Category Archives: Math

Two math tips

Two of the most important theorems in differential geometry are Gauss’s Theorem egregium and the Inverse function theorem. Basically the theorem egregium says that you don’t need to look at the shape of a two dimensional surface (say the surface of a walnut) from outside (e.g. from the way it sits in 3 dimensional space) to understand its shape. All the information is contained in the surface itself.

The inverse function theorem (InFT) is used over and over. If you have a continuous function from Euclidean space U of finite dimension n to Euclidean space V of the same dimension, and certain properties of its derivative are present at a point x of U, then there exists a another function to get you back from space V to U.

Even better, once you’ve proved the inverse function theorem, proof of another important theorem (the implicit function theorem aka the ImFT) is quite simple. The ImFT lets you know if given f(x, y, .. .) –> R (e.g. a real valued function) if you can express one variable (say x) in terms of the others. Again sometimes it’s difficult to solve such an equation for x in terms of y — consider arctan(e^(x + y^2) * sin(xy) + ln x). What is important to know in this case, is whether it’s even possible.

The proofs of both are tricky. In particular, the proof of the inverse function theorem is an existence proof. You may not be able to write down the function from V to U even though you’ve just proved that it exists. So using the InFT to prove the implicit function theory is also nonconstructive.

At some point in your mathematical adolescence, you should sit down and follow these proofs. They aren’t easy and they aren’t short.

Here’s where to go. Both can be found in books by James J. Callahan, emeritus professor of Mathematics at Smith College in Northampton Mass. The proof of the InVT is to be found on pages 169 – 174 of his “Advanced Calculus, A Geometric View”, which is geometric, with lots of pictures. What’s good about this proof is that it’s broken down into some 13 steps. Be prepared to meet a lot of functions and variables.

Just the statement of InVT involves functions f, f^-1, df, df^-1, spaces U^n, R^n, variables a, q, B

The proof of InVT involves functions g, phi, dphi, h, dh, N, most of which are vector valued (N is real valued)

Then there are the geometric objects U^n, R^n, Wa, Wfa, Br, Br/2

Vectors a, x, u, delta x, delta u, delta v, delta w

Real number r

That’s just to get you through step 8 of the 13 step proof, which proves the existence of the inverse function (aka f^-1). The rest involves proving properties of f^-1 such as continuity and differentiability. I must confess that just proving existence of f^-1 was enough for me.

The proof of the implicit function theorem for two variables — e.g. f(x, y) = k takes less than a page (190).

The proof of the Theorem Egregium is to be found in his book “The Geometry of Spacetime” pp. 258 – 262 in 9 steps. Be prepared for fewer functions, but many more symbols.

As to why I’m doing this please see

Help wanted

Just about done with special relativity. It is simply marvelous to see how everything follows from the constancy of the speed of light — time moving more slowly for a moving object (relative to an object standing still in its own frame of reference), a moving object shrinking (ditto), the increase in mass which occurs as an object begins to approach the speed of light, and how this leads to the equivalence of mass and energy. Special relativity is even sufficient to show how a gravitational field will bend light — although to really understand this, general relativity is required.

The one fly in the intellectual ointment is the Minkowski metric for the space time of special relativity. In all the sources I’ve been able to find, it appears ad hoc, or is defined analogously to the euclidean metric. I’d love to see an argument why this metric (time coordinates positive, space coordinates negative) must follow from the constancy of the speed of light. It is clear that the Minkowski metric is preserved under the hyperbolic transformation of space-time, but likely others are as well. Why this particular metric and not something else.

Consider the determinant function of an n by n matrix. It has a god awful mathematical form involving the sum of n ! terms. Yet all you need to get the (unique) formula are a few postulates — the determinant of the identity matrix is 1, the determinant is a linear function of its rows (or its columns), interchanging any two rows of the determinant reverses the sign of the determinant, etc. etc. This basically determines the (unique) formula of the determinant. I’d really like to see the Minkowski metric come out of something like that.

