So I would submit that people don't like math in their software for the same reason they don't want to exhaustively show its correctness by other means. In either case, it means a whole lot of work. But I'm sure it's popular to blame mathematicians for stating a problem in its full generality and pathological difficulty (if the attitude of physicists is any indication).

I thought programmers were fond of category theory

I have to give a talk

What subjects in math are yall curious about right now? I could use some inspiration

I wish. No, uncoverable contains some lines from R, and some lines not in R. Given that what we really want is to understand the partition of R and inverse R, the fact that uncoverable code exists at all is a major problem with using coverage as a testability metric.

An intuitive way of understanding this situation is that coverable lines are those lines we can prove either run or not run - meaning covered lines have been proven run, and uncovered lines have been proven not run. Uncoverable lines cannot be proven either way. In practice, uncoverable code is either evaluated at compile time, or it cannot be instrumented (or is not instrumented by choice, for convenience of the tool programmer).

Code review is essential, and given the mass adoption of the github workflow I think this opinion isn’t even controversial anymore. Code reviews are almost as good as pair programming for knowledge sharing, sober second thought, and code quality/standards peer pressure. But code reviews are much better for optional expert approval, automatic policy enforcement, and scheduling (since pair programming requires coordination of work schedules of multiple engineers, while code reviews are an offline process).

I might have lost some context. To be totally clear, code review and measuring test coverage are engineering practices, not tools. There are software tools that help you perform these tasks, but it’s still up to you to do them. It’s also up to you to choose the correct tools, if any, to do your job.

Git is the lingua franca of SCMs, so yeah, you should learn that one for sure.

Computer science is taught as a degree in applied discrete mathematics. You are required to take several proof based courses in all but the absolute worst schools, and are encouraged to take many others. Whether students retain a meaningful understanding of that material, or have occasion to apply it as professional software developers, is a separate discussion.

har

Read this, then decide what to do (you can ignore the title).

For example, investigating what Arnold was talking about here looks interesting.

Was not a fan of that really. I feel like his complaints of any substance aren't new, and otherwise much of it seems pointless.

For instance:



No it isn't. Math is not part of physics, nor could it be, nor does that make sense. I get he's being kinda rhetorical with this, but it's still wrong. Math is separate because it does other things from physics. That's why we put a qualitative difference on it.

He should say applied math is better than pure math, because I believe that's what's meant, and is also wrong.



Uh-huh



That's exactly what books like Stewart do, and then everyone complains about how it's not rigorous enough.

It's a constant cycle. Math is difficult, so everyone tries to blame the pedagogy for being at fault. So you always have complaints like this. Basically every undergrad loves having really strong opinions on how things "should" be done. Not that it's a bad thing, but people need a bit more perspective before prescribing overhauls to the education system.



But things like this is where he goes wrong. I agree many topics in math appear unmotivated when you approach them from a vacuum. But good teachers don't do that, and moreover he seems wrong to want to place all motivation in physics. We can understand the motivation for developing abstract algebra out of studying number fields, where factoring becomes harder to do. We don't need strict real-world applications for that.



I don't think it's really the case that a bunch of people mistake mathematical models for reality itself. If anything the complaint here is a philosophical one, anyway.



I don't even get the complaint here. It seems he's mad for the sake of being mad.



That's because that's what math does you idiot. That's not a fault of math, that's why it's math, and that's why it's not physics. This entire complaint is a waste of breath. If you want to do physics, you can go do physics, don't tell people in number theory that what they're doing is pointless because they haven't found a use for the twin prime conjecture in quantum physics.



So, what he's saying here is inaccurately accurate. What he said is a proper geometric interpretation of the determinant. But it also fails to capture the multilinear form properties, and how it relates to the wedge product and other stuff that's really useful in physics. The complaint here is a pedagogical one: yes, in intro linear algebra courses, this sort of intuition should exist, and I'm on board, but to try and dismiss the mathematical rigor behind appropriate definitions because it's not easy to intuit is just ego.

