Nearly twenty years after I graduated high school and my last calculus class, I still get that nightmare where I’m at the exam for a calculus course I somehow forgot to attend, or that I faked my way through with absolutely no idea what was going on. When I wake up, I have a hard time being sure it wasn’t all real.

But actual students *actually* working to grasp calculus, algebra or trigonometry can’t say for sure whether or not they are studying “real” stuff either. Even though so much of our world relies on math – from algorithms to rocket engineering to cash registers to mathematical equations describing real phenomena in the universe – there isn’t yet a consensus on whether math is actually objectively real or just some stuff humans invented.

Indeed, within the sub-field of philosophy of mathematics, mathematicians, philosophers and quantum physicists advance and argue about theories regarding the “realness” of numbers and the logical systems by which they are used in mathematics. The views on this range from “the universe *is* pure mathematics” to “mathematics is an internally-consistent logical construct with no relation to real things in the real world.” Much of the discussion depends on the historical development of mathematical thought and scientific understanding — but digging deeper into the question might challenge our assumptions about not only the nature of numbers, but the nature of the universe itself. Or it might inspire us to take up math.

Dr. Penelope Maddy, who is a professor emeritus of logic and philosophy of science and mathematics at U.C. Irvine, is a prominent American philosopher of logic, science and mathematics known for her work on mathematical realism. Simply put, it’s the idea that mathematics exists independently of human cognition and that we discovered math rather than inventing it. “Realism in Mathematics” was the title of Maddy’s first book, published in 1990, followed by “Naturalism in Mathematics” published in 1997, which explores mathematical naturalism. These days, she sees herself as having landed somewhere between the two extremes.

“In the days of Galileo and Newton, it wasn't unreasonable to regard mathematics as the language of the Great Book of Nature,” Maddy explained in an email interview with Salon. “But over the course of the 19th Century, developments in both mathematics and science undermined this view.”

Euclidean geometry remains “true” in one of a wide range of possible “abstract mathematical spaces.”

Geometry as developed by Euclid, she said, was once thought to be “a unique collection of undeniable truths about physical space.” But then non-Euclidean geometries were developed and so Euclidean geometry was reduced to one competing theory among many. When Einstein was able to hang his physical ideas about gravity (known as general relativity) on the mathematical structure provided by a different kind of geometry, Riemannian geometry, that could have been curtains for what was once considered Euclid’s undeniable truth about the real world.

Indeed, after Einstein, Euclidean geometry might be understood as a falsified theory of physical spacetime, Maddy told Salon. But, as she explains further in “Defending the Axioms: on the Philosophical Foundations of Set Theory,” her 2011 book about set-theory axioms (the fundamental assumptions that allow for mathematical proofs) mathematicians “rescued” Euclid. They now describe Euclidean geometry as, sure, not applicable to physical space, but not false overall — it remains “true” in one of a wide range of possible “abstract mathematical spaces.”

“Mathematical theories are protected from empirical falsification by positing a special realm of abstracta about which they remain true”, Maddy writes.

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Even at this point, Maddy told Salon, we could still see mathematics as conveying the literal truth about the world: “Mathematics may offer us many more options than we imagined, but the one that fits, fits perfectly. That comforting idea came under pressure when the burgeoning and wildly successful field of applied differential equations — the leading tool of physical sciences at the time (and still central) — turned out to be based on an effective 'smoothing out' of a complicated and chaotic atomic microworld.”

In “Defending the Axioms,” Maddy describes the way we’ve gone from mathematics being seen by Plato as an eternal form, like truth or beauty, to identifying math and science as almost one discipline, or at least equals working together, from the time of the scientific revolution onward, to beginning to accept that math and the real world might diverge significantly, thanks to the introduction of new mathematical concepts like n-dimensional spaces and negative numbers in the first half of the 19th Century. And then on to 20th Century empiricism in which the practices of the scientific method are brought to bear on mathematical questions.

Having used such thinking to dismiss empty philosophical language such as Heidegger’s “the Nothing itself nothings,” which Vienna Circle philosopher Rudolf Carnap ridiculed as a pseudostatement, Carnap’s fellow logical positivists now feared that because there are no conditions under which mathematics it could be tested empirically, it might not count as science. They concluded that math must be more linguistics than anything else, with its statements or axioms needing internal logic but not requiring any relationship with the outside world to be valid. And then – skipping a bunch of mathematical and philosophical history – here we are today, with our chaotic quantum mechanics microworld.

It’s up to applied mathematicians and physical scientists, Maddy said, to figure out which mathematical description of abstract structures *can* be adapted to capture which phenomena in the real world, to determine what sort of fudges and idealizations will allow for that adaptation, and to explain why or when it’s reasonable – "benign," as she put it — to make those fudges or idealizations.

Mind you, it’s not that quantum mechanics is too wacky to be described mathematically.

Carnap’s fellow logical positivists now feared that because there are no conditions under which mathematics it could be tested empirically, it might not count as science.

“Mathematized quantum mechanics is one of the most successful and precise physical theories we have,” Maddy emphasized. “What we don't understand, for now, is why it works so well. That is, what worldly phenomena are responsible for its success? Some people react to this by limiting the goals and capabilities of science — it's just an instrument for generating predictions, say — but this seems to me an overreaction. No doubt understanding the quantum world is a challenge, and it may even be that our human intellect can't do it, but that's no reason to conclude that it isn't latching onto something in the world.”

It’s just that, in her view, mathematics *also* includes descriptions of structures that, by contrast, do not latch onto something in the world. Plato, similarly, thought that in addition to describing real stuff in the real world, mathematics described "real" stuff that nevertheless didn’t exist as a physical object in the real world — abstract structures that exist outside time and space.

