‘Modern man has his ghosts and spirits too, you know.’
‘Oh, the laws of physics and of logic . . . the number system . . . the principle of algebraic substitution. These are ghosts. We just believe in them so thoroughly they seem real.’
–Robert Pirsig, Zen and the Art of Motorcycle Maintenance
It is a popular position among physicists that mathematics is what ultimately lies behind the universe. When asked for an explanation for the universe, they point to numbers and equations, and furthermore claim that these numbers and equations are the ultimate reality, existing objectively outside the human mind. This view is known as mathematical Platonism, after the Greek philosopher Plato, who argued that the ultimate reality consisted of perfect forms.
The problem we run into with mathematical Platonism is that it is subject to some of the same skepticism that people have about the existence of God, or the gods. How do we know that mathematics exists objectively? We can’t sense mathematics directly; we only know that it is a useful tool for dealing with reality. The fact that math is useful does not prove that it exists independently of human minds. (For an example of this skepticism, see this short video).
Scholars George Lakoff and Rafael Nunez, in their book Where Mathematics Comes From: How the Embodied Mind Brings Mathematics into Being, offer the provocative and fascinating thesis that mathematics consists of metaphors. That is, the abstractions of mathematics are ultimately grounded in conceptual comparisons to concrete human experiences. In the view of Lakoff and Nunez, all human ideas are shaped by our bodily experiences, our senses, and how these senses react to our environment. We try to make sense of events and things by comparing them to our concrete experiences. For example, we conceptualize time as a limited resource (“time is money”); we conceptualize status or mood in terms of space (happy is “up,” while sad is “down”); we personify events and things (“inflation is eating up profits,” “we must declare war on poverty”). Metaphors are so prevalent and taken for granted, that most of the time we don’t even notice them.
Mathematical systems, according to Lakoff and Nunez, are also metaphorical creations of the human mind. Since human beings have common experiences with space, time, and quantities, our mathematical systems are similar. But we do have a choice in the metaphors we use, and that is where the creative aspect of mathematics comes in. In other words, mathematics is grounded in common experiences, but mathematical conceptual systems are creations of the imagination. According to Lakoff and Nunez, confusion and paradoxes arise when we take mathematics literally and don’t recognize the metaphors behind mathematics.
Lakoff and Nunez point to a number of common human activities that subsequently led to the creation of mathematical abstractions. The collection of objects led to the creation of the “counting numbers,” otherwise known as “natural numbers.” The use of containers led to the notion of sets and set theory. The use of measuring tools (such as the ruler or yard stick) led to the creation of the “number line.” The number line in turn was extended to a plane with x and y coordinates (the “Cartesian plane“). Finally, in order to understand motion, mathematicians conceptualized time as space, plotting points in time as if they were points in space — time is not literally the same as space, but it is easier for human beings to measure time if it is plotted on a spatial graph.
Throughout history, while the counting numbers have been widely accepted, there have been controversies over the creation of other types of numbers. One of the reasons for these controversies is the mistaken belief that numbers must be objectively real rather than metaphorical. So the number zero was initially controversial because it made no sense to speak literally of having a collection of zero objects. Negative numbers were even more controversial because it’s impossible to literally have a negative number of objects. But as the usefulness of zero and negative numbers as metaphorical expressions and in performing calculations became clear, these numbers became accepted as “real” numbers.
The metaphor of the measuring stick/number line, according to Lakoff and Nunez, has been responsible for even more controversy and confusion. The basic problem is that a line is a continuous object, not a collection of objects. If one makes an imaginative metaphorical leap and envisions the line as a collection of objects known as segments or points, that is very useful for measuring the line, but a line is not literally a collection of segments or points that correspond to objectively existing numbers.
If you draw three points on a piece of paper, the sum of the collection of points clearly corresponds to the number three, and only the number three. But if you draw a line on a piece of paper, how many numbers does it have? Where do those numbers go? The answer is up to you, depending on what you hope to measure and how much precision you want. The only requirement is that the numbers are in order and the length of the segments is consistently defined. You can put zero on the left side of the line, the right side of the line, or in the middle. You can use negative numbers or not use negative numbers. The length of the segments can be whatever you want, as long as the definitions of segment length are consistent.
