It may not be coincidental that code written by mathematicians tends to be ... terse.
Why math is painful to read
11–20 of 106 posts
Re: Why math is painful to read
#12Really good point. That always annoys me with most formulas. I have to dig back several paragraphs (with no scope/indexing tool) to figure out what the individual variables mean.
And a lot of papers do not even bother to explain some variable but just assume the reader already knows it. One of the earlier commenters made this mistake very prominently enthusing over Maxwell's equations.
Yes of course the formula looks better shorter when you already know what it means. But that completely misses the point that it was written down to explain it to someone who doesn't already know what it means. If you already do you're the wrong target audience.
Longer variable names would help. Or for online paper just have a tooltip that opens the paragraph that explains what the variable mans when you hover over it would also help a lot.
Or maybe some color coding to easily find it (there was a recent link here recent explaining FFT which used this trick very successfully)
But I guess most Mathematicians don't bother because they only write for a small circle of colleagues anyways.
Re: Why math is painful to read
#13[deleted]
Re: Why math is painful to read
#14The real reason math is hard to read is that it's the distillation of a thousand winding ideas and failed attempts that took hundreds of hours of thinking and hundreds of pages of scribbling, all condensed into the most concise possible three pages of unmotivated line-by-line proof with no context or explanation.
In other words, math is a language for formalizing intuition which, when finally written, unfortunately removes all traces of intuition and leaves only the end result of the thinking, not the path that it took to arrive at the critical insight.
Re: Why math is painful to read
#15No, that's not why math is painful to read. The reason math is painful to read is because it's either obvious, in which case it's not painful to read, or because it contains new ideas that you need to work hard on before you can grok them.
There is no short cut. You can't write math like a novel, you can't read math like a novel. There are ideas that are deep and difficult, and unless you actually spend time working on the material you simply won't "get it."
Time and again I have explained to people something that's comparatively simple, and easily within their grasp. I've tried different approaches, different formulations, different arcs through the material, and time and time again the person has followed every step in detail, and just not "got it." There is no enlightenment, there is no sense of what's happening. It's only after they've invested the time that they have come to a sense of what's happening, and then the light dawns. Occasionally they say things like "Why didn't you explain it like that in the first place?" Then I point out that I did, but at that stage, they weren't ready for it.
It's like following a SatNav. You get to your destination having followed the turn-by-turn instructions, and yet, in truth, you really have no idea where you are. You have traveled from one local pocket to another local pocket, with no understanding of the space through which you've been.
Math is painful to read because you're trying to follow the turn-by-turn directions, instead of taking control of your own understanding and working to gain knowledge and insight, scope and intuition.
You won't understanding it without doing the work, and complaining that it's badly written is the smallest part of the problem.
Re: Why math is painful to read
#16It seems to me like the author approaches math looking for a concise & elegant programming language, but instead just finds...math.
The notion of scope as an integral part of mathematics is a particularly interesting suggestion. Mathematics is the study of abstract, quantitative truths (and falsehoods), and these truths, being universal, have universal scope (although you can constrain some truths as special cases of more general truths, but I doubt this is what the author had in mind). To consider this a problem indicates that the author is searching for a tool rather than a way of thinking.
"Math is such a cognitive overload of operators" because it succinctly describes a great deal of action. A program in a conventional language like C might take a hundred lines to describe what could be described in a handful, using some mathematical notation. If you want a programming example, just look at the average length of an APL program vs the average length of a Javascript or C program.
Math is inherently hard. So is programming. Not many people can do either very well. Even fewer can do both very well. The notations of math are meant to be flexible because math is a tool for discovering new ideas and analyzing their properties. Programs are meant to accomplish some real world task. Criticizing the language of one solely from the perspective of the other is not a useful activity.
Re: Why math is painful to read
#17Earlier quoted context omitted.
> 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 th…
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.
Re: Why math is painful to read
#18I think for many, the problem is pacing. When you're reading text, your eyes are moving down a line at least once a second. But reading an equation takes much longer. You have to stop your eyes and ponder every piece of it. It can help to copy it down on paper.
Re: Why math is painful to read
#19Hungarian notation for formulas would be terrible. It would make it that much harder to memorize, as well as increase the difficulty of seeing patterns between various formulas.
The real problem here is trying to force programming notation into mathematics notation. Programming variables emphasize what you're working with while mathematics variables emphasize ideas and relationships.
Re: Why math is painful to read
#20I 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 e…
> 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 th…
> 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.