Can anyone out there shed light on this or give me a link?

A Mathematical Near Death Experience

As I’ve alluded to from time to time, I’m trying to learn relativity — not the popularizations, of which there are many, but the full Monty as it were, with all the math required. I’ve been at it a while as the following New Year’s Resolution of a few years ago will show.

“Why relativity? It’s something I’ve always wanted to understand at a deeper level than the popularizations of it (reading the sacred texts in the original so to speak). I may have enough background in math, to understand how to study it. Topology is something I started looking at years ago as a chief neurology resident, to get my mind off the ghastly cases I was seeing.

I’d forgotten about it, but a fellow ancient alum, mentioned our college president’s speech to us on opening day some 55 years ago. All the high school guys were nervously looking at our neighbors and wondering if we really belonged there. The prez told us that if they accepted us that they were sure we could do the work, and that although there were a few geniuses in the entering class, there were many more people in the class who thought they were.

Which brings me to our class relativist. I knew a lot of the physics majors as an undergrad, but not this guy. The index of the new book on Hawking by Ferguson has multiple entries about his work with Hawking (which is ongoing). Another physicist (now a semi-famous historian) felt validated when the guy asked him for help with a problem. He never tooted his own horn, and seemed quite modest at the 50th reunion. As far as I know, one physics self-proclaimed genius (and class valedictorian) has done little work of any significance. Maybe at the end of the year I’ll be able to read the relativist’s textbook on the subject. Who knows? It’s certainly a personal reason for studying relativity. Maybe at the end of the year I’ll be able to ask him a sensible question.”

Well that year has come and gone, but I’m making progress, going through a book with a mathematical approach to the subject written by a local retired math prof (who shall remain nameless). The only way to learn any math or physics is to do the problems, and he was kind enough to send me the answer sheet to all the problems in his book (which he worked out himself).

I am able to do most of the problems, and usually get the right answer, but his answers are far more elegant than mine. It is fascinating to see the way a professional mathematician thinks about these things.

The process of trying to learn something which everyone says is hard, is actually quite existential for someone now 76. Do I have the mental horsepower to get the stuff? Did I ever? etc. etc.

So when I got to one problem and the profs answer I was really quite upset. My answer appeared fairly straightforward and simple, yet his answer required a long derivation. Even though we both came out with the same thing, I was certain that I’d missed something really basic which required all the work he put in.

One of the joys of reading math these days (at least math books written by someone who is still alive) is that you can correspond with them. Mathematicians are so used to being dumped on by presumably intellectual people, that they’re happy to see some love. Response time is usually under a day. So I wrote him the following

“Along those lines, you do a lot of heavy lifting in your answer to 3a in section 4.3. Why not just say the point you are trying to find in R’s world is the image under M of the point (h.h) in G’s world and apply M to get t and z.”

Now usually any mathematician I EMail about their books gets back quickly — my sardonic wife says that it’s because they don’t have much to do.

Fo days, I heard nothing. I figured that he was trying to figure out a nice way to tell me to take up watching sports or golf, and that relativity was a mountain my intellect couldn’t climb. True existential gloom set in. Then I go the following back.

“You are absolutely right about the question; what you propose is elegant and incisive. I can’t figure out why I didn’t make the simple direct connection in the text itself, because I went to some pains to structure everything around the map M. But all that was fifteen or more years ago, and I have no notes about my thinking as I was writing.”

A true mathematical (and existential) near death experience.

How to think about two tricky theorems and other matters

I’m continuing to plow through classic differential geometry en route to studying manifolds en route to understanding relativity. The following thoughts might help someone just starting out.

Derivatives of one variable functions are fairly easy to understand. Plot y = f(x) and measure the slope of the curve. That’s the derivative.

So why do you need a matrix to find the derivative for more than one variable? Imagine standing on the side of a mountain. The slope depends on the direction you look. So something giving you the slope(s) of a mountain just has to be more complicated. It must be something that operates on the direction you’re looking (e.g. a vector).

