This is my response to Phil Wilson's article on Constructivism in a recent Plus Magazine, in particular to the following paragraph (but please read the whole article!) where he talks about
'…how firmly entrenched is the realist view that mathematical objects exist independently of the human mind, "out there" somewhere just waiting to be discovered, even in our every day conception of objects as fundamental as the real numbers. Intuitionism is radically antirealist: antirealist in that it claims mathematical objects only come into existence once they are constructed by a human mind (a sad quirk of language that this is called "anti"realist), and radical since it seeks to recast all of mathematics in this light.'
We don't have to choose between the view that mathematical objects exist independently of the human mind and the view that they only come into existence once they are constructed by a human mind. There is a third approach: We think about mathematical objects as if they exist independently of the human mind. In particular, mathematicians have gotten away with pretending that all the digits of a real number exist all at once and proving theorems such as trichotomy based on that view, without running into contradictions. The justification is just that: it works.
This approach has the advantage that our brain has a whole system of thinking about physical objects. We use this system to think about other things such as Sherlock Holmes and pi and appointment schedules and it works quite well. It doesn't work perfectly: for example, physical objects change over time and affect each other, whereas we must think of mathematical objects as eternal and inert if our proof techniques are going to work properly. Indeed, it is thinking of the decimal digits of pi (for example) as "going toward infinity" that gets students into trouble with limits.
Even so, objectification, if that is the right word, has worked very well for mathematicians and we don't need to give it up, nor do we need to be Platonists — we need only act as if we are Platonists.
I wrote about this in several places:
A scientific view of mathematics
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These distinctions are not fruitful. The point is, Godel’s arguments are said to have logical content–even though he is said to have a constructivist orientation. We have to locate the constructivist intervention in Godel’s theorems, if we are to show clearly that they are constructivist in nature, that is, to show that they have no logical content.
People are quite inconsistent. They tend to half-sign off on constructivism (because they suppose that the “paradoxes” have logical content), but also insist on logical content in their arguments, even though we know that constructivists feel logical content is impossible (because argumentation qua argumentation leads to paradox).
So Godel retains his reputation because no one has been able to identify specifically where he inserts his constructivist intervention in order to avoid his argument ending in “paradox,”as he fears.
I’ve been able to locate this intervention in the constructivist argument of another constructivist: the “natural” coincidence Einstein inserts in the relativity of simultaneity.
But where is it in Godel? For that matter, where is it in the Pythagorean theorem.
It is pretty useless to quibble as this “third way” argument does. It’s off point. The point is, where is the constructivist intervention in Godel’s argument? Show me precisely where, as I have shown you precisely where it is in Einstein:
Ryskamp, John Henry,Paradox, Natural Mathematics, Relativity and Twentieth-Century …
papers.ssrn.com/sol3/papers.cfm?abstract_id=897085