Seventy-Three, Yes. Higally-Piggle, No. What Is Math Anyway?

What does it mean that seventy-three is a genuine number, but higally-piggle isn’t? Smithsonian Magazine delves into the old question: What is math? The answer is provided below. (See the original article for all the links.)

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 “real.” She added: “I know it’s real, because we all learn it in school… but who came up with this concept?” Pythagoras, she muses, “didn’t even have plumbing—and he was like, ‘Let me worry about y = mx + b’”—referring to the equation describing a straight line on a two-dimensional plane. She wondered where it all came from. “I get addition,” she said, “but how would you come up with the concept of algebra? What would you need it for?”

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

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 . . . wrote a two-page reply and said Cunningham had raised profound questions about the nature of mathematics “in a very deeply probing way.”

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 with—numbers, algebraic equations, geometry, theorems and so on—real?

Some scholars feel very strongly that mathematical truths are “out there,” waiting to be discovered—a position known as Platonism. It takes its name from the ancient Greek thinker Plato, who imagined that mathematical truths inhabit a world of their own—not 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 Emperor’s New Mind, he wrote that there appears “to 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 truth—a truth which has a reality of its own…”

Many mathematicians seem to support this view. The things they’ve discovered over the centuries—that 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 forever—seem 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.

“I 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,” says James Robert Brown, a philosopher of science . . . . “Working mathematicians overwhelmingly are Platonists. They don’t always call themselves Platonists, but if you ask them relevant questions, it’s always the Platonistic answer that they give you.”

Other scholars—especially those working in other branches of science—view 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 “outside of space and time” 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. . . .

Massimo Pigliucci, a philosopher at the City University of New York, was initially attracted to Platonism—but has since come to see it as problematic. If something doesn’t have a physical existence, he asks, then what kind of existence could it possibly have?

. . . 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 doesn’t have the answer, but he suggests that we grasp the truth of mathematical statements “with the mind’s eye”—in a similar fashion, perhaps, to the way that scientists like Galileo and Einstein intuited physical truths via “thought experiments,” 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 “brake” 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 number—and they don’t need physical triangles or pebbles for counting to make the case, just a nimble brain.

Meanwhile, notes Brown, we shouldn’t be too shocked by the idea of abstractions, because we’re used to using them in other areas of inquiry. “I’m quite convinced there are abstract entities, and they are just not physical,” says Brown. “And I think you need abstract entities in order to make sense of a ton of stuff—not only mathematics, but linguistics, ethics . . .”.

Platonism has various alternatives. One popular view is that mathematics is merely a set of rules, built up from a set of initial assumptions—what 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. . . .

But this view has its own problems. If mathematics is just something we dream up from within our own heads, why should it “fit” 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, “The Unreasonable Effectiveness of Mathematics in the Natural Sciences.” Wigner concluded that the usefulness of mathematics in tackling problems in physics “is a wonderful gift which we neither understand nor deserve.”

However, a number of modern thinkers believe they have an answer to Wigner’s 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: “The best answer that I can provide [to Wigner’s question] is that this ‘unreasonable effectiveness’ is actually very reasonable, because mathematics is in fact tethered to the real world, and has been, from the beginning.”

Carlo Rovelli, a theoretical physicist at Aix-Marseille University in France, points to the example of Euclidean geometry—the geometry of flat space that many of us learned in high school. . . . A Platonist might argue that the findings of Euclidean geometry “feel” universal—but they are no such thing, Rovelli says. “It’s 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 ‘natural thing’ that everyone should do,” he says. “. . . Remember ‘geometry’ means ‘measurement of the earth’, and the earth is round. We would have developed spherical geometry instead.”

Rovelli goes further, calling into question the universality of the natural numbers: 1, 2, 3, 4… 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 “only exists where you have stones, trees, people—individual, countable things,” he says. “Why should that be any more fundamental than, say, the mathematics of fluids?” If intelligent creatures were found living within, say, the clouds of Jupiter’s atmosphere, they might have no intuition at all for counting, or for the natural numbers, Rovelli says. Presumably we could teach them about natural numbers—just like we could teach them the rules of chess—but if Rovelli is right, it suggests this branch of mathematics is not as universal as the Platonists imagine.

Like Pigliucci, Rovelli believes that math “works” because we crafted it for its usefulness. “It’s like asking why a hammer works so well for hitting nails,” he says. “It’s because we made it for that purpose.”

