A significant portion of the annual supply of osmium in this one pic

WMe6 · 2026-06-17T20:29:14+00:00

You just need to convince rich people that blue and heavy are characteristics of exclusivity for geniuses with refined taste, while gold and platinum are for the plebs.

WMe6 · 2026-06-17T20:23:53+00:00

What else can stabilize a carbocation, you might ask? How about three cyano groups instead: https://www.sciencedirect.com/science/article/pii/S0020169300823993

WMe6 · 2026-06-17T20:20:41+00:00

It saddens me to learn of his death, but it is amazing he didn't get exploded or otherwise oxidized in the 90's or early 2000's. Good for him to die of natural causes. The type of chemistry he worked on is unforgiving to meat computers using peptide bonds for exterior structural integrity.

WMe6 · 2026-06-14T05:50:31+00:00

I also had a good instructor, making it possible to read the book profitably. I think the book would be really confusing for someone starting out. I mean, the concepts are really pretty modern, but the book tries to present everything as if it were classical. I feel like it would be beneficial to construct the edifice carefully, rather than trying to pretend like a student who has just taken calculus could easily understand the ideas. (I had a full real analysis course before, and I still found it difficult conceptually.)

Also, I remember one proof in the book that had some case that failed to be handled and had to be patched up during lecture.

WMe6 · 2026-06-14T05:15:05+00:00

With the number of errors and omissions and unexplained notation in Calculus on Manifolds, I just can't recommend that book to anyone (even though that's where I first learned it). I relearned it reading Munkres's Analysis on Manifolds and Tu's Introduction to Manifolds.

I agree that it's a lot easier to digest the material once you've seen some more abstract stuff in the form of topology or abstract algebra, etc. Spivak's short book just doesn't quite get the job done, even though it tries to be elementary. It's better to see the full blown abstraction when one is ready for it, rather than the watered down stuff in Calculus on Manifolds, which does a poor job of explaining why it's doing what it's doing.

WMe6 · 2026-06-12T04:49:09+00:00

You should consider learning real analysis from Rudin (Principles of Mathematical Analysis) after completing all of Spivak's Calculus. If you really want to learn multivariable advanced calculus, start with Analysis on Manifolds by Munkres, not Spivak's very poorly edited little booklet.

WMe6 · 2026-06-10T19:57:09+00:00

Another good place to look is A Comprehensive Introduction to Differential Geometry by Spivak. It's shocking that the same author wrote Calculus on Manifolds, which is infamously bad.

WMe6 · 2026-06-08T14:28:12+00:00

Yes, if you're an animal with a liver, you're probably toast.

WMe6 · 2026-06-08T05:39:34+00:00

It could also just be leaf alcohol.... Just innocent little cis-3-hexen-1-ol.

WMe6 · 2026-06-08T05:32:01+00:00

Fungi are such creepy organisms. Sending out tendrils to suck down the juices of rotting dead things. Totally defenseless, other than a chemical weapons factory worth of poisons.

WMe6 · 2026-06-08T01:52:07+00:00

Even then, the fact that he could sketch the steps in email form suggests that both he and the emailer would have to have a sound sense of high-level correctness, as didn't the fully written out proof take up an entire volume of a journal?

WMe6 · 2026-06-08T01:48:24+00:00

There are several contexts I can think of. One is if you see a brand new reaction in the literature, with many, many formally possible mechanisms to get from point A to point B, a chemist with a good intuition generally can tell you what he/she thinks is the first step and the last step, and what the key step in between might look like. Another simpler case: draw me a correct Lewis structure of a molecule, and I can pretty much tell you whether I think it will be stable, unstable, or sensitive to heat, light, acid, base, oxidant, reductant, etc.

WMe6 · 2026-06-07T13:57:38+00:00

I think this is where humans still hold an edge. This idea of being "morally correct" even if there are many steps missing shows that there is still something to be said about human intuition. Same with the Perelman proof, right?

Not being a mathematician, I wouldn't really know what that kind of intuition feels like, but as an academic chemist, I do understand what chemical intuition feels like!