Same with this:


Yes, the Cauchy axiomatic based approach to group theory is something of a misleading one, and betrays the history of why group theory was invented. But let's read at what he says:



Literally what he said he is just a restatement of the group axioms in terms of transformations. That's not "totally different", and he's saying something in a poor and roundabout way: what makes groups important isn't the group itself, but how the group acts on another object. But every algebra course I've ever taken paused to talk about this. So his complaint isn't coherent, it's really not, he's just finding more things to rant about.

Plus, for the typical undergrad, to go through the volume of material necessary to follow the historical rise of group theory is far much more work than a few weeks of trust that the axioms lead to important results.



And again, every differential topology book I've ever seen discusses this openly. John Milnor's lectures make this very clear. Literally all discussion I've ever seen about smooth manifolds says this, then lays out the axioms. It's just simply not true the portrait he's painting of math pedagogy.



Yeah, that's not the situation.

I'll be fair: he does hint at some truths, personally I believe undergrad curriculum could be better motivated, and spend more time on examples and concrete stuff. Sure, but he's putting the cart before the horse on a bunch of topics, and fails to see the sense of learning abstract materials before jumping in to examples.

Also, I'm not sure how that was supposed to help me pick a topic. Maybe I should go teach physics, from reading that?

For example: try watching and see if Milnor's presentation style matches up with Arnold's complaints.

As Sy said, mathematics is the art of the possible, right? I think the physicists understand this better than anyone else. Except maybe when they do things with math that aren't actually possible?

If Vladimir's excellent (in my opinion) rant isn't your cup of tea, I could only suggest reading Pierre Cartier's take on this theme.

mathematical constructivism bingo.

Sort of.

My original thought was to offer the calculus of variations as a suggested topic, which has an illustrious history in physics going back to Euler and Lagrange, through today in people working in (I think?) subjects involving differential geometry, and (suprise!) is one of the more beautiful and also useful things that mathematicians have produced. And it seems to have a lot to do with physics, even now that people use it for all sorts of things like computer vision or machine learning.

There's a deep and opinionated GTM on differential geometry by R.W. Sharpe, with the subtitle "Cartan's Generalization of Klein's Erlangen Program". Klein himself didn't hesitate to take inspiration from physics rather than reasoning from a deductive framework in the modern style of Hilbert. I could also add that Riemann's writings (I have read) are incredibly difficult to understand, and I bet this has something to do with his deep geometric intuition.

Just to be clear, Arnold was addressing the attitude that he had observed coming out of France. Milner, on the other hand, is a fantastic mathematician who (I believe) takes after Poincare (who basically invented modern dynamics and topology).

Please geometric intuition for a graph with a negative length cycle tia.

Edit: I mean, I’m not close enough to a mathematician or a physicist to know whether this is wrong. It sounds wrong, but I don’t know. However, I’m deeply suspicious of any argument that suggests, even in part, that experts should approach a field differently because of pedagogical concerns. That seems like an academic version of “won’t somebody think of the children”. Suggesting that the material should be focused on physical or intuitive areas is one step removed from suggesting that non-physical or non-intuitive areas shouldn’t be studied. If the abstract and formal is an area of overwhelming interest among academics there is probably a good reason for it.

You chopped off the full quote. What do you think he meant with the last sentence in this paragraph?

[quote=Vladimir Arnold]
Mathematics is a part of physics. Physics is an experimental science, a part of natural science. Mathematics is the part of physics where experiments are cheap.
[/quote]



Applied math doesn't exist according to Arnold:

[quote=Vladimir Arnold]
By the way, I shall remind you of a warning of L. Pasteur: there never have been and never will be any "applied sciences", there are only applications of sciences (quite useful ones!).
[/quote]

Most of what Arnold wrote reflects the times he came from. In the early 20th century there was far less separation between math and physics.

It's been said that mathematics tends to obliterate its own history. I would argue that you can't really appreciate that history without understanding physics. Of course that's not to say that we can't have specialists in the 21st century who only work in formal logic and are totally oblivious of this history. Whatever. I think you've mentioned that computer science has similarly branched out into multiple subjects.

