> Greek thinkers; most were extremely disinterested in observation. Of course, but they had at times supremely penetrative insight. Consider Democritus and the Atom. > Pythagoras believed in a highly idealized world "Number governs the Universe" is the shared thought. That school was a comprehensive order governing all aspects of life, including mores, hygiene, etc. Anyway, that school only knew of {N}. I am talking {Z} and harmonic systems: http://gdz.sub.uni-goettingen.de/dms/load/img/?PPN=PPN235993... http://www.wolframalpha.com/input/?i=e%5EiPI+%2B+1 To frame it by more contemporary terms, my philosophical views regarding Mathematics are likely close to Constructivism/Intuitionism: I reject the "excluded middle" [1] as a universal truth, and, more fundamentally consider the entire edifice of time-space (and all objects therein) a construct of mind, a side-effect of cognition. (Recall our previous conversation.) I assert that knowledge requires effort, at a fundamental level and is but the fruit of computation. While I personally like Ludwig far (far) more than Alan Turing, it would appear that Turing (an "empiricist" you should note) was correct and our dear Ludwig was merely philosophizing. [1]: Of course (!:) I have a wonderful proof 'showing' this, but the internets is too small to contain it ...