tag:blogger.com,1999:blog-4987609114415205593.post3796036243282652453..comments2024-03-28T13:40:26.497+00:00Comments on M-Phi: Deductive proofs: from premises to conclusion, or from conclusion to premises?Jeffrey Ketlandhttp://www.blogger.com/profile/01753975411670884721noreply@blogger.comBlogger7125tag:blogger.com,1999:blog-4987609114415205593.post-86740155876460973852014-05-24T14:41:14.136+01:002014-05-24T14:41:14.136+01:00Aristotle does seem to talk about deducing or infe...Aristotle does seem to talk about deducing or inferring premises through conclusions in the section on reciprocal proofs. I have a theory that induction is precisely this. Hope the grant approval board thinks likewise!<br /><br />David Bottingnoreply@blogger.comtag:blogger.com,1999:blog-4987609114415205593.post-28854175061981707982013-08-15T14:52:29.130+01:002013-08-15T14:52:29.130+01:00The same dichotomy between forward and backward pr...The same dichotomy between forward and backward proofs happens in type theory (in the computing tradition) where functional programmers normalize proofs written from premisses to conclusions, while logic programmers build proofs from conclusions to axioms.Valeriahttps://www.blogger.com/profile/01336528462208811726noreply@blogger.comtag:blogger.com,1999:blog-4987609114415205593.post-52868204019646866222013-06-13T08:53:27.175+01:002013-06-13T08:53:27.175+01:00Agreed! :) And thanks for the helpful illustration...Agreed! :) And thanks for the helpful illustrations, anonymous.Catarinahttps://www.blogger.com/profile/03277956118114314573noreply@blogger.comtag:blogger.com,1999:blog-4987609114415205593.post-62493456247801330102013-06-12T05:59:22.251+01:002013-06-12T05:59:22.251+01:00Well, at least I am not alone in thinking that (de...Well, at least I am not alone in thinking that (deductive) inference is the key concept in Frege's conception of logic:<br /><br />http://www-personal.usyd.edu.au/~njjsmith/papers/smith-freges-js-logic.pdf<br /><br />(I've been much influenced by my former supervisor G. Sundholm for my thinking about Frege.)<br /><br />You may quibble with my use of the term 'logicism' instead of 'logic', but other than that there are very respectable Frege scholars (to mention one more: Danielle Macbeth) who hold this interpretation. You may well disagree with them, but it's not a view that is obviously so 'completely wrong' as you claim.Catarinahttps://www.blogger.com/profile/03277956118114314573noreply@blogger.comtag:blogger.com,1999:blog-4987609114415205593.post-68642258183068709552013-06-12T02:21:07.617+01:002013-06-12T02:21:07.617+01:00There is much true here, but as far as logicism go...There is much true here, but as far as logicism goes, it is completely wrong. Logicism is sometimes an epistemological view, and sometimes a metaphysical view, but, either way, it is a view about what is based upon what, NOT a view about what is INFERRED from what. To interpret it the latter way is to think of it as a psychological view, and that could not be further from the intentions of either Frege or Russell.Anonymoushttps://www.blogger.com/profile/08532628015636927800noreply@blogger.comtag:blogger.com,1999:blog-4987609114415205593.post-57073063126080845532013-06-11T18:40:54.104+01:002013-06-11T18:40:54.104+01:00Yes, that's how it works. Math is done inducti...Yes, that's how it works. Math is done inductively; but presented (to students) deductively. This leaves the students -- even at the graduate level -- ignorant of how math is done.<br /><br />A couple of historical illustrations. Newton invented calculus in the mid to late 1600's. The Principia dates from 1687, so take that as the official date if you like. Now, Newton well understood that he could not make logical sense of the limit of the difference quotient (what he called the "fluxion"). It's an expression of the form 0/0, which makes no sense. But calculus worked, spectacularly so. It wasn't till the work of Weirstrass and other's in the 1800's that the logical definition of the limit was finally arrived at; and not till Zermelo, in the early 1900's, that we finally had a completely rigorous account of the real numbers and limiting processes starting from the axioms of set theory all the way up to calculus. This process took over 200 years! <br /><br />But today, in real analysis class taught to math major undergrads, we start from the axioms of set theory, construct the real numbers, and then rigorously prove the basic theorems of calculus. This is a complete inversion of how the subject was discovered and developed. But undergrads -- and, sorry to say, most philosophers -- take the axiomatic, deductive *presentation* and confuse it with the actual practice of mathematics.<br /><br />Another striking example is group theory. Mathematicians were trying to figure out how to solve polynomial equations. Abel showed that the 5th degree equation did not have a general solution. Galois showed that the underlying reason for this had to do with the mathematical structure of the set of permutations of the roots.<br /><br />It wasn't till much later that someone came along and defined a "group" as a set with a binary operation satisfying such-and-so axioms; and then was able to re-derive Galois's and Abel's proof from the axioms of group theory. <br /><br />Today, we teach undergrads that a group is such-and-so; then we spend the rest of the semester deriving consequences. The thoughtful undergrads wonder: How did they come up with these particular axioms? And they get the impression that math is about writing down axioms and mechanically deriving logical conclusions.<br /><br />But this is of course a complete inversion of how math is done. In math, you first suspect the theorem; then -- after centuries, sometimes -- you eventually figure out the right axioms.<br /><br />Math is done inductively and presented deductively. Math actually goes from theorems to axioms; not the other way round. Anonymousnoreply@blogger.comtag:blogger.com,1999:blog-4987609114415205593.post-36397017584097596312013-06-11T17:06:15.891+01:002013-06-11T17:06:15.891+01:00Perhaps what characterises the 'deductive jour...Perhaps what characterises the 'deductive journey' from premises to conclusion or from conclusion to premises is, nevertheless, unidirectional. By analogy, in physics one can evolve a system forwards in time, and then one can reverse the procedure, and evolve the system 'backwards' in time. Despite the terminology, one is always evolving the system unidirectionally irrespective of 'sense'. Anonymoushttps://www.blogger.com/profile/17159756589211653368noreply@blogger.com