The Reasonable Ineffectiveness of Defining Mathematics

The Reasonable Ineffectiveness of Defining Mathematics

Imagine meeting an extraterrestrial civilization and, for the sake of argument, let’s assume that you can somehow communicate with these beings. This civilization is a peculiar one in that it has no concept of what we call mathematics.

During some friendly conversation, you mention something about math.

Your curious alien companion stops you and asks, “What is this ‘mathematics’ that you speak of?”

“Well,” you begin, “mathematics is the study of…”

You pause.

What exactly is mathematics? How do you describe it?

This question has recently plagued me, and I fear I have no good answer. Fortunately, I’m not alone. Even Wikipedia recognizes that mathematics “has no generally accepted definition.”

As any good digital citizen would, I turned to Twitter to crowdsource an answer:

I didn’t put too much thought into my poll choices.

I chose Numbers because whenever I told people that I was studying mathematics, one of the most common responses was, “Oh? You must be really good with numbers!” (I’m not.)

Logic was meant to be a bit of a red herring. Mathematicians certainly use logic as a tool. But so do theoretical physicists and philosophers. Logicians study logic, and while there have been many logicians that study mathematics, the two pursuits have always been separate in my mind.

Finally, Quantitative ideas was a choice I threw in because it sounded reasonable. It’s general enough to capture a wide range of topics but still keeps the focus on numbers. This isn’t entirely accurate, though. Ask a topologist if their work is quantitative, and they’ll probably say, “Not really.”

To my surprise, Logic and Quantitative ideas received the same portion of votes. Those who answered Other commented with notions such as space, structure, and patterns.

What I hope that poll participants realized is that, to some degree, mathematics studies all of these things. Yet every answer to the question “What is mathematics the study of?” feels inadequate.

Something Beyond Science

While pondering the definition of mathematics, one thing that struck me is that it seems possible to complete this exercise for other scientific disciplines. For example, you could say that physics is the study of how the universe works. Chemistry is the study of what the universe is made of. Biology is the study of living organisms.

I’d be a fool not to point out that these “definitions” are simplistic. Every human endeavor is fraught with nuance. But, to a certain extent, you can somehow distill these disciplines into a single sentence. This doesn’t appear to be the case with mathematics.

Throughout history, mathematics has been placed on a pedestal and given an almost god-like significance. One example of this is the oft-quoted article titled The unreasonable effectiveness of mathematics in the natural sciences penned in 1960 by physicist Eugene Wigner.

In his article, Wigner expresses his awe of mathematics’ applicability to physics:

“The miracle of the appropriateness of the language of mathematics for the formulation of the laws of physics is a wonderful gift which we neither understand nor deserve.”

Despite my love for mathematics, I find Wigner’s attitude a bit strange and borderline fanatical. In my mind, it’s not surprising that one can describe the observable universe in the language of mathematics. After all, mathematics was born in human minds attempting to understand the things they observed.

Something Of This World

I’m no historian, but from what I’ve gathered, the advent of mathematics stems primarily from the need to measure quantity. As civilization formed and trade and economies developed, we humans needed to keep track of everything from land area to amounts of goods to currency.

Humans are an inquisitive bunch, though, and our capacity for abstraction is not a modern development. You can find examples in Indian texts from as early as 800 BCE of what we now call the Pythagorean Theorem.

The concept of number itself is a powerful abstraction. We don’t know who the first human was that recognized that three grains of sand and a group of three friends gathered together to have a shared three-ness. But we know that at some point, it did occur — likely a very long time ago and, I would guess, to multiple people independently.

What is clear to me, though, is that mathematics began with observations of our world. Necessity drove the invention of numerical systems and measurement. Curiosity carried the invention to new heights.

Something Greek, Or Not

Greek mathematics is widely considered the birthplace of mathematical proof. Most of the accounts of the history of mathematics I've read put the Greeks at the center of modern mathematical thought. Whether or not this narrative is accurate is up for debate.

The story goes that Thales of Miletus learned mathematics from the Babylonians and Egyptians. Later, Thales brilliantly applied logic and reasoning to explain mathematics earning him the title of the first person to apply deductive reasoning to geometry.

Some modern scholars are skeptical of the Babylonian influence on Thales’ mathematics. But the Babylonian and Egyptian influence on Greek society is hardly controversial. And, as pointed out by Karine Chemla in her prologue to the book The History of Mathematical Proof in Ancient Traditions, “several reasons suggest that we should be wary regarding the standard narrative.”

For one, it appears that sources have gone missing from the historical record. Chemla offers as one piece of evidence how remarks made by Henry Thomas Colebrooke, the first person to translate ancient Sanskrit texts to European languages, seem to have been ignored by later historians:

“Colebrooke read in the Sanskrit texts a rather elaborate system of proof in which the algebraic rules used in the application of algebra were themselves proved. Moreover, he pointed resolutely to the use in these writings of ‘algebraic proofs’. It is striking that these remarks were not taken up in later historiography. Why did this evidence disappear from subsequent accounts?”

So, Indian mathematicians — and quite possibly Babylonian and Egyptian mathematicians — were doing mathematical proofs before the Greeks. They just didn’t look like Greek proofs, and historical bias has all but erased their contribution to this part of the development of mathematics.

What we can gather, though, is that at some time, mathematics moved from accounting and engineering to something more like philosophy. The statement of a mathematical principle must be accompanied by a proof explaining why that principle is true.

Something Intangible

So what, then, is mathematics? Mathematics has exploded into something far beyond the geometry and algebra handed down to us from the ancients. Some of the objects studied by mathematicians are so abstract that they have no apparent example in nature.

This is where I have to come clean. I really don’t know what mathematics is. I know a few things that it is not, and a few things that it is in part.

The most significant difference I see between mathematics and sciences like physics and chemistry is that mathematics, although rooted in observations of our universe, is not concerned with material objects.

So perhaps mathematics is best described as the science of the immaterial universe — the universe of human thought. Just as physical science is divided into various disciplines — like astronomy or chemistry — so too can mathematics be broken down into things like geometry, algebra, and analysis.

In other words, I wonder if the word mathematics is more akin to the word science than it is physics. It is not a discipline in itself but encompasses many disciplines that share a common toolkit. Science has the scientific method, and mathematics has deductive logic.

Doesn’t this place mathematics into the realm of philosophy? Perhaps. But even philosophers struggle to explain mathematics. Stewart Shapiro, in the Oxford Online Handbook’s introduction to the philosophy of mathematics, has this to say about the trouble mathematics presents to philosophers:

“The conflict between rationalism and empiricism reflects some tension in the traditional views concerning mathematics, if not logic. Mathematics seems necessary and a priori, and yet it has something to do with the physical world. How is this possible? How can we learn something important about the physical world by a priori reflection in our comfortable armchairs? As noted above, mathematics is essential to any scientific understanding of the world, and science is empirical if anything is—rationalism notwithstanding.”

Is it this duality of rationalism and empiricism that plagues anyone who dares try to define mathematics? Maybe. I don’t know. I’m not sure I’ll ever know.

There is one thing I can say now that I’ve spent a couple of weeks thinking about the definition of mathematics. The ineffectiveness of any definition now seems to me to be perfectly reasonable.

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