Young babies count

At least according to an article in Scientific American.

Of course it's not clear that this is "counting" in any conventional sense. Additionally, some monkeys can count this well (and I think also many other mammals). So I'm not sure what conclusions can be drawn from this experiment.

What's the most implausible but true mathematical proposition?

Some would say the Banach-Tarski paradox but my attitude has always been that all bets are off when dealing with unmeasurable sets. I think the existence of a number satisfying the main property of Khinchin's constant is much less plasuble.

Addendum: According to this paper, even though the Khinchin property is true for almost all real numbers, nobody has explicitly found a number for which it is true! You can artificially construct a number for which it is true but that doesn't give you an explicit result, and anyway, that's cheating. Experimentally, π gives something near the Khinchin constant, but there's no proof of what happens in the limit.

Is Mathematics too Easy?

Over the years many people have felt that mathematicians have made life too easy for themselves by using axioms that seem much like the proverbial sledgehammer cracking open a nut. Look at the way analysts and algebraists will wield the Axiom of Choice when they can prove what they need with much weaker axioms. Wouldn't it be more enlightening to carry out these proofs using the weakest axiom system possible so we can see exactly what is needed to make them provable?

It turns out that this is more or less what reverse mathematicians do. They take theorems such as the well known Bolzano-Weierstrass theorem and try to figure out precisely how much mathematical machinery is required to prove them. There is no need to use all of ZFC, or even ZF, to prove such a result.

So reverse mathematicians sometimes start with something simpler like second order arithmetic. In its universe are integers and sets of integers, and nothing else. It can still talk about real numbers by encoding them as Cauchy sequences in the usual way but there is no way to encode subsets of the reals in this way. Maybe surprisingly, it is possible to represent continuous functions because the rationals are dense in the reals so continuous functions are defined by the values they take at rational numbers. It turns out that almost all classical mathematics can be encoded in this system even though it's much weaker than ZF.

But usually reverse mathematicians like to go even weaker still and work with recursive comprehension which is essentially the Peano axioms combined with induction and a restricted comprehension axiom. Using these weak tools it's still possible to prove things like the Intermediate Value Theorem, the uncountability of the reals and even results about metric spaces such as the Baire category theorem (but only for separable spaces).

One thing that has always intrigued me about number theory is how many theorems are proved by making excursions into analysis. For example, think about results in analytic number theory that can easily be stated using the language of Peano's axioms, and yet whose proofs require techniques like contour integration that make reference to infinite sized objects and limits. In 1988 Stephen G Simpson proved that such theorems can in fact be reduced to "primitive recursive arithmetic" and hence turned into "finitistic" proofs. On the other hand, I'm not sure that these proofs are necessarily going to give any insight into why the theorems are true. They probably end up being as unwieldy as Borbaki's definition of one.

An End to Coding Theory

Error correcting codes have inspired some beautiful mathematics. One of my favourite mathematical constructions, the binary Golay code, is an error correcting code.

The idea behind the theory is simple. Suppose there is a set of M messages, one of which you want to send over a noisy communication channel. If you simply send one of the messages it might be corrupted and interpreted as another. So instead you embed the set M inside a larger set C and send an element of C as a message. If the message is corrupted then they might receive a message in C but not in M. They'll know right away that the message has been corrupted. But even better, if the embedding of M in C is designed cunningly enough they'll be able to make a good guess as to what the original message was assuming some probability model for the corruption. You pay a price of course, the elements of C typically take longer to transmit than the elements of M, reducing the rate of data transmission.

There have been lots of proposed schemes over the years since Shannon started the field. They use a variety of methods from discrete mathematics and algebraic geometry. There are nice connections between the sporadic groups and coding theory. The Mathieu groups arise directly as symmetries of the Golay code and the Golay code can itself be derived from the Leech lattice whose symmetries are described by the Monster. A common feature of these codes is that they are linear. The set C is a vector space over F₂, the field with two elements. M is a subspace of this space. M is the kernel of an F₂-linear map represented by a matrix called the syndrome. If the syndrome, when applied to a message, gives zero, then it's likely that the message has arrived uncorrupted. Much of the work in coding theory has been about generating suitable syndrome matrices.

But in the sixties Robert Gallager looked at generating random sparse syndrome matrices and found that the resulting codes, called Low Density Parity Check (LDPC) codes, were good in the sense that they allowed messages to be transmitted at rates near the optimum rate found by Shannon - the so-called Shannon limit. Unfortunately the computers of the day weren't up to the task of finding the most likely element of M from a given element of C. But now they are. We now have near-optimal error correcting codes and the design of these codes is ridiculously simply. There was no need to use exotic mathematics, random matrices are as good as almost anything else. The past forty years of coding theory has been, more or less, a useless excursion. Any further research in the area can only yield tiny improvements.

A good place to learn about these codes is in the book by one of the 'rediscoverers' of Gallager's codes, David McKay. There's also the Wikipedia article. I originally heard of these codes from a friend of mine and was sceptical at first. Then I read this Science News article and I'm now a bit embarassed about my scepticism.

The Flyspeck Project

I just came across the Flyspeck Project. The problem they are dealing with is the verification of the Kepler conjecture. Although there is a potential proof nobody is 100% sure of its validity and it seems that the group asigned to verify it have run out of steam. So the Flyspeck Project seeks to build a complete formal proof using HOL Light, a computational framework for proving things. Unfortunately the guys behind the project expect it might take as long as twenty years to verify the potential proof. Let's hope that not too many interesting conjectures require this treatment.

Note that this is different from the four colour problem. For that problem, the proof required an enumeration that was too long and complicated to carry out by hand, so a machine carried it out. For this problem we have a slightly different situation: we already have a potential proof, but we need to reduce it to a completely formal form so that the entire thing can be verified to follow from axioms.

Biggest Test of Benford's Law?

Earlier this week a swapped a couple of emails with someone at Google involved with looking at web page statistics on the Google database. They probably have the one of the largestest corpus of textual data in the world and it's a mixture of data from a wide variety of sources. This makes it an ideal candidate for testing Benford's law and that's what I'm suggesting they do. The published tests I've seen so far are pretty small but it'd be fun to see the results from, say, a billion web pages. Coincidentally Boing Boing just had a linky to a New York Times story on it form 1998. Anyway...I asked the guy to email me if they carry out the test. It's probably pretty low down on their list of priorities. So of course I'll report back.

De Rham Cohomology and Computer Algebra

Computers are now good at a number of branches of mathematics. They've been performing symbolic integration, differentiation, summation and differential equation solving for decades. In recent years, with the advent of Grobner basis methods, they've become pretty competent at algebraic geometry. But one area has stood out in my mind as being unsuitable for computational methods and that's computing de Rham cohomology. In order to compute this we have to study spaces of differential forms on manifolds and these are large spaces. Given lots of pieces of differential form you can glue them to other bits of differential form using a partition of unity so in a sense you have a great freedom to mix and match forms. This is in stark contrast to algebraic geometry where spaces of functions frequently tend to be finite dimensional.

So I was surprised to find that in recent years computers have been calculating de Rham cohomology, and doing so using methods from algebraic geometry. In fact, functions to compute the de Rham cohomology of the complement of various algebraic varieties have been available in Macaulay2 for a while. There's a chapter of the Macaulay2 book available on the subject. The methods used are based on D-modules and discussed in the book Grobner Deformations of Hypergeometric Differential Equations.

I have no idea where the connection with hypergeometric series is coming from but it's interesting to see the connection with algebraic geometry. D-modules are rings with elements x_i and d_i sith the proprty that x_ix_j=x_jx_i and d_id_j=d_jd_i but that d_ix_j-x_jd_i=δ_ij. We can form ideals of these things and study them using Grobner basis methods. Given a smooth function we can form the ideal of elements of a D-module that annihilate the function by considering the d_i to be the differential operators d/dx_i. This gives a connection between the methods of algebraic geometry and differential geometry. But that's as far as my knowledge extends for now.

Geometry is Innate

It seems that geometry is innate according to a MSNBC. The study worked with the Mundurukú people from a remote region of the Amazon who lack vocabulary for geometric terms. Nonetheless they scored well on a variety of nonverbal tests. Judging by the pictures they were testing for the ability to identify connectedness, similarity and congruence. They also displayed an ability to use maps even though they are not a map-making culture.