Finally, I didn't mean to imply (as Arnold did) that mathematics ought to be taught this way or that, but only meant for Reid to consider the provocative position of Arnold in asserting the primacy of physics over mathematics, if only as a way to inspire him to pick a topic for his talk (like the calculus of variations), since tracing a subject back to its classical roots in this case would seem to make for an engaging topic that is surprisingly accessible. In fact, this accessibility largely has to do with pulling back some of the abstraction that mathematicians have since layered onto the subject, back to the 19th century where one could reasonably be expected to understand the entire subject at the level of rigor the times demanded. There's an argument to be had that this makes the subject more pedagogical, since it is more honest about the context in which the subject developed, but I won't try to argue that a more historical approach necessarily makes for better pedagogy in all cases.

For example, if I wanted to study geometric group theory, I would consider reading about combinatorial group theory, which is the old fashioned moniker for the subject, but I imagine not at all a bad literature to learn a few things from if you want to appreciate what the later, more abstract approach, did for the subject.

Incidentally, an early innovator of the subject of geometric / combinatorial group theory was Camille Jordan, whose proof of the famous "Jordan Curve Theorem" had been long dismissed as flawed, following a supposedly more rigorous proof by Oswald Veblen in the early 20th century.

However, when Thomas Hales proved the theorem through modern formal tools, he himself couldn't actually find a flaw with Jordan's original proof, nor could any of the mathematicians he contacted about it:

And just a paragraph or two down in Hales' history of the theorem, just read how the supposedly more "rigorous" program of Veblen fared.

I think that Arnold would cringe that you would pass physics off as mere "real-world application". Newton invented fluxions in the course of inventing modern physics. What Arnold is referring to here has more to do with geometric intuition as understood and used by physicists than it does with examples in a textbook used as carrots to motivate students suffer through the "abstract" or ostensibly useless pure mathematical part. In Arnold's mind, they are two sides of the same coin, and he is mostly reacting to those in mathematics who would relegate physics to "applied" mathematics and not an inexorable part of the subject.

Physics is not inexorable to mathematics.

You can play that game, sure. But you'd be missing an opportunity to learn about your subject by appreciating its history.

My argument is that this history is not even incidental, but essential. The only reason you can ignore it is because of abstraction. There's simply no way that mathematics would exist in any meaningful way without physics. In fact the same can be said to apply today, where a lot of cutting edge mathematics is inspired by modern physics.

I agree that, in many cases mathematics was inspired by physics, and knowing the history of your subject is important, and that physics and math go together well and should be done together. But that characterization is not really a good one. From what I know, and granted I'm not a historian, but my understanding is that nearly all of topology was developed on its own, within the realm of pure mathematics, with no relation to any specific physics problems, and post hoc has been found useful in many areas. In fact, topology is one of the more rapidly expanding subjects in applied mathematics.

The idea being presented is that Real World Problems -> Mathematical Approaches, the point is that there's also examples where Mathematical Approaches -> Real World Problems. It's just too restrictive and narrow to imply we should teach a subject in this direction when that's also an incorrect view.



And this is also part of it. The sort of changes required to bring physics and history into the discussion of math would increase the workload by a substantial amount. Given the amount of restrictions, it makes sense to maximize the "core content" at the sacrifice of a bit of context.

The people who really care will learn it anyway.

I also believe much of the stuff Einstein wrote about special and general relativity was inspired by the work done by pure mathematicians in hyperbolic geometry - work that came as a result of doubting what physicists assumed to be obvious, and proving results from first principles that lead to counter-intuitive results, and then rediscovering the value of this pure math work in physics.

Do you have something a bit more topology-related? I do appreciate the suggestion and will touch on it, but I think topology is pretty cool.

I was thinking of finding stuff in topological data analysis if you know anything about that stuff.

This is not true. Topology has been been related to dynamics since its early days:



http://www-groups.dcs.st-and.ac.uk/history/HistTopics/Topology_in_mathematics.html

Can I make it about all three of the things we've been talking about (topology, calculus of variations, and dynamics)?



http://www-groups.dcs.st-and.ac.uk/history/HistTopics/Topology_in_mathematics.html

Got it. I'm creating a list of priorities for things to learn, so I'll put this after Git and some other business practice ideas.



Sounds a whole lot simpler than I previously thought. Thanks for the pointers. Maybe when I get in full swing C++ development, I can start playing with these tools.

