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What is Mathematics?
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This is the final (for now) chapter discussing the philosophy of mathematics. In part one, I reviewed Pi in the Sky which is an overview of Math Philosophy which favors Platonism. In part two, I looked at new ideas about group think vs. the power of the individual mind. We eliminated the "mystery" of how math came to be, but we still did not answer the fundamental question, "What is Mathematics?"
In this chapter, we look in depth at the "brain math" theories, specifically how does the brain do mathematics. Basically, the brain does mathematics the same way the brain handles language. Which leads to the inevitable question: How come we can't do math as easily as we speak?
The linguist in me knows one answer: spoken language uses simpler grammars than mathematical grammars. Cognitive Psychologists will tell you another: mathematics generally has a higher level of abstraction than everyday language. (If you did not understand those last two sentences, don't worry, I will explain later.) There is, however, a much more surprising answer: We can do math as easily as we speak! Once we recognize what math really is, it becomes clear that many kinds of math come very naturally. Unfortunately, what we normally think of math (arithmetic and algebra) are not one of those topics that come naturally.
The point is that studying how our brains handle mathematics not only will help us understand what mathematics is, it also will help us find better ways of teaching mathematics.
Let's start this essay with a couple of logical problems.
Problem 1: There are four cards laying on a table. Each card has a number on one side and a letter on the other. Face up you see these symbols:
E K 4 7 The cards are printed according to the rule: If a card has a vowel on one side, it has an even number on the other side. Which cards do you have to turn over to be sure that all the cards satisfy the rule?
Got an answer? Are you sure it is right? Before I reveal the answer, let me give you another problem:
Problem 2: You are in charge of a party where there are young people. Some are drinking alcohol, some are drinking soft drinks. Some are old enough to drink some are under age. You are responsible for ensuring that the drinking laws are not broken, so you ask that they have their ID's on the table.
At one table there are four young people who may or may not be legal drinking age. One has a beer, and another has a coke, but their ID's are face down so you cannot see their ages. The other two have their ID's up, one is a legal drinking age, the other is under age, but both are drinking a clear fizzy liquid that may be either 7-up or vodka tonic.
Which ID's and drinks do you need to check to make sure everyone is legal?
The answer to both questions are the same, you check the first and the last. The two problems are in fact are identical with these substitutions: vowel = alcohol, consonant = soft drink, even number = legal drinking age, odd number = too young to drink. But, if you are like me (and most people) you probably found problem 2 much easier to answer than problem 1.
Why? Because the first problem is a higher level of abstraction than the second problem. It is similar to another study I saw in which two word problems were given to kindergarten and first graders:
There are nine birds and five worms. How many more birds are there than worms?
There are nine birds and five worms. How many birds do not get worms?
This was a study into how the difficulty of word problems has to do with how well they are written. It should come as no surprise that the students got the second one right more often then the first. The biggest problem people have with word problems is that they are not written very well, which makes me question whether or not learning to solve word problems does anything to help students to apply math to the real world. That is a question for another time.
In this essay we will be exploring the connection between language and mathematics. A good book on this topic came out just last year called The Math Gene by Keith Devlin. The paperback version came out in June. What follows is a brief summary of the main ideas found in this book. I will not go into detailed arguments that support these ideas, you will just have to read the book.
First, a quick review. This is not a book written only for math people. People who are interested in the current problems of math education would benefit as well. This is not a scholarly work, it is easy to read, without math, psychology, or educational jargon. Keith Devlin explains his ideas easily, and occasionally with humor, without getting too technical. On with the overview:
How the brain does math
After an introductory chapter, Devlin has two chapters devoted to what psychologists know about how our brain does math. It follows that old genetics vs. environment argument; some of our math ability comes naturally, though the majority is learned.
Chapter 2 is devoted to fairly new ideas about our natural ability to do basic math such as counting. It seems from birth we have the ability to understand the difference between one, two, three, and many. Psychologists have confirmed this through rather clever testing of infants as young as four weeks old.
For example, one test involved showing babies a series of puppets going behind a curtain. The testers show the babies a couple of puppets going one at a time behind a curtain. The curtain is then lifted. If there are one or three puppets revealed behind the curtain, the babies will stare longer at the puppets than if there are only two puppets, which is what the baby expects. Babies stare longer at things that do not make sense to them, so when they see two puppets go behind a curtain, they expect two puppets when the curtain is dropped.
