Hacker News Board Hacker News, browsed like a classic message board
Live data from Hacker News

Students’ insight proves that the local-global conjecture doesn’t hold

quantamagazine.org

Post reply on HN
Print view View original

91–100 of 107 posts

Re: Students’ insight proves that the local-global conjecture doesn’t hold

#91

Post by commonlisp94 »

vkazanov wrote: post #88
constantcrying wrote: post #55

Earlier quoted context omitted.

Nothing is more important for numerics than a veey good grasp of analysis.

Yup. But there's understanding and there's understanding. If all you have is a piece of paper then there's no way around analytical approach. People used to be unbelievably good at it, including the applied crowd, like engineers. Having matlab at hand makes it possible to save years of analytical tinkering and simplifications and approximations. It is useful to understand what happens exactly but we no longer need 3-…

> Having matlab at hand makes it possible to save years of analytical tinkering and simplifications and approximations.

And they use poor and imprecise approximations that they can't debug, because they just plug in numbers and hope for answer.

Re: Students’ insight proves that the local-global conjecture doesn’t hold

#92

Post by roywiggins »

denton-scratch wrote: post #64
anon291 wrote: post #47

Earlier quoted context omitted.

Godel's incompleteness theorems say that all mathematics that is complicated enough to encode basic arithmetic must rely on unproven assumptions. Unproven assumptions are the basis of all mathematics. For example, even numbers rely on very unproven assumptions. We assume, for example, that there is always a number following another number, but it is not at all obvious what that means.

> We assume, for example, that there is always a number following another number Assuming you're referring to natural numbers or integers, that's not an assumption: https://proofwiki.org/wiki/Natural_Numbers_are_Infinite

That proof assumes that "s(n)=n+1 is clearly an injection" - ie that every natural number has a unique successor, and uses it to prove something about the size of the set. But it doesn't prove there is "always a number following another number", it assumes it.

imho it's part of the definition of what the set of natural numbers even is. Peano arithmetic just has S(n) always exists and is injective as axioms.

Re: Students’ insight proves that the local-global conjecture doesn’t hold

#93

Post by commonlisp94 »

syndicatedjelly wrote: post #12
travisjungroth wrote: post #8

Earlier quoted context omitted.

All of those new discoveries involve running code. More people are learning to code, more libraries are being written, compute costs are going down. It’s hard to break ground on a “sit and think” problem, especially one that is approachable enough that it’s also understandable news when you solve it.

Here's your answer as to why these kinds of discoveries make it to the front page of HN.

The math literacy here is pretty poor when people imagine mathematicians to be solving college "calculus and algebra" problems.

Re: Students’ insight proves that the local-global conjecture doesn’t hold

#94

Post by vkazanov »

commonlisp94 wrote: post #90
vkazanov wrote: post #24

Earlier quoted context omitted.

Well, algebra and calculus have been around for a while. And then, Calculus took about 3 years for me to begin to scratch the surface... and realise I will never need it as numeric methods took over completely with the advent of cheap compute.

> I will never need it as numeric methods took over completely What if you want to make a numerical method for something you can't look up the recipe for?

I don't think the point is that nobody needs to understand the theory. The point is that 3-5 years of analysis might be an overkill in most of practical situations.

Numerical methods are quite well-studied by now. If you need a new method or a variation then you probably specialise in these things, and that's a different question.

Re: Students’ insight proves that the local-global conjecture doesn’t hold

#95

Post by vkazanov »

commonlisp94 wrote: post #91
vkazanov wrote: post #88

Earlier quoted context omitted.

Yup. But there's understanding and there's understanding. If all you have is a piece of paper then there's no way around analytical approach. People used to be unbelievably good at it, including the applied crowd, like engineers. Having matlab at hand makes it possible to save years of analytical tinkering and simplifications and approximations. It is useful to understand what happens exactly but we no longer need 3-…

> Having matlab at hand makes it possible to save years of analytical tinkering and simplifications and approximations. And they use poor and imprecise approximations that they can't debug, because they just plug in numbers and hope for answer.

Understanding that something is fundamentally an approximation is important. Users of these packages are well aware of the fact.

What point are you trying to make?

Re: Students’ insight proves that the local-global conjecture doesn’t hold

#96

Post by chongli »

rcme wrote: post #86
chongli wrote: post #45

Earlier quoted context omitted.

