Ms Algebra and Ms Geometry are two girls who look very different. No one is surprised. They have very different mothers. Like most women, both mothers care about the emotions and appearances. However, in most respects, they strikingly differ. And they realize that they do.The mother of Ms Algebra is called Ms Language. This old lady loves to talk and listen. She likes to exchange the information that can be communicated as a stream of bits. Thousands of years ago, people would employ Ms Language when they wanted to talk about their families and animals. Sometimes they needed to count the cattle. Ms Algebra realized that she needed some help. So she gave birth to Ms Algebra who did the arithmetics. Later, Ms Algebra has learned some number theory as well. Ms Language's family likes everything to be discrete. Everything must come in packages.
Ms Geometry comes from an entirely different household. Her mother is Ms Motion. She loves to dance, swim, and caress her children. Thousands of years ago, people used Ms Motion to learn how to hunt and grow plants, aside from other kinds of motion that will remain censored. They did a lot of work and some of it - such as the architecture needed to build their buildings - had to be more accurate. Ms Motion found out that she was too old for the job. So she gave birth to Ms Geometry. The little girl was just excellent in measuring distances and angles, in giving tools their perfect shapes, and in putting them to the right places.
But only the wisest men have been quietly aware of a dark and secret cloud looming on a distant horizon.
Ms Geometry and Ms Algebra are half-sisters. They share a father. A long time ago, people thought that the two families were independently created by the Creator. However, it was realized by Charles Darwin of Mathematics that all these ladies were related to each other. And Ms Geometry and Ms Algebra are closely related because they have the same father, the Spirit of Mathematics, or Mr Spirit for short. The newest research in biology actually suggests that only the male DNA carries the genetic information about the disciplines so Ms Algebra and Ms Geometry could actually be full-fledged sisters! ;-)
Boasting his profound perspective, Mr Spirit loves to penetrate deeply into things. He couldn't hide this obsession when he met Ms Language and Ms Motion, either. This text is OK for readers of all age groups so let us skip a few steps here.
Morphology of algebra and geometry
Algebra and geometry are superficially very different. Algebra seems to be focusing on discrete information, algorithms that get some input and spit out some output. Like language, this information may be phrased as a continuous stream of discrete packets.
Geometry seems to be closer to the world of experience and perceptions. The objects accessible to geometry tend to be smooth and continuous. No finite amount of words can ever express the exact information about your generic location. It seems that the continuous numbers - defining your location (distance from 3 points) and shapes - that seem to be the basic building blocks of geometry - cannot be translated into the discrete language.
It's not hard to understand why the people used to believe the Creation story in which the two families were completely unrelated. The families used to be organized in a hierarchy of increasingly continuous disciplines such as:
LanguageOnce we get to continuous numbers, the amount of geometry is increasing e.g. with the number of dimensions. You may think that the individual steps in the sequence above are different and largely independent portions of mathematics. At most, there exist some qualitative links in between the adjacent stairs.
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Arithmetics (adding integers), number theory (primes)
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Rational numbers
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Real numbers, algebra
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Pairs and triplets of real numbers, two- and three-dimensional geometry
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Higher-dimensional geometry
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Infinite-dimensional geometry, functionals, field theory, physical phenomena with all of their complexities
However, today we know better. The stairs in the staircase above are connected by many tunnels that often join non-adjacent or extremely distant stairs. First of all, what is the relationship between algebra and geometry?
Uncovering the features of father's DNA
Arithmetics is the kind of mathematics that first-graders are usually taught: it is about the integers and their addition, subtraction, multiplication, and division. In principle, these operations are not too difficult for professional mathematicians so they would waste a word if they used the term "arithmetics" in this way. Instead, they use it for portions of number theory.
Number theory is dedicated to integers. However, it focuses on their more complex properties, e.g. their being primes. To find out whether an integer is prime, you usually need to apply an algorithm. If an integer is inserted to an algorithm, the resulting program often leads to very different results, depending on the integer parameter. Number theory is all about these things.
