Tag Archives: infinity

The Mythos of Mathematics

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‘Modern man has his ghosts and spirits too, you know.’

‘What?’

‘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.

Uncertainty, Debate, and Imprecision in Mathematics

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If you remember anything about the mathematics courses you took in high school, it is that mathematics is the one subject in which there is absolute certainty and precision in all its answers. Unlike history, social science, and the humanities, which offer a variety of interpretations of subject matter, mathematics is unified and absolute.  Two plus two equals four and that is that. If you answer a math problem wrong, there is no sense in arguing a different interpretation with the teacher. Even the “hard sciences,” such as physics, may revise long-established conclusions, as new evidence comes in and new theories are developed. But mathematical truths are seemingly forever. Or are they?

You might not know it, but there has been a revolution in the human understanding of mathematics in the past 150 years that has undermined the belief that mathematics holds the key to absolute truth about the nature of the universe. Even as mathematical knowledge has increased, uncertainty has also increased, and different types of mathematics have been created that have different premises and are incompatible with each other. The value of mathematics remains clear. Mathematics increases our understanding, and science would not be possible without it. But the status of mathematics as a source of precise and infallible truth about reality is less clear.

For over 2000 years, the geometrical conclusions of the Greek mathematician Euclid were regarded as the most certain type of knowledge that could be obtained. Beginning with a small number of axioms, Euclid developed a system of geometry that was astonishing in breadth. The conclusions of Euclid’s geometry were regarded as absolutely certain, being derived from axioms that were “self-evident.”  Indeed, if one begins with “self-evident” truths and derives conclusions from those truths in a logical and verifiable manner, then one’s conclusions must also be undoubtedly true.

However, in the nineteenth century, these truths were undermined by the discovery of new geometries based on different axioms — the so-called “non-Euclidean geometries.” The conclusions of geometry were no longer absolute, but relative to the axioms that one chose. This became something of a problem for the concept of mathematical “proof.” If one can build different systems of mathematics based on different axioms, then “proof” only means that one’s conclusions are derivable from one’s axioms, not that one’s conclusions are absolutely true.

If you peruse the literature of mathematics on the definition of “axiom,” you will see what I mean. Many authors include the traditional definition of an axiom as a “self-evident truth.” But others define an axiom as a “definition” or “assumption,” seemingly as an acceptable alternative to “self-evident truth.” Surely there is a big difference between an “assumption,” a “self-evident truth,” and a “definition,” no? This confusing medley of definitions of “axiom” is the result of the nineteenth century discovery of non-Euclidean geometries. The issue has not been fully cleared up by mathematicians, but the Wikipedia entry on “axiom” probably represents the consensus of most mathematicians, when it states: “No explicit view regarding the absolute truth of axioms is ever taken in the context of modern mathematics, as such a thing is considered to be irrelevant.”  (!)

In reaction to the new uncertainty, mathematicians responded by searching for new foundations for mathematics, in the hopes of finding a set of axioms that would establish once and for all the certainty of mathematics. The “Foundations of Mathematics” movement, as it came to be called, ultimately failed. One of the leaders of the foundations movement, the great mathematician Bertrand Russell, declared late in life:

I wanted certainty in the kind of way in which people want religious faith. I thought that certainty is more likely to be found in mathematics than elsewhere. But I discovered that many mathematical demonstrations, which my teachers expected me to accept, were full of fallacies, and that, if certainty were indeed discoverable in mathematics, it would be in a new kind of mathematics, with more solid foundations than those that had hitherto been thought secure. But as the work proceeded, I was continually reminded of the fable about the elephant and the tortoise. Having constructed an elephant upon which the mathematical world could rest, I found the elephant tottering, and proceeded to construct a tortoise to keep the elephant from falling. But the tortoise was no more secure than the elephant, and after some twenty years of arduous toil, I came to the conclusion that there was nothing more that I could do in the way of making mathematical knowledge indubitable. (The Autobiography of Bertrand Russell)

Today, there are a variety of mathematical systems based on a variety of assumptions, and no one yet has succeeded in reconciling all the systems into one, fundamental, true system of mathematics. In fact, you wouldn’t know it from high school math, but some topics in mathematics have led to sharp divisions and debates among mathematicians. And most of these debates have never really been resolved — mathematicians have simply grown to tolerate the existence of different mathematical systems in the same way that ancient pagans accepted the existence of multiple gods.

Some of the most contentious issues in mathematics have revolved around the concept of infinity. In the nineteenth century, the mathematician Georg Cantor developed a theory about different sizes of infinite sets, but his arguments immediately attracted criticism from fellow mathematicians and remain controversial to this day. The central problem is that measuring infinity, assigning a quantity to infinity, is inherently an endless process. Once you think you have measured infinity, you simply add a one to it, and you have something greater than infinity — which means your original infinity was not truly infinite. Henri Poincare, one of the greatest mathematicians in history, rejected Cantor’s theory, noting: “Actual infinity does not exist. What we call infinite is only the endless possibility of creating new objects no matter how many exist already.”  Stephen Simpson, a mathematician at Pennsylvania University likewise argues “What truly infinite objects exist in the real world?” Objections to Cantor’s theory of infinity led to the emergence of new mathematical schools of thought such as finitism and intuitionism, which rejected the legitimacy of infinite mathematical objects.

