171–180 of 226 posts
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Because the square root of 2 exists.
How does it exist though? Does it exist because we have a notation for it, or because we know its definition? Does the number 2 itself even exist? What does it mean to say that the number 2 exists? Calculo, ergo sum?
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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…
To me at least, if you can write down a finite procedure that can produce a number to arbitrary precision, I think it is fair to say the number at that limit exists.
This made me think of a possible numerical library where rather than storing numbers as arbitrary precision rationals, you could store them as the combination of inputs and functions that generate that number, and compute values to arbitrary precision.
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You can make this statement for any dense subset of the reals, but we don’t because that would be silly. I think the conceit is supposed to be that analysis—and therefore the reals—is the “language of nature” more so than that we can actually find the reals using scientific instruments. To illustrate the point, using the rationals is just one way of constructing the reals. Try arguing that numbers with a finite decim…
I think it makes much more sense to make this statement for the rational numbers: It's the smallest field inside the real numbers that contains the naturals. So every subset that allows you to do your daily calculations contains the rationals.
I think teachers lie to children and say that decimals are just another way of representing rationals, rather than the approximation of real numbers that they are (and introduce somewhat silly things like repeating decimals to do it), which makes rationals feel central and natural. That’s certainly how it was for me until I started wondering why no programming languages come with rational number packages.
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…
But take one thing and then another: you have two things. That’s true whether or not anyone notices. Some mathematics is a human system of ideas, but some of it isn’t. Arithmetic reflects real patterns in the world. Logic, too, is not merely invention, it formalizes cause and effect. Numbers, in the Pythagorean sense, aren’t just marks on paper or symbols of order; they are the order inherent in reality, the ratios and structures through which the world exists at all.
At bottom, this debate is about the logos: what makes the universe intelligible at all, and why it isn’t simply chaos. When people say “math is real,” they mean it in the Platonic sense, not that numbers are rocks, but that they belong to the intelligible structure underlying reality.
God enters the picture not as a bolt-on explanation, but as the consequence of taking mathematical order seriously. If numbers and geometry are woven into reality itself, then the question isn’t whether math is real, it’s why the universe is structured so that it can be read mathematically at all. Call that intelligible ground the logos, or call it God; either way, it’s not an extra mystery but the recognition that reason and order are built into the world.
Calling math “just useful” misses the point. Why is the universe so cooperative with our inventions in the first place? The deeper issue is the logos: that the world is intelligible rather than chaos. That’s what people mean when they say math is real, not that numbers are physical things, but that the order they reveal is woven into reality itself.
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> 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). Citation needed. Especially since there are well-established math proofs of irrational numbers.
The argument is essentially that you can only measure things to finite precision. And for any measurement you've made at this finite precision, there exist both infinitely rational and irrational numbers. So it's impossible to rule out that the actual value you measured is one of those infinitely many rational numbers.
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.
God created the rational numbers. The universe requires infinite divisibility, i.e. a dense set. It doesn't require infinite precision, i.e. a complete set. Our equations for the universe require a complete set, but that would be confusing the map with the territory. There is no physical evidence for uncountable infinities, those are purely in the imagination of man.
> The universe [...] doesn't require infinite precision, Doesn't it though? What happens when three bodies in a gravitationally bound system orbit each other? Our computers can't precisely compute their interaction because our computers have limited precision and discrete timesteps. Even when we discard such complicated things as relativity, what with its Lorentz factors and whatnot. Nature can perfectly compute thei…
That doesn't follow. Nature can perfectly compute them because they are nature. Nowhere is it required to have infinite precision, spatial or temporal.
Certainly Turing and Godel showed that computation is not universal (complete). Looking at one element of number theory in isolation seems unproductive? Arithmetic space is staggeringly complex, but structured, layered. IMHO number theory is like a hall of mirrors without a defined interface with physics. See Yang Mills mass gap.
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…
Hard disagree. This is the problem with math disconnected from physics. The real world is composed of quanta and spectra, i.e. reality is NOT continuous!
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> You can't do rigorous calculus (i.e. real analysis) on rationals alone. Yep, but that wasn't my point. My point was that it is possible that all values in our universe are rational, and it wouldn't be possible for us to tell the difference between this and a universe that has irrational numbers. This fact feels pretty cursed, so I wanted to point it out.
You can make this statement for any dense subset of the reals, but we don’t because that would be silly. I think the conceit is supposed to be that analysis—and therefore the reals—is the “language of nature” more so than that we can actually find the reals using scientific instruments. To illustrate the point, using the rationals is just one way of constructing the reals. Try arguing that numbers with a finite decim…