Shameless promotion: I'm writing a book in which I introduce mathematics for programmers specifically. If you found yourself here you may be interested. https://jeremykun.com/2016/04/25/book-mailing-list
I'd be willing to spend a good 40 bucks on something like this (that may not seem like a lot but 40$ to a college student is a lot). Something that has a twinkle of a promise in teaching a programmer math is something I need in my life.
I just don't "get" what is being told to me when it's in the context of math-notation. As a result I do horrible in math classes in college. The only thing that was taught with a CS-Style notation was approximating using Newton's method. After I saw that it made sense, it was just a recursive method that zeroed in on that location.
Very little aside from that in my calc 1 class made sense. Probably limits, but that's it really. It didn't click as well.
I'd like to get to the point where I understand math concepts as well as I do most of the CS but I just think that's impossible at this point. It seems to be very "this is our garden, you stay in yours".
In anticipating a book like this, would you ideally want to learn the math context and notation (provided one goal of the book is to teach that), or skip that entirely in favor of the underlying ideas (as in your example with Newton's Method).
Also, would you be interested in being paid to read a chapter or two and provide feedback?
Yes exactly. It seems that no matter what you know math majors don't care. You need to walk the walk and talk the talk.
If you read through my comment history you will find me arguing with many mathematicians who think that math is just inherently harder and I must not understand it because I'm not smart enough. I think that the concepts and basic building blocks seem simple when explained outside the context of the current notation used.
I'd also like to be clear. I'm not saying math is easy, I'm just saying that it can't be impossible for my peasant brain to not be smart enough to understand the concepts of what's going on. I mean it's not Greek.... well... it currently is but it doesn't need to be!
The concepts are not individually hard, but they build on each-other (sometimes unnecessarily, but sometimes very helpfully). If you care about understanding the tools you should take the few years of study it takes to learn the notation to a level of fluency, or you won’t be able to read the vast literature written in that notation.
It’s like if you’re interested in 19th century piano music, but all your experience with reading music is in the form of guitar tablature, then you’re going to have a tough time with your study. You could conceivably find top-down videos of someone playing some of the music you care about, or convince someone to translate some parts of it into guitar tablature. But it would be a better use of your time to just learn to read standard music notation.
I'd read a few articles from your blog some time ago and I must say I really liked it! Especially those about automata and Turing Machines. I'll definitely consider buying your book!
You mention this is a "step up" from the Better Explained book, which was supposed to be an intro and went right over my head, still making lots of assumptions about what I should already know.
I'm a decent programmer, maybe above average, and I know I could understand this stuff if I found the right resources and put in the time, but I'm not sure what value it can provide me at this point.
When I look at Better Explained most of what I see is "a friendly guide to the logarithm" and "intuition for the law of cosines" (though Kalid does have some more advanced topics). I haven't seen his book so I can't comment on it.
My blog, on the other hand, tends to be more like "Here's an algorithm/theorem you can only find in research papers or graduate textbooks, proved, explained, and implemented." If you don't know proofs, most of my blog posts are too much too fast.
My book is trying to be in between the two. You have seen logarithms before and you know (or can look up) what similar triangles are. However, reading a typical math book is immediately too fast, the notation is too foreign, and the proofs seem to leave out a lot. My goal is to bring the reader in the fold w.r.t. notation and mindset and expectations (using programming analogies and leaning on the concepts you already understand well), showcase impressive applications in Python at the end of every chapter, and survey different areas of math relevant for software applications like machine learning and crypto.
I would be interested to know more about what specific topics you find difficult, maybe using Kalid's book as a reference?
I should also mention that I don't expect every programmer to find career value out of the book. I like math, I hear from a lot of programmers who want to like math more (for fun or side projects or work), and I feel like I'm in a good position to help them.
I just wanted to say that I have really enjoyed Math ∩ Programming over the past year. Lot of interesting reading material for my commute. Keep up the good work! (Signed up for the mailing list. Good luck!)
There are two books I know of that address an audience of programmers interested in mathematics: Klein's Coding the Matrix and Stepanov's From Mathematics to Generic Programming.
Have you looked at these two? It might help your writing process to find what readers of those books found confusing or frustrating in the presentation.
Thanks! I'll be looking into these. From a first glance, Coding the Matrix looks to be excellent and exhaustively complete, but also very verbose. From Mathematics to Generic Programming appears to be the kind of book I want to avoid. It seems to be more about a study of a few particular algorithms useful for generic programming than actually understanding the principles of mathematics.
A great book! It's very readable and more of an applied math book written with engineers and scientists in mind.