This is Platonism, and since most mathematicians believe the concepts they study are “real” — they can agree on precisely what a particular concept consists of in order to talk about it and work on it and describe as elegantly as possible —most mathematicians today are Platonist to at least some degree. On the other hand, they might really be *formalists*. Formalism, as set out by those math-saving logical positivists, and others, is where mathematics focuses on axioms — basic assumptions accepted as true, about which you can make inferences using specific rules, resulting in proofs, or theorems, deduced from inferences made about the axioms. The content of the axioms — their so-called “mathematical objects” — are whatever the axiom defines them to be.

Thus there’s no* need* for a relationship between those objects and the physical world, and the non-philosophers can happily pursue abstract mathematics without worrying about the issue. But such formalism — while having basically rewritten the practice of mathematics in the 20th Century — side steps rather than actually solving the philosophical problem of why there sometimes seems to be such a relationship, and sometimes does not. Maddy has pointed out that while mathematics were historically developed in an attempt to solve real-world problems, we tend to focus on the successes, forgetting how often mathematical answers to such problems have failed or or only been arrived at after a long process of trial and error.

Thus the fact that mathematics are sometimes useful in solving problems in the real world doesn’t really tell us anything about whether or not mathematical objects are necessarily objectively real.

Then again, what if the universe just* is* mathematics? The name Max Tegmark is associated with this extreme version of Platonism or of mathematical realism: what he called the ‘mathematical universe hypothesis’ in his 2014 book, “Our Mathematical Universe: My Quest for the Ultimate Nature of Reality.” He’s not fazed by those developments in math and physics since the 19th Century that Dr. Maddy described. Never mind mathematics’ "amazing effectiveness" at describing and addressing problems in the real world: the fact is, in his hypothesis, the real world *is* mathematical structures.

Tegmark, a Swedish-American astrophysicist who works at MIT and heads up a non-profit focused on the existential risk of advanced AI, explores this brain-melting idea in the context of another one: the existence of a nested series of multiverses. It can, however, it can be challenging to fellow scientists and theoreticians as well as the rest of us to grasp what statements like the following, from "Our Mathematical Universe," actually *mean*: “We’re on a planet in a galaxy in a universe that I think is in a doppelgänger-laden Level I multiverse in a more diverse Level II multiverse in a quantum-mechanical Level III multiverse in a Level IV multiverse of all mathematical structures.”

After some back-and-forth with Salon, Tegmark ultimately wasn’t able to make time for an interview, suggesting that our Level 1 multiverse is a busy place in human terms, however elegant its mathematical structures may be.

"Mathematics may offer us many more options than we imagined, but the one that fits, fits perfectly."

To U.K. polymath Raymond Tallis, a retired physician and author of numerous books of philosophy, fiction and science, the problem with the mathematical universe hypothesis isn’t, as some of Tegmark’s critics have variously argued, just one of clarity, falsifiability or tautology (circular arguments: all mathematical structures are real, and everything that exists mathematically has a real existence in the real universe… which is a mathematical structure.) Regardless of the dubious logical strength of his hypothesis, Tegmark is just missing something pretty important, in Tallis’ view.

As are Pythagoras (“All is number”), Aristotle, Plato and Galileo, all of whom saw in the principles of mathematics, in Aristotle’s words, “the principles of all things.”

What’s lacking, argues Tallis in an article on the bizarre effectiveness of mathematics, are those aspects of reality that “resist mathematization.” In a video interview with Salon, Tallis explained it very simply. Say you describe a table in the form of mathematical formulae. You know its dimensions, its weight or mass, perhaps the chromaticity and luminance of how its color might appear to an average human viewer (color being a prime example of a thing that exists more in the mind than in and of itself). You might describe it in terms of the atomic structure of the wood it’s made of. None of this will get you very far, though, Tallis said: “The table basically disappears and is replaced by its quantity.”

This sounds much like cosmologist Andrew Liddle, in his review of Tegmark’s book in Nature, writing: “It is impressive how far Tegmark can carry you until, like a cartoon character running off a cliff, you wonder whether there is anything holding you up.”

The consequence of these scenarios is “We drain the universe of qualities. But it gets worse,” Tallis said. Worse is that you get yourself into “a situation where you couldn’t tell the difference between the universe and nothing.”

Is it possible, though, that rather than being circular arguments or an argument that ultimately misses the point of what makes reality real, that we are challenged in grasping the idea that the universe *is* math — our instinctive reluctance to accept that a table might be adequately defined by mathematics — by our own cognitive limits, which make us insist on cosmically irrelevant qualities of this table of ours (in philosophy, these non-mathematical aspects of a thing are actually called its "secondary qualities"), rather than the important stuff — what we’d consider the more quantitative or mathematical aspects of a thing.

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So, might a true, meaningful description of that table actually be one that leaves out petty human concerns? After all, the mathematics that we use was derived by human brains, just as the technology we use to observe and understand the world is. Even the Platonists, who believe a mathematical concept is real if they can reach consensus about what it is, are working with brains limited by results of natural selection operating on natural variation in the environments in which we evolved.

“It doesn't follow [from the fact that we have such limits] that we can't work hard to see the world as clearly as we can,” Maddy explained. “Sometimes we can even see around some pretty basic cognitive structures.” As when Einstein pushed us to see that both time and space are relative, not absolute.

“Understanding quantum mechanics might require us to look around even more basic cognitive mechanisms,” Maddy hinted, “like the very idea of relatively stable objects with relatively stable properties. Kant would say we can't do it, but he would have said the same about Euclidean geometry.”

In the words of British mathematician Marcus du Sautoy: “It’s not a boring place to be, the mathematical world. It’s an extraordinary place; it’s worth spending time there.”

Whether or not that’s a real place. Or an imaginary place. Or maybe even every place.

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