I’m in love with abstractions, but I’m also deeply suspicious of them. I think there’s a very real danger that they lure us away from things that are realer and more important, or, worse, they infect us and shroud our visions so we no longer see reality, we see only our representations of it. I’ve written about this many times on this blog, and longtime readers may be a bit bored with it. I didn’t intend to write so much about this topic, I swear. It’s just that I keep discovering permutations of this basic problem all over the place.
Kill math sounds like something a sufferer of severe math phobia might say. But Bret Victor professes to be a math lover. He just happens to believe that “the power to understand and predict the quantities of the world should not be restricted to those with a freakish knack for manipulating abstract symbols.
” (I tend to agree.) To that end, he’s looking at various visual and non-symbolic ways to deal with mathematics. On that page, there’s a bunch of resonant quotes. Here’s what Oliver Steele has to say:
I think that mathematicians are those who succeed in figuring out how to think concretely about things that are abstract, so that they aren’t abstract anymore.
And further down, the programmer Alan Kay:
Jacques Hadamard, the famous French mathematician, in the late stages of his life, decided to poll his 99 buddies, who made up together the 100 great mathematicians and physicists on the earth, and he asked them, “How do you do your thing?” They were all personal friends of his, so they wrote back depositions. Only a few, out of the hundred, claimed to use mathematical symbology at all. Quite a surprise. All of them said they did it mostly in imagery or figurative terms.
Now this is really interesting. The world’s greatest mathematicians (at least at one point, in the middle of the 20th century) do not work in mathematical symbology? This seems to also tie into my worry about the prevalent but wrongheaded notion that language is a necessary condition of thought, even a determinant of what one can think (linguistic determinism). Hadamard’s book, Essay on the Psychology of Invention in the Mathematical Field, is online. Hadamard seems to as horrified by linguistic determinism as I am, and large portions of the book constitute a passionate defense of what one might call visual or, at least, non-symbolic thinking. Although I try to avoid the fallacy of appeal to authority, it was a sort of relief to see a mathematical authority validate my concerns. Here he’s quoting the psychologist Ribot, describing some of what I wrote about here:
It is what he calls the “typographic visual type” and consists in seeing mentally ideas in the form of corresponding printed words. The first discovery of this by Ribot was the case of a man whom he mentions as a well-known physiologist. For that man, even the words “dog, animal” (while he was living among dogs and experimenting on them daily) were not accompanied by any image, but were seen by him as being printed. Similarly, when he heard the name of an intimate friend, he saw it printed and had to make an effort to see the image of this friend. It was the same with the word “water,” and carbonic acid or hydrogen appeared to his mind either by their printed full names or by their printed chemical symbol. Being strongly surprised by that statement, the sincerity and accuracy of which were not to be doubted, Ribot later on observed that that case was by no means a unique one and similar ones were to be found in several people.
Moreover, according to Ribot, men belonging to that typographic-visual type cannot conceive how other people’s thought can proceed differently.
This seems like a sort of horror scenario to me, and Hadamard seems to agree. Elsewhere, he also seems to speak directly to my concerns from the beginning of this post:
Among philosophers, there seems, indeed, to be a certain tendency to confuse logical thought with the use of words. (…) I think, on the contrary, and so will a majority of scientific men, that the more complicated and difficult a question is, the more we distrust words, the more we feel we must control that dangerous ally and its sometimes treacherous precision.
One of the many mathematicians and physicists Hadamard asked about their thinking for his essay was everyone’s favorite token (and actual) genius, Albert Einstein. Here’s what he had to say about his own thought process:
The words or the language, as they are written or spoken, do not seem to play any role in my mechanism of thought. The psychical entities which seem to serve as elements in thought are certain signs and more or less clear images which can be “voluntarily” reproduced and combined.
There is, of course, a certain connection between those elements and relevant logical concepts. It is also clear that the desire to arrive finally at logically connected concepts is the emotional basis of the rather vague play with the above mentioned elements. But taken from a psychological viewpoint, this combinatory play seems to be the essential feature in productive thought before there is any connection with logical construction in words or other kinds of signs which can be communicated to others.
The above mentioned elements are, in my case, of visual and some of muscular type. Conventional words or other signs have to be sought for laboriously only in a secondary stage, when the mentioned associative play is sufficiently established and can be reproduced at will.
According to what has been said, the play with the mentioned elements is aimed to be analogous to certain logical connections one is searching for.
Einstein, according to himself, is not only a visual thinker, he’s a “muscular” thinker. And it makes sense, when you think of it, that the much older neural circuitry we evolved to deal with things like moving from here to there or looking around would take some of the brunt off the newer, higher cognitive circuitry in our brains. We were walkers and lookers long before we were mathematicians.
Activities that require motor skills—the cliché of being “in touch with and at home in” your own body—are, of course, ways to get back to the more primal, perhaps realer and more important things that abstraction sometimes clouds. And “muscular” or “motoric” thought really does differ from visual thinking. If you have any experience with sports, and in particular martial arts, I’m sure you know the huge difference between seeing something done, knowing the way it’s supposed to look, and being able to actually execute it with your own limbs.
For a while, I trained Brazilian Jiu-Jitsu (then I got lazy, then I moved to a town that doesn’t have a BJJ club). One of the things I appreciated about it was the way grappling takes away abstraction and forces you to consciously inhabit your body, and think about nothing more. For instance, one fact I learned that can easily be stated intellectually, but needs to be felt is this: the hips are really, really fucking important to motion. It’s easy to imagine, but it needs to be experienced. I’m a skinny dude, and I was outweighed by everyone at my club (the few women included). The one technique I got down pretty well was how to hold bigger guys down from the side control position. They’d shrug me off mount, because I didn’t have my balance down, but I knew how to neutralize the hips from side control, and if you don’t have mobility of the hips, you aren’t getting anywhere. (Which sounds like an intellectual fact, except it’s to do with the physical sensation of shifting body weight around etc. more than with principles of human physiology.)
I don’t know if Einstein would have found some sort of physics analog to that muscular fact.
Sep 17, 2011