Mathematicians are chronically lost and confused • Hacker News Board

Re: Mathematicians are chronically lost and confused

#161

Post by alex-g » Thu, Mar 06, 2014, 12:52 AM UTC

nnq wrote: post #154

alex-g wrote: post #130

Earlier quoted context omitted.

Right, this is how mathematics really works. But formalizations of mathematics may suffer from leaky abstractions. If we prove facts about numbers by compiling them into sets, and then using set-theoretic axioms, we might accidentally make it possible to prove things about numbers that are incorrect or meaningless.

Isn't this an abstraction problem that you solve by simply providing an "interface" or equivalent concept or access specifiers in oop like private/protected? All other modules that use the number module for applied math will just see an "interface" (let's call it GeneralNumber - as far as I know there are a few other alternate ways of defining numbers besides sets, right?), and the particular "implementation of numbe…

Yes, the interface idea is basically the right thing, but there are technical difficulties which aren't immediately obvious. For example, we'd like to be able to prove that two different implementations of the natural numbers are equivalent (one can do this mathematically, so if our system is going to handle general mathematics then it has to be capable of doing this). So we have to think quite hard about what this "equivalence" actually means. It's not enough to say that they satisfy the same axioms, because in general there can be all kinds of models for some set of axioms. It's closer to say that all number-theoretic statements about numbers-v1 are true of numbers-v2, and vice versa: but you can see that this is starting to get a bit hairy in terms of computable proofs.

A related problem, which speaks to the leaky abstraction issue, is "proof irrelevance". Typically, if I've proved something, it shouldn't matter exactly how I did it. But it turns out to be tricky to make sure that the proof objects in the system don't accidentally carry too much information about where they came from. Sure, you can define a way to erase the details, but you still have to prove that erasing doesn't mess up the deductive system.

None of this is insurmountable, but it's a glimpse into the reasons why encoding mathematics computationally is not trivial.

Re: Mathematicians are chronically lost and confused

#162

Post by neurobro » Thu, Mar 06, 2014, 1:44 AM UTC

DerpDerpDerp wrote: post #91

polymatter wrote: post #82

Earlier quoted context omitted.

I agree. To clarify, its not that the reader gets confused between the gas constant and the set of real numbers. The issue is never actually explicitly stating "R represents the set of real numbers" or "n is a natural number". At uni it once took me hours to work out that "." was used for function application in one particular paper. "." was also used for multiplication and (in some example code) had the usual object…

> The issue is never actually explicitly stating "R represents the set of real numbers" or "n is a natural number". It's at the start of like... every book ever. Pretty much any book on mathematics will start off with a fairly in-depth list of symbols. It's the math equivalent of expecting you know what a 'while' loop is when you go reading through the documentation for a library (reading a paper) - basic programming…

Yep, it's also a good practice for programming. Describe every single-letter variable in the README, which everyone obviously reads and memorizes before diving into the code. I still do this as a throwback to the days when we wrote code on parchment that cost a week's wages per square cubit.

Re: Mathematicians are chronically lost and confused

#163

Post by tel » Thu, Mar 06, 2014, 1:49 AM UTC

SilasX wrote: post #120

logicallee wrote: post #85

Earlier quoted context omitted.

The issue is that a lot of intuitive stuff is wrong. When you formalize, you remove the simple, intuitive explanation - but you make it much harder for you to remain wrong, if you are wrong - or to become wrong, if you started off right. As a simple explanation, consider the difference between explaining the Monty Hall problem - which might seem to be philosophical, open to interpretation - and coding it up. The mome…

That would be more convincing if the scientific papers were written in a way that make the point as clearly as the coded-up version of Monty Hall problem. In practice, it's more like they publish the assembly code and when you ask why they didn't do it in Python or something, they lecture you about the need for formal rigor.

I think there's generally selection bias about what part of an exposition makes the "a-ha" hit in two ways. First, your a-ha moment may not be the same as someone else's, but you're less likely to observe theirs. Second, your own a-ha is likely the product of a larger production than the moment itself of which you're most attuned to.

A good mathematical author must be guarding against both of these selection biases.

Re: Mathematicians are chronically lost and confused

#164

Post by pseut » Thu, Mar 06, 2014, 2:19 AM UTC

jorleif wrote: post #58

makmanalp wrote: post #55

Earlier quoted context omitted.

I've argued the same with a mathematician friend of mine. I hate academic papers because of their seemingly convoluted and backwards way of explaining things. His answer was that papers were not made to convey thoughts to laymen, they were made to communicate facts and proofs with as little ambiguity as possible, optimized for reading by other mathematicians. It's meant to be high bandwidth (hence the terse style and…

I think the process of peer review often causes papers to be optimized for being hard to critique rather than easy to understand. I have myself participated in writing papers where we decided to leave out some non-crucial but very useful detail just because it opens up too many questions and opportunities for critique.

The process of peer review also optimizes papers to be shorter than intended.

Re: Mathematicians are chronically lost and confused

#165

Post by j2kun » Thu, Mar 06, 2014, 2:25 AM UTC

acjohnson55 wrote: post #155

j2kun wrote: post #119

Earlier quoted context omitted.

