## What Is Math?

A teenager asked that age-old question on TikTok, creating a viral backlash, and then, a thoughtful scientific debate

By Dan Falk smithsonianmag.com September 23, 2020 Article Source: https://www.smithsonianmag.com/science-nature/what-math-180975882/

It all started with an innocuous TikTok video posted by a high school student named Gracie Cunningham. Applying make-up while speaking into the camera, the teenager questioned whether math is âreal.â She added: âI know itâs real, because we all learn it in schoolâŠ but who came up with this concept?â Pythagoras, she muses, âdidnât even have plumbingâand he was like, âLet me worry about y = mx + bâââreferring to the equation describing a straight line on a two-dimensional plane. She wondered where it all came from. âI get addition,â she said, âbut how would you come up with the concept of algebra? What would you need it for?â

Someone re-posted the video to Twitter, where it soon went viral. Many of the comments were unkind: One person said it was the âdumbest videoâ they had ever seen; others suggested it was indicative of a failed education system. Others, meanwhile, came to Cunninghamâs defense, saying that her questions were actually rather profound.

```
@gracie.ham
this video makes sense in my head but like WHY DID WE CREATE THIS STUFF
âŹ original sound - gracie
```

Mathematicians from Cornell and from the University of Wisconsin weighed in, as did philosopher Philip Goff of Durham University in the U.K. Mathematician Eugenia Cheng, currently the scientist-in-residence at the Art Institute of Chicago, wrote a two-page reply and said Cunningham had raised profound questions about the nature of mathematics âin a very deeply probing way.â

Cunningham had unwittingly re-ignited a very ancient and unresolved debate in the philosophy of science. What, exactly, is math? Is it invented, or discovered? And are the things that mathematicians work withânumbers, algebraic equations, geometry, theorems and so onâreal?

Some scholars feel very strongly that mathematical truths are âout there,â waiting to be discoveredâa position known as Platonism. It takes its name from the ancient Greek thinker Plato, who imagined that mathematical truths inhabit a world of their ownânot a physical world, but rather a non-physical realm of unchanging perfection; a realm that exists outside of space and time. Roger Penrose, the renowned British mathematical physicist, is a staunch Platonist. In The Emperorâs New Mind, he wrote that there appears âto be some profound reality about these mathematical concepts, going quite beyond the mental deliberations of any particular mathematician. It is as though human thought is, instead, being guided towards some external truthâa truth which has a reality of its ownâŠâ

Many mathematicians seem to support this view. The things theyâve discovered over the centuriesâthat there is no highest prime number; that the square root of two is an irrational number; that the number pi, when expressed as a decimal, goes on foreverâseem to be eternal truths, independent of the minds that found them. If we were to one day encounter intelligent aliens from another galaxy, they would not share our language or culture, but, the Platonist would argue, they might very well have made these same mathematical discoveries.

âI believe that the only way to make sense of mathematics is to believe that there are objective mathematical facts, and that they are discovered by mathematicians,â says James Robert Brown, a philosopher of science recently retired from the University of Toronto. âWorking mathematicians overwhelmingly are Platonists. They donât always call themselves Platonists, but if you ask them relevant questions, itâs always the Platonistic answer that they give you.â

Other scholarsâespecially those working in other branches of scienceâview Platonism with skepticism. Scientists tend to be empiricists; they imagine the universe to be made up of things we can touch and taste and so on; things we can learn about through observation and experiment. The idea of something existing âoutside of space and timeâ makes empiricists nervous: It sounds embarrassingly like the way religious believers talk about God, and God was banished from respectable scientific discourse a long time ago.

Platonism, as mathematician Brian Davies has put it, âhas more in common with mystical religions than it does with modern science.â The fear is that if mathematicians give Plato an inch, heâll take a mile. If the truth of mathematical statements can be confirmed just by thinking about them, then why not ethical problems, or even religious questions? Why bother with empiricism at all?

Massimo Pigliucci, a philosopher at the City University of New York, was initially attracted to Platonismâbut has since come to see it as problematic. If something doesnât have a physical existence, he asks, then what kind of existence could it possibly have? âIf one âgoes Platonicâ with math,â writes Pigliucci, empiricism âgoes out the window.â (If the proof of the Pythagorean theorem exists outside of space and time, why not the âgolden rule,â or even the divinity of Jesus Christ?)

The Platonist must confront further challenges: If mathematical objects exist outside of space and time, how is it that we can know anything about them? Brown doesnât have the answer, but he suggests that we grasp the truth of mathematical statements âwith the mindâs eyeââin a similar fashion, perhaps, to the way that scientists like Galileo and Einstein intuited physical truths via âthought experiments,â before actual experiments could settle the matter. Consider a famous thought experiment dreamed up by Galileo, to determine whether a heavy object falls faster than a lighter one. Just by thinking about it, Galileo was able to deduce that heavy and light objects must fall at the same rate. The trick was to imagine the two objects tethered together: Does the heavy one tug on the lighter one, to make the lighter one fall faster? Or does the lighter one act as a âbrakeâ to slow the heavier one? The only solution that makes sense, Galileo reasoned, is that objects fall at the same rate regardless of their weight. In a similar fashion, mathematicians can prove that the angles of a triangle add up to 180 degrees, or that there is no largest prime numberâand they donât need physical triangles or pebbles for counting to make the case, just a nimble brain.

