Tuesday, March 30, 2010

Rational Abstraction


When I discuss mathematics with non-mathematicians, in particular higher mathematics, I am often asked, “What does this have to do with the real world?” Sometimes even other mathematicians ask this question and it’s an entirely reasonable question to ask. With other sciences, it’s clear what they have to do with the real world. We may not run into quasars every day, but you can still see them if you point a telescope in the right direction. But when does anyone encounter the 248-dimensional Lie group E8? Or for that matter a real, perfect circle? Or the number three?

Let’s start with the number three, or numbers in general. We may not encounter the number three in real life, but we still encounter three apples, or three cups of flour or a gallant ship that sails around three times before sinking to the bottom of the sea. The number three is not an actual object, but a property that applies to all the above things in the same sense that yellowness is not an actual object, but a property that applies to lemons and bananas.

Essentially mathematics is the science of abstraction. If you have objects that satisfy certain properties and perform actions on them that satisfy other properties you will get objects that satisfy other properties. In mathematics, you study those based on the properties, and then you know that those rules will apply to any objects satisfying those properties. Since 2+3=5 we know that two apples plus three apples is five apples and also that two cups of flour plus three cups of flour is five cups of flour.

This seems simple enough, and most mathematicians would accept such an explanation. But there are two issues that challenge this notion. First of all, some of the more abstract mathematicians study concepts, like large cardinals, that have no known objects that represent them. Second of all, certain simple concepts, like a circle, can’t be represented perfectly.

In the case of a circle, we may encounter imperfect examples. Wheels, CDs and lenses all appear to be circular. But if you look closely at any of those objects, they will turn out to be a bit jagged at the edges. To be a circle every point on the edge has to be the same distance from the center, but in all of those objects one molecule will be a few microns closer than another, so they aren’t really circles.

So not only do we not encounter abstract circles in real life, but we don’t encounter objects that satisfy the property of “circleness”.

Platonism

There is a belief called Platonism, named after the Greek guy Plato. Platonism holds that the real world isn’t actually real, but just a shadow of a “real” real world, called the Platonic ideal. In this world, the number three does exist as an actual physical object and the three apples, three cups of flour and three times the ship sailed around are all just shadows of the Platonic ideal of three.

This is pretty much nonsense. The main argument for the existence of the “Platonic ideal” is that the concept of a circle exists, but none of the wheels, CDs, lenses or other representations of circles completely captures that representation exactly. Therefore there must be some place where the concept of three is represented exactly and this is the “Platonic ideal”.

The problem with this argument is every fiber of its being. It happens to be exactly the same as Thomas Aquinas’s fourth (and most laughably incorrect) “proof” of the existence of God. The only difference is that Aquinas used “goodness” as the abstraction instead of “circleness”.

By the same standard there must be a world consisting of a Platonic yellow and a Platonic smelly because there are no objects in the real world that perfectly represent said concepts.

In mathematics, however, people who otherwise take a rational materialist view of the world nonetheless accept Platonism because it provides a quickie explanation for the lack of a perfect circle. Besides, even if Platonism is wrong, it comes down to some fairly picky metaphysics.

Universal Truth

Nonetheless, that picky metaphysics reared its ugly little head in the form of the parallel postulate. One of the consequences of Platonism is that any mathematical statement must be either universally true or universally false. But things didn’t work out that way.

The parallel postulate is a rule of Euclidean geometry that states that given a line L and a point P there is a unique line going through P that doesn’t intersect L. Of Euclid’s original axioms for geometry, this one was the bulkiest, and many mathematicians tried unsuccessfully to prove it using Euclid’s other axioms.

It turns out that, the axiom was independent of the other axioms. There is another type of geometry called elliptical geometry in which there are no parallel lines at all and another called hyperbolic geometry in which there are multiple lines going through P that don’t intersect L.


So if the Platonic ideal really exists, and you have pure points and pure lines in this world, does the parallel postulate hold? If not, is projective geometry or hyperbolic the one true geometry?

In the case of elliptical geometry, one model it by considering the surface of a sphere to be your plane. Then the great circles on the sphere would be the lines. Of course two great circles will intersect in two points, so you would define the points to be pairs of antipodal points on the sphere. Antipodal points have an axis connecting them, and you can define the distance between two points to be the angle between their axes. With this notion of distance defined, you have the notion of a circle. Great circles are all contained in a plane, and the angle between those two planes is the angle between the two great circles.