In fact, says Rovelli, Wigner’s claim that mathematics is spectacularly useful for doing science doesn’t hold up to scrutiny. He argues that many discoveries made by mathematicians are of hardly any relevance to scientists. “There is a huge amount of mathematics which is extremely beautiful to mathematicians, but completely useless for science,” he says. “And there are a lot of scientific problems—like turbulence, for example—that everyone would like to find some useful mathematics for, but we haven’t found it.”

Mary Leng, a philosopher at the University of York, in the U.K., holds a related view. She describes herself as a “fictionalist” – she sees mathematical objects as useful fictions, akin to the characters in a story or a novel. “In a sense, they’re creatures of our creation, like Sherlock Holmes is.”

But [Leng says] there’s 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 today’s mathematicians discover are esoteric in the extreme, but in the end, math and science are closely allied pursuits . . . “Because [math] is invented as a tool to help with the sciences, it’s less of a surprise that it is, in fact, useful in the sciences.”

Given that these questions about the nature of mathematics have been the subject of often heated debate for some 2,300 years, it’s unlikely they’ll go away anytime soon.

Unquote.

Nope, anybody who reads this post (or the whole underlying article) has to accept that math depends on axioms or rules. They are invented, not out of the blue, but because they’re plausible and useful. Once you settle on the axioms or rules, discoveries can follow. Therefore, Platonism is wrong. Its competition, which is usually called “nominalism”, is right. I’m glad that’s finally settled.

On a related note, the National Public Radio site has a short article on time travel. The article cites research in physics that purports to show that paradox-free time travel is possible:

Researchers ran the numbers and determined that even if you made a change in the past, the timeline would essentially self-correct, ensuring that whatever happened to send you back in time would still happen. . . . In other words, a time traveler could make changes, but the original outcome would still find a way to happen — maybe not the same way it happened in the first timeline but close enough so that the time traveler would still exist and would still be motivated to go back in time. [According to one of the authors], “No matter what you did, the salient events would just recalibrate around you.”

I’m not qualified to judge the technical merits of this research, but I doubt it shows that paradox-free time travel is guaranteed in every case. If you went back in time and murdered your grandfather before he met your grandmother, and your grandfather didn’t have an identical twin or make a deposit at a sperm bank, I’m pretty damn sure you’re grandmother would have an insurmountable problem generating your precise DNA, whether or not “the numbers” say otherwise.

On yet another mathematical topic, America’s Electoral College is authorized to determine that 63 million counts more than 66 million. That’s why we all need to vote this year.

One-Fifth of “The Peripheral” by William Gibson

Although I was disappointed by William Gibson’s 2004 novel, Pattern Recognition, I began reading The Peripheral, his 2014 science fiction mystery novel, anyway. It was interesting but challenging.

I quote from a blogger, Patrick D. Joyce, who wrote about The Peripheral in 2015:

My own pleasure as a reader of that type of fiction is being left in the dark, confused, gradually putting it together — William Gibson

That’s exactly the kind of reader you have to be to enjoy William Gibson’s new novel.

Peripheral drops you into two separate futures, one near, one distant, without so much as a guide in either, much less some kind of portable universal translator. Nope, you’re on your own. And it gets bewildering at times.

Some way into The Peripheral, I looked at the book’s Wikipedia page to see if I understood the plot so far. I kind of did, but that’s when I discovered I was reading about two separate futures. A few chapters later, I wanted to remind myself who a particular character was. A search for “The Peripheral characters” turned up Mr. Joyce’s post, which includes helpful lists of “Characters in the Near Future” and “Characters in the Distant Future”, as well as a list of words Gibson made up.

That helped me get through the first 100 pages or so, which one reviewer called “uncharacteristically dense”. At that point, a police detective shows up in the distant timeline and asks a few of the characters to explain who they are and what’s been happening (which made Detective Inspector Ainsley Lowbeer — possessed of court-certified perfect recall — my favorite character after just one chapter).

The same reviewer said that after the “uncharacteristically dense first one hundred pages”, the book is “a super enjoyable read”. I’ll never know. I was intending to keep reading, but 400 more pages suddenly felt like a bridge — to the future — too far.