Starting out with the basics of state-space representations, it moves from 1D to 2D to 3D dynamics. The last section deals with the popular topics of chaos theory and fractals. But not from a simple pop-math perspective that you'll find in the math section of the book store. It's the real deal, except presented in a way that can be understood by non-mathematicians.
Strogatz is also coauthor of the paper that introduced small world networks. I really admire him for being so brilliant and at the same time capable of getting these topics across to everybody. I also recommend his book Sync.
Used this book during an undergraduate introductory course. I am now revisiting some of the chapters again, this is a great book! I think those interested in "data science" should definitely be exposed to some of the concepts introduced in it.
The reason I posted this article was that this is the second mathematician I've come across in two days to talk about the elegance and beauty of mathematics.
I wish I had seen some of this beauty from a younger age.
It goes to show how much the right teacher, especially early on, can have a profound impact on your life.
But I don't know that my younger self had the maturity to see the deeper beauty. It was only starting college that I had the necessary complexity of thought (combined with wonderful professors).
It takes a while to develop a taste refined enough to perceive these things. In the meantime, rote memorizing your multiplication tables and polynomial expansions might be a necessary evil to have foundations strong enough to later support that kind of more abstract, artistic reasoning.
This goes for other fields too. You have to practice your chords to play Bach, and learn your conjugations to write a novel.
> In the meantime, rote memorizing your multiplication tables and polynomial expansions might be a necessary evil...
I disagree on both ideological and technical terms. Won't students lose the "flow" as soon as they start memorizing? It seems like an anti-intellectual activity to memorize data or particular steps (e.g. (a+b)^2 = a^2 + 2ab + b^2). I think the further we stay from memorization the better the learner's experience will be.
Now for the technical objection. You said some degree of memorization might be a "necessary" evil, but I have a counter example: me. I've been doing math for the past 15+ years, but to this day I never learned the multiplication table. People are often surprised when I need 78 and I have to do 74 and add the result twice. "I thought you were a math person, and you don't know the multiplication table?" some will say... and I'm, like, yeah.
I've been doing math for the past 15+ years as well, and the nature of my education (French system) forced me to learn a number of things by rote. I'm very glad I now have these automatisms built in my brain and basic things come instinctively to me.
> Won't students lose the "flow" as soon as they start memorizing?
The core of my argument is that memorization will help them better find and stay in the flow in the first place, over the long term.
People wouldn't argue that having to learn your scales, chords, etc. gets in the way of creative piano playing, for instance; how and why is math different?
I can't argue with the music analogy—it's a very strong point.
My view is that pursuing fluency in the basic arithmetic skills as an intermediate goal towards higher understanding might not be a necessary step. Assuming the person learns a lot of math (like five years' worth) then they will be forced to develop fluency in the process of using the basic skills as building blocks for the more advanced topics. You're right that having useful "chunks" of memorized procedures would make learning the advanced stuff easier in the first place, but I think students could also develop arithmetic/algebra fluency "just in time" while learning the more advanced levels.
Indeed. It’s much better to practice basic arithmetic in the context of solving some harder problem than to just drill on worksheets of completely repetitive trivial problems for years on end.
Unfortunately, it takes much more teacher skill (e.g. to assess individual gaps in student understanding), doesn’t scale as well to large class sizes (ideally problems should to be catered to the ability level of each individual student), and isn’t as easy to assess with standardized tests.
who else think it could be better to explain such identities with geometric combinatorial diagrams ? at least at first step toward formal memorization.
The right curriculum is also important. I don't know why so much emphasis is placed on calculus. Why was my calculus class filled with nursing students? Why weren't they off studying statistics, logic, set theory or any other of the more applicable fields of mathematics?
To each his own. Education aside (which is often way too bad) your mind may only catch a subset of how math can be beautiful. Had to wander around 10 years on my own before finally see the value in complex planes, non linear abstractions, or even very simple geometric concepts you encounter at a young age (tan/cotan).
I agree. Most of mathematics is taught as calculations without motivation. If it was taught as the study of systems, and a toolbelt to understand the universe, more people would be passionate about it. All we can do is show the light to others.
Actually if you dig in the calculation techniques taught in primary school(Like multidigit addition/multiplication/ division), you would find some of them are actually quite clever ideas (O(log n) algorithms), but what almost all school do is to ask the student execute the algorithm like a computer, not how to come up with ideas like those...
> If it was taught as ... a toolbelt to understand the universe ...
While true, I think that this is still a bit abstract. Rather than "a toolbelt to understand the universe", I would say "a toolbelt to formally characterize problems in order to analyze and solve them".