This way, any two lines will intersect in exactly one point, any two lines will have exactly one line connecting them, any three points that aren’t in a line will have a unique circle connecting them. You get all the all of Euclid’s axioms, except that, since any two lines intersect in a point, there are no parallel lines.

The Platonist explanation is that since a sphere isn’t really a plane, a great circle isn’t really a line and a pair of antipodal points isn’t really a point, that projective geometry isn’t the real geometry. However, since real planes, lines and points, according to the Platonist, only exist in another world of which this is a shadow, who are they to say what really is a plane, a line or a circle?

There’s actually another model of elliptical geometry, in which there are only seven points and seven lines. The picture below is called the Fano plane:


The six lines and the circles are the lines. The dots are the points. You know longer have a notion of distance, but you can define a circle to be three points that aren’t in the same line. This way, any two lines intersect in exactly one point and any two points extend to exactly one line. You get all of Euclid’s axioms except for the parallel postulate and the axioms that explicitly deal with distance and angle measures.

Even weirder you can get a model by treating the six lines and the circle as the points and the points as lines.

All of these are models for some form of geometry, but they don’t satisfy all of Euclid’s axioms. If you’re a Platonist and insist that abstractions must refer to a singular object in another world, then only one of these models is the “correct” geometry.

If you’re a materialist, then which model of geometry to use depends on which real-life object you’re studying. If you’re trying to navigate around a city you use Euclidean geometry. If you’re flying a plane around the world, you use elliptical geometry. If you’re an ant crawling in someone’s pants you use hyperbolic geometry. The Fano plane has applications in coding theory. Given that all of these models have their uses, why should you fix one model as being the truth.

Abstractions of Abstractions

The development of non-Euclidean geometry in the 19th century was part of a general move towards more abstract math. It started in the middle ages when the Arab mathematicians came up with algebra, where you allowed variables to represent arbitrary numbers, an abstraction of an abstraction. It continued in the baroque period with Decartes’ analytic geometry which let one describe lines and points and planes with algebraic equations.

Eventually we had abstractions of abstractions of abstractions. We had the notion of geometry which represented the different models of geometry which represented shapes which represented objects. The same thing happened with algebra, as the notions of groups, rings and fields was introduced, which allowed one to come up with alternate systems of arithmetic similar to the alternate systems of numbers.

The earliest of these systems was the complex numbers, developed in the renaissance. Here, you had a number i, defined to be the square root of -1. Descartes, a Platonist, called this an imaginary number because there didn’t seem to be any real-life thing that modeled it. The numbers on the number line are now known as real numbers, while the imaginary numbers, real numbers times i, can form another axis in the complex plane.

The names real and imaginary have since become established as official names, even though both types of numbers have found practical applications.

By the 19th century, even more abstract notions developed. You had the quaternions, in which you have two other square roots of -1, j and k, such that ij=k, jk=i and ki=j, but ji=-k, kj=-i and ik=-j. Now multiplication is no longer commutative and, instead of all the numbers being on a line or a plane, they’re in four-dimensional space. Then you have the octonions which exist in eight-dimensional space and are non-associative as well.

You also had the notion of Boolean algebra. Here, instead of numbers of have statements. If you have two statements A and B you multiplication is equivalent to “A and B” and addition is equivalent to “A or B, but not both.” The number 1 would correspond to a statement that is always true, and the number 0 would correspond to a statement that is always false. This way you have associativity, commutativity and the distributive property, as well as the rules saying that 1 times anything is itself and 0 plus anything is itself.

However, you have two extra rules that are a bit weird. One of them is that anything times itself is equal to itself. The other is that anything plus itself is equal to zero.

This sort of stuff resulted in a split in Platonism, between the people who rejected the modern abstract mathematical developments, and the people who accepted them.

One of the people who rejected the new developments was Charles Dodgson, AKA Lewis Carroll. In the New Scientist article Alice’s Adventures in Algebra: Wonderland Solved Melanie Bayley points explains how most of the craziness in Alice’s Adventures in Wonderland was intended to mock the crazy new-fangled math. The idea of numbers that didn’t count anything was like “a grin without a cat”.

Of course, there are plenty of examples of grins without cats, such as grins on humans.

Set Theory

The Platonists who accepted the new developments tended to take on a more openly mystical attitude. For example Georg Cantor, a devout Lutheran, tried to tie the notion of infinity to God.

Cantor is known as the father of set theory. A set refers to a collection of objects. There’s a set consisting of the integers, a set consisting of the points on a circle, and a set that consists of a banana, a dog and the concept of forgiveness.

