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Lost in Math?

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Re: Lost in Math?

#31

Post by [deleted] »

tathougies wrote: post #17
dnautics wrote: post #3

I haven't been in the developer industry for too long, but excepting the haskell community, I would say that the way CS tends to treat math is as guardrails, as in, "you can't do that because of the halting theorem". "you might be butting up against computational complexity if you try doing it this way". "reconstruction of this data shard is impossible because you don't have enough points to determine the equation".…

> Your datastructure is never going to be more than 10-100 deep. Don't worry about it. Just write the most legible recursive algorithm, not the most performant. What? Try computing the 1000th fibonacci number.

[deleted]

Re: Lost in Math?

#32

Post by dkarl »

through_17 wrote: post #21
dkarl wrote: post #9

But complexity theory aims at describing the performance of A over the space of all problem instances and it does so by abstracting away from individual problem instances. I appreciate the effort to extend the story into CS, but I wonder if you have to be familiar with the particular work he's alluding to. The charge (as leveled against theoretical physics) is not that some people do pure mathematical work for the sa…

"the particular work" he is referencing is computational complexity theory; a long-running attempt to mathematically characterize which problems can be efficiently solved with computers and which cannot. I work in complexity theory, and this does make the objects of his criticism obvious to me. I'll try to explain below. Think of "problems" as abstract primitives: sorting, search, graph operations, optimization, etc.…

So, to summarize: the article criticizes researchers who focus only on worst-case complexity. While the theory is beautiful, we can point to many problems for which it does not accurately predict performance.

I still don't get it, because it's okay for some people to be working on purely theoretical problems, motivated by mathematical curiosity and aesthetics. Is there a lack of people working on more concrete problems, bridging the gap between theory and practice? Are there outstanding problems arising from practice that are ignored because supposed "applied" researchers don't actually care about applications?

I think this criticism is a little strange, because most complexity theorists also work on "average-case" complexity

That sounds almost as theoretical as worst case to me. I would expect "applied" complexity theory to provide a theoretical framework for how a practitioner can add information that they know about how their problem differs from the aesthetically ideal problems that arise in theory. Like, my factory floor is not a frictionless plane, aha, here's how you measure a "coefficient of friction," and here's a new equation where you can see how the coefficient of friction affects the results. Or, my dataset isn't a uniformly random blob of bits, so is there a statistical property I can measure that lets me estimate the probability of the working memory of my algorithm exceeding 2.4 times the size of the input size?

Re: Lost in Math?

#33

Post by [deleted] »

dkarl wrote: post #32
through_17 wrote: post #21

Earlier quoted context omitted.

"the particular work" he is referencing is computational complexity theory; a long-running attempt to mathematically characterize which problems can be efficiently solved with computers and which cannot. I work in complexity theory, and this does make the objects of his criticism obvious to me. I'll try to explain below. Think of "problems" as abstract primitives: sorting, search, graph operations, optimization, etc.…

So, to summarize: the article criticizes researchers who focus only on worst-case complexity. While the theory is beautiful, we can point to many problems for which it does not accurately predict performance. I still don't get it, because it's okay for some people to be working on purely theoretical problems, motivated by mathematical curiosity and aesthetics. Is there a lack of people working on more concrete proble…

[deleted]

Re: Lost in Math?

#34

Post by masteranza »

sls wrote: post #20
paulpauper wrote: post #12

But the seductive power of mathematical beauty has come under criticism lately. In Lost in Math, a book published earlier this year, the theoretical physicist Sabine Hossenfelder asserts that mathematical elegance led physics astray. Specifically, she argues that several branches of physics, including string theory and quantum gravity, have come to view mathematical beauty as a truth criterion, in the absence of expe…

In a recent blogpost [1], Hossenfelder responds to a review of her book. All but the first two paragraphs are basically a response to this idea. A brief excerpt: > In most cases, however, physicists are not aware they use arguments from beauty to begin with (hence the book’s title). I have such discussions on a daily basis. > Physicists wrap appeals to beauty into statements like “this just can’t be the last word,” “…

There's no need for anyone to read her book, because it's just a pile of garbage written by someone who doesn't have a clue about modern physics. Hossenfelder is an armchair physicist who failed in her discipline and started writing books for profit. The very idea of naturalness is basically the thing people hope the machines will someday do - namely think and be able to explain what is actually going on through fleshing out noise from the essential mechanisms.

