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What is Mathematics?
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"A mathematician is a blind man in a dark room looking for a black cat which isn't there." - Charles Darwin
This is the first of at least three chapters discussing the philosophy of mathematics. The philosophy of mathematics is at a crossroads right now. There are two major camps building. On one side are the old school mathematicians who see mathematics as a foundation of science. On the other side is a small but growing group of scholars made up of cognitive psychologists, linguists, and neural biologists (and some mathematicians as well) who see mathematics as a function of the brain.
While a comprehensive philosophical definition of mathematics is not really possible, philosophers have been working on it for millennia without success, these new neurobiological/ linguistic/ cognitive theories show promise in helping us understand how we learn and understand math. If we better understood how the brain handles math, we could find approaches to teach math more effectively.
I have two books in my possession which I plan on reading and reviewing in future chapters of this topic. One is What Counts: How Every Brain is Hardwired for Math by Brian Butterworth which is the first book I found (published 1999) to look into the connection between math and cognitive psychology. The other is by Keith Devlin whose book The Math Gene: How Mathematical Thinking Evolved and Why Numbers are Like Gossip (published 2000) looks to be at the very least a fun book to read. A third book I have found, but I have no plans yet to review is Where Mathematics Comes From: How the Embodied Mind Brings Mathematics into Being by George Lakoff, Rafael Núńez which from what I have seen in browsing at the bookstore and reading reviews may be the most comprehensive and controversial of the bunch.
But, before we explore these new brain-math theories, it would be helpful to learn about the classic math philosophies. The most popular are called Formalism, Inventionism, Intuitionism, and Platonism. All of which are covered in detail in John D. Barrow's 1994 book Pi in the Sky, which I shall now review:
The May 7, 2001 issue of Newsweek had as its cover story an article about how religious experiences may be explained as a function of the brain. At the same time scientists are looking into the possibility that math is also a basic function of the brain, as I mentioned in the introduction.
Coming full circle is John D. Barrow's 1994 book Pi in the Sky: Counting, Thinking, and Being which has as one of its primary themes the idea that mathematics has much in common with religion. Here is the intriguing first paragraph:
A mystery lurks beneath the magic carpet of science, something that scientists have not been telling, something too shocking to mention except in rather esoterically refined circles: that at the root of the success of twentieth-century science there lies a deeply 'religious' belief -- a belief in an unseen and perfect transcendental world that controls us in an unexplained way, yet upon which we seem to exert no influence whatsoever. What this world is, where it is, and what it is to us is what this book is about. (Op. cit. pg 1)
As Carl Sagan said, "incredible claims require incredible evidence," and I will cut to the chase and say that Barrow fails to prove his initial thesis. Math is not really a religion, nor is it built on a religious foundation. What Barrow does provide are numerous ideas that will inspire deep thoughts on the paradoxical nature of mathematics and its place in human knowledge.
For the next 100 pages, Barrow gives a brief account of the early history of mathematics. He begins at a time when math really was a religion, the time of Pythagoras.
Pythagoras believed that 'all is number' and that the world behaved mathematically. While many of Pythagoras's ideas have proven faulty, his cult of maqhmatikoz (matters learned) contributed the first three books of Euclid's Elements including the first known formal proof of the Pythagorean theorem (the theorem actually was known to the Egyptians and Babylonians way before Pythagoras). Their idea that everything has as its basis in number influenced Greek philosophy, especially Socrates and his student Plato.
At least three dialogues of Plato discuss the mystical nature of mathematics,
Meno, The Republic, and The Laws.
Click
here for an entertaining explanation of Plato's philosophy.
It should be noted that today very few mathematician takes Plato's views literally*, but it has become a powerful metaphor. It is why we label new theories in mathematics as "discoveries". Math is metaphorically described as existing in an independent world we can explore with our minds. This is mathematical Platonism, and it has existed longer than Christianity. So why Platonism has few true believers today, it is so ingrained in how we think about math that we cannot help thinking of new theories as discoveries instead of what they probably are: "creations".
