Comment by aroberge | Hacker News Reader

Comment by aroberge | Hacker News Reader

I could not disagree more with that post. Mathematical notation is to Python what English would be to Assembly Language: it is a high-level notation, designed see the forest rather than getting lost in the trees.

Take Maxwell's equations for example. After learning vector calculus and becoming familiar with its notation, you notice how much meaning you can extract at a quick glance from the four Maxwell's equations expressed in differential form. The purpose is not to obfuscate the meaning (like the author implies when he mentions the Obfuscated C contest) but rather to reveal/encapsulate as much information as possible in a way that is easy to understand at a glance. You want to know what would happen if magnetic monopoles existed? Easy to do by just adding a few extra terms that are obvious by symmetry of the equations.

And, if you go beyond vector calculus, you can see an even clearer picture by writing

    F = dA   (equivalent to the usual two homogeneous Maxwell Equations)
    d*F = *J  (equivalent to the remaining two)

> Mathematical notation is to Python what English would be to Assembly Language

Which is often the problem, it looks specific, but it really isn't; it's full of implicit assumptions about what the reader should know. It's not executable, because it's a language designed for being written by hand rather than executed by a computer.

However, the teaching of math would greatly benefit from an explicit executable form that makes no assumptions, i.e. a programming language. Gerry Sussman makes this case, he's pretty convincing.

You think that people learning math will find it less confusing if they have no context (which, in mathematics, is largely equivalent to "assumptions") for what they are learning and have to build their mathematical knowledge from first principles? Or am I misunderstanding your post?

I don't think anything, I was recalling what I heard Sussman say at a conference, which he makes clear in http://en.wikipedia.org/wiki/SICM. Though I saw him a good 8 years after the book, so I'm sure he'd grown some new opinions.

He is of the opinion that if you can't write the algorithm, you don't understand it; forcing students to write the algorithm aids in teaching them a real understanding. He spent quite a bit of time complaining about the vagueness of mathematical notation due to its implicit assumptions for the purposes of teaching and the superiority of executable code for the task.

On a side note, the speech was aimed at both engineers and programmers and the audience was a mix of both. He did a wow demo with a circuit on the overview, calculating all the voltages and resistances on the fly from chosen starting values that made all the programmers smile really big and all the engineers jaws drop. Everyone was impressed because he flew through it, but the programmers much less so because what he did was just how a programmer thinks, albeit it very fast. He was demonstrating to the engineers the benefits of thinking like a programmer. As he is both and teaches both, he clearly connected some dots that they generally don't.

He is of the opinion that if you can't write the algorithm, you don't understand it; forcing students to write the algorithm aids in teaching them a real understanding.

Absolutely.

I had memorized the notation for derivatives. Until I was able to realize how the notation tied to finite numerical approximations, I never really understood it.

If you don't know what I'm talking about, the right way to approximate (d^2/dx^2)(f(x)) is to write an operator d defined as d(f)(x) = f(x + dx) - f(x). And now your numerical approximation for the second derivative is (d(d(f))/(dx * dx))(x).

Now replace your d operator with the much better d(f)(x) = f(x + dx/2) - f(x - dx/2), and see how good your numerical approximations become!

The notation actually means something. When we switched from infinitesmals to limits, we lost sight of that. When you try to write the algorithm, you're going to be reminded of its meaning, fast.

> When you try to write the algorithm, you're going to be reminded of its meaning, fast.

Exactly, and forced to replace implicit assumptions with explicit instructions to make it work.

But that said, what about pure existence proofs?

There are algorithms that we can prove exist, but we have no idea how to find them.

The programming is a tool to aid learning, not replace all the other tools at your disposal. Use it when it helps.

That is fine for you, not everyone needs to know about Wiener path integral formulation of quantum mechanics ... which is based on functions that are almost continuous everywhere but nowhere differentiable (the set of functions that are differentiable and with which you are familiar is a set of measure zero i.e. it is comparable to using only integers instead of real numbers). The limit process, which you discard as not meaning anything, is essential to understand the Wiener path integral formalism. Fortunately, mathematicians and physicists did not "lose sight of that" ... and are able to go beyond what your limited understanding would limit you to. I don't mean this in a deragotary way - you obviously don't need to understand at a deeper level than you do; few people do ... but to those that need this deeper level, the mathematical formalism that you dismiss is, in fact, essential.

