Yep! When I was in high school I was self-learning physics and was learning calculus via Khan academy, which made me learn that “sometimes”, “somehow”, you can for example solve a differential equation by getting all the dx and x’s on one side and dy’s and y’s on the other. I was always expecting when I would study this stuff more rigorously in university, this would be explained at some point. To my great disappointment, the mandatory class on this subject for my CS degree “real analysis”, did not in fact clear up why this works in the slightest!? I don’t know if every real analysis class is this bad (My linear algebra courses were excellent in comparison), but mine was mostly focusing on making students adopt “rigor”[1], presenting definition after definition and theorem after theorem (for short theorems proofs were included, but with zero sub-text what the motivation or idea of the proof was. Complicated proofs were just skipped). Starting studying during Corona and not being able to ask professors questions like that directly certainly didn’t help. Somehow I forgot, though, that I never got these questions answered that I had hoped to get an answer to. Also, the quotient in df/dx still confuses me, and it might be due to my confusion about division? Looking at the wikipedia article you link, maybe I should take a look at differential forms?
As my secondary subject for my bachelor, I chose physics, but that mostly involved applying Euler-Lagrange to particular systems. I remember becoming really confused why this can work at this point and noticing that I can’t fit the explanation in my head (I am in fact still confused and would love if you can point me to useful resources!). I remember thinking why can you differentiate with respect to speed and treat acceleration etc. as not dependent on speed, weird. Clearly, when framing them all as a function of time, it made no sense (and still doesn’t). I was looking it up on Wikipedia and apparently “Differential Geometry” was the answer to my questions, so I got 2–3 textbooks on Differential Geometry (for physics), but just skimming, I couldn’t find something that made this click.
Probably I should have looked more aggressively for people to tutor me. Now with language models it feels worth taking another stab at this though. For example when I told GPT4-o the rough areas I am confused about (link to full chat): there are these people talking about scaling laws in the context of “scaling laws” in engineering, and it seems like I might be missing some prerequisites, because I’ve bounced off this topic a few times even though this seems extremely interesting. Which led it to introduce dimensionless analysis to me, and then I asked more questions and ultimately gave me some textbook recommendations: “Street fighting mathematics”, I had already heard praise for in “Biology by the numbers”, so then I checked it out and discovered this gold.
Another question I am still confused by: how does your choice of units affect what types of dimensionless quantities you discover? Why do we have Ampere as a fundamental unit instead of just M,L,T? What do I lose? What do I lose if I reduce the number of dimensions even further? Are there other units that would be worth adding under some circumstances? What makes this non-arbitrary? Why is Temperature a different unit from Energy?
Also, I notice all the formalizations of dimensional analysis that Terence Tao mentions here strike me as ugly? Is that me being stupid, thinking that there must exist something nicer?
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Which I found mostly annoying and trivial, but made me glade I had spent ~60h before university just practicing proving things. How to proove it is a great book!
I don’t know if someone else was making this particular mistake. I certainly find it quite tricky to describe these things with language unless I am extremely careful. I am still confused, but I do find your suggested pathway via transposons more plausible, and the story for why transposons fits into my head. I don’t think a correct story for why not epigenetic “bit-flips” currently fits properly into my head, and that correct story would have to be created in order to convince other people. Ideally, some testable predictions[1].
I don’t understand why this has to be such a dichotomy? What I hear you say is that these dynamic equilibria are too stable to be a root cause of aging. I think this is true in the case of X-inactivation[2], which is redundant across most of the X-chromosome. But as far as I can tell, the number of redundant parts that make up one “unit” that can bit-flip lies on a spectrum. Example: one methylated C is probably too unstable to encode something. But what if I have a local cluster of five of them where each one reinforces each other (because enzymes that change the methylation often check the methylation status of close-by C’s)? That’s the point of CpG islands, right? So some of these CpG islands are probably more stable than others, and some could be just stable/unstable enough to be a root cause of aging, no?
For example, my guess would be that the transposon story and the accumulating epigenetic chaos story make different predictions about which methylations would serve as an aging clock and how well. They would also predict different things if we compare aging in different species, because transposons and mitochondria are (quite?) universal in eukaryotes, while epigenetics between eukaryotes is quite different.
My understanding is X-inactivation almost never bit-flips in adults. If it were unstable on the timescale of years, you’d sometimes see one white hair in a spot where a cat has black hair, for example. Maybe it is more common in cells with faster turnover?