Integers may be added and multiplied and operations may usually be reverted. The operation that "undoes" addition is called called subtraction; you need to extend natural numbers (non-negative integers) to all integers, including negative ones (aside from zero that you should have known from a bit earlier), to make this operation universally well-defined. The operation that "undoes" multiplication is division. However, the latter often fails to return integers. X divided by Y is rarely an integer even if both X, Y are integers.
That's the first time when the family of Ms Language got inspired by Ms Motion's daughter. It is impossible to fairly divide 5 alive cows among 3 people. However, it's possible to divide 5 killed cows among 3 people. When cows are sadly converted to meat, its amount becomes a continuous quantity. The rational numbers such as 5/3 were invented soon after people figured out that they can kill animals and store the meat. ;-)
Soon afterwards, they also realized it was possible to divide the meat in more complicated ways than just "to N equal pieces". The amount of meat looked like a continuous quantity. The people could generalize the rational numbers, P/Q for integer P,Q, to real numbers. Real numbers could have been approximated by rational numbers arbitrarily exactly; but they didn't have to be thought of in this way. Instead, the real numbers became "new and independent real objects" that can be accessed in many new ways, too. The original integers were a subset of the rational numbers which are a subset of the real numbers.
N kilograms of meat occupy some space and there is a one-to-one correspondence between the amount of meat and the location of the other, open end of a pile of meat. So the generalized number of cows - the amount of meat - can be identified with the location, something that plays an important role in geometry.
Getting somewhat more complicated
Well, getting from integers to real numbers by the extension of the sets is way too easy for this blog, isn't it? ;-) Let's get just infinitesimally more complex.
Let us ask a simple question: if you only care about integers, and you don't really want to kill any cows or generalize the integers in the continuous way, can the real numbers ever be useful to you? The answer is Yes. You don't have to do too complex things to see that it is so.
Imagine that you only care about integers and you discover the Fibonacci sequence:
0, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, ...This sequence starts with 0,1 and each new entry is the sum of the two previous entries. But imagine you don't want to make 200 steps in order to calculate the 200th entry of the sequence. Can you engineer an explicit formula for the Nth element?
The answer is Yes.
Let me use a semi-advanced toolkit. The two most recent integers may be written as the column vector (x,y)^T. The step of adding a new entry is nothing else than the replacement of (x,y) by (y,x+y). It's like acting by the matrix M = ((0,1),(1,1)) on the column vector (x,y)^T. So far, all the numbers are integers.
However, we want to calculate M^{200}. How can we do it? Well, let's diagonalize M i.e. rewrite it as C.D.C^{-1} where D is a diagonal matrix. The entries in D are called the eigenvalues of D - which are the same numbers as the eigenvalues of M because M and D are conjugate to one another. When you do the calculation, you will find out that the eigenvalues are
lambda1,2 = [1+-sqrt(5)]/2In fact, to check that they're the right eigenvalues, we just need to check that the sum and the product of the two eigenvalues are equal to the right numbers. Their sum is 1, equal to the trace of M, as required: the sqrt(5) terms cancel. Their product is (1-5)/4 = -1, equal to the determinant of M. So it works.
If you also write down the matrix C, and realize that
MN = (C D C-1)N = C DN C-1,it will become trivial for you to multiply the original vector, (0,1)^T, by the matrix above. The formula for the Nth element of the Fibonacci sequence will therefore be
fibN = [ lambda1N - lambda2N ] / sqrt(5)The 0th element is chosen to be 0 while the 1st and 2nd elements are 1,1. You may check that the formula above works: the rational numbers in the numerator always cancel while the multiples of sqrt(5) always double and get divided by sqrt(5), giving you the right Fibonacci numbers.
So the Fibonacci numbers that are all about integers were calculated from numbers that depend on sqrt(5), namely from
lambda1,2 = +1.618034, -0.618034.Of course, these two eigenvalues are nothing else than the golden ratios. Note that 1.618034 and 0.618034 differ by one but their product is one, too. It shouldn't shock you that the asymptotic ratio of the adjacent Fibonacci numbers is the golden ratio.
