The Gambler’s fallacy is actually based on our experience

We don’t understand randomness very well. When asked to produce a random sequence we never produce enough repeating patterns thinking that they are less probable. This is the Gambler’s fallacy.  If heads come up 3 times in a row, the Gambler will bet on tails on the next throw   Why?  This reasoning is actually based on experience.

The following comes from a very interesting paper of a few years ago  [ Proc. Natl. Acad. Sci. vol. 112 pp. 3788 – 3792 ’15 ].  There is a surprising amount of systematic structure lurking within random sequences. For example, in the classic case of tossing a fair coin, where the probability of each outcome (heads or tails) is exactly 0.5 on every single trial, one would naturally assume that there is no possibility for some kind of interesting structure to emerge, given such a simple form of randomness.

However if you record the average amount of time for a pattern to first occur in a sequence (i.e., the waiting time statistic), it is longer for a repetition (head–head HH or tail–tail TT  (an average of six tosses is needrequired) than for an alternation (HT or TH, only four tosses is needed). This is despite the fact that on average, repetitions and alternations are equally probable (occurring once in every four tosses, i.e., the same mean time statistic).

For both of these facts to be true, it must be that repetitions are more bunched together over time—they come in bursts, with greater spacing between, compared with alternations (which is why they appear less frequent to us). Intuitively, this difference comes from the fact that repetitions can build upon each other (e.g., sequence HHH contains two instances of HH), whereas alternations cannot.

Statistically, the mean time and waiting time delineate the mean and variance in the distribution of the interarrival times of patterns (respectively). Despite the same frequency of occurrence (i.e., the same mean), alternations are more evenly distributed over time than repetitions (they have different variances) — which is exactly why they appear less frequent, hence less likely.

Then the authors go on to develop a model of the way we think about these things.

“Is this latent structure of waiting time just a strange mathematical curiosity or could it possibly have deep implications for our cognitive level perceptions of randomness? It has been speculated that the systematic bias in human randomness perception such as the gambler’s fallacy might be due to the greater variance in the interarrival times or the “delayed” waiting time for repetition patterns. Here, we show that a neural model based on a detailed biological understanding of the way the neocortex integrates information over time when processing sequences of events is naturally sensitive to both the mean time and waiting time statistics. Indeed, its behavior is explained by a simple averaging of the influences of both of these statistics, and this behavior emerges in the model over a wide range of parameters. Furthermore, this averaging dynamic directly produces the best-fitting bias-gain parameter for an existing Bayesian model of randomness judgments, which was previously an unexplained free parameter and obtained only through parameter fitting. We show that we can extend this Bayesian model to better fit the full range of human data by including a higher-order pattern statistic, and the neurally derived bias-gain parameter still provides the best fit to the human data in the augmented model. Overall, our model provides a neural grounding for the pervasive gambler’s fallacy bias in human judgments of random processes, where people systematically discount repetitions and emphasize alternations.”

Fascinating stuff

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Homology: the skinny

I’d love to get a picture of a triangulated torus in here but I’ve tried for hours and can’t do it.

Homology is a rather esoteric branch of topology concerning holes in shapes (which can have any number of dimensions, not just two or three.  It is very easy to get bogged down in the large number of definitions and algebra without understanding what is really going on.  I certainly did.

 

The following explains what is really going on underneath the massive amounts of algebra (chains, cycles, chain groups, Betti numbers, cohomology, homology groups etc. etc.) required to understand homology.

The doughnut (torus) you see just above is hollow like an inner tube not solid like a donut.  So it is basically a 2 dimensional surface in 3 dimensional space.  Topology ignores what its objects of study (topological spaces) are embedded in, although they all can be embedded in ‘larger’ spaces, just as the 2 dimensional torus can be embedded in good old 3 dimensional space.

 

Homology allows you to look for holes in topological spaces in any dimension.  How would you find the hole in the torus without looking at it as it sits in 3 dimensioal space.

