Friday, May 11, 2018

Reflecting on the "Cogito"

377 years and we're still baffled, René

Descartes was prepared to doubt the evidence of his senses (illusions, errors, dreams), but "we cannot in the same way suppose that we are not while we doubt of the truth of these things; for there is a repugnance in conceiving that what thinks does not exist at the very time when it thinks. Accordingly, the knowledge, I think, therefore I am, is the first and most certain that occurs to one who philosophizes orderly." (Wikipedia).

An argument which seems compelling .. yet curiously hard to formalise.

Never, ever trust the logicians to lead you closer to truth. We have:
|- thinks(descartes)                                    [assumption]
|- ∃x.thinks(x)                                             [existential introduction]
|- thinks(descartes) → ∃x.thinks(x)            [standard logic]
and with a bit more effort one can conclude that x here must, or could be descartes.

All the heavy lifting here is being done by the existential quantifier ∃, simply reflecting the mundane point in logic that an individual exists as part of its assertion. If you replace 'thinks' by 'walks' the argument works equally well (or badly).

Had they worked (harder) to attempt:
|- thinks(descarte) → exists(descarte)          (*)
(and is existence a predicate?) one would have confronted the central mystery. How do we formalise 'thinks' and 'exists'? What axioms support a theory which has (*) as a theorem?

As an AI person, I'd rephrase the "Cogito" as a robot problem. How would we design a robot which could convincingly assert Descartes's thesis?

The "Cogito ergo sum" is an internal reflection; by definition it doesn't relate to the outside world. In the jargon, it's metacognition. We immediately hit a problem. What is going on in a robot when it asserts "I'm thinking"?

Presumably it's thinking about something in particular, which we normally model 'without loss of generality' as a deduction within the robot's database of assertions and rules, its world-model.

So consider an inferential process occurring in the problem-solving layer of the robot and a concurrent metalevel representation of that inference. (An inference which is occuring, or maybe has just occurred?)

If P and Q are a couple of arbitrary facts while P→Q is a rule, then something like this?
thinking (P, Q, t1, t2) :- database([P, P→Q], t1), database([P, P→Q, Q], t2), t> t1.
Lacks conviction, don't you think? One is drawn to the byways of self-referential languages.*

I'm inclined to view the "Cogito" as more about consciousness, more about feeling self-aware than logic. Probably explains why we're as far away as ever from a compelling formalisation.

I notice I wrote about this in a similar fashion for back in 2011.


*  Reflective Prolog and "Reflection in logic, functional and object-oriented programming".


  1. Descartes probably cannot be formalised in First Order Logic. We might need to look at Second Order Logic (at least). Then in
    exists (Descartes) and thinking (Descartes) Descartes is not an individual, but a set (of (logical) individuals).

    Before long the first order theory also runs into Penrose-Godel issues too, but I think that the second order case might benefit from some new results in logical foundations. There is an interesting second order entity (originally found in Reverse Mathematics) with an invariant form and many invariant names e.g. WKL and Muchnik Degree 1. Its key defining property is the assertion of the existence of a special kind of dividing set.

    WKL has a (non-trivial) model theory and almost proves its own existence. It is transpiring that most mathematical ideas which are not Recursive (but nearly so) are linked to WKL: one example amongst many is that basic differential equations can have solutions shown to exist recursively (Picard Theorem); but many theorems in applied mathematics (E.g Theorem 1 in my book of Engineering Mathematics) asserting that Differential Equations have solutions are actually from the different Peano Theorem which require WKL to allow these solutions to actually exist.

    The reference to "Muchnik Degree 1" above refers to a generalisation of Turing Degrees needed to accommodate this construction; these results are only a few years old - and perhaps validate the Penrose concern that traditional Turing Theory is not rich enough to accommodate formalisation of Thinking.

    Another equivalent to Muchnik Degree 1 /WKL is the space of all completions of a given first order theory e.g. ZF Set Theory. This "space" could be viewed as the complete tree of all Godel extensions of e.g. ZF Set theory axioms: ie the "Multiverse" of all possible Sets, incorporating all possible new ZF Axioms.

    So Descartes is likely to be in there somewhere...

    1. There are so many roads to the 'Cogito' .. it's like a mirror, reflecting one's own interests .. .

    2. The expectation, though, is that all theses roads (at least the mathematical/logical/scientific ones) will converge.
      Interesting to read about the "Reflection Towers" (e.g. in LISP and Prolog) in one of the associated papers! I would need to do some work to see if I can relate this ( Tower infinity) to my comment above. Then we will see if all this can converge...


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