Bruno Marchal wrote: > > At 11:34 08/01/04 +0100, Georges Quenot wrote: > > >I am very willing (maybe too much, that's part of the > >problem) to accept a "Platonic existence" for *the* integers. > >I am far from sure however that this does not involve a > >significant amount of faith. > > Indeed. It needs an infinite act of faith. But I have no problem > with that ...
Unfortunately, it seems that some people do.
It seems, but it isn't. Well, actually I have known *one* mathematician, (a russian logician) who indeed makes a serious try to develop some mathematics without that infinite act of faith (I don't recall its name for the moment). Such attempt are known as "ultrafinitism". Of course a lot of people (especially during the week-end) *pretend* not doing that infinite act of faith, but do it all the time implicitly. You know an ultrafinitist cannot assert that he is an "ultrafinitist" without going beyong ultrafinitism. So perhaps only animals do not do that infinite act of faith, but IMO, most mammals does it in a sort of passive and implicit way. If you pretend to understand a statement like:
N = {1, 2, 3 ...}, or N = {l, ll, lll, llll, lllll, llllll, lllllll, ...},
then you do it. Words like "never", "always", "more", "until", "while", etc. have intuitive meaning relying on it. I have worked with highly mentally disabled people, and only with a few of them I have concluded that there was perhaps some evidence in their *non grasping* of the "simple" potential infinite. All finitist and all intuitionnist accept it. Second order logic and any piece of mathematics rely on it. Some people would like to doubt it but I think they confuse Arithmetical Realism with some substancialist view of number which of course I reject. (I reject substancialism even in physics, actually I showed it logically incompatible with the comp hyp). Fearing the death in the long run (as opposed of fearing some near catastroph) also rely on that faith in the infinite, at least implicitly. Some people believe that human are religious because they fear death, but it is the reverse which seems to me much more plausible: it is because we are religious (i.e. we believe in some infinite) that we are fearing death.
I am not sure how much I share that faith. As I mentionned, I am willing to but since I could not find some ground to support that willingness, I might be a bit agnostic too.
No problem. The point is that it is a nice and deep hypothesis which makes comp fun and extremely powerful. It is definitely among my working hypotheses.
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> Why there is no FAQ? Because we are still discussing the meaning of > a lot of terms ....
I saw some posts on tentative glossaries of acronyms. Maybe before complex terms, we should focus on "basic" ones like "universe". I would not be upset to encounter definitions for several possible senses of that word.
I don't think the word "universe" is a basic term. It is a sort or deity for atheist. All my work can be seen as an attempt to mak it more palatable in the comp frame. Tegmark, imo, goes in the right direction, but seems unaware of the difficulties mathematicians discovered when just trying to define the or even a "mathematical universe". Of course tremendous progress has been made (in set theory, in category theory) giving tools to provide some *approximation*, but the big mathematical whole seems really inaccessible. With comp it can be shown (first person) inaccessible, even unnameable ...
Bon week-end,
Bruno