 

Look at the figure.  Its full of intersecting lines.  It is an amazingly difficult to prove theorem that every 2 dimensional surface can be triangulated (e.g. points placed on it so that it is covered with little triangles).  There do exist topological objects which cannot be triangulated (but two dimensional closed surfaces like the torus are not among them).

 

The corners of the triangles are called vertexes.  It’s easy to see how you could start at one vertex, march around using the edges between them and then get back to where you started.  Such a path is called a cycle.  Note that a cycle is one dimensional not two.

 

Every 3 adjacent vertices form a triangle. Paths using the 3 edges between them form a cycle.  This cycle is a boundary (in the mathematical sense) because it separates the torus into two parts.  The cycle is one dimensional because all you need is one number to describe any point on it.

 

So far incredibly trivial?  That’s about all there is to it.

 

No go up to the picture and imagine the red and pink circles as cycles using as many adjacent vertices as needed (the vertices are a bit hard to see). Circle

Neither one is a boundary in the mathematical sense, because they don’t separate the torus into two parts.

 

 

Each one has found a ‘hole’ in the torus, without ever looking at it in 3 dimensions.

 

 

So this particular homology group is the set of cycles in the torus which don’t separate it into two parts.

 

 

Similar reasoning allows you to construct paths made of 3 dimensional objects (say tetrahedrons instead of two dimensional triangles) in a 4 dimensional space of  your choice.  Some of these are cycles separating the 4 dimensional space into separate parts and others are cycles which don’t.  This allows you to look for 3 dimensional holes in 4 dimensional spaces.

 

 

Of course it’s more complicated than this. Homology allows you to look for any of the 1, 2, . . , n-1 dimension holes possible in an n dimensional space — but the idea is the same.

 

There’s tons more lingo to get under your belt, boundary homomorphism, K complex, singular homology, p-simplex, simplicial complex, quotient group, etc. etc. but keep this idea in mind.

 

How a first rate mathematician thinks

For someone studying math on their own far from academe, the Math Stack Exhange is a blessing.  There must be people who look at it all day, answering questions as they arise, presumably accumulating some sort of points (either real points or virtue points).  Usually answers to my questions are answered by these people within a day (often hours).  But not this one.

“It is clear to me that a surface of constant negative Gaussian curvature satisfies the hyperbolic axiom (more than one ‘straight’ line not meeting a given ‘straight’ line). Hartshorne (Geometry: Euclid and Beyond p. 374) defines the hyperbolic plane as the Hilbert plane + the hyperbolic axiom.

I’d like a proof that this axiomatic definition of the hyperbolic plane implies a surface of constant negative Gaussian curvature. ”

Clearly a highly technical question.  So why bore you with this?  Because no answer was quickly forthcoming, I asked one of my math professor friends about it.  His response is informal, to the point, and more importantly, shows how a first rate mathematician thinks and explains things.  I assure you that this guy is a big name in mathematics — full prof for years and years, author of several award winning math books etc. etc.  He’s also a very normal appearing and acting individual, and a very nice guy.  So here goes.

” Proving that the axiomatic definition of hyperbolic geometry implies constant negative curvature can be done but requires a lot of work. The first thing you have to do is prove that given any two points p and q in the hyperbolic plane, there is an isometry that takes p to q. By a basic theorem of Gauss, this implies that the Gaussian curvature K is the same at p and q. Hence K is constant. Then the Gauss-Bonnet Theorem says that if you have a geodesic triangle T with angles A, B, C, you have

A+B+C = pi + integral of K over T = pi + K area(T)

since K is constant. This implies K area(T) = A+B+C-pi < 0 by a basic result in hyperbolic geometry. Hence K must be negative, so we have constant negative curvature.

To get real numbers into the HIlbert plane H, you need to impose "rulers" on the lines of H. The idea is that you pick one line L in H and mark two points on it. The axioms then give a bijection from L to the real numbers R that takes the two points to 0 and 1, and then every line in H gets a similar bijection which gives a "ruler" for each line. This allows you to assign lengths to all line segments, which gives a metric. With more work, you get a setup that allows you to get a Riemannian metric on H, hence a curvature, and the lines in H become geodesics in this metric since they minimize distance. All of this takes a LOT of work.