Edit: my apology for making an unwarranted assumption about what your level of understanding is.

I don't mean this in a deragotary way - you obviously don't need to understand at a deeper level than you do; few people do ...

Do you like to assume much?

I have a masters in mathematics. I likely understand a lot more than you give me credit for.

I do not "discard" the limit process. I'm saying that the d/dx notation, which we get from the old infinitesmal approach, has insights buried in it that most students miss. I certainly missed it for a long time.

Limits won for good reason - they were the first consistent approach to defining the derivative that mathematicians understood, and were a key step on the way to resolving the inconsistencies in previous definitions that Fourier series had revealed.

But limits are not actually "essential". I am personally aware of multiple completely rigorous formalizations of Calculus. The second most famous of which is Robinson's non-standard analysis, which has absolutely nothing looking like a limit in sight.

> However, the teaching of math would greatly benefit from an explicit executable form that makes no assumptions

I'm skeptical of this logic. I think nobody would be able to learn mathematics if they had to keep it all in their head at once. "Unobfuscated" maths, i.e. maths that has every single logical deduction written out from first principles so that you can (to use your words) "execute it", would be impossible to understand because, fundamentally, we are not computers.

Instead we learn some, we internalise it, and then we can appreciate the new notation that encapsulates what we have learnt before without exposing us to the underlying complexity that we simply don't need to solve higher problems.

Mathematics is beautiful. If anything, it's programming languages that have the catching up to do.

Edit: more generally, the blog post seems to be a case of "when all you have is a hammer, everything looks like a nail" -- if you spend all day programming, then it might make sense to try and relate mathematics back to something you're more comfortable with. That doesn't mean that it's the most sensible way for everyone else to learn mathematics.

> I'm skeptical of this logic. I think nobody would be able to learn mathematics if they had to keep it all in their head at once.

That's what functions are for.

> Instead we learn some, we internalise it, and then we can appreciate the new notation

Those new notations are functions, simplified abstractions of more complex processes that allow us to think in larger concepts and not sweat the details.

> Mathematics is beautiful. If anything, it's programming languages that have the catching up to do.

Perhaps, perhaps not.

Programming languages aren't explicit executable forms, they are abstractions. Unless you are working with machine code or assembly, you're getting the abstract ideas which get expanded by the compiler. What it sounds like you're advocating is to write out 3+3+3+3+3 instead of 3*5 (the latter is assuming).

They're executable abstractions, thus they are explicit. And no, that's not what I'm advocating at all. There is no assumption in 3 * 5; * is a defined function, if you need to know how it works you can look. What's important is that it's computable. Notation often isn't, too many implicit assumptions that rely on a trained eye to execute correctly.

What's that, we should write math in a language that computers can understand?

Knuth did that:

     \frac{\frac{1}{x}+\frac{1}{y}}{y-z}

And thank Don, he made the computer output something human readable.

It would be very tricky to make a computer understand what a TeX equation actually means in general, rather than just know how to typeset it. e.g.

   f(d) = \int_0^1 g(y) d y dy

(Meaning integral of (g(y)dy) w.r.t. y)

I couldn't possibly disagree more with your post. :)

The language of math requires exact precision which makes extracting what the series of abstract symbols "mean" incredibly difficult. Trying to learn new mathematical concepts from wikipedia is damn near impossible. Here are two examples.

b-spline: http://en.wikipedia.org/wiki/B-spline spherical harmonics: http://en.wikipedia.org/wiki/Spherical_harmonics

Trying to figure out what those walls of symbols mean requires quite a lot of focus. Dig up source code for either and it's orders of magnitude easier to understand. The images for spherical harmonics conveys more meaning and understanding in a fraction of a second than spending 3 hours with a piece of paper and the equations would.

The language of math sucks.

For the sake of an argument, let's assume you are right, and the language of math sucks.

Then, we have had centuries of mathematicians inventing ever changing notations and ending up with a notation that sucks.

If so, those mathematicians must be extremely stupid. Surely persons of average intelligence must be able to come up with better notations, and find proofs for stuff that is out of reach of those morons?

I do not rule out that notation can be improved. After all, it took millennia to move from Greek and Roman arithmetic systems to Arabic notation. There is no reason to assume we have reached a global optimum. However, I also find it unlikely that mathematical notations truly are stupid.

Your argument is like someone complaining about the difficulty of discriminating between "37" and "73" because it is way easier to see the difference between 37 and 73 scratches in a piece of wood.

