Clockbackward Essays

MATHEMATICS is often thought to be universally and unassailably true. I have even heard it argued that God, omnipotent though He may be, could not make math false even if He was impulsive enough to try it. Can mathematicians actually prove that math is true? If they cannot, does the fact that math is so useful in solving real world problems provide evidence of its truth? And, if mathematics is not true, then does that imply that conclusions drawn from it are faulty or suspect? These are some of the questions that I will try to address.

The first attempt we might make to prove that mathematics is true is to consider real world situations where mathematical equations seem to appear. Some examples are:

• If I have three red balls in a bag and add two more, the bag will then contain five red balls.

• If I am on a train traveling at three miles per hour and throw a ball at two miles per hour (measured with respect to the train) then the ball will be traveling at five miles per hour with respect to the ground.

•If I had three dollars worth of goods yesterday and borrow two dollars worth of goods from you today then I have five dollars worth of goods in my possession.

Each of these three situations seem to imply the equation 3+2=5, but do they actually prove that the equation 3+2=5 is true? One problem with drawing conclusions about mathematics from these examples is that the number ‘3′ is not the same as ‘three balls’ or ‘three hours’ or ‘three dollars’, and the operator ‘+’ is not the same as grouping together balls or combining velocities or aggregating wealth. It is true that 3+2=5 is typically an excellent model for each of these situations, but nonetheless the equation is not precisely equivalent to these situations. It is also true that when we group balls together (by, in this case, placing them in a bag) the procedure generally behaves as though we are performing addition. But now suppose that the objects we are grouping together are made of packed sand, or some other delicate and malleable substance. In this case, when we add new objects to our bag they will sometimes fracture and split into multiple objects, and occasionally multiple objects will even fuse together into a single object. The addition operator ‘+’ no longer models this situation well because when we place a new object in the bag it does not always increase the number of objects contained in the bag by one.

It is not terribly difficult to annihilate the relationship between the equation 3+2=5 and the other real world situations given above. For example, Einstein’s theory of relativity tells us (in contradiction to the more intuitive but less accurate equations of Newtonian mechanics) that when a person on a train which is moving three miles per hour (with respect to the ground) throws a ball at two miles per hour (with respect to the train), then the speed of the ball with respect to the ground is actually very slightly less than 5 miles per hour, not equal to 5 miles per hour. What’s more, if I had three dollars worth of goods yesterday and then borrow two dollars worth of goods from you today, the total number of dollars worth of goods that I have possession of will not necessarily be five dollars if the value of my original goods changed between yesterday and today (as can happen in real economic markets). What these examples show us is that the only reason to say that grouping balls or combining velocities or aggregating wealth encapsulates the idea of mathematical addition is because most of the time the addition operator ‘+’ provides a good model for these scenarios. We can no more conclude that 3+2=5 is a true statement simply because putting two balls into a bag that already has three balls generally produces a bag with five balls, then we can conclude that 3+2=5 is false simply because velocities have been proven not to add. In other words, while real world situations can motivate the equations of mathematics and provide justifications for applying them, they cannot prove that those equations are actually true.

We have stared at equations like 3+2=5 so many times in our lives that it can be difficult to consider them with fresh eyes in order to ask ourselves what it really is that they are saying. Clearly ‘3′, ‘+’ , ‘2′, ‘=’ and ‘5′ are not objects in the physical universe. You can go to the zoo and see three bears, or see the numeral ‘3′ printed on a sign, or perform arithmetic on paper using the symbol ‘3′, but nowhere in the universe can you find the actual (metaphysical) number ‘3′. This is hardly surprising, since ‘3′ is a concept or idea, not a physical thing. But this line of thought implies that 3+2=5 is a statement about the relationship among the concepts ‘3′, ‘2′, and ‘5′, and not a statement about physical entities that actually exist.

But how do we define the word “true” when it comes to relations among abstract concepts? One possible approach is to say that statements about abstract concepts are true if they follows as logical consequence of the definitions of the concepts themselves. This leads us to ask whether 3+2=5 and all other mathematical statements are simply true by definition as a consequence of our chosen definitions for ‘3′, ‘+’, ‘2′, ‘=’, ‘5′ and the other mathematical objects. Unfortunately, this question cannot be answered without further qualification. To begin with, what do we mean by “mathematical objects”, and how do we choose to define concepts such as ‘3′? Various authors have attempted to define mathematics by developing lists of axioms (which are simply assumed to be true) and then proving that the basic mathematical objects (e.g. integers) and theorems (e.g. a+b = b+a) follow from these axioms. Unfortunately, there are a variety of different ways that math can be axiomatized (i.e. built up from basic axioms). Some approaches use sets as the most basic objects (as is done in what is probably the most popular axiomatization, Zermelo-Fraenkel set theory), while others use Category Theory to provide the basic building blocks, and still other theories attempt to axiomatize only small portions of math, such as Euclid’s Axioms of planar geometry, Hilbert’s axiomatization of Euclidean Geometry and the Peano axioms for arithmetic. What is even worse (when it comes to deciding what is true) than having so many conflicting viewpoints for constructing math is that the axioms of these viewpoints are themselves not provably true. If you are willing to assume the axioms of math are “true” (whatever that means), then all of the resulting theorems that can be derived from those axioms are also true, but the axioms themselves must simply be accepted without proof in order for this process to work. As a matter of fact, if we could prove that the axioms were true then they would be called “theorems” and not “axioms”!

As convoluted as this discussion has become, matters get still murkier. Even those mathematicians who agree to rely on a single basic axiomatization (such as Zermelo-Fraenkel set theory) sometimes cannot agree on whether certain extra axioms (such as the continuum hypothesis, which concerns itself with the existence of sets of certain infinite sizes, or the axiom of choice which pertains to being able to select one element from each element of a set of sets) should be added or left out. And to top that off, mathematics (as defined by whichever axiomatization you like) has not even been proven to be consistent, meaning that no one has been able to mathematically demonstrate that the axioms of any single axiomatization do not contradict each other. In fact, Gödel’s 2nd incompleteness theorem shows that if mathematics is in fact consistent then it will not be possible to use math to prove that no inconsistencies exist!

In conclusion: numbers and other mathematical objects are simply concepts, and not things that are actually observable in the universe, so we cannot say that statements like 3+2=5 are true in the same way that we can say that the statement “massive objects exert forces on other massive objects” is true. We might like to think that mathematical statements are true by definition, but this idea is complicated by the fact that there is more than one way to axiomatize mathematics, and therefore more than one definition that we might choose in order to define numbers, operators and other mathematical objects. But even if there were truly only one way to axiomatize math, the axioms themselves would still not be provably true (they would only be assumed to be true), and hence it would hardly seem fair to then conclude that mathematical theorems are “true” in some objective and universal sense. These problems are compounded by the fact that we cannot prove that our commonly accepted mathematical axioms do not contradict each other, leading to a still deeper level on which to question the truth of mathematical statements. In the end, while it hardly seems fair to say that math is false, it is also not reasonable to conclude that math is true. Math is probably neither “true” nor “false” in the usual sense of those words, though it does undeniably provide extraordinarily useful models for making predictions about what will happen in our physical universe. This will perhaps seem less surprising if we remember that mathematics was not originally developed from the ground up using axioms, but rather piece by piece in order to find solutions to problems that appear in the real world (like those related to calculating the size of plots of land, counting money, measuring roads, tracking the movements of the stars, understanding heat flow in cannons, etc.). Hence, mathematical definitions were chosen by humans to model physical reality so that we could make useful predictions, not to encapsulate metaphysical truth, so really, why should we expect math to be true?