It's curious to note that the people of this culture weave baskets that appear to involve interesting geometry and whose design look anything but 'innate'. And though 'remote' they also wear glasses and Western style clothing. So clearly they do transmit 'geometric' culture, even if it's nonverbal, and they have had non-trivial contact with cultures that do have a strong geometric vocabulary.

Eleven Reasons to use Haskell as a Mathematician

As a mathematician, what programming language should you use when you need to do anything algorithmic? For some specific domains there are languages tailored for the task, for example Macaulay2 is great for some types of algebra. For general problems many turn to Mathematica or Maple. But although they are universal (in the sense of Turing) they're not great platforms for general purpose algorithms and can perform very poorly unless the feature set of these languages just happens to marry well with the problem you're trying to solve. These tools are also like the proverbial sledgehammer - you're using a giant engine to solve what can sometimes be a simple problem and you'll pay the price in terms of memory and CPU usage. If you want a general purpose programming language the answer is simple. There are eleven reasons why you should use Haskell. So if you're a mathematician who occasionally needs to write code, get started on a tutorial right away!

(1) Haskell is Functional

Functions are the bread and butter of mathematics so it makes sense to use a programming language in which functions are first class objects. It's not at all unusual for a mathematician to take a function of two arguments, say f(.,.), and construct from it a function of a single variable defined by f_x(y) = f(x,y). Computer scientists call such a thing a closure - it's a function constructed by supplying values for some, but not all, of the arguments of another functions. It's one of the most natural operations in the world and yet it's actually quite horrible to construct such a thing in languages like Fortran, C or C++. It's slightly cleaner in C++ because you can use objects and templates to simulate closures, but the syntax is still pretty hideous. (I know there are libraries that make this even easier - but they do so by implementing a functional language as a 'domain specific language' within C++, essentially making my point for me.) Here's how you do this in Haskell: you write "f x". The way it works is quite neat. Given sets A, B and C there is a natural bijection between the set of functions AxB->C and the set of functions A->C^B. So rather than write functions of multiple arguments Haskell function always take one argument and return the function that operates on the rest of the arguments. At first this might seem like a bad idea because functions of multiple arguments are now, in some sense, a second class object. But it's not a problem at all. To apply the function f to the arguments 1 and 2 we simply write "f 1 2".

To get an idea of how horrible it can be to work with functions without functional programming consider the code in Numerical Recipes in C. Most of their code is 'functional' in character in the sense that it is intended to perform operations on functions. But look at how you have to make use of horrible void * pointers to pass in those extra arguments to simulate closures.

(2) Haskell Insulates You

As a mathematician you're more interested in solving the problem than dealing with the idiosyncracies of the hardware and issues like memory management. Haskell does a great job of insulating you from the underlying hardware so that you don't have to worry about these things. Languages like Java aren't too bad at this either. C++ is terrible, if you decide you don't want to manage memory yourself you can use shared_ptr<> objects and turn your code into a horribly unreadable mess.

(3) Haskell Performs Well

It's surprising how well Haskell peforms. You might think "if every time I add a to b it has to construct a temporary intermediate object that represents the function that adds a to a number and then apply that function to b it's going to be slow". But that is not the way to look at things. Haskell doesn't simply execute code by working top to bottom, left to right through your source code. The compiler (itself written in Haskell) is pretty sophisticated and frequently performs comparably to C. It's generally not as fast as C, but it's usually in the same league. The Haskell compiler rips apart your code and reconstitutes it as an executable in ways that are quite surprising. In Haskell you can often write code using idioms that feel natural, but are generally considered slow in other languges, without performance impact. One example is in list handling. Often you want to return a temporary list of objects and perform some operation on them. In many languages there is an overhead in making such a list, only to take it apart again when you want to operate on the actual elements so you instead have to restructure your code so that no list is constructed. In Haskell the compiler will often figure out that your list is temporary and do the restructuring behind the scenes for you.

(3) Haskell is Declarative

Mathematicians like to spend their time saying what is the case rather than thinking in terms of sequences of operations. If b = f(a) and c = f(b) then it doesn't matter if you say c = f(b) and then b = f(a). Order is irrelevant. In an imperative language like C it matters to a large extent what order your code is written in because a statement like b = f(a) is an instruction. In Haskell you aren't giving a list of instructions so much as stating what is true. This fits much better with the way that mathematicians work. Haskell can take care of some of the details of precisely what order the operations are going to be performed in and hide those details from you.

(4) Haskell is Lazy

This goes hand-in-hand with Haskell being declarative. Haskell only computes what it needs to compute in order to get the answer you require. It tries not to do unnecessary work. Languages that aren't lazy like this are called 'strict' languages. In a strict language, say, if you required the first component of the cross product of a vector, you would typically compute all three components of the vector and throw away the second and third components. If your code needed to be efficient you'd probably end up writing special purpose code to compute just the first component of the cross product. You'd be unable to reuse your cross product code. Mathematicians don't want to do this. They want to say that x is the first component of the cross product because it's true statement and not worry about what exactly gets implemented when. This laziness is ubiquitous throughout all of Haskell.

One place that laziness really pays off is in working with lists. Haskell has absolutely no problem working with infinite lists as long as you don't actually try to look at all of the elements. Instead of evaluating all of the elements it simply retains enough information to construct as many elements as are needed. Computing the elementwise sum, say, of two infinite lists is completely straightforward. And performing computations with power series, say, where the later elements are functions of the earlier elements, is absolutely straightforward. You don't have to check to see if you've computed enough of the earlier elements to compute the later one. Haskell does all that for you. If you want to see the first ten elements of the final infinite list you can just ask for them. You don't need to decide in advance how many elements you want and build that into the code from the beginning.

(5) Haskell is good at Datastructures

Haskell is good at dealing with datastructures. Often mathematicians don't realise that they are using datastructures all the time. But they are. For example a group is a set equipped with functions to perform multiplication and inversion as well as a singled out element, the identitiy. Haskell makes it very easy to work with structures that are tuples of objects like this. It's also very easy to build one datastructure out of another and even specify rules about datastructures. For example a matrix is a two-dimensional array of objects. But if the objects form a ring then so does the matrix. This kind of statement is easy to express in Haskell making it well suited to working with algebraic structures. If you've defined formal power series over a ring and you've defined the ring of polynomials over another ring it's trivial to build the ring of power series over polynomials over the integers, say. What's more, the compiler can tell you when the structure you're trying to create doesn't make sense, for example if you were to try to multiply two matrices whose underlying elements can't be multiplied. A programming language like Mathematica is very poor at this as it doesn't really distinguish between datatypes. You'll only discover your error when the program is actually running and it tries to multiply the underlying objects. Haskell also uses type inference. You don't have to explicitly tell it the type of something because it can often figure it out for itself. If you see the number '1' in a textbook what does it mean? Is it an element of N, Z, R, Q, R[sqrt(2)], C or some other completely different algebraic structure? It could be any of these things depending on the context. The same is true in Haskell, it'll figure out what you mean by '1' using completely unambiguous rules. This means that you don't have to use a different symbol for '1' in different contexts, and yet you don't have to laboriously state what you mean every time you use it.

And rather than having an ad-hoc system of datastructure types Haskell's is algebraic.

(6) Haskell is Compact

Not in the topological sense. Haskell code tends to be much shorter than its counterpart written in other languages. There is little extraneous verbiage and the verbiage that does exist is very powerful. This is more than a space saving exercise. Consider the history of calculus in n-dimensions. When Maxwell first wrote down his equations of electrodynamics they took up most of a page. WIth the introduction of vector notation they were simplified. By using the notation of exterior calculus they were simplified to dF = 0, d*F = *J. This isn't just typographically simpler but conceptually simpler. You can see immediately how you might generalise to n-dimensions instead of three, you can see trivially that d*J=0 from d^2 = 0 and you can now perform countless computations in a few lines that previously would have taken dozens of pages. What's more the new notation makes it harder to accidentally write equations of motion that don't make physical sense because they automatically obey symmetries that are difficult to see in more complex notation. Haskell is just like this too. You can do a lot with a tiny amount of code. And if two blocks of code appear to do similar things you can often factor out the common part and make it a separate block of code in a way that's much harder than with just about any other language. This makes it much easier to generalise your code and have insights about it. This works well with the way mathematicians like to think about things.

