6 ms·
Nah, infinities don't exist. There are just recursive procedures that emit ever-increasing numbers of digits with no bounds checking. The diagonal proof is jus
by ludston 3y ago
Nah, infinities don't exist. There are just recursive procedures that emit ever-increasing numbers of digits with no bounds checking.
The diagonal proof is just arguing that two nested while(true) loops will run for longer than one. (And then we define this as "bigger" just to confuse undergraduates)
- snthpy 3y agoI like this. Nicely put!
- xanderlewis 3y agoWhy is “longer” any less confusing than “bigger”? Also, what has nested loops got to do with it? You can use one loop to generate the natural numbers, and a pair of nested loops to generate pairs of natural numbers (if you like, the rationals). But the diagonal proof doesn’t show that these will have ‘different’ run times — they’re in fact in bijective correspondence.
- nairboon 3y agoYou need another loop for each real number. A "proper" real number is in itself not finite. If you want to actually physically create a real number, for almost all of them, the process will never terminate.
- xanderlewis 3y agoYou need to define your terms more precisely to be able to make these kind of assertions. What do you mean by physically create? What do you mean by process? The answers depend on these very strongly.
- nairboon 3y agoCan you write down the complete decimal expansion of let's say sqrt(5) ?
- xanderlewis 3y agoNo; it’s infinitely long and non-repeating. Are you trying to refute my original comment? I think we’re on the same page.
- ludston 3y agoSince you can't write down this number, how can you generate a second number using this number? You'll never finish generating the first number in order to start the second. You can come up with a bunch of procedures that generate numbers indefinitely, and you can even define relationships between those procedures, but a procedure is not a number.
- xanderlewis 3y agoI don’t know what you’re talking about. Are you responding to something I’ve said?
- eigenket 3y agoI don't understand what you mean by this. What would it mean for an infinity or any number to "exist", does the number 2 "exist" somewhere in a way that infinities do not? In any case our best physical models right now are full of infinities. Space looks like it is infinitely big to an absurd degree of precision. The spectrum of the hydrogen atom contains two different infinities! A countable infinity of bound states and a continuum of free electron states at higher energies.
- rolisz 3y agoWell, I have two apples on my desk. I can't have an infinity of anything. In that sense 2 exists more tangibly than infinity. Our best physical models are just that - models. All models are wrong, some are useful. Infinity in physics is usually a shorthand for "it's so big I don't have to care about the edges"
- eigenket 3y agoI see, so the belief is that improved theories of physics will remove all the infinities we appear to see around us? I guess as religions go, that one is no worse than many, but there isn't really any evidence for it now. Space looks really infinite, the hydrogen atom spectra are very well described by infinite series.
- nairboon 3y ago> will remove all the infinities we appear to see around us? Can you actually show us one of these apparent actual infinities? All one can reasonably postulate is a potential infinity, but this is completely different from Cantors actual infinities.
- eigenket 3y agoThe spectrum of the hydrogen atom and the volume of the universe both appear to be actually infinite based on our best physical models.
- dlkf 3y agoYour critique of the use of ”bigger” in your strawman applies equally to your own use of ”longer” - neither loop will ever terminate. So in what sense is either one longer? The diagonal proof is arguing that there is no bijection from N to {0, 1}∞ , despite the fact that both are infinite. The sense in which the latter is ”bigger” is that there are always elements left over that are covered by N.
- ludston 3y ago> So in what sense is either one longer? Neither. That's my point. Literally any definition of something infinite can always be reduced to a procedure that recursively transforms or observes some prior state. To say that one of these functions can produce more distinct states than another is pointless, because the procedure that produces the most states will always be the one that you ran the most times. There is nothing observably infinite, since it would take infinite time to observe that any given thing was infinite. The only possible proof of infinity would be a machine that runs infinitely quickly. e.g. https://qntm.org/responsibility https://qntm.org/responsibility
- kbrkbr 3y ago> Literally any definition of something infinite can always be reduced to a procedure that recursively transforms or observes some prior state. Could you come up with or point to such a procedure for R (the reals)? As I understand the diagonalization argument you can do that for N, but not for R.
- ludston 3y agoYes? Take any natural or real number. If it has no decimal point add one. Then append any digit. Repeat as necessary.
- falserum 3y ago1, 1.1, 1.01, 1.001, … This will never reach 2, so it will not generate all real numbers. (Which was what parent was asking for, to recusively generate all R)
- empath-nirvana 3y agoMathematics is just an exercise in symbol manipulation according to a particular rule set. Mathematics is a _game_. Park Place exists in the context of a game of Monopoly and also exists in Atlantic City. The tokens in a game of Reversi don't correspond to anything in the real world. You might be playing it on a physical board of a particular size, but you could just as easily play it on an imaginary board, or on a piece of paper, or a computer, or in some imaginary board of infinite size. Whether or not either exists in "reality" doesn't matter in the context of the game, though. If the rules of mathematics allow for infinities, then they exist in the context of mathematics. Whether they exist in reality doesn't matter. You can still say things that are true about these imaginary objects in the context of the game that everyone is playing with these symbols, just like you can say things that are true about the game of Reversi, even one played entirely in the minds of two players on an imaginary, infinite board.
- ludston 3y agoIf you use "exist" to mean "is a symbol defined within some system" rather than the typical "has isomorphism with reality" then you can say literally anything exists. Unicorns exist because English has a word for them. So be it, but it is a useless definition of exists then isn't it?
- empath-nirvana 3y agoIf you're playing dungeons and dragons, then unicorns "exist" within the context of that game, and there are rules for interacting with them. If you're reading a fantasy novel, then unicorns exist within the context of that fictional universe. And because people enjoy playing Dungeons and Dragons and reading fantasy novels, that existence has real value both for those people and for the people creating those novels and games. And you can say that certain things are true about unicorns within the context of those games and fictional worlds. (For example, it is true that unicorns have horns). If it makes you feel better to put infinity into the same category of reality as "unicorns", then I think that's fine? It doesn't necessarily need to be the case that anything in mathematics corresponds to something that is physically manifested, and probably _most_ of mathematics does not -- not just infinities.