"A Mathematician's Lament"; programming in the fight against ambiguity

[gusl]

Lockhart's "A Mathematician's Lament" is a sad paper about the tragedy of K-12 math education. Lockhart starts with the metaphor of a musician's nightmare in which students are extensively trained in memorizing the names of notes, but not allowed to play them, let alone compose. (h/t Vikash)

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NerdWisdom talks about Sussman's new paper "The Role of Programming in the Formulation of Ideas", about why so many concepts in physics are difficult to learn.

<< The basic point is that our notation is often an absolute mess, caused by the fact that we use equations like we use natural language, in a highly ambiguous way. >>
<< Sussman and Wisdom do show how the ambiguous conventional notation can be replaced with unambiguous notation that can even be used to program a computer. >>

They focus on Lagrangian mechanics, but lest mathematicians feel smug, Sussman and Wisdom have similar things to say about Differential Geometry.

I hope someone formalizes things using a typed lambda calculus too.

Comments

[rdore.livejournal.com]

Completely unambiguous notation is often quite cumbersome.

I also feel the bottom up approach of just layering mechanical definitions down is often not the right way to build intuition. E.g. infinitesimals are very useful to think about even if you don't know that nonstandard analysis exists.

[jcreed.livejournal.com]

I agree that it's cumbersome to use a single uniform notation in practice --- even after reading parts of SICM I switched back to using ad-hoc and overloading-heavy "physicist notation" occasionally when doing quick computations, rather than being careful about which particular things were higher-order functions over what variables etc. But it was incredibly valuable for me while learning I think to have access to completely unambiguous formal definitions of concepts in a uniform language. And besides that I'm not sure what value actually ambiguous notation has in practice. Is this really what you mean?

[rdore.livejournal.com]

Well, I guess I had two separate points. One is the one you hit on, which is that it is often cumbersome in practice to use completely unambiguous notation. And I think this is true even if it's more than a quick calculation, if your audience is already comfortable enough with the topic that they should understand your shorthand. E.g. saying "we will work in model M", rather than having to explicitly superscript every statement with an M to indicate where it is relativized too.

I also think that when teaching a subject, some people offer meticulous definitions in lieu of intuition and bigger picture. And having to choose between the two, I would much rather someone give me the right intuitions first, and then fill in the definitions, rather than the other way around. If it were actually a situation where the definitions were unavailable that would indeed be very bad, but in my experience this is pretty rare. Whereas failing to give a good intuition is quite common.

[jcreed.livejournal.com]

Yeah, I think I basically agree with you. Avoiding subscripting is the kind of thing that I think of as "ad hoc" but not "ambiguous" --- it's not quite perfect and pristine and formally nice, but it's not actually unclear. You note somewhere that "oh yeah btw I'm just going to say M in this paper everywhere I actually mean M-sub-whatever" and it's okay.

I think you have a point also that meticulous definitions don't suffice. However, the big picture by itself also doesn't suffice. I don't necessarily want to read the meticulous definitions as my first introduction to a subject, but by gum I want them in an appendix so I can check that I'm understanding the fuzzy big picture correctly when I drift off track.