That mathematics is often described in metaphorical terms is not the problem. The real problem is that mathematics does not have a good description of what it really is except in metaphorical terms. Barrow describes it this way:
Just as our picture of the most elementary particles of matter as little billiard balls, or atoms as mini solar systems, breaks down if pushed far enough, so our most sophisticated scientific description in terms of particles, fields, or strings may well break down as well if pushed too far.
Mathematics is also seen by many as an analogy. But, it is implicitly assumed to be the analogy that never breaks down. Our experience of the world has failed to reveal any physical phenomenon that cannot be described mathematically. That is not to say there are not things for which a description is wholly inappropriate or pointless. Rather, there has yet to be found any system in Nature so unusual that it cannot be fitted into one of the strait-jackets that mathematics provides.
This state of affairs leads us to the overwhelming question: Is mathematics just an analogy or is it the real stuff of which the physical realities are but particular reflections?
This leads us to our first glimpse of the mysterious foundation of modern science. It uses and trusts the language of mathematics as an infallible guide to the way the world works without a satisfactory understanding of what mathematics actually is and why the world dances to a mathematical tune. (pp. 21-22)
The Central Problem of the History of Numbers
Approximately a third of the book is dedicated to the study of the history of numbers. Why? Because the best way to find out what mathematics is, is to look at where mathematics comes from. This is the primary source of contention in the debate over the philosophy of math. The history of mathematics seems our best chance at answering the following questions (from the book):
The new brain-math theories make a big deal of the fact that so many diverse and independent civilizations came up with symbolic numbers and arithmetic principles on their own. It is hinted that this is not possible unless math is something that comes natural to the human brain. Their answer to at least the first four questions would be "yes".
Barrow's own conclusions are not so simple, "Whilst we have seen that rudimentary counting systems were almost universal in the ancient world, they were not completely so." Furthermore he points out, "Whilst not every society could count, they could all speak. Language predates the origin of counting and numeracy" (pg. 102) This implies a notion of distinction for counting and number which developed solely as a feature of language. If you develop words for hot and cold, boy or girl, you are bound to develop words for one and two.
Barrow contradicts the brain-math theories all together by pointing out:
The existence of a natural human propensity for counting would lead one to expect that counting would spring up independently all over the globe. But we have seen that the rival picture, in which distinctive counting practices develop once in an advanced cultural centre and then spread to lesser societies, has much evidence to support it. (pg. 104)
Another contradictory point has to do with the development of higher math skills, which unlike counting and adding, is extremely rare. Barrow points out:
Having a notion of quantity is a long way from the intricate abstract reasoning that today goes by the name of mathematics. Thousands of years passed in the ancient world with comparatively little progress in mathematics. [...] It is not good enough to possess the notion of quantity. One must develop an efficient method of recording numbers... more crucially still, the adaptation of a place value system with a symbol for zero was a watershed. A good notation permits an efficient extension to the ideas of fractions and the operations of multiplication and division.[...] Again, we find these discoveries are deep and difficult; almost no one made them. (pp. 103-104)
Even more damaging evidence can be found in the fact that most ancient cultures lacked the ability to abstract numbers. Most ancient counting systems varied in vocabulary depending on what was counted. We do a little of that today with ordinals and other counting methods. Besides one, two, three, four we sometimes use first, second, third, fourth or mono (or uni), bi, tri, quad, or whole, half, third, quarter, or primary, secondary, tertiary, quaternary or singular, dual, trio, quad or solo, duet, trio, quartet, etc. (it is interesting to note that each of these English counting systems have different languages of origins with French, German, Latin and Greek all represented in one form or another) Many ancient languages are worse, having separate counting words for people and cattle, and time, etc. This made it hard to abstract oneness, twoness, etc. Without the ability to see numbers as abstract entities regardless of what is being counted, mathematical insight of any kind is very difficult at best.
