Ask HN: How did you learn math notation?

I have this same problem with programming, when I have to deal with code written by non-mathematicians. They tend to use all these stupid variables with more than one letter and that confuses the heck out of me.

This convention is used in a whole bunch of scientific fields, like quantum mechanics, chemistry, biology, mechanics, thermodynamics, etc.

It’s also very useful in how it relates to derivatives, which is a crucial concept in just about any kind of science you could care to mention.

So yes, there is a learning curve, but we write things this way for good reasons, most of the time.

Multiplication should be represented by a (thin) space in good typography, to avoid this sort of things. Not doing it is sloppy and invites misreading. Same with omitting parenthesis around a function’s argument most of the time (e.g. sin 2πθ instead of sin(2 π θ) ).

But as I say, immediately above the formula it says:

"For a pulse of width ∆t, the critical time ..."

So that really is saying exactly what that cluster of symbols means. There will be things like this everywhere as you read stuff. Things are rarely completely undefined, but you are expected to be reading along.

And you need to work. I just typed this into DDG:

"What does ∆t mean?"

The very first hit is this:

https://en.wikipedia.org/wiki/Delta_%28letter%29

That gives you a lot of context for what the symbol means, and this is the sort of thing you'll need to do. You need to stop, look at the thing you don't understand, read around in the nearby text, then type a question (or two, or three) into a search engine.

Please don't take this the wrong way. It is not meant to be demeaning, and it is not meant to be gatekeeping (quite the contrary!). But: If you do not know what a derivative is, then learning that that symbol means derivative (assuming that it does, I have not actually looked at what you link to) will help you next to nothing. OK, you'll have something to google, but if you don't already have some idea what that is, there is no way you will get through the paper that way.

I hope you take this as motivation to take the time to properly learn the fundamentals of mathematics (such as for example calculus for the topic of derivatives).

Let’s say you go on a journey, and the distance you’ve travelled so far is “x” and the time so far is “t”.

Then your average velocity since the beginning is x / t .

But, if you want to know your current velocity, that would be delta x divided by delta t .

The delta is usually used in a “limiting” sense - you can get a more accurate measurement of your velocity by measuring the change in x during a tiny time interval. The tinier the interval, the more accurate the estimate of current velocity.

What I’m talking about here is the first steps in learning differential calculus. You could look for that at kahnacademy.com. You might also benefit by looking at their “precalculus” courses.

Just keep plugging away at it, the concepts take awhile to seep in. Attaining mathematical maturity takes years.

> Maybe it's just a too advanced paper for my knowledge.

Maybe it is for now ... the point being that if you start at the beginning, chip away at it, search for terms on the 'net, read multiple times, try to work through it, and then ask people when you're really stuck, that's one way of making progress.

You can, instead, enroll in an on-line course, or night-school, and learn all this stuff from the ground up, but it will almost certainly take longer. Your knowledge would be better grounded and more secure, but learning how to read, investigate, search, work, then ask, is a far greater skill that "taking a course".

Others have answered your specific question about the delta symbol, but there are deeper processes/problems/questions here:

* Not all concepts or values or represented by a single glyph, sometimes there are multi-glyph "symbols", such as "Δt" in your example.

* When you see a symbol you don't recognise, read the surrounding text. The symbol will almost always be referenced or described.

* The notation isn't universal. Often it's an aid to your memory, to write in a succinct form the thing that has been described elsewhere.

* In these senses, it's very much a language more akin to natural languages than computer languages. The formulas are things used to express a meaning, not things to be executed.

* Specific questions about specific notation can be answered more directly, but to really get along with mathematical notation you need to "read like math" and not "read like a novel".

* None of this is correct, all of it is intended to give you a sense of how to make progress.

But I think the problem isn't the specifics of the "Δ", it's the meta-problem of believing that symbols have a "one true meaning" instead of being defined by the scope.

I agree that explaining the delta notation would be helpful, but that's like giving someone a fish, or making them a fire. They are fed for one day, or warm for one night, it's the underlying misconceptions that need addressing so they can learn to fish and be fed, or set on fire and be warm, for the remainder of their life.

But most will already be familiar with the family of goniometric functions such as sin and cos, there’s log and possibly exp and sqrt. There’s min and max; advanced math has inf and sup.

