"Because if you doubt that you're doubting, you're still doubting" - What is the analogous mathematical/logical expression to this sentence?

Question

In an answer here, the following was stated:

The essence of his [Descartes] argument is that you can doubt almost everything about the world, but you can't doubt that you're doubting. Because if you doubt that you're doubting, you're still doubting...

What is the analogous mathematical/logical expression to the last sentence?

To me it feels a little like a self-reference paradoxon, but I can't sort it out. The best, up to now, for me is to compare with a projector, which is an idempotent map p from a set E into itself (thus p∘p = p).

I doubt, that it doesn't have an analogy...

Answer 1

This might seem strange to you but I've always thought that the best mathematical analogue of the cogito is some fixed point theorem, let's say Brouwer's fixed point theorem (the point can be made with most of them.) In its simplest form this states that any continuous function f from a closed disc to itself has a fixed point. So the idea is that any way you choose to shift (in a "nice", i.e. continuous way) the points of the disc around at least one of them will always remain constant.

What does that have to do with the cogito? Well if you take the "closed disc" to be a human being (or the set of his beliefs) and your "function" to be the act of doubting, then even if doubting as it were "moves all points" i.e. makes you question everything, there will still be a "fixed point", namely the belief that you are, at this very moment in the process of doubting - doubting cannot as it were "move" that "point" (=the belief that you are doubting) - it is a fixed point of the doubting "function."

I might be stretching what it means for something to be an "analogue", but if not a direct analogue, fixed point theorems are, I think, still a nice way to conceptualize the cogito.

Answer 2

What is the analogous mathematical/logical expression to the last sentence?

Why would there be one?

Wittgenstein famously gave up the project of the Tractatus Logico-Philosophicus when he saw that there was no logical form to the Sraffa gesture; this led to the realization that there are all kinds of things which cannot be reduced to a mathematical or logical expression, and began the project of the Philosophical Investigations.

(As an aside: it is perhaps worth remembering that Descartes's project of a logically indubitable explanation of the world along geometrical principles, which you are referring to above, came to him in a dream.)

Answer 3

I doubt there are any direct analogs. Capturing the essence of "doubting" (which is a manifestation of human emotion(s)) mathematically or logically is a very tricky thing to do, if not impossible. For example, what is the mathematical or logical equivalent of anger, happiness, confusion?

At any rate, I believe the underlying process by which the conclusion in the last sentence of the quote is obtained does have a mathematical equivalent, and is, namely, the idempotent map you had referred to. But, that was a logical progression to begin with, wasn't it? ;-)

EDIT:

On second thought, here's a better mathematical equivalent. Consider a function which maps every element to itself. Or, more formally:

∀ x₁, x₂ ∈ R, function, f: R → R, x ↦ x;

Answer 4

I think it has less to do with maths and more to do with logic; I think a nice equivalent of doubting would be saying that it cannot be proven." I doubt of X" = ⊬X (with a bit of license) That would even make the 'doubt of doubt' unnecessary I doubt of everything= I cannot show things in general=∀X ⊬X. There's only two cases. Either that is true of that is false. But if it were true then there would be a contradiction, because there would be something provable (that same affirmation). If it isn't true, then there's at least something that we can prove. To be honest, there's still the case where the affirmation is true but you can't prove it (Gödel anyone?), but Descartes was looking for truths, not proofs