I’ve been reading a lot of hard core math lately (I’ll explain why at the end), along with Clayden et al’s new edition of their fabulous Organic Chemistry text. The level of sophistication takes a quantum jump about 2/3 of the way through (around pp. 796) and is probably near to the graduate level. The exercise is great fun, but math and orgo require quite distinct ways of thinking. Intermixing both on a daily basis brought home just how very different they are.

First off, the concepts in organic chemistry are fuzzy. On p. 796 the graph of the Karplus relationship between J splitting in NMR and the dihedral angle of the hydrogens being split is shown. It’s a continuous curve as the splitting is maximal at 180, zero at 90 and somewhat less than maximal at 0 degrees.

There is nothing like this in math. Terms are defined exactly and the logic is that of true, false and the excluded middle (e.g. things are either true or false). Remember the way that the square root of 2 was proved not to be the ratio of two whole numbers. It was assumed that it *could* be done, and than it was shown no matter how you sliced it, a contradiction was reached. The contradiction then implied that the opposite was true — if the negative of a proposition leads to a contradiction (it’s false) than the proposition must be true. Math is full of proofs like this.Or if you are trying to prove A implies B, proving the contrapositive ( not B implies not A) will do just as well. You never see stuff like this in orgo.

*There just aren’t that many concepts in organic chemistry*, even though the details of each and every reaction are beyond the strongest memory. The crucial points are to have the orbitals of the various atoms firmly in mind and where they are in space. This tells you how molecules will or won’t react, or how certain conformations will be stable (see anomeric effect). Entropy in physics is a very subtle concept, but pretty obvious as used by organic chemists. Two molecules are better than one etc. etc. Also you see these concepts over and over. Everything you study (just about) has carbon in it. Chair and boat, cis and trans, exo and endo become part of you, without thinking much about them.

Contrast this with math. I’m currently reading “Introduction to Topological Manifolds” (2nd. Edition) by John M. Lee. I’ve got about 34 pages of notes on the first 95 pages (25% of the text), and made a list of the definitions I thought worth writing down — there are 170 of them. Each is quite precise. A topological embedding is (1) a continuous function (2) a surjective function (3) a homeomorphism. No more no less. Remove any one of the 3 (examples are given) and you no longer have an embedding. The definitions are abstract for the most part, and far from intuitive. That’s because the correct definitions were far from obvious even to the mathematicians formulating them. Hubbard’s excellent book on Vector Calculus says that it took some 200 years for the correct definition of continuity to be thrashed out. People were arguing about what a function actually was 100 years ago.

As you read you are expected to remember exactly (or look up) the 170 or so defined concepts and use them in a proof. So when you read a bit of Lee’s book, I’m always stopping and asking myself ‘did I really understand what I just read’? Clayden isn’t at all like that — Oh that’s just an intramolecular Sn2, helped because of the Thorpe Ingold effect, which is so obvious it shouldn’t be given a name.

Contrast this with:

After defining topological space, open set, closed set, compact, Hausdorff, continuous, closed map, you are asked to show that a continuous map from a compact topological space to a Hausdorff topological space is a closed map, and that such a map, if surjective as well is an embedding. To get even close to the proof you must be able to hold all this in your head at once. You should also remember that you proved that in a Hausdorff space compact sets are closed.

No matter how complicated the organic problem, you can always look at the molecule, and use the fabulous spatial processing capacity your brain gives you. The interpretation of NMR spectra in terms of conformation certainly requires logical thinking — it’s sort of organic Sudoku.

I imagine a mathematician would have problems with the fuzzy concepts of organic chemistry. Anslyn and Dougherty take great pains to show you why some reactions fall between Sn1 and Sn2, or E1cb.