Another point to remember about derivatives is that they basically take something that looks bumpy (like a mountain), look very closely at it under a magnifying glass and flatten it out (e.g. linearize it). Anything linear comes under the rubric of linear algebra — about which I wrote a lot, because it underlies quantum mechanics — for details see the 9 articles I wrote about it in —

Any linear transformation of a vector (of which the direction of your gaze on the side of a mountain is but one) can be represented by a matrix of numbers, which is why to find a slope in the direction of a vector it must be multiplied by a matrix (the Jacobian if you want to get technical).

Now on to the two tricky theorems — the Inverse Function Theorem and the Implicit Function theorem. I’ve been plowing through a variety of books on differential geometry (Banchoff & Lovett, McInenery, DoCarmo, Kreyszig, Thorpe) and they all refer you for proofs of both to an analysis book. They are absolutely crucial to differential geometry, so it’s surprising that none of these books prove them. They all involve linear transformations (because derivatives are linear) from an arbitrary real vector space R^n — elements are ordered n-tuples of real numbers to to another real vector space R^m. So they must inherently involve matrices, which quickly gets rather technical.

To keep your eye on the ball let’s go back to y = f(x). Y and x are real numbers. They have the lovely property that between any two real numbers there lies another, and between those two another and another. So there is no smallest real number greater than 0. If there is a point x at which the derivative isn’t zero but some positive number a to keep it simple (but a negative number would work as well), then y is increasing at x. If the derivative is continuous at a (which it usually is) then there is a delta greater than zero such that the derivative is greater than zero in the open interval (x – delta, x + delta). This means that y = f(x) is always increasing over that interval. This means that there is a one to one function y = g(x) defined over the same interval. This is called an inverse function.

Now you’re ready for the inverse function theorem — but the conditions are the same — the derivative at a point should be greater than zero and continuous at that point — and an inverse function exists. The trickiness and the mountains of notation come from the fact that the function is from R^n to R^m where n and m are any positive integers.

It’s important to know that, although the inverse and implicit functions are shown logically to exist, almost never can they be written down explicitly. The implicit function theorem follows from the inverse function theorem with even more notation involved, but this is the basic idea behind them.

A few other points on differential geometry. Much of it involves surfaces, and they are defined 3 ways. The easiest way to understand two of them takes you back to the side of a mountain. Now you’re standing on it half way up and wondering which would be the best way to get to the top. So you whip out your topographic map which has lines of constant elevation on it. This brings to the first way to define a surface. Assume the mountain is given by the function z = f (x, y) — every point on the earth has a height above it where the land stops and the sky beings (z) — so the function is a parameterization of the surface. Another way to define a surface in space is by level sets: put z equal to some height — call it z’ and define the surface as the set of two dimensional points (x, y) such that f (x, y ) = z’. These are the lines of constant elevation (e.g. the contour lines) – on the mountain. Differential geometry takes a broad view of surfaces — yes a curve on f (x, y) is considered a surface, just as a surface of constant temperature around the sun is a level set on f(x,y,z). The third way to define a surface is by f (x1, x2, …, xn) = 0. This is where the implicit function theorem comes in if some variables are in fact functions of others.

Well, I hope this helps when you plunge into the actual details.

For the record — the best derivation of these theorems are in Apostol Mathematical Analysis 1957 third printing pp. 138 – 148. The development is leisurely and quite clear. I bought the book in 1960 for $10.50. The second edition came out in ’74 — you can now buy it for 76.00 from Amazon — proving you should never throw out your old math books.

An old year’s resolution

One of the things I thought I was going to do in 2012 was learn about relativity.   For why see  Also my cousin’s new husband wrote a paper on a new way of looking at it.  I’ve been putting him off as I thought I should know the old way first.

