Calculus

The first thing I remember was rockets. Fall 2014, I was doing rocket calculations and ran into integration as a thing I needed. [embed: rocket simulator program] That is exactly the right way for me to encounter math: not somebody announcing "today we begin the unit on integration," but some physical question becoming annoying enough that the missing mathematical operation starts glowing by its absence. I wanted to calculate what the rocket was doing and the quantities were changing continuously and suddenly there was this machinery for adding infinitely many little pieces of change together into a finite answer. Wait. WHAT.

By spring break 2015 I was reading a calculus book voluntarily. [verify book / photos / notes] I need to reconstruct the exact chronology because I remember the conceptual acceleration more clearly than the dates, but calculus cracked open the universe of math for me. Derivatives were already beautiful: local behavior compressed into a rate, geometry and dynamics touching each other through slope. Integration was even better because it felt like the inverse move, accumulating local structure into global quantity. Then the fundamental theorem says these are not just two useful tricks but dual views tied together. It is difficult to explain how satisfying this was without sounding like I am sexualizing mathematics, which, to be clear, I basically am. Oh my god.

Then multivariable calculus. Jesus christ. A scalar function stops living on one little line and suddenly the geometry has room. Partial derivatives, gradients, level sets, Jacobians, divergence, curl, line integrals, surface integrals. You can represent how some quantity changes in different directions and then build operators that extract exactly the structural fact you care about. The gradient is not just notation; it is the direction the function wants to climb locally. A Jacobian is not just a matrix somebody makes you calculate; it is the local linear behavior of a transformation. Once this stuff becomes intuition, papers later stop looking like hieroglyphics because you are not decoding the symbols from scratch. You recognize what move the author is making.

Differential equations made the world dynamic. Instead of describing a static relationship, you describe the rule for how a state changes and then ask what trajectories follow. This is basically the language I would later use everywhere: robotics, control, neural dynamics, physics simulation, learning systems, brain models. Laplace transforms were BEAUTIFUL because you could take the annoying temporal structure of a differential equation, move representations, and suddenly operations that were hard in one domain became algebraically tractable in another. Fourier analysis gave another version of the same revelation: representations are tools. A signal can be the same object and expose completely different affordances depending on the basis you look through.

I coordinated community-college courses around high school until I had Calculus I, II, III and differential equations by senior year. This was logistically annoying and emotionally fantastic. [embed: transcripts] There were entire mathematical neighborhoods opening faster than school had room to formalize them. I got addicted to pushing until the operators stopped being procedures and started being things I could feel. That is the part I think paid the biggest dividend later. Memorizing a transform table is useful for an exam. Thinking in terms of what the transform buys you is useful forever.

Years later, when I began reading ML papers, I often had this weird advantage where section 3 would contain the mathematical formulation that was supposed to intimidate non-specialists and I could just... read it. Not always. There are obviously enormous parts of math I do not know. But enough of the standard machinery was already internalized that I could spend cognitive budget on the architecture instead of prerequisites. More importantly, the old habit of asking "what affordance does this operator expose?" transferred beautifully. Why this norm? Why this projection? Why this kernel? Why this latent space? What becomes easy after the representation changes? I think that habit is part of why I ended up more interested in structured models than brute-force scale.

Math also supplied a kind of honesty that appealed to me spiritually. You could be wrong. Not sinful, not socially disapproved, just wrong in a way the structure itself could expose. The proof does not care who your elders are. The derivative does not change because your family is uncomfortable with the implication. I did not consciously use calculus as an escape from religion. At this point the two worlds coexisted. But one of them was teaching me habits that would eventually become corrosive to unjustified certainty.

[embed: rocket simulator] [embed: notebooks] [embed: college-credit transcript] [photo: calculus books] [optional interactive: equation snippets that still feel beautiful]

Some of the adults around me understood how much this meant to me. Mrs. Restivo was one of the teachers who saw the whole weird thing up close, and she became attached in my memory to the end of high school in a way that still makes me emotional and embarrassed at the same time.