I doubt you actually can do that and understand it all. A computer can do it, but I doubt you the human can do that and get a perfect picture of any non trivial program without making errors. Human math is a human language first and foremost, its grammar is human language which is used to define things and symbols. This lets us write things that humans can actually read and understand the entirety of, unlike a million lines of code or cpu instructions.

Show me a program written by 10 programmers over 10 years and I doubt anyone really understands all of it. But we have mathematical fields that hundreds of mathematicians have written over centuries, and people still are able to understand it all perfectly. It is true that a computer can easily read a computer program, but since we are arguing about teaching humans you would need to show evidence that humans can actually read and understand complex code well.

Someone hasn't read the C spec, with all its specified as undefined behavior.

Programs working on real systems is very different from those systems being formally specified. I suspect that if you only had access to the pile of documentation and no real computer system - if you were an alien trying to reconstruct it, for example - you'd hit serious problems.

Shall we be satisfied with the same low standards in a technical field, because it is how it is?

Hands-on users of math notation are complaining that it sucks. I'm not sure why a dismissive "works for me" is so often the default response.

I made this point in another comment, but I think it bears repeating and elaboration: Consistency isn't required (at least outside any single paper), but explicitness would be a tremendous boon.

Software incorporates outside context all the time, but it pretty much always does it explicitly (though the explicitness may be transitive, ie. dependencies of dependencies). Math papers often assume context that is not explicitly noted in the citations, nor those papers' citations, etc.

Instead, some of the context might only be found in other papers that cite the same papers you are tracking down. You sometimes need to follow citations both backward and forward from every link in the chain. And unlike following citations backward (ie. the ones each author considered most relevant), the forward links aren't curated and many (perhaps most) will be blind alleys (there also may be cycles in the citation graph, but these are relatively rare). But somehow you have to collect knowledge (or at least passing familiarity) with an encyclopedic corpus in order to at least recognize and place the context left implicit in any one paper in order to understand it.

It's maddening.

I often wish people were more like computers. It probably wouldn't make the world better but it would make it more comprehensible.

That's only really a good description for the most well trod areas, where people habe bothered to iterate. I think a more realistic statement would be:

"Math notation is the way it is because some mathematician found it sufficient to do and communicate math, and others found it tolerable enough to not bother to change."

Personally, though, my problem has always been where publications use letters and symbols to mean things that are just "known" in some subfield that isn't directly referenced. It's not a problem for direct back and forth communication during development, true, but it dramatically increases the burden on someone who wants to jump in.

That all said, it would still be quite nice if it was somehow more accessable. A lot of papers containing material that's probably actually quite standardizable remain opaque to me, and the notation invariably falls by the wayside if there's a code or language description available.

Many times, math notatons have been thought to be minimal, or most clear possibly, only to fall by the wayside

Whereas this notation serves domain specialists well, it still leaves people like me somewhat confused

A cheat sheet - even to the practical norms - would go a long way

CS professors writes just as incomprehensible math as everyone else, as you can see many here brings up examples of CS professors writing incomprehensible math in their papers.

Try to understand programming based on a programming lecture powerpoint, it is usually impossible.

Edit: Also you can't write code for what he is talking about in that lecture. Code cannot deal with infinities or continuous values. You'd get approximations which isn't the same thing, then you'd need to prove that those approximations are good enough which would have to be done without code anyway.

Haskell is unreadable to one who has not trained in it or similar languages ... why don't they make the syntax more readable? Or C++ with its modern templating ... why don't they change the syntax to make it more readable?

You might be tired of wandering into someone else's area of expertise and telling them:

You must change! You must make it more accessible!

Believe me, mathematicians are tired of non-mathematicians wandering up and saying:

Look! Computer programs are easy and intuitive and everyone can understand them, even without training! Make math like that!

Do you really believe that math notation is deliberately designed to make it hard for people untrained in math to learn how to use it? Do you really believe that no one has tried to make it more accessible?

Do you really believe you know more about why math notation is what it is than mathematicians and trained mathematics educators do?

I don't see why wouldn't it be some kind of local maximum. Maybe there are better ways, but they are sufficiently far away from current notation, that they aren't even thought about.