Honestly, I was going to rebut some of your stuff by appealing to differential equations having been largely inspired by physics, but when you look into it, a lot of the early work in calculus seems to have been inspired by geometry as understood by the Greeks. But suffice to say that mathematics would be a much less rich subject without differential equations.

But anyway, what you've described here sounds like the approach to mathematics and physics that Arnold would have advocated. It was the super abstract stuff that artificially filtered out the physical connection on purpose that he found so repugnant.

Point taken, but I still think that's a misrepresentation of the relationship a bit. It's certainly the case that not all important results and applications in topology rose out of immediate concern for physics.

If you know of something accessible to someone who knows nothing about dynamics or calculus of variations, sure.

We could also just search for stack exchange threads on the topic:

https://math.stackexchange.com/questions/280530/can-you-provide-me-historical-examples-of-pure-mathematics-becoming-useful

As an unrelated note based on that thread, "imaginary number" was an insult Descartes gave to the idea as he thought it was absurd to accept the notion.

Okay. The passage I quoted from Mac Tutor on the history of topology was meant to demonstrate the connection between topology and dynamics. As the passage from that post said, the calculus of variations makes its appearance through the beautiful subject of functional analysis. However, the subject of functional analysis and Hilbert spaces as developed in mathematical physics is extremely well-trodden territory, and the subject is so standard now (quantum mechanics has traditionally been formulated in terms of Hilbert spaces), that you are probably the wrong person to teach it, unless you really are interested in learning standard fare mathematical physics. Mathematics: It's Content, Meaning, and Methods is a classic book that covers all this stuff in various chapters (the relevant ones being calculus of variations, topology--including connections to dynamics and vector fields, and functional analysis) in a straightforward and accessible way that gets to the heart of the matter and also explains the history.

Instead of all this, though, I think a much better approach would be to take a geometric tact. One extremely elementary way to make inroads on the subject of calculus of variations is not through physics, but through geometry; specifically, through the isoperimetric problem, which goes back to the Greeks:



This is the first page of a long and interesting history of the problem approached through geometry, showing in fact that you can get away with drawing cute little pictures and imagining what happens when you contort its shape in your mind, and on page 12 we meet up with the analytic approach, due to Euler:

Then, if you want to read more about the interesting geometric methods of Steiner mentioned in those notes (among many other things!), with lots of interesting connections to convex and differential geometry, you can do no better than check out the beautiful book on Geometry by Marcel Berger.

I know well of this book, I'm a huge fan of it. In fact, since you posted it I just brought up a PDF to read the section on analyticity of functions, and the explanation was one of the most clear I've ever read.

Thanks for the links, I think the idea for this seminar/project is to find a specific paper, so something like a book isn't appropriate, but I may need to refer to one anyway.

The isoperimetric problem is probably better for an undergraduate audience. I guess you'll be talking to grad students, so maybe that paper I linked to by Victor Blasjo is too elementary, and, well, not very contemporary (but hey, it was edited by John Stillwell, so no surprise there).

Oh man, to rekindle this mathematics versus physics pissing war, look no further than page 12 (548) of Blasjo's paper on the isoperimetric problem, where he discusses Gromov's "flaky" intuitive physics-inspired proof.

In light of this, we might simply think of "physical intuition" as merely a "frontend" to the underlying dynamics and geometry that the universe "runs" on. Many users will never need to look at the underlying dynamics in their full mathematical glory, but instead are content to draw upon their intuitive understanding of their experience living in a world governed by these very dynamics. But nevertheless, because the underlying dynamics are sound (they exist reliably enough for us to live within them, after all), we can translate our intuition into more precise mathematics (but which still require testing to verify).

What I am saying is that Galileo and Newton were reverse engineering the underlying dynamics, working backward from their intuitive understanding, and setting up lots of experiments to get there (with pencil and paper being the cheapest such experiments ). But in the end, Fermat, Newton, and Leibnitz realized it would be easier to build a decompiler, which lets us draw geometric pictures from our brain's intuitive picture of the underlying the dynamics, and mechanically turns out the underlying dynamics in their native language (differential equations).