Ellipses aren't functions :) I think if everyone agreed that the goal is to teach critical thinking skills, and have the factual knowledge be a byproduct (and elementary facts are very easy to pick up if you have critical thinking skills), then it would make a world of difference. As to the motivations, after the students get going they don't need more real world motivation. They seem to be interested enough to ask t…

Ellipses can be functions, if plotted using polar coordinates :)

Touche. Do they teach polar coordinates in high schools these days? I vaguely remember fiddling around with the option on my calculator and plotting cardiods. Speaking of which, I got a nontrivial amount of programming experience by totally ignoring my math lectures and trying to write programs on my calculator.

Re: Mathematicians are chronically lost and confused

#166

Post by pseut » Thu, Mar 06, 2014, 2:27 AM UTC

pflats wrote: post #87

AimHere wrote: post #45

Earlier quoted context omitted.

>we have a system of mathematics built on a few basic principles (that you would not disagree with) and deductive reasoning I think it's even better than that; mathematicians don't necessarily care whether the reader 'agrees' with the axioms, or whether they're in any sense 'true' or 'false'. Mathematics is always of the form 'if these axioms are true, this theorem follows from it'. The real world and the notions whi…

Oh, I agree, that's the best part. Pick your rules: oh, you picked those seven? You've got a ring; here are your math rules! In that specific case, though, I figured I'd point out that the basic rules of math aren't usually things people squabble over. (Although I do enjoy a little mathematical philosophy from time to time.)

Even better is when, to everyone's surprise, those seven turn out to be relevant for describing real world phenomena.

Re: Mathematicians are chronically lost and confused

#167

Post by j2kun » Thu, Mar 06, 2014, 2:30 AM UTC

chalst wrote: post #26

michaelochurch wrote: post #15

Earlier quoted context omitted.

How are transfinite numbers "nonsensical"? When you get into infinity, you have two notions of "number" that diverge. Mathematical operations on them do different things. (For example, cardinal "exponentiation" is the power set; ordinal "exponentiation" is something different and smaller.) One is size , but proper subsets can have the same size at infinity (integers, even numbers, rationals). That's where Aleph-0 (ca…

Well put; a little quibble: these are the two notions of infinity that most interest set theorists, but there are many other notions of infinity in mathematics, e.g., 1. Representation of geometric entities "at infinity" in, e.g., the point at infinity from the projective sphere that allows straight lines to be treated as circles; 2. Infinitesimals; 3. Game-theoretic constructions of infinite numbers, e.g., in Conway…

A quibble of my own: a lot of the "infinity" constructions in mathematics only use infinity as a name. Projective geometry (1) is a good example of that. The formalization doesn't actually appeal to any sort of infinite quantities.

Re: Mathematicians are chronically lost and confused

#168

Post by pseut » Thu, Mar 06, 2014, 2:36 AM UTC

jmnicolas wrote: post #54

Before a few articles of this kind I never suspected there was such depth in Maths. There's already so much to learn in programming, but I'm sure I'd love to dive in Maths (without the pressure of school like "understand this or you're an idiot").

Without that pressure, you won't learn it. :)

Re: Mathematicians are chronically lost and confused

#169

Post by Crito » Thu, Mar 06, 2014, 3:09 AM UTC

logicallee wrote: post #159

Crito wrote: post #153

Earlier quoted context omitted.

Such that the contestant chooses 1 door, the host then opens 98, and they are given the opportunity to switch to the last remaining door? That's actually a pretty brilliant way of explaining it. With numbers like that the answer becomes much more intuitive.

Why 100? Why not just 5? Some people would 'get' it at five, some people at 100, and some people at a million. If you have to choose out of a million doors, and no matter what the host opens all but one of them, so that your prize is either behind the door you picked, or behind the other one -- then should you switch your choice? Well, obviously, you should - with a million doors, it becomes obvious that you have jus…

Yeah, I'm not wed to a particular number... I'm just observing that larger numbers seem to make it more intuitive than, say, 3 doors.

Re: Mathematicians are chronically lost and confused

#170

Post by zhte415 » Thu, Mar 06, 2014, 3:13 AM UTC

nly wrote: post #20

Secondary math education, for me in the UK, didn't deal with anything outside of elementary algebra, Euclidean geometry, some statistics, and relatively simple calculus. Nobody talked to us about imaginary or complex numbers, or bayes theorem, decision theory, or non-trivial mechanics problems until I was in college (age 16+). Nobody mentioned matrices, broader number theory or discrete transforms until I was in univ…

This is slightly tangential:

I'm a big fan of A Levels (my experience in late 90s) because of the depth they can bring. Between the ages of 16 and 18 I was exposed to group theory, n dimensional linear algebra, (basic) proofs, curvature, complex numbers, various series and convergence. It was very eye opening. And the same is true of virtually all disciplines. The insight into a discipline A Levels gives, at a quite young age, cannot be achieved with a broad syllabus where depth is diluted.