I've recently been on a drive to look for the source of the asymmetries I see in mathematics, and I ran into the idea of cointuitionism. First, let me remind you about intuitionism (as I know it):(adsbygoogle = window.adsbygoogle || []).push({});

Intuitionism is the school of logic that rejects the law of excluded middle -- they do not require [itex]P \vee \neg P[/itex] to be true.

This leads to some interesting properties. For example, it's possible for [itex]\neg P[/itex] to be false when [itex]P[/itex] is not true. More generally, [itex]\neg P \equiv \neq Q[/itex] does not mean [itex]P \equiv Q[/itex].

Intuitionists still retain the law of contradiction, though: [itex]P \wedge \neg P[/itex] is false.

It seems that when doing intuitionist logic, the thing to do is to take

"implies" as a fundamental operation, instead of "not" as one might do in Boolean logic. So, you formulate things in terms of "and", "or", and "implies".

Now, this strikes me as being asymmetric! The law of the excluded middle, [itex]P \vee \neg P[/itex], and the law of contradiction, [itex]P \wedge \neg P[/itex] are duals of each other. So, I wondered what we would get if we tried it in the other direction:

So, what I call the "anti-intuitionist" school of logic would reject the law of contradiction -- [itex]P \wedge \neg P[/itex] is not required to be false.

Working through the duality, it turns out the right fundamental operation to consider, in addition to "and" and "or" is "B does not imply A"

It strikes me that this might be appropriate for the philosophy of science, since the philosophical role of experiment is not to arrive at an implication, but to arrive at thedenialof an implication. In other words, an experiment is used as an attempt tofalsifyvarious theories -- in other words, to expose what the initial conditionsdo not imply.

Working further through the duality, it suggests another view on proof:

Normally, a proof consists of doing the following:

We start with a collection of statements, which we assume to be true.

Using (assumed true) implications, we infer additional statements which we add to the collection.

I.E. if we assume "A" and "A => B" are true, we can infer "B" is true.

By looking at the dual of this, it suggests that what we might want to consider doing is:

We start with a collection of statements, which we assume to befalse.

Using (assumed false) instances of "A does not imply B" (which I'll write as A%B), we infer additional statements which we add to the collection.

I.E. if we assume "B" and "A % B" are false, we can infer "A" is false.

Again, maybe this is useful to science, since, intuitively we can be more sure about what isnottrue than what is?

Okay, I'm done rambling! What do you think?

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# Source of the asymmetries I see in mathematics

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