(7) Haskell is Mind Expanding

Haskell draws on ideas from interesting branches of mathematics, for example Category Theory and Logic. And the logic part doesn't appear the way you expect - it appears via the Curry-Howard 'isomorphism' in the type system used to define datastructures. As mentioned above, Haskell code lends itself well to generalisations. You'll find all sorts of commonality between things that previously didn't seem similar, and you'll find ways to express that similarity in your code. One of my favourite examples is by Ralph Hinze. He doesn't just illustrate the
similarity between binomial tree and binary arithmetic, he actually shows how merging binomials and adding binary numbers are actually instances of the very same thing. Haskell encourages this kind of abstraction. Though I've wandered into the realm of computer science with talk of trees this is still the type of abstraction that mathematicians find interesting.

(8) Haskell Has No Side Effects

When you add two numbers you get the sum. You don't find that something else in the universe has happened as well (except maybe the temperature in the room goes up if it's a really hard sum). So functions in a programming language shouldn't have any side effects. In Haskell a
function does nothing other than return the value of its result. This means that it's easy to bring your mathematical experience to bear when analysing programs. (Almost) all of the usual rules for manipulating mathematical expressions apply so that, for example, if b = f(a) and c =
g(b,b), then c = g(f(a),f(a)). In many other programming languages you'd have to make sure that calling f twice didn't have some side effect that made it behave differently to calling it once.

(9) Haskell Eats Circular Definitions for Breakfast

In most imperative languages a line of code like x=f(x) means replace the value of x with f applied to the old value of x. In Haskell it means x equals f(x). It has no problem with you defining a variable in terms of itself and x=f(x) means that x is a fixed point of the function f. In practice it's not so simple, you can't just write x = 1-x^2 and expect it to figure out that x is the golden ratio. Haskell is a programming language not an equation solving package. But in my formal power series library I'm able to use a line like "e = 1+integrate e" to define the power series for exp x. See here for another example of using circular definitions to solve differential
equations.

(10) Haskell is Well Founded

What I mean is that Haskell is built on fairly solid mathematical foundations in lambda calculus and proof theory. It's not just an ad hoc language that has accreted features over the years. Everything is there for a reason and people have thought long and hard about the theoretical underpinnings of what went into it.

(11) Haskell is Pretty

Well, that's a matter of taste. But I find it very elegant.

That's 11. I think there's a rule against making lists with prime numbers of elements but I'm ignoring it.

To be fair I ought to point out that it's not all a bowl of cherries. Haskell isn't very good at dealing with arrays. In most programming languages it takes time O(1) to access the ith element of an n element array. In Haskell you can get this kind of performace if you don't mind using the various painful to use array operators available, or you can sacrifice performance, keep the elegance, and get O(log n) performance. On the other hand, you'd be surprised just how many useful matrix algortihms don't require random access to the ith element.

Because Haskell functions don't have side effects it doesn't make sense to perform an operation more than once because it'll just do the same thing each time. So there's no notion of a 'for' loop (well, maybe there is) in Haskell and instead you must find other ways to express iterative operations - ie. using 'higher order functions' or recusrion. But sometimes you feel like you just want to write the loop and be done with it. There's always another way of expressing the problem but when the iterative approach is the first thing to pop into your head it can occasionally be frustrating. And sometimes you'd like side effects.

Haskell comes with a library of functions called the 'standard prelude'. Unfortunately, whoever designed it knew just enough mathematics to be dangerous and made a complete hash of it. Instead of using a well known hierarchy of algebraic structures such as group->ring->field they defined all kinds of bizarre structures starting with something like a ring with a norm or valuation. And while the library supports complex numbers it's not flexible enough to support the Gaussian integers (probably a consequence of forcing rings to have a norm). All things that could have been implemented correctly in a few seconds if a mathematician had been consulted early on.

And the last caveat. I found learning Haskell hard. Let me put it in perspective: I'm 39. I've been programming since I was 11. I've been writing recursive code since about 15 and I've been using whatever approximations to functional programming are avalable to me in any language I use for almost that long (including torturing myself with C++ template metaprogramming). I've even implemented a couple of declarative languages, Prolog and SASL, the latter being a functional lazy language a lot like Haskell that I thought I'd investigate before Haskell. And I still found Haskell tricky to learn. But it was worth it. And I believe that the trickiness of the language is the kind of trickiness that mathematicians should be able to handle. (And it probably helps if you don't know any C or C++...)

A friend of mine, and I, have been writing all kinds of mathematical code and finding Haskell well suited to the task. Logic (eg. building analytic tableau as in Smullyan's book on first order logic), Group Theory (eg. the Schreier-Sims algorithm), Calculus (automatic differentiation), Combinatorics (via formal power series), Algebra (eg. Grobner bases) and now we're looking into investigating relationships between elliptic functions and modular forms. (Hmmm...my friend takes most of the credit here.) Haskell is great at all of this stuff even though it's not in any way an algebra or mathematics package.

And one list thing about Haskell. If x is an integer (called an "Integer" in Haskell) then x+1>x. It's nice to work over the integers rather than a cheap and nasty 2-adic approximation (though the cheap and nasty approximation is there if you want it.)

You're needed!

As reported in Business Week, and cited on Slashdot, here's a story about why mathematicians aren't so useless after all.

General Relativity for Dummies

No, really, for complete dummies. John Baez has just posted a description of Einstein's field equations that just about anyone can understand. So simple, even, that it can be described in words. And yet the formulation is equivalent to the field equations. It goes like this:

Given a small ball of freely falling test particles initially at rest with respect to each other, the rate at which it begins to shrink is proportional to its volume times: the energy density at the center of the ball, plus the pressure in the $x$ direction at that point, plus the pressure in the $y$ direction, plus the pressure in the $z$ direction.

More information is here. He also has a derivation of Newton's law of gravition here.

As Baez points out - at first it looks implausible because the Einstein field equation is a tensor equation and this looks like a scalar equation. But the statement must hold true for any set of test particles that are at rest relative to each other. So at any point it doesn't just give one constraint.

This is pretty neat stuff and it ought to go into all GR textbooks...

Self-Reference in the Real World

A sign. From Boing Boing,

Theorem Proving and Code Generation

According to the Curry-Howard isomorphism a type in a programming language is essentially the same thing as a proposition and an object or program of the type is a(n intuitionist) proof of the proposition. So an automatic proof generator should also be usable to generate objects or programs of a given type. This is exactly what Djinn does. It seems pretty neat.

(I only learned about the Curry-Howard isomorphism in the last couple of years but it's probably of general interest. So I recommend reading the Wikipedia article I link to if you haven't met it before. It's also probably not what you're currently guessing it is. It's not about program correctness but type correctness and it doesn't prove programs to be correct but shows how programs themselves are proofs of type correctness. In the language of the article, Djinn solves the "Type Inhabitation Problem".)

Rant: Just for kids?

Search Google News for stories about mathematics and you'll find that they are dominated by a particular subsection of our society: kids. It seems that the only people in the world doing mathematics are children. Very occasionally
you'll see a story about an adult, but then it's usually about an adult teaching children. Do a search on history and you'll see some actual history. Search on physics and you'll get stories about kids mixed in with stories about people doing actual physics. So why are there so few stories about adults doing mathematics?

It's not just in news stories. Look at Texas Instruments' and Hewlett Packard's web sites and you'll see that calculators are largely marketed towards kids. TI are much worse on this front. They have powerful calculators that can solve all kinds of real world problems and yet they're not being marketed as devices for solving such problems, they're about getting kids through exams. HP at least have a section for professionals. Even so, in the old days if an HP calculator had exchangeable faceplates it was so that you could label the keys with new functions, but now it's to change the colour to make it look cool. The only people working with numbers are clearly children.

On closer inspection I see that I was wrong. I missed a story. It's about an adult. No mention of children or education. Ah...it's about an adult who can't do mathematics.

So clearly no adults use mathematics. So why do we bother teaching children a skill they'll never use in adulthood?

Bees can fly after all!

According to this article it seems that bees can fly after all.

PS I'm wondering about the field of 'human aviation' that the article mentions. Am I lacking a superpower that normally comes standard with being human?