I knew that general relativity involved lots of math such as manifolds and the curvature of space-time.  So rather than read verbal explanations, I thought I’d learn the math first.  I started reading John M. Lee’s two books on manifolds.  The first involves topological manifolds, the second involves manifolds with extra structure (smoothness) permitting calculus to be done on them.  Distance is not a topological concept, but is absolutely required for calculus — that’s what the smoothness is about.

I started with “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. Eventually I got through a third of its 380 pages of text.  I thought that might be enough to help me read his second book “Introduction to Smooth Manifolds” but I only got through 100 of its 600 pages before I could see that I really needed to go back and completely go through the first book.

This seemed endless, and would probably take 2 more years.  This shouldn’t be taken as a criticism of Lee — his writing is clear as a bell.  One of the few criticisms of his books is that they are so clear, you think you understand what you are reading when you don’t.

So what to do?  A prof at one of the local colleges, James J. Callahan, wrote a book called “The Geometry of Spacetime” which concerns special and general relativity.  I asked if I could audit the course on it he’d been teaching there for decades.  Unfortunately he said “been there, done that” and had no plans ever to teach the course again.

Well, for the last month or so, I’ve been going through his book.  It’s excellent, with lots of diagrams and pictures, and wide margins for taking notes.  A symbol table would have been helpful, as would answers to the excellent (and fairly difficult) problems.

This also explains why there have been no posts in the past month.

The good news is that the only math you need for special relativity is calculus and linear algebra.  Really nothing more.  No manifolds.  At the end of the first third of the book (about 145 pages) you will have a clear understanding of

l. time dilation — why time slows down for moving objects

2. length contraction — why moving objects shrink

3. why two observers looking at the same event can see it happening at different times.

4. the Michelson Morley experiment — but the explanation of it in the Feynman lectures on physics 15-3, 15-4 is much better

5. The Kludge Lorentz used to make Maxwell’s equations obey the Galilean principle of relativity (e.g. Newton’s first law)

6. How Einstein derived Lorentz’s kludge purely by assuming the velocity of light was constant for all observers, never mind how they were moving relative to each other.  Reading how he did it, is like watching a master sculptor at work.

Well, I’ll never get through the rest of Callahan by the end of 2012, but I can see doing it in a few more months.  You could conceivably learn linear algebra by reading his book, but it would be tough.  I’ve written some fairly simplistic background linear algebra for quantum mechanics posts — you might find them useful.

One of the nicest things was seeing clearly what it means for different matrices to represent the same transformation, and why you should care.  I’d seen this many times in linear algebra, but seeing how simple reflection through an arbitrary line through the origin can be when you (1) rotate the line to the x axis by tan(y/x) radians (2) change the y coordinate to – y  – by an incredibly simple matrix  (3) rotate it back to the original angle .

That’s why any two n x n matrices X and Y represent the same linear transformation if they are related by the invertible matrix Z in the following way  X = Z^-1 * Y * Z

Merry Christmas and Happy New Year (none of that Happy Holidays crap for me)

The New Clayden pp. 1029 – 1068

p. 1034 — “Small amounts of radicals are formed in many reactions in which the products are actually formed by simple ionic processes.”  Interesting — how ‘small’ is small?  

p. 1036 — A very improbable mechanism (but true) given in the last reaction involving breaking benzene aromaticity and forming a cyclopropene ring to boot.  

p. 1043 — Americans should note that gradient (as in Hammett’s rho constant) means slope (or derivative if the plot of substituents vs. sigma for a particular reaction isn’t a straight line).  However we are talking log vs. log plots, and you can fit an elephant onto a log log plot.  It’s worth remembering why logarithms are necessary iin the first place.  Much of interest to chemists (equilibrium constants, reaction rates) are exponential in free energy (of products vs. reactants in the first case, of transition state vs. reactions in the second).

p. 1044 — Optimally I shouldn’t have to remember that a positve rho (for reaction value) means electrons flow toward the aromatic ring in the rate determining step), but should gut it out from the electron withdrawing or pushing effects on the transition state, and how this affects sigma, by remembering what equilibrium constant is over what for sigma, and rho), but this implies a very high working memory capacity (which I don’t have unfortunately).  I think mathematicians do, which is why I’m so slow at it.  They have to keep all sorts of definitions in working memory at once to come up with proofs (and I do to follow them).  

