Hacker News Board Hacker News, browsed like a classic message board
Live data from Hacker News

God created the real numbers

ethanheilman.com

Post reply on HN
Print view View original

141–150 of 226 posts

Re: God created the real numbers

#141

Post by Agraillo »

Eddy_Viscosity2 wrote: post #10
empath75 wrote: post #4

Earlier quoted context omitted.

> But the Cantor vision of the real numbers is just wrong and completely unphysical. They're unphysical, and yet the very physical human mind can work with them just fine. They're a perfectly logical construction from perfectly reasonable axioms. There are lots of objects in math which aren't physically realizable. Plato would have said that those sorts of objects are more real than anything which actually exists in…

The human mind can't work with a real number any more than it can infinity. We box them into concepts and then work with those. An actual raw real number is unfathomable.

I felt also something like this before. Also integers seem pretty close to the reality around us. One of their functions is to symbolically represent the similarity of objects (there might be a better way to put it). Like, if you see 5 sheep in one group and 6 in another, after that point they’re no longer just distinct sheep with unique properties - the numbers act as symbols for the groups. Real numbers still can work in the brain, but they're most distant from the world around us, at least when it comes to going from visual to conceptual understanding.

Re: God created the real numbers

#142

Post by edanm »

skybrian wrote: post #97
Kranar wrote: post #87

Earlier quoted context omitted.

What specifically doesn't sound right?

The claim is that every bb(n) is computable but I don’t think you can compute bb(6) without knowing which machines won’t halt. That doesn’t seem like a finite calculation? But given the answer, I suppose you could write a program that just returns it. This seems to hinge on the definition of “computable.” It’s an integer, so that fits the definition of a computable number. My mistake.

The main thing to make it clear is that BB(n) for a specific n isn't a function - it's a number. Just like Mult(10,4) isn't a function, it's a number (40).

So a specific BB(n) is just a number and is computable.

Re: God created the real numbers

#143

Post by omnicognate »

BeetleB wrote: post #93
omnicognate wrote: post #37

Earlier quoted context omitted.

As in why bother using real numbers in physics? Mostly because you need them to make the maths rigorous. You can't do rigorous calculus (i.e. real analysis) on rationals alone.

We don't need reals to make the math rigorous. Only to make the math a lot more tractable. I've solved multiple continuous value problems by discretizing, applying combinatorics to the techniques, and then taking the limit of the result - you of course get the same result if you had simply used regular integration/differentiation, and it's a lot easier to use calculus than combinatorics. But the point is the "rationa…

It's more about derivation of theorems than calculations.

Computation can only use rationals, and of course can get arbitrarily close to an answer because they are dense in the reals.

However, the entire edifice of analysis rests on the completeness axiom of the reals. The extreme value theorem, for example, is equivalent to the completeness axiom; the useful properties of continuous functions break down without it; the fundamental theorem of calculus doesn't work without it; Etc. So if the maths used in your physics (the structure of the theory, not just the calculations you perform with it) relies on these things at all, you're relying on the reals for confidence that the maths is sound.

Now you could argue that we don't need mathematical rigour for physics, that real analysis is a preoccupation of mathematicians, while physicists should be fine with informal calculus. I'm not going to argue that point. I'm just pointing out what the real numbers bring to the table.

Here's Tim Gowers on the subject: https://www.dpmms.cam.ac.uk/~wtg10/reals.html

Re: God created the real numbers

#144

Post by znkr »

zamalek wrote: post #50

When I truly grokked complex numbers, I felt as though real numbers were a lie - though I would now say that it was a convenient omission. There are many things that are more naturally described using complex numbers - waves (which much of reality boils down to) immediately come to mind. Even if something does align better with real numbers, it's still just x+0i. Maybe I'll change my mind ~when~ if I finally grok qua…

> Even if something does align better with real numbers, it's still just x+0i

Beware, it’s not always useful to work in complex numbers, you sometimes want to do something different for reals and complex numbers. The prime example here is complex analysis. Defining differentiation is based on limits, on the complex plane there are a lot more directions to approach a limit vs just two on the real line. This has some interesting implications. For example, any function differentiable on the complex plane is infinitely differentiable.

Re: God created the real numbers

#145

Post by cernocky »

getnormality wrote: post #89
cernocky wrote: post #64

Earlier quoted context omitted.

Ideas are real in the way rocks are if we are concerned with their informational being. They are real informationally - ideas and math participate in forming the world. Nowadays, LLMs, Search and other apps probably affect the world even more than any common rock. Which is more real?

I don't know what is meant by the informational being of a rock.

one way to think about rock is to acknowledge it as an informational entity. an entity which is likely more passive then lets say a human or an app, yet by simply being part of the environment, it changes what can be done in the environment. if it wasn't there and if it didn't had a certain shape, the opportunities of other actors in the environment would surely be different. after all rock can be used as a tool and even as a computer. if its still not intuitive, think about Aeolian Harp which is a passive statue, yet a musical instrument, or think how you could encode a perceptron or a simple neural net into a stone (through which a water or air would flow for example). now, even if any ordinary rock doesn't exactly encode neural net, it should be more clear that it still affects information flow. does it help?

Re: God created the real numbers

#146

Post by moi2388 »

lo_zamoyski wrote: post #140
moi2388 wrote: post #138

Earlier quoted context omitted.

If something can exist theoretically but not practically, your theory is wrong. But go ahead and divide a length an infinite number of times then. And actually infinite, not as it tends to infinity.

Take the time to understand the subject matter, because your first sentence doesn't makes sense.

It does.

Re: God created the real numbers

#147

Post by jojomodding »

moi2388 wrote: post #127

I am a finitist and constructionist at heart. Sure, mathematical abstractions and infinite structures are fun to play around with.. But go ahead and actually provide me the list of all naturals. You can not. Ever.

But for any list of naturals you give me I can give you one with more natural numbers on it.

Re: God created the real numbers

#150

Post by ndsipa_pomu »

tempodox wrote: post #148

You mean, she didn’t create the complex numbers?

She must have as the universe seems to run on rules that need negative and complex numbers (also irrationals). Just because humans took a while to discover them doesn't mean that they weren't always existing.
Post reply on HN