This is part 6 of an ongoing series on mathematical culture.

I’d say math is some kind of Dynamicland, in the sense that it is part of the fabric of reality, to make sense of reality itself. For example, when you read Nonlinear Dynamics and Chaos by Steven Strogatz, he introduces some mathematical ideas to precisely capture nonlinear movements. I think he calls them deterministic mechanisms that are unpredictable: the movements are produced by some deterministic mechanism, but you cannot predict how it’s going to move. That is something that really blew his mind, and he wanted to investigate it further. One concept he introduces is phase space, which basically lays open the entire behavior of that system, to put it in a vague way.

Another example is the Dihedron algebra, which has some non-obvious ideas that only emerge after you spend weeks thinking in it and watching what people do with it. The dual numbers live inside this algebra, and it’s the environment the complex numbers are born from. When you dig around in it, the complex numbers simply fall out, because they’re a sub-algebra.

Or, for example, when you compare that algebra with the algebra of polynomials that are modulo x + 1, you can see that they are isomorphic.

To come back to the dual numbers: they are of the form α + βε. What Wildberger did was put that into a polynomial, p(a + bε) with p = αⁿ. Then you get the expression (a + bε)ⁿ, and after applying the binomial expansion and some algebraic manipulations and simplifications, you can see how the derivative falls out of it.

If p = αⁿ, then

\[p(a+b\varepsilon) = (a+b\varepsilon)^n = a^n + \binom{n}{1}a^{n-1}b\varepsilon + \binom{n}{2}a^{n-2}(b\varepsilon)^2 + \cdots + (b\varepsilon)^n\]

and since ε² = 0, we have

\[p(a+b\varepsilon) = \underbrace{a^n}_{p(a)} + \underbrace{n a^{n-1}b\varepsilon}_{Dp(a)}\]

So math can be a breeding ground for new math when you play around in it,Mathematicians have marveled at versions of this for a long time; Wigner’s “unreasonable effectiveness” is the outward-facing version, and mine is the inward-facing one. and the math provides you with tools to investigate the math itself. It’s like this unfolding process, right? You create the sense-making environment as you’re making sense of the math, and reality gives you the tools to do that. Isn’t that completely nuts? And Dynamicland is this very meta-environment, which promises to let you make sense of the math in Dynamicland. It tries to be the most general environment where ideas become physical, shared, and directly manipulable in a room with other people.

Update

I feel like I have arrived at a point at which I’m able to riff off of some mathematical ideas. Like I’m really investigating some mathematical structure. I’m not just doing exercises. I’m actually investigating some mathematical thing and it feels like a conversation that is unfolding. I’m picking up a language and I’m creating compared to mechanically executing some algorithm, which has little to do with math. And it feels like I’ve been doing it this week for the first time. It feels like I’m doing real math research for the first time in maybe five years or maybe my entire life. Everything that happened before that wasn’t math. The two months I spent drilling exercises in Math Academy? That wasn’t math. That had nothing to do with math. So, I think the change happened after I spent a lot of time in Dynamicland and becoming familiar with its philosophy. Even though I don’t have access to Dynamicland itself, just immersing myself in this world through its website, which is better than nothing. I think that made me absorb some of its ideas subconsciously and apply them to other domains. That’s kind of my feeling. And it has affected my relationship to math.

But the last two days I didn’t do more math because developing mathematical ideas is incredibly exhausting. I can only do it one step at a time. As Terence Tao would put it, I’m using the wastebasket. So, if an idea or structure is too complicated and abstract for me, I put it aside. I’m not trying to build on top of vagueness and confusion. I’m not trying to build on top of some really abstract thing that I don’t understand. I can’t force myself how I’m going to emotionally respond to a problem that I’m confronted with. Am I only going to be confused? Will I be able to pick something up? Well, that is out of my control. I can create an environment in which I can feel at ease where I don’t feel anxious. I feel calm so that my attention can go where it wants to go. So, that is usually outside of the classroom. And the exam and exams because these are not environments that are favorable for research or creative thinking. They don’t support it. Quite the opposite.