Does inductive reasoning lead to knowledge?
in [Logic]
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From: Marshall on 8 Jan 2010 02:43 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. 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. 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. 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. Let us ask a question about the rules of Monopoly: is it correct for the rules to have $200 be the amount to collect when you pass Go? The question is absurd, because Monopoly *is* a game, an intellectual exercise, and something that bears only coincidental resemblance to actual real estate transactions. We can take any arbitrary rule of Monopoly and change it freely, and there is no basis to label the change correct or incorrect. Let us ask a question about mathematics: is it correct to say that addition is commutative? The answer is: yes, it is correct. If math were merely a game, an intellectual exercise, then there were no basis for calling any part of it correct or not. Marshall
From: dorayme on 8 Jan 2010 02:47 In article <hi6gu1$b7n$3(a)news.eternal-september.org>, Les Cargill <lcargill99(a)comcast.net> wrote: > Marshall wrote: > > On Jan 7, 6:15 pm, "J. Clarke" <jclarke.use...(a)cox.net> wrote: > >>>> In math, which is really a logic game, the > >>>> axioms don't necessarily have any basis in the physical universe, > >>> It is an open question whether it is not *just* a logic game. There > >>> are semantics. And it is not clear what "having a basis in the > >>> physical universe" really means. > >> No, it is not an open question. Mathematics is a game, an intellectual > >> exercise, any relation that it bears to practical reality is purely > >> coincidental. > > > > That's bullshit. > > > > > > Marshall > > > Not really. > Really deep Les! Stafford would be so proud of your succinctness. You two share a desk in the basketweaving class? -- dorayme
From: J. Clarke on 8 Jan 2010 03:20 dorayme wrote: > In article <hi6gu1$b7n$3(a)news.eternal-september.org>, > Les Cargill <lcargill99(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. >>> >>> >>> Marshall >> >> >> Not really. >> > > > Really deep Les! Stafford would be so proud of your succinctness. You > two share a desk in the basketweaving class? Funny how you consider anybody who actually knows anything about math to be "basketweaving".
From: Errol on 8 Jan 2010 03:49 On Jan 7, 6:11 pm, jbriggs444 <jbriggs...(a)gmail.com> wrote: > On Jan 7, 7:43 am, Errol <vs.er...(a)gmail.com> wrote: > > > On Jan 7, 12:23 pm, Michael Gordge <mikegor...(a)xtra.co.nz> wrote: > > > Seeing that axiomatic means "self evident', an axiomatic certainty is > > one that you do not have to check up because you already know what the > > answer will be. > > To my mind, that's a pretty childish notion of "axiomatic". It's the > one I was taught in grade school. It's the one I was forced to > unlearn in order to understand formal systems in mathematics. > > In the context of physics it's still a bit childish. You don't stop > checking an conjecture because you _know_ what the answer will be. > You stop checking when it's more work to check it again than the > resulting increase (or decrease!) in certainty is likely to be worth. > > Arguably, that works out to just about the same thing. At some point > your confidence in a conjecture is so high that you simplify things by > treating it as if it were absolute fact. But... Next thing you know, > your instruments get better, somebody checks again and darned if your > certainty wasn't misplaced. > > Can you give an example of something "self evident". > > Is the parallel postulate "self-evident". > Is the axiom of choice "self-evident". > Is the negation of the axiom of choice "self-evident". > > Are any of these three things "true"? > > > I can say "Any 5 digit positive integer starting with 9 will always be > > greater than any 5 digit positive integer starting with 7." > > > That is an axiomatic certainty, because I do have to play around with > > my calculator to check whether it is true or not. I know it is true. > > I would call it a deductive certainty. "Theorem" for short. It can > be deduced (aka proven) in a formal system within which certain > simpler and more general things are taken as axiomatic. > > In order to even make the statement in question you're pulling in a > signicant bit of well understood mathematics. The phrase "5 digit > positive integer starting with 9" pulls in the notion of "integer". > And the term "greater than" pulls in the notion of an order and of a > default ordering relation for the integers. You're also apparently > implying and pulling in the notion of simple decimal notation and a > big-endian digit ordering convention. Well over half the work in > proving this "axiomatic certainty" would likely be involved in filling > in defaults and specifying an environment within which you can > formally phrase it so that it is amenable to proof. > > The standard mathematical notion of integer is often formalized using > systems which are equiconsistent with other systems within which > various of the underlying axioms are negated. In particular, if the > axiom of infinity is negated it follows that there no such set as "the > positive integers" from which to select your "5 digit positive > integers starting with 9" and your supposed "axiomatic certainty" is > ill-formed on its face. > > [That's overstating things a bit. I do regard your statement as being > both meaningful and "true". It's provable in ZF. And with some > slight rephrasing, it's provable in ZF even with the axiom of infinity > negated. Just because the set of all positive integers does not exist > as a set would not mean that there isn't a set of just the "5 digit > integers". Indeed ZF-I+~I is able to prove the existence of such such > a set] > > If your notion of axiomatic certainty includes "follows from the > axioms" then we're good. The above claim is an axiomatic certainty. > Not because it's obvious. Not because you know it to be true without > looking. But because it follows from the axioms. Thank you for your well considered reply. I understand that I used a simplified usage of 'axiomatic' but I believed it relevant to the context and the spirit of the message and poster I was replying to. I also agree that it follows from the axioms rather than that I know it to be true. I took the liberty of bridging them to illustrate the point to the person I was replying to.
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 |