This is excellent. Thank you for taking the time to write it.
I don't know what is it about math -- especially when it involves manipulation of symbols as opposed to pictures or lay language -- that turns off so many people.
The fact that so many software developers "don't like math" is ironic, because they're perfectly happy to manipulate symbols such as "x", "file", or "user_id" that stand in for other things every day. The entirety of mathematical knowledge is very much like a gigantic computer language (a formal system) in which every object is and must be precisely defined in terms of other objects, using and reusing symbols like "x", "y", "+", etc. that stand in for other things.
Perhaps the issue is motivation? Many wonder, "why do I need to learn this hard stuff?" If so, the approach taken by Rachel Thomas and Jeremy Howard at fast.ai seems to be a good one: build things, and then fill the theoretical holes as needed, motivated by a genuine desire to understand.
> I don't know what is it about math -- especially when it involves manipulation of symbols as opposed to pictures or lay language -- that turns off so many people.
The biggest turn off about math is the way people are taught math.
Most people are taught math as if it's an infinite set of cold formulas to memorize and regurgitate. Most students in my statistics class didn't know where and when to use the formulas taught in real life; they only knew enough to pass the tests. Students who obtain As in Algebra 2 hardly know where the quadratic formula comes from (and what possibly useful algebraic manipulation could you do if you can't even rederive the quadratic formula?). It's not just math, I've been in a chemistry class where the TA was getting a masters in chemistry and yet she taught everyone in my class a formula so wrong that if interpreted meant that everytime a photon hits an atom, then an electron will be ejected with the same energy and speed as the photon. This is obviously wrong but when I pointed it out, everyone thought I was wrong because "that's not what it says in the professor's notes" (later, the professor corrected their notes). In my physics class, the people who struggled the most are the ones who tried the least to truly grasp where the formulas come from. I don't blame them, it's the way most schools teach.
> build things, and then fill the theoretical holes as needed, motivated by a genuine desire to understand.
I totally agree.
Source:
My experience with tutoring people struggling with math for the past eight years. I used to like math then I got to college where 95% of people don't understand the math they're doing and thus can't be creative with it; this includes the professors who teach math as the rote memorization of formulas. Yeah, call me arrogant, but I have found it to be true in my experience. I strongly believe the inability to rederive or truly grasp where things come from destroys the ability to be creative and leads to a lack of true understanding. But everyone believes they understood the material because they got an A on the exam. I'll stop ranting on this now.
This is exactly my experience! I breezed through math all the way until I got to calculus because I was excellent at rote memorization. Calculus made me realize that I didn't really understand most of what I had learned for the past several years.
Years later, I'm trying to relearn math, but I'm taking the exact opposite approach. No calculator, no rote memorization, just reading about the concepts and thinking about what they mean until I can do the manipulations in my head. When I do practice problems, I don't care so much about the specific numbers, but about my ability to understand what's happening to each part of an equation, what the graph looks like, etc.
The impression I get is that many people want to be system designers stringing together pieces to create systems to solve problems (part of the motivation might be that it is easier to extract economic value from such integrated solutions, rather than better functioning pieces).
The problem is that in an immature field that's still evolving, the components are not yet well-understood or well-designed, so available abstractions are all leaky. However, modern software engineering is mostly built on the ability to abstract away enormous complexity behind libraries, so that a developer who is plumbing/composing them together can ignore a lot of details [1]. People with that background now expect similarly effective abstractions for machine learning, but the truth is that machine learning is simply NOT at that level of maturity, and might take decades to get there. It is the price you pay for the thrill of working in a nascent field doing something genuinely uncharted.
"Math in machine learning" is a bit of a red herring. We hear the same complaints about putting in effort to grok ideas in functional programming, thinking about hardware/physics details, understanding the effects of software on human systems [2], etc. Fundamentally, I think a lot of people have not developed the skill to fluidly move between different levels of abstraction, and a variety of approximately correct models. And to be fair, it seems like most of software engineering is basically blind to this, so one can't shift all the blame on individuals.
Deep learning, in particular, is a trade today. If we want to be generous we can call it an "experimental science"... but my perception is that only a minority of papers in the field actually deserve that moniker.
(Speaking as a deep learning practitioner with expertise in a narrow domain.)
I don't know what is it about math -- especially when it involves manipulation of symbols as opposed to pictures or lay language -- that turns off so many people.
