Re: Foundations of Constructive Mathematics

Frank J. Lhota wrote:
John Jones wrote:
Frank J. Lhota wrote:
> Constructive mathematics asserts that the existence of a mathematical
> entity can be proved only by providing a method for constructing
> that entity.

What's wrong with that sentence? What entity exists prior to its proof? and if it doesn't exist prior to it's proof then are you saying that something can be proved into existence? Is 'proof' the right word to use?

First of all, let me apologize for the lateness of my reply. I've been having problems with my newsgroup server.

Let me clarify. Consider a statement of the form

(Exists x).P(x)

In classical mathematics, it is possible to prove a statement like this without actually finding an x such that P(x). For example, one could prove this statement by assuming that

(For all x).~P(x)

and obtaining a contradiction. Such proofs often provide no clue as to how to actually find an x with the desired properties.

Constructive mathematics does not accept a statement such as

(Exists x).P(x)

without identifying an x such that P(x).

There are several schools of constructive mathematics. I have a special interest in the Russian school, due to its intimate connection with computability.

I don't know if you have only rephrased it.

The point I was making was that there is no significant distinction between proof and existence. A mathematical entity that exists is proved. I can't go about proving the existence of a particular mathematical entity without already assuming that mathematical entity.
.

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