Granted, I think get the impression we might be overloading the term "mathematics".
And it seems to me that you learned the concept of functions through programming first but that there's not evidence that you couldn't have learned it from mathematics at first. Functions are a pretty easy concept, so I think it's pretty easy to introduce it from a variety of points of view. So I'm not sure the anecdote backs up any argument about learning programming makes mathematics easy.
I am of the opinion that programming can be used to explore and learn mathematical ideas and am a big proponent of that, but that is something different than "I know how to program so mathematics will be easy now".
So my question is: Did you study analysis and would you credit programming in helping you get good at it?
As a student that currently learns Analysis and linear algebra, it is far more complex and abstract than calculus. It is not neccessarily harder to learn but different. And adapting to this paradigma takes time (and effort).
It is similiar to learning Assembley as a Python developer. Knowing Python will help with Assembley. But the levels of abstraction are obviously different and will require a lot of learning.
Where is calculus "often taught" without "theorems/proofs that are used to build up calculus"?
Analysis discusses less well behaved functions and spaces than these.
Unfortunately, since engineering students outnumber math majors by a large margin, the departments cater to them and not the math majors. The latter study analysis in later years - usually third or fourth.[1] Universities with strong math programs may offer it in the first or second year.
[1] I just checked my undergrad's program. They begin taking analysis in their 4th year, and it's offered only one semester a year!
I was taught Calculus in high school, years 11 and 12 (two optional senior years for those that want more education prior to university, etc rather than transfer to a trade apprenticeship or leave school altogether).
That was a mixture of basic theory and proof and some computation (eg: derive an expression for the volume of intersection of two pipes at right angles, etc).
University went straight in to heavy analysis and foundations (for Math 100 - math for the serious (math, physics, hard chemistry)), with seperate streams for "casual math" - engineering, business, law, etc.
Engineering math prepped people for calculating dynamics and kinetics with vary loads, masses, thrusts, harmonic forces, network propagation, mesh computations, etc.
This was Australia in 1980.
This is insane to me. In the UK about ten years ago, on literally the first day of my degree, my first class was real analysis. Yes, it was the easy stuff like proving sequences and series converge, various things about continuous functions, but we learnt how to prove it all and the exam was all about proving various things. And we built up to harder stuff as the year went on.
What is even happening if math majors aren't studying analysis until their fourth year?
Sorry I'm so incredulous, it's just that I literally don't know what I would've been studying if analysis had been delayed so much.
We are; the challenge is there is a near infinite difficulty in both "programming" and "mathematics" and at a certain point, the skill sets diverge greatly.
I’d be curious what people’s coolest math tricks are that they’ve used at work. I did like one Taylor Series expansion and felt cool for a week.
Why is most STEM research produced in Eastern Europe, the US, or China?
This really depends on what metrics you are using and has infinite room for gaming.
US undergrads (both native and foreign born) absolutely crush all of these nondescript places you are suggesting are better, and graduate programs make the gap even wider.
A non trivial fraction of those are educated K12 in the US system.
The US was a neat idea we had but it went a little off-kilter around the second half of Pennsylvania.
After you pull out HPYSM, these are not ridiculous rankings either.
Georgia Institute of Technology is 38th in the world by research impact, with the top ten or twenty being your standard prestige universities. (Elsevier)
If the US is so bad, why is a poorer southern state competitive with far more prestigious institutions from other countries you would perceive as better?
This is ahead of University of Tokyo, Urbana Champaign, etc.
To specifically respond to just that one point, New England is in many ways an unusual region so I don’t think we should use the scores from here to come to any particular conclusions about the country in general.
This shouldn’t be taken as a response to your other points, most of which seem reasonable enough or at least I don’t know anything in particular about them.
Sorry for being a bit reactionary there, sometimes a certain opinion is common enough that you respond like it's being stated due to similar comments being a prelude to it.
https://www.edweek.org/teaching-learning/calculus-is-the-pea...
It is possible that the majority of students forget it though, I tutored calculus for a bit and it didn’t always seem like their hearts were in it.
> I would not sell yourself short on being able to understand.
I have a master's in mathematics and have continued to take courses while working full-time. It's just that I'm more than aware of the amount of mathematics I do not know compared to an active graduate student in mathematics and above. Or maybe you meant the royal you.
But no I don't think experience doing boot camps and churning out React apps and gluing together APIs would help with learning mathematics.
Go learn a language that seriously challenges you like Haskell.
I will agree it’s been more difficult learning math than programming for myself as well: but that’s because math is geared and targeted for people who like doing symbolic logic by hand. Math people think we’re a level below them (we are in some ways from a working perspective) so they tend to write off complaints like this as us just wanting to make math more like programming.
