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I have been reading up on Real Analysis on my own via Terrence Tao's Analysis I. It's a great book, but it has left me uncertain about the nature of the pure real numbers.

My question can be best understood as the following contradiction. Fact 1: We can never faithfully expresses irrationals like $\sqrt{2}$ in the sense that we can never write out its infinitely long decimal representation. We shall always have some truncation error. Fact 2: In the real world, we do see these irrational numbers showing up (for example, you can construct a $1$-$1$-$\sqrt{2}$ right angled triangle on a piece of paper).

So, is nature capable of finding the exact value of $\sqrt{2}$ with zero truncation error to make the ends of such a triangle meet (and to create a perfect circle, irrational in question being $\pi$, of course), or is it also just an approximation down to the smallest possible particle of matter whatever that might be, after which it is quantized (and thus has holes like the rationals; and it is just a $1$-$1$-$\frac{99}{70}$ triangle in reality)?

Mridul Gupta
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For your fact 1: we cannot write a decimal expression of a number doesn't imply we cannot express that number accurately. Essentially, each number (or anything else in math) is a symbol. It is the structures you assign to the symbol that make that symbol meaningful to you. For your case a "faithful" expression of $\sqrt2$ can just be writing $\sqrt2$ literally. Nobody require you to write it as a super long decimal, and that is totally necessary.

For your fact 2: actually, you cannot "construct" some math object in "real world". Drawing a triangle on a paper produces some physical object (e.g. arrangements of ink molecules) that represents a triangle for you, but that is never a triangle itself.

Remark. In Tao's Analysis he defines a real number like $\sqrt2$ as the equivalence class of rational Cauchy sequences converging to that number. Fundamentally such an equivalence class is a set of functions from positive integers to formal expressions of integers involving Tao's $/$ notation, and each integer is a construction supported by Peano axioms (or von Neumann's ordinal construction, etc.). How can you draw or create these "functions", "sets", "axioms" in "reality"?

Nuaptan
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  • I did have it in mind that even $0$, $1$, $\frac{99}{70}$ are symbols, and $\sqrt{2}$ is the perfect representation of $\sqrt{2}$ in this framework. But, the construction of $\mathbb{N},\mathbb{Z},\mathbb{Q}$ is much more simpler than that of $\mathbb{R}$, requiring much less thinking of convergence of infinitely many rationals, and I can see a correspondence for that in nature at least when counting things, or dividng things in proportions. But I cannot see nature "doing convergence" of rational Cauchy sequences to get $\sqrt{2}$. For this purpose, I like Planck length (@Goaki↓) – Mridul Gupta Nov 05 '24 at 10:08
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Math is not nature, In physics we have a thing called Planck length, you cant have somthing that is smaller than 1 planck length, so, if you try to measure the side of an perfect 1-1-√2 triangle, you wouldn't get √2, because you cant measure the infinte part that is smaller than planck length.

This is not a contradictory, math tries to show us a "perfect" world, nature isnt perfect (it us but not perfect in the sense of math).

Goaki.
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  • Would we get $1$ perfectly though through measurement? Because if we cannot measure any length perfectly, then it all looks meaningless to me. I walked 5kms today? Who knows, maybe my path can be approximated by an infinite staircase of length 10kms? Forgive my brevity, but I'm experiencing an existential crisis . – Mridul Gupta Nov 05 '24 at 10:13
  • @MridulGupta If you had the best measuring tape you would be unable to distinguish between 1 [cm] and 1 + 0.5 planck length [cm], plack length is so small that in 99.9999% of cases it dosent matter (it is equals to 0.000000000000000000000000000000000016 meters) – Goaki. Nov 05 '24 at 11:17