A Game of Logic

I've been thinking about a game which I haven't named yet. The rules are simple. The playing board is a logical expression built from the binary connectives 'and' and 'or', the logical constants 'true' and 'false', and atoms 'A', 'B', 'C', ... For example (A and B) or C. To make things easier I'll use abuttal for 'and' and + for 'or'. The previous example is just AB+C.

There are two players, False and True, who take turns. A legal move for False consists of replacing all occurences of one atom with 'false' and you can guess what True's moves are. True wins if the expression evaluates to 'true' with the obvious interpretation of the symbols I've given and False wins if the expression evaluates to false. There is no possibility of a tie.

For example consider the game (A+B)(C+D)(E+F). This is a win for True whoever plays next. Any time False sets an atom to 'false' True merely has to set the other atom in the same factor to 'true' making the factor 'true'. Eventually all of the factors will have the value 'true'.

So here are some questions:

1. Has this game already been studied? If so, what's it called?
   
2. Is there a nice way to evaluate any position in the game? A partial answer is given by some papers by Anshelevich on the subject of Hex. All Hex positions are positions in this game and many interesting Hex positions and subproblems can be approximated well by simple positions in this game. (Note that a good strategy for this game might not be a good strategy for Hex because a Hex game corresponds to a very large expression.)
   
3. Can you assign Conway numbers to positions in this game? What is the value of 'true'?
   
4. What happens if you allow the unary operator 'not' and allow the two players to replace an atom with either 'true' or 'false'.
   
5. What other games, besides Hex, are special cases of this game? (All combinatorial games I guess.) What games are special cases in an interesting way?
   

Opinion: A Farewell to Angles

This story was on Slashdot recently. While the guy might be seen as a bit of a crackpot I'm coming round to his point of view. As a worker in the field of computer graphics I'm frequently faced with software that performs operations like finding the angle, theta, between two normalised vectors, by computing the inverse cosine of the dot product. And yet later down in the code the angle is used only as the argument to further trig functions. Trig functions are expensive to compute. If you know the cosine and the quadrant of an angle then you immediately also know the value of all of the angle's other trig functions by some elementary computations involving the four elementary operations and square roots. These are far cheaper to compute than trig functions. And yet programmers frequently insist on working with the angles themselves because they are familiar with them.

As an extreme example: I recently optimised a block of someone else's code in which a point (x,y) was rotated through an angle π around (0,0) by converting it to polar coordinates, adding π to the angle and then converting back to rectangular coordinates! (Incidentally, this block of code was hard-coded for this one particular rotation, and there were similar blocks of code for π/2 and -π/2 rotations.)

It seems to me that people learning computer graphics need to learn that there are many operations that can be performed without using angles. There needn't be a reflex action to convert any dot product to an angle. Converting to an angle this way may also have unfortunate numerical inaccuracies, especially when the angles are converted back to trig functions again later in the code. Maybe I'll put together a document with a list of things you can compute given the sines, cosines and tangents of angles without having to evaluate a single trig function.

The issue is that the word 'angle' has two meanings. One sense of the word 'angle' is a set of equivalence classes of intersections between two lines. The pair of intersecting lines (A,B) and the pair (C,D) are said to be equivalent if there is a rigid motion of space that transforms A to C and B to D. There are a number of easily computed invariants we can associate with these classes, but the most popular one is called 'the angle' and is often horrible to compute as it typically requires the use of transcendental functions. Unfortunately, as soon as people think 'angle' they also think 'the angle' and are led to use these functions unnecessarily. (It's worse than this, 'the angle' properly lives on S¹ but people work with the cover of S¹, R, leading to troublesome ambiguities.) It seems to me that other invariants besides 'the angle' ought also to be taught. And that is what this book seeks to do, despite its crackpot trappings.

Quadratic Residues and Acoustics

Here are some articles on quadratic residue diffusors to improve sound quality in your home. And an old classic: golden section stranding.

I hear that numerology can also be used to tell your fortune.

A Fabulous Biplane

Just read this paper on the (11,5,2)-biplane. A biplane is like a projective plane except that each pair of points is contained in precisely two lines and each pair of lines intersects at exactly two points. It's an exceptional object in the sense that a whole slew of exceptional objects can be constructed from it: the S(5,6,12) and S(5,8,24) Steiner systems, the binary and ternary Golay codes and the Mathieu groups.

I know it's all kid's stuff but I never studied (finite) groups beyond a first course and so little things please me!

This post brought to you by the number six...

The number 6 has an amazing property singling it out as unique among all of the integers. In fact, back in 1945 JA Todd was so amazed by this property he published a paper called "The Odd Number 6". What I have to say about it comes from chapter 6 of the book "Designs, Codes and their Links" by Cameron and van Lint. The chapter is called "A property of the number 6".

The simplest expression of this property is this: "There are six objects constructed in a canonical way on a set of 6 points. 6 is the only number for which this is possible." This is pretty vague so I'll quote the theorem:

Six is the only natural number n for which there is a construction of n isomorphic objects an an n-set A, invariant under all permutations of A, but not naturally in one-to-one correspondence with the points of A.

Pretty amazing result! Except, embarassingly, I'm a bit confused about this because I'm not sure what "naturally" means here. Come to think of it, I'm not sure what "isomorphic" means here either. I don't remember ever doing a course on "objects" and their isomorphisms. The authors make the meaning of "natural" clearer over the page in a category-theoretical definition:

the category whose objects are the n-element sets, and whose morphisms are the bijections between them has a non-trivial functor to itself if and only if n=6

Which is all very well except I don't know what "non-trivial" means here. Or at least I think I know what it means and I can construct counterexamples. My brain must be working sub-par today. Ho hum.

Anyway, there is one version of this property whose statement is absolutely unambiguous to me: the permutation group S_n has an outer automorphism only for n=6. In fact S₆ acts on the set of 6 objects in two inequivalent ways. Interestingly a bunch of 'exceptional' constructions follow from this: the Hoffman-Singleton graph, the projective plane of order 4, the Steiner system S(5,6,12).

This reminds me a little some other outer automorphism that also leads to an exceptional object: the "triality" outer automorphisms of Spin(8) which lead to a well known exceptional object - the octonions.

Hmmm...I just discovered that John Baez has written his own musings on the number six.

Alternative Logic

As mentioned a while back, this is an article on "Alternative Logic" I wrote for K5 a while back. I've made some tiny changes to it, in particular some claims that probably reflect my frustration with understanding some logic texts more than anything else. It's intended as a "pop science" level article on alternatives to classical logic. I'm in no way an expert on logic but I think it's mostly correct. I do have some worries about my explanation of an intuitionistic proof but I provide links for people who want a more accurate picture. I just wrote this for fun.


Introduction

In a previous installment you've seen some of the alternatives to numbers that exist. But can we do something even more radical? How about going to the very foundations of mathematics itself and replacing logic itself? That might sound a little unusual, I mean logic's logic, you can't choose what you want to be true can you? Well actually there are plenty of alternatives to familiar classical logic, so many that I can't hope to do justice to them in one article. Nonetheless, I can still try to throw out a few teasers and hope you are motivated to check out the references and links to see what I'm really talking about.

Before talking about alternative logics I must talk a little about classical logic, the familiar logic used by mathematicians, philosophers and others. What do we mean by logic? In the old days it simply meant reasoning correctly. That was pretty vague. These days the subject has been formalized and there are two aspects to consider.

On the one hand consider the sentences "All men have less than 3 legs" and "John is a man". From this we can conclude "John has less than 3 legs". Note the very important fact that we don't really need to know what men or legs are to make that deduction. If we were told "All X are P" and "x is an X" then we know "x is P" regardless of what x, X and P are. So we can think of deductions as mechanical operations on sentences. In fact we can consider sentences to be nothing more than strings of characters and proofs to be strings of sentences where each sentence follows from some previous ones by application of a rule. These rules are called deduction rules. Of course we also need some starting points to which we can start applying rules. Those are called axioms. An example deduction rule is the one that says if you have derived A and have derived "A implies B" then you can derive B. In fact that rule has a name, modus ponens. If you can derive a statement from the axioms then the derivation is called a proof. Sometimes, however, you may hypothesise something without being sure of it. You can still apply the deduction rules to get a derivation but it's not a proof until you can eliminate the hypothesis. In fact some systems of logic have very convenient rules for eliminating hypotheses.