1. They explain all abbreviations at the top.

2. There is a lenghty text explaining the formula.

3. It's mathematically pretty easy if you know partial differentation.

Also scientists and engineers write pretty horrible code....

Programs don't do this, why do you expect every math paper to do it?

> Link to external resources with explanations!

This is called a bibliography, every book that isn't so old that it is the definition and paper includes one. In many textbooks there are also appendices which cover (some of) the foundational material. And most include sections (often in the front and back covers) that show the symbols and their names, if not their definitions.

But I remember sketching an algorithm to someone and just inventing notations on the fly as I did so, knowing that they would simply be ways to remember the underlying ideas.

Even so, at 1st year undergrad the notations used in Mathematical Physics vary from those used in Introductory Graph Theory, and again from Real Analysis. But once the reader knows the underlying semantics, the actual notation is mostly a non-issue (as you know).

Of course you still wont be able to understand most math papers written by pure mathematicians, but it should be fine for whatever you need in CS. I know all the topics on that page, it is just a very fleshed out math major.

It's formula 1.12 at the start of section 1.5 on this page converted into a Scheme representation, in section 1.5.2.

Of course, even that isn’t quite the standard notation; it’s using a less ambiguous notation which they invented for the book. From the preface (https://mitpress.mit.edu/sites/default/files/titles/content/...):

---

Classical mechanics is deceptively simple. It is surprisingly easy to get the right answer with fallacious reasoning or without real understanding. Traditional mathematical notation contributes to this problem. Symbols have ambiguous meanings that depend on context, and often even change within a given context.¹ For example, a fundamental result of mechanics is the Lagrange equations. In traditional notation the Lagrange equations are written

d/dt ∂L/∂q̇ⁱ − ∂L/∂qⁱ = 0.

The Lagrangian L must be interpreted as a function of the position and velocity components qⁱ and q̇ⁱ, so that the partial derivatives make sense, but then in order for the time derivative d/dt to make sense solution paths must have been inserted into the partial derivatives of the Lagrangian to make functions of time. The traditional use of ambiguous notation is convenient in simple situations, but in more complicated situations it can be a serious handicap to clear reasoning. In order that the reasoning be clear and unambiguous, we have adopted a more precise mathematical notation. Our notation is functional and follows that of modern mathematical presentations.² An introduction to our functional notation is in an appendix.

Computation also enters into the presentation of the mathematical ideas underlying mechanics. We require that our mathematical notations be explicit and precise enough that they can be interpreted automatically, as by a computer. As a consequence of this requirement the formulas and equations that appear in the text stand on their own. They have clear meaning, independent of the informal context. For example, we write Lagrange’s equations in functional notation as follows:³

D(∂₂L ∘ Γ[q]) − ∂₁L ∘ Γ[q] = 0.

The Lagrangian L is a real-valued function of time t, coordinates x, and velocities v; the value is L(t, x, v). Partial derivatives are indicated as derivatives of functions with respect to particular argument positions; ∂₂L indicates the function obtained by taking the partial derivative of the Lagrangian function L with respect to the velocity argument position. The traditional partial derivative notation, which employs a derivative with respect to a “variable,” depends on context and can lead to ambiguity.⁴ The partial derivatives of the Lagrangian are then explicitly evaluated along a path function q. The time derivative is taken and the Lagrange equations formed. Each step is explicit; there are no implicit substitutions.

---

    (define ((Lagrange-equations Lagrangian) q)
      (- (D (compose ((partial 2) Lagrangian) (Gamma q)))
         (compose ((partial 1) Lagrangian) (Gamma q))))

And it has the advantage that you can run it immediately after typing it in, assuming you have a coordinate path to pass to it. They immediately go to a concrete example:

    (define ((L-free-particle mass) local)
      (let ((v (velocity local)))
        (* 1/2 mass (dot-product v v))))

    (define (test-path t)
      (up (+ (* 'a t) 'a0)
          (+ (* 'b t) 'b0)
          (+ (* 'c t) 'c0)))
    
    (((Lagrange-equations (L-free-particle 'm))
      test-path)
     't)
    ⇒ (down 0 0 0)

They then give another example, symbolic this time:

    (show-expressi...