Idea: People hate math because math is not a language, but everybody says it is.
Gallileo said that the “book” of nature “is written in the language of mathematics.” But mathematics isn’t a language. Mathematical signs don’t have meanings like linguistic signs. They have rules that govern how you can move them around and replace them with other signs. Mathematical signs are like pieces on a chessboard, not like words.1
I know, Wittgenstein said that (most) linguistic signs have their meaning based on their use in “language games.” But if by that he meant that linguistic signs function like pieces in a game of chess, he was just wrong. Or, rather, he was confusing linguistic signs with mathematical ones.
The problem is that no one teaches students to think of mathematical signs like pieces in a game. No one teaches kids to play math. So, students get stuck trying to understand or read mathematical problems as if they were learning a foreign language. But math isn’t a language. So, instead of having fun playing mathematical games and solving mathematical puzzles, students end up miserable.
For more on the various types of signs, see my article in The New Yearbook for Phenomenology.
1. See Edmund Husserl, Logical Investigations, trans. J. N. Mohanty (Routledge, 2001), Investigation I, section 20, pp. 210-11.
Funny the timing of this note. I am just reading a long article in the most recent Philosophia Mathematica about Gian-Carlo Rota, a mathematician primarily. (I was his note-taker for his Combinatorics course at MIT in the 1960s.) He was also a phenoenologist, so of course I thought of you last night. I’ll forward a copy of the article after I’ve finished reading it. Rota makes your point, although this article makes a careful distinction between Rota’s and Husserl’s epistemological anti-foundationalism. I’m assuming that your article cited above is a spin-off from your dissertation, so maybe I can be spared reading it.
BTW, I listened to a podcast of yours for the first time yesterday, the one on death, reincarnation & resurrection. I didn’t recognize your voice. It was much more basso than I remembered. Myabe it is computer-enhanced? After all, what is reality in a digital world? Maybe you hadn’t fully reached puberty when you were at Messiah?
Hah! It’s funny what a recording one’s voice does to it. I think it has something to do with the fact that mics pick up what one’s voice sounds like from a much shorter distance than normal conversation happens at. And yes, I have been known to enhance my own vocals digitally “to compensate for a poor recording environment.” I don’t remember if I had gotten comfortable enough with my recording setup by ep. 10 to feel okay about not doing so. 🙂
I think my dissertation director worked with Rota. Small world!
Also, yeah, the article is a development of part of my dissertation. A better-supported argument and a clearer, more-developed theory, but the basic idea was already there in the dissertation.
I am sorry to say that I disagree with you. I pains me to do this, so I choose to only halfway disagree. Back in the history of Mathematics, when math equations were written out rather than in the symbolic methods we use currently, it would be much easier to see the formal and structured language of math. See Euclid’s postulates for examples of this type of mathematical language and the building of one statement off another to construct proofs. Perhaps this is why Galileo would say something like that in his age. The symbols we use now make a paragraph long statement take only a short line of space. I make this point because of the Ian Stewart book I read recently called In Pursuit of the Unknown. He mentions math’s humble beginnings.
Now that I so rudely disagreed with you, I will take part of it back. I do think that since math has moved so far and divorced itself from linguistic objects recognizable as language, modern math may no longer be just a language as we define them today. Matt Parker in his book Things to Make and Do in the Fourth Dimension, makes exactly the same point as you regarding mathematicians having fun trying to solve and understand how something works. Forcing students to interact with math simply by rearranging equations misses out on the fun and power of math. I agree that if we want people to see and understand math, we need to show them the fun of it rather like showing someone how to write an enjoyable story with language instead of just how to parse and move around parts of a sentence. Hopefully we can still be friends after this terrible argument.
I still love you, Jeremy. And you make excellent points. It’s CRAZY to see the pre-symbolic math people had to work with. All word problems all the time! Thank goodness for symbolisms!
I think the shift from a more linguistic approach to a more symbolic/gamey approach also happened logic, when mathematicians shifted logic out of its Aristotelian setting into a symbolic mode. And that shift is what gave rise to computers, and hence gave rise to our shared college education, and hence our everlasting friendship!
Also, Matt Parker is awesome!
A propos Jeremy Keever’s comments, but with my usual rambling.
The *second* article in the journal I cite above is about a proposed axiomatic system for Special Relativity. (Only 7 axioms, in a three-sorted logic, to be precise!) The authors argue that something is gained by axiomatization because now one can have a better idea about what really is at stake in Special Relativity. The physics becomes more than just a collection of experiments. One can make and test predictions — predictions which the logical manipulation of the axioms suggest, or more precisely in some sense the axioms force on us. (“If you love our axioms, you’ll love our theorems.”) Of course that’s what Galileo did, whom you cite, Micah, although not very favorably. I disagree with you when you say that mathematics is (just) a game.
Of course one can do almost all of Calculus I by manipulating syntax according to the algebraic rules of a game … except for the “word problems.” And that’s precisely the point: translating the words to a syntax is the hard part. And that requires meaning. A good Calculus I course uses pictures along with words, even pictures *before* words.
Lest you think that the Special Relativity axioms are just the usual Minkowski equations about mass and time — more syntax–, the primitives of the axioms are things like Observers, Photons, InertialBodies, and an (algebraic) Field.
Jeremy is right. Math and logic are more than syntax. But math and logic are also more than a game. Like any good abstraction, they allow us to get at the essence of things. (I’m using the word “essence” as a layman. No fair throwing Latin at me: ens, res, aliquid, you philosopher you. Or German: Fundierung, in Husserl’s sense.)
When I dialog with atheists, I feel like saying: “Mathematics exists, therefore there is a God. Q.E.D.” I don’t know what holds Eugene Wigner back from that conclusion in his essay (available for free on-line) “The unreasonable effectiveness of mathematics in the natural sciences.” Mathematics is not a game like chess; it’s a game like physics or chemistry or biology or … life itself.
More on Rota. You might be interested to know that Rota had inscribed on his tombstone “mathematician-philosopher,” which is what Galileo had inscribed on his. Yes, the order is important. Neither saw themselves as philosophers primarily. Rota once called himself the “Galileo of Combinatorics.”
OK, one more thing. I review one of Rota’s books here, for the curious. https://dl.dropboxusercontent.com/u/2354407/ROTA.HTM
This is all awesome! And you’re right that math isn’t *just* a game. It’s a game connected up to the real world, like Ender’s Game. (It’s just that one can follow the rules/syntax of the game while the game is being played, whether or not one thinks about what all the symbolic moves mean for the real world.) Thanks for all the references too!
I get RSS feeds on your posts, but not on comments to your posts. Can micahtillman.com do the latter? It’s a pain to remember to go out to get the comment list to see if the dialog continues.
… but I will because I love you.
I agree about the comments. I would have missed out on yet another learning opportunity from the erudite Dr Chase. Thankfully I just had to find out what you thought of my comments and I learned some things as a result. I will have to check out the axiomatic approach and have my mind expanded.