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Lost in Math?

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Re: Lost in Math?

#51

Post by bitL »

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…

It's like you have some problems in graph theory that are NP-Complete/Hard, so in theory you are out of luck. But if you restrict yourself to a certain type of graphs, or you change your problem from e.g. computing a chromatic number on a certain type of graphs to instead decide if for a given K such K-coloring exists, suddenly there is a polynomial algorithm. And practically-wise, you might be fine with just K={2,3,4,5} and couldn't care less about other values of K for your practical case; so while in general you can't solve it, in practice you have a fast algorithm you run 4x in a row and are done.

Re: Lost in Math?

#52

Post by neokantian »

roenxi wrote: post #23
throwawaymath wrote: post #13

> About 10 years ago, in the wake of the 2008 financial crisis, the Nobel Laureate economist Paul Krugman made the same point with respect to economics and mathematics in an influential article titled "How Did Economists Get It So Wrong?" His main answer was: mistaking mathematical beauty for truth. "As I see it," wrote Krugman, "the economics profession went astray because economists, as a group, mistook beauty, cla…

It is possible to be too critical about a magazine article, but it can't be stated enough that the evidence that economists are abusing maths is weak. Most of the evidence points to the economists abusing assumptions , which is hardly a mathematics problem. Most assumptions can lead to elegant math. The biggest problem in modern economics as practiced is the tacit assumption that because practically all people would…

> Most of the evidence points to the economists abusing assumptions, which is hardly a mathematics problem.

Agreed. Economics is about the real world. Therefore, it has to be empirical. That means that axiomatically deriving conclusions from assumptions is not legitimate in economics. Still, in the context of empirical knowledge, we have only two usable methods: the scientific or the historical one. Economics cannot be validated by testing experimentally. Hence, economics cannot possibly rest on a sound method. Therefore, economics is fundamentally not a legitimate academic discipline.

Re: Lost in Math?

#53

Post by kevinventullo »

salty_biscuits wrote: post #28

If you count all the work in AI/ML then the criticism has been overwhelmingly in the other direction, i.e. to much "just trying stuff to see what happens" and not enough "really understanding what is going on". Always seemed like a weak criticism to me honestly. You can advance theory, or you can advance through experimental insight. Neither is the right or wrong path, just whichever seems like the best way to make p…

Yeah, I feel ML would have been a more apt analogy than complexity theory. For some problems it's an ugly, "brute force" approach that works really really well.

Re: Lost in Math?

#54

Post by endgame »

BucketSort wrote: post #50
mywrathacademia wrote: post #43

Earlier quoted context omitted.

Isn't the book you linked a bit dated?

The main concepts are presented well, but people have complained about it being dated. What in particular do you find outdated?

I saw someone confused on haskell-cafe yesterday because it doesn't cover Applicative so their Monad instance was invalid.

Re: Lost in Math?

#55

Post by cma »

throwawaymath wrote: post #13

> About 10 years ago, in the wake of the 2008 financial crisis, the Nobel Laureate economist Paul Krugman made the same point with respect to economics and mathematics in an influential article titled "How Did Economists Get It So Wrong?" His main answer was: mistaking mathematical beauty for truth. "As I see it," wrote Krugman, "the economics profession went astray because economists, as a group, mistook beauty, cla…

> On the other hand I haven't seen that in finance. Highly exotic (read: "beautiful") mathematics is extremely rarely used in financial engineering. Pricing derivatives is decidedly mundane work compared to the brain-meltingly abstract mathematics deployed in high energy particle physics research.

Not sure if they stick to mundane math, but Renaissance Technologies did pretty well hiring mathematicians and theoretical physicists.

Re: Lost in Math?

#56

Post by drilldrive »

Coming from a mathematics background, I personally do not care so much for the 'beauty' of mathematics, and am moreso interested in clarity of properly abstracting and insight to the resulting formal theory. I feel that physicists care more about such intuitive ideas than anybody else.

Regardless, you only can become lost in math if you have bad premises. Mathematics is a relative subject, abstracting the arbitrary of reality to axioms. And if the axioms do not hold, the theory is bunk. It will always be the case that more granularity is required in real-life situations. Mathematics is precise and sound; it's not gospel.

Re: Lost in Math?

#57

Post by speedplane »

drilldrive wrote: post #56

Coming from a mathematics background, I personally do not care so much for the 'beauty' of mathematics, and am moreso interested in clarity of properly abstracting and insight to the resulting formal theory. I feel that physicists care more about such intuitive ideas than anybody else. Regardless, you only can become lost in math if you have bad premises. Mathematics is a relative subject, abstracting the arbitrary o…

> Coming from a mathematics background, I personally do not care so much for the 'beauty' of mathematics, and am moreso interested in clarity and insight.

Beauty is a subjective emotion, it's not so easy to quantify. You may not find mathamatics beautiful due to it's clarity and insight, but many others would precisely because of that.

And it's not just math. Beautiful paintings, architecture, legal arguments, religions, etc., are often believed to be beautiful because of the clarity and insight they provide as well.

Re: Lost in Math?

#58

Post by mikorym »

drilldrive wrote: post #56

Coming from a mathematics background, I personally do not care so much for the 'beauty' of mathematics, and am moreso interested in clarity of properly abstracting and insight to the resulting formal theory. I feel that physicists care more about such intuitive ideas than anybody else. Regardless, you only can become lost in math if you have bad premises. Mathematics is a relative subject, abstracting the arbitrary o…

> And if the axioms do not hold, the theory is bunk.

You assume the axioms hold. That is what an axiom is: an assumption.

> Mathematics is precise and sound; it's not gospel.

We do think that mathematics is precise. We don't really know if it is sound (unless I am missing something). For example, what is a set? It is not a "collection of things". Rather, it is an object in some mathematical setting.

The topic of whether mathematics is "correct" is something else. We can all go out and build a DIY logical machine from scratch and see for ourselves that the mathematics we use give the results that we expect. Applied mathematics in this sense concerns itself with mathematics that suitably describe real situations.

Re: Lost in Math?

#59

Post by mikorym »

throwawaymath wrote: post #13

> About 10 years ago, in the wake of the 2008 financial crisis, the Nobel Laureate economist Paul Krugman made the same point with respect to economics and mathematics in an influential article titled "How Did Economists Get It So Wrong?" His main answer was: mistaking mathematical beauty for truth. "As I see it," wrote Krugman, "the economics profession went astray because economists, as a group, mistook beauty, cla…

I agree totally. A lot of people tried to bury the 2008/2009 crisis in philosophy, whereas there is a simple "model" to described what happened: bad debt.

Re: Lost in Math?

#60

Post by mikorym »

> the theoretical physicist Sabine Hossenfelder asserts that mathematical elegance led physics astray

I feel the other way around: Applied mathematicians and physicists led the pure mathematicians astray. But I say this for a different reason. I feel that physics has become convoluted with a plethora of theories where this same beauty is interpreted wildly differently by different people. In other words, people all have different ways of thinking, different notions of beauty, and ultimately, this manifests into different (competing) notions in physics. These may even be equivalent notions and they may embody the desired perspective on beauty, but instead of consolidation there is extended differentiation.

My point is that physics has become ugly exactly because of physicists' ignorance towards mathematical elegance in favour of personal beauty. I don't think physics can become consolidated without exactly a stark appreciation for elegance.

IMO this is why category theory took so long to start appearing in physics: The physicists are caught up in their own idea of beauty rather than the mathematical tradition of finding the minimal sufficient proofs and theories (which I call elegance).

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