> >> >> Are any of these things constructed theoremetrically from > >> >> demonstrably true assumptions? > >> > > >> >What does a "constructed theoremetrically" mean in English? > >> > >> Probably exactly what it says. > > > >As I do not find any definition of "theoremetrically" anywhere, I assume > >that it means nothing at all. > > Well, hell, Virgil, I just coined the term to describe the logical > metric associated with the demonstration of theorems more artfully > with considerably fewer words.
In other words, I assumed correctly. > > >> >> > > >> >> > I don't see any > >> >> >sense in asserting that the axioms of Euclidean geometry or of group > >> >> >theory are true or wrong. > >> >> > >> >> But you see plenty of sense in assuming the truth of what you can't > >> >> demonstrate true or false? > >> > > >> >If one cannot assume anything without proofs or demonstrations, one > >> >can never get started at all. > >> > >> Getting started is way different from getting finished. What you're > >> talking about is getting started; what I'm talking about is getting > >> finished by demonstrating the truth of initial axiomatic assumptions. > > > >How does one come up with the ur-truths on which to found one's initial > >axiomatic assumptions? > > Well here if the ur-truths are actually demonstrably true they > represent more than mere assumptions. But getting from mere > assumptions to ur-truths is an interative regressive process of > determining what and how things can be proven true other than by > consistency with aximatic assumptions which of course don't prove them > true at all absolutely but only relatively.
In other words, and more briefly, there are no absolute ur-truths in mathematics.
> However apart from this I have to suggest it > is possible to stumble on an initial assumption which must be true > because its tautological alternatives must be false and its > tautological alternatives must be false on thoroughly mechanical > grounds.
In what sense are you using the word "tautological? none of the meanings I can find in any dictionaries make sense in your context. > > >And I know of no way to do that. > > That's true. > > >I have always regarded something for nothing as pie in the sky. > > > >> Conventional mathematical methodologies just get started and then > >> assume everything is honky dorry for no better reason than they got > >> started. > > > >One has to start somewhere to get anywhere. You seem to think it > >possible to get something from nothing. I do not. > >> > >> >> > And this does not mean that axioms of this kind are arbitrary > >> >> > assumptions. There seem to be very few axiom systems that lead to > >> >> > interesting theories. > >> >> > >> >> Arbitrary or not they're still assumptions. > >> > > >> > > >> >Without any of which one has nothing to start with. > >> > >> Once again getting started is not the same as getting finished. > > > >When you claim to be able to justify an initial set of axioms as somehow > >being "true", I have to ask "How do you know they are true?" > > Because mechanically possible alternatives are false.
For example? Outside of the trivially tautologous statements of formal logic like "if P then P" or "P or not P", I was not aware that there were any such axioms possible. > > >Where does your truth come from? > > The tautology.
Which one? > > >> >Unless, like Jefferson, et al, you take certain truths to be self > >> >evident. > >> > >> Exactly my point. Jefferson had his self evident truth and every > >> dictator in history theirs. Hard to tell the chaff from the wheat > >> without a scorecard. > >> > >> >But then, again like Jefferson, et al, you must state what those self > >> >evident truths are, so that we can know what we start with. > >> > > >> >I find that in matters mathematical there are no self evident truths. > >> >One must assume something in order to get anywhere. > >> > >> And if what one assumes is not true? > > > >That is precisely my point! By what criteria can one determine that? > > Self contradiction.
Do you mean that every statement that is not self-contradictory is true? That would mean that two statements which are not self contradictory but are mutually so, could both be "true". > > >What I say that there are no absolute a priori criteria. > > Well if you're talking prior to any experience at all I'd agree. But > the sense in which I use "a priori" is strictly "logically consistent > with and demonstrable of".
But there is nothing at the very beginning to be logically consistent with or demonstrable of. > > >> This whole discussion is about > >> whether contemporary mathematical set theory is true or false not > >> whether it's assumed to be true. > > > >And I am asking once more, how one tells the difference. > > And the answer is that tautological alternatives to what's true are > self contradictory.
There is that ambiguous "tautological alternatives" again which as far as I can see means nothing. So I find your response non-responsive.
> There is nothing in set theory which shows that > tautological alternatives are self contradictory
There is nothing in set theory or logic which explains what "tautological alternatives" even are, so it is not surprising one cannot show much of anything about them.
> no "non set theory" > which demonstrates alternatives to set theory are impossible.
Some versions of category theory are a sort of non-set theory, and there are those who expect eventually to develop viable alternatives to set theory along lines suggested by abstract category theory.