Thursday, December 09, 2010

Rambling About Actual Infinites

One of my Logic students, Lindsay, asked some questions about the philosophy of mathematics. I pointed her to Stanford Encyclopedia of Philosophy's article. Now I'm reading it.

In the early 20th century three non-platonistic accounts of mathematics were developed: logicism, formalism, and intuitionism. It was unacceptable to think that numbers had an actual ideal existence in Plato's realm of the Forms.

I'm reading down to the section on intuitionism. "According to intuitionism, mathematics is essentially an activity of construction. The natural numbers are mental constructions, the real numbers are mental constructions, proofs and theorems are mental constructions, mathematical meaning is a mental construction, … . Mathematical constructions are produced by the ideal mathematician, i.e., abstracted from contingent, physical limitations of the real-life mathematician. But even the ideal mathematician remains a finite being. She can never complete an infinite construction, even though she can complete arbitrarily large finite initial parts of it. (An exception is made by Brouwer for our intuition of the real line.) This entails that intuitionism to a large extent rejects the existence of the actual (or completed) infinite; mostly only potentially infinite collections are given in the activity of construction."

Aha! The good old Craigian-Hilbertian "impossibility of an actual infinite." Or, perhaps not actually Hilbert's idea, since intuitionism's rejection of an actual infinite is based on the mathematician as a finite being who is, ipso facto, unable to complete an infinite series.

Hilbert was a "formalist." Natural numbers are not mental constructions as the intuitionists claimed, but are symbols. Note: if natural numbers have no ontological status then they are something like mental constructs, or maybe symbols. For Hilbert,symbols are abstract entities, "but perhaps physical entities could play the role of the natural numbers. For instance, we may take a concrete ink trace of the form | to be the number 0, a concretely realized ink trace || to be the number 1, and so on. Hilbert thought it doubtful at best that higher mathematics could be directly interpreted in a similarly straightforward and perhaps even concrete manner."

In that case, on Hilbert, an actual infinite is impossible because if numbers are symbols which can in principle be represented by physical entities, one could have a potential infinite but of course not an actual infinity of, say, ink traces.

See here, e.g.: "The question of how exactly Hilbert understood the numerals is difficult to answer. They are not physical objects (actual strokes on paper, for instance), since it must always be possible to extend a numeral by adding another stroke (and, as Hilbert also argues in “On the infinite” (1926), it is doubtful that the physical universe is infinite)."