The Math Myth • Hacker News Board

Re: The Math Myth

#201

Post by MrManatee » Sat, Sep 03, 2016, 5:43 PM UTC

iopq wrote: post #94

MichaelBurge wrote: post #55

Earlier quoted context omitted.

https://en.wikipedia.org/wiki/L%C3%B6wenheim%E2%80%93Skolem_...

I didn't understand anything on that page.

This is not the easiest thing to explain briefly, but let's give it a shot anyway.

There are several ways of defining real numbers, and one of them is the axiomatic definition. Real numbers are defined by a list of axioms they must satisfy. These would include, among others:

(1) If x and y are reals, then x + y = y + x. (2) If S is a nonempty subset of reals with an upper bound, then S has a least upper bound.

There is a crucial difference between these two. In (1) the variables x and y only quantify over reals, but in (2) the variable S quantifies over subsets of reals. We say that (1) is a first-order axiom and (2) is a second-order axiom. Actually, among all of the axioms of real numbers, (2) is the only one that is second-order. Therefore, it is natural to ask: can we rid of it?

No, we cannot. Löwenheim-Skolem theorem says that if we only have first-order axioms, then it is impossible to distinguish between countable and uncountable sets - even if we have an infinite number of first-order axioms. In particular, this means that if we try to define real numbers using only first-order axioms, then the definition cannot even capture the basic fact that there is an uncountable number of reals.

From here on, there are two roads you could take. If you're like me, then you just accept that real numbers cannot be defined using first-order axioms. By my standards, any definition that only uses first-order axioms cannot be a satisfactory definition of the real numbers.

But some people don't want to accept definitions that are not based on first-order axioms. And this is not as crazy as it might sound. First-order axioms are very nice from a theoretic point of view. For example, with first-order axioms it is absolutely clear what it means to prove something based on those axioms. With second-order axioms, the situation is a lot hairier.

Re: The Math Myth

#202

Post by Bahamut » Sat, Sep 03, 2016, 5:43 PM UTC

hibikir wrote: post #191

treehau5 wrote: post #157

Earlier quoted context omitted.

For software engineers specifically -- every time you are coding formalized programmatic logic, you are using math. If you took a logics and proofs course, it would help you formalize logic better. Every time you write a "for" loop you are essentially using summations. There is a book called Concrete Mathematics and one of the primary authors is Donald Knuth, basically it's "Programmers math" and in my opinion, would…

You seem to describe a very different world of programming than the one I live in. I have no dislike for logic, formal proofs, category theory and such: I can live in that world just fine. However, almost none of the problems I've ever had to dealt with in my 15 years programming made me wish for large amounts of that: I've used bits here and there, but going all in never made any sense. The hard part of most tasks I…

I disagree, as someone coming from a math background - from no formal CS training, but 4 years of math at a top graduate program, I fully believe that a large percent of my peers from there could be extremely successful developers. In fact, I ran into a fellow former peer once at a conference who joined Google after exiting academia - he is currently on the Angular team. To take myself as an example, I have been very successful as well, currently being a lead developer who mentors/helps other developers on a near daily basis.

Formal mathematics training isn't necessary to be successful - many of us know many people who have picked up the intuition that those formally trained understand at a deep level & have been able to use it to great advantage. However, I argue that more training in areas of logic & reasoning would serve people well in all facets of their lives, including helping people be successful in software development over the long term. I know that I use my knowledge I gained from decades of mathematics to code strong abstractions faster than just about everyone else I have worked with so far (I've made mistakes too, but nothing terribly costly & unrepairable) - it helps me nearly every day I code & learn new things.

Re: The Math Myth

#203

Post by haddr » Sat, Sep 03, 2016, 5:45 PM UTC

I think it has something to do with the "theatre paradox": when someone stands up from her seat, then finally everybody needs to stand up in order to see the show. If we start having a surplus of people with degree, then everybody starts to look for "harder" degrees, better universities or just higher degrees (phd). And you need to have one in order to be successful. Side effects? Look at Google for instance: "They can hire the very best people — so everyone is overqualified." [1]

[1] https://www.quora.com/Working-at-Google-1/What-is-the-worst-...

Re: The Math Myth

#204

Post by munificent » Sat, Sep 03, 2016, 5:45 PM UTC

"You don't use X the majority of the time." is only a compelling argument to not learn X if the minority of the time where you do use it isn't that important.

Most people will spend very little time giving first aid, controlling a vehicle in dangerous weather, resolving serious relationship discussions, negotiating important deals, or doing cost/benefit analysis of large purchases.

However, in each of those cases, the tiny fraction of time where they do those is so important, it's still worth preparing for them. It may be that most people rarely use math, but when they do, they use it on important enough things to still warrant teaching them.

Re: The Math Myth

#205

Post by soberhoff » Sat, Sep 03, 2016, 5:50 PM UTC

yequalsx wrote: post #56

NhanH wrote: post #20

Earlier quoted context omitted.

