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Some stuff I found interesting about number theory research

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Re: Some stuff I found interesting about number theory research

#111

Post by joppy »

webmaven wrote: post #61
rocqua wrote: post #28

Earlier quoted context omitted.

I think the reason for this black-boxing in software and mathematics is quite different. In top-level mathematics its that you have no choice but to treat results as black-boxes because the complexity otherwise is just too high. Its the only way forward, and mathematicians would prefer not to take this approach. In software development, black-boxing is a way of accelerating development. It allows people to abstract a…

But "accelerating development" and "lowering complexity" are, if not synonymous, then at least highly correlated.

In research maths, there are some results you use as a black box because even though you could re-derive the result, it’s just a ton of work and you’re glad someone else has done it. For example perhaps it involves some very tedious case-by-case checking, or a large and long inductive argument. Then there are some results you use as a black box because it would take you (in this case you already have a PhD in the field, so let’s say at least 8 years of field-specific education) literally years of full-time study to become familiar enough with the underpinning technology to even follow the statement of the proof, let alone deeply understand it yourself.

I think there are very few, if any, things in software development that have an analogue in that second category.

Re: Some stuff I found interesting about number theory research

#112

Post by rocqua »

webmaven wrote: post #87
rocqua wrote: post #82

Earlier quoted context omitted.

The difference is in why. Mathematicians do it out of necessity because they are at the edge of complexity they can handle. Developers do it because it is convenient. Developers have a choice.

I think you may be underestimating how much "because they are at the edge of complexity they can handle" is a driver of mainstream SE. In particular, as a factor in the popularity of frameworks and libraries. I'm not taking a stance like "if it was hard to write it should be hard to use", but quite a lot of software only seems to exist in order to hide complexity away in black boxes, often in trivial ways.

Any software dev worth their salt can drop down 1 level in the stack and be productive, if maybe less so. I'd wager most software devs could drop down 2 levels and fully grok everything that is happening.

Re: Some stuff I found interesting about number theory research

#113

Post by jaksmit »

Biganon wrote: post #105
jaksmit wrote: post #97

If anyone is interested in number theory and would like to help some incarcerated people that are working on a number theory paper: I'm a director of the Prison Math Project: http://prisonmathproject.org/ We have a few advanced mathematicians in jail who don't have access to computers. They need help with formatting their work in LaTeX and stuff like that to make it ready for publication (credit offered). They have p…

Very interesting project. BTW the Photos link leads to a 404 error and a Hello world.

thanks and thanks for the headsup!

Re: Some stuff I found interesting about number theory research

#114

Post by sidr »

ska wrote: post #103
smallnamespace wrote: post #99

Earlier quoted context omitted.

Knowing and using a thing and proving it are (usually) different skills.

True; but so are understanding a thing well and being able to use it. I wasn't really talking about proof-based vs. not though - it's true that an analysis course will be more proof based (than Calculus) because that is also a skill you are expected to be developing. However most calculus graduates, even the ones at the top of their class, still have at best a somewhat superficial understanding of the set of real num…

The thing is, you really don't neeed to consider non-Riemann integrable functions if you're actually trying to integrate or measure things. At some point, trying to find non-Borel measurable, but Lebesgue measurable sets serves no practical purpose other than being written down in an analysis textbook. More power to pure mathematicians and their students who enjoy playing these games, but the more applied folks can get around those "problems" by just making some niceness assumptions about your model (like f is Riemann integrable!) and moving on to the more pressing issues in their fields.

Re: Some stuff I found interesting about number theory research

#115

Post by ska »

sidr wrote: post #114
ska wrote: post #103

Earlier quoted context omitted.

True; but so are understanding a thing well and being able to use it. I wasn't really talking about proof-based vs. not though - it's true that an analysis course will be more proof based (than Calculus) because that is also a skill you are expected to be developing. However most calculus graduates, even the ones at the top of their class, still have at best a somewhat superficial understanding of the set of real num…

The thing is, you really don't neeed to consider non-Riemann integrable functions if you're actually trying to integrate or measure things. At some point, trying to find non-Borel measurable, but Lebesgue measurable sets serves no practical purpose other than being written down in an analysis textbook. More power to pure mathematicians and their students who enjoy playing these games, but the more applied folks can g…

Perhaps I was unclear, but we seem to be agreeing.

Re: Some stuff I found interesting about number theory research

#116

Post by webmaven »

rocqua wrote: post #112
webmaven wrote: post #87

Earlier quoted context omitted.

I think you may be underestimating how much "because they are at the edge of complexity they can handle" is a driver of mainstream SE. In particular, as a factor in the popularity of frameworks and libraries. I'm not taking a stance like "if it was hard to write it should be hard to use", but quite a lot of software only seems to exist in order to hide complexity away in black boxes, often in trivial ways.

Any software dev worth their salt can drop down 1 level in the stack and be productive, if maybe less so. I'd wager most software devs could drop down 2 levels and fully grok everything that is happening.

There is a difference between a Python web developer dropping one level down to examine and debug the SQL the ORM is producing, and dropping one level down to examine and debug a performance problem in the CPython implementation.

Re: Some stuff I found interesting about number theory research

#117

Post by webmaven »

joppy wrote: post #111
webmaven wrote: post #61

Earlier quoted context omitted.

But "accelerating development" and "lowering complexity" are, if not synonymous, then at least highly correlated.

In research maths, there are some results you use as a black box because even though you could re-derive the result, it’s just a ton of work and you’re glad someone else has done it. For example perhaps it involves some very tedious case-by-case checking, or a large and long inductive argument. Then there are some results you use as a black box because it would take you (in this case you already have a PhD in the fie…

> I think there are very few, if any, things in software development that have an analogue in that second category.

Sure, if you're positing that the average developer has a PhD in computer science, or software engineering.

But for the average software developer, gaining an understanding of the implementation details of their preferred programming language, or relational database, or browser HTML rendering engine, etc. (or any of a half dozen or so other systems or components they regularly use as black boxes), would in fact require a similar level of effort.

For any given X, an "X Developer" can mean a developer using X, or a developer of X. The latter is where the analogies lie.

Re: Some stuff I found interesting about number theory research

#118

Post by daxfohl »

daxfohl wrote: post #100
yellowcake0 wrote: post #98

Earlier quoted context omitted.

There is no way that AI will be better than us at math in a couple of decades.

Yeah I think I agree now. They'll probably be better at proving straightforward things that don't require any big new insights, but I don't think machines will ever be any good at determining what kinds of new concepts will be interesting to define and explore. Like a machine isn't going to develop calculus just for fun from base principles.

Actually maybe I take that back. I mean, it took how many centuries for humans to invent calculus? Maybe AIs left to their own devices in a reinforcement-learning context could do better.

The interesting thing though, is given free reign to prove whatever they want, they may go off and develop some entirely new field of mathematics and proving really deep stuff, but we just wouldn't recognize it as interesting. Like imagine if such an AI existed 200 years ago and invented Turing machines and proved P != NP, but it wasn't all that great at solving polynomials for whatever reason. We'd have probably thought it was all rubbish and threw it away.

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