A basic inherent "number sense" is not just a human trait, it has been confirmed in mammals and birds as well. There is an old story about a bird that was nesting in a tower, and the owners of the tower wanted to get rid of it. When they sent someone into the tower to capture the bird, the bird would leave to a nearby tree and would not return to the tower until the man left. So they came up with a clever plan of sending two men into the tower, then soon after one man left, but the bird stayed in the tree and would not return to the tower until it saw both men leave: it saw two men enter, and it waited until both men were gone. They tried again with three, same thing. They tried with four, same thing. Finally, it got confused with five men entering the tower and four men leaving, the fifth man was able to capture the bird.
I can attest to the fact that this number sense also exists in cats. I have a cat, and sometimes I give her some kitty treats just before I leave (this distracts her so I can go out without her darting outside). If I toss three treats on the floor with my cat watching, she will systematically go after all three treats. If I toss five, she will get three easily, then she has to sniff around and search for the others; she knows there are more than three. If she finds one, she may stop looking, because she lost track of how many I threw.
So the number sense is a naturally occurring ability of knowing one, two, three, and many. It is an instinct developed for survival. Many primitive human languages actually only have words for one, two, three, and many. Remnants of this can be found in English, like the ordinal numbers first, second, and third. Later ordinals fourth through twentieth all have identical endings, which makes the first three ordinals unique.
Language ability and math ability
As language reflects the number sense, it is necessary to anything higher in mathematics. That is to say, anything beyond one, two, three, and many requires language and counting ability to understand. This is the focus of chapter three of the book: the close ties between the brains ability to handle language and the brains ability to do higher math.
Devlin gives some interesting examples of this connection. He relates the story of meeting a math wizard, one of those people who can do large calculations in his head. Before the demonstration, he asked that the air conditioning be turned off, because the noise it makes interfered with his ability. He does the lightning calculations by hearing them in his head, and certain kinds of noises hampered this process. Devlin relates:
We all use the human ability to remember a spoken linguistic pattern when we learn our multiplication table. We learn by reciting the table over and over. Even today, forty-five years after I "learned my tables", I still recall the product of any two single digit numbers by reciting that part of the table in my head. I remember the sound of the number words spoken, not the numbers themselves. Indeed, I believe the pattern I hear in my head is precisely the one I learned when I was seven years old! (pg. 59)
Devlin is right on the money. I must confess that whenever I multiply three by any number greater than five, in my head I sing the chorus to "Three is a magic number" from Multiplication Rock.
More evidence can be found studying bilingual learners. Mathematical ability seems to be tied to what language you learn it in. Devlin mentions a study done with bilingual Russian-English speakers. They were taught various mathematical concepts, some in English, some in Russian. They were then given a timed test of these concepts, some of the questions in English, some in Russian. The results were that if they were given a concept in one language, and tested on it in another, it took them longer to do the question because they had to translate the concept in their head.
Having once been fluent in Spanish, I can attest to this fact. When you are fluent in a second language, you develop the ability to think in that language. If some asks you a simple question like "Como esta usted?", you can answer "Muy bien, gracias" without doing any translating into English. Math concepts don't work that way. If someone asks me "Cual es seis por siete?", I cannot come up with the answer "Cuarenta y dos" without thinking in my mind "six times seven is forty two" in English.
While Devlin does not get into this controversy, I could not help thinking that Bilingual Education, at least as it applies to math, may be a major disservice to students. For those who do not know, Bilingual Education is the practice of teaching native foreign students basic topics (math, reading, social studies, etc.) in their native language while they learn English as a second language. If we do better in math when we do it in the language we are taught in, then Bilingual Education students will struggle with math in the English classroom. Just a thought.
One last interesting point about math and language. Devlin points out that some languages are more suitable to mathematical thinking than others. Chinese is maybe the best language for mastering math:
Doing arithmetic, and in particular learning multiplication tables, is simply easier for Chinese and Japanese children, because their number words are much shorter and simpler – generally a single, short syllable such as the Chinese si for 4 and qi for 7.