When it comes to fiction, I’m fine with mystery. Intriguing, in general, is better than obvious. (The same applies to non-documentary movies.) When it comes to life, however, clarity is, in general, better than obscurity. A work of fiction, therefore, is subject to a rough aesthetic calculation (actually, so is a work of non-fiction). How do the interesting intrigue, the boring obviousness, the helpful clarity and the confusing obscurity add up? 

In the case of The Peripheral, there is quite a bit of interesting intrigue and just about zero boring obviousness. There is some helpful clarity — for instance, a view of what the future might be like — but way, way too much confusing obscurity. For me, in the first 100 pages, the pleasurable intrigue and clarity outweighed the painful obscurity. But I decided that, having now had some pleasurable exposure to Gibson’s version of the future, another 400 pages wasn’t going to add much more to the experience. The painful obscurity (all these hard-to-follow conversations and descriptions) would outweigh any more pleasurable intrigue and clarity.

I should add that, to my mind, fiction always starts with a problem. Does whatever pleasure I get out of reading this outweigh the fact that the events described didn’t happen? Just as with a work of non-fiction, I always ask myself why I’m spending time on this. Fairy tales can be fun, but I’m prejudiced in favor of reliable information. So, for example, a description of the weather in a novel might be very well-written, but it will make me wonder if I really care about a breeze that never blew or rain that never fell.

There was one thing about The Peripheral that I especially liked though. The Chinese, who are apparently far ahead of what’s left of the human race, have invented a kind of time travel. It’s not the kind that allows people to travel back and forth in time. Nor is it the paradoxical kind in which you can bump into yourself or kill your own grandfather. This kind of time travel is a two-way information connection (so it’s sounds and images that are traveling). 

Gibson’s idea is that it’s possible to establish a communication channel with the past. There’s a computer server in the future that allows this. It wasn’t clear (from the first 100 pages) how the people in the past were able to communicate with the future, considering that they didn’t have the Chinese technology back then, but maybe the 2115 Chinese were clever enough to somehow identify past technology they could connect to, like somebody’s old 2015 computer. 

Anyway, the best part is that when you open a link to the past, it creates what’s called a “stub”. This is a new timeline that branches off. The communication you have with the past is with this new, separate timeline, not your own timeline. That means you can’t interfere with what happened in your past. Your own past stays the way it was, meaning your present stays the same too (your future will be different, of course, because you just did something really cool with the Chinese server.

Except that now you’ve created a different world that will eventually lead to god knows what future for the people over there. Basically, you’re playing at being God, inventing a new universe every time you connect with the past. It’s a nice way to get around the weirdness of time travel, if you don’t mind creating a world in which there might be untold suffering. But who knows? Maybe that new world will be one in which William Gibson’s counterpart chooses clarity over obscurity.

Time Travel: A History by James Gleick

There are two principal topics in this book: time travel and time. Since time travel is fiction, the history of time travel presented in the book is the history of ideas about time travel, mostly ideas expressed in novels like H. G. Wells’s The Time Machine, short stories like Robert Heinlein’s “By His Bootstraps” and movies like The Terminator. Time travel can be fun to think about, and ideas about time travel are suggestive of what people have thought about time, but I quickly lost interest in the topic. So I ended up skimming those sections of the book.

On the other hand, Gleick’s discussion of time itself was worth reading. He covers both physics and philosophy, and does an excellent job explaining complex, competing ideas about time. For example:

You can say Einstein discovered that the universe is a four-dimensional space-time continuum. But it’s better to say, more modestly, Einstein discovered that we can describe the universe as a four-dimensional space-time continuum and that such a model enables physicists to calculate almost everything, with astounding exactitude, in certain limited domains. Call it space-time for the convenience of reasoning….

You can say the equations of physics make no distinction between past and future, between forward and backward in time. But if you do, you are averting your gaze from the phenomena dearest to our hearts. You leave for another day or another department the puzzles of evolution, memory, consciousness, life itself. Elementary processes may be reversible; complex processes are not. In the world of things, time’s arrow is always flying.

It’s an interesting question whether the calculations of the physicists are so accurate because the universe really is a four-dimensional space-time continuum. And is the passage of time some kind of illusion, like many physicists believe? Gleick leans toward time being quite real and physicists taking their models a bit too seriously. I think this would have been a better book if he spent more time on the physics and philosophy and less time on the fiction.