This comes at a very opportune time in my life. when, my ward enters their HS -Junior year and they are taking SVC. Question to the Author(in case they are monitoring this thread), is it appropriate for a High-schooler with just an intro to Python?
IMHO a HS student learning Calculus should first learn the subject with pen and paper for a while before programming. It is important to work through the problems and think about the fundamental concepts involved, rather than thinking about the syntax involved in coding their solutions. Practising with pen and paper helps to internalize the subject matter better.
Makes perfect sense. And the parents suggestions for books is what I needed. Fun summer project. Learn svc and Mvc together with my Rising junior. What can go wrong.
I'm going to say a strong no to that. The programming parts are fine, but on a quick skim, the mathematics is written in a way they will likely find very confusing unless they already know calculus and leave them thinking they are bad at maths whereas actually the author is just not really trying to explain the maths to someone who doesn't already know it.
For example look at the diagram on[1]. It has unlabled axes of a shaded L-shaped box with a curve going through it and then it is followed by a bunch of equations where he derives the formula for integration by parts using a set of parametric equations with multiple substitutions etc. I know what integration by parts is and how it works really well. This is possibly the most confusing way you could possibly derive the formula and/or explain it, and this diagram really adds absolutely nothing unless you already know and understand the concept. If you have seen the exceptional diagrams and illustrations and clarity of explanation in a book like "Calculus" by James Stewart the contrast is so stark it really jumps out.
The normal way of explaining integration by parts starts the way the author does (with the product rule for derivitives) and shows a few examples of taking the derivitive of things using the product rule so you get an intuition for what the form of the resulting antiderivitive looks like. You then go through the derivation of the formula for integration by parts by using the product rule for derivitives, integrating both sides and splitting the resulting integral[2]. Its very clear and easy to follow. And if they want to actually help the student to do this they teach something like the tabular/"DI" method so the student isn't tearing their hair out getting the signs mixed up when integrating by parts multiple times.
[2] My own notes on this derivation which I took when learning this are here. Note that this isn't me really trying to explain it to a beginner - it's just my personal notes but it's still a lot easier to follow than the example given https://publish.obsidian.md/uncarved/3+Resources/Public/Inte...
That's awesome. Very happy you found it useful. Bear in mind of course that while I do try to get it right, there will be errors etc. If you find it helpful there is a lot more to come - I have a bunch more notes that I need to edit and make new diagrams for but I should be publishing soonish, and then I am learning all the time. Please let me know if there's anything that you found confusing/hard to understand[1] or if you see any errors.
[1] because that indicates I don't understand it well enough myself and I need to work on it more and then fix the note.
That's awesome. Hope your studies are going well and you achieve what you are hoping for. I'm finding it fantastic so far I have to say. Super-interesting and always more to discover.
I self taught myself calculus at that age (11th grade), obviously had a hard time at first, and persevered by sampling a good half dozen different books till I found one that would make things click. These are my credentials for making the following recommendation:
Get the book Quick Calculus by Kleppner and Ramsey. Nothing else I used came close, for developing intuition about the concepts when they're fresh and new.
Once they have mastered that, any good book will do. James Stewart's is fantastic but so big and thick you may need to use it as a resource to intelligently select appropriate readings and problems from as they go, rather than just start at page 1 and assign every problem. But the main thing is to really get the basics of what a derivative and integral and limit are from the getgo, and nothing I tried compare to Quick Calculus for that.
Using this Julia book or some other similar book to select exercises based on your appropriate judgment from it to supplement Stewart would also be very cool if your student has an interest in programming. Let them complete then in python if they like, python should be fine. I recall implementing a numerical integrator and a symbolic differentiator in python at the same age, as a way of consolidating my knowledge of calculus, and found both to be very valuable and fun exercises. Symbolic differentiation seemed like magic especially, but it was just a matter of parsing and then continually adding rules for each rule I learned in math.
Thanks for taking time to answer. This is why I keep coming back to HN. For concrete recommendations based on one’s own experience. Ordered an used Quick calculus.
Perhaps one thing to say which may be useful, is that ‘calculus’ can mean a lot of different things, and the meaning can vary between courses as well as between schools and countries. When I was in high school, we did calculus in the last two years, and it was similar to the earlier years: you learned a bunch of algorithms for manipulating symbols, some of which required a bit of intuition for some step (make some substitution, for example), and you learned a few facts (about slopes and areas) to be able to solve word problems. The first thing we learned was a part of the differentiation algorithm: the derivative of x^n (where n is constant) is n x^(n-1). At university, we called the course analysis, though I believe other places may have called the same thing calculus, and it was about defining various concepts and proving properties about them. We may have derived the above rule in class (I don’t remember) but I think anyone would have been able to derive it from the definitions and proven properties. Such courses typically follow a tripartite structure: firstly sequences, series, and their convergence to limits; secondly continuity of functions and limits of functions; thirdly differentiation and integration (and maybe some version of Taylor’s theorem). This course structure hasn’t changed that much since Cauchy introduced a lot of the ‘modern’ definitions in a textbook (obvious exceptions are integration – known as ‘Riemann integration’).
I don’t know what sort of course this one is. It starts with limits so maybe more the latter kind, but I don’t really see how Julia would be very useful there. My best guess about the suitability of the course is that it may assume mathematical maturity that you may not have, rather than anything about programming. Mathematical maturity tends to mean being able to cope with precise definitions and abstract concepts without deducing a load of wrong things, as well as being able to follow the kind of argument that is a mathematical proof.
In North America, traditionally Calculus is where you learn how to use the techniques to solve problems, and analysis is where you actually learn how to prove that it all is true.
But I have to correct you on the history. Essentially none of the material in a modern analysis course is the same as in Cauchy's day. You identify integration as coming after. But Cauchy didn't have a definition of a limit, axioms for the real numbers, a construction of the real numbers, set theory, the modern idea of open and closed sets, and so on and so forth.
No, epsilon-delta in the modern form was invented by Karl Weierstrass in 1861.