To be fair, 0.9999... = 1 is not quite basic. You need to know things like infinitesimals, the distinction between value and representation of numbers etc.

Property of real numbers: between distinct real numbers is at least one other number. Now try to find a decimal representation of a number bigger than 0.9999999... but less than 1.0. You clearly can't. They must be equal. No need for infinitesimals.

I'd argue that the distance is 0.00000... = 0 and that distance = 0 implies that the points are equal.

Re: The Math Myth

#206

Post by ianai » Sat, Sep 03, 2016, 5:52 PM UTC

__s wrote: post #38

Relevant Carmack tweet: https://twitter.com/ID_AA_Carmack/status/767911253763170304

Decent application of high school mathematics is still better than many can muster.

Re: The Math Myth

#207

Post by rm445 » Sat, Sep 03, 2016, 5:57 PM UTC

ar0 wrote: post #43

I don't buy the sports analogy with which he argues that it is "self-serving nonsense" if people state that mathematics education trains your general problem solving skills. His argument that soccer players should only play soccer seems not to be anchored in reality: Of course professional soccer players spend a lot of time in the weights room or go running to enhance their general strength and stamina [1]! They do n…

Allegedly (association) football coaches seventy years ago would make players train without the ball all week, on the grounds that they would be keener to actually play football come Saturday. Of course they ended up under-skilled. My point is that one should find the combination of training that gets best results.

In my engineering career, successfully solving technical problems has generally consisted of working out what basic techniques solve an approximation of the problem and leaving it at that. I would say first-year undergrad level rather than 8th-grade, but definitely not the most complex mathematics I've ever looked at. Apparently being able to put together any sort of solution from scratch is relatively rare.

I do think problem-solving could be better taught. And schoolkids should definitely learn more about finance and statistics. Going on, the OP's stance seems fairly objectionable, but it's hard to disagree that employers use success in maths-heavy degrees as a proxy for selecting who may be best at a technical job. It seems like a fairly good filter, but it probably leads to injustice in certain cases, and the credential-chasing and learning less-necessary things may be inefficient.

Re: The Math Myth

#208

Post by dkarapetyan » Sat, Sep 03, 2016, 6:10 PM UTC

My current project is using GLPK to do some basic mixed integer programming to optimize AWS spot instance allocation. If I had not taken linear algebra, calculus, and a few courses in linear programming the idea would not even have crossed my mind that I could use mixed integer programming to solve the spot allocation problem. That's the first half. The second half can be considered a problem in control theory because it requires taking the new allocations and gracefully transitioning from the old set of allocations.

You can go even further and say that the whole thing would be even better if I understood more about stochastic processes and could potentially model the spot market and make predictions ahead of time to simplify the control problem and get ahead of the price fluctuations. Saying all you need is Excel and 8th grade is in the words of one famous physicist "not even wrong".

If you're in an engineering discipline then the more math you know the better.

Re: The Math Myth

#209

Post by geoka9 » Sat, Sep 03, 2016, 6:13 PM UTC

alejohausner wrote: post #95

soVeryTired wrote: post #70

Earlier quoted context omitted.

Limits use a construction that's pretty similar to an infinitesimal. The epsilon-delta definition of a limit is no joke for students.

Oddly enough, I never understood the epsilon-delta description of limits until I read David Foster Wallace's book on infinity. All through my degree in math I was taught about things without learning the historical context that created those things.

Getting a degree in math without understanding epsilon-delta is quite an achievement! I am serious.

Re: The Math Myth

#210

Post by berntb » Sat, Sep 03, 2016, 6:16 PM UTC

Bahamut wrote: post #202

hibikir wrote: post #191

Earlier quoted context omitted.

You seem to describe a very different world of programming than the one I live in. I have no dislike for logic, formal proofs, category theory and such: I can live in that world just fine. However, almost none of the problems I've ever had to dealt with in my 15 years programming made me wish for large amounts of that: I've used bits here and there, but going all in never made any sense. The hard part of most tasks I…

I disagree, as someone coming from a math background - from no formal CS training, but 4 years of math at a top graduate program, I fully believe that a large percent of my peers from there could be extremely successful developers. In fact, I ran into a fellow former peer once at a conference who joined Google after exiting academia - he is currently on the Angular team. To take myself as an example, I have been very…

I'll believe that there is a correlation of more than 0.5 to 0.7 between good at software development and good at abstract math. But hardly higher -- there are also some other talents needed, I don't know what.

I saw a few people that had no problems with the math courses in CS but that never really got the programming part. (No names or details, people might be recognized.) But this is of course anecdotal.

(My surprise over this was similar to when I realized that rationality and intelligence didn't have as high correlation as I expected. I know some intelligent people which are Truthers and believe in other conspiracy theories. Others can't reason when their emotions are involved.)