It is a lot easier to build a model that satisfies the axioms. Since the axioms are categorical (all models are isomorphic), you can prove theorems in the model. Doing axiomatic proofs in geometry can be grueling if you insist on justifying every step by an axiom or previous result. When I teach geometry, I try to treat the axioms with a light touch."

I responded to this

"Thanks a lot. It certainly explains why I couldn’t figure this out on my own. An isometry of the hyperbolic plane implies the existence a metric on it. Where does the metric come from? In my reading the formula for the metric seems very ad hoc. "

He got back saying —

"Pick two points on a line and call one 0 and the other 1. By ruler and compass, you can then mark off points on the line that correspond to all rational numbers. But Hilbert also has an axiom of completeness, which gives a bijection between the line and the set of real numbers.

The crucial thing about the isometry group of the plane is that it transitive in the sense of group actions, so that if something happens at one point, then the same thing happens at any other point.

The method explained in my previous email gives a metric on the plane which seems a bit ad-hoc. But one can prove that any two metrics constructed this way are positive real number multiples of each other. "

Logically correct operationally horrible

A med school classmate who graduated from the University of Chicago was fond of saying — that’s how it works in practice, but how does it work in theory?

Exactly the opposite happened when I had to do some programming. It shows the exact difference between computer science and mathematics.

Basically I had to read a large textEdit file (minimum 2 megaBytes, Maximum 8) into a Filemaker Table and do something similar 15 times. The files ranged in size from 20,000 to 70,000 lines (each delimited by a carriage return). They needed to be broken up into 1000 records.

Each record began with “Begin Card Xnnnn” and ended with “End Card Xnnn” so it was easy to see where each of the 1000 cards began and ended. So a program was written to
l. look for “Begin Card Xnnn”
2. count the number of lines until “End Card Xnnn” was found
3. Open a new record in Filemaker
4. Put the data from Card Xnnnn into a field of the record
5. Repeat 1000 times.

Before I started, I checked the program out with smaller sized Files with 1, 5, 10, 50, 100, 200, 500 Cards.

The first program used a variable called “lineCounter” which just pointed to the line being processed. As each line was read, the pointer was advanced.

It was clear that the runTime was seriously nonLinear 10 cards took more than twice the time that 5 cards did. Even worse the more cards in the file the worse things got, so that 1000 cards took over an hour.

Although the logic of using an advancing pointer to select and retrieve lines was impeccable, the implementation was not.

You’ve really not been given enough information to figure out what went wrong but give it a shot before reading further.

I was thinking of the LineCounter variable as a memory pointer (which it was), similar to memory pointers in C.

But it wasn’t — to get to line 25,342, the high level command in Filemaker — MiddleValues (Text; Starting_Line ; Number_of_Lines_to_get) had to start at the beginning of the file, examine each character for a character return, keep a running count of character returns, and stop after 25,342 lines had been counted.

So what happened to run time?

Assume the LinePointer had to read each line (not exactly true, but close enough).

Given n lines in the file — that’s the sum of 1 to n — which turns out to be n^2 + n. (Derivation at the end)

So when there were 2*n lines in the file, the runtime went up by over 4 times (exactly 2^2 * n^2 + 2n)

So run times scaled in a polynomial fashion k * n lines would scale as k^2 * n^2 + k * n

At least it wasn’t exponential time which would have scaled as 2^k

How to solve the problem ?

Think about it before reading further

The trick was to start at the first lines in the file, get one card and then throw those lines away, starting over at the top each time. The speed up was impressive.

It really shows the difference between math and computer science. Both use logic, but computer science uses more

Derivation of sum of 1 to n.

Consider a square n little squares on a side. The total number of squares is n^2. Throw away the diagonal giving n^2 – n. The number of squares left is twice the sum of 1 to n – 1. So divide n^2 – n by 2, and add back n giving you n^2 + n

Entangled points

The terms Limit point, Cluster point, Accumulation point don’t really match the concept point set topology is trying to capture.

