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God created the real numbers

ethanheilman.com

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Re: God created the real numbers

#221

Post by andrewla »

dullcrisp wrote: post #108
andrewla wrote: post #39

Earlier quoted context omitted.

I loosely identify with the schools of intuitinalism/construtivism/finitism. Primary idea is that the Law of the Excluded Middle is not meaningful. So yes, generally not starting with ZFC. I can't speak to "truth" in that sense. The skepticism here is skepticism of the utility of the ideas stemming from Cantor's Paradise. It ends up in a very naval-gazing place where you prove obviously false things (like Banach-Tars…

Won’t the reals we can construct by any computation be enumerable? What measure can they have if not zero?

Yes, they have measure zero. So the question becomes whether "measure" is a useful concept at all. In my opinion, no, it is not. It's just another artifact of non-constructive and meaningless abstractions. Many modern courses in analysis skip measure theory except as a historical artifact because the gauge integral is more powerful than the Lebesgue integral and doesn't require leaving the bounds of sanity to get there.

Re: God created the real numbers

#222

Post by gpm »

andrewla wrote: post #220
gpm wrote: post #79

Earlier quoted context omitted.

Yeah but if you look down the axis of rotation you will have a perfect (to many decimal places anyways) circle... which was the demand.

> to many decimal places anyway > > The idea of arbitrary precision is intrinsically broken in physical reality. There is no contradiction here.

Yeah I was just responding to the 64bit float thing, people overestimate floats.

Re: God created the real numbers

#224

Post by moi2388 »

griffzhowl wrote: post #217
moi2388 wrote: post #216

Earlier quoted context omitted.

Sure they do. Cantors argument, infinite sums, mathematicians keep using infinite lists as if they do exist. Let’s not even begin about the axiom of choice. Or transcendental numbers.

You're also treating the natural numbers as infinite if you think there can't be a finite list containing all of them

Depends. Not the way I look at them. But this discussion is getting off topic (my own fault), so perhaps we should continue it somewhere else.

Re: God created the real numbers

#225

Post by empath75 »

Eddy_Viscosity2 wrote: post #106
SabrinaJewson wrote: post #22

Earlier quoted context omitted.

I don’t know about you, I can work with it just fine. I know its properties. I can manipulate it. I can prove theorems about it. What more is there? In fact, if you are to argue that we cannot know a “raw” real number, I would point out that we can’t know a natural number either! Take 2: you can picture two apples, you can imagine second place, you can visualize its decimal representation in Arabic numerals, you can…

You can hold a two in your head, but you can't hold a number with infinitely many decimal places. Any manipulations you do with the real 2 are done conceptually whereas with the natural 2, its done concretely.

You absolutely cannot visualize the concept "2" in your head. You can only visualize "two things" or the word "two" or the symbol "2", but the actual concept "2" is just as abstract as a real number.

You might say, I can imagine 2 apples, but I can't imagine pi apples, but you could just as easily imagine unrolling a circle with a diameter of 1, and you have visualized "pi" just as well as you can visualize 2 apples.

Re: God created the real numbers

#226

Post by ngcc_hk »

Any elaboration to help us know the structure around us is useful and interesting. Might lead to useless results like even and odd numbers. Still say using bijection and diagonal rules illuminate something about our thinking of number system. And might be we have inconsistent maths but nevertheless still interesting.
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