In my opinion, the best parts of the cartesian method are:

- doubt as a tool for advancing understanding, i.e this part of the rule 1: “never to accept anything for true which I did not clearly know to be such […] to comprise […] so clearly and distinctly as to exclude all ground of doubt”,
- and this part of rule 3, taken out of context, seen as a belief, a state of mind of the researcher: “by commencing with objects the simplest and easiest to know, I might ascend […] to the knowledge of the more complex”

This is, in a nutshell, that part of the scientific method which apply even to mathematicians (who don’t do experiments). If there is any need to tell, I strongly believe in the scientific method, although I recently understood that I have problems with that parts of the cartesian method which I now think will become obsolete, due to better and faster communication among scientists and due to the confrontation with big data (which will require techniques adapted to the recognition of the fact that reality might be complex enough so that a model of it does not fit wholly into one human brain).

The following are just doubts and questions which might be naive or based on ignorance, but nevertheless I write them here, in the hope of informed comments.

Remember again I am a mathematician, so maybe I disappoint some readers (I hope not my readers) by saying that I have no problem with Cantor diagonal argument for the proof of the fact that the set of reals is uncountable. What makes me feel less comfortable is the impression, which might be false, that famous theorems in logic, like Godel, or Turing, show in fact that there are limits to the enumeration part of the cartesian method. Am I right? To make a comparison, suppose I’am a physicist with infinite powers and I say: I made the experiment of counting all them Turing machines and I get results which are contradictory with older experiments. I deduce then, by the scientific method, that the counting procedure itself, any I might think about, is flawed, because all the other parts have been checked independently. It means I am not allowed to use the part of the cartesian method which comprises enumerations if I want to understand the reality. Reality is like this, point. If I count then I am led astray, like, as another comparison, if I suppose that a quantum object has a arbitrarily well localized position and moment, then I am led astray. (Not that I think there is any connection between logic theorems based on Cantor diagonal argument and quantum mechanics.)

So, what is left of logic if counting arguments are eliminated? Anybody knows? I hope so.

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* UPDATE: * Gromov has now a sequel to his ergobrain paper, which I commented before, see Gromov’s Ergobrain and How not to get bored, by reading Gromov and Tao. The sequel is called “Ergostuctures, Ergologic and the Universal Learning Problem: Chapters 1, 2.“. My interest is stirred by this new term “ergologic”, which may be of some interest to the readers of some posts from this blog, namely those regarding the cartesian theater and/or the graphic lambda calculus.

Looks like self-promotion, but that’s my blog, so I’ll say it: my graphic lambda calculus works without needing names for variables (or any names at all), also without enumerations, this is one of those things I am trying to push here.

Likewise, our brains, as well as any brains, no matter how simple, don’t work by the cartesian method, don’t solve problems, neither in parallel nor sequentially, and don’t set tasks. They are certainly not computers, although I believe we shall arrive to understand them very well sometime, scientifically. But that’s vastly out of my competence, so I shall stick to, for example, geometry, which has as the most important message that geometric properties are those which are independent on naming or enumerating (i.e. chart-independent). Finally, I think the question from the end of the post – what is left of logic if counting arguments are eliminated? – is a subject which may be explored with mathematics tools, leaving aside (for the moment at least) any meta-issues. (I agree with you, for example, that the whole of set theory is just an effort to understand better Parmenides, which eventually does not advance past the difficulties explained by Plato, but my mind is not clear enough to dare to discuss at the meta- level, so I just advance this little problem.)

The conclusion of your almighty physicist is a bit premature. Philosophically speaking, he should spend more attention to the conditions of his activities. In order to understand “any -1″ conceptual framework, you have to step out of it. In case of the scientific method, this regards the use of symbols.

Famously, Wittgenstein accused Russell to commit a mistake in his principia precisely regarding the use of symbols in the context of thinking about the role of logic. (In my opinion, Cantor never ever understood that there is an issue at all).

Considering counting, the physicist interestingly says: “any I might think about”, probably meaning, “any I am able to think about”. Which does means that his limitation could not tell anything.

Second, Goedels issue was not the issue of falsity. Similar to Wittgenstein, and later Quine, the only thing he proved is that logic can’t be taken as a means for talking about truth values insofar those are claimed to be part of the world.

Goedel’s message is: If there is a closed and such complete formalism, we can’t proof its correctness.

In another words: open systems could proof their correctness up to a certain point, but only by introducing a factual self-contradiction which comes as some kind of extension. That extension must be, of course, more abstract than its subject.

And in my opinion, this precisely describes the state and the process of science as we meet it today. If you like, the method of science in the critical perspective. Even Popper missed it, tough he is not terribly far away from it.

Back to the issue with the symbols: Logicians and Mathematician very well can engage in playing around with symbols. No problem. The challenge arises in applying mathematics. For logical symbols are empty, it is the procedure of assignment (empirical stuff to symbols) that introduces the above mentioned extension. Hence, any case in which we apply logic we can NOT apply it as logic, but only as kind of quasi-logic. And the “quasi” is determined by the embedding culture. Which finally explains the limitations of your (no so) almighty physicist

cheers

monnoo