Does inductive reasoning lead to knowledge?
in [Logic]
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From: chazwin on 9 Jan 2010 06:35 On Jan 9, 12:19 am, 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. Er--- yes they are. They have to be. Who else thinks them, writes them and uses them? God? > > Suppose I want to investigate two-element algebras. > How many unary functions are possible? I claim that > there is only a single correct answer to that question: > four. This can be established by simple case analysis. > > If axioms are what it's all about, please demonstrate > so. Show me how choosing some axioms that are > 'accepted' as truth can make the right answer come > out three. Ideally you will also show how accepting > those axioms also makes case analysis come up with > the answer three. Axioms are statements that are accepted as truth, all conclusion formed from axioms are also accepted as true. none of this relies on sensory evidence and as such is not real. There are no integers, straight lines, or spheres in nature. These are all constructs of the human imagination. > > Marshall
From: chazwin on 9 Jan 2010 06:42 On Jan 9, 3:53 am, Patricia Aldoraz <patricia.aldo...(a)gmail.com> wrote: > On Jan 8, 9:27 pm, chazwin <chazwy...(a)yahoo.com> wrote: > > > Hume pointed out this problem in the early 18th C, but despite these > > difficulties the last 200 years+ has used inductive methods to design > > all the wonders of technology and we have reached to moon. > > If "inductive methods" merely means "whatever scientists do when they > do the non-deductive bits of their work" naturally it is true. But > what use is this. That is not any answer to the problem of induction. > > > Whilst we > > have to always be aware that our inductive knowledge is a question of > > probabilistic truth. > > Nor is this. The problem of induction is not a search for 100% > probability. It is a search for anything over 50%. Until this is > understood, it is impossible to understand what so troubled Hume and > modern versions of the problem Hume's own answer to the problem of induction was mitigated skepticism. Which means that although law-like statements built from induction are not 100% certain, if they work they it is possible to have a pragmatic belief in them until they are replaced by something more accurate or probable. Thus Newtonian physics woks in most cases. But Hume would have approved of the change in science offered by Einstein.
From: chazwin on 9 Jan 2010 06:42 On Jan 9, 5:26 am, "J. Clarke" <jclarke.use...(a)cox.net> wrote: > dorayme wrote: > > In article <cWI1n.2906$%P5.1...(a)newsfe21.iad>, DanB <a...(a)some.net> > > wrote: > > >> Marshall wrote: > > >>> if math is just a game, then > >>> what basis is there for claiming anything > >>> like "correctness" for any particular mathematical > >>> statement? > > >> Axioms that are 'accepted' as truth. > > > Why the qualifying quotes? > > > The point is that in a game, truth does not figure prominently. In > > maths and physics, truth is a bigish player. > > In math "truth" is whatever you develop logically from your axioms. Some > sets of axioms lead to interesting games, some don't. And some sets lead to > games that are actually useful. But none of it is real.
From: chazwin on 9 Jan 2010 06:46 On Jan 9, 6:51 am, Marshall <marshall.spi...(a)gmail.com> wrote: > On Jan 8, 9:26 pm, "J. Clarke" <jclarke.use...(a)cox.net> wrote: > > > > > > > dorayme wrote: > > > In article <cWI1n.2906$%P5.1...(a)newsfe21.iad>, DanB wrote: > > >> Marshall wrote: > > > >>> 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. > > > > Why the qualifying quotes? > > > > The point is that in a game, truth does not figure prominently. In > > > maths and physics, truth is a bigish player. > > > In math "truth" is whatever you develop logically from your axioms. > > Gödel's first incompleteness theorem proved that idea false. > > Marshall No, you misunderstand. It is a complete affirmation of this position. The incompleteness is an expression of the unknown elements which lie outside the axioms and the determinate conclusions that are drawn from those axioms.
From: chazwin on 9 Jan 2010 08:39
On Jan 9, 7:46 am, "Androcles" <Headmas...(a)Hogwarts.physics_r> wrote: > "Marshall" <marshall.spi...(a)gmail.com> wrote in message > > news:43e98d31-7748-4755-b5dc-e2389bbc4c0b(a)m26g2000yqb.googlegroups.com... > On Jan 8, 10:05 pm, DanB <a...(a)some.net> wrote: > > > Marshall wrote: > > > > Show me how choosing some axioms that are > > > 'accepted' as truth can make the right answer come > > > out three. Ideally you will also show how accepting > > > those axioms also makes case analysis come up with > > > the answer three. > > > Accepting an axiom determines results. > > I accepted no axioms, yet was able to determine the > answer "four" to my question of how many unary functions > are possible for a two-element algebra. This is > a simple counterexample to you assertion. > > Also note that G del's first incompleteness theorem proved > the existence of formulas that are true, yet do not > follow from any axioms. > > > Are you getting this? > > I'm getting that you completely failed my simple challenge > to show an arbitrary result following from some axioms, > nor did you apparently even understand it. I'm getting > that you have nothing to offer in the way of an argument > other than "look it up." > > Marshall > ============================================ > a = b (given) > a^2 = ab (multiply both sides by a) > a^2 - b^2 = ab-b^2 (subtract b^2 from both sides) > (a+b)(a-b) = b(a-b) (factorize) > a+b = b (divide both sides by a-b) > b+b = b (substitute b for a) > 2b = b (addition of b +b ) > 2 = 1 (divide by b) > Hence a = 2b since a = b. > 2b+b = b (substitute 2b for a) > 3 = 1 > > The axioms were followed, the result was arbitrary. > An arbitrary result followed from some axiom. > What we have is a well-known exception for which > the axiom of division fails. Interesting. |