| ▲ | dr_dshiv 4 days ago |
| So what is the length of the diagonal of a unit square, if not square root of 2? It can’t be rational—how is that rationalized by Wildberger? |
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| ▲ | alexey-salmin 4 days ago | parent | next [-] |
| I don't know about Wildberger specifically, but an interesting point is that only a countable subset of real numbers can be described like that. Polynomials with integers coefficients are countable and so are their roots, which means almost all real numbers are transcendental. We think we study the real numbers but it seems we can't even have a system to express them. And indeed, that's not even a limitation of algebraic systems: any notation over a finite alphabet can only express a countable set of distinct objects which amounts to nothing when real numbers are concerned. I'm not a finitist, but I do find it curious that we approach mathematics by inventing a more-than-infinite set of objects that's impossible to fully grasp. I don't see it as a bad thing though, I also love Complex Analysis and many people (and some mathematicians even) denounce them for being imaginary. My impression is that transcendental numbers are as imaginary as are imaginary numbers, it's just we don't notice. And they're obviously still useful as are the complex numbers. |
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| ▲ | drdec 4 days ago | parent | prev | next [-] |
| If you do not accept that space is infinitely divisible, then the diagonal of a unit square does not actually exist in the space. |
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| ▲ | adrian_b 3 days ago | parent [-] | | A long time has passed since the paradoxes of Zeno of Elea, so now there really is no reason for not accepting that space is infinitely divisible. The error of Zeno of Elea was that he did not understand the symmetry between zero and infinity (or he pretended to not understand it). Because of this error, Zeno considered that infinity is stronger than zero, so he believed or pretended to believe that zero times infinity is infinity, instead of recognizing that zero times infinity can be any number and also zero or infinity. For now, there exists no evidence whatsoever that the physical space and time are not infinitely divisible. Even if in the future it would be discovered that space and time have a discrete structure, the mathematical model of an infinitely divisible space and time would remain useful as an approximation, because it certainly is simpler than whatever mathematical model would be needed for a discrete space and time. | | |
| ▲ | drdec 2 days ago | parent [-] | | > For now, there exists no evidence whatsoever that the physical space and time are not infinitely divisible. What is your evidence for it? You want to make a claim about something being infinite, it is up to you to provide evidence. > recognizing that zero times infinity can be any number and also zero or infinity. This statement makes no sense in formal mathematics. Multiplication is a function, which means for each set of inputs there is one output. I imagine you are trying to say something about limits here, but the language you are using is very imprecise. |
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| ▲ | numpy-thagoras 4 days ago | parent | prev [-] |
| Read Wildberger if you want to know what he thinks. I can tell you that it is the output of a function, not a distinct entity that exists on its own independently of the computation. The whole point is that as a theory for the foundations of mathematics, you do not need to assume numbers with infinitely long decimal expansions in order to do math. |
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| ▲ | birn559 4 days ago | parent [-] | | > I can tell you that it is the output of a function, not a distinct entity that exists on its own independently of the computation. Could you elaborate? What is the output of that function if not an entity in it's own? Having studied math with philosophiy minor long time ago I am curious. | | |
| ▲ | numpy-thagoras 3 days ago | parent [-] | | It's part of a dependency relation, the function computes and produces an output that we call sqrt(2). On the other hand, using the axioms of ZFC, one can say any real number exists without having a function to compute it, or a proof to construct it. For an ultrafinitist, or any finitist for that matter, we say that you only need the minimum of ingredients to produce math -- you do not need to assume anything over and above that, as it's not even helpful in the verification process. So assuming only finitely many symbols and finitely many numbers, I can produce what we call sqrt(2). We only ever verify it numerically and finitely anyways. We can never reach decimals at infinite ordinals. So it makes no sense to say, "Hey I assume transfinitely many entities, and my assumption says these numbers exist even though the proofs and decimal expansions are only ever finite." |
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