On the other hand - what does it mean for a sentence to be true? Well for a sentence like "Fred has red hair" you can find out by checking directly. But what about compound sentences like "Fred has red hair and Jane has black hair". Here are two propositions conjoined by "and". What you can do is test each sentence individually and assign a truth value: 0 if it's false and 1 if true. In fact we can define a function v (called a valuation), acting on sentences, that maps them to 1 if true and 0 if false. We can then use the binary operation, &, defined by the following table, to determine the truth value of the conjunction:

& |0 1

--+----- eg. 1 & 0 = 0

0 | 0 0

1 | 0 1



| | 0 1

--+----- eg. 0 | 1 = 1

0 | 0 1

1 | 1 1



~ | 0 1

--+----- eg. ~0 = 1

. | 1 0

If the result is 1 then the conjunction of the two sentences is true. Formally we're saying v(A and B)=v(A) & v(B). This is taking a truth theoretical view ie. that the truth of "A and B" depends on the truth value of A and the truth value of B. Similary we test the truth of "A or B" using the | operator so v(A or B)=v(A) | v(B) and v(not A)=~v(A). These operations are examples of logical connectives and the functions &, | and ~ are collectively known as boolean operations.

There is one more logical connective to mention: implication. v(A implies B) is v(A) -> v(B) where A -> B is defined to be ~(A & ~B). In other words to tell whether or not "A implies B" is true we look at the truth of A and B and the result is false only if A is true and B is false. Note this means that if we can find any inconsistency in our logic, ie. any proof of something provably false, then we can prove anything, because a false statement implies anything and using modus ponens we can then deduce anything we want. So it just takes one little inconsistemcy to bring the whole system crashing down.

By the way, sometimes I'll be a bit lazy and, for example, use & and 'and' interchangeably. Really they are different things: one combines sentences and the other combines truth values, but it should be clear what is going on from the context.

I should also mention the quantifiers for all and there exist. They also have a truth theoretical interpretation but it's quite technical so I'll leave it out.

Use these connectives and throw in a few axioms and you get what's called predicate calculus. (If you leave out the quantifiers you have the simpler system propositional calculus instead).

So we have two sides to logic: the syntactical side which involves proving things by the application of mechanical rules applied to sentences and the semantic side which is about how you decide if a sentence is actually true or valid. But does proving something make it true? Well if this is the case for a type of logic it's said to be 'sound'. And if everything true is also provable it's said to be 'complete'. When logicians consider alternative ways of doing logic soundness and completeness are among the first things they look for. And while there is good reason for sticking with logics that are sound, it is well known that there are many logical systems that are incomplete. However I'll leave discussion of Gödel's incompleteness theorems for another time.

Note. My notation is a little nonstandard - I am limiting myself to a subset of HTML that I think is viewable by most browsers. If your browser is text only you may be having trouble with even this limited subset.

Multivalued Logic

How can we modify logic then? Well there's an obvious thing to do to start with. I mentioned the truth values 0 and 1. How about throwing in some alternative values? 1/2 seems like an obvious choice. This is exactly what Lukasiewicz did in around 1917. One of his motivations was that he was able to sidestep Russell's paradox with it. But if you were following what I was saying above then that's just half the story - the semantic side. So in the thirties an axiomatization (ie. a set of axioms and deduction rules) for this system was also devised introducing new connectives corresponding to necessity and possibility. We can define new logical operators called M and L with M(1/2)=1 and L(1/2)=0. If X is a proposition MX can be interpreted as meaning "maybe X" and LX as "necessarily X".

But there are some problems with this kind of logic. Lukasiewicz defined ~x=1-x so ~(1/2)=1/2. He also defined 1/2 | 1/2 as 1/2. So we find that for some X, X | ~X is 1/2. To many people that simply doesn't feel right, they argue that X | ~X should have a truth value of 1.

Pressing on regardless we can generalize even further to allowing truth values to take any number between zero and one. One way to to this is with Fuzzy Logic but another approach is via Post logics. If you really want to you can even consider probability theory as a branch of logic where probabilities are real-valued truth values.

Relevance Logic

Now consider what I said above about implication. I basically said that implication has a truth theoretical interpretation - that you could determine the truth value of "A implies B" by knowing the truth values of A and B. But does that really capture what is meant by "implies"? If "A implies B" really were the same as "not both A and not B" then any false proposition would imply anything you like. For example, my car has air bags. So according to classical logic it's correct to say "if my car had no air bags the moon would be made of cheese". But what does my car have to do with the moon? There seems to be something fishy about a system of logic that allows you to make deductions from premisses that are completely irrelevant. So some logicians take issue with the idea that "A implies B" should be defined truth theoretically. In fact, some logicians argue that it's so obvious that you can't define implication truth theoretcially and that only several years of brainwashing (eg. a university course in mathematics) could convince someone otherwise.

This is where relevance logics come in. Relevance logic (pioneered by Dunn) attempts to define conditions under which A is relevent to B, for example by tagging the explicit use of A in the demonstration of B from A. There is an interesting consequence of saying that A has to be relevant to B to prove B. We might find that even if we have managed to prove one false statement we can't prove some others. This is different to what I said about classical logic above where just one false statement brings the whole system crashing down. So relevance logics can be robust against inconsistency making them what's called paraconsistent. Some logicians consider this to be an advantage. For example, at several points in history physicists have had inconsistent laws of physics (for example just before a scientific revolution) to work with and yet have still reasoned succsfully. But there are also other approaches to logic that deal with this issue.

Intuitionistic Logic

You may remember I ended the Alternative Numbers article with a quote from Kronecker. He had a hard time believing in some of the constructions mathematicians were using. In particular he disliked the Law of the Excluded Middle (LEM) which says that either X or not X. In other words that either a proposition is true or its negation is true. There's no middle ground between these two possibilities. It's often used by mathematicians like this: suppose I'm having a bit of trouble proving X. Instead I can suppose that not X is true and see where that gets us. If I find that it leads to a contradiction then I know something is up because in classical logic it is assumed that you cannot prove a false statement. If I did every step of the derivation correctly then the original hypothesis must have been false. There is no "middle way" where the original hypothesis might have been half true. If it leads to contradiction it must have been false. (This is known in the trade as Reductio ad Absurdum.) But here's the issue for people like Kronecker: X might be a proposition saying that a certain mathematical object exists. Using LEM you might be able to prove it exists without actually showing how to find it. It's a bit like getting something for nothing and mathematicians use it all the time. But Kronecker didn't like it at all, it seemed like cheating. He wasn't alone.

Another mathematician, Brouwer (probably more famous for his fixed point theorem), also had trouble with LEM, and he started devising an alternative form of logic called Intuitionistic logic. Actually Brouwer was dead against formalism of the type that logicians use, but nonetheless someone came along after Brouwer and formalized his logic! It turned out that this led to a particularly elegant type of logic.

Part of the intuitionist's problem is that it's possible to prove "A or B" without having a proof of A or a proof of B. To an intuitionist a proof of "A or B" is a proof, either of A, or of B. In fact - that's just how intuitionists define a proof of "A or B". When you do this it suddenly become a lot harder to 'cheat'.

Consider what is meant by "A implies B". To a classical logician it's just "not (A and not B)". But this just doesn't seem to capture the "intuition" of what implication means. A better idea might be something along these lines: if you know A implies B, then as soon as you know A then you know B. In other words the statement "A implies B" should give you the means to prove B the moment you've proved A. So you can imagine "A implies B" as a recipe for converting a proof of A into a proof of B. In fact this is exactly how intuitionists view implication. A proof of "A implies B" is a function that takes as input a proof and spits out another. If the input is a proof of A then the output should be a proof of B. This is radically different to the classical view. By making implications a type of function we connect logic with set theory and with subjects like lambda calculus. And there's another really cool thing going on: mathematicians use the symbol -> in many different ways. Among them are two: they use it to mean both implication and a function mapping (eg. f:A->B means f maps from set A to set B). In other words, by strange coincidence we've turned a proof into a function and we use the same symbol for both. It turns out there is a whole branch of mathematics that deals with arrows! It's called category theory and it provides a beautiful integration of everything I have mention in the last two paragraphs.

