## Tag Archives: Topological manifolds

### Why math is hard (for me) and organic chemistry is easy

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.

In particular, to understand gravity, a la relativity, you have to know how mass bends 4 dimensional space-time. This means you must understand curvature on manifolds, which means you must understand smooth manifolds, which means that you must understand topological manifolds which is why I’m reading Lee’s book.
So perhaps when the smoke clears, I might have something intelligent to say to my classmate.