Does inductive reasoning lead to knowledge?
in [Logic]
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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
From: dorayme on 9 Jan 2010 16:48
In article <hi95ed$li4$4(a)news.eternal-september.org>, Les Cargill <lcargill99(a)comcast.net> wrote: > Patricia Aldoraz wrote: > > On Jan 9, 3:35 am, John Stafford <n...(a)droffats.net> wrote: > > > >> Text analysis is not trivial. Learn up. > > > > It is trivially related to this thread. > > It's also threadily related to the trivial :) > Good point! <g> (I wonder if Stafford will use my above phrase "Good point!" as data in his textual analysis. Along with the symbolic grin? Will he use the grin as a separate datum or will he use the combined phrase? I suggest separate would be more valuable. Does Patricia use the same symbol? Oooo! This must be terribly exciting for him. Has he solved The Desert Problem yet?) -- dorayme |