Just for fun, when your humble correspondent was 10 years old or so, he didn't know much about the matrix calculus. But he could find the formula for the Fibonacci numbers by a generalization of complex numbers that involved the so-called umaginary unit "u". Unlike "i" that satisfies "i^2=-1", the "u" relevant for the Fibonacci problem obeys "u^2=1+u". One can show that the real and umaginary part of "u^n" are the Fibonacci numbers "fib_{n-1}, fib_{n}", respectively.
However, "u^n" can also be explicitly calculated if you write "u" as a combination of numbers "p1,p2" where "p_i = a_i + u b_i" that satisfy "p1^2=p1", "p2^2=p2" (and the same for higher powers). By the way, the fact that you can find two "idempotent" elements (different from 1) that square to themselves - they play the role of the ((1,0),(0,0)) and ((0,0),(0,1)) matrices - shows that the umaginarily complex numbers are not really irreducible, unlike complex numbers. But their algebra was still useful.
Fine, the punch line is that even if you care about integers only - such as those in the Fibonacci sequence - real numbers such as the golden ratio or the square root of five are very useful and important to understand your integers in their entirety. Recall that the text Why complex numbers are fundamental began with a very analogous story: complex numbers are useful and important even if you only care about real solutions of (cubic and other) equations with real coefficients.
Functions and graphs
There exists one more relationship between algebra and geometry that could be classified as trivial: functions have their graphs.
A function may be viewed as a part of algebra. For example, you may calculate polynomials out of their argument. Such a calculation may be left to a calculator. The procedure is a systematic, "linear" sequence of operations - or an Al Gore Rhythm - that ends with a result - a typical job for the girls from Ms Language's family.
However, a function - e.g. the polynomial function - may also be represented by a graph. You may mold a mirror of this shape and Ms Motion's daughter may go skating on the shape. They will have a lot of fun. However, the function expressed by the algebraic expressions - a polynomial written in terms of the letters and digits - carries the same information as the shape that you can draw or caress.
Smooth functions that are not polynomial functions may be translated into the algebraic form, too. For many of them, you may simply generalize the polynomials to infinite "power expansions". Any natural enough function has both geometric aspects (the graph) as well as algebraic aspects (e.g. the information about all the coefficients of a power expansion).
It's not just the object (function) itself that can be mapped from one family to the other. As calculus shows, detailed features of the object may be given interpretations on both sides, too. These two interpretations "feel" different because the mothers differ - we think that talking is something very different from dancing. However, the propositions are fundamentally identical when it comes to their father; they're isomorphic. For example, a graph of the function has points where the tangent is horizontal. Algebraically, you may find the value of x where this occurs by demanding f'(x)=0 where f' is the derivative of the original function.
Of course, when you do such things, it becomes obvious that geometry may be naturally algebraized. Points in n-dimensional space correspond to n-tuples of real numbers. The distances between points may be calculated from the Pythagorean theorem. Angles and other things may be calculated as well.
Complex numbers and angles
When you are constructing this "calculational" picture of geometry, you will encounter one pleasant surprise. All the geometric questions that have something to do with angles will look much clearer in terms of complex numbers. That's because
exp(ix) = cos(x) + i sin(x).The powers - and exp(x) known as the exponential function is nothing else than the x-th power of "e=2.71828" - used to be thought of as games that only Ms Algebra and other daughters of Ms Language, such as Ms Loan, Ms Refinancing, and Ms Radioactive Decay, enjoy. Ms Geometry seemed to prefer other games, such as the seesaw and the carousel - games where something periodically wobbles or rotates along a circle.
However, the beautiful formula above shows that these games are actually identical. The seemingly unbounded exponential function - that used to grow in an unlimited way - may also become periodic if you insert an imaginary argument. If a "mechanical calculator" is asked to literally calculate the exponential of an imaginary number, it ends up with a result whose real and imaginary parts are the cosine and sine of the argument (with i omitted), respectively.