Cantor focused his mental energies on concepts of the infinitely large, but another idea in mathematics was also controversial — that of the infinitely small, the “infinitesimal.” To give you an idea of how controversial the infinitesimal has been, I note that Cantor himself rejected the existence of infinitesimals! In Cantor’s view, the concept of something being infinitely small was inherently contradictory — if something is small, then it is inherently finite! And yet, infinitesimals have been used by mathematicians for hundreds of years. The infinitesimal was used by Leibniz in his version of calculus, and it is used today in the field of mathematics known as “non-standard analysis.” There is still no consensus among mathematicians today about the existence or legitimacy of infinitesimals, but infinitesimals, like imaginary numbers, seem to be useful in calculations, and as long as it works, mathematicians are willing to tolerate them, albeit not without some criticism.

The existence of different types of mathematical systems leads to some strange and contradictory answers to some of the simplest questions in mathematics. In school, you were probably taught that parallel lines never meet. That is true in Euclidean geometry, but not in hyperbolic geometry. In projective geometry, parallel lines meet at infinity!

Or consider the infinite decimal 0.9999 . . .  Is this infinite decimal equal to 1? The common sense answer that students usually give is “of course not.” But most mathematicians argue that both numbers are equivalent! Their logic is as follows: in the system of “real numbers,” there is no number between 0.999. . . and 1. Therefore, if you subtract 0.999. . .  from 1, the result is zero. And that means both numbers are the same!

However, in the system of numbers known as “hyperreals,” a system which includes infinitesimals, there exists an infinitesimal number between 0.999. . .  and 1. So under this system, 0.999. . .  and 1 are NOT the same! (A great explanation of this paradox is here.) So which system of numbers is the correct one? There is no consensus among mathematicians. But there is a great joke:

How many mathematicians does it take to screw in a light bulb?

0.999 . . .

The invention of computers has led to the creation of a new system of mathematics known as “floating point arithmetic.” This was necessary because, for all of their amazing capabilities, computers do not have enough memory or processing capability to precisely deal with all of the real numbers. To truly depict an infinite decimal, a computer would need an infinite amount of memory. So floating point arithmetic deals with this problem by using a degree of approximation.

One of the odd characteristics of the standard version of floating point arithmetic is that there is not one zero, but two zeros: a positive zero and a negative zero. What’s that you say? There’s no such thing as positive zero and negative zero? Well, not in the number system you were taught, but these numbers do exist in floating point arithmetic. And you can use them to divide by zero, which is something else I bet you thought you couldn’t do.  One divided by positive zero equals positive infinity, while one divided by negative zero equals negative infinity!

What the history of mathematics indicates is that the world is not converging toward one, true system of mathematics, but creating multiple, incompatible systems of mathematics, each of which has its own logic. If you think of mathematics as a set of tools for understanding reality, rather than reality itself, this makes sense. You want a variety of tools to do different things. Sometimes you need a hammer, sometimes you need a socket wrench, sometimes you need a Phillips screwdriver, etc. The only true test of a tool is how useful it is — a single tool that tried to do everything would be unhelpful.

You probably didn’t know about most of the issues in mathematics I have just mentioned, because they are usually not taught, either at the elementary school level, the high school level, or even college. Mathematics education consists largely of being taught the right way to perform a calculation, and then doing a variety of these calculations over and over and over. . . .

But why is that? Why is mathematics education just about learning to calculate, and not discussing controversies? I can think of several reasons.

One reason may be that most people who go into mathematics tend to have a desire for greater certainty. They don’t like uncertainty and imprecise answers, so they learn math, avoid mathematical controversies or ignore them, and then teach students a mathematics without uncertainty. I recall my college mathematics instructor declaring to class one day that she went into mathematics precisely because it offered sure answers. My teacher certainly had that much in common with Bertrand Russell (quoted above).

Another reason surely is that there is a large element of indoctrination in education generally, and airing mathematical controversies among students might have the effect of undermining authority. It is true that students can discuss controversies in the social sciences and humanities, but that’s because we live in a democratic society in which there are a variety of views on social issues, and no one group has the power to impose a single view on the classroom. But even a democratic society is not interested in teaching controversies in mathematics — it’s interested in creating good workers for the economy. We need people who can make change, draw up a budget, and measure things, not people who challenge widely-accepted beliefs.

This utilitarian view of mathematics education seems to be universal, shared by democratic and totalitarian governments alike. Forcing students to perform endless calculations without allowing them to ask “why” is a great way to bore children and make them hate math, but at least they’ll be obedient citizens.