Mathematics is a strange and wonderful mystery, that grows more strange and wonderful the more we learn about it. Our first introduction to mathematics in school is through arithmetic, a pragmatic system of accounting and quantification, first developed for applications such as counting sheep, for recording debts or payments, for counting elapsed time, to quantify distances and areas for travel and surveying, and to predict the motions of the heavenly bodies, and thereby the seasons. These are of course but a tiny sample of the innumerable other applications where arithmetic comes in handy. The remarkable thing about math is that all of these varied and diverse problem domains can be addressed using a single general purpose conceptual tool. All these diverse problem domains are quantified and calculated using the same system of numbers, although the details of that system can vary from one culture to the next.
If we are to trace the origins of mathematics, we must focus on the part that is common to all number systems across all cultures and historical periods, because the common root of mathematics is likely to give a good indication as to that component of mathematics which is innate, a genetic heritage, as opposed to a cultural heritage passed down from generation to generation. Lakoff & Núñez (2000) trace the origin of mathematics to an instinctive form of counting called subitizing, in which you can instantaneously ‘see’ the number of things if the number is small, less than about five to seven items. This is the way you recognize the number of dots on dice.
You don’t have to count the individual dots, you can ‘see’ the numerical quantity immediately, in one glance, even if you had never been taught their names. Surely this is the origin of mathematics, the ability to see a difference in quantity, even if those numbers have no name, and even if that quantification only works for small numbers. Human cultural mathematics takes this innate numerical instinct and extends it to express much larger numerical quantities than can be counted by subitizing. But the cultural extension to innate math adheres to the same basic rules that govern the first few numbers, and many of the extensions to the natural numbers, such as the concept of zero, of negative numbers, fractions and rational numbers, irrational and imaginary numbers, are such a natural extension by the same rules, that they almost seem obvious after the fact, as if they had been there all along waiting to be discovered, as soon as the problems that they resolve are first encountered.
But the concept of numbers runs much deeper than a simple one, two, three. Before you can even think of counting, you have to understand the concept of countable things. Whether you are counting people, coins, or pebbles on a beach, you must decide on the set of things you want to count, and what requirements you should establish to qualify for inclusion in that set. How big can a pebble be before it counts as a rock instead of a pebble, and how small before it counts as a grain of gravel or sand? The answer depends on why you are counting them, that requires a fine mathematical judgment. Counting requires that all the members of the set can be considered equivalent, interchangable, they each count for exactly one. This is so natural and obvious an assumption (to us humans) that we don’t even realize that we are making it. But it is an artifact of the human mind, not of the natural world, that things come in identical countable units. In reality pebbles are not at all identical, each one has its own unique size and shape and color, and the distinctions between rock and pebble and gravel are overlapping and indefinite. The fact that we can make ourselves see a set of pebbles as identical units is itself a conceptual feat that is a prerequisite to learning to count them.
Lakoff explains the foundations of mathematical thinking with the concept of schemas. For example, before counting pebbles on the beach, you must first see the beach, and see the pebbles on it, and you must select which pebbles you wish to count based on some criteria that determine which pebbles belong in your countable set. If counting to merely demonstrate the concept, you might choose a dozen or so pebbles of a certain size or color, and set them apart from the rest, drawing an encompassing circle with your mind, distinguishing these pebbles as the set to be counted. For the process of counting, you must imagine another group, or imaginary circle, into which you either move the pebbles one by one, or perhaps you imagine that circle to engulf the pebbles one by one as you are counting, moving them from the set of pebbles to be counted, to the set of pebbles already counted, while keeping a tally of the total by the familiar sequence 1, 2, 3, … . All of the thought before the actual counting begins, is wordless thought that we did not have to learn in school, we knew instinctively how to judge whether this pile of pebbles is greater in number than that one. We didn’t need to know how to count for that. We don’t need to know numbers for forming a schema in our mind. However it is impossible to do any real mathematics without first forming a schema in our mind. The schema is the framework that gives meaning to the math. It is what poses the question in the first place, and what interprets the meaning of the results at the end. That is the real mathematical thinking beyond mere arithmetic, the framing of the problem to be solved, the selection of the algorithm to be used to solve it, and understanding the significance of the results after the counting is done. The counting itself is the simplest part of the problem – so simple that even a stupid computer can do it. But conceptualizing the situation and seeing the schema in it, is in fact the real mathematical part of the task. The rest is merely arithmetic.
Another schema that Lakoff cites as a guiding metaphor for numbers is the concept of pacing off a distance with steps of equal size. This relates to the idea of the number line, numerical value depicted as spatial extension in one dimension. This opens the possibility for negative numbers, when pacing backward beyond the origin, and it also presents the unitary interval as a continuum, infinitely sub-divisible into fractional sub-intervals. Again, the actual counting of the paces, the choice of names to call the numbers, or whether to record them in binary or hexadecimal, or using Arabic or Roman numerals, is trivial compared to the real act of mathematical thought, which is seeing the land that requires surveying, knowing which dimensions to measure, making the measurements, and then understanding the significance of the paced-off quantities. The greater part of mathematical thought is not the simple arithmetic that we first learn in school as math. Real mathematical thought is an embodied process, one that cannot be meaningfully separated from our direct experience of our body located in the world.
Lakoff is right in identifying the origin of math in practical hands-on problem solving and basic conceptualization of reality. But the problem with identifying schemas as the origin of mathematics is that the notion of a schema is very vague and abstracted. We are all familiar with the experience of forming and using a schema, but we have no idea how we are actually doing it, or what a schema even is. Lakoff leaves the story of the origins of mathematics hanging at this point. This is the frontier of the terra incognita at the root of mathematics. From here on down we have no idea how we do the most primal mathematics, constructing for ourselves a schema of the world. What does that even mean?
The profound difference between schematic conceptualization and arithmetical calculation has come into sharper contrast in this era of the digital computer, a mechanism that is capable of millions of arithmetical operations per second, and of computational algorithms of fantastic complexity, with virtually perfect reliability and reproducibility, far beyond anything we can accomplish with pencil and paper. The computer is a very useful tool for the mathematician, to plot his mathematical thoughts and make them visible on the screen, And yet the digital computer is totally incapable of even the most primitive mathematical thought. The computer is completely incapable of conceptualizing a schema, or understanding the significance of the quantities that it calculates. It seems that there are two starkly contrasting aspects of math, one which is thoroughly understood, which can be performed by stupid machines even better than by humans, the other that we all do unconsciously and instinctively virtually every waking moment, but have no idea how we do it or even what it is we are doing. The focus of this book is on that other aspect of mathematical thought, what it is, and how it works, and ultimately, how we could build an artificial mind that is capable of real mathematical thought, and with it, a spatial consciousness and sense of self-existence like our own.
Continued Chapter 2: The Schema as a Mental Image