Once you appreciate this important result, you will be naturally led to consider the coordinates in any two-dimensional Euclidean plane to be real and imaginary parts of complex numbers. You will realize that all natural - the so-called holomorphic - functions of such complex variables (those that can be "mechanically extrapolated" from ordinary functions of real variables) have the property that they preserve the angles. You will rediscover the two-dimensional conformal field theory and other things. Complex numbers know a lot of secret things about the angles - and especially about the ways how the angles may be preserved.
Primes and Riemann zeta function
You may be left unimpressed by the results above. After all, you may always imagine that the information carried by a complex number is a pair of real numbers. And all the results about the complex exponential may be translated into a less elegant - but possible - form that only uses the cosines and sines.
However, there exist results that show that the complex functions of complex variables also know about seemingly completely "discrete" aspects of arithmetics and number theory - such as the distribution of prime integers. Of course, the main function that is responsible for this secret "tunnel" is the Riemann zeta function.
zeta(z) = sum(n=1...∞) (1 / nz)This function may be defined for a complex value of z and it's just some particular shape, generalized to the complex plane. It is summed over all integers, so it can't know anything about numbers' being primes, can it?
Well, it can. It knows a lot about them. Why? Each value of n that we are summing over may be factorized as a product of primes
n = 2n2 3n3 5n5 7n7...where the exponents n2,n3,n5,n7... are non-negative integers. Effectively, instead of summing over all positive integers n, we may be summing over all infinity-tuples of non-negative integers n2,n3,n5,n7 (the numbers that follow "n" are all the primes). Note that the number of positive integers is the same as the number of infinity-tuples of non-negative integers - a crazy self-reproduction that is possible with infinite sets. ;-)
However, when you rewrite the zeta function as this sum over n2,n3,n5,n7..., the partial sums over each of these infinitely many variables can be easily done - because they're the geometric sums. At the end, you will be able to express the zeta function in a new way,
zeta(z) = product(p=2,3,5,7...) [1 / (1-p-z)].In this form, it's clear that zeta(z) knows about the primes p. Independently of that, it can be seen that the probability that a random huge integer comparable to N is prime is equal to 1/ln(N). Otherwise, the distribution seems random and free of obvious waves and conspiracies. This statement may be quantified by saying that the number of primes smaller than N never deviates from the integral of 1/ln(N) by more than a particular function of N that is much smaller than 1/ln(N).
Surprisingly, this "discrete" statement about the counting of primes may be translated to an extremely simple proposition about zeta(z). It is true if and only if zeta(z) can only be zero if z is either real (a negative even integer where zeta(z) is easily seen to vanish) or 0.5+it where t is real (the so-called non-trivial zeros).
The proposition that zeta(z) doesn't vanish for any other values of z is the Riemann Hypothesis and it is arguably the prettiest $1 million Millennium problem of the Clay Institute.
Witten, Chern, Simons, and knots
Using the zeta function, we could study the properties of all primes "simultaneously" by a complex function that can also be defined by other formulae - formulae that seemingly know nothing about primes.
A number's being prime is a discrete piece of information. There exist qualitatively analogous, discrete questions about various objects - for example knots. You may ask whether two knots can be continuously deformed to one another or disentangled, aside from similar questions.
In the case of the zeta function, we were interested in the discrete property (being a prime) of a discrete object - an integer. In this case, we're looking at a discrete property of a continuous object (knot) - a curve in the three-dimensional space. So we need a somewhat more important "complex function" that will replace the zeta function. We need some "functionals" - functions whose arguments are functions themselves (but the resulting values are ordinary complex numbers).
What we really need is the Wilson loop that goes along the curve of the knot. Its expectation value and the correlators of such knots know everything about the inequivalent ways to knit them and disentangle them, especially if you compute these correlators in a simple enough quantum field theory called the Chern-Simons theory (whose Lagrangian density is the Chern-Simons three-form).
Such insights were clarified primarily by Edward Witten and it was one of the main pieces of research that earned him the Fields medal.
With this working example how the adventurous discrete homework problem - the disentangling of a knot - may be translated to a seemingly mechanical and continuous procedure - the evaluation of an object in a quantum field theory, it won't be too hard for you to bravely try to find the algebraic representation of any combinatorial problem of a similar kind.
