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Mathematicians are chronically lost and confused

j2kun.svbtle.com

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Re: Mathematicians are chronically lost and confused

#171

Post by rahimnathwani »

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…

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+)

For the benefit of those not familiar with the UK school system:

- Compulsory education goes up to the age of 16 (roughly 11 'grades')

- Many people continue at school until around 18 (an additional 2 years, called the 'sixth form')

- These final 2 years of school can be taken at a secondary school (~= high school in the US) or at a college (which may specialise in these 2 years only, or may go beyond). This college is not the same as a university, although there are some areas where they may compete for the same students

- Most people who enter university as undergraduates do so after completing these additional 2 years, and the exams which partially determine university acceptance are taken when people are about 17 or 18

Re: Mathematicians are chronically lost and confused

#172

Post by j2kun »

nnq wrote: post #17
michaelochurch wrote: post #4

Earlier quoted context omitted.

I'm of the mind that there is immense value in being able to figure out difficult proofs. Absolutely. However, the rabbit hole is very deep. Many papers make leaps from one sentence to the next that, if you're not familiar with the field, can take a couple days to figure out. Even then, real world proofs are informal and therefore not air-tight. They're close enough, almost always, but there's a reason why a mathemat…

> most of it can't be done that way and remain useful to humans (like assembly language, it's too low-level for most applications). Sincere question (I'm not a mathematician): why can't it be done that way?! On top of an assembly language you can create a higher level language and on top of that an even higher level one, and it is airtight , it has to be or the code won't compile or will throw a runtime exception, th…

This post ignores the central driving force behind why mathematicians do mathematics: mathematics was created by humans for their own pleasure. It's a coincidence that they find such useful applications in the real world, and mathematicians mostly don't care (it's like a bonus if someone finds a nice application of your work, and maybe it provides you more grant money).

In this light, why would any mathematician want to replace peer review with machine checking? It would result in two undesirable things: 1) you'd have to submit papers in a machine-checkable format, which very few people enjoy writing because it's not a natural language to do high-level reasoning in. 2) you'd lose the chance for others to tell you whether they think your research is interesting or not, relate it to other work, give their own conjectures or ask about clarification, which is the real reason for the peer review system. Overly complex proofs with subtle mistakes are few and far in between.

Similarly, why would we want to offload the joy of finding proofs and discovering new techniques to computers? That's like proposing we have computers produce all art and music. The process of creation and discovery is largely what makes mathematics worth doing!

Re: Mathematicians are chronically lost and confused

#173

Post by voltagex_ »

charlieflowers wrote: post #107
vargas84 wrote: post #93

Earlier quoted context omitted.

Would anyone be interested in seeing a "Rap Genius for academic papers" to address the third paragraph of parent comment? To those who aren't familiar with Rap Genius, I basically just mean a place where academic papers can be annotated and explained in plain language by the community. Would you read/contribute?

I think that would be fantastic . I have envisioned that kind of thing often. Something like "autodidact.stackexchange.com" (which doesn't exist), for example. The best explanations could be voted up. For bigger topics, it would not be explanations you're voting on, but rather "learning roadmaps" -- a recommended set of materials in a recommended order that is designed to help one grok the topic. For academic papers,…

Learning roadmaps would be huge, especially for programming languages.

e.g. "I want to create an IRC client in XYZ language"

simple i/o -> client-server/sockets -> simple parsers -> simple client (it's been a long day, this example sucks)

Re: Mathematicians are chronically lost and confused

#174

Post by ufo »

nnq wrote: post #17
michaelochurch wrote: post #4

Earlier quoted context omitted.

I'm of the mind that there is immense value in being able to figure out difficult proofs. Absolutely. However, the rabbit hole is very deep. Many papers make leaps from one sentence to the next that, if you're not familiar with the field, can take a couple days to figure out. Even then, real world proofs are informal and therefore not air-tight. They're close enough, almost always, but there's a reason why a mathemat…

> most of it can't be done that way and remain useful to humans (like assembly language, it's too low-level for most applications). Sincere question (I'm not a mathematician): why can't it be done that way?! On top of an assembly language you can create a higher level language and on top of that an even higher level one, and it is airtight , it has to be or the code won't compile or will throw a runtime exception, th…

We do have automated theorem checkers that follow your idea: from a very simple set of axioms you can build higher level lemmas and proofs that are all mechanically verified by a small trusted logical code. Its actually a very rich and fruitful research area that also happens to live quite close to computer science! Many of these proof checkers are actually just programming languages with a very fancy type system.

However, I don't think we will ever see math move to a totally computer-verified schema like you mentioned:

* First of all, its very hard to write the rigorous proofs that can satisfy a computer. Do you know that feeling when you know that your program will run just fine but the type checker is being picky about a technicality and wont accept it, meaning you need to refactor lots of stuff to appease it? Mechanically verified proofs are like that but the typechecker is even pickier!

* Often, there is a big disconnect between the important mathematical ideas and the logical foundations they build upon. For example, differential calculus has a very simple intuition (rates of change, areas under curves, etc) and can quickly biuld into applications, but its actually pretty hard to build it from basic principles. In fact, it took hundreds of years until mathematicians finally figure that part out and in the end the foundations were totally different from what they were when they started. If you tried to model this in a computer, the effect would be having to rewrite all the foundational proofs and breaks tons of interfaces and higher level lemmas as people started to learn about all the corner cases in the system.

