Counterintuitions and the well-ordering theorem
in [Logic]
|
Prev: Solutions manual to Entrepreneurship 1e Bygrave Zacharakis
Next: Solutions manual for Engineering Mechanics Statics (12th Edition) by Russell C. Hibbeler
From: Rupert on 29 Nov 2009 19:56 On Nov 25, 4:12 am, WM <mueck...(a)rz.fh-augsburg.de> wrote: > On 24 Nov., 04:10, Rupert <rupertmccal...(a)yahoo.com> wrote: > > > > > > > On Nov 21, 1:14 am, WM <mueck...(a)rz.fh-augsburg.de> wrote: > > > > On 20 Nov., 13:37, stevendaryl3...(a)yahoo.com (Daryl McCullough) wrote: > > > > > Bill Taylor says... > > > > > >stevendaryl3...(a)yahoo.com (Daryl McCullough) wrote: > > > > >You admit doubts about the O.S. of sets; I presume > > > > >(maybe wrongly?) you have no, or at least much lesser, > > > > >doubts about the O.S. of natural numbers. > > > > > I don't see any big difference between the two. The set > > > > of naturals and the set of reals are both abstractions. > > > > I don't understand in what sense either exists, other > > > > than exists as a coherent topic of study. > > > > None of them does exist other than as a name and a wrong, i.e., self- > > > contradictory idea. We can write sequences of symbols that allow us to > > > talk about numbers and to manipulate numbers. That's all that exists - > > > and it's enough to do mathematics. Everything else is a useless object > > > for useless Fools Of Matheology. > > > > Regards, WM > > > That's not such an uplifting take on it. If you're really smart and > > study really hard, then you can learn to write down symbols in ways > > that a small handful of other people find interesting and worthy of > > respect, but which just about everyone has no hope of understanding > > and couldn't care less about, and fortunately the government is > > prepared to subsidise it. > > > We like to think that the story is more uplifting than that.- Zitierten Text ausblenden - > > Others like to use hashish. I don't recommend either of them. > > Regards, WM- Hide quoted text - > > - Show quoted text - Well, obviously I am not interested in your recommendations about either issue except to the extent that you can offer reasons... The fact that a certain worldview puts what you are doing with your life into a depressing perspective is not in itself a reason for rejecting that worldview, I grant you, but there are serious difficulties with the contention that mathematics is nothing more than a game with meaningless symbols, notably: (1) Can this position even be coherently and consistently maintained? If you believe that every mathematical proposition is just a meaningless string of symbols, then presumably statements such as "this formal theory is consistent" are also meaningless and lack a truth-value... Isn't it a bit difficult to stick to such a position sincerely and consistently? (2) What are we to make of the remarkable effectiveness of mathematics in helping scientists to achieve their goals? How is that to be accounted for on the view that mathematics is nothing more than a meaningless game with symbols?
From: Nam Nguyen on 29 Nov 2009 22:47 Rupert wrote: > If you believe that every mathematical proposition is just a > meaningless string of symbols, then presumably statements such as > "this formal theory is consistent" are also meaningless and lack a > truth-value... Isn't it a bit difficult to stick to such a position > sincerely and consistently? You've mis-characterized the nature of mathematical reasoning. The game of manipulating of meaningless strings of symbols is there to act as the _grammar_ of sentences that would express ideas, intuitions, etc... You could talk about the "applications" of reasoning and the rules of inference may have been extracted from our intuitions of the applications, but reasoning itself doesn't (and shouldn't) have anything to do with meaning of symbols or strings of which! And that's the position one should consistently stay with (and it's not that difficult as you might have feared.) As for the purported 'truth' of "this formal theory is consistent", there are statements which is meaningful but can't be assigned a truth value. But that's expected. > (2) What are we to make of the remarkable effectiveness of mathematics > in helping scientists to achieve their goals? By try-and-errors, like physics and other sciences. > How is that to be > accounted for on the view that mathematics is nothing more than a > meaningless game with symbols? You've mixed up 2 different paradigms. There's genuinely meaningless game of symbols. And there are _reasons why_ we play such a meaningless game. The two aren't the same!
From: Marshall on 29 Nov 2009 23:56 On Nov 29, 7:47 pm, Nam Nguyen <namducngu...(a)shaw.ca> wrote: > > And that's the position one should consistently stay with (and it's not > that difficult as you might have feared.) Such condescension! What arrogance! Marshall
From: Nam Nguyen on 30 Nov 2009 00:05 Marshall wrote: > On Nov 29, 7:47 pm, Nam Nguyen <namducngu...(a)shaw.ca> wrote: >> And that's the position one should consistently stay with (and it's not >> that difficult as you might have feared.) > > Such condescension! What arrogance! As it's often said when hungry eat thirsty drink, when one needs to stop the other kind of arrogance one might have to be arrogant! (For what it's worth it takes Riddick a "bad" guy to stop the Lord Marshall the other bad guy).
From: George Greene on 30 Nov 2009 07:54
On Nov 29, 3:15 am, Keith Ramsay <kram...(a)aol.com> wrote: [quoting DMC] > |If I say "Tom is the tallest person in the > |room", I've identified Tom by a quantification over a set that > |includes Tom as a member, so it's impredicative. However, that > |use of impredicativity is harmless because Tom existed prior > |to the definition. The definition is just selecting Tom, not > |creating him. Mathematical impredicative definitions are similarly > |harmless if you believe that the objects exist independently of > |their definitions. We can make this distinction sharper. Suppose we're just talking about natural numbers. If I say, "Among all the numbers written on the board, S is defined as the largest", then there is no problem; but if I say, "Among all the numbers written on the board, S is the one that is their sum", then S can't even exist at all unless it is the only non-zero number written. The sum is created from/after the numbers written on the board, whereas the maximum in some sense isn't; it does not change with changing values of the smaller numbers as long as they remain smaller. |