever bizarre means you recruit to prove it, so long as they are consistent, if you prove it it must be true.
  9. So, according to Hilbert, what we need to do is pretend Cantor's paradise exists and show, using finitary methods, that that pretense is consistent.
  10. Even though the theory is far from infinitary, we have a formal proof system that mimics it which involves nothing more than finite manipulation of finite strings of symbols. Don't add the infinite, add the symbolism for it and interpret what that symbolism purports to refer to as "ideal elements."
  11. In order to carry out Hilbert's program for the justification of set theory we need to show that set theory is consistent, so that we can conclude on generally acceptable grounds that it is a useful tool for arriving efficiently at meaningful (contentual, finitary, intuitionistically valid) conclusions.
  12. No one will accept the conclusion that set theory is consistent unless it is proved to their satisfaction, and the proof of consistency must therefore be finitary.
  13. At first, it seems hopeless to give a finitary proof that set theory is consistent, after all, set theory is far from a finitary theory and it mentions quintessentially nonfinitary objects. However, and this is Hilbert's chief insight, which leads people to call his view formalism, is that, since set theory can be axiomatized, a proof of consistency just needs to show that if we start with a certain finitistically specified collection of quasiconcrete objects ("axioms") and transform them according to certain finitistically given rules ("inference rules"), we cannot get to the string (for example) $\varempty\in\varempty.$ (This works because, given one contradiction, everything follows: Proof: Suppose we can prove $A$ and $\lnot A$. We prove $B$, for example, as follows: $A\lor\lnot A$, so $A\or\lnot A\lor B$. Since $\not A$, $\lnot A\lor B$. Since $A$, it follows that $B$. Thus, the problem of proving consistency, even for a highly infinitary theory, is an elementary combinatorial problem, the sort we often solve using finitary methods.

Problems

  1. Intuitionists do not, in fact, accept Hilbert's finitary mathematics, though Hilbert occasionally and his students, for example, von Neumann, often, regard finitary and intuitionistically valid as synonymous. Here's why: Finitary mathematics is quasiconcrete, but intuitionism is all about mental constructions. Brouwer said something like, "The intuitionist thinks that the truth of a mathematical theorem lies in the mind of the mathematician, the formalist, on paper." I don't think this is serious, though some people do. After all, the sequences of quasiconcrete symbols have mental counterparts. Either the two camps could agree to disagree and recognize that a proof of one sort provides what is needed to give a proof of the other sort, or they could agree to fasten on to the mental version of symbol pushing.
  2. The more famous and devastating problem is that G&246;del's second incompleteness theorem is generally accepted to show that no consistent mathematical system that includes primitive recursive (finitary) arithmetic can prove its own consistency, let alone the consistency of a stronger system.
    1. Thus, even if we can get consistency proofs, they will be from stronger to weaker systems. As Ed Nelson says, that would be a bit like having a mafioso on trial call a bigger mafioso as a character witness.
    2. Some have denied that primitive recursive arithmetic is all of (or part of) finitary arithemetic. That leaves open the possibility that someone will discover a really strong finitary technique that would let us prove the consistency results we want.
      1. Tait has argued, in a widely cited article, for the generally accepted thesis that finitary = primitive recursive.
      2. Some have argued that the ordinal induction principles are in fact finitary. The arguments only persuade the converted.
      3. What G&246;del showed is that a theory cannot prove its own consistency for a very specially formulated notion of consistency. It is easy to give a definition that it is intuitively plausible to take to be a definition of consistency such that every system can easily prove its own consistency. Unfortunately all known such definitions are no good for Hilbert's purpose. However, we cannot rule out the existence of a surprising definition of consistency that does the job, as Detlefsen has argued in detail.

-- ProfessorShaughanLavine? - 31 Jan 2005