This was an important development because it allowed one to describe all of mathematics in a single language. You represent shapes as subsets of a coordinate plane. You can represent functions as sets of ordered pairs with the input in one coordinate and the output in the other. You can then represent algebraic concepts by with a set of numbers and a set of functions on the numbers.

Set theory also provided a new way to think of infinity mathematically. The number of elements of a set is known as its cardinality. To show that two sets have the same cardinality, all you need is an invertible function from one to another.

Because infinity was something that couldn’t exist in real life, Cantor viewed the study of infinity as a religious endeavor. His mathematical opponents, the Platonists Poincaré and Brouwer, shared a similar view, but interpreted that to mean that set theory was not real math. They were the fathers of the intuitionist school of mathematics, which held that math was only legitimate if it could be perceived. Something only exists if we can find it and show it to people.

Cantor changed his mind, however, when he discovered there were different sizes of infinity. This is known as the Diagonalization Theorem, which showed that there are more real numbers than there are integers. Furthermore, for any size of infinity, you can come up with a bigger one, so there is no absolute infinity. Cantor then declared that mathematical infinity was potential infinity as opposed to the religious notion of actual infinity, then he went crazy before he could elaborate.


Out of this came two more notions similar to the parallel postulate: the axiom of choice and the continuum hypothesis. The continuum hypothesis states that there are no sizes of infinity between continuous infinity of the real numbers and the countable infinity of the integers. It turned out that, like the parallel postulate, this was independent of the other axioms of set theory.

In order to have a definitive ordering of the different infinities, you need the axiom of choice, which says that if you have a collection of sets there is another set consisting of one element from each set in the collection. So, if you have a bunch of pairs of socks, you can choose one sock from each pair. If you have a finite number of pairs of socks, this is pretty clear, but if you have an infinite number of pairs of socks, it would take forever to choose one sock from each pair.

The axiom of choice also has other counter-intuitive consequences, such as the Banach-Tarski paradox, which says that you can cut a sphere into five pieces and piece them together to get two spheres of the same size. The axiom of choice is also independent of the other axioms of set theory, so the intuitionists declared that it was false.

More recently, Platonists have declared that the axiom of choice is true, but the continuum hypothesis is false. Ed Witten even went so far as to claim that there is exactly one size of infinity between countable and continuous, because the models of set theory that use that assumption look prettier than the others.

Formalism

In response to these issues, the mathematician David Hilbert came up with a mathematical philosophy called formalism. Formalism holds that mathematics is just the manipulation of formal symbols and any meaning of those symbols is supplied by our minds. Essentially this was saying that we each have a Platonic ideal in our own heads. If Platonism is like a mathematical Aquinas’s Catholicism, then formalism is like a mathematical mixture of solipsism and Unitarianism.

Formalism alleviated the most obvious problems with Platonism. If the different independent axioms had no explicit meaning then you didn’t have to worry about whether they were true or not. It didn’t matter whether infinity existed, because it was just a symbol. Is elliptical, Euclidean or hyperbolic geometry correct? They’re all just games with slightly different rules. They’re all legitimate.

However, like Platonism, formalism insists on separating mathematic from the material world. As such it carries its own mystical baggage which causes problems. The biggest problem with formalism: applied mathematics exists. Three apples are three apples whether or not we decide to give them the meaning of three.

One of the more extreme formalists, the number theorist G. H. Hardy, wrote a book called A Mathematician’s Apology in which he lamented that his mathematical research had no bearing on real life, but took comfort in the fact that, unlike physics, chemistry and biology, his work could never be used in warfare. After his death, his research in number theory turned out the be the basis for cryptography, which was used in warfare.

Poincaré vs. Hilbert

Hilbert’s formalism meant that he was a strong proponent of set theory, because it allowed for a language in which all of mathematics could be described, while rejecting Cantor’s openly mystical interpretations. When Poincaré and his followers tried to keep set theory from being accepted at universities, Hilbert came in set theory’s defense. This resulted in the biggest philosophical debate in the history of modern mathematics.

Because Poincaré was a Platonist and Hilbert a formalist, there is a mistaken view that this was a debate between Platonism and formalism. As a result many historians of mathematical philosophy present these as being the only two views one could take.

For instance, Mario Livio’s book Is God a Mathematician? provides a well-written and well-researched account of mathematical philosophy, but ultimately succumbs to its horrible title. According to Livio, the central question in mathematics is “Is God a mathematician?” If you answer yes, you’re a Platonist. If you answer no, you’re a formalist. But what if you don’t believe in God? The mere act of asking the question “Is God a mathematician?” forces people to choose between two different types of mysticism, and neglects to consider any materialist explanation.


