How many real numbers exist? New proof moves closer to an answer • Hacker News Board

Re: How many real numbers exist? New proof moves closer to an answer

#81

Post by schoen » Thu, Jul 15, 2021, 5:23 PM UTC

Ivoirians wrote: post #60

sillysaurusx wrote: post #39

Earlier quoted context omitted.

This is an excellent answer. I have a mostly unrelated question. In the case of the Collatz conjecture, it seems all but proved: > If the conjecture is false, it can only be because there is some starting number which gives rise to a sequence that does not contain 1. Such a sequence would either enter a repeating cycle that excludes 1, or increase without bound. No such sequence has been found. There are lots of hist…

They've verified that the Collatz conjecture holds for all numbers up to ~2^68, but that's precisely 0% of all the numbers that need to be checked. But more importantly, the goal of (pure) mathematics isn't to declare truths. If you had a machine from God himself that outputted True or False for theorems you put in, that wouldn't demotivate (pure) mathematicians from doing the work they're doing. Understanding the re…

Though note that that True/False machine could also quickly lead to the discovery of actual proofs, for example by making it extremely efficient to search for counterexamples. "The Collatz conjecture is true for all n ≤ some_huge_number." Also perhaps by making it extremely efficient to search for correct proofs in a lexicographically ordered list of proofs. "The shortest valid proof of the Collatz conjecture in the list of all proofs in my formalism is before proof number some_huge_number."

Although I think the basic version of the machine is "just" the first Turing jump oracle

https://en.wikipedia.org/wiki/Turing_jump

-- it depends on how you formalize the inputs to the machine, right? -- so maybe mathematicians would still be busy afterward. :-) Maybe the machine is an oracle with infinite Turing degree?

Re: How many real numbers exist? New proof moves closer to an answer

#82

Post by intuitionist » Thu, Jul 15, 2021, 5:25 PM UTC

pavpanchekha wrote: post #4

Great article. Since it looks like a lot of folks are interested in this article, some extra background. First, what is forcing? The article actually has a great description of ultrapowers (a key part of the construction) but it goes by a little fast, so you might like Tim Chow's "A beginner's guide to forcing" [1] which does a good job not only laying out the mathematical details at a high level, but also really cle…

Maddy’s work is great and worth reading (even as an antirealist!), but it’s worth keeping in mind that debates about foundations are basically irrelevant to the working lives of the large majority of mathematicians. If you’re studying, say, extremal graph theory or the Langlands program or low-dimensional topology, the cardinality of the continuum is simply not relevant.

Re: How many real numbers exist? New proof moves closer to an answer

#83

Post by fighterpilot » Thu, Jul 15, 2021, 5:26 PM UTC

thom wrote: post #27

bmitc wrote: post #20

Earlier quoted context omitted.

We know that the cardinality of the natural numbers is less than the cardinality of the real numbers. The Continuum Hypothesis, which is a long unsolved problem, states that there are no sets with cardinality between the two. The posted article states that this new result strengthens the case against the hypothesis, that is that’s it’s probably false. All of this is nuanced but is important to mathematics and philoso…

Why does “the set containing the natural numbers and a sandwich” not have cardinality between the two?

For the same reason that integers have the same cardinality as only odd numbers. You can create a 1:1 mapping between them.

Re: How many real numbers exist? New proof moves closer to an answer

#84

Post by question000 » Thu, Jul 15, 2021, 5:26 PM UTC

The only thing this proves is that mathematics is a soft science, where concepts like "number" and "infinite" are subjective.

There are obviously infinite numbers, if you think there's a finite number of numbers, take that number and add one to that. QED

Re: How many real numbers exist? New proof moves closer to an answer

#85

Post by bordercases » Thu, Jul 15, 2021, 5:27 PM UTC

sillysaurusx wrote: post #69

svat wrote: post #57

Earlier quoted context omitted.

