Does inductive reasoning lead to knowledge?
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From: Les Cargill on 8 Jan 2010 03:52 Marshall wrote: > On Jan 7, 9:47 pm, Les Cargill <lcargil...(a)comcast.net> wrote: >> Marshall wrote: >>> On Jan 7, 6:15 pm, "J. Clarke" <jclarke.use...(a)cox.net> wrote: >>>>>> In math, which is really a logic game, the >>>>>> axioms don't necessarily have any basis in the physical universe, >>>>> It is an open question whether it is not *just* a logic game. There >>>>> are semantics. And it is not clear what "having a basis in the >>>>> physical universe" really means. >>>> No, it is not an open question. Mathematics is a game, an intellectual >>>> exercise, any relation that it bears to practical reality is purely >>>> coincidental. >>> That's bullshit. >> Not really. > > Yes, it really, truly, deeply is. That's why the claim isn't merely > wrong, but bullshit. > I think the actual story is richer than that. > Axioms are not arbitrary; not the ones anyone pays any > attention to. Sure the choice of what glyph to use for > addition and what glyph to use for equality is an > arbitrary human invention, but that's notation, not > mathematics. > > Mathematics begins with counting up fingers and > rocks and so forth, which of course is a physical > reality, and with the observations of physical laws > that counting must obey. From the designation of > a starting point, be it zero or one, and the observation > of the fact that we can count one more past anywhere > we have counted so far, we get the successor function. > The progression from successor to addition, multiplication, > etc. is not an arbitrary human choice but an inevitable > consequence of the physical universe. The fact that > addition and multiplication are commutative and > associative is not an "intellectual exercise" but > a physical law. > But take hyperbolic geometry - It too *happens* to be physically useful, but it wasn't initially conceived in that way, SFAIK. > Do you imagine someone just woke up one morning > and said "hey, I'm going to write down some > arbitrary game rules and call it group theory?" No, > group theory arose after decades, centuries of > observing commonalities among various formalisms. > Won't argue there. The history of constructs of mathematics vary. I still take the thesis to be principally anthropological - how it was done varies. > The resemblance that math bears to physics is not > coincidence, but rather an aspect of the Church-Turing > thesis. What we can compute, what we can calculate, > is exactly determined by what manipulations we can make > to forms; those manipulations are exactly those which > the laws of physics allows. We say that numbers are > abstract, but what that means is precisely that they > are a simplified, generalized form of what happens > in the physical world. It doesn't mean that they are > somehow unmoored from physical reality. That is > ridiculous. > For one, computability is congruent with but not exactly equivalent to completeness. When we calculate Chaitin's Omega, we arbitrarily set a time limit for "never terminates". And Godel Incompleteness doesn't require an example to be true - the diagonalization simply proves they exist. > Let us ask a question about the rules of Monopoly: > is it correct for the rules to have $200 be the amount > to collect when you pass Go? The question is > absurd, because Monopoly *is* a game, an intellectual > exercise, and something that bears only coincidental > resemblance to actual real estate transactions. We > can take any arbitrary rule of Monopoly and change > it freely, and there is no basis to label the change > correct or incorrect. > > Let us ask a question about mathematics: is it > correct to say that addition is commutative? The > answer is: yes, it is correct. If math were merely > a game, an intellectual exercise, then there were > no basis for calling any part of it correct or not. > > But here are most certainly binary operators that are not commutative, much as inversion of the fifth postulate brings about hyperbolic geometry. What I see is that a mathematics which has no bearing on physical reality *at all* is entirely possible. It's just a matter of someone doing it. I mean, if people can formalize Klingon, anything is possible. > Marshall -- Les Cargill
From: Les Cargill on 8 Jan 2010 04:03 J. Clarke wrote: > dorayme wrote: >> In article <hi6e7n0r1(a)news7.newsguy.com>, >> "J. Clarke" <jclarke.usenet(a)cox.net> wrote: >> >>> You do some grad work in mathematics and physics and get back to us. >>> As things stand you need more background than can be provided via >>> USENET to get to where you understand the issues under discussion. >> >> You are a rude and insolent man with vague half-baked ideas > > No, I'm a rude and insolent man who has taken those graduate courses and > have some idea how mathematics really works. > Actually, it's a pretty difficult question, and it's always possible I could read something tomorrow that would change my mind*. There's also an anthropic component to the thing - any totally abstract mathematics is much less likely to be seen by me than something I'd have an actual use for. *but it would have to be one whale of a "something". <snip> -- Les Cargill
From: Errol on 8 Jan 2010 04:58 On Jan 8, 7:29 am, Michael Gordge <mikegor...(a)xtra.co.nz> wrote: > On Jan 8, 6:24 am, PD <thedraperfam...(a)gmail.com> wrote: > > > > > I know what you're asking. I asked you whether Euclid's fifth > > postulate is a postulate or not. > > What are the premises? Are the lines converging or they parallel? They > cant be both. Euclid never postulated the converse case of parallel lines, within his fifth postulate, only diverging lines. He said, 'If a straight line crossing two straight lines makes the interior angles on the same side less than two right angles, the two straight lines, if extended indefinitely, meet on that side on which are the angles less than the two right angles.' The converse of the fifth postulate is where the interior angles add up to 180 (parallel lines) > > You do realize there are no lines in reality, they are mind dependent > and only matter to man's survival, when a problem of matter / survival > is solved with them. > > MG Ok! Now you can stop dodging the question. I want to read what PD is getting at. Say yes or no
From: J. Clarke on 8 Jan 2010 04:46 Les Cargill wrote: > J. Clarke wrote: >> dorayme wrote: >>> In article <hi6e7n0r1(a)news7.newsguy.com>, >>> "J. Clarke" <jclarke.usenet(a)cox.net> wrote: >>> >>>> You do some grad work in mathematics and physics and get back to >>>> us. As things stand you need more background than can be provided >>>> via USENET to get to where you understand the issues under >>>> discussion. >>> >>> You are a rude and insolent man with vague half-baked ideas >> >> No, I'm a rude and insolent man who has taken those graduate courses >> and have some idea how mathematics really works. >> > > Actually, it's a pretty difficult question, and it's always possible I > could read something tomorrow that would change my mind*. There's also > an anthropic component to the thing - any totally abstract mathematics > is much less likely to be seen by me than something I'd have an > actual use for. > > *but it would have to be one whale of a "something". Whether inductive reasoning leads to knowledge is a good and difficult question, but that's not the point I was addressing. I'm seeing "axiom" tossed around here by people who clearly don't understand how the term is used in mathematics.
From: chazwin on 8 Jan 2010 05:29
On Dec 27 2009, 11:19 pm, Michael Gordge <mikegor...(a)xtra.co.nz> wrote: > On Dec 28, 1:40 am, chazwin <chazwy...(a)yahoo.com> wrote: > > > though the universe is > > deterministic (which I have always maintained), > > You have always denied free will? You have always denied that people > have or can make a choice between good and bad right and wrong? > > MG Free-will is an illusion. We are all bound by nature. We are not outside of cause and effect. |