As a result, most ancient counting systems are simply incapable of being used to describe higher level math. Algebra was not possible until the advent of zero as a place holder in a numbering system. Numbering systems have been around thousands of years before India came up with a place value system of counting needed for higher levels of mathematics. In other words, it was a miracle that high level math was ever developed. Looked at this way, there is a sense that mathematical truth exists independently of the brain, and that there are mathematical truths that the brain cannot comprehend. (I previously reviewed a book called Uncommon Sense by Alan Cromer that says similar things about the development of science.)
It would be negligent of me as a reviewer not to point out that Barrow's history of counting is far from comprehensive, nor are his conclusions solidly proven to the point of being free from alternate interpretation. The books that I will be reviewing later also contain their own telling of the history of numbers, again not comprehensive, with understandably different conclusions. This is the central problem of the history of numbers, there simply is no singular way to interpret the evidence.
Having made these conclusions about the history of math and counting, Barrow proceeds to tackle the philosophy of math which covers the last two thirds of the book. This is divided into four chapters, each covering a major philosophy of mathematics: Formalism, Inventionism, Intuitionism, and Platonism. After a definition and brief history, Barrow proceeds to punch holes in each philosophy. None of the schools of thought come out unscathed, including his obvious favorite Platonism. Here is a brief summary of each:
Formalism
Formalism is the view that mathematical statements are not about anything, but are rather to be regarded as meaningless marks. The formalists are interested in the rules that govern how these marks are manipulated. Mathematics, in other words is the manipulation of symbols. The fact that (a + b) + c = a + (b + c) is simply a rule of the system. The principle protagonist of this philosophy is David Hilbert.
Unfortunately, this philosophy was proven unfit by Gödel's incompleteness theorem. I described this conflict between Hilbert and Gödel in my Greatest Math Mistake of the Century essay. Gödel's theorem suggests that the truth of a mathematical statement cannot merely consist in its proof from a set of axioms. Hence, Formalism has been defeated.
A break off of Formalists school is the Logicists school, championed by Bertrand Russell and Gottlob Frege, who sought to show that are knowledge of mathematical truth was as certain as our knowledge of logical truth. They attempted to define mathematics in the language of logic. Their efforts resulted in some important ideas, such as the relationship between number theory and set theory, but ultimately this enterprise was found faulty as well due to paradoxes such as Russell's Paradox, an important principle of set theory which could not be based on logic.
Inventionism
Inventionism, also sometimes called Constructivism, holds that true mathematical statements are true because we say they are. Mathematicians do not discover mathematics, as the Platonists claim, they invent new mathematics. This is actually pretty close to the brain-math philosophy, so I will not go into detail. However, here are Barrow's arguments against this philosophy:
We have explored the case for regarding mathematics as a human invention, shaped primarily by the structure of the human mind and its particular ways of processing and organizing information, and responsive to the ways of human society and culture. The products of human thinking must necessarily be fallible at some level. On this picture we do not discover mathematics 'out there'; it need not exist in the absence of mathematicians and the form it takes is strongly associated with our own genetic makeup. ...
To pursue the inventionist philosophy is to make mathematical truth dependent upon time and history. We are forced to an anti-Copernican stance which sees mathematical truth changing with the evolution of the human mind. Inventionism is a wonderful philosophy for the arts and humanities where we see the fruits of imaginative subjectivity; their nature and practice contrast so drastically with that of mathematics that the objective element seems to have failed to be adequately incorporated into this view of mathematics. Inventionism fails to provide insight into the fact that Nature is best described by our mental inventions in those areas furthest divorced from every day life and from those events that directly influence our evolutionary history. In the end, one cannot help but feel that humanity is not really clever enough to have 'invented' mathematics. (Ibid pp. 176-177)
In other words, if mathematics was invented, how is it possible that mathematics explains the real world so well? The history of science tends to support this view. For example, Non-Euclidean geometry was discovered in the 19th Century separately by Gauss and Riemann. No one saw any use for these geometries until Einstein based his General Theory of Relativity on them. Real space is Non-Euclidean in shape and operation. If the inventionist view is held, then Non-Euclidean Geometry should have been invented as a result of the General Theory of Relativity not the other way around. This is a central problem in the philosophy of Mathematics: Mathematics explains too well the nature of reality as new scientific discoveries are found.