I can tell you at least part of it, from my subjective perspective. I tend to "think" in a very verbal fashion and I instinctively try to sub-vocalize everything I read. So when I see math, as soon as I see a symbol that I can't "say" to myself (eg, a greek letter that I don't recognize, or any other unfamiliar notation) my brain just tries to short-circuit whatever is going on, and my eyes want to glaze over and jump to the stuff that is familiar.
OTOH, with written prose, I might see a word I don't recognize, but I can usually work out how to pronounce it (at least approximately) and I can often infer the meaning (at least approximately) from context. So I can read prose even when bits of it are unfamiliar.
There's also the issue that math is so linear in terms of dependencies, and it's - in my experience - very "use it or lose it" in terms of how quickly you forget bits of it if you aren't using it on day-in / day-out basis.
My way of dealing with this is to treat symbols as proxies for the verbal concept, rather than just some letters. As an example, when I see "E = mc^2" I read (energy-of-object) = (mass-of-object) * (speed-of-light)^2 and not "Eee equals Em Cee Square". Another great idea I use a lot when writing/reading is David Mermin's 2nd rule (verbalize the damn equation!) [1].
It's sad that many mathematical resources do not make a careful effort of helping someone reason verbally. I guess this is partly due to the fact that most people who are skilled in the subject and write about it prefer equational reasoning (for lack of a better word) to verbal reasoning! In my experience as a physics instructor for non-STEM majors, this might be one of the biggest impediments for otherwise intelligent people trying to learn math/physics.
Very good point about reading/verbalizing symbols and notation. Once you know what they mean, they are super useful for expressing complex concepts precisely and compactly, but when you're getting started they look like an alien language...
This is why I'm using Anki to memorize the Greek alphabet, and to keep basic algebraic (h.s. algebra that is) stuff in mind. It might seem like a small thing, but remembering the various rules for factoring, working with fractions, dealing with exponents / root, etc. is not easy when you don't do math all the time.
> The fact that so many software developers "don't like math" is ironic, because they're perfectly happy to manipulate symbols such as "x", "file", or "user_id" that stand in for other things every day. The entirety of mathematical knowledge is very much like a gigantic computer language (a formal system) in which every object is and must be precisely defined in terms of other objects, using and reusing symbols like "x", "y", "+", etc. that stand in for other things.
I don’t find it ironic, because I wouldn’t expect engineers to make good mathematicians implicitly (nor vice versa). There is some similarity between math and programming, but there is also a collossal amount of dissimilarity that makes them different things entirely.
For example, notation and terminology in mathematics is not actually rigorous. It’s highly context dependent and frequently overloaded (take the definition of “normal”, the notation of a vector versus a closure, or the notation of a sequence versus a collection of sets). As another example, consider that beyond the first few courses of undergraduate math you’re wading into a sea of abstraction which you can only reason about. There is no compiler flag to ensure your proof is correct in the general case, and you don’t have good, automatic feedback on whether or not the math works. In this sense, the entirety of mathematical knowledge is actually very much not like a formal computer language.
Beyond that, the ceiling of complexity for theoretical computer science or applied mathematics is far higher than programming. It’s not so much motivation (though that can be an issue too), it’s that learning the mathematics for certain things simply takes a vast amount of time. Meanwhile a professional programmer has to become good at things that mathematicians and scientists don’t have to care about, like version control or the idiosyncrasies of a specific language.
They're really orthogonal disciplines, for much the same reason that engineering isn't like computer science. There is a world of difference between proving the computational complexity of an algorithm and implementing an algorithm matching that complexity in the real world.
> Meanwhile a professional programmer has to become good at things that mathematicians and scientists don’t have to care about, like version control or the idiosyncrasies of a specific language.
Really depends on what kind of mathematician or scientist you are to be honest though. How good is someone's data analysis of an experiment if they can't reproduce it? Or if they've got 6 different versions of an application with 100k lines of code in a single file, each labelled "code_working(1).f90", "code_not_working.f90", etc... These are real problems with what people actually do in science; software development skills are poor and people do things badly.
There're organisations like Software Carpentry globally and the Software Sustainability Institute in the UK which exist to try and promote some thought about developing software as researchers, and making the software sustainable in the long term rather than letting it die every time a PhD student leaves.
That's essentially my point. Programming and mathematics are so different from each other that, without special effort, a professional in one domain shouldn't be expected to be meaningfully better than average in the other.
This applies in both directions most mathematicians and scientists have such poor version control and development hygiene because mathematics doesn't imbue them with any special insight about how to be an engineer.
