There is an old mathematical joke that goes like this:

A dairy farmer is interested in increasing his milk production. He goes to the local university, and seeks the advice of three experts: a biologist, a pharmacist, and a mathematician.

The biologist comes back with her report first. “Well,” she says, “I’ve gone over your numbers, and I am optimistic. If you you begin a selective breeding program now, in 10 years you should be able to increase your milk production by around 15%.”

The pharmacist was next to submit his report. “Well,” he says, “I’ve gone over your numbers, and I am optimistic. If you start administering bovine growth hormones and antibiotics now, in one year you should be able to increase your milk production by around 50%.”

The mathematician was the last to submit his report. “I’ve got great news for you!” he says. “I can increase your milk production by 500% before the end of the week!”

The farmer is incredulous. “How are you going to do that?” he asks.

“Quite simply,” the mathematician replies. “First, we must assume a spherical cow…”

At the risk of ruining the joke by over analysis, I present a question: why is this joke funny?

To someone who is not overly familiar with mathematics, the joke is probably interpreted as a jab against mathematicians. In essence, the joke reduces to, “Haha! Look at the silly mathematician! He is so disconnected from the world that he thinks that there are spherical cows!” The joke is funny because mathematicians are often thought of as holing up in their ivory towers, utterly apart from the real world.

But if this is the meaning of the joke, why do so many mathy folks think that it is so funny? ^{1}To borrow a phrase… Are they in on the joke, and laughing at themselves along with everyone else? or is there something deeper going on?

I suspect that there is something more to this joke. It relates to a subtle but important variation in how the joke can be interpreted. To an outsider, the joke is about the quirkiness of mathematicians. To an insider, the joke is about the quirkiness of mathematics itself.

If you ask a mathematician what mathematics is, he or she might reply that it is the study of patterns, and that the ultimate goal of mathematics is to deduce truth. Unfortunately, this definition raises almost as many questions as it answers. Chief among these is, “What is meant by ‘truth?'”

In the natural sciences (physics, chemistry, biology, &c.), truth is normally understood empirically and inductively. Scientists in these fields observe nature, and formulate rules (called *hypotheses*) for how they think nature works. They then devise tests to see how well these hypotheses explain natural phenomena. If a hypothesis passes a large number of tests, then it is provisionally accepted as being true, at least until such time as some new test proves it otherwise.

There are a couple of important things to note about the scientific understanding of truth. First and foremost is the notion that *nothing* is absolutely known to be true. All truth is accepted only on a provisional basis, with the understanding that anything that is believed to be true now may be shown to be false at some future time. All truth is tentative.

And second, truth is understood empirically. This means that truth is always understood to relate to the real, observable world. If something is true, then we should be able to see it, or touch it, or taste it, or otherwise access it with our senses. Truth is not arrived at through reason or argument, but rather through observation of the universe.

In our daily lives, we often understand truth in a way that is similar to the scientific understanding. We accept statements as being true if they align with our observations of the world, and reject them as false if they do not. Of course, in this context the truth is often heavily swayed by convincing arguments and the fallibility of human cognition, but the general principle is the same: truth is contingent upon how well it matches observable reality.

In mathematics, truth has a very different meaning. For instance, in mathematics, the truth is absolutely knowable. There is no such thing as tentative truth in mathematics. Either something is provably true, or it is provably false.^{2}This overstates the case a little bit: there are mathematical statements that are *believed* to be true, but which are incredibly difficult to prove, and there are classes of statements which cannot be proven to be either true or false. However, both of these categories of statements involve some pretty deep mathematics, and are far beyond the scope of this post.

Mathematical truth is built upon assumptions, called *axioms*. An axiom is generally a very simple, very basic statement which is taken to be self-evidently true. Mathematicians assume that their axioms are true, and proceed from that assumption. Truth is then a consequence of logical argument from the axioms. The problem is that there is no way of verifying that the axioms are true. In fact, mathematicians can, and often do, declare alternative sets of axioms to be true, and proceed to argue from those axioms.^{3}Consider, for example, spherical geometry or hyperbolic geometry. Both of these systems of geometry are different from the geometry that is taught in high school (Euclidean geometry) in only one assumption. Yet by changing just a single axiom, what is true and what is false is dramatically changed.

This means that mathematics is basically a giant game of “What if…” Every single statement of truth in mathematics begins with the phrase “*If* all of our axioms are true, *then*…” This implies that mathematics is very abstract. Mathematicians can assume that literally anything is true, then argue about what this implies. They are not beholden to the rules that govern the universe, and are free to experiment with different assumptions to their hearts’ content. Of course, while truth is only a matter of which axioms one assumes from the start, once those axioms are assumed, the truth of further statements can be proved and verified absolutely.

This, of course, brings us back to the joke that opened this post. Unlike the biologist and the pharmacist, who are bound by empirical observation of the universe in which they live, the mathematician is free to make any assumptions that he so desires. The mathematician’s reply may as well read, “If there were such a thing as a spherical cow, then…” Thus, as mentioned above, the joke is not about how disconnected *mathematicians* are from reality, but how disconnected *mathematics itself* is from reality.