A theorem is just a statement which has been proved. Proofs are logical deductions that begin with a set of assumptions (called the hypothesis of the theorem) and follow a sequence of valid steps to reach a result (called the conclusion of the theorem). Without any assumptions at all, you have nowhere to go. There’s nothing you can conclude if you begin by assuming nothing.

However, in order to be a theorem, it's generally required that the hypothesis be consistent with the axiomatic system, which is not possible if the hypothesis can be proven false.

Axioms can just be thought of as baseline assumptions that are included by the terms in your hypothesis. If your theorem is about some vector space V, then you’re assuming that vector spaces exist, and also assuming all of the building blocks do as well (fields all the way down to natural numbers and set theory).

Re: Students’ insight proves that the local-global conjecture doesn’t hold

#97

Post by dragonwriter »

chongli wrote: post #96
rcme wrote: post #86

Earlier quoted context omitted.

However, in order to be a theorem, it's generally required that the hypothesis be consistent with the axiomatic system, which is not possible if the hypothesis can be proven false.

Axioms can just be thought of as baseline assumptions that are included by the terms in your hypothesis. If your theorem is about some vector space V, then you’re assuming that vector spaces exist, and also assuming all of the building blocks do as well (fields all the way down to natural numbers and set theory).

You aren't assuming existence (math is abstract, none of it relies on things existing), you are assuming certain definitions of vector spaces, etc.

Re: Students’ insight proves that the local-global conjecture doesn’t hold

#98

Post by chongli »

dragonwriter wrote: post #97
chongli wrote: post #96

Earlier quoted context omitted.

Axioms can just be thought of as baseline assumptions that are included by the terms in your hypothesis. If your theorem is about some vector space V, then you’re assuming that vector spaces exist, and also assuming all of the building blocks do as well (fields all the way down to natural numbers and set theory).

You aren't assuming existence (math is abstract, none of it relies on things existing), you are assuming certain definitions of vector spaces, etc.

Existence is a very common term in mathematics. It doesn't mean existence like a table or a piece of paper. It means that a mathematical object could (in principle) be constructed without leading to a contradiction.

Re: Students’ insight proves that the local-global conjecture doesn’t hold

#99

Post by constantcrying »

vkazanov wrote: post #88
constantcrying wrote: post #55

Earlier quoted context omitted.

Nothing is more important for numerics than a veey good grasp of analysis.

Yup. But there's understanding and there's understanding. If all you have is a piece of paper then there's no way around analytical approach. People used to be unbelievably good at it, including the applied crowd, like engineers. Having matlab at hand makes it possible to save years of analytical tinkering and simplifications and approximations. It is useful to understand what happens exactly but we no longer need 3-…

Solving integrals (beyond simple cases) has never been all that useful, simply because most integrals don't even have a solution. You do it in calculus to get a feeling for the subject.

If you want to calculate a numerical solution to a novel analytic problem you absolutely need a good understanding of analysis. Being able to plug things into Matlab is only useful if Matlab implements a good solver for that problem.

>It is useful to understand what happens exactly but we no longer need 3-4 years of calculus depths.

Indeed. We need 5+ years of analysis now.

Re: Students’ insight proves that the local-global conjecture doesn’t hold

#100

Post by anon291 »

denton-scratch wrote: post #89
anon291 wrote: post #85

Earlier quoted context omitted.

The statement 'there are infinitely many natural numbers' is not the same as 'there's a natural number following every other natural number'. In particular, the real numbers are infinite, but there is no unique real number following another real. Moreover, the natural numbers are typically defined axiomatically, either directly or via a set-based representation. Either way, you run into the same issue which is that e…

Real numbers != Natural numbers. Every natural number has a 'next'; there is no 'next' for a real number.

> Every natural number has a 'next'

Again... an axiomatic statement that is not well defined. What does it mean for something to have a 'next'. Shouldn't it be the case that if something has a next, then it can be named, identified, and perhaps even written down in some manner? Yet, by the same axioms, there certainly exist natural numbers that we cannot write down simply because there are not enough atoms in the universe that could be used to write them down on or with.

So basically, we have a conundrum, we say something exists after some other, yet for sure such a thing cannot be identified in any meaningful way, and its existence is just some conjecture that can never be proven. In what way is such a number distinguishable from any of the other infinite numbers that are supposedly greater than it? Since none can be written and all we can really say is that it's greater than whatever other number we have, one again questions whether or not the statement 'every natural number has a next' is truly well defined.

Post reply on HN