Cauchy's approach was to define an infinitesmal as a sequence approaching 0, and then to use the old infinitesmal definition of a derivative. This almost works. One of the places where it falls apart is defining the chain rule when the inner derivative is 0, because the sequence may wind up with 0/0 infinitely often.
But it is from this definition that we get Cauchy sequences today.
As a post-Calculus introduction to Julia and other similar mathematical computing languages, this doesn't seem like a bad thing to do. Understanding how to use one of these at a high level is pretty useful, and any time I have to do real math for work, Mathematica makes things a lot easier. R and Matlab are also big in the space, but iPython notebooks with SymPy might also work (although SymPy is comparatively underpowered in comparison).
I imagine a language with decent support for macros would be much much more ergonomic for symbolic computation. Even ignoring all the other pros & cons, just this one reason would push one strongly towards Julia, Lisp(s), Mathematica, etc for symbolic computation.
Sympy is a poor tool to learn because it simply doesn’t scale to problems one most often encounters, even in schooling. Frankly, CAS are so general and unintelligent that problems with well known and elegant closed-form solutions, when presented to a system like sympy, result in an output which is often not even human readable - thousands of algebraic terms for instance to describe the equations of motion for a simple double pendulum.
Personally I have found that the best tools are 1) a firm grasp of elementary calculus (differential, series, and integral) and 2) exposure to simple numerical methods that apply to a broad range of problems.
Armed with this knowledge, in my opinion, Julia is a far superior language to python and its package ecosystem has a brighter future. Indeed, the language has been built with a focus on mathematical modeling and efficient numerical computation. It is a more natural starting point for those with an interest in mathematical modeling and engineering science, and will serve anyone who learns it better than python+numpy+sympy+scipy.
As someone who's used Python but not Julia - when you say package ecosystem has a brighter future, does that include not needing several versions of Julia installed on my PC?
Julia has a bunch of little niceties for mathematics:
* Prefixing a variable with a scaler will implicitly multiply, as in standard math notation (e.g. 3x would evaluate to 6 if x is 2)
* Lots of unicode support. Some feel almost gratuitous (∈ and ∉ are operators that work as you would expect) but it's pretty nice to have π predefined (it's even an irrational datatype) and to use √ as an operator (e.g. √2 is a valid expression and evaluates to a float). It's not just that Julia supports these constructions but provides a convenient way to get them to appear.
* This is a little less relevant for calculus but vectors and matrices are fist class types in Julia. Entering and visually parsing matrices is so much easier in Julia than in Python.
m = [1 2 3;
4 5 6;
7 8 9]
vs.
m = [[1, 2, 3], [4, 5, 6], [7, 8, 9]]
Transposition is a single character operator ('). Dot product can be done with the dot operator (m ⋅ n). A\b works as it does in matlab.
* Julia supports broadcasting. It also has comprehensions but with broadcasting I personally find much less need for comprehensions.
* Rationals are built-in with very simple syntax (1//2))
I quite like and use Julia but wish there was a language mixing the best aspects of Julia and Swift (which I think can be done without many compromises, i.e. it would be a better language overall).
Some things I don't like about Julia:
- array.mean().round(digits=2) is more readable than round(mean(array), digits=2)
- Poor support for OOP (no, pure FP is not the optimal programming approach).
Well, Julia also has a pipeline operator built in so:
array |> mean |> x->round(x; digits=2)
will do what you want and is arguably more readable than dot notation.
Not supporting OOP is very much intentional. It has structs and multiple dispatch (which has proven to be insanely useful in Julia's domain of scientific computing) so about the only thing you lose is the namespacing that dot syntax facilitates.
That's not more readable, you're just switching a few characters.
Any language that has Uniform Function Call Syntax (UFCS) [1] lets you replace wrapped function calls with linear calls like that, it's clearly easier to read, and much easier to write, specially when you consider programmers' tooling (in case it's not obvious why: you can type the name of a symbol and a '.' and the IDE shows you the available functions... this works also for free functions that take the symbol's type as the first argument as well.... that could work for the pipe operator as well, of course, that's why I said that the two things are basically equivalent, just with slightly different notation).
is how mathematicians have thought for hundreds of years now. They do so in this way, because it's indeed easier to parse, and philosophically it correctly represents what is happening mathematically.
> array.mean().round(digits=2)
is how a subset of software engineers have thought about computation in the last couple of decades.
Technology should whenever possible adopt already existing conventions, because technology is created to serve the user, and not vice versa.
> is how mathematicians have thought for hundreds of years now.
"Because tradition" is not a good argument. We can and should strive for better.
The modern OOP-derived style of `array.mean().round(digits=2)` is just as exact in its specification, and arguably more semantically precise as well as it organizes the meaningful bits to be together.
Also I think you don't account for how much mathematical notation has changed over the years. The `round(mean(array), digits=2)` structure is relatively recent, early 20th century I believe, and the result of active reforms.
> mathematical notation has changed over the years.
I agree. Math notation has changed in response to problems mathematicians have faced, and as their thinking have evolved. It should not change because tool builders for mathematicians - when the tool builders in most cases are not even professional mathematicians - decide some other notation is better.
> array.mean().round(digits=2)
might be good (it is not) when you are operating on one object. As soon as you have a binary operation (say multiplication), this notation completely breaks down. Because the operation is often symmetric, or "near" symmetric, meaning it is not so different as to write something as ridiculous as x.product(y).
But more importantly, (common/dominant) mathematics is functional in nature. It's decidedly not object oriented. Given a matrix, M, doing something like M.eigenvalues() is vomit-inducing because eigenvalues is not a property of a matrix M. For matrices, its a map from a set of linear transformations (of which M is merely one item) onto a set of numbers. It is its own thing, separate from M.
In programming it is typical to have an “eig” function, or something of the sort, which returns the eigenvalues of a matrix. eig(M) seems no less vomit inducing than M.eigenvaues(). Usually when mathematicians want eigenvalues they write something like “Given a matrix M with eigenvalues \lambda” and the eigenvalues just show up.
