how my philosophy homework turned into a rant

[gusl]

WARNING: The following is not very interesting.


Philosophy of Mathematics

Discuss: "Realism-in-truth-value leads to a normative philosophy of mathematics"

A normative philosophy of mathematics is one which prescribes how one should do mathematics.

A mathematician is a realist-in-truth-value iff he believes that forall statements phi about mathematics (phi \/ ~phi).

Such a mathematician is likely to believe in the following norm: the goal of a mathematician should be to determine whether phi or ~phi is the case for whatever phi they are working on. Thus realism in truth-value automatically leads to this norm.

How is such a mathematician to deal with the Continuum Hypothesis, given that both CH and ~CH are consistent with core mathematics? One possibility is to have method of deciding which one is true, e.g. intuition; but he may even simply acknowledge that we don't know and can never know which of phi or ~phi is the true one, while still maintaining that exactly one of them is true.

To such a mathematician, intuitionistic mathematics is overly restrictive, because to him, all proofs of ~~phi are actually proofs of phi. The structure of intuitionistic mathematics, however, may still be an interesting object of mathematical study; and as in all other areas of mathematics, our realist-in-truth-value will attempt to determine where phi or ~phi is the case for each question under investigation phi. And in this sense, he would be no different from intuitionists, who tend to do their meta-reasoning classically while keeping a straight face.

Comments

[dusc.livejournal.com]

So many ways "(phi \/ ~phi)" can fail (dropping the "about mathematics" clause), things understood by the mathmatically disinclined. Subjectivity ... Scope ... paradox ... I'm sure there are others.

I don't think I've ever been called an "intuitionist" before. "realist", "objectivist", "clown", "idiot" and "fool" all made the list long ago. And, I do keep a straight face, at least until I get to the conclusion!

Logical riddle: How can "I am lieing" have a single determined T/F value, and what must it be, when it exists?

[gustavolacerda.livejournal.com]

Intuitionism is the logic of provability, if phi is interpreted as "phi is provable".

The negation is interpreted as "leads to a contradiction".

phi \/ ~phi is not always the case, e.g. when both phi (i.e. phi is provable) and ~phi (it is provable that phi leads to a contradiction) are consistent with what we know.

So we have to be careful with the disjunction.

We no longer have the classical rule:

~~A
---
A

Adding either excluded middle or double-negation-elimination or Peirce's law to intuitionistic logic will give you classical logic (i.e. in this context, all 3 are equivalent).