Does inductive reasoning lead to knowledge?
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From: PD on 12 Jan 2010 10:49 On Jan 11, 8:45 pm, Marshall <marshall.spi...(a)gmail.com> wrote: > On Jan 11, 3:05 pm, PD <thedraperfam...(a)gmail.com> wrote: > > > > > On Jan 11, 8:53 am, Marshall <marshall.spi...(a)gmail.com> wrote: > > > On Jan 11, 6:47 am, PD <thedraperfam...(a)gmail.com> wrote: > > > > On Jan 8, 6:19 pm, Marshall <marshall.spi...(a)gmail.com> wrote: > > > > > On Jan 8, 8:19 am, DanB <a...(a)some.net> wrote: > > > > > > 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. > > > > > > That's supposed to be the basis? Just that noun > > > > > phrase by itself? > > > > > > And anyway, axioms themselves also come from > > > > > somewhere. They are not just arbitrary creations > > > > > of man. > > > > > Why yes, yes, they are. > > > > Your post almost demands the response "Oh no > > > they are not." But that wouldn't be much use, > > > would it? > > > > Do you feel the same way about the natural numbers? > > > Are they an arbitrary creation of man? > > > > Do you have any argument you'd care to supply to > > > justify your position? > > > It's really a matter of definition more than anything, as far as I > > know. > > I guess what you mean by that is, it depends on what the > definition of "arbitrary creation of man" is? Please clarify. > > > The natural numbers are a concept, but I don't think they are an > > axiom. On the other hand, there is a set of axioms (the Peano axioms) > > that are used to rigorously define them. Notice that one of the Peano > > axioms is precisely the one that ensures the *mathematical* sense of > > induction will work at all. > > I wasn't suggesting that the natural numbers are an axiom; > of course they are not. I was asking if you think the natural numbers > are an arbitrary creation of man. It is harder to engage with > your position when you don't clarify what it is. Ah, well, in that case, I'd say that yes, natural numbers are an arbitrary creation of man. However, many of the concepts that man devises are in some ways reflections of reality, because those concepts are aimed to categorize reality or describe it. I'd be hard pressed to point to nature, though, and say "Look, here is the boundary in nature between natural numbers, rational numbers, irrational numbers, and complex numbers." > > It has been proposed on this thread that math is just a game > with no significance or utility, except by coincidence (this is > bullshit.) It has been suggested that axioms are the source > of truth in math, by fiat; Goedel's first incompleteness theorem > proves this false. > > Here the question is whether axioms come from man or > from nature. It seems to me that (relative to the axioms > of the natural numbers at least) this question depends > on whether the natural numbers themselves are man-made > or not. If they are not, then the axioms that they obey > are not, since the axioms derive from the numbers and > the operations on them. They are man-made, perhaps, but they do reflect something in the reality that surrounds us. > > > I'll reiterate one of the examples I've cited in this thread: Euclid's > > fifth postulate. Now, either that is an arbitrary creation of man or > > it has some undeniable objective truth. > > I do not accept this dichotomy; it is much too impoverished to > capture what is going on. > > > But if the latter, then there > > is a serious problem with Riemannian geometry, which disbands the > > validity of the fifth postulate. Since both of those systems seem to > > have equally good applications not only in the mind but to real life, > > it seems difficult to say that they are both objectively true. On the > > other hand, if they are arbitrary creations of man, then it makes > > sense how the geometries that stem from them both have value. > > Since I don't accept you dichotomy, I don't accept this argument; > it depends on the earlier dichotomy. > > All that is to be gained from a discussion of Euclid's fifth is > to observe that, if a consistent theory doesn't decide a > formula one way or the other, then we may add that > formula or its negation and get a new, consistent theory. > This tells us nothing about where axioms come from. > > Marshall
From: PD on 12 Jan 2010 10:50 On Jan 11, 9:20 pm, Patricia Aldoraz <patricia.aldo...(a)gmail.com> wrote: > On Jan 12, 10:05 am, PD <thedraperfam...(a)gmail.com> wrote: > > > The natural numbers are a concept, but I don't think they are an > > axiom. > > An axiom says something, the natural numbers simply exist. > > > I'll reiterate one of the examples I've cited in this thread: Euclid's > > fifth postulate. Now, either that is an arbitrary creation of man or > > it has some undeniable objective truth. > > These are not the only possibilities. Elaborate, please.
From: PD on 12 Jan 2010 10:51 On Jan 12, 12:58 am, Michael Gordge <mikegor...(a)xtra.co.nz> wrote: > On Jan 11, 11:45 pm, PD <thedraperfam...(a)gmail.com> wrote: > > > I didn't see a yes or no anywhere in your answer, > > Thats encouraging, all you need do now is to expand non-contradictory > identification into your own ideas, e.g. lines that are parallel do > not converge, intersect and or diverge no matter how far their > journey. Still lacking a "yes" or "no" answer to a yes or no question. Is this a form of passive-aggressive behavior that is comfortable for you?
From: J. Clarke on 12 Jan 2010 10:30 jmfbahciv wrote: > Nam Nguyen wrote: >> Marshall wrote: >>> >>> It has been proposed on this thread that math is just a game >>> with no significance or utility, except by coincidence (this is >>> bullshit.) >> >> Mathematics is a game of the mind. > > Which can be written down on paper. > >> Whether or not that has any utility >> or significance, or that is by coincidence, or that is "bullshit" >> doesn't matter, to the fact that it's just a game. >> > Have you done any cost analysis lately? Or materials design? Or > built a bridge? Or figured out the load of the roof on your house? That sort of thing uses one branch of mathematics that coincidentally has real-world utility. This is a small subset of the totality of mathematics.
From: dorayme on 12 Jan 2010 14:50
In article <sCV2n.3785$ZB2.3547(a)newsfe13.iad>, Nam Nguyen <namducnguyen(a)shaw.ca> wrote: > dorayme wrote: > > Well, I don't know how you would work it so that one could see that > > there are no primes between 1 and 21 and such things without changing > > the *meanings* of the words used. > > Keep the "meanings"; just change the frameworks, axioms, models, or a > combination of. > You mean like "A pig can fly" can be changed into a falsity by a Martian with different "frameworks, axioms, models", but who means exactly what we mean by "pig", and "fly" and "is". What an intriguing suggestion! > > > >>> That does not mean it is > >>> just a game nor that it is a coincidence that mathematics is useful to > >>> us. > >> It's a misconception that games in general have to be useless to human > >> beings. > > > > Who is falling for that misconception. Neither of us! But the question > > that is relevant is what sort of use. Mere use because it reduces stress > > levels is not much relevant! > > The point is mathematics is still just a game, even though it might be a > useful one by no coincidence. The point is that it is not always just a game then. -- dorayme |