As usual, the motivation for any topological concept (including this one) lies in the real numbers.

1 is a limit point of the open interval (0, 1) of real numbers. Any open interval containing 1 also contains elements of (0, 1). 1 is entangled with the set (0, 1) given the usual topology of the real line.

What is the usual topology of the real line? (E.g. how are its open sets defined) It’s the set of open intervals) and their infinite unions and their finite intersection.

In this topology no open set can separate 1 from the set ( 0, 1) — e.g. they are entangled.

So call 1 an entangled point.This way of thinking allows you to think of open sets as separators of points from sets.

Hausdorff thought this way, when he described the separation axioms (TrennungsAxioms) describing points and sets that open sets could and could not separate.

The most useful collection of open sets satisfy Trennungsaxiom #2 — giving a Hausdorff topological space. There are enough of them so that every two distinct points are contained in two distinct disjoint open sets.

Thinking of limit points as entangled points gives you a more coherent way to think of continuous functions between topological spaces. They never separate a set and any of its entangled points in the domain when they map them to the target space. At least to me, this is far more satisfactory (and actually equivalent) to continuity than the usual definition; the inverse of an open set in the target space is an open set in the domain.

Clarity of thought and ease of implementation are two very different things. It is much easier to prove/disprove that a function is continuous using the usual definition than using the preservation of entangled points.

Organic chemistry could certainly use some better nomenclature. Why not call an SN1 reaction (Substitution Nucleophilic 1) SN-pancake — as the 4 carbons left after the bond is broken form a plane. Even better SN2 should be called SN-umbrella, as it is exactly like an umbrella turning inside out in the wind.

Norbert Weiner

In the Cambridge Mass of the early 60’s the name Norbert Weiner was spoken in hushed tones. Widely regarded as a genius tutti the assembled genii of Cambridge, that was all I knew about him aside from the fact that he got a bachelor’s degree in math from Tufts at age 14. As a high school student I tried to read Cybernetics, a widely respected book he wrote in 1948, and found it incomprehensible.

Surprisingly, his name never came up again in any undergraduate math courses, graduate chemistry and physics courses, extensive readings on programming and computation (until now).

From PNAS vol. 114 pp. 1281 – 1286 ’17 –“In their seminal theoretical work, Norbert Wiener and Arturo Rosenblueth showed in 1946 that the self-sustained activity in the cardiac muscle can be associated with an excitation wave rotating around an obstacle. This mechanism has since been very successfully applied to the understanding of the generation and control of malignant electrical activity in the heart. It is also well known that self-sustained excitation waves, called spirals, can exist in homogeneous excitable media. It has been demonstrated that spirals rotating within a homogeneous medium or anchored at an obstacle are generically expected for any excitable medium.”

That sounds a lot like atrial fibrillation, a serious risk factor for strokes, and something I dealt with all the time as a neurologist. Any theoretical input about what to do for it would be welcome.

A technique has been developed to cure the arrhythmia. Here it is. “Recently, an atrial defibrillation procedure was clinically introduced that locates the spiral core region by detecting the phase-change point trajectories of the electrophysiological wave field and then, by ablating that region, restores sinus rhythm.” The technique is now widely used, and one university hospital (Ohio State) says that they are doing over 600 per year.

“This is clearly at odds with the Wiener–Rosenblueth mechanism because a further destruction of the tissue near the spiral core should not improve the situation.” It’s worse than that because the summary says “In the case of a functionally determined core, an ablation procedure should even further stabilize the rotating wave”

So theory was happily (for the patients) ignored. Theorists never give up and the paper goes on to propose a mechanism explaining why the Ohio State procedure should work. Here’s what they say.

“Here, we show theoretically that fundamentally in any excitable medium a region with a propagation velocity faster than its surrounding can act as a nucleation center for reentry and can anchor an induced spiral wave. Our findings demonstrate a mechanistic underpinning for the recently developed ablation procedure.”