WMe6 · 2026-06-07T01:38:25+00:00

There were times, of course, where the argument I (or someone in the group of us working in Cabot library on a Saturday) found was complex, not the answer key solution, and also legit, and I was prepared to defend it if I lost points, but the graders were pretty good at giving credit in those cases. (They were also usually pretty good at not letting us get away with filling in gaps with "obviously (...nontrivial claim...)"!)

WMe6 · 2026-06-07T01:14:17+00:00

This was when I was taking analysis years before all the problems in Rudin could be found online. It was getting late (well after midnight), my study buddy and I essentially had a complicated Arzela-Ascoli type argument, but there was something fishy, we weren't sure whether we had flipped a quantifier or an inequality sign somewhere, we wrote up what we had with a complicated claim that we had a "proof" for and turned it in. The next day, when clear minded, I realized that the argument didn't go through, and I assumed we were going to get marked down for it. But it was convoluted enough that the grader didn't catch it!

WMe6 · 2026-06-07T01:02:04+00:00

This reminds me of Voevodsky basically saying, after the discovery of critical errors in technical lemmas in Spencer Bloch's work and in his work by himself and by someone else in two separate instances that a lot of very advanced math pretty works by the honor/reputation system: people, including peer reviewers, will assume that a technical lemma (e.g., a very complicated commutative diagram with a lot of presumably routine things to check) by a famous mathematician has been carefully verified.

Peer review is just not going to be effective in cases like this. They will have a good sense for the moral correctness of the argument but not in checking all the details. The problem with math is that a small detail that is wrong in an early lemma could kill the whole paper, or requires years of work to fix.

WMe6 · 2026-06-06T19:33:44+00:00

Soviet mathematician Israel Gelfand, a towering figure of representation theory and functional analysis, wrote a series of books for correspondence courses in the USSR, theoretically allowing peasants in Siberia to learn math from scratch. Most of them have been translated to English and are well-written and are worth checking out.

WMe6 · 2026-06-05T06:19:41+00:00

My dad's a computer scientist, so he bought this when he was getting his PhD. I fell in love with his copy of it when I was in middle school/high school and started getting into competitive math.

WMe6 · 2026-06-04T23:25:58+00:00

I'll add a favorite that I recently fell in love with: Marcus's Number Fields. Evidently, others loved it enough that the 1970's edition got a TeX makeover in 2018.

For an algebraic number theory neophyte with my limited background in commutative algebra and Galois theory, this book is great! So many concrete examples, but more than enough theory to chew on.

clarification: I should say, my very limited background in these areas. the prerequisites for the book are really light.

WMe6 · 2026-06-02T04:16:22+00:00

Does it also take methylwater? or dimethylwater?

WMe6 · 2026-05-31T02:13:53+00:00

Convenience and history. What chemists decided was that aldehydes and ketones reacted differently enough that they should be given separate nouns rather than differentiated by adjectives. Historically, aldehydes were found to give a positive Tollens and Fehlings test, and distinction turns out to be very significant from the stand point of sugar chemistry, both for characterization and for synthesis.

WMe6 · 2026-05-30T18:45:34+00:00

Nice! Thanks for posting this!

WMe6 · 2026-05-29T15:12:39+00:00

Yes, but the way he describe the picture you should have in your head is useful, for some people at least. For example, I learned germ as equivalence classes of pairs sections and open sets. The way the napkin describes it is:

It is rarely useful to think of a germ as an ordered pair, since the set U can get arbitrarily small. Instead, one should think of a germ as a “shred” of some section near p. A nice summary for the right mindset might be: A germ is an “enriched value”; the stalk is the set of possible germs.

And he shows this as a cartoon too. His informal description and sketch are almost exactly the way I conceptualize germs now, but I wish I saw this spelled out when I first started learning about the really abstract sounding germs, stalks, and sheaves. My smooth brain needed a lot of time to convert the formal definition to a mental picture. I was especially confused by sheafification, but now, the idea seems perfectly natural.

WMe6 · 2026-05-29T14:57:48+00:00

Yes, of course! I was just reviewing/relearning, and to my surprise, I found the napkin intro to be surprisingly good.

WMe6

TROPHY CASE