Maths: invented or discovered?
Is Mathematics Discovered or Invented?
Robert J. Marks II July 9, 2023
Some think math is invented. (See an article by Peter Biles.) Evidence, though, points towards discovery. Simultaneous mathematical discovery supports this viewpoint. Many mathematical breakthroughs are sometimes independently reported by two or more mathematicians at roughly the same time. The most famous is the simultaneous discovery of calculus by Isaac Newton and Gottfried Wilhelm Leibniz. Newton was secretive about his discovery and shared his results with only a few members of the Royal Society. When Leibnitz published his discovery of the calculus, Newton charged him with plagiarism. Today, historians agree that the discoveries were independent of each other.
Some Other Examples
Here are some other lesser-known examples of simultaneous discovery.
The Papoulis-Gerchberg Algorithm (PGA). The PGA is an ingenious method for recovering lost sections of functions that are bandlimited. (I describe the PGA in detail in my Handbook of Fourier Analysis.) The PGA was first reported by Athanasios Papoulis1 but was first published in an archival journal, independently, by Gerchberg2. The discoveries occurred independently of each other.
The Karhunen–Loève Theorem, independently discovered by Kari Karhunen3 and Michel Loève4, showed that certain random processes could be represented as an infinite linear combination of orthogonal functions, analogous to a Fourier series.
Non-Euclidean Geometry. Euclid published Elements circa 300 BC. His work wonderfully established Euclidean geometry. It was only in the first half of the 19th century that three men — J´anos Bolyai, Carl Friedrich Gauss, and Nikolai Lobachevsky, independently discovered non-Euclidean geometry. Jenkovszky et al.5 note: "The striking coincidence of independent discoveries… after more than two thousand years of stagnation, may seem almost miraculous."
Space-Variant Processing. Here's a personal example. During my graduate work, I developed a method for performing general space-variant processing. My advisor, John F. Walkup, found out that the same method was simultaneously discovered at Stanford by his PhD advisor's research group. Rather than competing, we agreed to publish all of our findings in the same issue of the journal Applied Optics.6-7
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