ed298-ak@violet.berkeley.edu.UUCP (05/26/87)
In article <3978@sdcc3.ucsd.EDU> ma188saa@sdcc3.ucsd.edu.UUCP (Steve Bloch) writes: >>a contradiction is reached because we have taken as certain the hypothesis >>that nothing can be known for certain. Therefore, it must be so that you can >>know something for certain. >> >Did Descartes really say that? I'm disappointed. Let X, in the >following sentence, be "nothing can be known for certain." >The truth of X does not imply that anybody KNOWS X. It seems to be that the above argument misses the point of Descartes. It is merely that X is true, but that he, Descartes, showed it to be true. If the mind deduce from things that are aleady known to a valid conclusion, then the nature of deduction allows you to "know" the conclusion. Descartes can be faulted from another flaw. If we assume that nothing can be known for certain, then we don't know for certain that contradictories are incompatible. So the logical engine Descartes wants to use also breaks down, and nothing can be deduced. However, denying that logic works is a very unconfortable position to be in. So in some sense of "pragmatic", I think that Descartes argument is basically sound. Edouard Lagache School of Education U.C. Berkeley lagache@violet.berkeley.edu
dhesi@bsu-cs.UUCP (05/27/87)
In article <3722@jade.BERKELEY.EDU> lagache@violet.berkeley.edu (Edouard Lagache) writes: >In article <3978@sdcc3.ucsd.EDU> ma188saa@sdcc3.ucsd.edu.UUCP (Steve Bloch) writes: [a self-inconsistent discussion of self-inconsistent reasoning by Descartes] Doesn't the paradox exist because we force it to exist despite knowing better? It would be more accurate to say, "Only one thing can be known for certain, and that is that nothing else can be known for certain." Although I doubt that even that is universally true, for "certainty" itself depends on what we define it to be. Consider a set that contains itself, and contains every other set that does not contain itself. Where's the paradox? Paradoxes arise when you choose an inconsistent set of axioms. "Doctor, it hurts when I do that!" "Don't do that!" -- Rahul Dhesi UUCP: {ihnp4,seismo}!{iuvax,pur-ee}!bsu-cs!dhesi
lambert@mcvax.UUCP (05/28/87)
In article <728@bsu-cs.UUCP> dhesi@bsu-cs.UUCP (Rahul Dhesi) writes: > Doesn't the paradox exist because we force it to exist despite knowing > better? I did not see any paradox. As I interpreted Descartes' argument, he was saying: There is at least one thing we can know for certain, namely that it is absurd to claim as certain knowledge that nothing can be known for certain. > Consider a set that contains itself, and contains every other set that > does not contain itself. Where's the paradox? Here: Call that set S. We have x in S <=> x = S or (x != S and x !in x). (*) Now let T consist of the elements of S, except S itself. So x in T <=> x in S and x != S <=> (x = S or (x != S and x !in x)) and x != S <=> x != S and x !in x. In particular (putting x:=T), T in T <=> T != S and T !in T. By the construction of T, we know that T != S (since S in S but S !in T), and so T in T <=> T !in T. An axiom set that allows the construction of S but at the same time prevents the construction of T would have to be pretty weird. > Paradoxes arise when you choose an inconsistent set of axioms. "Doctor, > it hurts when I do that!" "Don't do that!" This misses the whole point that there is no agreed way to make sure that a set of axioms is consistent. Is ZFC consistent? If you say "Yes, that has been proved", then what about the reasoning used in consistency proofs of ZFC? How do you know with mathematical certainty that such methods cannot "prove" inconsistent systems consistent? Patient: Doctor, it hurts when I try to give consistent axiomatic foundations for mathematics! Doctor Brouwer: Don't try to axiomatize mathematics! Doctor Wiener: Stop worrying! It is just a natural thing on the way to mathematical adolescence we have all had. Just concentrate on the beauty of ZFC, my child, and soon you will outgrow these qualms. -- Lambert Meertens, CWI, Amsterdam; lambert@cwi.nl
ma188saa@sdcc3.UUCP (05/29/87)
In article <3722@jade.BERKELEY.EDU> lagache@violet.berkeley.edu (Edouard Lagache) writes: >In article <3978@sdcc3.ucsd.EDU> ma188saa@sdcc3.ucsd.edu.UUCP (Steve Bloch) writes: >>>a contradiction is reached because we have taken as certain the hypothesis >>>that nothing can be known for certain. Therefore, it must be so that you can >>>know something for certain. >>The truth of X does not imply that anybody KNOWS X. > It seems to be that the above argument misses the point of Descartes. > It is merely that X is true, but that he, Descartes, showed it to be > true. If the mind deduce from things that are aleady known to a > valid conclusion, then the nature of deduction allows you to "know" > the conclusion. But X is not already known, it's assumed. The assumption was originally "Assume there is nothing that can be known for certain," NOT "Assume it is known for certain that nothing can be known for certain," which would be obvious nonsense. He, Descartes, did NOT show X to be true (which would indeed give him the right to say it was known), he showed (or tried to) something else UNDER THE CONDITION that X was true. The mere assumption that X is true does not allow him to conclude (within the scope of the assumption) that X is known for certain. Or am I misunderstanding you?