The grammatical rules for building up number words in Chinese and Japanese are also much easier than in English or other European languages. For instance, the Chinese rule for making words for numbers past ten is simple: 11 is ten one, 12 is ten two, 13 is ten three, and so on, up to two ten for 20, two ten one for 21 two ten two for 22, etc. Think how much more complicated is the English system. (It’s even worse in French and German, with their quatre-vingt-dix-sept for 97 and vierundfünfzig for 54.) A recent study by Kevin Miller showed that language differences cause English speaking children to lag a whole year behind their Chinese counterparts in learning to count. By the age of four, Chinese children can generally count up to 40 American children of the same age can barely get to 15, and it takes them another year to get to 40 How do we know the difference is due to language? Simple. The children in the two countries show no age difference in their ability to count from 1 to 12. Differences appear only when the American children start to encounter the various special rules for forming number words. The Chinese children, meanwhile, simply keep applying the same ones that worked for 1 to 12. (American children often apply the same rules, but they find they have made a mistake when they try to use words like twenty-ten and twenty-eleven.)
In addition to being easier to learn, the Chinese number word system also makes elementary arithmetic easier, because the language rules closely follow the base-10 structure of the Arabic system. A Chinese pupil can see from the linguistic structure that the number "two ten five" (i.e., 25) consists of two 10s and one 5). An American pupil has to remember that "twenty" represents two 10s, and hence that "twenty-five" represents two 10s and one 5. (Pg. 65)
This is a pretty good explanation why Asian students are better at math than American and European students; it is easier for them.
What is Mathematics?
So if math is closely tied to language, but at the same time simple math is instinctual, then what came first: math ability or language ability? It seems obvious that we need to speak in order to do math, but the existence of a "number sense" seems to contradict that. Devlin's own answer is that math came first, but before he explains how this is possible, we have to answer that nagging question that we have been struggling with for three months, namely What is Mathematics?
Devlin's short one sentence answer is that mathematics is the science of patterns. He attributes the definition to W.W. Sawyer's 1955 book Prelude to Mathematics:
For the purposes of this book we may say , "Mathematics is the classification and study of all possible patterns." Pattern is here used in a way that not everyone may agree with. It is to be understood in a very wide sense, to cover almost any kind of regularity that can be recognized by the mind. Life, and certainly intellectual life, is only possible because there are certain regularities in the world. A bird recognizes the black and yellow bands in a wasp; man recognizes the growth of a plant follows the sowing of a seed. In each case a mind is aware of the pattern (Sawyer, Prelude to Mathematics, pg. 12)
When we think of what a "pattern" is, we usually thinking wallpaper or floor tiles, but as it refers to mathematics we are referring to a property, a category, a type, or a kind of something. Pattern here refers to relationships between things, not the things themselves.
This is a very broad definition of mathematics, but a fairly accurate one. Every topic of mathematics has to do with patterns or relationships, though the reverse is probably not true: there are patterns and relationships which are not mathematical (though they certainly fall into some branch of science). This is also a good explanation of why mathematics is "the Queen of Science", because no scientific discipline is complete without a study of patterns of behavior and interactive relationships of whatever is being studied, and most of these patterns can be expressed in a mathematical language.
Devlin gives more specific examples that are interesting, but which I have no time to go into here. One of the great things about a book like this is I now have a few essay topics to follow up with in the future.
On the Origin of Mathematics
As I pointed out in part 1, all books dealing with the philosophy of mathematics inevitably give a history lesson to support this theory. This book is no exception, but where Pi in the Sky started with primitive counting methods that existed 10,000 years ago, The Math Gene goes back to Homo Erectus, some 3.5 million years ago. Devlin does this to support his controversial theory that mathematical ability may have developed prior to linguistic ability.
While this theory is intriguing, I am not convinced it is true. Devlin himself acknowledges the fact that there is no way to prove (or disprove) the theory's accuracy, thus it is purely an academic argument, a philosophical idea, with no hard facts to back it up. Lack of evidence aside he makes a convincing case. Here is a terribly simplified summary of the theory (which in fact covers the last half of the book):
1. Language follows a rather complex pattern which no primate can duplicate. (Some may argue that some apes have demonstrated sign language or picture languages, in truth apes have only demonstrated an ability for "protolanguages", which while an impressive feat by itself, it is not as complicated as all human languages.) Strangely enough, and this is a prominent theory of linguistics, all human languages have virtually identical grammars, which hints at some common origin at some unknown part of our development. Devlin discusses this point at length, because it is necessary to understand his theory.
2. Early humans did actually have some basic skills necessary for mathematical development. From the book:
We have already seen that many creatures have a number sense and that this sense yields obvious survival benefits, ranging from recognizing which tree has the most fruit to knowing whether your group is outnumbered in a potential confrontation.
