Uncertainty, Debate, and Imprecision in Mathematics

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.

Belief and Evidence

A common argument by atheists is that belief without evidence is irrational and unjustified, and that those arguing for the existence of God have the burden of proof.  Bertrand Russell famously argued that if one claims that there is a teapot orbiting the sun, the burden of proving the existence of the teapot is on the person who asserts the existence of the teapot, not the denier.  Christopher Hitchens has similarly argued that “What can be asserted without evidence can also be dismissed without evidence.”  Hitchens has advanced this principle even further, arguing that “exceptional claims demand exceptional evidence.”  (god is not Great, pp. 143, 150)  Sam Harris has argued that nearly every evil in human history “can be attributed to an insufficient taste for evidence” and that “We must find our way to a time when faith, without evidence, disgraces anyone who would claim it.”   (The End of Faith, pp. 25, 48)

A demand for evidence is surely a legitimate requirement for most ordinary claims.  But it would be a mistake to turn this rule into a rigid and universal requirement, because many of the issues and problems we encounter in our lives are not always rich with evidence.  Some issues have a wealth of evidence, some issues have a small amount of indirect or circumstantial evidence, some issues have evidence compatible with a variety of radically different conclusions, and some issues have virtually no evidence.  What’s worse is that there appears to be an inverse relationship between the size and importance of the issue one is addressing and the amount of evidence that is available.  The bigger the question one has, the less evidence there is to address it.  The questions of how to obtain a secure and steady supply of food, water, and shelter, how to extend the human lifespan and increase the economic standard of living, all have scientific-technological answers backed by abundant evidence.  Other issues, such as the origins of the universe, the nature of the elementary particles, and the evolution of life, also have large amounts of evidence, albeit with significant gaps in certain details.  But some of the most important questions we face have such a scarcity of evidence that a variety of conflicting beliefs seems inevitable.  Why does the universe exist?  Is there intelligent life on other planets, and if so, how many planets have such life?  Where did the physical laws of the universe come from?  What should we do with our lives?  Will the human race survive the next 1000 years?  Are our efforts to be good people and follow moral codes all in vain?

In cases of scarce evidence, to demand that sufficient evidence exist before forming a belief is to put the cart before the horse.  If one looks at the origins and growth of knowledge in human civilization, belief begins with imagination — only later are beliefs tested and challenged.  Without imagination, there are no hypotheses to test.  In fact, one would not know what evidence to gather if one did not begin with a belief.  Knowledge would never advance.  As the philosopher George Santayana argued in his book Reason and Religion,

A good mythology cannot be produced without much culture and intelligence. Stupidity is not poetical. . . . The Hebrews, denying themselves a rich mythology, remained without science and plastic art; the Chinese, who seem to have attained legality and domestic arts and a tutored sentiment without passing through such imaginative tempests as have harassed us, remain at the same time without a serious science or philosophy. The Greeks, on the contrary, precisely the people with the richest and most irresponsible myths, first conceived the cosmos scientifically, and first wrote rational history and philosophy. So true it is that vitality in any mental function is favourable to vitality in the whole mind. Illusions incident to mythology are not dangerous in the end, because illusion finds in experience a natural though painful cure. . . .  A developed mythology shows that man has taken a deep and active interest both in the world and in himself, and has tried to link the two, and interpret the one by the other. Myth is therefore a natural prologue to philosophy, since the love of ideas is the root of both.

Modern critics of traditional religion are right to argue that we need to revise, reinterpret, or abandon myths when they conflict with new evidence.  As astronomy advanced, it was necessary to abandon the geocentric model of the universe.   As the evidence for evolution accumulated, it was no longer plausible to believe that the universe was created in the extremely short span of six days.  There is a difference between a belief formed in the face of a scarcity of evidence and a belief that goes against an abundance of evidence.  The former is permitted, and is even necessary to advance knowledge; the latter takes knowledge backward.

