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| ▲ | Nevermark 4 days ago | parent | next [-] | | My comment is on whether the total class of reals are in nature. Given the theme and title of the article. Given one of the primary (but often not emphasized) properties of the class of reals, is that it contains and is actually dominated by the un-constructible reals, an argument that the total class of reals isn't represented in nature seems unremarkable to me. Un-constructible reals are a highly exotic abstract concept. You can never actually identify or operate on one. I do believe all the constructible reals (what most people think of when they think of real numbers), are likely to be found in nature/reality. It is a mishap of terminology, that the larger class, which is unnecessary for middleschool math, or any math most people will ever encounter, has a pithy name "real". While the practical concept, that directly correlates with arithmetic, algebra, calculus, diff eq, etc., has the unwieldily title of "unconstructible reals". So in order to avoid having to talk about constructible vs. unconstructible numbers to kids who shouldn't need to care, we use the pithy term and actually throw unconstructible reals into the mix, where it wasn't necessary at all. We tell kinds how the reals have a higher-order uncountable infinite cardinality, relative to integers. Which is true, but gives the impression that cardinality is somehow a concept necessarily linked to algebraic and other practical numbers that we introduce at the same time, which it is not. The set of constructible reals has the same countably infinite cardinality as the integers. | | |
| ▲ | griffzhowl 4 days ago | parent | next [-] | | But why think of any numbers as being "in nature"? And what does that really mean? Numbers and other mathematical concepts are used to describe and reason about physical systems. More or less everyone agrees on that much. Why make the further claim that some of these mathematical concepts or objects are "real"? There isn't any one-to-one mapping between numerical concepts and physical systems. Even for a finite collection of physical objects, we could associate with it a number, which is the number of items in the collection, but we could also associate with it another number, which is the number of possible combinations of items of the collection. It depends on our interests and what we want to do. Even grouping items into particluar collections is dependent on the goals we might have in some situation. We might choose to group items differently, or just measure their total mass. More generally, any physical magni9tude can be associated with an arbitrary number, just by a choice of unit. We use numbers, and mathematical concepts more generally, in many different ways to reason about physical systems and in our technical constructions. I don't see why we need any more than that, and to say that some mathematical concepts are "real" while some are not. > I do believe all the constructible reals (what most people think of when they think of real numbers), are likely to be found in nature/reality. Can you explain what it would mean to "find a constructible real in nature"? Maybe we just have different ideas about how this would be spelt out | | |
| ▲ | Nevermark 3 days ago | parent [-] | | > But why think of any numbers as being "in nature"? And what does that really mean? That there may be structures in nature that are 1-to-1 with any given (constructible) mathematical concept. Anywhere there is conservation of quantity we get addition and subtraction. Anywhere quantity can be looked at from two directions we get reversibility, i.e. positive and negative perspectives of the quantity. Multiplication happens anywhere two scalar values operate on each other, or orthogonal quantities create a commutative space between each other. We find reversibility, associativity, commutativity, and many more basic algebraic structures appearing with corresponding structures in physics. And more complex algebra where simpler structures interact. Wherever there is a dependency between constraints applying, we have logical relationships. So that is what I mean about mathematical structures appearing in nature. Numbers/quantities just being subset of those structures. > Can you explain what it would mean to "find a constructible real in nature"? Maybe we just have different ideas about how this would be spelt out My emphasis is really that we don't/won't find un-constructible reals. I WEAKLY claim (given that reality increasingly looks likely to extend beyond our universe, and more conjecturally, may be infinite), that any given constructible math structure has a possibility of appearing somewhere. Perhaps all constructible math appears somewhere. However, that is the weaker claim I would make. I more STRONGLY claim that un-constructible mathematical structures are highly unlikely to have counterparts. Which includes un-constructible reals. The un-constructible real invented by Cantor, was a value r, which has infinite decimal digits, but with no finite description. No algorithm to even generate. It is an interesting concept, but an un-instantiatable (even in theory) one. Infinite information structures immediately present difficult problems just for abstract reasoning. How corresponding structures might exist and relate in a physical analogue isn't something anyone has even attempted, as far as I know. -- I don't think I am saying something controversial. If there is anything surprising in what I am saying, it is that I am addressing the fact that reals got defined in a way that includes un-constructible reals. The only implication most people know about that, is that the cardinality of reals is greater than the cardinality of integers. But what might be very surprising for many, is that the cardinality of constructible reals, every possible scalar number we could ever calculate, measure or apply, is in fact the same as the integers. A distinction/insight that seems highly relevant and useful when dealing with instantiatable math, physics and computation. | | |