If you don’t know what working memory is, here’s a link –  

Here are a few literature references 

        [ Proc. Natl. Acad. Sci. vol. 106 pp. 21017 - 21018 '09 ] This one is particularly interesting to me as it states that differences among people in working memory capacity are thought to reflect a core cognitive ability, because they strongly predict performance in fluid inteliigenece, reading, attentional control etc. etc.  This may explain why you have to have a certain sort of smarts to be a mathematician (the sort that helps you on IQ tests).  

       [ Science vol. 323 pp. 800 - 802 '09 ] Intensive training on working memory tasks can improve working memory capacity, and reduce cognitively related clinical symptoms.  The improvements have been associated with an increase in brain activity in parietal and frontal regions. 

I think there are some websites which will train working memory (and claim to improve it).  I may give them a shot. 

Unrelated to this chapter, but Science vol. 337 pp. 1648 – 1651 ’12, but worth bringing to the attention of the cognoscenti reading this –as there is some fascinating looking organometallic chemistry in it.  This is a totally new development since the early 60′s and I look forward to reading the next chapter on Organometallic chemistry.   Hopefully orbitals and stereochemistry will be involved there, as they are in this paper.  Fig 1 C has A uranium atom bound to 3 oxygens and 3 nitrogens, and also by dotted bonds to H and C.

p. 1050 — The unspoken assumption about the kinetic isotope effect is that the C-D and C-H bonds have the same strength (since the curve of potential energy vs. atomic separation is the same for both — this is probably true — but why?    Also, there is no explanation of why the maximum kinetic isotope effect is 7.1.  So I thought I’d look and see what the current Bible of physical organic chemistry had to say about it. 

Anslyn and Dougherty (p. 422 –> ) leave the calculation of the maximum isotope effect (at 298 Kelvin) as an exercise.  They also assume that the force constant is the same.  Exercise 1 (p. 482) says one equation used to calculate kinetic isotope effects is given below — you are asked to derive it 

kH/kD = exp [ hc (vbarH – vbarD)/2KT }, and then in problem #2 plug in a stretching frequency for C-H of 3000 cm^-1 to calculate the isotope effect at 298 Kelvin coming up with 6.5

Far from satisfying.  I doubt that the average organic chemist reading Anslyn and Dougherty could solve it.  Perhaps I could have  done it back in ’61 when I had the incredible experience of auditing E. B. Wilson’s course on Statistical Mechanics while waiting to go to med school (yes he’s the Wilson of Pauling and Wilson).   More about him when I start reading Molecular Driving Forces. 

On another level, it’s rather surprising that mass should have such an effect on reaction rates.  Bonds are about the distribution of charge, and the force between charged particles is 10^36 times stronger than that between particles of the same mass. 

p. 1052 — Entropy is a subtle concept (particularly in bulk thermodynamics), but easy enough to measure there.    Organic chemists have a very intuitive concept of it as shown here.

p. 1054 — Very slick explanation of the inverse isotope effect.  

Again out of context — but more chemistry seems to be appearing in Nature and Science these days.   A carbon coordinated to 6 iron atoms ( yes six ! ! ! ) exists in an enzyme essential for life itself — the plant enzyme nitrogenase which reduces N2 to usable ammonia equivalents for formation of amino acids, nucleotides.   No mention seems to be made about just how unusual this is.  See Science vol. 337 pp. 1672 – 1675 ’12. 

p. 1061 — The trapping of the benzyne intermediate by a Diels Alder is clever and exactly what I was trying to do years ago in a failed PhD project — see

p. 1064 — In the mechanism of attack on epichlorohydrin, the reason for the preference of attack on the epoxide isn’t given — it’s probably both steric and kinetic, steric because attack on the ring is less hindered — the H’s are splayed out, and kinetic, because anything opening up a strained ring should have a lower energy transition state. 