Sure, my point is that even the math-y programming languages and libraries don’t map very well to how mathematicians seem to describe eigenvalues.
Personally I think of them as a characteristic of the matrix (but I just write numerical software, I’m not a mathematician) and implement them as the result of a process. So I don’t really see any reason to favor either syntax or call one more mathematical.
I'm not the GP, but I'm sorry you felt gatekeeped. I don't think GP conveyed "real mathematicians should do FP", but rather something else, so I'll try my best to share what I understood from that comment. Hope this makes you feel better.
Even though the notation changed over the years, the paradigm of "numbers as immutable entities and pure functions" has been the dominant way that math is presented, compared to something like "encapsulated objects that interact with each other via sending messages".
I don't think this has to be this way, and I do think that "real math" can also be laid out assuming principles of OOP. However, I do suspect it's the way it is because the laws of nature are unchanging, in contrast to the logic of a business application.
Because Julia is a tool with the target audience of mathematicians and scientists, I think it's a sensible decision to embrace the usual way of thinking, as opposed to presenting a relatively different way of thinking which steepens the learning curve. Not because data and functions are fundamentally better than OOP, but because it's more pragmatic for the target audience.
I understood the other poster correctly. This view of mathematics is as limiting and ignorant as the other poster’s.
The entire field of theoretical computer science, to which functional programming and type theory is closely tied, is a branch of mathematics. The Church-Turing thesis which gives both to our field equates the two at a very fundamental level. Questions about type theory, programming language design, and programmer ergonomics are fundamentally about math and applied math.
Maybe you and the other poster have in mind specific fields of math, but then you need to make claims for why those fields are sufficiently different as to be exempt from applicability of any of the advances in notation observed in other fields.
Your implicit assumption that you can divide computer science into a different bucket from “real math” is incorrect, and gatekeeping.
As I said though, I don’t think this is a profitable debate to have here on HN.
> Maybe you and the other poster have in mind specific fields of math,
Yes, because this is about Julia, I assumed we are talking about the specific fields of math that happen to be commonly used in mathematical and scientific computing, such as the ones learned in university math and science courses.
> Your implicit assumption that you can divide computer science into a different bucket from “real math” is incorrect, and gatekeeping.
It is regretful that we had a miscommunication. I agree with you that computer science belongs into the same bucket as "real math". The thing is, in the context of Julia it is not easy to read "math" as "math, but not only the domains that Julia is concerned in, but really all fields of math, including theoretical computer science". At least thanks to your comment, I see what you mean more clearly, and I think it'll help some other potential readers as well.
> but then you need to make claims for why those fields are sufficiently different as to be exempt from applicability of any of the advances in notation observed in other fields.
I'm curious to know specifically about the specific advances of notation observed in other fields. By this you mean dot notation for method application? I'm unsure if `a.method(b).anotherMethod(c)` is more advanced than `a |> method(b) |> anotherMethod(c)` (Edit: or `(anotherMethod(c) . method(b))(a)`) notation-wise.
I'd like to add another point worth mentioning. When I read the other poster's post, I was thinking about the semantics of OOP, specifically encapsulated objects and messages. I may have misinterpreted the other poster, but I hope it at least makes my other messages a bit clearer. (I think it's a little unwieldy to use Matrices that encapsulate its state for Linear Algebra, for example.)
I used to miss "OOP" in Julia. The funny thing about the dot notation is that it makes the limitations of single dispatch seem natural. Once you get used to multiple dispatch and its benefits, the dot notation seems limiting.
`x.plus(y)` dispatches based on (runtime) type of x (polymorphism) (single dispatch), and might call different functions based on the (compile time) type of y (function overloading).
In Julia, it would be `plus(x, y)`, and it can dispatch based on the runtime type of both x and y (or determine the correct method to call at compile time already, if the types of x and y can be determined then).
Virtual method dispatch isn’t always a good thing though. It’s a tradeoff and typically you really want one or the other. C++ makes both available for that reason, although C++’s syntax is horrible.
It's multiple dispatch though, not virtual dispatch. If I understand correctly, virtual dispatch means the method is picked at runtime. In Julia, the compiler is able to pick the correct method at compile time, even when dispatching on all arguments of the function.
As a person who respects many programming paradigms including OOP and FP, I beg to disagree. Specifically, I'm assuming you're talking about OOP the paradigm, not only the dot notation, for which the pipe operator is a viable alternative.
> The modern OOP-derived style of `array.mean().round(digits=2)` is just as exact in its specification, and arguably more semantically precise as well as it organizes the meaningful bits to be together.
I would have agreed with you if I were developing business applications. However, for the domain of math and science, every piece of material that I have seen assumes the paradigm of numbers as immutable entities and functions, not the paradigm of mutable encapsulated objects and interacting messages. To a mathematician or scientist the semantics of a `Vector` (as data) and a `mean` function that takes the `Vector` as input should be more natural than a black-box `Vector` object that can receive a `mean` method.
In some cases, applying a limited set of basic operations might make up a significant part of development time, but in my experience most of the time is spent designing algorithms, parsing function input and managing the flow of vectorized expressions across function boundaries. And that is generally more efficient in the context of multiple dispatch with efficient loops.
For example, writing a function that works on scalars, then mapping or broadcasting it over array input is very good for productivity (and is efficient in Julia). On the other hand, designing an inherently array-based algorithm is often a headache that one must contend with for the sake of performance in numpy.
So, if you design somewhat complex or novel algorithms, I don't think object-dot notation, or the lack thereof, has a significant impact on productivity.
This doesn't work, at least for me on 1.10. `round(; digits=2)` would have to produce a function. But Julia doesn't have automatic currying, so that only happens if someone has implemented a special-case version (e.g. (==)(3) returns a partial function that tests for equality with 3.)
> Poor support for OOP (no, pure FP is not the optimal programming approach).
Julia is not really a pure functional programming language, though it does support that style if one desires.
