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Lost in Math?

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Re: Lost in Math?

#61

Post by mikorym »

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…

Exactly. Mathematics is the study of which statements follow from which assumptions.

I have difficulty what the term "economics" means, but usually my best intuition is to regard it as modelling of economic phenomena rather than engineering economy from theory.

Re: Lost in Math?

#62

Post by bonoboTP »

tikhonj wrote: post #37
UncleMeat wrote: post #22

Earlier quoted context omitted.

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, i…

It's common. My lecturer in the first lecture gave a motivation why we learn about this saying:imagine your boss proposes you compute , then you - informed by this lecture - will recognize it belongs to an infeasible complexity class and can tell your boss it won't be possible.

We did learn the definitions but "worst case" was never highlighted as such, it was naturally assumed. The closest we got was the discussion that constant factors may matter for small problems and O analysis masks those differences.

Re: Lost in Math?

#63

Post by chopin »

BucketSort wrote: post #11
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…

This has been a common point of complaint in Physics since Einstein, and probably before. The idea that you can come up with ideas about how the universe works in the absence of experimentation doesn't sit right with many scientists. There are many scientists, like Edward Witten, that are working on things that are purely mathematical at the moment, and to some it seems like an inbred mathematical fantasy. In their d…

>In their defense, this is why what they do is called theoretical physics.

In my opinion, theoretical physics is about explaining observable phenomena in a falsifiable way (I am with Popper here). Otherwise an omnipotent god would be an equally good explanation for a given phenomenon.

Re: Lost in Math?

#64

Post by chopin »

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.

I agree as it is important to know your boundaries. But I also like the GP prefer legible over performant code whenever it is obvious that I am not going to need more performance. In almost all cases I encountered the necessity to optimize code the reason was I/O bound. I don't recall any instance of algorithmic performance being a problem.

Re: Lost in Math?

#65

Post by BucketSort »

endgame wrote: post #54
BucketSort wrote: post #50

Earlier quoted context omitted.

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.

Oof that is bad. I admit that I only used it to learn the basic concepts that just did exercises and read the prelude documentation ( which is fantastic ). Thanks for letting me know.

Re: Lost in Math?

#66

Post by BucketSort »

chopin wrote: post #63
BucketSort wrote: post #11

Earlier quoted context omitted.

This has been a common point of complaint in Physics since Einstein, and probably before. The idea that you can come up with ideas about how the universe works in the absence of experimentation doesn't sit right with many scientists. There are many scientists, like Edward Witten, that are working on things that are purely mathematical at the moment, and to some it seems like an inbred mathematical fantasy. In their d…

>In their defense, this is why what they do is called theoretical physics. In my opinion, theoretical physics is about explaining observable phenomena in a falsifiable way (I am with Popper here). Otherwise an omnipotent god would be an equally good explanation for a given phenomenon.

I consider their role to be the people that are exploring mathematical models of phenomena that may or may not exist, so when the regular physicists find something new, there's some mathematical precedent they can use. Physics is about explaining "observable phenomena in a falsifiable way," theoretical physics is broader than that. I think they both have their place. I don't know why people put them at odds against each other, I thought they have been shown to have a long and fruitful relationship ( the experimentalists and the mathematicians ). The relationship between Faraday and Maxwell should be the shining example that shows how the two sides of understanding nature balance each other.

Re: Lost in Math?

#67

Post by Nasrudith »

neokantian wrote: post #52
roenxi wrote: post #23

Earlier quoted context omitted.

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…

There is a difference between non-repeatable in same state and non-scientific. Applying absolute standards of rigor is ironically also unscientific.

We know that hyperinflation is a way to screw over an economy utterly. It can and will fail and in the best case be the equivalent of dissolving the currency and going bankrupt.

The most benign form of it that may not techically count would involve massive growth as well and the devaluation wouldn't be a pathology but a reflection that yes, a well honed spear, flint knives, a badket, and a few carved bone pieces of jewelry may have been respectable wealth for nomadic hunter-gathers but aren't really worth anything compared to even the contents of a jalopy in the great depression.

Just a steel knife or pot would be grand artifacts because they are better in performance than anything else they could find.

That their old currency isn't worth anything is reflective of the fact that past production has been rendered obsolete and the old goods are worth little.

Re: Lost in Math?

#68

Post by tathougies »

chopin wrote: post #64
tathougies wrote: post #17

Earlier quoted context omitted.

> 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.

I agree as it is important to know your boundaries. But I also like the GP prefer legible over performant code whenever it is obvious that I am not going to need more performance. In almost all cases I encountered the necessity to optimize code the reason was I/O bound. I don't recall any instance of algorithmic performance being a problem.

Tco has little to do with timing performance and all to do with keeping code legible in functional languages.

Re: Lost in Math?

#69

Post by tmountain »

BucketSort wrote: post #49
tmountain wrote: post #42

Earlier quoted context omitted.

I'm mostly in agreement with you with the caveat that I've seen lots of "throwaway code" turn into production code that folks end up depending on. These days, if something seems like it has even a remote chance of being adopted into a production system, I'll try to make sure that it's on a solid foundation. Also, in terms of learning Haskell and its cousins, my advice is to start building stuff right away. It's easy…

It is actually quite important to read up on Haskell if you don't have a background in modern FP. It isn't like other languages where you just see what's different from what you already know. There's a lot of conceptual stuff that helps understand what's going on. I actually started by jumping in myself, so I just wrote programs that were quite bad in Haskell terms. I usually recommend jumping into a language right a…

I should clarify that by "jumping in", I'm suggesting getting to a structural foundation that includes monads, applicative functors, monads, and their ilk, and then moving into writing code. I say this because I spent a few years learning this stuff without doing anything remotely practical, and I think that's too long.

Re: Lost in Math?

#70

Post by drilldrive »

mikorym wrote: post #58
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 whet…

>We don't really know if [mathematics] is sound. My apologies, I was hoping to be concise. I meant to say that mathematics is sound relative to the assumptions, which is exactly correct. But if the assumptions do not hold in reality (and they never quite do), then the theory as a whole is slightly off. I mean to say that mathematics is never 'correct' with reference to reality, but of course is always 'correct' with reference to the axioms.
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