Does inductive reasoning lead to knowledge?
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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.
From: M Purcell on 7 Jan 2010 11:24 On Jan 7, 7:45 am, jmfbahciv <jmfbahciv(a)aol> wrote: > M Purcell wrote: > > On Jan 6, 5:24 am, jmfbahciv <jmfbahciv(a)aol> wrote: > >> M Purcell wrote: > >>> On Jan 5, 5:54 am, jmfbahciv <jmfbahciv(a)aol> wrote: > >>>> M Purcell wrote: > >>>>> On Jan 4, 6:27 am, jmfbahciv <jmfbahciv(a)aol> wrote: > >>>>>> M Purcell wrote: > >>>>>>> On Jan 3, 6:48 am, jmfbahciv <jmfbahciv(a)aol> wrote: > >>>>>>>> John Stafford wrote: > >>>>>>>> <snip --piggy-backing another post> > >>>>>>>> I got to the library and looked up that induction-reasoning > >>>>>>>> web site. I had planned to watch myself think while doing > >>>>>>>> the test. Didn't happen. I popped out the answer to each > >>>>>>>> without thinking. > >>>>>>> You believe thinking is a physical activity unnecessary for answering > >>>>>>> questions? > >>>>>> Are you really trying to be ignorant? > >>>>> Are you really trying to watch yourself think without success? > >>>> I am successful. I was paid very well to do this kind of thing. > >>> Paid by whom to do what kind of thing? > >> the company who hired me and to be able to watch how people > >> do things which included how they did their thinking when > >> solving their problems. Often, we hadn't shipped yet, so > >> I used myself as the guinea pig in anticipation of how > >> our customers would act and think. > > > And you assumed everybody thinks like you do? > > Of course not. I simply am very good at being able > to figure out how other people think. > > > > > > > > >>>>>>>> If you call the process for finding those solutions > >>>>>>>> inductive reasoning, then I have to conclude that > >>>>>>>> inductive reasoning is in the hardware. I would > >>>>>>>> not use the word reasoning at all for that kind > >>>>>>>> of brain processing. > >>>>>>> The generalization of this test to all inductive reasoning is > >>>>>>> inductive reasoning. Apparently you are still not thinking. > >>>>>> There is a huge difference between conscious thinking and > >>>>>> automated thinking. One plans your survival the other > >>>>>> ensures you survive to carry those plans. > >>>>> A difference between survival and survival? > >>>> In business, it's called short-term and long-term. You > >>>> appear to be limited to short-term only. > >>> How did you arrive at this conclusion? > >> Your inability to think long term is clear from the nonsense > >> you type. > > > What is your reasoning to support this conclusion? > > The way you don't understand what is written in this thread. What is written in this thread that I don't understand? > You don't appear to have any critical analysis techniques. Do you judge by appearances or just judge? > Your habit of ignoring reality is part of it; it's called > cognitive dissonance. Gotta ignore something, how do you define reality? This happens when a person has an > inability to think in long-term scenarios. So my misunderstanding, appearance, and disinterest indicate immediate concerns?
From: M Purcell on 7 Jan 2010 11:30 On Jan 7, 8:11 am, jbriggs444 <jbriggs...(a)gmail.com> wrote: > 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. I'll accept that definition.
From: PD on 7 Jan 2010 11:38 On Jan 7, 2:17 am, Michael Gordge <mikegor...(a)xtra.co.nz> wrote: > On Jan 7, 2:43 am, PD <thedraperfam...(a)gmail.com> wrote: > > > Ah, so I see the problem. You *assert* that anything that is certain > > must be derived from sensory evidence. > > Nope, you are a liar, I said, "certainty (as against an axiom) > required the non-contradictory identification and integration of > evidence, of sensory evidence" And the distinction in meaning from what I said is what? > > > And that therefore "axiomatic > > certainty" is, by virtue of your assertion, a contradiction in terms. > > Nope by YOUR definition of axiom being something accepted without any > evidence. That IS the definition of axiom. > > > OK, let's take an example. Let's use Euclid's Fifth Postulate. Is that > > certain or not? > > What is the sensory evidence? Shrug, if its not matter then it doesn't > matter. OK, so are you saying that Euclid's Fifth Postulate is not a postulate? If that is so, then why do you suppose it is called his fifth postulate? Is this a case of someone pointing to a zebra and calling it a zebra and you saying, "But by MY definition, that isn't a zebra, it's an anteater"? > > MG
From: J. Clarke on 7 Jan 2010 12:20
jmfbahciv wrote: > J. Clarke wrote: >> jmfbahciv wrote: > > <snip> >> >> FWIW, I think that everyone interested in this topic might want to >> read some Hume and some Popper--they both had goes at the question >> of the validity and utility of inductive reasoning, and Popper I >> understand discusses it specifically in the context of the scentific >> method. I don't know their work beyond that so can't suggest any >> readings--they're on my list but there's a lot in front of them. >> > > Popper is on my list. I'm not so sure about Hume since I've noticed > that it's the name used in their name-dropping to cause me to worship > the ground they trod on. I'm still trying to understand politics; > it doesn't help that I've been allergic to the subject all my life > :-). > > These people don't name-drop Popper as often. Do you have any > idea why this happens? Not really. Maybe it's that Hume is more famous. |