In truth, I find this argument to be weak. It does not take into consideration the nature of science and scientific progress. I will explain my views in detail in part 2, but let me just say that Inventionism has more credibility than at first glance. When you take into account cultural considerations and scientific discovery and evolution in general, it is easy to see Barrows point, "one cannot help but feel that humanity is not really clever enough to have 'invented' mathematics." But, when you look deeper into how the process of discovery works in the scientific community, you can see direct parallels with the math community, and it turns out humanity is clever enough after all.
Intuitionism
Barrow's chapter on Intuitionism is probably the most interesting in the entire book, despite the fact that intuitionism is probably the weakest of all the mathematical philosophies. The easiest way to define intuitionism is that it is the corollary of logicism. The Logicists want to define mathematics in the language of logic. The intuitionists want to define logic in the language of mathematics.
Barrow credits the school of intuitionism on 19th Century Dutch mathematician Luitzen Brouwer, though other books take the philosophy back to 18th Century German philosopher Immanuel Kant. Barrow defines intuitionism as follows:
Because of [Brouwer's] concern about the uncertain and subjective influence of the mind upon our mental constructions, he sought to found mathematics in a conservative manner as possible, upon the smallest and surest island of those intuitions which he believed we all share. For Brouwer, the island consisted of the natural numbers 1,2,3,... and simple counting processes. From this basis he defined mathematics to be the edifice that can be constructed from them by step-by-step deductions using a finite number of steps. (pg. 185)
If you do not understand this last sentence, neither did I at first. The intuitionists basically are trying to separate mathematics from the fallacies of the human thought process. Sometimes we humans have a tendency to jump to conclusions ahead of step by step logic. The intuitionists are basically against these jumps to conclusions. The problem with ruling out jumping to conclusions, is that you are forced to rule out acceptable means of mathematical proof like reductio ad absurdum, also known as proof by contradiction. Many important mathematical proofs, like the irrationality of the square root of 2 are based on such proofs, so are proofs about the nature of infinity. Brouwer got around this by introducing a three valued logic system: true, false and 'not proven'.
The consequences of following Brouwer's demand that mathematics consist only of those statements that can be constructed in a finite number of steps from the properties of the natural numbers are very great. It produces a mathematics that is far smaller in extent, far more limited in power, and far more predictable than the conventional mathematics which employed a two-valued logic in which every statement was either true or false. The mathematics according to the intuitionists was just a part of the ocean of mathematical truths that were accepted by other mathematicians. Any truth of intuitionism would be a truth of traditional mathematics but not necessarily vice versa. Three-valued logic produces a host of changes to traditional assumptions. Tautologies of two-valued logic need not be tautologies in three-valued logic. An obvious example is the principle of the excluded middle itself (that is, 'a statement is either true or false'). (pg. 186)
Barrow goes on in this chapter to write about the nature of proof, and as I said before this is one of the most interesting sections in the book. Especially entertaining are the stories of Ramanujan, the Indian math protégé who came up with brilliant number theories after reading a study guide to the Cambridge entrance exams, as well as Cantor and his wild theories about infinity, and the computer that proved the four color conjecture. The ideas of what constitutes proof are too wide to be restricted as the intuitionists would have it.
Platonism
Like Sherlock Holmes' saying that once you eliminate the impossible, whatever is left no matter how improbable must be the truth. The shortcomings of Formalism, Inventionism, and Intuitionism lead us to take a new look at Platonism. Barrow explains Platonism this way:
Plato's philosophy of mathematics grew out of his attempts to understand the relationship between particular things and universal concepts. What we see around the world are particular things -- this chair, that chair big chairs, little chairs, and so on. But the quality they share -- let's call it 'chairness' -- presents a dilemma. It is not itself a chair and unlike all chairs we know it cannot be located in some place or at some time. But that lack of a place in space and time does not mean that 'chairness' is an imaginary concept.