Let me add in addition that in order to learn to use the fun tools in the toolbelt, one must first grasp the basic ones. This (in my limited experience teaching lower level math courses) is another obstacle in teaching math: (continuing with the tool analogy) it would be quite boring to spend 3 months studying screwdrivers, another 3 months on wrenches, etc. without applying these tools to any "fun" problems.
So here's the dilemma. People who see the beauty and elegance of mathematics are always going to go into something that exercises that, and will never go into teaching.
Why would anyone who is seriously good at lots of math go into teaching? Especially in the USA, UK, or other countries where teaching has such a low status. In many cases teachers are genuinely despised.
Why would anyone good at math ever go into teaching? No money, no status, no respect, no flexibility, no control, crushing hours, crushing workload ... why would anyone do that?!?
No wonder students are never exposed to the real beauty and elegance of mathematics.
Because one way to appreciate beauty is to show others. One of my math teachers in school had no kids, lived frugally. He could have retired a long time before I ran into him.
Instead, he gave me extra classes and put me on the math team. He spent lots of his own time to show me the interesting bits while also making sure I wouldn't get anything less than the top mark on the exams.
You're right though. It's not enough to rely on passionate people, there's not enough of them. I mean I like math too, and I'm not gonna be a teacher. You can't even send your own kids to school on a teacher's salary.
No teacher has time to do that. The workload is insane. Work from 7:30 to 3:30 in school. Then after school stuff with some kids for an hour, then staff meetings and a cup of tea (maybe a shit if you've got time). Commute home. 6pm cook and eat dinner. Bath kids, put to bed. 8pm plan lessons for next day, try to find some beauty in maths that works for the A* kids and the F kids and is in the curriculum.
Yeah. That's not happening often. Schools are broken.
If your degree/passion is for something like literature, history or music then teaching high school is, relatively speaking, one of the better jobs you can get where you can focus on your subject and use your degree. If someones degree/passion is in math, physics or programming then they simply have so many more options that the chances they end up in teaching is much lower.
You can be, just not in school - only at university level and above, when some time ago someone figured out a trick to combine research work with teaching responsibilities. This works well as long as there is no significant market demand for a given discipline.
Mathematics is the study of objective truth in a universe that has certain rules. How could knowing the universe in all its glory be boring? Wish they taught this rather than calculation at school. It's very uninspiring to not realize this and do math.
In fact, natural numbers are the basis for almost all of mathematics. And natural numbers are a manifestation of counting. Information theoretically, to count, one needs two dimensions (or degrees of freedom.) Space for storage of the number. And time to increment the storage medium, again and again, at different points. You could use two dimensions of space instead, and you'd just draw two orthogonal lines..and say the area inside the rectangle is the product.
So if counting is fundamentally related to the relationship between two dimensions, then so is addition since addition is repeated/recursive counting. And multiplication is repeated/recursive addition.
Once you realize this, physics seems less fundamental than mathematics, when it comes to understanding the universe.
Multiplication and addition are related in the way that f(x) and repeat(f(x), y) are related. They distribute over eachother and have certain properties.
So..the primes, the chaotic relationship between the modularity of addition over multiplication and the modularity of multiplication over addition, is due to the relationship between our universe's dimensions.
Want to study the universe at the most raw level? Mathematics. Want to study it a little higher? Physics. Want to study it at a higher level..maybe closer to the phaneron? Biology/neurochemistry.
Want to study it at the thinking level? Zen buddhism.
Choose your pick. The causal chain of human to universe doesn't discriminate. The whole chain could be the "real" reality. Pick your choice. It's all beautiful. It's all mystifying and enigmatic. Just do it.
>Once you realize this, physics seems less fundamental than mathematics, when it comes to understanding the universe.
But space and time are natural, ie. you mention a physical interpretation. And a paradox one at that, because you are explaining counting (numbers) by counting (dimensions).
> Want to study the universe at the most raw level?
I'd rather offer that mathematics is about formalization of what you know, while physics is about exploration of what you don't know. Maths is intrinsic, abstract and relies on intuition. Physics is extrinsic, concrete and relies on experimentation. These separations are arbitrary and not useful, if they don't serve a purpose. Why divide like that at all? If you talk hierarchy, I'd say logic is higher up there.
To wrap it up, what's that got to do with bad teaching? I'd suppose that deductive reasoning cannot be taught, because it is learned before language is acquired, then used to acquire language. Maybe, memory can be trained, and curiosity can be stimulated, but school is hardly the right setting to inspire creativity, when most of the time there talking is forbidden.
