Does inductive reasoning lead to knowledge?
in [Logic]
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From: jmfbahciv on 7 Jan 2010 10:49 M Purcell wrote: > On Jan 6, 11:48 am, PD <thedraperfam...(a)gmail.com> wrote: >> On Jan 6, 10:42 am, M Purcell <sacsca...(a)aol.com> wrote: >> >> >> >> >> >>> On Jan 6, 7:47 am, PD <thedraperfam...(a)gmail.com> wrote: >>>> On Jan 5, 4:31 pm, M Purcell <sacsca...(a)aol.com> wrote: >>>>> This make so much more sense than just cussing and if you stopped >>>>> using the Kantian label you would sound almost sane. I believe PD's >>>>> point was that an axiomatic certainty does not exist in physics. >>>>> Although increasing entropy and the consistancy of the speed of light >>>>> come close, they must be verified and reverified by observation. In >>>>> science, every "certainty" is subject to repeated verification. >>>> Your examples are cases of INFERRED statements. They are indeed >>>> postulates, but they are far from certain. As you say, they are only >>>> provisionally accepted as long as their consequents are consistent >>>> with experimental observation. >>> Indeed, they are inferred from observations with the assumption our >>> observations may not be valid. >> Well, they are general rules that are inferred from a known set of >> examples which may not be complete. >> The best example here is the principle of relativity, which was one of >> Einstein's postulates. It was known to hold for Newtonian mechanics in >> an obvious way, but it was not obviously true for the laws of >> electrodynamics. Einstein asked the question, "But what if the >> principle of relativity IS true for ALL laws of physics? What would be >> the implications of that?" Thus the principle of relativity was an >> inferred general rule, which led to the deduction of testable >> consequences, which in fact turned out to match measurement, lending >> credence to the inferred postulate. > > Yes, testable conclusions can be deduced from observations. > >>> But this requirement of repeated >>> verification seems to indicate inductive reasoning. >>>> As another example, Einstein firmly believed in the principle of >>>> locality, which is why he had such great difficulty with quantum >>>> mechanics. Jon Bell codified that belief into a firm prediction as a >>>> means to test it, and Alain Aspect did the experimental test which >>>> showed that Einstein's beloved postulate, the principle of locality, >>>> was simply wrong. >>> Which goes to show that nobody is perfect and quantum mechanical >>> concepts such as wave-particle duality are very unintuitive. >> Well, I wouldn't say it quite that way. Quantum mechanics isn't all >> that unintuitive (at least to those who have practice with it). But >> Einstein just had a very strong hunch that the principle of locality >> was a good rule of nature --- he arrived at that conclusion by some >> process of induction. So Einstein could put his finger on where the >> conflict was between his inductions and quantum mechanics. It just >> turned out in this case that Einstein's induction was wrong. > > Wave-particle duality is not commonly experienced. Wrong. /BAH
From: jmfbahciv on 7 Jan 2010 10:51 Patricia Aldoraz 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. Oh, good grief. You don't even have high school math in your background. /BAH
From: PD on 7 Jan 2010 10:58 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?
From: jbriggs444 on 7 Jan 2010 11:11 On Jan 7, 7:43 am, Errol <vs.er...(a)gmail.com> wrote: > On Jan 7, 12:23 pm, Michael Gordge <mikegor...(a)xtra.co.nz> wrote: > > Seeing that axiomatic means "self evident', an axiomatic certainty is > one that you do not have to check up because you already know what the > answer will be. To my mind, that's a pretty childish notion of "axiomatic". It's the one I was taught in grade school. It's the one I was forced to unlearn in order to understand formal systems in mathematics. In the context of physics it's still a bit childish. You don't stop checking an conjecture because you _know_ what the answer will be. You stop checking when it's more work to check it again than the resulting increase (or decrease!) in certainty is likely to be worth. Arguably, that works out to just about the same thing. At some point your confidence in a conjecture is so high that you simplify things by treating it as if it were absolute fact. But... Next thing you know, your instruments get better, somebody checks again and darned if your certainty wasn't misplaced. Can you give an example of something "self evident". Is the parallel postulate "self-evident". Is the axiom of choice "self-evident". Is the negation of the axiom of choice "self-evident". Are any of these three things "true"? > I can say "Any 5 digit positive integer starting with 9 will always be > greater than any 5 digit positive integer starting with 7." > > That is an axiomatic certainty, because I do have to play around with > my calculator to check whether it is true or not. I know it is true. I would call it a deductive certainty. "Theorem" for short. It can be deduced (aka proven) in a formal system within which certain simpler and more general things are taken as axiomatic. In order to even make the statement in question you're pulling in a signicant bit of well understood mathematics. The phrase "5 digit positive integer starting with 9" pulls in the notion of "integer". And the term "greater than" pulls in the notion of an order and of a default ordering relation for the integers. You're also apparently implying and pulling in the notion of simple decimal notation and a big-endian digit ordering convention. Well over half the work in proving this "axiomatic certainty" would likely be involved in filling in defaults and specifying an environment within which you can formally phrase it so that it is amenable to proof. The standard mathematical notion of integer is often formalized using systems which are equiconsistent with other systems within which various of the underlying axioms are negated. In particular, if the axiom of infinity is negated it follows that there no such set as "the positive integers" from which to select your "5 digit positive integers starting with 9" and your supposed "axiomatic certainty" is ill-formed on its face. [That's overstating things a bit. I do regard your statement as being both meaningful and "true". It's provable in ZF. And with some slight rephrasing, it's provable in ZF even with the axiom of infinity negated. Just because the set of all positive integers does not exist as a set would not mean that there isn't a set of just the "5 digit integers". Indeed ZF-I+~I is able to prove the existence of such such a set] If your notion of axiomatic certainty includes "follows from the axioms" then we're good. The above claim is an axiomatic certainty. Not because it's obvious. Not because you know it to be true without looking. But because it follows from the axioms.
From: John Stafford on 7 Jan 2010 11:14
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. Axiom do not 'reside' anywhere, however the definition and application of axioms can be different in certain _domains_, and each domain can have different systematic methods and qualities. |