So why am I doing this? Of course there’s the why climb Everest explanation — because it’s there, big and hard, and maybe you can’t do it. That’s part of it, but not all. For reasons unknown, I’ve always like math, even though not terribly good at it. Then there’s the surge for the ego should I be able to go through it all proving that I don’t have Alzheimer’s at 74.5 (at least not yet). Then there is the solace (yes solace) that math provides. Topology is far from new to me in 2011. I started reading Hocking and Young back in ’70 when I was a neurology resident, seeing terrible disease, being unable to help most of those I saw, and ruminating about the unfairness of it all. Thinking about math took me miles away (and still does), at least temporarily. When I get that far away look, my wife asks me if I’m thinking about math again. She’s particularly adamant about not doing this when I’m driving with her (or by myself).

The final reason, is that I went to college with a genius. I met him at our 50th reunion after reading his bio in our 50th reunion book. I knew several self-proclaimed geniuses back then, and a lot a physics majors, but he wasn’t one of them. At any rate, he’s still pumping out papers on relativity with Stephen Hawking, and his entries in the index of the recent biography by Kitty Ferguson take up almost as many entries as Hawking himself. He’s a very nice guy and agreed to answer questions from time to time. But to understand the physics you need to really understand the math, and not just mouth it.

## Comments

I understand what you are saying. I once spent a considerable amount of time studying George Simmons’s excellent book “Introduction to Topology and Modern Analysis” and remember how understanding the Heine-Borel theorem required an understanding of the entire mathematical machinery developed until that point. In a book on topology it’s impossible to skip chapters or theorems. I also remember the experience as intellectually very rewarding and hope to do it again.

Not sure that you need to understand topology to understand relativity. Einstein could do without it and he developed relativity using only good old fashioned 19th century math. Topological methods were first introduced into general relativity by Roger Penrose in the 60s.

I think it’s a matter of taste. Newton developed his laws of motion without calculus (which he developed at the same time). Would you be satisfied that your understanding of them pre-calculus was as good as it was afterwards?

Smooth manifolds have a lot more applications than just relativity. Essentially smoothness is another structure placed on a topological manifold, allowing you to do calculus on it. Curvature, like the derivative is essentially a calculus concept.

Since I’m retired and have the time (if not the brain), I’m going to give it a shot.

Not sure you’re correct about topology, Penrose and the 60’s. If I get through it all, I’ll let you know.

Thanks, as always, for commenting.

It’s really interesting that you mention Newton. The great astrophysicist S. Chandrasekhar wrote a fantastic book called “Newton’s Principia for the Common Reader” toward the end of his life. In he rederived all of Newton’s results using modern methods…and concluded that in most cases Newton was more elegant and simpler. That’s not an argument against modern methods, just an acknowledgement that the masters were often really good.

Regarding Penrose, my knowledge of topology in relativity is as preliminary as yours; I got this fact from Kip Thorne’s “Black Holes and White Dwarfs” which may be the best popular account of relativity and its application to astrophysics. Highly recommended. There’s also an interesting comparison of the British and American education systems (Thorne claims that at least in the 60s British students were more mathematically competent) in the context of Penrose’s contributions.

For what it’s worth, the construction of those mathematical definitions is the result of, in part, trial and error. Mathematicians knew they had some concept which needed expression, and tried different ways to try to capture just what was wanted, and to not include what was not wanted. The developing edge of mathematics includes a lot of experimenting with, “what do we

wanta Closed Glurbil Strand to be?”You’d probably never finish a topology book in any reasonable time fiddling with every definition, but it can be illuminating to try it for a few topics. Imagine trying a different definition for a topological embedding, and see if that makes some proof impossible to work out, or makes it possible to prove that something clearly

notan embedding is, or vice-versa.(This sort of vary-the-definition is easiest to do with the definitions of limits and continuity, or at least I’m most familiar with the ways to send an alternate definition into something disastrously wrong. But my recollection is that the definitions of limits and continuity is the first point where analysis gets really impenetrable, and experimenting got me through that point.)

I find myself explaining to students a lot that “organic chemistry is analog, not digital”. Each variable is like a knob that you turn and tune to a different value; and all the variables interact with each other.

I’ve also been finding inspiration and joy in going back through my college level astrophysics textbook – nothing like the hard-core math you describe, but stimulating nonetheless. So many wonderful things to explore in this life.