The fact we don’t have a nice and intuitive way of writing math via a keyboard is proof that these two fields, which should be tightly coupled, are not on the same page.
Simply not true. Most mathematical statements, e.g. proofs of existence have no relation to a "program".
>math is geared and targeted for people who like doing symbolic logic by hand.
No, it is not. There is absolutely nothing interesting about symbol manipulation, it is always the least interesting part of a proof. It usually is the part the author handwaves away, while focusing on the actual interesting parts, the idea behind the proof and the possible intuitions for them. Textbook might include them, but as training and because you need to be more explicit when teaching.
Okay. What is a proof other than a step by step explanation for why something’s true? You’re getting caught up in “program” when it’s objectively the case that all math follows a series a steps. A lot of those steps are “handwavy” I’ll give you that. That’s not relevant to what I said though.
>there is nothing interesting about symbol manipulation
Cool. You missed my entire point again. Math education is geared towards a certain set of people who pick up on (and gain an interest in) the language of math. Only once you get to higher level math do you even start to get alternative visualizations etc (at which point you’ve weeded out a ton of people who would have benefited from e.g. visualizing numbers as groups of shapes). There are a million ways to teach math and we’re leaving a lot of people behind. That was my point
Cooking instructions are a series of steps as well. Would you also claim cooking, mathematics and programming really are all the same thing?
It seems to me that it is an important point though.
> Even a non-constructive proof of existence for x can be significant in a programmatic context
Which has no relation whatsover to proofs and programms being the same thing.
There is a reason Turing is considered one of the greatest minds to ever live. He didn’t just invent a concept. He invented a completely new branch of science. We would’ve gotten there eventually but his idea to solve the Entscheidungsproblem using his machine was such a step ahead of the times that we christened him the father of an entire science.
There is no doubt that Turing was bright, very bright indeed, but next you'll be claiming he cracked the Enigmna Code or something.
Much (most?) of math is quite different from it. Proving that there is a well ordering of the reals, and simultaneously proving that it is impossible to show you such an ordering: Very different from skills needed in programming.
This is actually a problem of interest to me, so I’ve definitely been exposed to it and the limits of modern computation. But I’m not speaking strictly about the modern day CPU.
The set of computable real numbers is countable and thus has measure zero. In other words, almost all real numbers are non-computable, and almost all has an exact definition.
So you can’t compute the real numbers unless you’re meaning something else by computing.
That doesn't make any sense. What do you mean by that?
> I agree it’s been more difficult learning math than programming for myself as well: but that’s because math is geared and targeted for people who like doing symbolic logic by hand.
Mathematics is not about symbolic logic. Mathematics is the study of idealized structures, their properties, and their relationships. Symbols are just a convenient shorthand. They are not the mathematics in and of themselves.
> The fact we don’t have a nice and intuitive way of writing math via a keyboard is proof that these two fields are not on the same page.
What does that have to do with anything? Although LaTeX and the like are pretty decent at it, there's a lot of things we can't do easily via a keyboard. Why is that a constraint on anything or relevant?
Not trying to be provocative, but I honestly have little idea what you're talking about.
The only areas in CS that come close to that in terms of difficulty are discrete math/algorithms related ones and the CS chairs that are involved in such research also tend more towards the applied side of those topics. Really theoretical discrete math stuff resides in math departments most of the time.
Still, having also taken some discrete math classes, in terms of difficulty those somewhat paled in comparison to differential geometry/topology/abstract algebra kind of stuff. I can't even imagine how difficult it must be to be doing research in those areas.
>It seems to me that math-first people would probably find C to be an abomination.
I certainly don't. You might do so if you wanted programming to be an expression of pure mathematics, but I do not think that is the right approach. C does well for what it is an abstraction over an underlying, real machine and thinking of it as an abstraction is the right thing.
Someone may be led into thinking you're countering the OP.
But what level of programming and mathematics are you comparing here though? because college-level algebra and calculus is really not that hard imho (once it "clicks" for you, but it's the same for programming), and if we are comparing math as in what you see in a BSc/Msc of Mathematics (or research-level) then I agree it's hard but you have to compare for an equivalent level of programming.
> simple fact that shows this is the amount of programmers who have no formal training in engineering or computer science but we’re able to self-teach the concepts. The same cannot be said of mathematics
I would blame it more on the fact that programming is a very useful tool for people outside Computer-Science, it has very direct applications and you can monetize it very easily, so it's very likely that they might want to learn it, however, rarely you see someone deciding to take Calculus just for the sake of it if they never bothered with it in College.
Overall I agree with you, but personally I find math more difficult, most people probably do but I don't think it's inherently more difficult, it's just that people are less used to study it.