It's actually a "multiple dispatch" language... thinking of OOP as single dispatch, Julia is conceptually OOP on steroids -- but to benefit from the full expressiveness, the programming style looks different from the typical OOP syntax/patterns one is used to.
having spent my PhD writing in Julia, coming back to object oriented programming felt like programming with a hand tied to my back. many operations just make more sense dispatching on multiple arguments rather than just one, and. cramming methods into a class felt unnatural
You can overload your own types to get a lot of those things (poor support for OOP is harder but you can sort of emulate it with traits). Overloading `getproperty` for the former case.
Of course it's not built in, which I understand is annoying if that's your preferred coding style. I personally am sad that really good traits aren't encoded in the language.
Interesting remark about poor OOP support, when multi-methods and their interaction with protocols are two cool CLOS features yet to be fully embraced mainstream.
interesting, i was thinking of sharping my maths skill by programming. Have you tried something like lean, agda etc if so how do they fair against apl or j for doing maths
I've tried Idris. J and APL are great for quick analysis once you know them. I leave J open on my desktop for engineering and stats. Frink too for effortless units work. Haskell and its ilk are good for math. Programming to learn math works really well for some, since you work through a problem incrementally, especially with dynamic interpreters and REPLs. Pluto with Julia or Jupyter with Python or Mathematica too.
it's very hard to argue anything is "better" than Wolfram/Mathematica when it comes to symbolic stuff, after all, most theoretical physicists use Mathematica for their professional work.
But for (entry-level) learning and possibly pivoting to application, Julia is delightful to use and can transit into some symbolics and numerical. Besides, it's free and open source.
Unless you're working on undergraduate integrals then it's pretty useless for advanced stuff without writing your own library. By that point you may as well write it in a language that's more performant and cheaper.
> Unless you're working on undergraduate integrals then it's pretty useless for advanced stuff without writing your own library. By that point you may as well write it in a language that's more performant and cheaper.
Hard disagree. Mathematica's symbolic dexterity makes abstract reasoning (with equations/expressions) very easy. Think of it as the companion tool for anyone doing pages of algebra that would go into a paper, or form the backend for some code.
The numerical capabilities of Mathematica are passable but nothing fancy. Once you have the math figured out, you might even want to reimplement "in a language that's more performant and cheaper." But I haven't see anything come close to Mathematica for convenience of symbolic reasoning -- not just as a technology (lisp is pretty good) but as a ready-for-use product.
I agree that only some physicists use Mathematica. But I haven't really seen it being used it for calculus. Maybe some differential equations.
But mostly for symbolic algebriac manipulation. I used it during my phd to work with groups. Instead of having to calculate stuff by hand, you can just ask Mathematica to do it. Also lots of stuff with tensors in GR is so easy to do in Mathematica.
I think it's fair to say that most math/physics people use mathematica from time to time, but largely for different things than they use other programming languages for. It's very good as a CAS, but it's a pretty bad programming language for things that don't have analytical solutions.
It isn't. Mathematica is very much a niche product in academia.
The people who would get most out of it are students, but for some god forsaken reason universities don't support them.
I was in a pilot class with Mathematica back in 2006 and the review of the class were _all_ 5 stars and students on average got 10% higher marks in all other subjects they took that year.
They didn't run the course again.
Sagemath is now equally good if a teacher defines a DSL for the students to use in a class.
The only "problem" with sagemath is that it is based on Python. The rationale is that Python is easy to start using and widely known. This is the usual "make it easy for newcomers" trap.
For the mathematical constructs we care about in symbolic programming, I have found Python's syntax and Sage's menagerie of objects awful to use. Initially you feel comfortable, but when you want to do some real work, it gets horribly in the way. The Wolfram language, a LISP variant, is less familiar and harder for a newbie to learn but it is vastly superior for actual work.
The only "problem" with sagemath is that it is based on Python. The rationale is that Python is easy to start using and widely known. This is the usual "make it easy for newcomers" trap.
For the mathematical constructs we care about in symbolic programming, I have found Python's syntax and Sage's menagerie of objects awful to use. Initially you feel comfortable, but when you want to do some real work, it gets horribly in the way. The Wolfram language, not a LISP variant, is less familiar and harder for a newbie to learn but it is vastly superior for actual work.
> There are many examples of integrating a computer algebra system (such as Mathematica, Maple, or Sage) into the calculus conversation. Computer algebra systems can be magical. The popular WolframAlpha website calls the full power of Mathematica while allowing an informal syntax that is flexible enough to be used as a backend for Apple’s Siri feature. (“Siri what is the graph of x squared minus 4?”) For learning purposes, computer algebra systems model very well the algebraic/symbolic treatment of the material while providing means to illustrate the numeric aspects. These notes are a bit different in that Julia is primarily used for the numeric style of computing and the algebraic/symbolic treatment is added on. Doing the symbolic treatment by hand can be very beneficial while learning, and computer algebra systems make those exercises seem kind of redundant, as the finished product can be produced much easier.
TL;DR: they want the student to actually do some of the work that Mathematic magics away.
>Why is Julia better suited than any other language?
Because it is a language specifically targeted for doing numerical analysis.
Python, which is it's main "competitor" has a notoriously poor syntax for mathematical operations with miniscule standard library support for mathematics, a very limited type system and abysmal runtime performance. Julia addresses all of these issues and gives a relatively simple to read language, which often closely resembles mathematical notation and has quite decent performance.
Numpy has absolutely awful syntax compared to Julia or MATLAB. It is fundamentally limited by pythons syntax, which was never meant to support the rich operations you can do in Julia.
I have done quite a bit in both languages and it is extremely clear which language was designed for numerical analysis and which wasn't .
>Python's type system is pretty rich and allows overloading all operators
The language barely has support for type annotations. Duck typing is actually a very bad thing for doing numerical analysis, since you actually care a lot about the data types you are operating on.
Pythons type system is so utterly inadequate that you need an external library to have things like proper float types.
You're just repeating that it's bad without any argument.
Numpy's syntax is the same as MATLAB's.
Both MATLAB and Julia also use dynamic typing, though it is true Julia does have support for ad-hoc static typing through annotations in a way that is stricter than Python's.