No amount of empirical data is ever "good enough" for the mathematical standard of proof. (It may be enough for mathematicians to "believe" something in some informal sense, but not enough to consider it "proved".) You can see some examples at the answers to these questions; maybe at least one of them will be interesting to you: - https://math.stackexchange.com/questions/514/conjectures-tha... - https://math.stackexc…

That was another question in the back of my mind -- famously, the four color theorem was proved by computers through exhaustive analysis (checking every possibility). At the time, it was controversial as a "proof" since it didn't really take the usual form of a proof. I've often wondered "Why can't we do something like that, but for all instances of things like the Collatz conjecture?" Of course, it's computationally…

You're adhering to certain misconceptions for no good reason. An exhaustive check of all the cases in the model that supports the four-color conjecture is the "empirical" (here, meaning experimental) data required to falsify conjectures. And even physicists don't deny the value of falsification in narrowing down models to truth. But I think they are much more loose in what they are willing to stipulate as true if it means advancing other conclusions.

Furthermore mathematicians do accept certain kinds of probabilistic arguments, but they need to take a form of proving that an object will be guaranteed to have a certain property as a parameter on the object approaches a statistical limit. So partial evidence can sometimes point to the possible truth of a mathematical conjecture, but you are still burdened with the requirement of deductive demonstration if you desire logical certainty.

The tautological notion you take of math is a half-truth. On the one hand, models in math seek to be sound over the objects they specify. This is opposed to physics where deliberate simplifications must be made to make your idealizations of a system manageable or relevant. At the same time, no mathematical system is complete. So there will be truths that require other axiomatizations to access via proof. This is one source of mathematical creativity requiring judgment beyond following tautologies.

Re: How many real numbers exist? New proof moves closer to an answer

#86

Post by chaoticmass » Thu, Jul 15, 2021, 5:29 PM UTC

jostmey wrote: post #17

> "Not all infinities are equal" In other words, there are different categories of infinite, and it might be inappropriate to represent infinity with just one symbol! This article is about how many types of infinity might exist. I was taught there is countably and uncountably infinite. Integers are countably infinite because the number of integers between any two numbers if finite. Real numbers are uncountably infini…

https://www.youtube.com/watch?v=SrU9YDoXE88

I really like how VSauce covers this topic as well.

Re: How many real numbers exist? New proof moves closer to an answer

#87

Post by Ivoirians » Thu, Jul 15, 2021, 5:31 PM UTC

sillysaurusx wrote: post #76

Ivoirians wrote: post #60

Earlier quoted context omitted.

They've verified that the Collatz conjecture holds for all numbers up to ~2^68, but that's precisely 0% of all the numbers that need to be checked. But more importantly, the goal of (pure) mathematics isn't to declare truths. If you had a machine from God himself that outputted True or False for theorems you put in, that wouldn't demotivate (pure) mathematicians from doing the work they're doing. Understanding the re…

> They've verified that the Collatz conjecture holds for all numbers up to ~2^68, but that's precisely 0% of all the numbers that need to be checked. This is the crux of what has me looking like a fool to every mathematician in the thread, but I don't mind: why is 2^68 0% of the "numbers that need to be checked"? From a physicist standpoint, you can do a lot with numbers from 0 to 2^68. After all, 64-bit floats are q…

>Why is there no value in that?

Because it's uninteresting. The point of pure math is not to be "useful", it's to be interesting. The Collatz Conjecture is (as yet) a completely useless result. Like I said, if God himself came down and told the world "The Collatz Conjecture is true", all we'd get is a useless piece of trivia. "The Collatz Conjecture is true for the first 2^68 natural numbers" is even more worthless than that. Maybe it'd be useful if we had an application for it, but for context, many pure mathematicians are quite derisive at the idea that their work should have practical applications.

Here's a digression, a simple math problem. If you take a checkerboard and remove two opposite corner squares, can you tile the remaining 62 squares with 31 dominoes?

