Am I a crank?
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From: Lester Zick on 1 Sep 2006 14:53 On Fri, 01 Sep 2006 09:08:10 -0400, "Jesse F. Hughes" <jesse(a)phiwumbda.org> wrote: >schoenfeld.one(a)gmail.com writes: > >> Jesse F. Hughes wrote: >>> schoenfeld.one(a)gmail.com writes: >>> >>> > Definitions can be false too (i.e. "Let x be an even odd"). >>> >>> That is not what one usually means when he says "mathematical >>> definition". A mathematical definition is a stipulation that a >>> particular phrase means such-and-such. >>> >>> Like: A /group/ is a set S together with a distinguished element e and >>> an operation *:S x S -> S such that blah blah blah >>> >>> But what you're doing is different. You are specifying that a >>> variable should be interpreted as a certain kind of number, namely an >>> even odd. Even though there is no such thing as an even odd, however, >>> this is not false. How could it be false? It's an imperative, >>> telling the reader to do something (namely, assume that x names an >>> even odd). >> >> It is false as it contradicts itself. The statement 'x is an even odd' >> is an alias for a sequence of (first order) logical propositions >> containing at least one contradiction. > >"x is an even odd" is not the same sentence as "let x be an even odd." >The former is truth-bearing (and false) and the latter is imperative >and hence has no truth value. Not much truth value just lots of falseness. ~v~~
From: fernando revilla on 1 Sep 2006 10:50 > On Thu, 31 Aug 2006 19:20:15 EDT, fernando revilla > <frej0002(a)ficus.pntic.mec.es> wrote: > > >DontBother(a)nowhere.net wrote: > > > >> So how exactly do definitions differ from > >> propositions? > > > >In the posts of Jack Markan > > I can't recall posts of Jack Markan on the subject. > > > and > Virgil, you have the > >answer. > > No, what I have from Virgil at least is the bald > assertion that there > are certain kinds of statements called definitions > which are not > subject to proof and certain kinds of statements > called theorems which > are subject to proof but no indications whatsoever of > the difference > between the two which renders one demonstrable and > the other not > except that neomathematikers are too lazy or stupid > to be taxed with > demonstrations of truth in the case of definitions > but nevertheless > want to enter them in the record and use them as if > they were true. > > > If you give a name to something with > sense, > >you have a definition. > > And what if you give a name to something with no > sense? > > > Nobody say " a > horse carriage or > >motor carriage that may be hired for short journeys" > > >usually we say " cab ", that is a definition, an > agreement > >( time is gold ! ). Propositions are not agreements. > > Both definitions and propositions are however > statements. > > ~v~~ Sorry, surely you are being honest in this discussion but I can guess that there are a lot of problems of communication even in a non mathematical language. Regards. Fernando.
From: Lester Zick on 1 Sep 2006 15:00 On Fri, 01 Sep 2006 09:49:03 +0300, Aatu Koskensilta <aatu.koskensilta(a)xortec.fi> wrote: >Lester Zick wrote: >> On 31 Aug 2006 10:54:03 -0700, "MoeBlee" <jazzmobe(a)hotmail.com> wrote: >> >>> schoenfeld.one(a)gmail.com wrote: >>>> Definitions can be false too (i.e. "Let x be an even odd"). >>> That's not a definition. That's just a rendering of an open formula >>> whose existential closure is not a member of such theories as PA. >> >> Which really clears things up for us, Moe. > >MoeBlee's answer might be considered somewhat more obfuscated than >necessary. We can rephrase it without reference to all this formal stuff >and simply say that "let x be an even odd" is not a definition, it's >just a shorter version "let x be a natural number and further assume >that x is both even and odd" which is neither a proposition nor a >definition. It's probably the beginning of a - relatively boring! - >proof in which it is established that there is no natural that is both >even and odd. > >Definitions in mathematics often appear in the following forms > > "An X such that P will be called a Q" > "By a X we mean a Y" > "When P(X,Y) we often say that X glurbles Y" > ... > >One might quibble and say that a definition of that kind might be false >if in fact no one will call an X such that P a Q; or if, in fact, the >devious author doesn't really mean a Y by X; or if "we" will not, in >fact, often say that X glurbles Y when P(X,Y); and so forth. However, >such objections ignore the role claims such as above have in >mathematical language, acting as they do essentially as stipulations, >analogously as one might say "let's call whoever it was who committed >this heinous crime 'John Doe'". Is a stipulation a statement? Is a definition a statement? Is a theorem a statement? Then my question remains as to what the difference is between or among statements such that one can be true or false and others not? Obviously the answer is that there is no such difference. It really doesn't matter the form of the statement or abstruse reformulations. They all entail predicate combinations and predicate interrelations which are either true, false, or ambiguous. ~v~~
From: Virgil on 1 Sep 2006 15:14 In article <00ugf2t19b2dg4u25v8q7rfr5s03762j38(a)4ax.com>, Lester Zick <dontbother(a)nowhere.net> wrote: > No, what I have from Virgil at least is the bald assertion that there > are certain kinds of statements called definitions which are not > subject to proof and certain kinds of statements called theorems which > are subject to proof but no indications whatsoever of the difference > between the two which renders one demonstrable and the other not One had hoped that Zick had enough wits to distinguish between "It is the case that 'A' equals 'B' " and " Let 'A" represent 'B' in the following". But apparently not.
From: Virgil on 1 Sep 2006 15:20
In article <2bugf25frrhqsvk64ll0naeqsd4o58qk8k(a)4ax.com>, Lester Zick <dontbother(a)nowhere.net> wrote: > On 31 Aug 2006 16:34:11 -0700, "MoeBlee" <jazzmobe(a)hotmail.com> wrote: > > >fernando revilla wrote: > >> In the posts of Jack Markan and Virgil, you have the > >> answer. > > > >You have one or two answers there. There may be others. > > None of which seem to highlight the reasons one kind of statement is > demonstrable and the other not except that neomathematikers prefer it > that way. Those who choose to work in mathematics soon learn to distinguish between " 'A' is equal to 'B' " as a proposition and "Let 'A' represent 'B' in what follows" as a definition. We are sorry to note that Zick seems incapable of learning how to make that distinction without our help. Perhaps with a bit of practice... |