The New Clayden pp. 931 – 969

p. 935 — I don’t understand why neighboring group participation is less common using 4 membered rings than it is using  3 and 5 membered rings.  It may be entropy and the enthalpy of strain balancing out.  I think they’ve said this elsewhere (or in the previous edition).   Actually — looking at the side bar, they did say exactly that in Ch. 31.  

As we used to say, when scooped in the literature — at least we were thinking well.

p. 935 — “During the 1950′s and 1960′s, this sort of question provoked a prolonged and acrimonious debate”  – you better believe it.  Schleyer worked on norbornane, but I don’t think he got into the dust up.  Sol Winstein (who Schleyer called solvolysis Sol) was one of the participants along with H. C. Brown (HydroBoration Brown).

p. 936 — The elegance of Cram’s work.  Reading math has changed the way I’m reading organic chemistry.  What you want in math is an understanding of what is being said, and subsequently an ability to reconstruct a given proof.  You don’t have to have the proof at the tip of your tongue ready to spew out, but you should be able to reconstruct it given a bit of time.   The hard thing is remembering the definitions of the elements of a proof precisely, because precise they are and quite arbitrary in order to make things work properly.  It’s why I always leave a blank page next to my notes on a proof — to contain the definitions I’ve usually forgotten (or not remembered precisely).

I also find it much easier to remember mathematical definitions if I write them out (as opposed to reading them as sentences) as logical statements.  This means using ==> for implies | for such that, upside down A for ‘for all’, backwards E for ‘there exists, etc. etc. There’s too much linguistic fog in my mind when I read them as English sentences.

       So just knowing some general principles will be enough to reconstruct Cram’s elegant work described here.  There’s no point in trying to remember it exactly (although there used to be for me).   It think this is where beginning students get trapped — at first it seems that you can remember it all.  But then the inundation starts.  What should save them, is understanding and applying the principles, which are relatively few.  Again, this is similar to what happens in medicine — and why passing organic chemistry sets up the premed for this style of thinking. 

p. 938 – In the example of the Payne rearrangement, why doesn’t OH attack the epoxide rather than deprotonating the primary alcohol (which is much less acidic than OH itself).

p. 955 – Although the orbitals in the explanation of why stereochemistry is retained in 1,2 migrations are called  molecular orbitals (e.g. HOMO, LUMO) they look awfully like atomic orbitals just forming localized bonds between two atoms to me.  In fact the whole notion of molecular orbital has disappeared in most of the explanations (except linguistically).  The notions of 50 years ago retain their explanatory power.  

p. 956 — How did Eschenmoser ever think of the reaction bearing his name?  Did he stumble into it by accident? 

p. 956 — The starting material for the synthesis of juvenile hormone looks nothing like it.  I suppose you could say its the disconnection approach writ large, but the authors don’t take the opportunity.   The use of fragmentation to control double bond stereochemistry is extremely clever.   This is really the first stuff in the book that I think I’d have had trouble coming up with.  The fragmentation syntheses at the end of the chapter are elegant and delicious.

On a more philosophical note, the use of stereochemistry and orbitals to make molecules is exactly what I mean by explanatory power.  Anti-syn periplanar is a very general concept, which I doubt was brought into being to explain the stereochemistry of fragmentation reactions (yet it does).  It appears over and over throughout the book in various guises.