It certainly has the ring of post hoc propter hoc about it.

The strangeness of mathematical proof

I’ver written about Urysohn’s Lemma before and a copy of that post will be found at the end. I decided to plow through the proof since coming up with it is regarded by Munkres (the author of a widely used book on topology) as very creative. Here’s how he introduces it

“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.”
”

I’m not going to present the proof just comment on one of the tools used to prove it. This is a list of all the rational numbers found in the interval from 0 to 1, with no repeats.

Munkres gives the list at its start and you can see why it would list all the rational numbers. Here it is

0, 1, 1/2, 1/3, 2/3, 1/4, 3/4, 1/5 . . .

Note that 2/4 is missing (because 2 divides into 4 leaving a whole number). It would be fairly easy to write a program to produce the list, but a computer running the program would never stop. In addition it would be slow, because to avoid repeats given a denominator n, it would include 1/n and n-1/n in the list, but to rule out repeats it would have to perform n-2 divisions. It it had a way of knowing if a number was prime it could just put in 1/prime, 2/prime , , , (prime -1)/n without the division. But although there are lists of primes for small integers, there is no general way to find them, so brute force is required. So for 10^n, that means 10^n – 2 divisions. Once the numbers get truly large, there isn’t enough matter in the universe to represent them, nor is there enough time since the big bang to do the calculations.

However, the proof proceeds blithely on after showing the list — this is where the strangeness comes in. It basically uses the complete list of rational numbers as indexes for the infinite number of open sets to be found in a normal topological space. The proof below refers to the assumption of infinite divisibility of space (inherent in the theorem on normal topological spaces), something totally impossible physically.

So we’re in the never to be seen land of completed infinities (of time, space, numbers of operations). It’s remarkable that this stuff applies to the world we inhibit, but it does, and anyone wishing to understand physics at a deep level must come to grips with mathematics at this level.

Here’s the old post

Urysohn’s 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 https://luysii.wordpress.com/2012/09/11/why-math-is-hard-for-me-and-organic-chemistry-is-easy/.

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?

Tensors yet again

In the grad school course on abstract algebra I audited a decade or so ago, the instructor began the discussion about tensors by saying they were the hardest thing in mathematics. Unfortunately I had to drop this section of the course due a family illness. I’ve written about tensors before and their baffling notation and nomenclature. The following is yet another way to look at them which may help with their confusing terminology

First, this post will assume you have a significant familiarity with linear algebra. I’ve written a series of posts on the subject if you need a brush up — pretty basic — here’s a link to the first post — https://luysii.wordpress.com/2010/01/04/linear-algebra-survival-guide-for-quantum-mechanics-i/
All of them can be found here — https://luysii.wordpress.com/category/linear-algebra-survival-guide-for-quantum-mechanics/.

Here’s another attempt to explain them — which will give you the background on dual vectors you’ll need for this post — https://luysii.wordpress.com/2015/06/15/the-many-ways-the-many-tensor-notations-can-confuse-you/

To the physicist, tensors really represent a philosophical position — e.g. there are shapes and processes external to us which are real, and independent of the way we choose to describe them mathematically. E. g. describing them by locating their various parts and physical extents in some sort of coordinate system. That approach is described here — https://luysii.wordpress.com/2014/12/08/tensors/

Zee in one of his books defines tensors as something that transforms like a tensor (honest to god). Neuenschwander in his book says “What kind of a definition is that supposed to be, that doesn’t tell you what it is that is changing.”

The following approach may help — it’s from an excellent book which I’ve not completely gotten through — “An Introduction to Tensors and Group Theory for Physicists” by Nadir Jeevanjee.

He says that tensors are just functions that take a bunch of vectors and return a number (either real or complex). It’s a good idea to keep the volume tensor (which takes 3 vectors and returns a real number) in mind while reading further. The tensor function just has one other constraint — it must be multilinear (https://en.wikipedia.org/wiki/Multilinear_map). Amazingly, it turns out that this is all you need.

