Finitary

The greatest discovery in the history of human civilization was the finitary proof. That is what allows people to demonstrate truths in a finite number of logical steps from axioms, and to be 100% certain about the result.

This concept was only discovered once, as far as we know, by the ancient Greeks. The best early exposition was Euclid's Elements in about 300 BC. There was independent mathematical progress in India, China, and elsewhere, but nothing like the logic of Elements. They had arithmetic, but not genuine mathematics. The method was formalized with symbolic logic, and ultimately perfected by Kurt Goedel and others in the 1930s. They showed that all of mathematics was reducible to the axiomatic method and finitary proofs. See articles on Logicism and History of logic for more info.

Mathematical proof should not be confused with scientific proof, which means empirical evidence to support hypotheses, or legal proof, which underlies our justice system. These non-mathematical proofs are not finitary because they allow the possibility that additional evidence could undermine the conclusion.

Finitary proofs might involve infinite sets or even the uncountable real numbers. It might even involve infinite ordinal and cardinal numbers. But there has to be an airtight and unassailable finite sequence of logical steps.

A detailed reduction of mathematics to formal logic was published in Principia Mathematica by Alfred North Whitehead and Bertrand Russell in 1913. Putting all of mathematics on finitary foundations is often called Hilbert's program. For the history, see the Stanford SEP article and Hilbert's 1926 essay. Hilbert appears to have hoped to find an algorithm for deciding any mathematical question in a finite number of steps. That proved to be impossible, but the vision of axiomatic foundations and finitary proofs does include all of known mathematics, and his program had many successes.

Kurt Goedel proved in 1931 that there are some limits to what can be done with computable axiom systems and finite proofs. Proving the consistency of a nontrivial computable system requires stronger axioms. This is quantified by ordinal analysis, where countably infinite ordinals measure the strength of a system. These are computable if they are less than the Church-Kleene ordinal. In order to decide Goedel statements, it can be argued that the finitary concept should be relaxed to included transfinite induction up to a computable ordinal. These ordinals are infinite, but can be defined in finite terms, and the proofs are still finite. It turns out that much weaker systems are sufficient to prove all the known and interesting mathematics, and that has been established by the field of reverse mathematics.

While all mathematical proofs are finite in length, they may or may not use infinite quantities. The word finitary is also used to describe arguments restricted to finite quantities.

A proof using infinities might be called soft. Sometimes a proof can avoid infinities and just use finitary arguments. Finitary arguments are not any more or less valid, but they sometimes yield more precise info. That more precise info is called hard analysis, as opposed to soft analysis. An elementary example of this distinction is in the proof of infinitely many primes. The theorem can be restated as a method for finding a new prime number, given any finite set of primes. For more advanced explanations of finitary arguments, see Terry Tao's blog, and for advanced examples, see The quantitative Morse theorem or Tao's finitary pigeonhole principle and finitary invariant subspace problem. He even has a finitary version of the Banach-Tarski paradox, one of the most counter-intuitive uses of infinity.

The words hard and soft here do not mean difficult and easy. The distinction is more like the distinction between quantitative and qualitative. The qualitative argument might be easier, but the quantitative argument is harder in the sense of being more rigid and precise. The advantage of hard (quantified) knowledge was expressed by Lord Kelvin:

I often say that when you can measure what you are speaking about, and express it in numbers, you know something about it; but when you cannot express it in numbers, your knowledge is of a meagre and unsatisfactory kind; it may be the beginning of knowledge, but you have scarcely, in your thoughts, advanced to the stage of science, whatever the matter may be.

Likewise, within mathematics, the finitary arguments of hard analysis can be more satisfactory than soft analysis.

A debate over the meaning of actual infinity goes back to Aristotle. C.F. Gauss wrote in 1831:

I protest against the use of infinite magnitude as something completed, which is never permissible in mathematics. Infinity is merely a way of speaking, the true meaning being a limit which certain ratios approach indefinitely close, while others are permitted to increase without restriction.

That is, mathematicians use infinities all the time, but the usage is really a way of speaking about finitary arguments.

Mathematics is all about making logical deductions from axioms. It is indispensable to science, which is all about formulating theories and testing hypotheses. And both mathematics and science are essential to the technology that makes modern civilization possible.

Finitary. It is why we are not still in the Iron Age.

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