You can probably write a program that can exhaustively churn through all the possible arrangements of dominoes in a checkerboard, and it'll spit out the answer (it's "no"). But is that interesting? No. This is a boring fact, "if you take a checkerboard and remove the two opposite corners you can't tile the remaining squares with 31 dominoes". No one cares about that.

But, here's a proof that this is true. If you look at the colors of each square on a checkerboard, there are 32 black squares and 32 white squares. When you remove the two opposite corner squares, you're removing two squares of the same color. So you have 30 black squares and 32 white squares left (or the converse). Meanwhile, every domino takes up one black square and one white square. So no matter how you place 31 dominoes, they should cover 31 black squares and 31 white squares. Therefore, we've proven the tiling is impossible.

That's somewhat interesting. You have an easily understandable argument for why the fact is true, and you have an application of a method (here, invariants) for looking at other math problems. Plus, it's kinda fun and satisfying and "elegant" to solve a problem like this. The proof is much, much more interesting than knowing the answer to the problem. Hopefully this helps convey that.

Re: How many real numbers exist? New proof moves closer to an answer

#88

Post by btilly » Thu, Jul 15, 2021, 5:32 PM UTC

As a constructivist I'll be over in the corner that says that there are only a countable number of real numbers, and the unimaginable number of unimaginable infinities that classical mathematics insiste exists is all made up nonsense. That, in fact, things that can't ever be named, even in principle, don't actually exist.

What is interesting is that as shocking as constructivism may be, there is no logical flaw in it. If classical mathematics is consistent, then so is constructivism. And "there exists" is a whole lot more meaningful in constructivism.

Re: How many real numbers exist? New proof moves closer to an answer

#89

Post by dfdz » Thu, Jul 15, 2021, 5:32 PM UTC

fighterpilot wrote: post #13

sillysaurusx wrote: post #12

Earlier quoted context omitted.

It took many years to understand that! It’s not a flippant question at all. The answer is, it doesn’t matter. And that’s the joy of it. It wasn’t until I got into ML that I learned the value of doing unimportant work. When you’re free to think about inconsequential matters very seriously, you end up discovering so many useful things. It was how I independently rediscovered what is apparently called the Discrete Hartl…

> it doesn’t matter It doesn't matter yet . Applications of pure math happen downstream decades or centuries later. It's hard to predict the impact.

To follow up on this comment, I think that math should be viewed as a giant tree structure or pyramid

At the bottom are all sorts of applications of linear algebra and optimization (calculus) are related to machine learning, deep learning (AI), physics, engineering etc.

One level up there are all sorts of ideas in algebra and mathematical analysis that are leading to new ideas in linear algebra and optimization (calculus)

Going one level higher, there are ideas in mathematical logic and set theory that are improving our understanding our algebra and analysis.

If you look at a specific result at the very top of the pyramid as ask why does it matter? It is difficult to give a good answer. Clearly we want to base of the pyramid, but do we need to keep building it up? Should we stop at a certain level?

Re: How many real numbers exist? New proof moves closer to an answer

#90

Post by lupire » Thu, Jul 15, 2021, 5:33 PM UTC

rssoconnor wrote: post #65

pavpanchekha wrote: post #4

Great article. Since it looks like a lot of folks are interested in this article, some extra background. First, what is forcing? The article actually has a great description of ultrapowers (a key part of the construction) but it goes by a little fast, so you might like Tim Chow's "A beginner's guide to forcing" [1] which does a good job not only laying out the mathematical details at a high level, but also really cle…

> but at the same time there is no "right answer"—all of these axioms, after all, are independent of ZFC, and it's "ok" to add any of them. Minor quible: Just because a potential axiom is independent of ZF(C) doesn't make it necessarily "okay" to add. Potential axioms can be unsound, for example if they prove new / untrue Sigma_1 statements. As an example, even in the likely circumstance that ¬Con(ZFC) is independent…

why is that "unsound"? what's wrong with an unwritable natural? Almost all reals are unwritable.