Numpy does provide all the traditional floating-point types.
That is just plain false. Can you try reading the numpy documentation? They have an article about exactly that topic and even say that numpys Syntax is more verbose.
If you read that documentation you would also realize that numpy has a different approach to Arrays than both MATLAB and Julia.
No, that is not the only difference. I literally linked you an article in the numpy documentation which goes over dozens differences, large and small. I encourage you to read the section about the different numpy types, those concepts don't exist in MATLAB, where everything is an n-dimensional array and matrices as such don't exist as a special object.
There are also many differences between Julia and MATALB, although Julia obviously borrowed a lot from MATALB, for the obvious reason that MATLAB had a similar design goal in it's language.
That doesn't change that python inherently won't overcome the fact that it wasn't designed for numerical analysis, while both MATLAB and Julia were (although those also had somewhat different design objectives themselves, especially when it comes to the role of functions).
And that documentation just says all differences are minute, e.g. no matrix operators and zero-based indexing.
The matrix type exists because the operators don't exist.
Any good general-purpose programming language can enable any domain-specific language to be embedded within itself through its type system as a library. Numpy embeds array programming in Python, and arguably that approach is much more flexible than building separate domain-specific languages.
At this point you're just grasping at straws and systematically modding down my posts because facts don't agree with your beliefs.
It doesn't say they they are minute, because obviously they aren't. The fact alone that numpy has fundamental datatypes which don't exist in MATLAB should be enough to make it functionally distinct.
Seriously please read the article and try to actually use these tools. You would very quickly realize just how different numpys approach is to MATLAB. Again, they don't even operate on the same types. MATLAB only knows the n-dimensional array, that data type doesn't exist in numpy where the array has quite a different fundamental structure, which becomes pretty clear if you read the article.
I get that someone with little experience in both languages might be confused, purely looking from the outside they might seem similar, but if you are familiar with both you realize that they are very different and require you to approach implementations from different directions.
Ultimately numpy was designed to fit into python, which will always limit it compared to a DSL.
You are mostly correct, though I want to point out that N-dim arrays are different from matrices. In Matlab everything is a matrix, unless it is a higher-dimensional array. This means that any 'scalar value' is actually a 1x1 matrix, which you can verify by calling the `size` function on scalars, vectors, and matrices, but higher-dimensional arrays it do not fit this pattern. In fact, N-d arrays are the 'odd man out' in Matlab. The article you link to is actually a bit inaccurate in this regard.
I must agree with the other poster that there are key differences between numpy and Matlab (and Julia).
All 1D/2D arrays in both Matlab and Julia come endowed with linear algebra semantics, so that `A*B` is a matrix product, while in numpy it is not, you have to create an actual `numpy.matrix`, which is a different class. Matlab is not consistent, though, so exp(A) in matlab will work element-wise, as will numpy (both for array and matrix), while julia returns a matrix exponential. The same difference goes for other functions like sqrt, sin, etc.. In Matlab, reduction functions typically work column-wise, so for example sum(A) returns the sum of each column as a 1xN matrix, while in julia and numpy the sum functions will return a sum over all the elements as a scalar. The same goes for max/min, etc.
Both Python and Julia have actual scalars, and with numpy, both have actual vectors, while Matlab has neither. Numpy's matrices, on the other hand are, conceptually, vectors of vectors (even though they are stored as contiguous memory) meaning that for example iteration behaves very differently in numpy than in Julia. Numpy iterates row-by-row, Julia iterates element-by-element. Matlab iterates column-by-column, though I rarely see that used in the wild.
Indexing, too, is different (and I skip the zero-vs-one-based stuff): indexing (with a single index) into a 2d array in numpy produces a row-vector, indexing into a julia or matlab 2d array produces an element (even though iteration in matlab produces column, as mentioned). Another point is that numpy slices return views, while the other two languages return copies (or, actually, matlab returns views that are copied if you try to modify them...)
Numpy arrays have restrictions on the sort of types they will accept. Elements must either be built-in numpy types, or the catch-all `object`, which is basically an untyped container that holds pointers to `anything`. In contrast, everything is a matrix in Matlab, so even user-class objects are held in homogenously typed matrices. Julia arrays will allow anything you like. Built-in types, user-types, homogenously, abstractly or heterogeneously typed (unions). And bitstypes are stored inline, not as pointers.
There are many differences, both semantic and practical, both between matlab and numpy, matlab and julia, and julia and numpy, this is just off the top of my head.
(Actually, I should have gone more into the syntax stuff, constructor syntax is way different in numpy, as is the way you do operations, with numpy using `obj.method()` to a large extent, unlike matlab and julia, which use regular function call syntax for array operations.)
I forgot to mention the difference in function passing, the fact that Matlab passes arguments by value (unless it's a `handle` class) makes it really hard to do in-place transformations, as in passing an array to a function and modifying it, since the modifications are not visible outside the function.
I didn't even mention the dot operator syntax (.*,.^,./) used in Matlab, while numpy uses only implicit broadcasting. On the other hand, numpy can partially leverage map, filter, comprehension (though with performance loss). Julia has both much expanded dot-syntax and (multidimensional) comprehensions/map/filter with full performance.
What else? Matlab indexes with end, while numpy just leaves it open (e.g. 3:). Numpy allows negative indices, not Matlab. n:m is both a standalone range and an indexing expression in Matlab, not in numpy. Also numpy uses open ranges, Matlab closed ranges. And A[2:2:10] has dramatically different meaning in Matlab and numpy.
I would argue that default element-wise operations and 0 based indexing can lead to a lot of cognitive overhead vs the "more math-y" notation of MATLAB or Julia. And overloaded operators like block matrix notation or linear solves are much closer.
The type system in Julia is perhaps even better than MATLAB for some linear algebra operations (a Vector is always a Vector, a scalar is always a scalar). But the Python ecosystem is often far superior to both, and MATLAB has toolboxes which don't have equivalents in either other language. Julia has some uniquely strong package ecosystems (autodiff to some extent, differential equations, optimization), but far fewer.
