hythloday, godel, penrose 1ofX

• axioms can't be proved mathematically (with formalism)
• people still want to know how we find them
• and how we can confirm them as truly well found
• people still want to know how we justify them
• to see if we really can
• I don't see anything at odds, it's just that WITHIN mathematics the rule, of necessity, is assume these things.  Assume induction.  Look, induction is better than that, it's not merely assumed, we're forced to "assume it" but it's not like assuming something else, say "fooduction" will do just as well.  
• I have long thought we find the axioms through trial and error, namely, through evolution
• Penrose thinks there is another way, which I see as quite possibly similar, that is, the quantum computational methods of "trial and error" or "parallel modeling" in which multiple models can be simulated at once.
• he thinks I'm wrong about G(R).  G(R)
• Penrose on Platonism
• platonism... plato said we didn't have to assume, we had direct access to the truth of the axioms, they were essentially a priori and true-in-themselves
• your relations-in-math diagram is nice... but there is no representation of these foundationalist issues... it's just "formal systems"... iow, "some formal system"... but some people look into the detail of that, and that's where the problem is.  
• when I talk about ideas that turn out not true, I'm comparing all things we think to induction... if we "just assume induction" and "just assume, say, racial superiority"... sorry, induction is cool that latter is bogus... and so far, induction is solid as a rock... but just "assumed"... 
• you have argued against Penrose's examples as with chess, and space filling... well sorry, I think he had a point...  why even bother with that... it's easier and the same exact subject to ask ourselves how we know induction is true... it seems true, it works fabulously well, practically, to assume it's true (you get math and lots of formal logic of use)... it's a bit unsatisfactory to say "we don't know that's true"... that I know an apple can be nutritious more than I know that induction is true.  Or even deduction.
• I'm sorry, I don't feel I am competant to defend penrose's argument against yours... I think penrose makes his point well... which means if I try, I'll probably just be trying to show you how you've misunderstood the issue.   If you want me to try to explain his explanation, ok, but I couldn't do it better than him.  I could run through it and tell you what I think it means. While I don't believe in all of this, if you are familiar with complexity theory, you'll see all he's talking about is especialy NP-hard type things, that computers suck at but humans HAVE been able to formalize.
• You reject Penrose but all he is saying is that human, heuristic methods, work better than analytic ones, even in finite games like chess where one might think analytics was not only sufficient but superior.
•  one way to talk about this is to ask... can an assumption be made computationally?  it would seem not.  it's posited, and from a logical point of view it's out of thin air.  But really it's out of something... I'd say trial and error of evolution as a default, Penrose says, out of the platonic realm which is accessible in quantum computation.
• (16:11) it sounds like you say you don't think there is some way to FOUND the axioms.  So you think they are without foundation.  They are held by... faith?!??!  please no. Right, if just defining as true is ok, then why can't I define anything I want to be true?  
• note: trial and error answers this... you assume as true and if it fails a trial, you say, nope.
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