Even if there was no intelligent life at all in the whole universe, if there were no humans, or other thinking creatures, mathematics would still exist, wouldn't it? Of course no one would ever find out about mathematics' existence, but its truths would just be THERE... Isn't that magnificiant?
We didn't make up mathematics. It just exists and doesn't require any atoms or whatever... Do you think it is something divine?
I find the notion of fictionalism in mathematics utterly perplexing. From what I understand of it, it seems that fictionalism is the thesis that mathematics is a created fiction, and that there is no mathematical truth separate from the relevant fiction. On this view, it seems, mathematical statements -- such as 2 + 2 = 4 -- are analogous to statements like “Humbert Humbert is infatuated by Dolores Haze.” But how can this be right? Does this mean I can construct a mathematical fiction in which, e.g., 2 + 2 = 5? On the fictionalist account, I can’t see why we ought to prefer, say, a mathematics in which 2 + 2 = 4 over one where 2 + 2 = 5 unless the former captures some inherent truth that the latter misses.
I guess there is the following difference between ordinal and cardinal numbers: while zero is a cardinal number, there is no ordinal number that corresponds to it: it makes no sense to talk about a (or the) "zeroth" something. Curiously enough, I think that there are many occasions where it is meaningful to talk about negative ordinal numbers. If I am considering a sequence of weeks, for instance, and only the weeks after some moment have some relevant feature, it will probably be reasonable to number those weeks with positive ordinals and to numer the previous weeks with negative ordinals. What do you think?
Is this for philosophers, mathematicians, or logicians? But here goes:
Given that the decimal places of pi continue to infinity, does this imply that somewhere in the sequence of numbers of pi there must be, for instance, a huge (and possibly infinite) number of the same number repeated? 77777777777777777777777777... , say?
If Pi goes on forever, you might think it must be. After all, if you checked pi to the first googol decimal places you obviously would't find an infinite number of anything. Try a googlplex! Still nothing.
But we haven't scratched the surface, even though the universe would have fizzled out by now. If pi's decimal places go on forever, there may be, (not just 77777777777777... or 1515151515151) but all of them, in all combinations, forever. After all, you only have to say "You've only checked a googolplex. There's still an infinite number to to check. The universe is long gone, but pi goes on and on."
Philosophers, mathematicians, logicians, any ideas?
How good does one need to be in mathematics to do good work in philosophy of mathematics? Does one need to be able to *do* original math research, or just read and understand math research, or neither? Or does the answer depend on the topic within philosophy of math? If so, which topics are those in which math knowledge is most useful, and in which is it least useful?
I am often confused by the rhetorics of physicists that their theory "came from mathematics". I remember the physicist, Brian greence tell the story of paul dirac discovery of anti-matter by pure a priori manipulation of mathematics. I see this to be very confusing, because i often imagine mathematics as being a priori, and necessary without any connection to the real world. That is, i can always imagine possible worlds( or universes) governed by different mathematical expressions, or descriptions. Does it follow that every mathematical expression/description describes our universe? Obviously not. With paper, and pencil, we could probable describe any universe with any arbitrary number of dimension of space, but does it follow that our universe has arbitrary number of spatial dimension? Obviously not. The use of mathematics seems to be good in formulating regularities of nature( laws of nature), and to extract the implication of those laws. It makes me wonder why physicists would say their theory comes...