But before you take up intuitionist logic you should be warned. Brouwer had carried out some great mathematical work in his life. But when he became an intuitionist he regarded much of his work as being no longer of value. He had to reject things that other mathematicians thought were obviously true. Eventually other mathematicians came to see Brouwer as a little crazy! On the other hand intuitionist logic is closely tied up with Constructive Mathematics (not to be confused with a similarly named educational fad) which insists that you always construct things you claim exist. This means that it goes hand in hand with computer science where people's programs are usually expected to produce a result - not just say a result exists.

Second and Higher Order Logic

Earlier I briefly mentioned quantifiers. An obvious question about them is this: when I say "for all x", what kinds of x are we talking about? If you answer this question then you know what the basic 'objects' are that your logic deals with. For example, in mathematics we generally deal with sets and when we say "there exist an x such that..." we really mean "there exists a set x such that...". In fact mathematicians generally build everything out of sets using a set of axioms called ZF (and maybe some more axioms too - like the Axiom of Choice).
For example, as I mentioned in the Alternative Numbers story, mathematicians build the ordinal integers out of sets.

But what about properties of sets or numbers? How do we talk about those as opposed to the underlying sets themselves? Mathematicians have several different ways to talk about properties. For example, if you want to talk about primality you can just talk about the set of prime numbers. As ZF can talk about sets, by considering primality through the set of primes we have just turned primality into something ZF can talk about. But sometimes there are properties we want to discuss where we can't form a set. For example consider the property of being a set. If we were to convert this into a set then we'd need to have a set of all sets.
But ZF doesn't allow this (because of hairy problems like the Russell paradox). So how can we talk about a property like that?

One approach is to modify logic by extending our universe to include properties. This is what is known as Second Order Logic - as opposed to the usual logic that is first order.
Using Second Order Logic you can say things like "there is a property such that if two sets have this property then so does their union". You simply can't say this in First Order Logic.
But there are some difficult issues associated with First Order Logic. Logicians like to prove things, not just in their logical systems, but about their systems. It's actually very hard to prove anything about Second Order Logic. There are also some other tough issues like deciding when two properties are the same. In fact Quine believed that Second Order Logic isn't even Logic. And even though Second Order Logic appears to have greater expressive power mathematicians actually get along fine with First Order Logic. Every time they make a statement about properties there's always a workaround (sometimes quite inelegant) for expressing something that's just as useful in First Order Logic. By the way, you can also generalize further, so for example in third order logic we can talk about properties of properties.

Modal Logic

When I was talking about multi-valued logic I mentioned the concepts of necessity and possibility. These are called 'modalities' by logicians and a logic that deals with this kind of concept is called a modal logic. There are many different types of modal logic but I'll mention four: alethic logic, deontic logic, epistemic logic and temporal logic. An alethic logic is one like that mentioned above. Operators for necessity and possibility are introduced.
These operators are very closely related in that if it is not necessarily the case that X is not true then X is possibly the case and vice versa. We can write this, using the above notation, as MX=~L~X. Let's think about the semantics of this for a moment. What does it mean to say Bush is a Democrat is false not not necessarily false? If it's false, it's false. Well when logicians are thinking about the semantics of modal logic they often work with 'possible worlds'. Although it is true that Bush is a Republican we can imagine a world in which he is a Democrat (well I think I can anyway). Only when something is true in all possible worlds is it considered necessarily true, otherwise it's just plain true (this is not strictly the definition used but it's close enough for now). Can you imagine a world in which 1=2?

Deontic logic deals with what you ought to do. Introduce the symbol O that means 'ought'.
Then OX means X ought to be the case. Just like in alethic logic we can consider the 'dual' operator ~O~. ~O~X means It is not true that it ought not to be the case that X. In other words it means X isn't forbidden. Epistemic logic deals with belief and knowledge.
For example KX means X is known to be true. KX->X but the converse isn't true. Among other things epistemic logic can be used to study cryptographic protocols in a formal way. Lastly temporal logic introduces time into logic with operators like H so that HX means hitherto X. In temporal logic whenever you ask about the truth of something you need to say at what time you are talking about. HX then means X is was always true up to the time specified. Here's an exercise: what does ~H~ mean in English?

Quantum Logic

Are our logical principles necessarily correct or are they an empirical fact that we have discovered about the universe? Once upon a time Euclidean geometry was seen as necessary. It just seemed obvious that parallel lines were always the same distance from each other. But along came Lobachevsky with the notion that there were different types of geometry and then came Einstein who showed that the universe itself was actually described by one of them. So maybe logic will go the same way. One philosopher who argued this was Hilary Putnam. Just as General Relativity changed geometry he said that Quantum Mechanics changes logic and that in fact we should be using Quantum Logic!

So how do you have to modify logic to make it Quantum? It's difficult to go into the details but I can give a taste of it. Instead of truth values 0 and 1 truth values are subspaces of vector spaces. If you don't know what that means I'll try to give an example. A vector space is a flat space that extends out to infinity and has a chosen point in it called the origin or zero.
A subspace is a space contained in that space with the same chosen point. For example a 2D plane can be thought of as a vector space and the lines going through the origin are examples of subspaces. We can now define how the logical connectives work: given a valuation v v(A and B) = means the intersection of v(A) and v(B). v(A or B) means the smallest subspace containing v(A) union v(B). v(not A) means the space perpendicular to v(A) (in particular not(V), where V is the entire vector space, is the set {0} containing just the origin and not({0}) is the whole of V).

So doing logic is now doing geometry. The boolean & operation, for example, is replaced by vector space intersection. If a proposition turns out to have truth value {0} then we can consider it to be false and if the truth value is the whole of V then it can be considered true. But what about all those in-betweens? Well they correspond to those weird situations in quantum mechanics where you simply can't know two things at the same time. For example because of the Heisenberg uncertainty principle it simply doesn't make sense to say "this electron has mass m and momentum p". Quantum Logic perfectly captures these aspects of quantum mechanics.

(Aside for physicists: truth values aren't strictly the subspaces but the operators projecting onto these subspaces. A projection, P, has the property that P²=P. So the eigenvalues of P are 0 and 1. This means that Quantum Logic really is the quantization of logic - we have replaced truth values with non-commuting Hermitian operators whose eigenvalues are the results we want!)

So do physicists use Quantum Logic? Well, no actually. Part of the reason is that it's a victim of its own success. It works so well and fits perfectly with quantum mechanics that it doesn't actually say much that's new. So physicists get along fine without it. But philosophers of science do occasionally make use of it. (By the way, if you thought it was weird that truth values in Quantum Logic were geometrical, then maybe I'd better not mention that one way of defining a valuation for Intuitionistic Logic is through topology!)

Linear Logic and Non-monotonic Logic

Looks like I still have space to briefly mention some other types of logic. There are still some aspects of logic that have the potential to be varied. For example in every system we've looked at above we are able to take propositions and deduce further propositions from them.
As you apply the deduction rules there is an ever increasing set of theorems that you have proved. Consider the propositions "all birds can fly" and "a robin is a bird". From these we may conclude "robins can fly". However, if we also have the proposition "a penguin is a bird" we might conclude that penguins can fly. It looks like the proposition "all birds can fly" needs to be amended. Yet it doesn't seem entirely right to completely throw away that proposition - it appears to have some value. One approach is through default reasoning. By default, in the absence of other information, birds fly. Logics that can handle this type of reasoning are called non-monotonic. The name comes from the fact that if you have more information, eg. penguins don't fly, you can end up being able to prove less, eg. you can no longer use the proposition "all birds can fly" in the default way. Default logic is of interest to practitioners of AI as it gives a way to codify some aspects of 'common sense'.

You can even 'lose' theorems in one form of logic - Linear Logic. Suppose you are describing a vending machine. You put in one dollar and get either a Coke^TM or a Sprite^TM. If you have one dollar you can get one or the other but not both. So the proposition "I have one dollar" can be converted into either "I have a Coke^TM" or "I have a Sprite^TM" but once it has you can't do it again unless you have another dollar. Linear Logic is something like this and certain types of deduction rule actually 'consume' their premisses as you use them. Linear Logic can be useful for describing systems with limited resources.