Really, the debates between Poincaré and Hilbert was about intuitionism, not Platonism. Hilbert was defending the Platonist Cantor from the Platonist Poincaré, and he had Platonists Bertrand Russell, Alan Turing and Kurt Gödel on his side.

With regard to the specific question of whether set theory is a legitimate branch of mathematics, Hilbert was completely and utterly correct. Set theory is unavoidable in modern mathematics. Intuitionist mathematics is still studied, but only because is has been formalized using set theory and is treated as another model of mathematics, just like the three different types of geometry.

Does this mean that Hilbert’s formalism was correct? No, but it means Hilbert was a pretty cool guy, despite his formalism. Does this mean Russell’s Platonism was correct? No, but it means Russell was a pretty cool guy, despite his Platonism. Does it means Poincarés Platonism is wrong. Yes, it does mean his intuitionist interpretation of Platonism is wrong. Poincaré is still a cool guy, who did a lot of great mathematics, though, despite his particularly wrong-headed brand of Platonism.

Turing vs. Einstein

To look at another example, Alan Turing, the father of computing, was a Platonist, while Albert Einstein, the father of relativity was a formalist. They said so themselves? But is that what they really believed? Einstein was a physicist and regularly used math in his physics, so it would seem odd that he would identify with a philosophy that claims math is just meaningless symbol manipulation. Similarly, Turing’s promotion of Artificial Intelligence was based on the idea that intelligence reduced to the ability to be able to do mechanical symbol manipulation, which puts him at odds with Platonism.

Unfotrunately, the portrayal of Platonism and formalism as the only two mathematical philosophies means that people have to choose between two forms of mysticism, so they end up choosing whichever one seems to be the lesser evil.

In the case of Einstein, his work in relativity resulted in a model of space-time that was based on four-dimensional elliptical geometry. Elliptical geometry can be modeled with Euclidean geometry where objects change size. For instance, if you model the surface of the Earth using Euclidean geometry, you would get a map, rather than a globe, and Greenland would be really big. So, as people move away from the equator, they get bigger and that’s why they don’t realize just how big Greenland is. This is a pretty unhelpful model, but it can be formalized.

Poincaré advocated this in the case of relativistic space-time. His argument was that the mathematical calculations involved in Euclidean geometry are simpler than the calculations in elliptical geometry so, by Occam’s razor, Euclidean geometry was the one true geometry. Einstein, being a physicist, realized that the physics calculations in the elliptical model were simpler than those for the Euclidean model, because they didn’t involve people changing size whenever they go to Greenland. Because of this, Einstein identified with formalism.

Similarly, Turing’s main encounters with formalism were through Ludwig Wittgenstein, who took the notion to its post-modern extreme, arguing that there was no such thing as truth. As such Turing identified with Platonism.

Mathematics is based on rational thought, and in order to do it you have to have to have some sort of materialist understanding of things. For the most part mathematicians who identify with Platonism or formalism just use those ideas to fill in holes that they don’t have time to explain. While Livio’s Is God a Mathematician? and in the extreme of ludicrousness, Neal Stephenson’s novel Anathem portray mathematics as being dominated by an endless battle between those two views, most mathematicians don’t give the matter much thought.

Even Poincaré, one of the more extreme Platonists, nevertheless did a lot of important math that was entirely at odds with his philosophy. The recently solved Poincaré conjecture is concerns four-dimensional spheres, and it’s proof depended on formalizing topology with, amongst other things, set theory.

Dialectics

So how does one explain the lack of a perfect circle while still accepting that the real material world is, in fact, the real material world, and that made-up magic circle-worlds are, in fact, not real? If Platonism relies on made-up magic circle-worlds and formalism says that a circle has no real meaning, what does that leave. For that we can look to some other Greek guys, like Heraclitus.

Heraclitus is the father of a philosophy known as dialectics. This is based in the idea that the material world is always in a state of flux. Heraclitus’s most well-known saying is “No man ever steps in the same river twice, for it's not the same river and he's not the same man.” This is because the particles of the river and the man have all moved or been replaced with other particles.

This sounds like empty mysticism, but it’s fundamentally true. However, just the moving of water or atoms isn’t going to alter the fundamental character of the river or the man. So in some other sense, the sense most people would accept, of course you can step in the same river twice. But eventually, when the river dries up or the man dies, the quantitative changes result in a qualitative change.