In addition to number sense, living in trees, with all that swinging from branch to branch, demands a good three-dimensional spatial sense, and survival on the open plains requires a two-dimensional spatial sense, including the ability to judge distance. I doubt that anyone would call such abilities geometric, but they are necessary prerequisites – the first steps, if you will – for geometric ability, the beginnings of a ”geometry sense” analogous to number sense.
Another necessary component of mathematical thinking is an awareness of cause and effect Since contemporary primates all appear to have such a capacity, it is reasonable to suppose that their (and our) ancestors in the forests and on the savanna likewise realized – if only in a restricted sense – that one thing can cause another.
Thus, as much as seven million years ago there were brains having some of the capacities necessary for mathematical thought. This is not to imagine that those early primates possessed anything like ”mathematical ability.” We have no reason to suppose that they had any capacity for reflective or abstract thought at all. Rather, my point is that mathematical thinking, as practiced today, makes use of mental capacities that were developed hundreds of thousands, and in some cases millions, of years ago. Doing mathematics does not require new mental abilities, but rather a novel use of some existing capacities. (Of course, strictly speaking, being able to use those existing capacities in a new way does in fact constitute a ”new mental ability.”) (pp. 179-180)
3. The ability to use language seems to have come out of nowhere about 200,000 years ago, yet our brains were developing steadily for 3,500,000 years prior. How could we evolve an ability to speak before we were capable of speaking?
Far more likely is that one development set the scene for the other: that language arose as a by-product of some more fundamental ability, the one that drove the initial brain growth over a 3,500,000 year period.
So what is that ability? If what I have been saying is correct, the answer should be staring us in the face, so obvious that we fail to see it, so commonplace and accepted that we do not pause to regard it as anything out of the ordinary. Something we simply take for granted. An ability we have had from the moment we were born. An ability that lies behind language.
Any ideas? Here’s another clue. It was when I realized what this key ability must be that I knew everybody has the math gene – the ability to do mathematics.
If you are still puzzling over the matter, then almost certainly you are a victim of a view of the human mind propagated by the famous French philosopher Rene Descartes during the seventeenth century. The modern incarnation of Descartes’ view is that the mind is a computing machine, which thinks by following a progression of discrete logical steps. According to this view, the key to understanding all of the other things people do with their minds – such as recognizing faces or understanding stories – is to express those mental activities in terms of logical rules.
As I argued in my book Goodbye Descartes, the assumption that all mental processes can be captured as logical rules is false. Moreover, it is precisely the falsity of that assumption that explains the failure of the many attempts to program digital computers to recognize scenes, handle natural language, and exhibit artificial intelligence. At the end of Goodbye Descartes, I proposed an alternative view of the human mind as a device for recognizing patterns – visual patterns, aural patterns, linguistic patterns, patterns of activities, patterns of behavior, logical patterns, and many others. Those patterns may be present in the world, or they may be imposed by the human mind as an integral part of its view of the world. (pp. 186-187)
In a way, this is a throw back to the Artificial Intelligence problem I mentioned in part 2. Computers have the reputation to do math very well, and for most kinds of math, this is true. But, the ability to recognize patterns in general, is something computers have difficulty doing.
The important connection you need to make here is that if the human mind is a "device for recognizing patterns", and mathematics is "the science of patterns", then our brain's primary functions are mathematical in nature.
4. If what Devlin implies is true, then the skills needed to learn language are greater than skills needed to learn math. Math skills, at least in the case of pattern recognition, developed necessarily ahead of linguistic skills.
5. If math skills are more basic than language skills, then why are we not all gifted at doing math? Because not all math skills are basic. Devlin makes a list of skills necessary for math ability:
The abilities necessary to develop language are 1 to 5, 6 to 9 came later (in fact, according to Pi in the Sky, they developed much later, only 3000 years ago or so.)
6. The principle ability that stands in our way from becoming good at mathematics is #7, the ability to handle abstraction; 6, 8, and 9 all require 7. As demonstrated at the very beginning of this essay, problems involving lower levels of abstraction are simpler than higher levels of abstraction. Devlin lists four levels of abstraction:
Level 1 abstraction is where there is really no abstraction at all. The objects thought about are all real objects that are perceptually accessible in the immediate environment.
Level 2 abstraction involves real objects that the thinker is familiar with, but which are not perceptually accessible in the immediate environment.
Level 3 abstraction [are] the objects of thought which may be real objects that the individual somehow has learned of but has never actually encountered, or imaginary versions of real objects, or imaginary versions of real objects, or imaginary variants of real objects, or imaginary combinations of real objects.