Laws of Form

Phew! We're coming near to the end. So to wind down I'm going to end with a very simple logic from the book Laws of Form by G Spencer-Brown. Now most mathematicians are pretty sure that this is the work of a crackpot, nonetheless the first part introduces a fun idea that people have managed to get mileage out of. To introduce this type of logic you really have to break the most fundamental rule of logic - that sentences be strings of characters.
In this type of logic there is only one symbol, a circle. (Actually, Spencer-Brown didn't use circles but this notation is equivalent and is in a similar spirit.) It doesn't have to be a precise geometric circle, it could be a squiggly circle. This is known as a distinction.

As I can't use pictures I'll use ASCII art - the net effect being that I'll be using strings of characters anyway.

Here are some circles: () and o. What can you do with a circle? Well you can put one next to the other like this:

o o.

According to Spencer-Brown the rule is that whenever you see that you can replace it with o.
I.e.

o o = o.

The other rule is that if you see one circle directly inside another they cancel.
I.e.

(o) = .

That's a circle within a circle on the left. (Use your imagination!)
Notice there's nothing at all on the right hand side of that '=' sign! Here are some example derivations:

((o)) = o and (((o)(o)o)o) = ((o)o) = (o) = .

Now you can throw in some unknowns and start proving theorems like that
for any x and y, ((x))=x and y y = y. It turns out that you can represent propositional calculus using these symbols.
Have fun doodling!

Further Reading

I learned a tremendous amount from The Blackwell Guide to Philosophical Logic. When I needed more detail on classical logic I used Notes on Logic and Set Theory and Computability and Logic. Otherwise I used the many resources available on the web some of which I have linked to above. Incidentally I learned about Laws of Form from a British magazine Personal Computer World in 1982.

Final Note
I've had to make many sacrifices in order to make this article of a reasonable length. I've had to be a little vague or sloppy in places. So please use the references and links if you are interested in anything I say. Unfortunately I've probably committed the extra sin, in places, of being incorrect too.

Quantum Mechanics and the Fourier-Legendre Transform over a Semiring

Update: see also my more recent notes on this subject.

Consider the two semirings: (R,+,*) and (R',min,+) where R' is R with a positive infinity value added in. The former is just the real numbers added and multiplied in the usual way and in the latter we use the operations min (which gives the lower of its two arguments) and + instead. There are many formal similarities shared by these structures. For example we have the distributive laws a*(b+c) = a*b+a*c and a+min(b,c) = min(a+b,a+c). Zero acts as the identity for + and we have a+0 = a and infinity is the identity for min with min(a,infinity) = a.

It turns out that the former semiring can be viewed as a quantum version of the latter semiring. In particular we can frequently take statements from quantum mechanics and consider them to be statements in a more general semiring rather than over (R,+,*). When we interpret these more general statements in the semiring (R',min,+) they turn out to say things about classical mechanics.

Consider for example the Hamiltonian formulation of classical mechanics. This essentially says that dynamical systems evolve in such a way that the integral of the Lagrangian is minimised. In other words, the integral of the Lagrangian is the min of its value for all possible paths. In quantum mechanics we no longer have systems taking the minimum but in a sense they take all paths. To compute physical quantities we must instead use the Feynman path integral to integrate over all paths. The factor we must integrate is essentially the exponential of the Lagrangian. In classical mechanics we have the min over an infinite set, in quantum mechanics we have the sum (ie. integral) over the same set. See here for a recent paper on this subject.

Another surprising analogy between these two semirings arises we we try to transfer the concept of the Fourier transform to (R,+,*) to (R',min,+). It's not obvious how to interpret the exponential function in (R',min,+) but it turns out that the natural choice is to consider the ordinary linear functions (in the conventional sense) to be the correct analogue. If we then replace the exponentials with linear functions in the definition of the Fourier transform and replace the integral with an infinite min what we end up with is another familiar transform: the Legendre transform. So the Fourier transform and the Legendre transform may be interpreted as the same thing, just over different semirings.

The analogy carries quite far. In classical mechanics the Legendre transform converts between the Lagrangian and Hamiltonian formulations of classical mechanics. So it changes equations of motion written in terms of (generalised) position into equations written in terms of momentum and vice versa. In quantum mechanics the Fourier transform does much the same thing: the Fourier transform of a wavefunction in space gives the wavefunction in momentum space. Sean Walston has a paper on this. I'm not sure he was aware of the semiring connection when he wrote that.

In summary we have:

(R',min,+)	(R,+,*)
min(a,infinity) = a	a+0 = a
a+0 = a	a*1 = a
a+infinity = infinity	a*0 = 0
Classical	Quantum
integral	minimisation
x -> k*x	x -> exp(k*x)
Legendre Transform	Fourier Transform
Hamiltonian principle of least action	Feynman path integral

I never did understand the Legendre transform. It always seemed like this strange thing plucked out of nowhere. So it's amazing to see that in some sense it is the 'right' thing to study and is as natural as the Fourier transform. Fascinating stuff!

Are Men Innately Better than Women at Mathematics?

No, argues Elizabeth Spelke. In the linked paper (which isn't yet peer reviewed) she looks at a great wealth of published data and comes to the conclusion that the differences can all be explained by socialisation. There's also an old debate between Spelke and Pinker over at Edge.

Tetris and matrices over semirings

Here's a position in Tetris:

  X    X
XXXX XXX X

We can describe this as a list of heights (1,1,2,1,0,1,1,2,0,1).
Here's a Tetris piece:

   Y
 YYY
 Y

and here's what you get when you drop one on the other

   Y
 YYY
 YX    X
XXXX XXX X

We get a new list of heights (1,3,3,3,0,1,1,2,0,1). This has failed to capture the 'overhang' in the 4th column but lets just think about Tetris pieces dropped from the top so overhangs make no difference. What is the relationship between the old and new list of heights?

It turns out that you can get one from the other from a single matrix 'multiplication' - the catch is that we have to work over the semiring (R union infinity,min,+) instead of the more usual semiring (well, actually field) (R,+,*).

Let hn be the initial height in the nth column and hn' be the height after dropping the new piece. Think about how low the new piece made of Ys can fall. If the left most column of the Ys is in contact with the X's then it's lowest point ends up at height h2+1. If he middle column is in contact then the lowest point ends up at h3. And if the right column is in contact the lowest point ends up at height h4. So the lowest point ends up at height min(h2+1,h3,h4). So we can now write down expressions for the new height: h2' = min(h2+1,h3,h4)+1, h3' = min(h2+1,h3,h4)+1, h4' = min(h2+1,h3,h4)+2. We can rewrite this as:

h2' = min(h2+2,h3+1,h4+1)

h3' = min(h2+2,h3+1,h4+1)

h4' = min(h2+3,h3+2,h4+2)

Now let's switch to the semiring (R union infinity,min,+) (with 'additive' identity OO = infinity). Use the notation (+) = min and (*) = +. Then we can write:

h2' = 2(*)h2 (+) 1(*)h3 (+) (1)*(h4)

h3' = 2(*)h2 (+) 1(*)h3 (+) (1)*(h4)

h4' = 3(*)h2 (+) 2(*)h3 (+) (2)*(h4)

We can now write this in matrix form h' = Mh where h is

(  0 OO OO OO ...)
( OO  2  1  1 ...)
( OO  2  1  1 ...)
( OO  3  2  2 ...)
( ...  ...    ...)
( ...          0 )

So corresponding to each Tetris piece in a particular column we have a matrix. To find the effect of dropping multiple pieces we just multiply the matrices in the semiring in turn.

Turns out this has all types of applications - for example resource allocation in a queing system. I'll leave you to figure out how falling testris pieces might represent jobs submitted to a queing system and how the columns might represent types of resource.

Here's an easy exercise: If M is the matrix of a piece, what piece is the transpose M^T the matrix of?

I learnt this from this paper by Gaubert and Max Plus. (I think Max Plus is a relative of Bourbaki.)

Many algorithms are 'linear' in this semiring. Another example that seems quite different is the dynamic programming algorithm for approximate DNA string matching. I tried to work out what the transpose matrix did (ie. what the adjoint algorithm was) and then use the code reversal method I mentioned earlier to see what new algorithm I obtained was. It was another approximate string matcher but this time it worked through the strings in the opposite direction. (Note how each operation in this description is essentially of the form a = max(some terms added to constants).)

The 100 Greatest Theorems

Here is the list, presented at a conference in 1999. I think that like IMDB we also need a list of the 100 worst theorems. Any suggestions for candidates?

