eli_sennesh comments on Deliberate Grad School - Less Wrong

22 Post author: Academian 04 October 2015 10:11AM

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Comment author: [deleted] 04 November 2015 02:13:00PM -1 points [-]

Insofar as logic consists in information-preserving operations, the non-physically-applicable parts of math still correspond to the real world, in that they preserve the information about the real world which was put into formulating/locating the starting formal system in the first place.

This is what makes mathematics so wondrously powerful: formality = determinism, and determinism = likelihood functions of 0 or 1. So when doing mathematics, you get whole formal systems where the theorems are always at least as true as the axioms. As long as any part of the system corresponds to the real world (and many parts of it do) and the whole system remains deterministic, then the whole system compresses information about the real world.

Comment author: TheAncientGeek 06 November 2015 08:35:09PM 1 point [-]

Insofar as logic consists in information-preserving operations, the non-physically-applicable parts of math still correspond to the real world, in that they preserve the information about the real world which was put into formulating/locating the starting formal system in the first place.

Whereas the physically inapplicable parts don't retain real-world correspondence. Correspondence isn'ta n intrinsic, essential part of maths.

Comment author: [deleted] comment score below threshold [+] (14 children)

Comment author: entirelyuseless 09 November 2015 01:40:59PM 3 points [-]

Transfinite mathematics is very interesting and currently has no correspondence to the physical world, at least not in any way that anyone knows about. And you can make the argument that even if there is a correspondence, we will never know about it, because you would have to be sure that actual infinities exist in the physical world, and that would seem pretty hard to confirm.

Comment author: IlyaShpitser 09 November 2015 03:01:05AM * 2 points [-]

What do cardinals correspond to?

I suppose it's about the language, not about the model. Still.

Comment author: [deleted] 10 November 2015 11:48:37PM -2 points [-]

Screw it. I'll just go do a PhD thesis on how abstraction works, and then anyone who wants to actually understand can read that.

Comment author: IlyaShpitser 11 November 2015 12:57:42AM * 0 points [-]

A lot of pure math takes the form of: "let's take something in the real world, like 'notion of containment in a bag' and run off with it." So it's abstracting, but then it's not about the real world anymore. There are no cardinals in the real world, but there are bags.

Comment author: [deleted] 11 November 2015 02:57:12AM 0 points [-]

So it's abstracting, but then it's not about the real world anymore.

Yes it is. It still consists in information from the real world. The precise structure was chosen out of an infinite space of possible structures based precisely on its ability to generalize scenarios from "real life".

Consider, for instance, real numbers and continuity. The real world is not infinitely divisible -- we know this now! But at one time, when these mathematical theories were formulated, that was a working hypothesis, and in fact, people could not divide things small enough to actually find where they became discrete. So continuity, as a mathematical construct, started out trying to describe the world, and was later found to have more interesting implications even when it was also found to be physically wrong.

Comment author: RichardKennaway 11 November 2015 09:34:40AM 2 points [-]

I think you're engaging in deepities here. It is clearly true that all of mathematics historically descends from thoughts about the real world. It is clearly false that all of mathematics is directly about the real world. Using the same words for both claims, "mathematics is about the real world", is the deepity.

So continuity, as a mathematical construct, started out trying to describe the world, and was later found to have more interesting implications even when it was also found to be physically wrong.

That is news to me. Physicists, even fundamental physicists, still talk about differential geometry and Hilbert spaces and so on. There are speculations about an underlying discrete structure on the Planck scale or below, but did anyone refound physics on that basis yet? Stephen Wolfram made some gestures in that direction in his magnum opus; but I read a physicist writing a review of it saying that Wolfram's idea of explaining quantum entanglement that way was already known not to work.

Comment author: [deleted] 13 November 2015 03:25:04AM 0 points [-]

I think you're engaging in deepities here. It is clearly true that all of mathematics historically descends from thoughts about the real world. It is clearly false that all of mathematics is directly about the real world. Using the same words for both claims, "mathematics is about the real world", is the deepity.

I'm actually only using it for the former.

Comment author: gjm 12 November 2015 04:35:53PM 0 points [-]

You may be thinking of Scott Aaronson's review of "A new kind of science".

Comment author: IlyaShpitser 11 November 2015 07:26:39AM 2 points [-]

Ok, so you would say inaccessible cardinals, versions of the continuum hypothesis etc. are "about the real world."

At this point, I am bowing out of this conversation as we aren't using words in the same way.

Comment author: gjm 12 November 2015 04:48:16PM -1 points [-]

I think it's at least arguable that there are plenty of cardinals in the real world (e.g., three) even though there very likely aren't the "large cardinals" that set theorists like to speculate about.

(Of course there are also cardinals and cardinals.)

Comment author: IlyaShpitser 12 November 2015 05:01:02PM 0 points [-]

What was the point of writing that? Do you think I was talking about "3"?

Comment author: gjm 12 November 2015 11:37:48PM 0 points [-]

I (1) was genuinely unsure whether you were asserting that numbers (even small positive integers) are too abstract to "live" in the real world -- a reasonable assertion, I think, though I thought it probably wasn't your position -- and (2) thought it was amusing even if you weren't.

Comment author: RichardKennaway 09 November 2015 01:56:00PM 3 points [-]

Sure, you can come up with a formal system that bears no correspondence to the real world whatsoever. Mathematicians just won't consider it very interesting most of the time.

They call it "pure mathematics".

Comment author: TheAncientGeek 09 November 2015 10:09:59AM 2 points [-]

But that's not at all relevant The existence of unphysical maths is a robust argument against the theory that mathematical truth is true by correspondence to the physical world. The interestingness of such maths is neither here nor there.

eli_sennesh comments on Deliberate Grad School - Less Wrong

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