Tensors are named by the number of vectors (written V) and dual vectors (written V*) they massage to produce the number. This is fairly weird when you think of it. We don’t name sin (x) by x because this wouldn’t distinguish it from the zillion other real valued functions of a single variable.

So an (r, s) tensor is named by the ordered array of its operands — (V, …V,V*, …,V*) with r V’s first and s V* next in the array. The array tells you what the tensor function must be.

How can Jeevanjee get away with this? Amazingly, multilinearity is all you need. Recall that the great thing about the linearity of any function or operator on a vector space is that ALL you need to know is what the function or operator does to the basis vectors of the space. The effect on ANY vector in the vector space then follows by linearity.

Going back to the volume tensor whose operand is (V, V, V) and the vector space for all 3 V’s (R^3), how many basis vectors are there for V x V x V ? There are 3 for each V meaning that there are 3^3 = 27 possible basis vectors. You probably remember the formula for the volume enclosed by 3 vectors (call them u, v, w). The 3 components of u are u1 u2 and u3.

The volume tensor calculates volume by ( U crossproduct V ) dot product W.
Writing the calculation out

Volume = u1*v2*w3 – u1*v3*w2 + u2*v3*w1 – u2*v1*w3 + u3*v1*w2 – u3*v2*w1. What about the other 21 combinations of basis vectors? They are all zero, but they are all present in the tensor.

While any tensor manipulating two vectors can be expressed as a square matrix, clearly the volume tensor with 27 components can not be. So don’t confuse tensors with matrices (as I did).

Note that the formula for volume implicitly used the usual standard orthogonal coordinates for R^3. What would it be in spherical coordinates? You’d have to use a change of basis matrix to (r, theta, phi). Actually you’d have to have 3 of them, as basis vectors in V x V x V are 3 places arrays. This gives the horrible subscript and superscript notation of matrices by which tensors are usually defined. So rather than memorizing how tensors transform you can derive things like

T_i’^j’ = (A^k_i’)*(A^k_i’) * T_k^l where _ before a letter means subscript and ^ before a letter means superscript and A^k_i’ and A^k_i’ are change of basis matrices and the Einstein summation convention is used. Note that the chance of basis formula for tensor components for the volume tensor would have 3 such matrices, not two as I’ve shown.

One further point. You can regard a dual vector as a function that takes a vector and returns a number — so a dual vector is a (1,0) tensor. Similarly you can regard vectors as functions that take dual vectors and returns a number, so they are (0,1) tensors. So, actually vectors and dual vectors are tensors as well.

The distinction between describing what a tensor does (e.g. its function) and what its operands actually are caused me endless confusion. You write a tensor operating on a dual vector as a (0, 1) tensor, but a dual vector is a (1,0) considered as a function.

None of this discussion applies to the tensor product, which is an entirely different (but similar) story.

Hopefully this helps

Spot the flaw

Mathematical talent varies widely. It was a humbling thing a few years ago to sit in an upper level college math class on Abstract Algebra with a 16 year old high school student taking the course, listening with one ear while he did his German homework. He was later a double summa in both math and physics at Yale. So do mathematicians think differently? A fascinating paper implies that they use different parts of their brain doing math than when not doing it. The paper has one methodological flaw — see if you can find it.

[ Proc. Natl. Acad. Sci. vol. 113 pp. 4887 – 4889, 4909 – 4917 ’16 ] 15 expert mathematicians and 15 nonMathematicians with comparable academic qualifications were studied (3 literature, 3 history 1 philosophy 2 linguistics, 1 antiquity, 3 graphic arts and theater 1 communcation, 1 heritage conservation — fortunately no feminist studies). They had to listen to mathematical and nonMathematical statements and decide true false or meaningless. The nonMathematical statements referred to general knowledge of nature and history. All this while they were embedded in a Magnetic Resonance Imager, so that functional MRI (fMRI) could be performed.

In mathematicians there was no overlap of the math responsive network (e.g. the areas of the brain activated by doing math) with the areas activated by sentence comprehension and general semantic knowledge.