> the Python ecosystem is often far superior to both
What would you say does the Python ecosystem have that the others are missing? The obvious thing is pytorch/tensorflow/.... I can't think of much else though.
>The obvious thing is pytorch/tensorflow/.... I can't think of much else though.
Julia has Flux, which is great. To create a neural network with Flux, you write the forward operation however you want and just auto differentiate the whole thing automatically and get a trainable net.
It is extremely flexible and intuitive.
What Julia lacks is everything which isn't numerical analysis though.
I agree that Flux.jl is very cool but you can't use it to train a SOTA LLM, can you? The Julia ecosystem does not have the resources to even attempt building a viable alternative to Pytorch.
I'm still curious what concrete things you mean that Julia is missing. Maybe I'm so stuck doing numerical analysis, can't think of anything else...
>I'm still curious what concrete things you mean that Julia is missing.
Basically everything not related to numerical analysis. The problem isn't that there isn't an option which does work, but that those options aren't fully developed and mature to actually use for a commercial project.
E.g. look at GUI libraries, the native ones are just wrappers around other libraries and poorly documented. There is a single web framework, but is that supported enough that you would bet your entire company on it being continually supported?
If you are doing anything which isn't related to numerical analysis, Julia isn't your first choice, it also isn't your second or third choice. And I think that is fine, clearly Julia wants to be a good at a particular thing and has achieved that.
> and just auto differentiate the whole thing automatically
Not so simple as it sounds. A lot of speed advantages of Julia comes from doing in place ops on arrays and Zygote does not work with in place updates. So you end up copying and then at that point Python + PyTorch becomes a superior choice as Python is the de facto the universal language.
Pytorch and tensorflow is pretty big. It implies all state of the art research code is not written in Julia, so it is a non-starter for any neural network based project.
That is just total nonsense. You can of course do neural network research without pytorch and tensorflow. Julia is especially good, with Flux being the most flexible Neural Network library which exists.
Pytorch and tensorflow are important and a very good reason to use python or C++, but those two being unavailable is more impactful for the industry, of course your research might need them, but it might very well not.
I disagree on the zero-based indexing complaint. Indeed, the fact that Julia indexed from 1 is the sole reason I will never use an otherwise great language. I can’t comprehend how people came to the conclusion this was a good idea.
Julia being a mostly functional non-systems-programming language, where memory addressing is usually in the background, I think you'd find you rarely use numeric-literal indexing and you were getting upset over nothing.
The typical looping over i from 1 to length(a) is not idiomatic in Julia, where looping over eachindex() or pairs() abstracts away any concern over index type (OffsetArray can have an arbitrary first index). There's firstindex() and lastindex(); bracket indexing syntax has begin and end special keywords. Multidimensional arrays have axes().
"But what about modular arithmetic to calculate 1-based indexes?"
Julia has mod1(), fld1(), etc., which also provide a hint to anyone reading the code that it might be computing things to be used as index-like objects (rather than pure arithmetic or offset computation).
The strict way to handle indexing, I suppose, would be to have distinct 1-indexed (e.g. Ind1) and 0-indexed (e.g. Ind0) Int-based types, and refuse to accept regular Ints as indexes. That would eliminate ambiguity, at the cost of Int-to-Ind[01] conversions all over the place (if one insists on using lots of numeric literals as indexes). And how would one abstract away modular arithmetic differences between the Ind0 and Ind1 types? By redefining regular mod and div for Ind1? That would be hella confusing, so that abstraction probably isn't viable.
> The strict way to handle indexing, I suppose, would be to have distinct 1-indexed (e.g. Ind1) and 0-indexed (e.g. Ind0) Int-based types, and refuse to accept regular Ints as indexes.
I would be happy if unsigned integer UInt would start indexing from 0, whereas for signed integer it would start from 1. Also a module level parameter where preference on whether a literal integer shall be interpreted as Int or UInt would alleviate many usability issues there.
It seems that a lot of programmers have trouble distinguishing between indexing and offsets.
This is probably due to legacy of C language which conflates those two, possibly for performance and simplicity reasons. Same goes for case sensitivity in programming languages, again a legacy of C...are Print, print and PRINT really different functions, is that really what you intended???
Anyway, indexing is enumerating elements and it naturally starts at 1 for the first element and goes to Count for the last.
Offsets are pointer arithmetic artifact where first element is at offset of 0 for most but not all data structures.
Then you get "modern" languages like Python where they adopt indexing from 0 (no offsets since there are no pointers) and range where start is inclusive and end is exclusive...very natural behavior there. Oh, and lets make it case sensitive so you never know if it's Print, print, PRINT...or count, Count and COUNT until you run for 20 min and then crash.
I'm confused that you're bringing in case sensitivity into the discussion. For simplicity, early systems could not be case sensitive because they didn't have a large enough character set.
The comment about crashing due to spelling error really should be addressed by some form of static analysis, not by making name lookups more relaxed, in my opinion.
You think mathematicians had it all wrong for centuries?
Incidentally, I find it amusing that in Europe, the ground floor is zero, while in the US, it's one. (A friend of mine arrived at college in the US and was told that her room was on the first floor. She asked whether there was a lift as her suitcase was quite heavy...)
> You think mathematicians had it all wrong for centuries
As a mathematician, yes, I do. Pretty much everything becomes simpler if you treat zero as the first natural number.
> Incidentally, I find it amusing that in Europe, the ground floor is zero, while in the US, it's one.
I really think it makes no sense to number the ground floor as 1. This means if you want to know the height of the nth floor, you have to multiply the height of a story by (n-1).
There are a ton of other cases where you have to needlessly subtract 1 when people use 1-based indexing. To name a few
- Dates: why did the 21st century start on 1/1/2001? Why are the 1800s the 19th century? It doesn’t make any sense. If we indexed from zero, today would be 5/0/2023 (5 months, 0 days, 2023 years since the common era), in the 20th century. It all becomes so easy and intuitive.
