111–120 of 141 posts
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You can't split a quark, partial quarks doesn't exist. In fact, singular quarks can't exist, if you try to pull quark out of nucleus, it produces another quark to pair with. Quarks can be destroyed in particle accelerators collisions but those aren't components. Also, all of the components of an atom, electrons and nucleus, have mass.
top quarks exist as lone quarks and decay before they can pair
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Does half of something have a limit? Not by its definition. Same thing with addition or multiplication. All of these only work with some concept of infinity. We could redefine "half" to mean "half of whatever you're talking about until you get to some arbitrary limit", but doing that to all of arithmetic is going to wind up in a very odd place.
Half of something has a value, and that value is not infinity. You need to be more specific about how exactly do you get infinity from the fact that half of something has a value.
If you accept that every natural number has a successor which is a natural number, and no two natural numbers have the same successor, and that there’s no loops (e.g. by saying that there’s a total order on natural numbers and that any natural number is less than its successor), then there can’t be a finite collection which is all the natural numbers.
You could say “there’s no collection which has all the natural numbers”, which, ok, how do you want to talk about things true of all natural numbers then?
Formulating descriptions of physics without the axiom of infinity (or, without something to play the role of the real numbers) is super icky. You, in practice, can’t do any significant mathematical physics in an ultrafinitistic approach.
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Rejecting infinity is a purely philosophical stance that doesn’t teach us anything about reality. There is a big difference between “infinity doesn’t exist” and “infinity doesn’t exist physically”. I should also add that the resolution of zeno’s paradox in the form of calculus where and infinite set of steps can occur in a finite time (or infinite set of distance can span a finite total distance) is conceptually very…
> There is a big difference between “infinity doesn’t exist” and “infinity doesn’t exist physically”. Is there? I think one could make a decent case for "nothing exists which doesn't exist physically[1]". [1] https://plato.stanford.edu/entries/physicalism/ EDIT: you could even probably claim "nothing exists which isn't physically measureable " which may or may not be a stronger claim depending on your point of view.…
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> Infinity is a mathematical symbol we can observe. This is like confusing the map for the territory. Symbols live in syntax (like the syntax of programming languages), while mathematical concepts live in semantics. Infinity is not a symbol, it's not ∞. ∞ is the symbol we use to represent infinity.
The number 42 is also a mathematical symbol we can observe. (Or two symbols, depending on how you want to define symbol). You can observe the symbol. You can observe 42 of some object, 42 sheep for example. You can observe a pie chart, or an actual pie, with 42% of it missing. You can observe a plank of wood that is 42 inches or centimeters long. But you can't observe 42 itself. It is not like a hill on a map, where…
42+1 = 43, 42 + 1 ≠ 42, ∞ + 1 = ∞
Infinity plays by very different rules than numbers.
> To Zeilberger, believing in infinity is like believing in God. It’s an alluring idea that flatters our intuitions and helps us make sense of all sorts of phenomena. But the problem is that we cannot truly observe infinity, and so we cannot truly say what it is. When the author says we cannot truly observe infinity, what does that mean? Infinity is a mathematical symbol we can observe. We can't observe infinitely ma…
In analysis, when we write "the limit as x goes to infinity" this translates into a logical statement like "for all x, there exists some y > x such that ..." I don't really see anything conceptually difficult or contradictory here.
I don’t understand, and I hope it’s just bad writing. Certainly you can build a branch of mathematics without an axiom of infinity, and that’s fine, it’s math over finite sets. However, an axiom of infinity is independent, it doesn’t contradict anything in standard formalizations, and so it doesn’t make sense to say “infinity is wrong”. He may think the axiom of infinity isn’t satisfied by our real physical world, bu…
Sad that the article doesn't mention wildberger (coincidentally similar last name), an (in)famous math youtuber that's been mentioned on HN several times before. He has a "rational trigonometry series" an approachable way to see how math would work in an ultrafinite setting.
Hn loves to dismiss him as a crank, which I think is overly harsh.
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Half of something has a value, and that value is not infinity. You need to be more specific about how exactly do you get infinity from the fact that half of something has a value.
Not from “that half of something had a value”, but from “that half of any thing has a value”. If you accept that every natural number has a successor which is a natural number, and no two natural numbers have the same successor, and that there’s no loops (e.g. by saying that there’s a total order on natural numbers and that any natural number is less than its successor), then there can’t be a finite collection which…
There's an entire branch of math for that: https://en.wikipedia.org/wiki/Constructivism_(philosophy_of_...
The notion of "believing in" axioms is absurd ... as absurd as believing in the rules of chess and disbelieving the rules of checkers. Each set of rules or axioms forms a system (possibly degenerate if the axioms are inconsistent). The rules, axioms, and systems aren't "true" or "false" -- that's a category mistake. Studying the systems resulting from the Peano axioms or ZFC is a worthwhile endeavor. Studying the systems resulting from finitist axioms may well be too, but the nonexistence of infinities in the latter doesn't mean that they don't exist in the former--that's crackpottery. Mathematics has room for both sorts of systems.
Which axiomatic systems best model the world is a different matter. Now we're in an empirical realm, where there are observations, evidence, facts. And observational reports are necessarily finite, so even if there are "real" infinities they can't be demonstrated. But "all models are wrong", so both infinite and finitist axiom systems might serve as good approximations.
Likewise with computer systems--all actual computer systems are finite state machines, but it's convenient and useful to model them as Turing Machines that allow for both infinite non-halting systems and finite halting systems.
And since both this medium and I are finite, I will stop there.