Today we have reached the point at which science is attempting to answer some very large questions, and science is running up against the limits of what is possible with observation, experimentation, and verification.  Increasingly, the scientific imagination is developing theories that are plausible, but have little or no evidence to back them up; in fact, for many of these theories we will probably never have sufficient evidence.  I am referring here to cosmological theories about the origins of the universe that propose a “multiverse,” that is, a large or even infinite collection of universes that exist alongside our own observable universe.

There are several different types of multiverse theories.  The first type, which many if not most cosmologists accept, proposes multiple universes with the same physical laws and constants as ours, but with different distributions of matter.  A second type, which is more controversial, proposes an infinite number of universes with different physical laws and constants.  A third type, also controversial, arises out of the “many worlds” interpretation of quantum physics — in this view, every time an indeterminate event occurs (say, a six-sided die comes up a “four”), an entirely new universe splits off from our own.  Thus, the most extreme multiverse theories claim that all possibilities exist in some universe, somewhere.  There are even an infinite number of people like you, each with a slight variation in life history (i.e., turning left instead of turning right when leaving the house this morning).

The problem with these theories, however, is that is impossible to obtain solid evidence on the existence of other universes through observation — the universes either exist far beyond the limits of our observable universe, or they reside on a different branch of reality that we cannot reach.  Now it’s not unusual for a scientific theory to predict the existence of particles or forces or worlds that we cannot yet observe; historically, a number of such predictions have proved true when the particle or force or world was finally observed.  But many other predictions have not been proved true.  With the multiverse, it is unlikely that we will have definitive evidence one way or the other.  And a number of scientists have revolted at this development, arguing that cosmology at this level is no longer scientific.  According to physicist Paul Davies,

Extreme multiverse explanations are therefore reminiscent of theological discussions. Indeed, invoking an infinity of unseen universes to explain the unusual features of the one we do see is just as ad hoc as invoking an unseen Creator. The multiverse theory may be dressed up in scientific language, but in essence it requires the same leap of faith.

Likewise, Freeman Dyson insists:

[T]he multiverse is philosophy and not science. Science is about facts that can be tested and mysteries that can be explored, and I see no way of testing hypotheses of the multiverse. Philosophy is about ideas that can be imagined and stories that can be told. I put narrow limits on science, but I recognize other sources of human wisdom going beyond science. Other sources of wisdom are literature, art, history, religion, and philosophy. The multiverse has its place in philosophy and in literature.

Cosmologist George F.R. Ellis, in the August 2011 issue of Scientific American, notes that there are several ways of indirectly testing for the existence of multiple universes, but none are likely to be definitive.  He concludes: “Nothing is wrong with scientifically based philosophical speculation, which is what multiverse proposals are.  But we should name it for what it is.”

Given the thinness of the evidence for extreme multiverse theories, one might ask why modern day atheists do not seem to attack and mock such theorists for believing in something for which they cannot provide solid evidence.  At the very least, Christopher Hitchens’s claim that “exceptional claims require exceptional evidence” would seem to invalidate belief in any multiverse theory.  At best, at some future point we may have indirect or circumstantial evidence for the existence of some other universes; but we are never going to have exceptional evidence for an infinite number of universes consisting of all possibilities.  So why do we not hear of insulting analogies involving orbiting teapots and flying spaghetti monsters when some scientists propose an infinite number of universes based on different physical laws or an infinite number of versions of you?  I think it’s because scientists are respected authority figures in a modern, secular society.  If a scientist says there are multiple universes, we are inclined to believe them even in the absence of solid evidence, because scientists have social prestige, especially among atheists.

Ultimately, there is no solid evidence for the existence of God, no solid evidence for the existence of an infinite variety of universes, and no solid evidence for the existence of other versions of me.  Whether or not one chooses to believe any of these propositions depends on whether one decides to leap into the dark, and which direction one decides to  leap.  This does not mean that any religious belief is permissible — on issues which have abundant evidence, beliefs cannot go against evidence.  Evolution has abundant evidence, as does modern medical science, chemistry, and rocket science.  But where evidence is scarce, and a variety of beliefs are compatible with existing evidence, holding a particular belief cannot be regarded as wholly unjustified and irrational.