- Mathematical foundations: If we are using set theory to encode mathematics, how is the number 1 defined? As a set containing itself? This leads to paradoxes in many cases. As the set containing a different single element? Then we can call its contents 0.
- Musical intervals: why do two thirds (ok, major and minor, but I’ll gloss over that) make a fifth? Does 3+3=5? The fact that musical intervals index from 1 significantly increases the cognitive burden for music theory. It becomes much easier when we index from 0.
- Birthdays: Age is correctly indexed from zero, but it may seem counterintuitive that your first birthday is the day you are born. So when is your second birthday? The day you turn 1, of course. The word “third” sounds like 3, so it seems reasonable to me to introduce a new ordinal here (I like “toward”).
- Computing: languages that use 1-based indexing are obscuring what is actually going on; they generally just subtract 1 internally from the user’s input. They have to, since indexing from 0 is fundamentally more efficient at the hardware level. 4 bits can only store 15 possible addresses if you throw away 0.
These are just a few examples and by no means an exhaustive list. Conversely, I have yet to know of a single instance where 1-based indexing makes more sense or simplifies things (aside from being more compatible with legacy features of our society).
After a while, when you think deeply about it, you start to feel that 0-based indexing is something closer to a fundamental truth, rather than simply a convention. Indeed, I propose that the only reason people find 0-based indexing counterintuitive is due to social conditioning.
>I can’t comprehend how people came to the conclusion this was a good idea.
If you look at a textbook for numerical analysis likely the algorithms will be 1-indexed. Julia is a language primarily for writing down mathematics, it would be quite silly not to use mathematical conventions.
Curious whether likely is indeed correct. Certainly the textbook my numerical analysis course had wasn't because remember playing with indices to get clean Julia code.
Yes, python exposes a limited list of names that map to operators, like __add__, __sub__, __ne__, __or__, etc. The list is large enough that it covers most normal usage, but the semantic meaning of the operators is normally fixed, so creating your own more specialized operators with new meanings is not very convenient (I mean, in principle, you can overload __xor__ to do something inventive, but it's probably a bad idea to veer from the established meaning of 'exclusive or'.)
In contrast, julia provides a nearly endless list of operators to use for whatever you like, allowing you to adopt mathematical operators from your field of study, for example.
Also, I am not aware of any type promotion mechanism in python, which, in Julia, makes e.g. binary operators on different types convenient to implement. The inherent multiple dispatch also seems more natural than the __roperator__ stuff you need to do in python.
I also clicked around and felt the formatting and progression of the book was a bit confusing, but found some of the Julia features intriguing. (like “postfixing” allowing the same pencil notation of f’ and f’’ et al)
In my opinion, and experience, the best “calculus book” is Learn Physics with Functional Programming which only relies on libraries for plotting, and uses Haskell rather than Julia.
> Why is Julia better suited than any other language?
Julia is known as a “programming language for math” and was designed with that conceit steering a lot of its development.
Explicitly it supports a lot of mathematical notation that matches handwritten or latex symbols.
Implicitly they may be referencing the simplified (see Pythtonic) syntax, combined with broad interoperability (this tutorial uses SymPy for a lot of the heavy lifting), lots of built in parallel computing primitives, and its use of JIT compilation allowing for fast iteration/exploration.
The very short story is that it's a language somewhat similar to Python (and, likewise, easy-to-use), but with much richer syntax for expressing mathematics directly.
It also has richer notebooks. The key property is that in python notebooks, you run cells. In julia notebooks, it handles things like dependencies. If I change x, either as a number or a slider, all the dependencies things update. You can define a plot, add sliders, and it just works.
(Also: I'm not an expert in Julia; most of my work is in Python and JavaScript. I'm sure there are other reasons as well, but the two above come out very clearly in similar courses)
Julia inherited a lot of MATLAB DNA like standard library linear algebra and 1 based indices. In addition it's got very commonly used Unicode support and intentionally can look like pseudocode in many cases.
Multiple-dispatch is the right abstraction for programming mathematics. It provides both the flexibility required to create scientific software in layers (science has a LOT more function overloading compared to any other field.), and the information required to compile to fast machine code.
3. Robust plotting and visualization capabilities, which are integral (I f*cking swear no pun intended) to understanding calculus, the foundational purpose of calculus being to find the area under the curve. (something I really wish they told me day one of Pre-Calc)
4. This one is just a vague feeling, but the fact that Python's most robust Symbolic Regression package, SymPy, relies on running Julia in the background to do all the real work suggests to me that Julia is somehow just superior when it comes to formulas as opposed to just calculations. IDK how, though.
Funnily enough I just started to re-learn math (linear algebra, calculus and statistics) because I want to learn ML.
While I learn various things I also try to use them in simple Python implementations (to my shame, I never use Python before). It's quite nice how you can rotate vectors, plot functions using mathplotlib. I can draw things by hand but not as nicely.
One has to be a little careful with designing courses like this. They are most likely to be of interest to people who already know, at least to some extent, both (i) calculus and (ii) programming. That is, the (presumable) target audience — those who are learning either of these subjects — is not really ready to take in such a class.
Anecdotally, my personal attempts at incorporating only slightly exotic CASes (Maxima or Sagemath) into calculus courses were met with tepid response at best. Part of the issue was, I believe, that freshmen are rarely interested in setting up software for a non-CS course.
That being said, for slightly higher-level classes it can work quite well as an optional component — I've had really good results with Python projects in an ODE course. Python not being a niche language certainly helped, too.
This is a common challenge in teaching stats where students have to learn statistical concepts and using statistical software simultaneously. In the end, I think it is worth the challenge and beats having students calculate everything by hand & look up values in tables.
> "I've had really good results with Python projects in an ODE course."
Which textbook did you use for this course? What were the reference material? Could you share with me anything about this course? Lecture notes, code, slides, books, anything.
Related: if you use Emacs it has the Calc package which supports computer algebra. I recently published an interface for Calc that makes it significantly easier to use and wrote about it here. http://yummymelon.com/devnull/mathing-in-emacs-with-casual.h...