For example, you may study 4-dimensional manifolds and find the Donaldson polynomials. For Calabi-Yau three-folds, you will find the Gromow-Witten and the closely related Gopakumar-Vafa invariants.
If we reduce the complexity of the objects a bit: for inequivalent p-dimensional "non-contractible" holes (submanifolds surrounding them) in d-dimensional manifolds, you may ask the daughters of Ms Motion (Ms Geometry et al.) and they will cut the manifold into patches and try to find and count the holes mechanically.
However, you will also be able to transform most of the important questions about the holes to the research of differential forms, their cohomology (which is purely algebraic and seemingly mechanical, something that Ms Language's family likes to do), K-theory, and at least in a subset with an "elevated physical relevance", all these things will be naturally unified in string/M-theory.
String theory will offer you many new ways how to look at the old questions, especially if you take its "dualities" into account. Just like the Riemann zeta function connected primes with a complex function, and Chern-Simons theory connected Wilson loops with knots, the string dualities always connect two theories with concepts that are usually unrelated (in most cases, both sides are superficially "continuous"), but there exist unsuspected links that actually imply that the theories on both sides are completely equivalent.
Obviously, string theory is the deepest network of tunnels that connect arbitrarily distant stairs of the staircase in ways that almost no one would consider possible some time ago.
At any rate, these tunnels exist whether or not you like them. They're real and you should better learn them and use them if you want to find something about any particular stair in the staircase. Any sufficiently important and universal discrete, algorithmic game can be interpreted as a shadow of much more grandiose, continuous mathematical objects that has many more additional shadows. And these continuous mathematical objects may be studied using many new types of "eyes" that are possible because of the fertility of the Spirit of Mathematics.
If someone wants to understand the world purely in terms of discrete objects and deny that their deep logic always involves continuous objects, then he or she has misunderstood the whole progress of mathematics and physics in the last 500 years.
And that's the memo.
A few additions
Let me only mention a few additional concepts that wanted to be a part of the story above but the natural flow of the story has kept them out.
I wanted to discuss the ways how various operations with numbers and functions may be realized in the geometric and algebraic frameworks. For example, two functions may be multiplied "geometrically" but you may also convert them to power expansions and determine the power expansions of their product "algebraically" from the coefficients.
Also, I wanted to mention the spaces of functions and minimization of a functional at these spaces - the action - as a way to find the classical equations of motion. The minimization of the action may be interpreted as the "more geometric" way of describing the right solution while the solutions to the particular differential equations is the "algebraic approach".
Originally, many more examples of methods of algebraic geometry - the daughter of the lesbian love between Ms Algebra and Ms Geometry - were supposed to be described in the article. The planned material contained the correspondence between homologies (cycles, submanifolds) and cohomologies (closed mod exact differential forms). However, that would make it way too technical for an article that begins with rational numbers and the Fibonacci sequence. Moreover, some of these objects, including the functions counting black hole microstates in string theory, are described at various other places of this blog.
Finally, I was originally planning to discuss methods to generalize both algebra and geometry in various ways - adding new types of numbers and objects we may multiply, adding new aspects of geometry such as noncommutativity or the p-adic geometry. The more we generalize algebra and geometry, the more self-evident their shared father becomes.
Quantum mechanics - and quantum gravity - is responsible for the most well-known examples of such generalizations. From one viewpoint, quantum mechanics tries to reshape classical physics in a way that is as discrete as possible (discrete eigenvalues of the angular momentum or, in some contexts, energy). However, it is also replacing discrete objects such as structureless point-like particles by much more continuous objects such as wave functions (and wave functions), thus making things seemingly more continuous than ever before. The result of these seemingly "opposite" trends is actually the very same quantum mechanics.
The idea that the discrete (and algebraic) concepts are separated from the continuous (and geometric) concepts is an artifact of the naive intuition that has to be superseded if we want to understand the reality at a deep enough level - for example, if we want to understand its quantum features. The more advanced and the more quantum our understanding is, the more self-evident and inevitable the algebraically-geometric relationships (and the father's DNA) become.