Re: Mathematicians are chronically lost and confused

#175

Post by rachellaw »

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…

I went through the UK education as well, did you do just E math? You can take Additional Math along with it, that covers complex/imaginary numbers as well as Poisson sequencing/binomial theorem.

The thing is, I use very little abstract math in programming. Doing analytic philosophy actually did more than mathematics. The math stuff I do in programming is super basic stuff like trigonometry (sometimes trig with differentiation if I want to be fancy and use inflexion on curves) for GPS triangulation etc. I've never had to do anything more complex than that

Re: Mathematicians are chronically lost and confused

#176

Post by rachellaw »

nly wrote: post #77
ronaldx wrote: post #64

Earlier quoted context omitted.

> 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. I've been a secondary math teacher in the UK and I want to defend this point a little. The job of a secondary math teacher at this level is to teach everyone math, particularly including a majority who don't have a strong interest and won't go on t…

I wasn't aiming to criticise the particular set or scope of topics taught at each stage, more so the fact there's no provision to ensure students are aware of what branches of mathematics even exist. There's no formal introduction or grounding or broad strokes. It's like teaching science where there's no innate notion of biology, chemistry and physics as three separate disciplines... or teaching history without putti…

If you took additional math along with basic/elementary math, they do go into more detail about the branches of mathematics. There's also further mathematics and combined mathematics, both are A level subjects.

I think basic math focuses more on arithmetic than math theory, which could explain your frustrations a little.

Re: Mathematicians are chronically lost and confused

#177

Post by rachellaw »

acjohnson55 wrote: post #28

I've felt this is the case for a long time. A lot of people have a smooth experience in math for years until they hit their first serious discontinuity. That could happen anywhere: times tables, fraction arithmetic, two-step equations, geometric proofs, radicals, limits, or maybe even college math. The reaction is nearly universal though. The person thinks, "holy crap, I guess I'm actually not good at math", anxiety…

that is a really fantastic point

I thought I was good at it until I hit 17-18, doing advanced mathematics. I really didn't understand Taylor series, and I just froze up on the calculations. You fall behind, and then the class just moves forward and it stops being fun anymore

I don't know the solution, but as you said - I went back into it later and it was much easier. Catching up those years inbetween was hard though!

Re: Mathematicians are chronically lost and confused

#178

Post by rahimnathwani »

jpfr wrote: post #150
CatMtKing wrote: post #145

Earlier quoted context omitted.

Hmm, remind me to write a handwriting-recognition program to search through that.

Someone already did exactly that. http://detexify.kirelabs.org/classify.html

This is truly awesome. It's just like using a Chinese dictionary with handwriting input (like Pleco or Nciku).

I don't use LaTeX but now I'm looking for an excuse :)

Re: Mathematicians are chronically lost and confused

#179

Post by hessenwolf »

pcmonk wrote: post #51
hessenwolf wrote: post #40

Earlier quoted context omitted.

I regularly explain my PhD thesis on napkins. The explanation involves twins, one with a broken nose; fish; GPS satellites; and a tractor driving in a field. Somehow this holds a crowd better than non-linear 72-dimensional space, and isometric and rigidity matrices.

But could they (1) extend your results or (2) have enough of an understanding of it to apply it to their work? The benefit of a napkin explanation is that they understand it well enough to know whether it would be useful/interesting to learn the issue better. If they really want to apply your results, though, then they'll need a better understanding. It's true that math papers aren't good for giving a surface-level i…

Very valid point. I suppose I just think we could make room in scientific papers, especially in the digital age, for an explanation of how the idea came about. I spent most of my early research career trying to find out how mathematicians got their ideas in the first place, because no amount of learning mathematics seemed to teach that.

Re: Mathematicians are chronically lost and confused

#180

Post by nnq »

j2kun wrote: post #172
nnq wrote: post #17

Earlier quoted context omitted.

> most of it can't be done that way and remain useful to humans (like assembly language, it's too low-level for most applications). Sincere question (I'm not a mathematician): why can't it be done that way?! On top of an assembly language you can create a higher level language and on top of that an even higher level one, and it is airtight , it has to be or the code won't compile or will throw a runtime exception, th…

This post ignores the central driving force behind why mathematicians do mathematics: mathematics was created by humans for their own pleasure. It's a coincidence that they find such useful applications in the real world, and mathematicians mostly don't care (it's like a bonus if someone finds a nice application of your work, and maybe it provides you more grant money). In this light, why would any mathematician want…

I understand what you say. But my interest is actually into AI, so the same way you find pleasure in creating and understanding interesting proofs, I find pleasure in trying to think about how the human mind works and how to make artificial forms of intelligence that share the properties of human mind but not the damn limitations of the "biological hardware" and of mortality. And I also think that in fields like AI "proof by engineering" is the only way to move forward - we'll have to build them or 'evolve them' in order to prove that they can be built (I know, this is not a line of thinking that mathematicians enjoy :) )

I'd love to see AI evolving to the level at which they create art and music or analogues of these (for their own pleasure and maybe for that of humans too, if the pleasures are compatible).

I know, you'd have to share my belief that strong AI is possible (and really close too I think) and that that minds are nothing but machines themselves and that all forms of intelligence will be comparable, regardless of whether the hardware will be "biology" or "technology", "natural" or "artificial" (in a few hundreds years I think we'll even cease to make the difference between these concepts, they will appear synonyms to our non-human or not-so-human-anymore descendents, just some linguists will identify different etymologies of them) :)

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