Meanwhile, notes Brown, we shouldnât be too shocked by the idea of abstractions, because weâre used to using them in other areas of inquiry. âIâm quite convinced there are abstract entities, and they are just not physical,â says Brown. âAnd I think you need abstract entities in order to make sense of a ton of stuffânot only mathematics, but linguistics, ethicsâprobably all sorts of things.â

Platonism has various alternatives. One popular view is that mathematics is merely a set of rules, built up from a set of initial assumptionsâwhat mathematicians call axioms. Once the axioms are in place, a vast array of logical deductions follow, though many of these can be fiendishly difficult to find. In this view, mathematics seems much more like an invention than a discovery; at the very least, it seems like a much more human-centric endeavor. An extreme version of this view would reduce math to something like the game of chess: We write down the rules of chess, and from those rules various strategies and consequences follow, but we wouldnât expect those Andromedans to find chess particularly meaningful.

But this view has its own problems. If mathematics is just something we dream up from within our own heads, why should it âfitâ so well with what we observe in nature? Why should a chain reaction in nuclear physics, or population growth in biology, follow an exponential curve? Why are the orbits of the planets shaped like ellipses? Why does the Fibonacci sequence turn up in the patterns seen in sunflowers, snails, hurricanes, and spiral galaxies? Why, in a nutshell, has mathematics proven so staggeringly useful in describing the physical world? Theoretical physicist Eugene Wigner highlighted this issue in a famous 1960 essay titled, âThe Unreasonable Effectiveness of Mathematics in the Natural Sciences.â Wigner concluded that the usefulness of mathematics in tackling problems in physics âis a wonderful gift which we neither understand nor deserve.â

However, a number of modern thinkers believe they have an answer to Wignerâs dilemma. Although mathematics can be seen as a series of deductions that stem from a small set of axioms, those axioms were not chosen on a whim, they argue. Rather, they were chosen for the very reason that they do seem to have something to do with the physical world. As Pigliucci puts it: âThe best answer that I can provide [to Wignerâs question] is that this âunreasonable effectivenessâ is actually very reasonable, because mathematics is in fact tethered to the real world, and has been, from the beginning.â

Carlo Rovelli, a theoretical physicist at Aix-Marseille University in France, points to the example of Euclidean geometryâthe geometry of flat space that many of us learned in high school. (Students who learn that an equilateral triangle has three angles of 60 degrees each, or that the sum of the squares of the two shorter sides of a right-triangle equals the square of the hypotenuseâi.e. the Pythagorean theoremâare doing Euclidean geometry.) A Platonist might argue that the findings of Euclidean geometry âfeelâ universalâbut they are no such thing, Rovelli says. âItâs only because we happen to live in a place that happens to be strangely flat that we came up with this idea of Euclidean geometry as a ânatural thingâ that everyone should do,â he says. âIf the earth had been a little bit smaller, so that we saw the curvature of the earth, we would have never developed Euclidean geometry. Remember âgeometryâ means âmeasurement of the earthâ, and the earth is round. We would have developed spherical geometry instead.â

Rovelli goes further, calling into question the universality of the natural numbers: 1, 2, 3, 4âŠ To most of us, and certainly to a Platonist, the natural numbers seem, well, natural. Were we to meet those intelligent aliens, they would know exactly what we meant when we said that 2 + 2 = 4 (once the statement was translated into their language). Not so fast, says Rovelli. Counting âonly exists where you have stones, trees, peopleâindividual, countable things,â he says. âWhy should that be any more fundamental than, say, the mathematics of fluids?â If intelligent creatures were found living within, say, the clouds of Jupiterâs atmosphere, they might have no intuition at all for counting, or for the natural numbers, Rovelli says. Presumably we could teach them about natural numbersâjust like we could teach them the rules of chessâbut if Rovelli is right, it suggests this branch of mathematics is not as universal as the Platonists imagine.

Like Pigliucci, Rovelli believes that math âworksâ because we crafted it for its usefulness. âItâs like asking why a hammer works so well for hitting nails,â he says. âItâs because we made it for that purpose.â

In fact, says Rovelli, Wignerâs claim that mathematics is spectacularly useful for doing science doesnât hold up to scrutiny. He argues that many discoveries made by mathematicians are of hardly any relevance to scientists. âThere is a huge amount of mathematics which is extremely beautiful to mathematicians, but completely useless for science,â he says. âAnd there are a lot of scientific problemsâlike turbulence, for exampleâthat everyone would like to find some useful mathematics for, but we havenât found it.â

Mary Leng, a philosopher at the University of York, in the U.K., holds a related view. She describes herself as a âfictionalistâ â she sees mathematical objects as useful fictions, akin to the characters in a story or a novel. âIn a sense, theyâre creatures of our creation, like Sherlock Holmes is.â

But thereâs a key difference between the work of a mathematician and the work of a novelist: Mathematics has its roots in notions like geometry and measurement, which are very much tied to the physical world. True, some of the things that todayâs mathematicians discover are esoteric in the extreme, but in the end, math and science are closely allied pursuits, Leng says. âBecause [math] is invented as a tool to help with the sciences, itâs less of a surprise that it is, in fact, useful in the sciences.â

Given that these questions about the nature of mathematics have been the subject of often heated debate for some 2,300 years, itâs unlikely theyâll go away anytime soon. No surprise, then, that high school students like Cunningham might pause to consider them as well, as they ponder Pythagorean theorem, the geometry of triangles, and the equations that describe lines and curves. The questions she posed in her video were not silly at all, but quite astute: mathematicians and philosophers have been asking the same imponderables for thousands of years.