That said, the math we're talking about (that is, the math necessary for understanding, say, the sequence of transformations that make up a convnet) lies far below the ceiling of complexity you mention.
I'm not sure symbol reuse and overloading are as much of an issue. I've run into people who are quite proficient with Perl and routinely use complicated regular expressions who say they didn't like math growing up.
> The fact that so many software developers "don't like math" is ironic, because they're perfectly happy to manipulate symbols such as "x", "file", or "user_id" that stand in for other things every day.
`user_id` says what it is; something like `β` does not. It's more like reading minified JavaScript than literate programming. Math notation is frequently horribly overloaded and needlessly terse.
Math education from the undergraduate level on is fairly horrible and not communicated well. Just go read the typical calculus textbook and realize that they reference a lot of stuff that no pre-calculus student would typically know, such as proof by induction, lemmas and so on. The textbooks are written to the professors, not the actual students.
Various non-intuitive concepts are handwaved, the foundations skipped over and students then start struggling because they don't understand the foundation of what they are trying to learn. Reading from the textbook is fairly useless and it ends up being used as a problem set source.
I argued to a few math professors about teaching things like calculus with the textbooks referencing concepts that were not actually taught until 5 classes later is a bad idea.
In return I got a shrug of indifference telling me that's just the status quo and the status quo is OK.
I attended a not-superb high school in rural Missouri, and we studied proof by induction in 11th grade, before calculus in 12th. (Only over the naturals, but that's enough to get the flavor...) Lemmas came in 10-grade geometry, although frankly I may not understand what you mean because that's not really a full "concept", just kind of an arbitrary detail. That was the early 90s, though, so perhaps standards have slipped.
Of course, colleges should cater to a range of preparatory educations, but the textbooks you're talking about are pitched at the correct level for somebody.
My recollection is vague, since it was quite a while ago, but there were other things that weren't introduced in my high school curriculum that I remember my calculus textbook containing. I'm glad your school taught you proof of induction and other such things although. I also remember talking to my classmates about how near indecipherable our calculus textbook was.
First semester of college, I took multivariate. That was kind of a mess, because instead of a textbook we got a "bound" compilation of our professor's notes that he was trying to turn into a textbook. It has been some time, but I kept lots of math books from college, and I specifically threw that thing out. My recollection is that it was pretty much worthless.
Well, I think the difference between developer symbols and math symbols is that developer symbols are lot more google-able. I can google a line of code, but it's hard to google a crap ton of greek letters and summations. Even if I did manage to parse it into a google search somehow, I probably wouldn't get any meaningful results.
Also, for me personally, it's just such a drag to learn all the notation. After the fact, I've always thought, "Wow, that's all this means?" but while I'm learning, I feel helpless. It doesn't feel like I have any way to google it. My professors never actually want to sit down and explain it to me. All the pages of math equations always look so intimidating. It's just such a drag.
I don’t enjoy math and I simply don’t have the intuition for it. Every time I attempt to do math in my head my brain groans and says “It’s the 21st century, jackass. Use a calculator.”
I do, however, have a talent for language.
The reason I am a good developer is because I can communicate with different machines through different programming languages in the same way I can communicate with different people through different human languages.
I have tried as of late to learn math in an attempt to contextualize it as language - the language of the universe, really - but it is far more of an uphill climb for me than JavaScript or Chinese.
I don't think we "don't like math", but in my case, I just need an accelerated version of the "math" that I need without the deep-dive.
Here's a crazy idea that machine learning might one day help with software engineers understanding algorithms and data structures.
You write some code to traverse a list or something and do some naive sorting, or maybe you're "everyday way of doing some operations on your lists is inefficent". I want some cool machine learning where I can submit my code and it does analysis.
I don't know what is it about math -- especially when it involves manipulation of symbols as opposed to pictures or lay language -- that turns off so many people.
The fact that so many software developers "don't like math" is ironic, because they're perfectly happy to manipulate symbols such as "x", "file", or "user_id" that stand in for other things every day. The entirety of mathematical knowledge is very much like a gigantic computer language (a formal system) in which every object is and must be precisely defined in terms of other objects, using and reusing symbols like "x", "y", "+", etc. that stand in for other things.
Perhaps the issue is motivation? Many wonder, "why do I need to learn this hard stuff?" If so, the approach taken by Rachel Thomas and Jeremy Howard at fast.ai seems to be a good one: build things, and then fill the theoretical holes as needed, motivated by a genuine desire to understand.