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.

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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........

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