Does inductive reasoning lead to knowledge?
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From: DanB on 8 Jan 2010 10:46 PD wrote: > On Jan 7, 11:29 pm, Michael Gordge<mikegor...(a)xtra.co.nz> wrote: > I'm just trying to get a yes or no answer out of you. You seem to be > challenged by simplicity. His forte is 'ewe' baby talk. Don't expect more.
From: Marshall on 8 Jan 2010 10:47 On Jan 8, 12:52 am, Les Cargill <lcargil...(a)comcast.net> wrote: > 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. I dunno; bullshit is pretty rich to begin with. > > 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. What its initial conception was has no bearing on what its actual nature is. > > 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. I'm not sure what you're saying here--perhaps you're saying you still think math is like Monopoly? What about this statement: "The history of the theories of physics 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. It requires an example to exists to be true. And what is the nature of that existence? Did it spring into being when man first began to think about the problem? > > 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. This does not address what I said in any way. > 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. You are confusing the matter of arbitrary choices with necessary physical realities. I tried to head off this confusion when I mentioned math notation, but apparently it didn't work. You mention language. Yes, the sequences of phonemes that make up the words of English are arbitrary, and a human construction, as is English syntax. Does it then follow that Chomsky's hierarchy was something that Chomsky made up, in the same way that the inventor(s) of Klingon made it up? This hypothetical math with no bearing on physical reality, it has axioms, yes? Do these axioms satisfy the theorems of proof theory? Can the axiomatic method be escaped by fiat? Or again I ask: if math is just a game, then what basis is there for claiming anything like "correctness" for any particular mathematical statement? Games do not have that property. Marshall
From: DanB on 8 Jan 2010 11:19 Marshall wrote: > > Or again I ask: if math is just a game, then > what basis is there for claiming anything > like "correctness" for any particular mathematical > statement? Axioms that are 'accepted' as truth.
From: chazwin on 8 Jan 2010 11:20 On Jan 8, 2:48 pm, jmfbahciv <jmfbahciv(a)aol> wrote: > J. Clarke wrote: > > jmfbahciv wrote: > >> J. Clarke wrote: > >>> jmfbahciv wrote: > >> <snip> > >>> FWIW, I think that everyone interested in this topic might want to > >>> read some Hume and some Popper--they both had goes at the question > >>> of the validity and utility of inductive reasoning, and Popper I > >>> understand discusses it specifically in the context of the scentific > >>> method. I don't know their work beyond that so can't suggest any > >>> readings--they're on my list but there's a lot in front of them. > > >> Popper is on my list. I'm not so sure about Hume since I've noticed > >> that it's the name used in their name-dropping to cause me to worship > >> the ground they trod on. I'm still trying to understand politics; > >> it doesn't help that I've been allergic to the subject all my life > >> :-). > > >> These people don't name-drop Popper as often. Do you have any > >> idea why this happens? > > > Not really. Maybe it's that Hume is more famous. I doubt that Hume is more famous than Popper, but Hume is a long neglected philosopher that is having a bit of a revival. ~Part of Hume's problem is that he was eclipsed by Kant. Kant threw a bucket of terminology at problems set out by Hume but this terminological warfare is looking more and more silly as the Romantic Nineteenth Century which promoted it is being left behind. Kant is no substitute for a good dose of Humean skepticism. > > Or this year's PC fad of the congenscenti is Hume. > > /BAH
From: jbriggs444 on 8 Jan 2010 11:20
On Jan 7, 2:54 pm, dorayme <doraymeRidT...(a)optusnet.com.au> wrote: > In article > <762f63fb-184d-4547-9a88-c13b015e9...(a)k17g2000yqh.googlegroups.com>, > > PD <thedraperfam...(a)gmail.com> wrote: > > On Jan 6, 11:52 pm, Patricia Aldoraz <patricia.aldo...(a)gmail.com> > > wrote: > > > On Jan 6, 9:38 am, John Stafford <n...(a)droffats.net> wrote: > > > > > Methinks PD is a mathematician in which axiomatic certainty can occur. > > > > Axioms do not reside in mathematicians, they reside in systems. > > > What do you mean by a "system"? You mean like an algebraic system or > > in a car engine? > > Oh, you are actually coming back to do more than abuse me personally! > What happened? Did you have a religious experience? > > The matter is quite trivial and it was a light remark of mine in > response to a stupid looking (perhaps grammatically stupid) sentence > from Stafford. When philosophers talk about axioms it is generally to > indicate statements that have a special status, often ones that generate > a system or tree of consequences, the statements themselves not being a > consequence of anything else in the system. It just seemed odd to me to > think that axioms were somehow in mathematicians themselves. [I'm not really trying to argue with you here, quite. More, making a pedant point] Be careful. Your summarization of the philosophical view does not appear to be coherent as written. As I read that summary, it contains two claims: 1. Axioms have a system or a tree of consequences. Well, that's certainly not an attribute unique to axioms. Lots of statements have consequences. Further, suppose that there is _any_ statement that has no consequences. There is nothing preventing us from positing that statement as an axiom. On the other hand, suppose that If there is no such thing as a statement without consequences. Then (1) is a pretty silly defining characteristic, having no exemplars. 2. Axioms are not themselves consequences of anything. (More formally, they are not provable in the formal system that remains after their removal as axioms). It is not forbidden to have a formal system defined so that certain of the axioms are redundant in the sense that they follow from the other axioms. It seems clear that one could quite easily cook up a formal system in which _all_ of the axioms are redundant. It is not always clear whether axioms are redundant in this sense. After all, people tried for quite a long time to prove the parallel postulate before finally demonstrating that it is independent of the other axioms of euclidean geometry. Your original point is taken. Regardless of any of this, an axiom is rarely discovered written on a slip of paper found within a mathematician. |