Does inductive reasoning lead to knowledge?
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From: jmfbahciv on 9 Jan 2010 10:42 chazwin wrote: > 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. No, it does not. Do you really think that science is based on a popular vote? <snip> /BAH
From: J. Clarke on 9 Jan 2010 10:37 chazwin wrote: > On Jan 9, 5:26 am, "J. Clarke" <jclarke.use...(a)cox.net> wrote: >> dorayme wrote: >>> In article <cWI1n.2906$%P5.1...(a)newsfe21.iad>, DanB <a...(a)some.net> >>> wrote: >> >>>> Marshall wrote: >> >>>>> if math is just a game, then >>>>> what basis is there for claiming anything >>>>> like "correctness" for any particular mathematical >>>>> statement? >> >>>> Axioms that are 'accepted' as truth. >> >>> Why the qualifying quotes? >> >>> The point is that in a game, truth does not figure prominently. In >>> maths and physics, truth is a bigish player. >> >> In math "truth" is whatever you develop logically from your axioms. >> Some sets of axioms lead to interesting games, some don't. And some >> sets lead to games that are actually useful. > > But none of it is real. That depends on how you define "real" and at what level you're looking at it. If you mean that none of it is an exact representation of the physical universe then I'm inclined to agree with you.
From: DanB on 9 Jan 2010 11:08 jmfbahciv wrote: > DanB wrote: >> Marshall wrote: >>> >>> Or again I ask: if math is just a game, then >>> what basis is there for claiming anything >>> like "correctness" for any particular mathematical >>> statement? >> >> Axioms that are 'accepted' as truth. > > No. Axioms lay a premise. then you can construct > geometries and algebras based on that premise. Yes, I just used different language, but said the same thing. After all a premise is a premise. Now I may have missed it but until now I had not seen this discussion move into the realm of algebras and geometries.
From: Marshall on 9 Jan 2010 11:44 On Jan 9, 3:46 am, chazwin <chazwy...(a)yahoo.com> wrote: > On Jan 9, 6:51 am, Marshall <marshall.spi...(a)gmail.com> wrote: > > > > > On Jan 8, 9:26 pm, "J. Clarke" <jclarke.use...(a)cox.net> wrote: > > > > dorayme wrote: > > > > In article <cWI1n.2906$%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. The first incompleteness theorem shows that in any consistent formal system that is at least as strong as ordinary arithmetic, there are formulas that are ***true*** but not provable from the axioms. Not "unknown elements" but true formulas. So, no, it is not the case that truth derives solely from axioms. Marshall
From: Marshall on 9 Jan 2010 11:45
On Jan 9, 5:39 am, chazwin <chazwy...(a)yahoo.com> wrote: > 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. How is that interesting? It's just an obfuscated divide-by-zero error. The claim that an "axiom of division fails" is silly. Marshall |