Level 4 abstraction is where mathematical thought takes place. Mathematical objects are entirely abstract; they have no simple or direct link to the real world, other than being abstracted from the world. (pg. 121)
These levels of abstraction seem to parallel Plato's worlds pointed out in part 1 of this essay: The real world, the objective world, the world of the imagination, the world of mathematical thought.
We can all handle levels 1 to 3, it is only at level 4 that many get confused. Level 3 abstraction is necessary to speak, thus we can all do it. Level 4 abstraction does not come naturally, it has to be learned and practiced. This is what is standing in our way from being good at math.
The key to being able to think mathematically is to push this ability to "fake reality" one step further, into a realm that is purely symbolic -- level 4 abstraction. Mathematicians learn how to live in and reason about a purely symbolic world. (By "symbolic world" I don't mean the algebraic symbols that mathematicians use to write down mathematical ideas and results. Rather, I mean that the objects and circumstances that are the focus of mathematical thought are purely symbolic objects created in the mind.) Although it does not require a different kind of brain to deal with this world, it does involve considerable mental effort. All mathematicians can solve the four cards problem if they put their minds to it. But, like everybody else they find it harder than the party problem. (pg. 123, emphasis mine)
The point being, which I have been trying to stress since April's essay, is that everyone can do math. It just takes a certain amount of effort on the learners part to practice.
How Numbers are like Gossip
The ability to do level 4 abstraction, or what Devlin calls "the math gene" is something our brain developed when it developed language skills about 200,000 years ago, but it lied dormant for about 195,000 of those years because we did not need it. Then the Chinese, Egyptians, and Babylonians all started using and developing math almost simultaneously. "The same cannot be said of language. Human beings found a particularly important use for language the moment it arrived, and they have been using it for that purpose ever since.", says Devlin. "What is that purpose? Answering that question will lead us to the explanation of why the language gene and math gene are one and the same."
The answer is that we gossip. Sociologists have done studies that say at least half of all our conversations are gossip about other people. Devlin explains that this is just human nature, and in fact it plays a vital role in our survival. So, gossip is not necessarily a bad thing.
As abstraction goes, gossip is actually pretty high. It is speculation and imagination about people who are not around us. Sometimes about people we never even met. So gossip exists between level 3 and even level 4 abstraction.
Not only do we acquire and maintain this vast amount of information about others, we can all reason about their lives. We can have opinions on the actions of others, we can understand, explain, and pass judgment on things they do, we can guess or predict what they will do next. Again, not only can we do all of these things, we do do them, and without effort
Now look back at the picture of mathematics I portrayed in Chapters 4 and 5. Mathematics studies the properties of, and relationships between, various objects, either real objects in the world (more accurately, idealized versions of those real objects) or else abstract entities that the mathematician creates.
I think we now have our answer as to what key capacity of the human brain enables some of us to do mathematics, and how (and why) our brain acquired that capacity long before mathematics came onto the scene. We have, in short, identified ”the math gene.” We have discovered the secret that enables mathematicians to be able to do mathematics: a mathematician is someone for whom mathematics is a soap opera.
I should stress that I am not referring to the mathematical community but to mathematics itself. The ”characters” in the mathematical soap opera are not people but mathematical objects – numbers, geometric figures, groups, topological spaces, and so forth. The facts and relationships that are the focus of attention are not births and deaths, marriages, love affairs, and business relationships, but mathematical facts and relationships about mathematical objects. Are objects A and B equal? What is the relationship between objects X and Y? Do all objects of type X have property P? How many objects of type Z are there? These are the kinds of questions that interest the avid devotee of the soap opera we call mathematics. (pp.260-261)
Now of course there are differences between everyday gossip and mathematics, mathematics is more exact and demanding. But, the point again is clear: anyone can learn math if they are willing to make the effort to do so. It requires a certain amount of rote learning and memorization and other things kids hate and teachers do not like to make their kids go through. Education's tendency to move away from rote learning and practice is yet another reason why math skills are declining.
Mathematicians are not born with an ability that no one else possesses. Practically everybody has "the math gene," just as practically everyone is born with two legs. We have it because the features of our brains that enable us to do mathematics are the same ones that allow us to make sense of the world and the people in it: the large repertoire of types that we use to classify the world and the syntactic structure of the human acquired when it became able to think [abstractly]. (pg. 267)
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