It's not that mathematicians are stupid, they just that they have backwards compatibility problems.

The notation of mathematics is centuries old. It's very hard to take notation away from the set that must be learned, but easy to invent more.

Eventually old constructs can fall out of favor, but that is a long and slow process.

This is hardly true. Mathematical notation changes all the time-in fact, inconsistency is a significant problem, because Mathematicians love to experiment with notation, definitions, etc.

Seriously, just look at a Mathematical logic book from 50 years ago vs. one today. You won't even recognize half the words, let alone the symbols.

> If so, those mathematicians must be extremely stupid.

Not necessarily. Before modern times mathematicians worked alone and didn't need a notation that others would understand. Newton, for example -- his calculus notation wasn't very good, and over time Liebniz' notation ended up being used instead (completely apart from the issue of who invented the calculus).

> Surely persons of average intelligence must be able to come up with better notations, and find proofs for stuff that is out of reach of those morons?

This doesn't support the thesis that mathematicians are stupid. Most mathematicians only care that they understand their own notation, and give less weight to its comprehensibility to others. And some mathematicians in the distant past deliberately used obscure notation to conceal their work from others.

I would look to educators to come up with a more comprehensible system of notation, not necessarily those at the frontiers of mathematics.

Liebniz was a mathematician.

Newton was a member of the Royal Academy. They communicated with each other.

Apart from the fact that you mean the "royal society", yes, both true. They also came into conflict about priority with respect to the calculus. And surprisingly enough, that debate rages on.

I agree, it's a fun exercise to try and come up with a "better" form of notation that works in the general case. I think that most people will find that modern mathematics notation is really good stuff.

Wikipedia is not designed to be a pedagogical repository; arguing that the mathematical language sucks because a terse summary in wikipedia is not enough to understand the material is missing the point.

Would you be able to calculate the probability of finding an electron within a certain distance from your pretty pictures? I would be able to do so fairly easily from those equations. What about computing special functions in higher dimensionalal systems? The generalization from equations is straightforward; our minds can not comprehend higher dimensions in pictures.

As you wrote, yes it requires a quite a lot of focus ... but that's the price one has to pay in order to be able to build models that describe the world and allow us to make predictions. That is not to say that adding pictures or explanatory sentences is unnecessary; but you can't replace the equations by only a combination of words and pictures and be able to make predictions (such as, for example, in quantum electrodynamics where predictions and experimental results agree to better than one part in 10^10 ... equivalent to something like 1 mm precision in measure the distance from coast to coast of the US) in a better way.

Scientific theories are those that are testable. You can't pretend to understand a scientific theory if you can't use your knowledge of it to make quantitative predictions. You need math for that.

> Wikipedia is not designed to be a pedagogical repository; arguing that the mathematical language sucks because a terse summary in wikipedia is not enough to understand the material is missing the point.

Nicely worded. I think the real issue is that many people want mathematics to be easy to read and understand. Unfortunately, much of mathematics is not easy to read nor is it easy to understand regardless of how it is written. Mathematics, like many other things, often requires a lot of intellectual scaffolding to be built from the bottom up. There is no shortcut to the top.

Both of your examples are new to me as well, however I can make sense of topics discussed. Just to use your example, the first equation for B-splines defines it as a function from a tuple of ordered reals to real vector of dimension d. The real difficulty with math is not so much the symbols as they give a precise but very legalistic way of defining things but the underlying concept or intuition that it represents, the deep semantics if you will. The best analogy is law. The way a lawyer or judge defines things will be different from the way a lay person would due to the difference in precision. The descriptions will be technically correct but will make you second guess on what it really means. I wouldn't go so far as to say it sucks but math is definitely scientific legalese.

1. I came into this thread expecting to find a math expert of some sort defending their "programming language". Thank you for not letting me down!

2. It depends on context. The people writing and manipulating mathematical formulae LOVE the short hand I'm sure... but for consumers of the information perhaps it's less than ideal.

This thread would be more informative if the post author had said _what_ he was trying to read. Most math is written to communicate with other mathematicians, so if he grabbed some random paper off of ArXiv then it's not at all surprising he found it hard to read. It also makes a difference if he's trying to actually learn about a subject, or just to apply some formulas without necessarily understanding him. Math is written differently for specialist and beginner audiences, with the very terse "CRC Handbook of X" off on some other axis.

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