The stuff she writes on her blog are so stupid it's beyond belief. First she criticizes theorists (who write far more reasonable and far less philosophical articles than she) for using advanced mathematics and exploring various possibilities, a minute later she criticizes experimenters for trying to build a better particle collider. She's a person who would gladly see the resources flowing to the less skilled wannabe-physicists with no real knowledge, because she's one of them.

Re: Lost in Math?

#35

Post by tmountain »

doppelganger27 wrote: post #14
tmountain wrote: post #4

Earlier quoted context omitted.

Me too. Listened to a podcast recently where they were talking about things like, "once you know what a monoid is, you start seeing them everywhere". I've tried to express these benefits to coworkers recently. Leveraging ideas from math allows you to take advantage of many decades of research and provides a structural foundation that's substantially more robust than things you might find in the gang of four book or o…

What's the name of the podcast?

The Haskell Cast. This is the episode I was referring to. It's pretty good.

https://www.youtube.com/watch?v=O3EjNRgypXg

Re: Lost in Math?

#36

Post by calf »

sls wrote: post #20
paulpauper wrote: post #12

But the seductive power of mathematical beauty has come under criticism lately. In Lost in Math, a book published earlier this year, the theoretical physicist Sabine Hossenfelder asserts that mathematical elegance led physics astray. Specifically, she argues that several branches of physics, including string theory and quantum gravity, have come to view mathematical beauty as a truth criterion, in the absence of expe…

In a recent blogpost [1], Hossenfelder responds to a review of her book. All but the first two paragraphs are basically a response to this idea. A brief excerpt: > In most cases, however, physicists are not aware they use arguments from beauty to begin with (hence the book’s title). I have such discussions on a daily basis. > Physicists wrap appeals to beauty into statements like “this just can’t be the last word,” “…

I find Hossenfelder problematic because if you look at the rhetoric the blog post itself is relying on intuition and in essence a beauty argument in order to make a sociological point. I think Hossenfelder has a point about the state of professional physics, but the way this is communicated is unclear and lacks this rhetorical introspection as well. It's why her posts draw so much attention; the strong assertions, the arguing, and how the unrigorous writing itself obscures this.

Re: Lost in Math?

#37

Post by tikhonj »

UncleMeat wrote: post #22
dkarl wrote: post #9

But complexity theory aims at describing the performance of A over the space of all problem instances and it does so by abstracting away from individual problem instances. I appreciate the effort to extend the story into CS, but I wonder if you have to be familiar with the particular work he's alluding to. The charge (as leveled against theoretical physics) is not that some people do pure mathematical work for the sa…

SAT is NP-Complete. In principle, this means that SAT solvers don't scale. If we stopped here then we would never have developed symbolic execution. It turns out that SAT and SMT solvers do scale for lots of real world inputs. Cook's proof is amazingly elegant and powerful but fails to inform real development.

SAT solvers only scale sometimes. Complexity theory is a starting point for understanding why this "sometimes" is inevitable, but it never even implies that NP-complete problems are never tractable. Complexity theory gives us an understanding of how powerful and expressive SAT is which very much does inform real development.

Arguing that "SAT is NP-complete and therefore useless" is not misusing complexity theory, it's misunderstanding complexity theory—not misunderstanding some deep result or non-trivial consequence of complexity theory, but misunderstanding the fundamentals that are covered in the first lecture of the first class on the topic.

Re: Lost in Math?

#38

Post by masteranza »

lalalandland wrote: post #26
andrewmatte wrote: post #19

I have long felt like "beauty" in mathematics is just oversimplification. Ironic that intelligent mathematics types get caught up in what could be analogous to socially hurtful stereotypes. I am going to follow the author.

Is't a large part of the history of physics about doing away with wrong assumptions based on beauty? Circular orbits of planets etc.