A Certain 24th Root

This paper by Heninger, Rains and Sloane (that's Sloane as in the famous Encyclopedia) is interesting. It's about the conditions required for a formal power series with integer coefficients, to be the nth power of another such series. In particular it focuses on theta functions for lattices.

The theta function of a lattice is simply the sum of x^u.u for all u in the lattice. Clearly it is a formal power series with integer coefficients if the lattice is such that u.u is always an integer. These are often pretty amazing functions, for example for certain types of lattice they are modular forms. It turns out that the theta function for the Leech lattice is a power of 24. More amazingly: this is a new result!

It's clear that the theta function is the generating function for the number of lattice elements of a given length. So if it's a 24th power you'd expect this to have a combinatorial explanation too. Unfortunately the proof isn't combinatorial. So even though the 24th root appears to be counting something, nobody knows what it is that is being counted!

Anyway, Sloane and friends had fun trawling through their sequence database trying to find other sequences that were powers. Another interesting one they found was the exponential generating function for the number of partitions of {1,...,n} into an odd number of ordered subsets. It turns out this is a perfect square. This is begging for a combinatorial explanation but it hasn't yet been found. I'll have to see if I can reproduce the square root of this series with my (unfinished) Haskell code for formal power series.

And of course this paper is interesting to me because of the prominent role of the number 24 :-)

Of Partitions and Primes

I must have been hiding under a rock or something. I've only just found out about a big mathematical result about the partition function, p(n). p(n) is the number of ways of writing n as distinct (unordered) sums of integers. p(4) is 5 as 4 = 4+1 = 2+2 = 2+1+1 = 1+1+1+1.

First a digression: There's a bizarre formula for p(n) due to Rademacher and you can see it here. Note the n-1/24 subexpression. You might wonder where that 1/24 comes comes from. Actually it has a neat physical interpretation. In bosonic string theory you have a string which can oscillate with different harmonics. Given a string with a certain amount of energy there are different ways the energy could be shared out between the different harmonics. In quantum mechanics the amount of energy that can go into each harmonic is 'quantised' so that it is proportional to an integer. In addition the energy of a single quantum of energy in the nth harmomic is proportional to n. What this means is that if you have m₁ quanta in the lowest harmonic, m₂ in the second and so on then the total energy in the system, in suitable units, is E=m₁+2*m₂+3*m₃+... In other words the number of different states with energy E is just the partition function p(E). In practice it's a little more complex because strings oscillate in a high dimensional background space so in fact there are harmonics in different directions. But essentially it's a similar counting problem. What I've actually described is a type of conformal field theory. Bosonic string theory (which is a 26D theory) is basically 24 copies of this system.

The interesting twist with quantum string theory is that when you consider the minimum energy you can put into each harmonic it isn't zero. You can argue this from the Hesenberg uncertainty principle, it can't be at rest because then you'd know its position and momentum simultaneously. So it must have non-zero energy. You can treat each harmonic independently so each harmonic has some energy. In fact they each have a minimum of half a quantum. (This is standard stuff.) So the correct formula is really E=(m1+1/2)+2*(m2+1/2)+3*(m3+1/2)+... The catch is that when you try to add up you get a divergent series. But this is no problem for physicists as you can sum the series using the methods outlined here. In fact, the 'zero-point' energy is essentially 1/2(1+2+3+...) so zeta regularisation gives -1/24. So the partition function of n is actually a function of the energy of a conformal field theory in its nth excited state. As I've obsessed about before, this is the same 24 that is the dimensionality of the Leech lattice, Golay code and lots of other good stuff. It's all connected mysteriously! Somehow this information about the quantum theory of strings is already encoded in the properties of the partition function discovered 70 years ago even though the partition function is defined by the elementary properties of numbers. This is also why I think String Theory is cool even if it might be useless as a physical theory. Even if we wanted to give up the string theory 'research programme' today it'd still be there hidden away inside the mathematics we study, regardless of whether we think that mathematics has physical applications.

I'll stop there as I don't even pretend to understand the connections. But do check out the proof of Rademacher's formula via Farey sequences and Ford circles. It's in a few analytic number theory books (eg. Apostol's Springer text on the subject) and it's one of the more beautiful proofs out there.

Anyway, that's all irrelevant. What's relevant is that the partition function has neat congruence properties. Ramanujan proved p(5m+4) is a multiple of 5 for all natural numbers m. He also showed p(7m+5) is a multiple of 7 and p(11m+6) is a multiple of 11. Many years ago Dyson explained the 5 and 7 cases using what was called the "crank conjecture" and then later this was extended to 11. But in the last couple of months Mahlburg has proved that the crank conjecture applies for all primes giving a way to produce such congruences for any prime. This is a pretty major result. People have been investigating these congruences at least since Ramanujan's day but now we have a handle on all of them simultaneously. More detail is posted here and this brings to the end a long chapter in number theory. Note, by the way, how often the number 24 appears through out this paper. :-)

Update: I've edited this so it makes slightly more sense.

Adjoints

I keep starting papers and not finishing them. Both reading and writing, but this time I mean writing. But my latest is short enough that the first draft must be practically the final draft so once my colleagues have suggested improvements I'll see if I can submit it to a journal before the end of the month. Anyway, the inspiration behind this paper was the realisation that you can transform code that computes linear functions into its adjoint by writing the code 'backwards' and taking the adjoint of each line. I could explain what I mean but I found a web site that explains it all. I found an application to digital filtering where I was able to transform a useful algorithm into another by essentially writing it backwards and reversing all of the loops. But the general method of computing the adjoint of code seems so useful I assume it's commonplace, but I can't find a single paper on transforming code in this way, just the web site I mentioned above. It's closely connected to reverse mode automatic differentiation but more general as it applies to any linear operator, not just the action of the derivative on the tangent space.

OK, I will explain. Consider the line of code, in C++, a += b. We can write this in matrix form as

(a) = (1 1) (a)
(b)   (0 1) (b)

If we take the transpose of the matrix we get

(a) = (1 0) (a)
(b)   (1 1) (b)

which is the same as the code b += a. More generaly, any block of code that performs linear updates to variables can be converted to its adjoint by writing the steps backwards and transforming each line of code like this. A more complex example (exercise) is a = 2*a+3*b-4*c; b = 2*a which maps to a += 2*b; b = 0; b += 3*a; c -= 4*a; a *= 2. If your code has loops, eg. iterates over arrays of variables, you just run the loops backwards. The application for me is that there is a published fast algorithm for some linear operator but I need the adjoint. Surprisingly the adjoint of the fast algorithm can actually turn out to be a whole new fast algorithm.

Incidentally, I asked the journal editor for the expected turnaround time for the decision on whether to publish. He responded with the mean, standard deviation, extrema and quartiles. If only people like plumbers, doctors and cable installers could respond in this way. "If I attempt to fix this leak it has a 50% chance of taking 1 hour to fix, a 25% chance of taking 2 and there's a 5% chance I won't get it done today" rather than "I'm charging by the hour and it'll take as long as it takes unless there's a complication in which case it'll take longer".

Infinitesimals

I'm currently reading this article on the history of the infinitesimals. My interest in the subject comes from a subject I've mentioned earlier: automatic differentiation. Implementing infinitesimals as an extension to the real numbers is straightforward in modern programming languages but for some reason it's not well known. They give a nice efficient way to compute derivatives of functions and remove removable singularities from functions.

One of the earliest ways to construct the real numbers from the rationals was as an equivalence class of sequences of rational numbers. Two sequences are equivalent if the difference tends to zero. In Non-Standard Analysis, developed by Robinson, this definition is tweaked slightly so that the sequences are equivalent if they are equal on a 'large' subset of the sequence. The trick is to define 'large' appropriately leading to the concept of an ultrafilter. This results in a set *R which is an extension of the usual set of reals R but also contains infinitely small and large numbers. There's also a nice 'metatheorem' that goes with this construction showing that 'reasonable' theorems about *R are also theorems about R. This basically gives us license to use infinitesimals in our proofs and still be sure that our results make sense as propositions about real numbers. So many of Leibniz's old 'proofs' using infinitesimals, say, can now be reinterpreted as perfectly valid proofs.

Anyway, you don't need all this machinery to do automatic differentiation, just an algebraic structure with elements whose square is zero.

Incidentally, Gödel used ultrafilters to 'prove' the existence of God.