| ▲ | griffzhowl 2 days ago | parent [-] | | > That there may be structures in nature that are 1-to-1 with any given (constructible) mathematical concept. There may be, or there may not be. I don't think we can make a definitive argument either way without perfect knowledge of the structure of the physical world, or at least some part of it. What we have are mathematical models of physical systems that are valid to within some error margin within some range of parameters. The ultimate structure of the physical world is so far unknown, and may be forever unknown. Any actual physical situation is too complex for us to fully analyze, and we can only make our mathematical models work (to within some error) when we can simplify a physical system sufficiently. I think I understand better though what your main point is: that whatever physical theories or models we might have, the unconstructible reals won't be an essential part of it, i.e. even if we have some physical theory or model whose standard formulation might be committed to unconstructable reals, we could always reformulate it into a predictively equivalent model which doesn't have this commitment. Is that fair? That might be true for all I know. I'm not sure how to evaluate it (IANA mathematical physicist). It seems plausible though. There's the example of synthetic differential geometry, which has a different conceptual basis to the standard formulation of differential geometry, and at least suggests that you can't a priori rule out the possibility of alternate formalizations of any mathematical structure. I don't know enough about it to say whether or not it postulates something equivalent to unconstructible reals, it's just something that came to mind as maybe being along the lines of your point of view https://ncatlab.org/nlab/show/synthetic+differential+geometr... | | |
| ▲ | Nevermark 2 days ago | parent [-] | | > I think I understand better though what your main point is: that whatever physical theories or models we might have, the unconstructible reals won't be an essential part of it, Yes, that is a good way to put it. Unconstructible reals are a major and interesting "what if". What if there were numbers that had no finite relation to other numbers? It is a great idea, from a mathematical boundary pushing way. So abstract we can never do anything not abstract with it! > even if we have some physical theory or model whose standard formulation might be committed to unconstructable reals, we could always reformulate it into a predictively equivalent model which doesn't have this commitment. Is that fair? But unconstructible structures can never be reformulated as constructible by definition. That would mean they were constructible. We can never define a specific unconstructible real. But anyone who manages to create an interesting systems theory that uses them, with dynamics that constructible math can't match, would have created a major work of mathematical art! |
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| ▲ | NoahZuniga 4 days ago | parent | prev [-] | | While the observable universe is in some sense finite, as far as I know it is definitely still possible (based on our understanding of science) that the universe is infinite (and has infinite matter in every direction). I'll use this definition: a constructible real is a real that you can describe uniquely in a purely mathematical way (and prove that your description identifies a unique real). IE the positive solution to x^2=2 identifies a unique real. Also, the first positive solution to sin x. Now if you accept that the universe is infinite (and has infinite matter in every direction), you could get representations of un-constructible reals in your universe. One pretty contrived way to get an un-constructible real from this infinite universe is this: start at earth with some velocity, lets say 0.01c in some direction. Start with r = "0." Every second, find the closest particle. Take the amount of meters (rounded down) that this particle is away from you and append it to r. So if after one second the closest particle is 145m away, r becomes 0.145. If after another second the closest particle is 0.14m away, r becomes 0.1450 The value this process converges to could be un-constructible. | | |
