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God created the real numbers

ethanheilman.com

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Re: God created the real numbers

#151

Post by tempodox »

getnormality wrote: post #36

All math is just a system of ideas, specifically rules that people made up and follow because it's useful. I'm so used to thinking this way that I don't understand what all the fuss is about, mathematical objects being "real". Ideas are real but they're not real in the way that rocks are. Whenever there's a mysterious pattern in nature, people have felt the need to assert that some immaterial "thing" makes it so. But…

Praised be therefore William of Ockham.

entia non sunt muliplicanda praeter necessitatem.

Thou shalt not multiply entities beyond necessity.

https://en.wikipedia.org/wiki/William_of_Ockham

Re: God created the real numbers

#152

Post by tempodox »

ndsipa_pomu wrote: post #150
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.

Agreed. I was just wondering about the reductive title, since the reals are a strict subset of the complex.

Re: God created the real numbers

#153

Post by AIorNot »

adrian_b wrote: post #139
dudinax wrote: post #118

Earlier quoted context omitted.

We do not know whether work time and space are continuous

What we know is that we use mathematical models based on the continuity of work, time and space (and on the discreteness of matter and electricity) and until now we have not seen any experiment where a discrepancy between predicted and measured values could be attributed to the falseness of the supposition that work, time and space are continuous. Obviously this does not exclude the possibility that in the future som…

Umm SpaceTime is likely NOT to be fundamental or continuous

https://youtu.be/GL77oOnrPzY?si=nllkY_E8WotARwUM

Also Bells Therom implies no locality or non realism which to me furthers the nail on the coffin of spacetime

Re: God created the real numbers

#154

Post by metalman »

I cant agree less. Taking the basic concept of dividing the universe into things that are real, and constructs and blowing it up into a whole hiarchy of values , putting numbers at the top, is only a construct, built on constructs. The current state of science is one where there are a few islands where things work internaly, like how elements are created through certain processes in stars, it's mostly good, but there are exceptions, and then there is a gulf, a vast hand wavy conjectural gulf between that and exactly where the helium came from, or when, or if it did. It is the eternal always almost of bieng poised on the brink of a revelation thats great for suckering in a few more to toil and grind away at the secrets of reality, which they do, and it is from those endevours that some of the best patentable ideas come from.

Re: God created the real numbers

#155

Post by vessenes »

NoahZuniga wrote: post #19
andrewla wrote: post #3

I'm an enthusiastic Cantor skeptic, I lean very heavily constructivist to the point of almost being a finitist, but nonetheless I think the thesis of this article is basically correct. Nature and the universe is all about continuous quantities; integral quantities and whole numbers represent an abstraction. At a micro level this is less true -- elementary particles specifically are a (mostly) discrete phenomenon, but…

You know it wouldn't be possible for us to tell the difference between a rational universe (one where all quantities are rational numbers) and a real universe (one where you can have irrational quantities). The standard construction for the real numbers is to start with the rationals and "fill in all the holes". So why even bother with filling in the holes and instead just declare God created the rationals?

To a mathematician saying god created the integers is the same thing as saying god created the rationals: there’s a bijection between the rationals and integers.

I’m not convinced that we could have our current universe without irrationals - wouldn’t things like electromagnetism and gravity work differently if forced to be quantized between rationals? Saying ‘meh it would be close enough’ might be correct but wouldn’t be enough to convince me a priori.

Re: God created the real numbers

#156

Post by Xcelerate »

Everyone likes to debate the philosophy of whether the reals are “real”, but for me there is a much more practical question at hand: does the existence of something within a mathematical theory (i.e., derivability of a “∃ [...]” sentence) reflect back on our ability to predict the result of symbolic manipulations of arbitrary finite strings according to an arbitrary finite rule set over an arbitrary finite period of time?

For AC and CH, the answer is provably “no” as these axioms have been shown to say nothing about the behavior of halting problems, which any question about the manipulation of symbols can be phrased in terms of (well, any specific question—more general cases move up the arithmetical hierarchy).

If it’s not reflective in this precise sense, then the derivation of, e.g., a set-theoretic ∃ in some instances has no effect on any prediction of known physics (i.e., we are aware of no method of falsification).

Re: God created the real numbers

#157

Post by jakeinspace »

mattxxx wrote: post #116
ysofunny wrote: post #78

Earlier quoted context omitted.

unless you're a mathematician then maths is really THE absolute best description available of language and nature. but non-mathematical minds will simply wonder and be amazed at how "maths explains the world", a clear indication that somebody is not thinking like a mathematician. > Whenever there's a mysterious pattern in nature, people have felt the need to assert that some immaterial "thing" makes it so. But this j…

> unless you're a mathematician As a young math researcher, my mentor definitely did not believe that Math was the absolute descriptor of the universe. You can definitely imagine a scenario where the world does not operate perfectly mathematically correct though Math still exists - as an abstract separate entity. You can do this such that everytime you recognize a new quirk in the world, then you can invent some new…

At one point, many people would have said that quantum field randomness is non-mathe

Re: God created the real numbers

#158

Post by akssri »

Theory of Computation wasn't around when all this "exciting" stuff was developed in Mathematics. Given their non-constructive nature "real" numbers are unsurprisingly totally incompatible with computation.

Chaitin has a great paper on this and shows how Cantor's constructions were reflected a half-century later by Turing. https://arxiv.org/abs/math/0411418

Except of-course, while "hyper-Turing" machines that can do magic "post-Turing" "post-Halting" computation are seen as absurd fictions, real-numbers are seen as "normal" and "obvious" and "common-sensical"! It was amusing sometime back to see people pooh-pooh the likes of Hava Siegelmann for being funded for their "super-Turing" machines with "real-number" computation, without realizing that the core issue is the "real"-number itself!

I've always found this quite strange, but I've realized that this is almost blasphemy (people in STEM, and esp. their "allies", aren't as enlightened etc. as they pretend to be tbh).

Some historicans of mathematics claim (C. K. Raju for eg.) that this comes from the insertion of Greek-Christian theological bent in the development of modern mathematics.

Anyone who has taken measure theory etc. and then gone on to do "practical" numerical stuff, and then realizes the pointlessness of much of this hard/abstract construction dealing with "scary" monsters that can't even be computed, would perhaps wholeheartedly agree.

edit: The post has a great link to a note on Cantor's theology,

https://apeironcentre.org/theology-of-georg-cantor/

Re: God created the real numbers

#159

Post by spyrja »

tomrod wrote: post #123
AIPedant wrote: post #67

Earlier quoted context omitted.

I am confused why you think the exactness of integers and rationals is unphysical. "This egg carton has 12 eggs" is a (boring) physical statement. "You can make 1/3rd of a carton of eggs without cutting an egg" also seems perfectly physical to me. Your problem with zero-point-three-repeating is a quirk of decimal representation, not a mystical property of 1/3. Egg cartons might sound contrived but the reals don't nec…

I take a unit square. It's diagonal is a real number but not rational.

OK, but surely only because the exact value of 1 exists in the first place.

Re: God created the real numbers

#160

Post by spyrja »

SabrinaJewson wrote: post #22
Eddy_Viscosity2 wrote: post #10

Earlier quoted context omitted.

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 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…

Or maybe we can know them equally well? The function f(x) = x(0^(sin(πx)^2)) for example "requires" infinities, but only returns integer values.
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