Comparable levels of programming, what we expect of CS juniors, are regularly picked up by "the guy who is good with Excel" in office settings as it's mostly a function of experience and exposure, not theory.
And now my worthless anecdotal evidence: I self taught myself into professional programming and it was a simple matter of banging my head against a wall until shit started working. The feedback loop, "did the thing crash or not", permitted me to learn on my own. I wouldn't even begin to understand how to self-teach myself Stokes Theorem or some shit, and have zero ability to author the proofs required to reach the conclusions higher level mathematics are built on.
Input it into a proof assistant, and rely on the same sort of feedback "does the computer accept your proof, or get stuck". The hard job of formalizing stuff for this purpose has seen significant progress, e.g. by the Lean mathlib project.
In programming I knew I needed to sort a list or find a most efficient path because some practical problem I was trying to solve demanded that I do that. Frequently I had a basically crap but working independent solution before I learned the names "EWD" or "A*". I independently discovered that I needed virtual interfaces (before I knew them by that name, "I wish pointers to parent classes could call implementations in subclasses") and then discovered language facilities for polymorphism and OOP.
Without formal or at least guided instruction I would never think to move towards or discover "I wonder if there's a relationship that makes these double integrals of curls of vector fields easier to solve for".
Programming has a high coupling between necessity, experience, and theory. In mathematics that coupling is much, much, much looser. Self learners in programming regularly re-discover and re-implement, typically less efficiently, all sorts of fundamentals of CS. The equivalent in mathematics rarely happens post-algebra.
For example, take any good static analyzer that implements abstract interpretation. It generally works using Galois connections, which is just abstract algebra.
Dijkstra's algorithm or A* came pretty early in the history of CS. It would be fair to compare their difficulty to something similar in mathematics, say some basic results in Euclidean geometry.
If you're discussing pure CS, the thing you can write down in a book and for which a computer is a largely theoretical device, sure CS is mathematics.
If we're talking about the practical reality that CS majors in America today are trying to achieve, and their undergraduate programs are trying to prepare them for, that's becoming a working programmer and has very little relation to mathematics.
But so much of programming isn't. It doesn't require deep understanding, static analysers are advanced level things that are way beyond what many working programmers ever encounter.
I can't say I've ever seen "Real math" myself.
Second, even if you wanted to understand the proof of a theorem, doing it with a proof assistant is an atrocious way to go about it.
As a math teacher who disagrees with the premise of the GGP post ("This couldn’t be more wrong. … I’m sorry, but mathematics is orders of magnitude more intensive and difficult than most programming"), and thinks that any good programmer can learn mathematics—of course there are code wodgers out there who don't really understand their craft of programming, and so can't translate that knowledge to facilitate an understanding of mathematics—I think I also disagree with this. I've never tried it, but I can't imagine someone learning about Stokes's theorem in anything like this way. One of the many axes along which I imagine this failing are that the state of human readability in proof assistants is, well, let's say it's less well developed than the, cough, stellar state of the art in compiler error messages.
But, more importantly, you can, at least in principle, know every single step in a proof of Stokes's theorem without understanding in any real sense why it's true—and a proof assistant in particular will force you into the weeds of minutiae that absolutely do not help to build any intuitive picture—and, even if you manage in the process to piece together that understanding of why it's true, you will never thereby gain an understanding of why it's interesting (e.g., among other things, its connections to physics and the entrée it offers to differential geometry).
What if we had something similar for mathematics?
Rapid feedback, error messages, maybe even linters and highlighting for the "mathematical syntax".
I've though about this before and I think tools like this could unlock math for a lot of people, and also increase the effectiveness of professional mathematicians.
When learning math / seeing other learning math I've noticed that simple errors such as typos often slow down or hinder understanding of the subject.
The linter in mathematics is whether the other mathematician (whoever you're proving to) knows what you mean. If you're locked into a rigidly defined syntax, an obvious line of questioning is: what's not expressible in this syntax?
I fear that by the time the tooling was agreed on, built, and taught in schools, you'd have something like APL, which might be cool to code in, but from which the mathematical conversion would have moved on a while ago. Efforts like that, after all, are how math becomes engineering.
Consider, for instance, Russel's theory of types, which was interesting math at the time and now strikes the student with an engineering background as "pretty much just Java" (or any "normal" statically typed language).
I can't quite pin it down, like with a definition, but I'm tempted to say that if it has a linter it's not math anymore, even if it once was.
After you get the hang of the system, you can play with the interactive theorem prover behind it: Lean [2]. There's also plenty other interactive theorem provers (Coq, Isabelle, HOL, Mizar, Metamath, ...) but Lean has a lot of traction amongst mathematicians at the moment.