The number line is a great mental tool, but it does not objectively exist, outside of the human mind. Neglecting this fact has led to paradoxes that confounded the ancient Greeks and continue to mystify human beings to this day. The first major problem arose when the Greeks attempted to determine the ratio of the sides of a particular polygon and discovered that the ratio could not be expressed as a ratio of whole numbers, but rather as an infinite, nonrepeating decimal. For example, a right triangle with two shorter sides of length 1 would, according to the Pythagorean theorem, have a hypotenuse length equivalent to the square root of 2, which is an infinite decimal: 1.41421356. . . This scandalized the ancient Greeks at first, because many of them had a religious devotion to the idea that whole numbers existed objectively and were the ultimate basis of reality. Nevertheless, over time the Greeks eventually accepted the so-called “irrational numbers.”
Perhaps the most famous irrational number is pi, the measure of the ratio between the circumference of a circle and its diameter: 3.14159265. . . The fact that pi is an infinite decimal fascinates people to no end, and scientists have calculated the value of pi to over 13 trillion digits. But the digital representation of pi has no objective existence — it is simply a creation of the human imagination based on the metaphor of the measuring stick / number line. There’s no reason to be surprised or amazed that the ratio of the circumference of a circle to its diameter is an infinite decimal; lines are continuous objects, and expressing lines as being composed of discrete objects known as segments is bound to lead to difficulties eventually. Moreover, pi is not necessary for the existence of circles. Even children are perfectly capable of drawing circles without knowing the value of pi. If children can draw circles without knowing the value of pi, why should the universe need to know the value of pi? Pi is simply a mental tool that human beings created to understand the ratio of certain line lengths by imposing a conceptual framework of discrete segments on a continuous quantity. Benjamin Lee Buckley, in his book The Continuity Debate, underscores this point, noting that one can use discrete tools for measuring continuity, but that truly continuous quantities are not really composed of discrete objects.
It is true that mathematicians have designated pi and other irrational numbers as “real” numbers, but the reality of the existence of pi outside the human mind is doubtful. An infinitely precise pi implies infinitely precise measurement, but there are limits to how precise one can be in reality, even assuming absolutely perfect measuring instruments. Although pi has been calculated to over 13 trillion digits, it is estimated that only 39 digits are needed to calculate the volume of the known universe to the precision of one atom! Furthermore, the Planck length is the smallest measurable length in the universe. Although quite small, the Planck length sets a definite limit on how precise pi can be in reality. At some point, depending on the size of the circle one creates, the extra digits in pi are simply meaningless.
Undoubtedly, the number line is an excellent mental tool. If we had perfect vision, perfect memory, and perfect eye-hand coordination, we wouldn’t need to divide lines into segments and count how many segments there are. But our vision is imperfect, our memories fallible, and our eye-hand coordination is imperfect. That is why we need to use versions of the number line to measure things. But we need to recognize that we are creating and imposing a conceptual tool on reality. This tool is metaphorical and, while originating in human experience, it is not reality itself.
Lakoff and Nunez point to other examples of metaphorical expressions in mathematics, such as the concept of infinity. Mathematicians discuss the infinitely large, the infinitely small, and functions in calculus that come infinitely close to some designated limit. But Lakoff and Nunez point out that the notion of actual (literal) infinity, as opposed to potential infinity, has been extremely problematic, because calculating or counting infinity is inherently an endless process. Lakoff and Nunez argue that envisioning infinity as a thing, or the result of a completed process, is inherently metaphorical, not literal. If you’ve ever heard children use the phrase “infinity plus one!” in their taunts, you can see some of the difficulties with envisioning infinity as a thing, because one can simply take the allegedly completed process and start it again. Oddly, even professional mathematicians don’t agree on the question of whether “infinity plus one” is a meaningful statement. Traditional mathematics says that infinity plus one is still infinity, but there are more recent number systems in which infinity plus one is meaningful. (For a discussion of how different systems of mathematics arrive at different answers to the same question, see this post.)
Nevertheless, many mathematicians and physicists fervently reject the idea that mathematics comes from the human mind. If mathematics is useful for explaining and predicting real world events, they argue, then mathematics must exist in objective reality, independent of human minds. But why is it important for mathematics to exist objectively? Isn’t it enough that mathematics is a useful mental tool for describing reality? Besides, if all the mathematicians in the world stopped all their current work and devoted themselves entirely to proving the objective existence of mathematical objects, I doubt that they would succeed, and mathematical knowledge would simply stop progressing.