| ▲ | Nevermark 3 days ago | parent [-] | | I love that example! I would argue that is still a constructible real. Only practical issues make calculating that value difficult. Since we are instances of physical constraints ourselves, just because we can't do a particular measurement, directly or indirectly, doesn't make a value un-constructible in the mathematical sense. (Also side noting, that we handle superposition/quantum collapse explicitly, by actually generating many alternate counts, or an expression that covers all the counts.) Note that your "algorithm" was finitely statable, and that its "data", consists of a finite number of particles (in any given superposition). But if I were going to argue for an un-constructible number with a physical counterpart, your thought experiment is a good starting point! | | |
| ▲ | NoahZuniga 2 days ago | parent [-] | | > Note that your "algorithm" was finitely statable, and that its "data", consists of a finite number of particles (in any given superposition). Well if the universe is actually infinite, the amount of data in the number this process approaches is infinite. > I would argue that is still a constructible real. That is what I was going for. I was trying to think up a construction that leads to uncountably many reals, but the construction I gave doesn't really work. Consider a different situation: Start with r = 0. (a number in binary) Look for an unstable radioactive isotope. Wait for its half time. If it decays within its half time, concatenate 1 to r. Else concatenate 0. Look for another radioactive isotope and repeat. The number this process approaches could be real number between 0 and 1 (including both bounds). Is the resulting number constructible? | | |
| ▲ | Nevermark 2 days ago | parent [-] | | > Is the resulting number constructible? That's a good one. I am going to say it absolutely is. Then acknowledge why others may feel very strongly that it isn't. So that's quantum mechanics, which from a field theory standpoint is completely deterministic. It just appears non-deterministic to us, because we are also superpositions. We are quantum structures too. And our field would keep splitting in two, at each measurement/decision point, so our total quantum field would remain completely predictable. But, it is true that each of our superpositions would have the experience of a completely random set of digits, going off to infinity. But, despite it adding additional physics and not explaining any more, some physicists seem to still think that there is a real collapse, not just an already explainable experience of collapse, of quantum fields. So, I think it is fair to say that if that was true, then truly unconstructible events would be happening. There would be no way to form an expression or algorithm to ever predict the flow of digits, even in principle. So you nailed the best possibility for it that I can think of. And this is a little circular, but between collapses adding a new phenomenon with no additional explanatory power (Occam's Razor be damned!), and the magic event decisions, are why I don't believe collapses happen. Collapses don't just imply that a magical event decision is made whenever we set up some careful experiment with one particle, but that all possible event situations in space-time, even in us, are constantly being magically decided as we are exposed to information about them, all the time. Given virtual particles are constantly frothing around even in empty space, this means that all of space-time is constantly flooding us with an unimaginable amount of magically created information. The magic bandwidth would be insane. One magical fundamental physical constant seems implausible to me. But 10^(very very big number) of magical decisions animating all of our universe and us every pico-second? Well, that would just be ... unconstructible! | | |
| ▲ | NoahZuniga a day ago | parent [-] | | > So that's quantum mechanics, which from a field theory standpoint is completely deterministic. This interpretation of reality, at least how I stand it, seems like the correct one to me. (At least that's how it feels to me.) But this combined with: > I am going to say it absolutely is. Means that all reals in [0, 1] are constructible, and as a result of that, all reals by modifying the starting value of r to ie 1. instead of 0., or 10. (2. in decimal). | | |
| ▲ | Nevermark a day ago | parent [-] | | I am not sure what your last statement means. Constructible reals are continuous over [0, 1] in that there are no gaps between any interval between constructible reals [r1 r2] that are not filled by more constructible reals (and in fact, the cardinality of the constructible reals within any interval is the same, i.e. countably infinite, in a fractal way). So there is no obvious motivation for anything but constructible reals from that standpoint. Unconstructible reals were invented (or at least used) by the mathematician Cantor to explore ideas about different infinities. A "real" number with infinite decimal digits but not any finite description lets him create a number space larger than the constructible reals, a larger infinite cardinality. So there is nothing missing in a [0 1] constructible interval. Or to put it another way, constructible numbers are closed. There is no sqrt(-1) situation requiring unconstrucible numbers to fill, like the square root of -1 required imaginary numbers (or geometric algebra dimensions) to fill. But [0 1] contains the higher infinity of unconstructible reals in it, if you want. But I am unaware of any claim that they solve any problems by being included, other than exploring interesting puzzles related to unconstructible numbers as interesting ideas in themselves. |
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