Perhaps I Got Lucky Never Learning Math
I don't feel like a scientist. When I imagine a scientist, I imagine roughly the same things that I imagined as a kid. A man in a lab coat laboring away. Or a man in a tweed coat scribbling notes on a chalkboard. I imagine someone like George Dantzig with two problems and a notebook and nothing else but his mind, thinking and scratching at paper until he advanced his field, and subsequently the human race. I am a scientist, in theory. I have published at least some original research. I've contributed to the design of drugs that may someday become real medicine for sick people. But there was never a moment where I felt like I was anything more than a pattern matching monkey who got really good at knowing which button to press and when.
One of the major reasons I suspect that I have never felt like a scientist is because while my peers were all in public or private high schools building a formal knowledge base that involved algebra and trigonometry and chemistry, I was not. I was instead homeschooled, and for reasons I'm not sure I ever really want to get into publicly, my homeschooling was very very unstructured. My last year of completing any sort of actual curriculum was fifth grade. For a few years after that I copied answers out of the answer books to pass the couple classes I technically completed, and by highschool, I had pretty much stopped altogether, doing only 1-2, typically soft courses per year, and barely completing those. What did I do with my time instead? I memorized the backs of baseball cards, I took apart old computers and pieced them together to try and build something strong enough to emulate Super Smash Bros on the PC, and voraciously read vintage young adult fiction like Tom Swift or G.A. Henty novels.
I didn't realize it at the time but I abhorred constraints. As a kid, it seemed like there were always constraints. They of course, did not announce themselves to me as constraints. Instead they manifested in the form of something like the K9 Web Blocker blocking me from downloading music, my best computer only had an AGP slot instead of PCI-E, or my parents' credit card and hard "no shooting games" rule blocking me from buying Battlefield. These were constraints to be solved, and I became adept at solving them. Always having just a little less than I needed, always having to think just a little bit outside the box, that conditioned my brain in a way that would later benefit me.
Neither of my parents had really gone to college, unless you count the one semester my mother did for journalism, but they knew that I needed to go to college because that's the way the world was. Because all of their friends who had were planning their retirements, and well, my parents were not. So I got pushed into community college with no idea of what I wanted to do. First I took a computer programming intro class, and a biology class. The programming class was a breeze - nothing in that was new, but the biology class was actually all new. And while it was just a basic intro class with no lab, all just memorization, it first exposed itself to me as a menagerie of constraints. An incredibly complex system, built from the simplest parts, and built to accommodate all of the various constraints.
That was when I decided I wanted to be a scientist. But there was a problem in the way - I needed to know math. And the last math class I passed unambiguously without cheating was basic algebra. So I took pre-calculus at the community college. About 6-8 weeks into the semester, after failing to pass a single test, I stopped showing up. I'll never forget the feeling of hopelessness I had sitting in those tests not even knowing where to start to solve a problem. It wasn't that they were too difficult for me - maybe they were, but that's not even what was being exposed. What was being exposed was a complete and total lack of intuition about how to think about problems in mathematical terms.
The professor, a kindly older lady who I never once asked for help (d'oh), arguably saved all semblance of the life I now know by grading me a WP instead of an F, even though I stopped showing up and never said a word to her about it. That WP let me get into an actual flagship state university, where I, with renewed vigor, enrolled in pathobiology (despite not really understanding what it was, I just saw "forensics" and thought that sounded cool). I loaded my first semester with entirely gen-eds to get them out of the way. This included pre-calculus again.
This time, I really did try hard. I even went to office hours. But I was being taught by a graduate student who had no incentive to actually identify what it was I was struggling with (not that it was in any way his fault), and combined with being in ROTC which had 5 AM wake-ups on the same days as 8 PM math exams, it just didn't work. That same sinking feeling came back. I'd do the homework somewhat passably; I'd follow along in lectures somewhat passably, but then the exam questions would come and they would be posed in some sort of "real world" problem that I needed to first translate to something I could mathematically reduce. I simply did not have the intuition to do that, even if I did understand the formulas. In my defense, this time I didn't quit, I went to every single class and exam until the final (which I foolishly skipped due to thinking it was mathematically impossible to pass. It was, but I didn't understand the magic of grade inflation yet and was right on the cusp.) There was no WP or even W. Just an F. And it fit right in with the rest of my grades that semester.
I muddled my way through college, and through grad school, but I always had this hanging over my head. I'd learn math with great difficulty, and my eyes would glaze over at formulas, and descriptions of formulas (let x be blah - why can't we just use normal language?). Eventually someone in my life would think it was valuable enough to drag me through the math to the point that I could at least understand it enough to apply in specific contexts, and that was how I'd get by.
The side effect of this, however, was that no matter how far I got in my scientific field or career, I still would feel like an imposter, not a scientist. Because without exception, every problem I'd ever work on would eventually hit a wall of math. Every idea I'd ever have would eventually be confronted by the necessity to express things mathematically. At best I could apply - I could never truly create. I felt akin to Jacopo Belbo in Eco's Foucault's Pendulum. A master of his field in almost every aspect, but fundamentally incapable of creation. I would never advance the world forward, let alone my scientific discipline.
In light of recent news, however, I am beginning to wonder if I was actually spared a huge chunk of my time. Mathematic "discovery" (perhaps output is a better word? Many, perhaps all, of these are not technically discoveries in the traditional sense, but rather surfacing of pre-existing but unsynthesized knowledge) by LLMs is increasing at an unbelievable pace. We went from it simply never having occurred, to now happening more quickly than I can keep up with:
It's not an absurd prediction to make that any remaining unsolved mathematical problems in a few years will simply be computationally irreducible, at least from inside of this universe. Think Archimedes with his "Give me a place to stand and I shall move the earth." (rough paraphrase).
Look - I'm not going to sit here and say that learning math was a waste of time for anyone that actually learned it. But it's very very rare for me to do anything math-adjacent without a calculator. In the last ~12 months, I have gone to LLMs with a number of math-adjacent problems in my field, and have seen LLMs verifiably solve them with my prompting. It took no actual knowledge of math, syntax, or formula. Instead all it took is the intuition that I had apparently developed since those first attempts in undergrad, to decompose a problem into something vaguely resembling mathematics. "I want to do this" or "I want to solve for this" and "we need to consider these aspects" is apparently enough to use a modern calculator. And of course, these are non-deterministic, but then again, I'm also non-deterministic. And yes, I have no easy way to evaluate the output, but if I ask my calculator the square root of 231, I also have no easy way to evaluate the output. I can sort of guess (1010 is 100, 2020 is 400, 15*15 is somewhere in the middle, so if the answer is 15-ish it's probably right), but I can also sort of guess at LLM outputs. And if you have a verifiable goal that you can check whether it's correct or not simply by whether it works, then there's a limit to how much you need to know.
I don't know whether this is a good thing or not. But the same way the calculator let me live in a totally applied space where I could find the diagonal of a square without needing to know how to solve a square root, LLMs let me apply, for instance, new refinement algorithms in cryo-EM without needing to have intimate knowledge of every term in the CTF equation.
I still don't feel like a scientist. I don't know exactly what this weird transitional state I occupy is. Power user? Hacker? Tinkerer? But for the first time in my life, I feel almost unbridled in my ability to create. I can take ideas, and test them, whereas before I was excluded. I cannot tell if we are entering the age of the generalist, or instead if we are entering the age of the specialist? I cannot tell if we are coming upon the age of the creative, or instead, the age of the critic? It used to be that ideas and technical ability were a necessary coupling to advance. And while that will certainly remain in some disciplines and cases, at least right now, ideas are king. The gate is unguarded.