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Mathematicians are chronically lost and confused

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Re: Mathematicians are chronically lost and confused

#91

Post by DerpDerpDerp »

polymatter wrote: post #82
simias wrote: post #61

Earlier quoted context omitted.

For me the problem is that math papers/articles lack a proper API documentation. By that I mean that it's very hard for me to understand what a lot of symbols mean because mathematicians (and physicians) love to use single letters to name various concepts and functions. Worse, they also like to use the same symbol to denote different things in different fields. I'm sure it's extremely convenient to have a shorthand w…

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 literacy is assumed.

Re: Mathematicians are chronically lost and confused

#92

Post by DerpDerpDerp »

makmanalp wrote: post #55
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'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 still think we can have our cake and eat it too, but I'm not sure. I think if the purpose is merely to transmit proofs and axioms unambiguously, I think we can have a language that performs just that and nothing else. I think stuff like this exists, but I don't know why it isn't the standard to publish with it.

Math papers are written with high compression using standard tables of translations to reduce the processing load when trying to manipulate several things at once.

Re: Mathematicians are chronically lost and confused

#93

Post by vargas84 »

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…

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?

Re: Mathematicians are chronically lost and confused

#95

Post by nnq »

nbouscal wrote: post #24
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…

I think it can in theory, but the reason it hasn't been done in practice is basically that mathematics is too damn big. Look at Whitehead and Russell's original attempt to do basically exactly what you're asking for: it took them a few hundred pages to build up enough machinery to prove that 1+1=2.

I never got to why this kind of work was stopped.

I only heard the same argument (this is the phrasing from Wikipedia but everybody seems to say something along these lines):

>"However, in 1931, Gödel's incompleteness theorem proved definitively that PM, and in fact any other attempt, could never achieve this lofty goal; that is, for any set of axioms and inference rules proposed to encapsulate mathematics, there would in fact be some truths of mathematics which could not be deduced from them."

...so what?! The fact that there can be truths that can't be deduced from an "assembler language" just means that the system will sometimes just say something like "error: no proof in the current model database for 'fact x' found" and then the mathematician will just add "consider 'fact x' proven as in the defined in modelXYZ" (a model that can have a totally different logic than the current one - think of a model as a library written in a completely different programming language, in software analogy), taking responsibility for the fact the equivalence of the concepts 'fact x in current model' and 'fact x in modelXYZ'.

The long term goal would be unification of as many of the models as possible (even with, what I understand from Godel, as the impossibility of total unification - if something is proved to be impossible, it doesn't mean you can't get great benefits by always getting asimptotically closer to it) preventing such "forced equivalences", but it would still be a working system in the meantime. And more importantly, I guess, the system will make the "forced equivalences" obvious, and label them as problems for mathematicians to solve.

Re: Mathematicians are chronically lost and confused

#96

Post by DerpDerpDerp »

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…

Because until very, very recently, we simply haven't had the means to compute any proofs that mattered.

Rendering even relatively simple high level concepts in to low-level structures and then deriving them in the basic steps takes massive amounts of resources. At least one proof took over 15,000 pages of text (had you printed it).

A secondary problem (which we've been working on for a couple decades now) is that we lack the machinery to translate the results back in to something human readable. At some level, mathematics is about the ability of humans to understand the relationships between things, so computer proofs that only the computer understands don't really help us - especially if we can't relate them meaningfully to our other knowledge.

I mean, people have been trying for hundreds or thousands of years - it's just kind of a hard problem.

Re: Mathematicians are chronically lost and confused

#97

Post by DerpDerpDerp »

nnq wrote: post #95
nbouscal wrote: post #24

Earlier quoted context omitted.

I think it can in theory, but the reason it hasn't been done in practice is basically that mathematics is too damn big. Look at Whitehead and Russell's original attempt to do basically exactly what you're asking for: it took them a few hundred pages to build up enough machinery to prove that 1+1=2.

I never got to why this kind of work was stopped. I only heard the same argument (this is the phrasing from Wikipedia but everybody seems to say something along these lines): >"However, in 1931, Gödel's incompleteness theorem proved definitively that PM, and in fact any other attempt, could never achieve this lofty goal; that is, for any set of axioms and inference rules proposed to encapsulate mathematics, there wou…

Part of the problem is, how do you know you can't prove it rather than just that you haven't proved it, without some kinda of metatheory framework?

Re: Mathematicians are chronically lost and confused

#98

Post by anonymousDan »

vargas84 wrote: post #93
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…

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 would find that interesting. Like others above have noted, a lot of papers are written for reviewers, and not with the intention of making it easy for non-experts in the field to understand the concepts involved. It would even be useful to be able to post questions/comments about the parts you don't get and have people direct you to relevant background reading.

Re: Mathematicians are chronically lost and confused

#99

Post by nnq »

alex-g wrote: post #53
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 is a very good thought. Some current projects are trying to develop computable mathematical foundations in a more structured way. Homotopy type theory ( http://homotopytypetheory.org/ ) is one example that has a lot of buzz around it just now, but automated theorem proving has been trying to work with higher-order concepts for ages now. In the classical approach of "compiling" everything into sets/logic/etc., yo…

> We want to be able to add two numbers and get another number, but we don't want to be able to intersect two numbers as if they were sets, even if they happen to have been built out of sets.

Can't we do this in current mathematics?! I mean, no physicist or engineer ever thinks of numbers as sets, even if you are the kind of physicist that reads and understands mathematical proofs.

Re: Mathematicians are chronically lost and confused

#100

Post by anonymousDan »

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…

Apparently there have been some interesting advances in this area recently:

http://cacm.acm.org/magazines/2014/2/171675-a-new-type-of-ma...

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