Urysohn’s Lemma

“Now we come to the first deep theorem of the book,. a theorem that is commonly called the “Urysohn lemma”.  . . .  It is the crucial tool used in proving a number of important theorems. . . .  Why do we call the Urysohn lemma a ‘deep’ theorem?  Because its proof involves a really original idea, which the previous proofs did not.  Perhaps we can explain what we mean this way:  By and large, one would expect that if one went through this book and deleted all the proofs we have given up to now and then handed the book to a bright student who had not studied topology, that student ought to be able to go through the book and work out the proofs independently.  (It would take a good deal of time and effort, of course, and one would not expect the student to handle the trickier examples.)  But the Uyrsohn lemma is on a different level.  It would take considerably more originality than most of us possess to prove this lemma.”

The above quote is  from  one of the standard topology texts for undergraduates (or perhaps the standard text) by James R. Munkres of MIT. It appears on  page 207 of 514 pages of text.  Lee’s text book on Topological Manifolds gets to it on p. 112 (of 405).  For why I’m reading Lee see

Well it is a great theorem, and the proof is ingenious, and understanding it gives you a sense of triumph that you actually did it, and a sense of awe about Urysohn, a Russian mathematician who died at 26.   Understanding Urysohn is an esthetic experience, like a Dvorak trio or a clever organic synthesis [ Nature vol. 489 pp. 278 - 281 '12 ].

Clearly, you have to have a fair amount of topology under your belt before you can even tackle it, but I’m not even going to state or prove the theorem.  It does bring up some general philosophical points about math and its relation to reality (e.g. the physical world we live in and what we currently know about it).

I’ve talked about the large number of extremely precise definitions to be found in math (particularly topology).  Actually what topology is about, is space, and what it means for objects to be near each other in space.  Well, physics does that too, but it uses numbers — topology tries to get beyond numbers, and although precise, the 202 definitions I’ve written down as I’ve gone through Lee to this point don’t mention them for the most part.

Essentially topology reasons about our concept of space qualitatively, rather than quantitatively.  In this, it resembles philosophy which uses a similar sort of qualitative reasoning to get at what are basically rather nebulous concepts — knowledge, truth, reality.   As a neurologist, I can tell you that half the cranial nerves, and probably half our brains are involved with vision, so we automatically have a concept of space (and a very sophisticated one at that).  Topologists are mental Lilliputians trying to tack down the giant Gulliver which is our conception of space with definitions, theorems, lemmas etc. etc.

Well one form of space anyway.  Urysohn talks about normal spaces.  Just think of a closed set as a Russian Doll with a bright shiny surface.  Remove the surface, and you have a rather beat up Russian doll — this is an open set.  When you open a Russian doll, there’s another one inside (smaller but still a Russian doll).  What a normal space permits you to do (by its very definition), is insert a complete Russian doll of intermediate size, between any two Dolls.

This all sounds quite innocent until you realize that between any two Russian dolls an infinite number of concentric Russian dolls can be inserted.  Where did they get a weird idea like this?  From the number system of course.  Between any two distinct rational numbers p/q and r/s where p, q, r and s are whole numbers, you can  always insert a new one halfway between.  This is where the infinite regress comes from.

For mathematics (and particularly for calculus) even this isn’t enough.  The square root of two isn’t a rational number (one of the great Euclid proofs), but you can get as close to it as you wish using rational numbers.  So there are an infinite number of non-rational numbers between any two rational numbers.  In fact that’s how non-rational numbers (aka real numbers) are defined — essentially by fiat, that any series of real numbers bounded above has a greatest number (think 1, 1.4, 1.41, 1.414, defining the square root of 2).

What does this skullduggery have to do with space?  It says essentially that space is infinitely divisible, and that you can always slice and dice it as finely as you wish.  This is the calculus of Newton and the relativity of Einstein.  It clearly is right, or we wouldn’t have GPS systems (which actually require a relativistic correction).

But it’s clearly wrong as any chemist knows. Matter isn’t infinitely divisible, Just go down 10 orders of magnitude from the visible and you get the hydrogen atom, which can’t be split into smaller and smaller hydrogen atoms (although it can be split).