When you replace the concept of 'chairness' with the concepts of number like 'threeness', you start to see Plato's point. Three is not a physical object it is a universal concept, like 'chairness'.
Plato's approach to these universals was to regard them as real. In some sense they really exist 'out there'. The totality of his reality consisted of all the particular instances of things together with the universals of which they were examples. Thus the particulars that we witness in the world are each imperfect reflections of a perfect exemplar or 'form'. (pg.25)
The view as pointed out earlier is this: Mathematics exists. It transcends the human creative process, and is out there to be discovered. Pi as the ratio of the circumference of a circle to its diameter is just as true and real here on Earth as it is on the other side of the galaxy. Hence the book's title Pi in the Sky. This is why it is thought that mathematics is the universal language of intelligent creatures everywhere. Yet, Platonism is far from a perfect philosophy of mathematics, and Barrow agrees:
The Platonic picture is one of those ideas that at first seems eminently simple and unencumbered by abstraction, but which is infested with all manner of problems. It is a philosophy of mathematics that is popular amongst physicists, but less so among mathematicians, and is almost universally regarded as an irrelevance by consumers of mathematics like computer scientists, psychologists, or economists. One immediate consequence of the abstract world that it introduces is the downgrading of the meaning of traditional mathematical objects like numbers. Whereas the Pythagoreans saw them as symbols imbued with irreducible meaning, Plato sees them as empty vessels whose significance lies entirely in the relationships they have with other symbols. (pg. 255)
Let us take a long hard look at what the Platonist is asking us to believe. We must have faith in another 'world' stocked with mathematical objects. Where is this world and how do we make contact with it? How is it possible for our mind to have an interaction with the Platonic realm so that our brain state is altered by that experience? (pg.272)
Barrow goes on to discuss Platonic views in detail. The most interesting idea is what Platonist mathematics has to say about Artificial Intelligence (it does not think it is really possible). The final conclusion of Platonism is one of near mysticism. Barrow writes:
We began with a scientific image of the world that was held by many in opposition to a religious view built upon unverifiable beliefs and intuitions about the ultimate nature of things. But we have found that at the roots of the scientific image of the world lies a mathematical foundation that is itself ultimately religious. All our surest statements about the nature of the world are mathematical statements, yet we do not know what mathematics "is" ... and so we find that we have adapted a religion strikingly similar to many traditional faiths. Change "mathematics" to "God" and little else might seem to change. The problem of human contact with some spiritual realm, of timelessness, of our inability to capture all with language and symbol -- all have their counterparts in the quest for the nature of Platonic mathematics. (pg. 296-297)
Ultimately, Platonism also is just as problematic as Formalism, Inventionism and Intuitionism, because of its reliance on the existence of an immaterial world. That math should have a mystical nature is a curiosity we are naturally attracted to, but ultimately does not really matter. Platonism can think of a mathematical world as an actual reality or as a product of our collective imaginations. If it is a reality then our ability to negotiate Platonic realms is limited to what we can know, if it is a product of our collective imaginations then mathematics is back to an invention of sorts. True or not our knowledge of mathematics is still limited by our brains.
Do there exist mathematical theorems that our brains could never comprehend? If so, then Platonic mathematical realms may exist, if not then math is a human invention. We may as well ask, "Is there a God?" The answer for or against does not change our relationship to mathematics. Mathematics is something that we as humans can understand as far as we need.
In Part II, I will look at a recent experiment in understanding how a group of individuals can work together to achieve a level of success that far exceeds what any single person is capable of doing. This experiment demonstrates how the math community could "invent" mathematics capable of complimenting science so well.
*Two famous mathematicians who do take Platonism seriously are Kurt Gödel, a colleague of Albert Einstein and one of the most famous mathematicians of the 20th century, and Roger Penrose the current Rouse Ball Professor of Mathematics at Oxford University and a colleague of Stephen Hawking. It is fascinating to speculate that the often publicized debates between Hawking and Penrose are very similar to the ones Einstein and Gödel had privately fifty years earlier.
If you are interested in Platonism, you might want to visit Platonic Realms web site.
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