Life is short, so one should read only the books that give you a new perspective on the world life and all that. This is a central tenet of my reading these days. Considering that I have probably lost more than half of my active life already, books with repetition of trivialities bother me.
I stumbled on Paul Lockhart while going through the black hole of learning maths. His essay mathematician’s lament is very famous. But not a lot of people know or have read his two books, measurement and arithmetic.
I did my first cursory reading of both of these books. Spent more time with arithmetic (the newer book) than measurement. But the reading is cursory in the sense that I was trying to get the overall sense of his arguments and few big picture ideas from them. I plan to follow up now with two detailed readings, one to actually solve the problems and one to sort of appreciate where I agree with the methods and where I disagree. (This is another technique I guess I heard about in some podcast, reading thrice to get full understanding). But this plan may never materialize unless I invent a way to stretch days to 48 hours or become a hermit.
I came out a lot wiser even through the cursory readings though. First Lockhart’s passion for the subject is contagious. Secondly, he thows around some tricks or big picture ideas that can radically change your perspective about math. One might even know about these ideas but you really see their importance when you hear from him. Here are a few that have captivated me in no particular order.
- Numbers as creatures. Lockhart says that he thinks of numbers as some sort of creatures with some properties (say prime-ness or even-ness). This POV suddenly changes maths from a study of inorganic objects which all looked similar (to me so far) to study of some living creatures.
- Structures of mathematics (numbers, shapes & curves, procedures for measuring things) are about conveying information. For example, numbers (counting and comparisons) are the most basic kind of information that you can convey. So choose representation of structure to convey the information in meaningful way. And no single representation is the best. This POV again liberates you from accepting some set stones to using whatever that floats the boat (example being some radix/bases in counting can be better for conveying certain information).
- Related to point 2, Lockhart keeps hammering about the distinction between the structure and the representation. The number 3 as represented in the decimal system by a hindu symbol vs the creature with certain properties representing threeness.
Profound Ideas I don’t Understand
There are many ideas in the measurement book that I have not grasped at all. I guess I need to sit down and actually solve the problems on pen & paper to really appreciate whats going on. But I am quite sure that he is trying to convey something really profound, something completely out of the box. In no particular order, things I need to dvelve deeper into,
- All the conics like parabola, hyperbola are just circles from another perspective. What are the implications of this?
- Few hints about non-euclidian geometries and relativity at the start of chapter on calculus where he says there is no reason for space to be defined only by straight lines.
- Importance of Leibneitz calculus, and solving the problems in geometry through differential equations (I know about the other POV, that problem of studying motion is same as problem of studying shape/geometry). Probably this is entrypoint to differential geometry.
- He spends some time explaining why modern (algebraic) maths perspective is more elegant and useful and what are the limits of classical/geometric perspective and how it kept greeks from making progress.
Disagreeing and Agreeing with Lockhart
There’s a underlying theme on which I disagreed with Lockhart a lot. Or so I thought. He advocates maths as purely artistic persuit and highlights at many places that the practical applications of this art in real world are incidental or they don’t interest him. He is fundamentally an inward looking mathematician.
I’m pedestrian at maths and my interest in maths is purely utalitarian. I have been going deeper into maths since I read Bret Victor’s (ironically named) project kill maths. He says that just the way reading (or coding maybe) is a skill that fundamentally changed civilization, maths can also be a fundamental skill. But for that maths needs a new interface rooted in trying out things and simulations rather than symbolic manipulations. So I’m interested in maths only towards solving some interesting real world problem as long as maths can give me such an interface. My fundamental disagreement with world at large is only on how many problems maths can help solve. I belive the number to be very high.
I thought I can never appreciate Lockhart’s inward looking obsession for maths. But when I compare it with another world: programming languages, I can start to see parallels. Though I can see programming languages just as tools to solve real world problems, I just can’t stop at that. For example, I gravitate towards simplicity of Lisp/Clojure or novel but simple ideas of Julia language. I have a distress towards more heavy industrial OOP garbage. So there’s an aesthetic sense for ideas in something that should only be a tool to solve real world problems. And I do know that this aesthetic sense is purely personal. Maybe I’m just not old enough or haven’t got deep enough into maths to have similar feelings for it!