Mathematics as thought
&gt;Forget executability for now but even just having a convenient way of parsing a formula, havings its notation explained to you automatically.. that would be so important<p>&gt;To me, math notation is a programming language...<p>That&#x27;s the thing though - <i>it&#x27;s not</i>. It&#x27;s a <i>language</i>, but not a programming language. It&#x27;s not strict. It&#x27;s by humans, for humans.<p>Why is it so clunky sometimes? Because explaining things is not easy. People are trying their best, but in the end, they get together after the conference talks over a glass of beer and go &quot;Well, here&#x27;s what&#x27;s really going on there&quot;.<p>Have you ever been in a state where you <i>know</i> what you want to say, but just <i>can&#x27;t find the right words</i> for it? That&#x27;s the perpetual state of mathematical writing.<p>&gt;You probably agree that the vast majority of math papers are never read by more than two people, the original author and the reviewer.<p>Unlikely. Usually, there are groups of people who get together at conferences and talk about what they do. Mathematicians rarely work in isolation.<p>&gt;Every paper could be like a programming library<p>In a way, they are -- but the hardware is your brain. You can&#x27;t make the computer do the work for you -- no more than we could improve on <i>this very comment</i>. In the end, the paper is communicating ideas.<p>Yes, there&#x27;s work on people formalizing math to turn proofs into computer programs. The result is machine-verifiable, but unreadable - as is often the case with code anyway; without documentation, it&#x27;s not easy to understand what the code is doing.<p>&gt;Then let&#x27;s read the API docs to get which final formulas are usable to me now that you wrote this paper.<p>But math is not about formulas. Often it&#x27;s about concepts, constructions, patterns, ways of looking at things.<p>&gt;We need to find a way to write math more efficiently on the computer now so we can get better and faster at it<p>I don&#x27;t think tooling is the bottleneck, really. We have LaTeX, which is easy enough to use, in my opinion.<p>----------------
That said, one person that would agree with you is Stephen Wolfram.<p>Mathematica Notebooks are pretty much exactly what you describe: they are executable papers, where you can mix text with computations and code, etc.<p>One reason they are not the norm is that Mathematica is a product that costs $$$ (although the engine is free with RaspberryPi, and Notebook reader is free, IIRC). They are going the way of the cloud now, though.<p>The competing FOSS solution is SAGE math: <a href="http:&#x2F;&#x2F;www.sagemath.org&#x2F;" rel="nofollow">http:&#x2F;&#x2F;www.sagemath.org&#x2F;</a> - but it&#x27;s more of code-for-math than the concept you describe. Mathematica hits it on the nose.<p>It&#x27;s been around for a while, but still didn&#x27;t really catch on.<p>Part of it is proprietary format, part of it is inertia -- but part of it is that it&#x27;s often not what the authors want.<p>What the authors want is <i>tell a story</i>, not create an executable object. The symbols are just crutches.<p>I think the <i>real</i> problem is that the story-telling aspect is thrown away in many papers, leaving the place only for the result. People don&#x27;t like showing the dirty work, the unfruitful steps, their thinking that brought them there. The informal, gritty stuff. (They leave it for beers after the talks).<p>But that&#x27;s not how it used to be. I was trying to learn about quaternions one day, and found the original lectures by Hamilton, their inventor. It read like a novel. That&#x27;s how math writing should be. It degenerated in the last 100 years or so, but it&#x27;s coming back to life now, I think.<p>&gt;Whoever sits at these prestigious journals in my opinion is already being paid for this very job: to make sure that papers are accessible.<p>HAAHAHAHAHAHAHAH. HA. HA.<p>Sorry, my friend, let me ruin your world view here.<p>First, effectively, <i>nobody</i> sits in the journals. Mathematicians write papers, other mathematicians review them - voluntarily. The journals are often little more than matchmakers. That&#x27;s why we are having a revolution of sorts now: people are starting to ask why we need the journals in the first place. And some people outright believe that we don&#x27;t - that ArXiV (the website where mathematicians put their papers without review) is enough.<p>Secondly, <i>nobody gets paid</i>. Mathematicians don&#x27;t get paid to write papers, reviewers don&#x27;t get paid to review. There is an immense pressure to publish, but it&#x27;s not like one gets paid per paper.<p>If you mean the <i>publishers</i> that host the papers - their purpose is to make money off subscriptions, and that&#x27;s about it. #downwithelsevier<p>Why do people still try writing good papers? Because they want the ideas in those papers to spread. Why do papers still suck? Because explaining something clearly is <i>hard</i>.<p>&gt;Also I don&#x27;t think that a math notation which has symbols that don&#x27;t even exist on ascii keyboards should be acceptable.<p>And everyone should just speak English. Увы, увы, было бы довольно печально жить в таком мире.<p>&gt;I will read your link.<p>Please please please come back here and share your thoughts when you do! Can&#x27;t wait to hear them.