Nature of Math
The saga continues! Is math a science? Is math discovered or invented? I thought I had a solid answer on the former question, but Alas! Mathematics Strikes Back!
The debate on these two questions were brought front and center in my History of Math class. The class was divided on this issue with a large portion of the class believing math was a science. A small portion believed math is completely separate from science. A small few were still agnostic on the issue. The class was less divided over whether math is discovered vs invented. So, I'll start there.
Math Discovered or Invented?
Discovered. Definitely. As our class has discussed and from what Pickover has discussed in The Math Book (blog post), there are math milestones that were worked out by different people simultaneously, but independently from each other (1). Credit for Calculus has been given to both Isaac Newton and Gottfried Wilhelm Leibniz (1). Another example was when hyperbolic geometry was being worked out by three different mathematicians at the same time: Janos Bolyai, Nikolai Lobachevskii, and Carl Friedrich Gauss (4). If math was invented then it would just be a random collection of concepts that were pulled out of thin air by the imaginations of great mathematicians, and it would be hard to make the case that these simultaneous creations by different mathematicians occurred by coincidence. How could they just happen to have the (almost) exact same creation at the same time? That would be like Leonardo DaVinci and Michelangelo painting the exact same Mona Lisa at the same time. Don't Even.
Furthermore, mathematics is far too ordered, structured, and interconnected to be randomly invented. In BBC Horizon's documentary, The Last Theorem, a connection was established between three seemingly unrelated math topics: elliptic curves, modular functions, and Fermat's Last Theorem (5). To prove Fermat's Last Theorem, Andrew Wiles had to prove that modular forms and elliptic curves are the same, because when someone believed that Fermat's Last Theorem was false, those equations looked like an elliptic curve that wasn't modular (5). In the end, Andrew Wiles proved Fermat's Last Theorem was true by showing elliptic curves and modular forms were the same (5).
The reason behind the order, structure and interconnectedness in mathematics is logic. Jordan Ellenberg in his book, How Not to Be Wrong: The Power of Mathematical, describes math as using common sense, and applying rational thinking or logic to problems (the problems don't have to be mathematical) (4). Eugenia Cheng, in a presentation at Grand Valley State University, explained that "mathematics is the logical study of logical things" (6). Logic is the underlying pinnings that unites all of mathematics together.
A part of mathematics logical structure are axioms. Ellenberg describes axioms as self-evident rules that form a system of mathematics (4). They are used to manipulate elements in the system to prove logical statements about the system, and these logical statements are based on truth tables, the language of logic. When Janos Balyai decided to remove the Parallel Postulate from Euclid's Axioms, he worked out a new geometry - hyperbolic geometry (4). He found that the parallel postulate was neither contradictory, which would make Euclid's geometry false, nor mandatory, which would make it the only possible geometry (4). Hyperbolic geometry is just another geometric world in addition to Euclid's geometry or, more broadly speaking, mathematical space in mathematics (3). When another world was discovered with a different set of axioms, the nature of axioms changed (3). Axioms are more like hypotheses that lay the framework for the world one is in (3). But even though there is an important difference between hyperbolic and Euclidean geometry, there is overlap because there is overlap of axioms between the two worlds and the logic that is used to prove statements in all of mathematics is the same. Another example is with integers and rings. Integers are a specific type of ring, and depending on which axioms are used, proven statements (theorems) in the integers can be used or slightly modified to prove similar results in rings.
Furthermore, when I think of something that is invented, I think of art and games. When children come up with a new game, they create their own rules. Most of the time, these rules aren't logical. Take Monopoly for example. Why do players get free money whenever they pass Go? That doesn't make sense or even apply to the real world. It's a whimsical rule, and mathematics are not governed by whimsical rules.
Mathematics are governed by logical rules, specifically truth tables and axioms. Axioms, which are the foundation of proof and mathematical theorems, can be compared to rules of a game. With games, rules can be changed and the game can still be played. To continue with the Monopoly example, if the rule of getting money when passing Go disappears, all the other elements of the game remain. But in math, the rules, or axioms can't just change or disappear and leave the theorem or concept fundamentally different. If they do, all the mathematical concepts that relied on those axioms and theorems crumbles (4 and 5). This is the key difference between games (rules without logic) and math (rules with logic). Axioms and truth tables, the logical rules, makes math a discovered endeavor, and rules without logic are of human invention.
Lastly, mathematics is not a solely human invention. If math is truly a human invention, then only humans could do it, understand it, and come up with it. But primates and other animals can be trained to count, while a type of Saharan Ant knows the number of steps and the lengths of their stride to find their way back to their nest (1). But as someone in the class mentioned, if math is discovered, then it has to be found in nature, which would make it a science. And that brings us to the next question.
Is Math a Science?
I was a firm believer that math is not a science. Now, there is room for doubt. I have written a detailed blog post on this topic, so I will only give a brief description of what I already discussed. The pursuit of science is to discover how natural processes work, while the pursuit of math is discovering relationships with numbers. Math is found and can be applied to nature. In fact discoveries in math started with the ancient Greeks describing our 3-D world with geometry (3). But even though math can be found or be used to describe nature, there are abstract ideas that transcends the physical world. The example I gave before was a one-dimensional line expanding to infinity. Furthermore, math's capability to describe our natural world with accuracy does not make it the reason why it works (2). Just because math is accurate in modeling natural phenomenon, it does not mean it's the explanation behind the process (2). There could be a completely different reason that makes the process act that way (2). Accuracy does not necessarily make something true (2).
But then I came across the Mathematical Universe Hypothesis. Max Tegmark is the man who came up with the hypothesis (1). According to Clifford Pickover's explanation of the hypothesis, "...our physical reality is a mathematical structure and that our universe is not just described by mathematics - it is mathematics" (1). If this hypothesis is proven to be true, then mathematics is not just something that can be applied and describe our natural world with accuracy - it is the foundation, the reason, and the process behind how everything in our natural world works. And that would make math a science....and it would make science provable with mathematical proof. But at this point, the Mathematical Universe Hypothesis is just a hypothesis, and the divide between science and math, nature and number, clear. And I can continue believing, a little shakily, that math is not a science.
(1) Pickover, Clifford A. The Math Book. Sterling Publishing Co., Inc. New York, NY. 2009.
(2) Wigner, Eugene. "The Unreasonable Effectiveness of Mathematics in the Natural Sciences." College Course 9. http://www.dartmouth.edu/~matc/MathDrama/reading/Wigner.html
(3) Space (mathematics). Wikipedia. https://en.wikipedia.org/wiki/Space_%28mathematics%29
(4) Ellenberg, Jordan. How Not to Be Wrong: The Power of Mathematical Thinking. The Penguin Press. New York, NY. 2014.
(5) Simon Singh and John Lynch. Fermat's Last Theorem Documentary. BBC Horizon. 1996.
(6) Eugenia Cheng's Presentation at Grand Valley State University. Quotation from MathHombre's Tumblr page: http://mathhombre.tumblr.com/post/129844731594/eugenia-cheng-the-logical-witty-charming-and