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.
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.