There are no limits to the math you an do with this. There is mathlib [3], the main mathematical library. It covers a lot of undergraduate material [4], and plenty of stuff beyond that [5]. The community has even covered some state of the art research math in Lean [6a, 6b].
You are very welcome to hang out on the leanprover zulip [7]] and ask questions about the Natural Number Game or anything else that is Lean-related.
[1]: https://wwwf.imperial.ac.uk/~buzzard/xena/natural_number_gam... [2]: https://leanprover-community.github.io/ [3]: https://github.com/leanprover-community/mathlib [4]: https://leanprover-community.github.io/undergrad.html [5]: https://leanprover-community.github.io/mathlib-overview.html [6a]: https://github.com/leanprover-community/lean-liquid [6b]: https://www.nature.com/articles/d41586-021-01627-2 [7]: https://leanprover.zulipchat.com/
Programming works with manmade abstractions, carefully designed to have very few interactions, keep mutable state contained, and to have all the parts structured in a hierarchy without recursion.
In math you have systems of equations that all reference each other. And they all happen at the same time because there's no steps and time or lines of code just 5 equations that all interact.
The atomic pieces are much larger and tied together and it inherently seems to require being able to fit nontrivial ideas in your head.
Programming lets you design a complex architecture one tiny piece at a time without considering other pieces.
I think Maths is harder because the abstractions are higher level, have less intuitive bases, the feedback loop is longer and doesn’t have robust testing. I don’t think the abstractions themselves are that much harder in general, but getting an intuition and doing anything useful (correctly) at a really high level of abstraction is quite difficult.
Not every programming language has a concept of time. Sometimes that's a good thing, it depends on what you're used to. Math arguably has less mutable state than most typical programing languages.
Programming is child's play compared to undergraduate mathematics taught in math departments. It's important that you take a module from the math department, not from a physical science or engineering department if you want to experience what it is like.
One thing you might learn in math is to avoid making overgeneralized statements that you can’t support.
A valid substitution in your statement for “programming” is writing a compiler. And for “undergraduate mathematics taught in math departments”, basic differential calculus.
Yet we regularly teach smart high school students and first-year undergraduates calculus, and almost never try to teach them to write a compiler, contradicting your proposition.
But what do I know? I’m just a dumb programmer. I can’t read your mind, so maybe you had something a little more specific you wanted to say.
High school students are taught plug-and-chug calculus where one uses rules and formulae without any real understanding of the underlying subtleties that make calculus work.
You can get hired after a brief boot camp, although it's not common.
A useful amount of math is like, ordinary differential equations in engineering school, since apps have taken over most use cases for simpler math.
The only direct use is to learn to access the "New way of thinking" math people talk about, and even that seems harder than making detailed to do lists.
One could have a valid proof to a theorem, but there's the human element of having people understand and accept the result.
That said, I think when people speak of math, they speak of its application to the real-world, not the proofs.
[1] - From the Wikipedia page for the axiom, Jerry Bona has an amusing quote: "The axiom of choice is obviously true, the well-ordering principle obviously false, and who can tell about Zorn's lemma?"
I think it very much depends on who the people are. I'm a math teacher, and, when I speak of math, I definitely don't just mean its applications to the real world. I definitely think programming—in the sense of thinking about the craft, not just cudgelling the computer into doing what one wants—is good preparation for learning the proof-theoretic arts of mathematics.
The ebook is pay what you want.
Anything that's just algorithmic steps would be done by a computer, so the useful math people want to learn requires some kind of insight or new mode of thought, or advanced methods that have not been made into an app yet.
I suspect when people say math is easy, they mean arithmetic and pre-algebra, and they're not concerned with practical applications at all, just the idea of general education and new ways of thought, and who knows if the level of math they are talking about is even enough to indirectly do much in daily life.
Closer to the right kind of math: https://sites.math.rutgers.edu/~zeilberg/GT.html
> Math is a lot easier to pick up after you know how to program. In fact, if you're a halfway decent programmer, you'll find it's almost a snap.
This couldn’t be more wrong. Mathematics is the hardest thing I have ever done. I’m sorry, but mathematics is orders of magnitude more intensive and difficult than most programming. A simple fact that shows this is the amount of programmers who have no formal training in engineering or computer science but we’re able to self-teach the concepts. The same cannot be said of mathematics, which requires deep, dedicated study. Most programmers I know know very little mathematics, and it’s not like I’d claim I know a lot either. I’ve forgotten more than I know.
He even mentions how little math he took, so I’m not sure he’s an authority on the subject. Most of his post is just surface level platitudes. I’m generally confused why I see his posts referenced so frequently.