It’s also clearly wrong as far as quantum mechanics goes — while space might not be quantized, there is no reasonable way to keep chopping it up once you get down to the elementary particle level.  You can’t know where they are and where they are going exactly at the same time.

This is exactly one of the great unsolved problems of physics — bringing relativity, with it’s infinitely divisible space together with quantum mechanics, where the very meaning of space becomes somewhat blurry (if you can’t know exactly where anything is).

Interesting isn’t it?

The New Clayden pp. 877 – 908

p. 878 — “The transition state has 6 delocalized pi electrons and thus is aromatic in character”.  Numerically yes, but the transition state isn’t planar, and there is all sorts of work showing how important planarity is to aromaticity. 

p. 881 — It seems to me that the arrow is wrong in the equation at the bottom. Entropy should increase when a Diels Alder product is broken apart, and since deltaG = deltaH  - T * deltaS heating the product should break it apart not cause it to form.  I guess the heat shown is required to increase molecular velocity so that collisions result in reaction.   Enough kinetic energy will blow anything apart (see Higgs particle).

p. 890 — “It is not cheating to use the regioselectivity of chemical reactions to tell us about the coefficients of the orbitals involved.”    I do think that this sort of thing is  cheating when you use the regioselectivity of chemical reactions as an explanation.  They are adding nothing new.  A real explanation predicts new phenomena, the way the anomeric effect does, for example.  You should contemplate the point at which a description of something becomes an explanation (e.g. epistemology).   It’s not the case here, but it was the case for Newton’s laws of gravitation.  Famously he said Hypotheses non dingo (“I frame no hypotheses”).  It appears in the following

I have not as yet been able to discover the reason for these properties of gravity from phenomena, and I do not feign hypotheses.

Yet his laws of gravity were used to predict all sorts of events never before seen, so they are explanatory in some sense.  

This sort of thing is just what a neurologist experiences learning functional neuroanatomy (e.g.  which part of the nervous system has which function).  Initially almost all of it was developed by studying neurologic deficits due to various localized lesions of the brain and spinal cord.  There’s a huge caveat involved — pulling the plug on a radio will stop the sound, but that isn’t how the sound is produced.  People with lesions of the occipital lobe lose the ability to see in certain directions (parts of their visual fields).  Understanding HOW the occipital lobe processes sensory input from the eyes has taken 50 years and is far from over.  

p. 892 — Unfortunately the rationale behind  the Woodward Hoffmann rules isn’t covered, so it appears incredibly convoluted and arbitrary.  Read the book — “The Control of Orbital Symmetry” which they wrote.  Also, unfortunately, the description of the rules uses the term ‘component’ in two ways.  At step two butadiene and the dienophile are each considered a component, as they are in steps 3, 4, and 5, then the two are mushed together into a single component in step 6. 

p. 894 — I haven’t been looking at the animations for a while, but those of the Diels Alder type reactions are incredible, and almost sexual.  You can rotate the two molecules in space and watch them come together and react.

p. 894 –”Remember, the numbers in brackets, [ 4 + 2 ] etc., refer to the numbers of atoms.  The numbers (4q +2)s and (4r)s in The Woodward Hoffman (should be Hoffmann) refer to the numbers of electrons.”  This is so very like math, where nearly identical characters are used to refer to quite different things.  Bold capital X might mean one thing, italic x another, script X still another.  They all sound the same when you mentally read them to yourself.  It makes life confusing. 

p. 894 — The Alder ene reactions — quite unusual.  The worst thing is that I remember nothing about them from years ago.  They must have been around as they were discovered by Alder himself (who died in 1958).  They produce some rather remarkable transformations, the synthesis of menthol from citronellal being one.  I wonder if they are presently used much in synthetic organic chemistry. 

p. 900 — How do you make OCN – SO2Cl, and why is it available commercially?

p. 904 — The synthesis of the sulfur containing 5 membered ring of biotin is a thing of beauty.  It’s extremely non-obvious beginning with a 7 membered ring with no sulfur at all. 

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.

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