In the case of a circle, wheels, CDs and lenses are circles up to some limitations. Those limitations don’t matter if you’re trying to make a bicycle or a CD player or a telescope. They are just small quantitative deformations. When a wheel gets bent too much those quantitative deformations result in a qualitative deformation and it ceases to be sufficiently circular for the purpose of making a bicycle. Similarly, if you had a different purpose for your wheel, those deformations would have more and more of an impact.

There may be no absolute infinity anywhere but we can get arbitrarily large numbers. Even though there are only a finite number of particles in the universe, there is still the potential for other universes that have their own particles.

Dialectics on its own isn’t inherently materialistic just as yellowness on its own doesn’t inherently taste like lemon. When you combine the two you get dialectical materialism. This way you can treat mathematics as an actual science while still acknowledging its contradictions.

So, is elliptical, Euclidean or hyperbolic the one true geometry? There is no one true geometry. Different physical notions are modeled by different types of geometry. The surface of the earth is modeled by elliptical geometry, while short-distance navigation is modeled by Euclidean geometry. Eventually the longer the distance you travel the less accurate Euclidean geometry is. Eventually quantity turns to quality and you can no longer use Euclidean geometry, and have to use elliptical geometry.

Does this mean that we are only supplying arbitrary meaning to the symbols used to describe the different types of geometry, as the formalists would say? No. Which mathematical model you would use depends on which physical activity is being performed, it doesn’t matter whether people are doing it or machines or even just the potential for people or machines.

What Does This Have to Do With the Real World?

So what about the more abstract mathematical theories? What about the 248-dimensional Lie group E8. What about those really large sizes of infinity and the continuum hypothesis? Are there any physical objects that can be said to be modeled by those? Yes there are. To see this one should look more closely at the formalist explanation.

The formalists say that mathematics is just symbol manipulation being done in our heads. This is not entirely true, but all mathematic done by humans does involve humans using their heads to manipulate symbols. Those heads and symbols are all part of the physical world. So while the mind games that the formalists talk about are not the only meaning of mathematics, they are a meaning of all mathematics. And mathematics is rational abstraction. And abstractions are ways of representing lots of things in a single concept.

When the alternate geometries were developed there was developed, there was no expectation that the math involved would eventually be used in relativity theory. When abstract algebra was developed, there was no expectation that it would eventually be used in computers. The abstract mathematicians of today don’t necessarily know how their math will be applied, and they don’t necessarily care. However the mere fact that math can be studied, no matter how abstract, means there is the potential, somewhere somehow, for it to be applied.

1 comment:

  1. If i read this correctly you identify with the dialectical materialism view of mathematics. I would dearly like you to expand on this, because I am not getting a good sense of what you mean by it from this article.

    I think the question of 'What is mathematics?' is one of the great philosophical questions. Your point about maths being the science of the abstract is wonderfully expressed. I wish more teachers would have an understanding of this, so that they valued the study of maths, above and beyond the 'real world' examples that they seek in order to make it somehow relevant to students.

    I have been reading John D Barrow's overview of this subject from his book The World Within the World (1988). He puts forward four interpreations of mathematics; Platonism, formalism, Conceptualism, and Intuitionism. Where does dialectical materialsim fit in these options?

    I used to be a Platonist. There is something attarctive about thinking that studying mathematics gives you a view into another world - so to speak. Kind of like thinking alchemy is cool.

    But in leaving that aside, I have not really been able to pin down where I now stand. Which of course makes the question interesting!

    If the basis of maths was in counting the goat herd, then we seem to get it from our experience of the material world. An experience that our evolved brains have developed in order to survive in the world. In a similar way, our brains impose the concept of colour. Colour being an abstract concept, and a useful one, survival wise, to evolve.

    So in this line of thinking we get (some) maths from our experience of the universe. All fine so far. But how do we explain the more abstract notions of mathematics? Does our reasoning sense come from a need to have the ability to predict the future - or where a an object of prey is likely to be next in order for us to catch it - also stem from our evolutionary history. And could this faculty of reason then be the source of the more abstract concepts of maths, such as imaginary numbers and the Mandelbrot set?

    The question for me seems to be morphing into the question of - How do we get our ability to think in abstract terms?

    Anyways, thanks for your blog. If you do have the time to elaborate further on dialectical materialism, or to point me towards some further reading I would appreciate it.

    Regards from Australia.


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