Am I a crank?
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From: Lester Zick on 31 Aug 2006 15:58 On Thu, 31 Aug 2006 13:00:40 -0600, Virgil <virgil(a)comcast.net> wrote: >In article <ub6ef21d9o1m22bbhok6d0knbu12v0dt5d(a)4ax.com>, > Lester Zick <dontbother(a)nowhere.net> wrote: > > >> So how exactly do definitions differ from propositions? > >Definitions are requests to let one thing represent another, whereas >propositions are declarations that something is true. So propositions can be true based on definitions which are not true? Hmm, curiouser and curioser. ~v~~
From: Lester Zick on 31 Aug 2006 16:00 On Thu, 31 Aug 2006 13:04:44 -0600, Virgil <virgil(a)comcast.net> wrote: >In article <fd6ef2lhai4j05a73goceh4tveovu0lcvb(a)4ax.com>, > Lester Zick <dontbother(a)nowhere.net> wrote: > >> On 31 Aug 2006 18:47:17 +0300, Phil Carmody >> <thefatphil_demunged(a)yahoo.co.uk> wrote: >> >> >schoenfeld.one(a)gmail.com writes: >> >> Definitions can be false too (i.e. "Let x be an even odd"). >> > >> >Nonsense. It appears you are unaware of the use of the word >> >'vacuous' in mathematics. Probably due to the matching state >> >of your brain cavity. >> >> So definitions in modern math are not true? > >Nor false. A definition is merely a request to allow one thing to >represent another. >Even if that other thing does not exist, one can at worst only decline >the request. So now neomathematics is anthropomorphic too? My what a fantastic beastie indeed. ~v~~
From: Lester Zick on 31 Aug 2006 16:01 On Thu, 31 Aug 2006 12:20:55 -0600, Virgil <virgil(a)comcast.net> wrote: >In article <1157024892.614341.149030(a)e3g2000cwe.googlegroups.com>, > schoenfeld.one(a)gmail.com wrote: > >> Lester Zick wrote: >> > On 30 Aug 2006 05:01:52 -0700, schoenfeld.one(a)gmail.com wrote: >> > >> > > >> > >Han de Bruijn wrote: >> > >> schoenfeld.one(a)gmail.com wrote: >> > >> >> > >> > Then there is no experiementation. Mathematics is not an experimental >> > >> > science, it is not even a science. The principle of falsifiability does >> > >> > not apply. >> > >> >> > >> Any even number > 2 is the sum of two prime numbers. Now suppose that I >> > >> find just _one_ huge number for which this (well-known) conjecture does >> > >> _not_ hold. By mere number crunching. Isn't that an application of the >> > >> "principle of falsifiability" to mathematics? >> > > >> > >Falsifiability does not _need_ to apply in mathematics. In math, >> > >statements can be true without their being a proof of it being true. >> > >Likewise, they can be false. >> > >> > Except apparently for definitions. >> >> Definitions can be false too (i.e. "Let x be an even odd"). > >That definition is not false, as it does not say that any such thing >exists. Nor is it true. It is merely impossible to fulfill. Arbiter dicta are often difficult to fulfill but we do the best we can anyway. ~v~~
From: Lester Zick on 31 Aug 2006 16:02 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. ~v~~
From: Lester Zick on 31 Aug 2006 16:12
On 31 Aug 2006 10:45:23 -0700, "MoeBlee" <jazzmobe(a)hotmail.com> wrote: >Lester Zick wrote: >> So how exactly do definitions differ from propositions? > >A common approach for formal languages is that definitions are I didn't ask what common approaches are, Moe, and I don't care what common approaches are. I asked what the difference is between definitions and propositions. It's a simple question and what I asked for was the answer. If you can't give the answer then extemporaneous speculations on approaches of others won't produce the answer either. >definitional axioms, which differ from non-definitional axioms as >definitional axioms satisfy the criteria of eliminability and >non-creativity, which is to say that defintional axioms provide that >any formula using the defined term has an equivalent formula not using >the defined term (i.e., the definitional axiom only provides for >abbreviation) and the definitional axiom does not provide for theorems >couched without the defined term that are not theorems without the >definitional axiom anyway. > >For a mere understanding of such formal systems, it is not necessary to >determine what are propositions, but rather what are formulas and what >are sentences. However, in an informal understanding of such formal >systems, of course, such definitional axioms are regarded as >propositions. But they are a special kind of proposition in the way I >just described: they provide merely for abbreviation. > >In the most strict technical sense, definitional axioms are true in >some models and false in other models. However, though that is, very >strictly speaking, true, it is also extremely pedantic to dwell upon. >Since definitions are not substantive (in the sense that they are only >abbreviatory), we regarded them as stipulative rather than amenable to >evaluation for truth and falsehood, which evaluation would allow >ourselves to lose sight of the abbreviatory role of definitions and get >us bogged down in extreme pedanticism that is, for working purposes, >irrelevent to the abbreviatory and stipulative nature of defintions. > >For example, whether we define '@' (as, say, the typographic shape to >represent the 3rd 2-place function symbol of the formal language) or we >define '#" (as, say, the typographic shape to represent the 4th 2-place >function symbol of the language) to denote the binary operation of a >group is not important. It is stipulative and virtually beyond all >reasonableness to argue about as being correct or incorrect or true or >false. > >MoeBlee ~v~~ |