> Zen buddhism
Sure, nihilism, or religion overall, is like an escape when you abandon all hope and try to recollect new hope. It is admitting you don't know anymore and going back to square one.
if you think mathematics relies on intuition..you haven't done much mathematics unfortunately. the field is deeply paradoxical once you get into it deeply
If there exists an absolute reality it only make sense to be non-objective [nirguna brahman in advaita] nothingness, in contrast to objective nothingness which conceptually is nihilism.
Mathematics can be used to study the universe at the objective level but do not make the mistake of treating numbers as ineffable, existing in some platonic realm outside of space/time/causation or talk about the "real" reality and mathematics together. Through mathematics we learn more about the limits of our own model building/cognition than any secrets of the universe.
> [don't] talk about the "real" reality and mathematics together
That stopped me just at the right moment, but then you went and still did it:
> Through mathematics we learn more about the limits of our own model building/cognition than any secrets of the universe.
This is illogic, assuming we are part of the universe. Ignoring that last part, as you wanted, I get the reference. It comes down to saying, through learning about models we learn more about models. I agree if the stress is on learning, because to learn is part of the root of the word mathematics.
The real kicker nowadays is learning about learning, the mind, neurology, etc. directly.
I first came across an alternative view in Godel, Escher, Bach, where Hofstadter was writing about Euclid's parallel postulate and the development of non-euclidean geometry. The latter could have been accepted sooner, he suggests, if people were not hung up on the idea that geometry was about physical space, rather than a derivation from axioms. Ironically, it turned out later that physical space is not euclidean.
It's not very helpful to study the ideas that form in human brains, though, since so many of them are nonsense. It would largely be a study of misconceptions and biases.
I think you could just as well have said "I believe our ideas are often formed on paper, when we write therm down. And that writing is part of the physical world. So I think the schism you presented would be void."
Does this get us into the question of whether anything mathematical exists before anyone (or anysomething) first thought of it, and exists even when no-one is currently thinking of it? If you answer yes, then mathematics has an existence independent of thought. If you answer no, then that conditional existence presumably extends to physics as well, especially if Max Tegmark is right about math being the reality of physics. In that case, then it seems to follow that the universe only exists when it is being thought about, a conclusion that appears to have a bootstrap problem...
I think I can avoid the whole issue by observing that I can imagine things that do not physically exist, and they are not made to physically exist by dint of a conceptual representation of them physically existing in my brain.
Yea, mathematics is more fundamental than physics, in the sense that it would be possible for different universes to exist with different physics, but it's not clear at all that universes could exist with different mathematics. What would that even mean? You may like Max Tegmark's idea that the universe is math: https://arxiv.org/abs/0704.0646
Do you know what the uni~ i universe means? It means only (one-ly). How could there be multiple? That's just non-sense.
The parent implied counting, not the natural numbers, need two dimensions. But even in Set Theory you need at least one level deep nested sets to build the natural numbers. Qunatification is the (a?) difference between zero and higher order logic, I suppose.
> Arguments from etymology are invalid, because words are not obliged to be backwards compatible.
That's how you end up with loads of confusing homonyms, which the parent almost admited to. I mean, sure an etymologic argument can be insufficient or even wrong.
My point is, words are only labels - pointers to concepts. You can't prove facts about objects from just their labels, nor do the labels have causal power over reality. Universe being derived from "uni" doesn't force the concept of universe to be a singleton.
Proliferation of homonyms is another topic altogether; it is an issue, but it's about creating barriers to communication.
So, extending the analogy, we've as well been dividing the universe? If you go with quantum mechanics, you also have to consider the Entanglement that, if I understand correctly, posits non-locality, so your argument of spacial division is still nonsense to me. To be fair, quantum theory might as well be nonsense to me. I wouldn't know, it's over my head. But multiverse is stuff of sci-fi and theoretical physics. Theoretical physics is mostly maths, indeed. So I agree to a point, I just contest the schism that's between maths and physics, because it's not constructive.
In my particular comment, I meant that THE Universe (proper noun) could have different physics. Max Tegmark uses the term a bit differently, but he's quite clear about what he means.
As for your second paragraph, I really don't have any idea what you two are on about.
I'm not really interested in metaphysics, hence my opposition to the claim.
>As for your second paragraph, I really don't have any idea what you two are on about.
I didn't quite understand the parent, but he clearly related to Computational Complexity of operations on Countable Sets. I happen to study that subject at the moment, so I am taking this discussion as an opportunity to test my understanding.
but I didn't read him and it's pretty arrogant to assume I had to. Sure, I hadn't to butt in on the discussion, but I had to read the comment to find out I didn't want to read it, first. So I felt I had to say something. I mean, I thought it was informative, but obviously I didn't hit the right tone.
If you actually read my post more closely you would see I said any two orthogonal dimensions, that could be two spatial dimensions..not necessarily including time.