The same brain networks were activated by all types of mathematical statement (geometry, analysis, algebra and topology) as opposed to nonMathematical statement. The areas activated were the dorsal parietal, ventrolateral temporal and bilateral frontal. This was only present in the expert mathematicians (and only to mathematical statements) These areas are outside those associated with language (inferior frontal gyrus of the left hemisphere). The activated areas are also involved in visual processing of arabic numbers and simple calculation. The activated areas in mathematicians were NOT those related to language or general knowledge.

So what’s wrong with the conclusion? The editorialist (pp. 4887 – 4889) pointed this out but I thought of it independently.

All you can say is that experts working in their field of expertise use different parts of their brain than they use for general knowledge. The nonMathematicians should have been tested in their field of expertise. Shouldn’t be hard to do.

High level mathematicians look like normal people

Have you ever had the pleasure of taking a course from someone who wrote the book? I did. I audited a course at Amherst from Prof. David Cox who was one of three authors of “Ideals, Varieties and Algorithms” It was uncanny to listen to him lecture (with any notes) as if he were reading from the book. It was also rather humbling to have a full professor correcting your homework. We had Dr. Cox for several hours each weak (all 11 or 12 of us). This is why Amherst is such an elite school. Ditto for Princeton back in the day, when Physics 103 was taught by John Wheeler 3 hours a week. Physics 103 wasn’t for the high powered among us who were going to be professional physicists (Heinz Pagels, Jim Hartle), it was for preMeds and engineers.

Dr. Cox had one very useful pedagogical device — everyone had to ask a question at the beginning of class, Cox being of the opinion that there is no such thing as a dumb question in math.

Well Dr. Cox and his co-authors (Little and O’Shea) got an award from the American Mathematical sociecty for their book. There’s an excerpt below. You should follow the link to the review to see what the three look like along with two other awardees. http://www.ams.org/publications/journals/notices/201604/rnoti-p417.pdf. Go to any midsize American city at lunchtime, and you’d be hard pressed to pick four of the five out of the crowd of middle aged men walking around. Well almost — one guy would be hard to pick out of the noonday crowd in Williamsburg Brooklyn or Tel Aviv. Four are extremely normal looking guys, not flamboyant or bizarre in any way. This is certainly true of the way Dr. Cox comports himself. The exception proving the rule however, is Raymond Smullyan who was my instructor in a complex variables course back in the day– quite an unusual and otherworldly individual — there’s now a book about him.

Here’s part of the citation. The link also contains bios of all.

“Even more impressive than its clarity of exposition is the impact it has had on mathematics. CLO, as it is fondly known, has not only introduced many to algebraic geometry, it has actually broadened how the subject could be taught and who could use it. One supporter of the nomination writes, “This book, more than any text in this field, has moved computational algebra and algebraic geometry into the mathematical mainstream. I, and others, have used it successfully as a text book for courses, an introductory text for summer programs, and a reference book.”
Another writer, who first met the book in an REU two years before it was published, says, “Without this grounding, I would have never survived my first graduate course in algebraic geometry.” This theme is echoed in many other accounts: “I first read CLO at the start of my second semester of graduate school…. Almost twenty years later I can still remember the relief after the first hour of reading. This was a math book you could actually read! It wasn’t just easy to read but the material also grabbed me.”
For those with a taste for statistics, we note that CLO has sold more than 20,000 copies, it has been cited more than 850 times in MathSciNet, and it has 5,000 citations recorded by Google Scholar. However, these numbers do not really tell the story. Ideals, Varieties, and Algorithms was chosen for the Leroy P. Steele Prize for Mathematical Exposition because it is a rare book that does it all. It is accessible to undergraduates. It has been a source of inspiration for thousands of students of all levels and backgrounds. Moreover, its presentation of the theory of Groebner bases has done more than any other book to popularize this topic, to show the powerful interaction of theory and computation in algebraic geometry, and to illustrate the utility of this theory as a tool in other sciences.”