Except that you've got the point completely upside down! At first people didn't believe in heliocentrism because it used circular orbits, and instead used to believe that the universe is better described from the geocentric point of view which was one ginormous mess from the mathematical point of view. Then people realized that the beautiful addition of conical curves solves the problem completely. I say 'beautiful' because clearly you don't have a sense of mathematical beauty. Mathematical beauty is not only the simplicity of the solutions, but also it's robustness and fluid way in which it fits into the rest of our knowledge. The real simplification that came from it is far deeper than any 'circular orbits' picture you imagine. Naturalness and search for beauty can never be proved wrong. Those principles can perhaps only be applied prematurely, before enough data is collected.

Re: Lost in Math?

#39

Post by through_17 »

dkarl wrote: post #32
through_17 wrote: post #21

Earlier quoted context omitted.

"the particular work" he is referencing is computational complexity theory; a long-running attempt to mathematically characterize which problems can be efficiently solved with computers and which cannot. I work in complexity theory, and this does make the objects of his criticism obvious to me. I'll try to explain below. Think of "problems" as abstract primitives: sorting, search, graph operations, optimization, etc.…

So, to summarize: the article criticizes researchers who focus only on worst-case complexity. While the theory is beautiful, we can point to many problems for which it does not accurately predict performance. I still don't get it, because it's okay for some people to be working on purely theoretical problems, motivated by mathematical curiosity and aesthetics. Is there a lack of people working on more concrete proble…

Let me clarify: average-case does not just mean "over the uniform distribution", though that is of course one distribution of inputs we can study (mostly with applications to cryptography). Average-case complexity can be studied over any family of distributions.

Finding ways to characterize hardness and easiness in terms of underlying structure to those input distributions is exactly what average-case complexity tries to accomplish. The "holy grail" is to classify problems over efficient distribution of inputs. That is, first make the fairly reasonable assumption that the family of formulas you actually run (say) SAT-solvers on came from an efficient computation ("nature", or people coming up with problems). Then, identify properties of those distributions that you can measure and "blame" intractability on.

For example, there's a huge amount of work on these types of parameters for SAT instances. See: https://people.csail.mit.edu/rrw/backdoors.pdf

This type of work could, eventually, directly inform practice.

And of course I think it is okay for some researchers to work on purely theoretical or worst-case problems. Insights from worst-case complexity are often useful in solving the more difficult problems of average-case complexity; it is useful to know where and how the theories diverge.

Reconsidering, I guess I just don't understand the article at all. Maybe he's arguing that TCS doesn't have this problem because we have a hierarchy of theories, where worst-case complexity inspires average-case complexity inspires parameterized average-case complexity which could someday be used in the real world. Over very long timescales (think, centuries) I think complexity theory will produce practical insights.

Re: Lost in Math?

#40

Post by gjm11 »

paulpauper wrote: post #12

But the seductive power of mathematical beauty has come under criticism lately. In Lost in Math, a book published earlier this year, the theoretical physicist Sabine Hossenfelder asserts that mathematical elegance led physics astray. Specifically, she argues that several branches of physics, including string theory and quantum gravity, have come to view mathematical beauty as a truth criterion, in the absence of expe…

> No one has argued that beauty holds precedent over truth

Here's Paul Dirac, writing in 1963:

"it is more important to have beauty in one’s equations than to have them fit experiment. [...] It seems that if one is working from the point of view of getting beauty in one’s equations, and if one has really a sound insight, one is on a sure line of progress."

Here's John Schwarz, one of the pioneers of string theory, explaining why he and a collaborator kept working on it in the early 1970s after quantum chromodynamics turned out to be a better way of dealing with the strong nuclear force and before it emerged as a promising approach to quantum gravity:

"We felt strongly that string theory was too beautiful a mathematical structure to be completely irrelevant to nature."

The idea that string theory is "so beautiful it must be right" is, I think, mostly a strawman -- you hear critics of string theory taking it down, rather than advocates of string theory talking it up -- but the idea that beauty is a reliable guide to truth in physics isn't so strawy.

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