Holistically, I prefer MOOCulus. However, the value-add of Calculus with Julia is large. If the two could integrate somehow... the key thing about MOOCulus is the writing quality is better (much less verbose), and the integrated exercises means kids follow it closely. It's also very refined, from a lot of classroom use.
If it were forked and Julia-enhanced, that would be a very big step up, though. As would the addition of applications.
* The usability issues are 95% CSS issues, and would require a good UX designer. They're distracting for the first 10 minutes, but really need a fix.
* I've had students go through two semesters of courses. We ran into two minor errors. We fixed them. In my experience, the content is rather bug-free.
Contributing a fix takes all of five minutes. Hit "edit" at the top right, make the change in the github UX, and submit a PR.
The quality of your bug report here leaves much to be desired.
Are you talking about the Devyn / Riley dialogue at the top? If so, many of those appear designed to highlight misconceptions which are then addressed in the text. See "vicarious learning" for research evidence on how this works (and in practice, students seem to like the dialogues too).
While not a Matlab nor Julia user, I think you may be neglecting the nearly 40 years of code, toolboxes, and countless examples of common engineering problems solved in Matlab. Engineers tend to be more of a practical sort than developers, and just want to apply a known solution to a problem than mess around with newish software languages.
This is true, but even engineers see the advantages of Julia. My engineering school has went from almost pure Matlab usage to many key engineering courses switching to Julia due to its simplicity and friendliness.
It's also SOTA for many engineering applications, particularly for acausal modelling and scientific machine learning (see https://sciml.ai), which has led to big companies like Pfizer adopting it [1]. And for engineers writing novel libraries, it clearly has a strong edge. See for example the work by NASA's JPL [2, 3], the FAA [4] or the CliMa project [5].
Tools are valid regarding a context (when you have a hammer everything is a nail etc.). In the industrial context, Julia is not valid. In the research/education, it is valid. It does not mean that Julia will never be relevant in research.
Our upcoming JuliaCon 2024 has a significant number of industrial talks and a minisymposium. ASML uses Julia quite widely in a definitively industrial context.
What I mean is as a worker or student, I want to know if the tools I spend time on will be relevant in the industry. And I think I have more chance to meet companies working with MATLAB than Julia.
But if I want to work at asml Julia might be a good fit ofc.
One enormous difference is that MATLAB is paid, which, ironically, makes it much much easier to use in a corporate context.
For MATLAB it isn't just toolboxes, but also integrations with other tools. So it depends on what you are trying to do, but if your problem is taking in data, computing, putting out data, Julia can absolutely compete with MATLAB, even on the most practical "I really just want this to just work and look at the results", level.
Being paid does not make it easier to use in a corporate environment, that is nonsense. I specifically tried to ween my Mech E.’s off Matlab because I don’t want to pay for Matlab licenses for our company.
The big win for Matlab seemed to be there are pre-existing solutions/examples for hundreds of common engineering problems that only need to be tweaked for the problem at hand. Engineers like this. They seem pretty happy and productive in Matlab too, so I finally gave up and decided it was my best interests to let them use it.
>Being paid does not make it easier to use in a corporate environment, that is nonsense.
No, it isn't. It is the single biggest issue in getting it into your corporation. I can easily get a 10k Euro PC if I had any reasonable need for it which I could coherently explain to my boss. Getting Julia installed on it would take months of debates with IT and tens of thousands in internal expenses.
It is the same reason why corporations use Redhat, there is nothing about Redhats Linux which makes it inherently superior to e.g. Debian, definitely nothing that would justify the price. But corporations still willingly pay for it.
There are multiple reasons for this. First is support, if you aren't paying someone, there is nobody who will support you. Secondly is liability this is enormous, MATHWORKS is willing to take legal responsibility for their mistakes. That is a very big reason people use MATLAB, in some areas it is basically the only option for that reason. Thirdly paid software is easy to understand to everyone else, you wouldn't imagine how hard it is to tell people that something good is actually free and in fact so free that you can do with it whatever you want. If you aren't part of an enormous organization this might be hard to understand, but for most people free means "trial version" or "scam".
Yes. It's like giving a glass of ice water to somebody in hell (with apologies to Steve Jobs, who said that about iTunes on Windows, back when iTunes was not the mess it turned into later).
Mind you, I have huge respect for Cleve Moler and Matlab and all they have accomplished (making LINPACK and EISPACK and all that easily accessible). And they've worked hard to overcome the limitations of initially having only one datatype: a matrix.
But Julia as a modern general purpose language is just so much more pleasant to work with, while retaining nearly all of Matlab's power.
In some ways – it's much faster, it generally has a lot of features that make it better as a programming language, it's much easier to parallelize. As someone who's written both professionally, I would say Julia is usually the better choice today.
But there is a huge array of existing algorithms that are already implemented in languages like matlab or stata that may not have corresponding equivalents. If what you want is one of these, it's often hard to justify using another language (though in practice I've usually found it pretty easy to port matlab/python/stat to Julia.)
It depends on whether you are depending on specific MATLAB toolboxes or integrations.
If you don't, Julia will feel good and be a more than adequate replacement. It has a somewhat similar syntax and many of the Array niceties of MATLAB. Additionally it has a good type system and a much better handling of "general purpose" programming, e.g. it isn't weird about where functions are.
"These notes may be compiled into a pdf file through Quarto. As the result is rather large, we do not provide that file for download. For the interested reader, downloading the repository, instantiating the environment, and running quarto to render to pdf in the quarto subdirectory should produce that file (after some time)."
I find it interesting that after all these year of little development on LuaJIT, and active development in Julia - that LuaJIT is still faster (while also being general purpose).
LuaJIT is only appreciably faster on text ingestion and printing in those benchmarks. Julia is also plenty general purpose, just targeted (initially at least) towards the numerical community. LuaJIT is fantastically impressive though.
I find it interesting that after all these years of HN, commenters still ignore the main topic of the OP (a calculus book) to focus on the language it's written in
