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Is Mathematics Discovered or Invented?

Photo credit: John Moeses Bauan via Unsplash.

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 Theoremindependently 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

Einstein’s Shoulders

In the context of the argument for discovery, some inventions can curiously be considered discovered rather than invented. Isaac Newton famously said that “if I have seen further [than others], it is by standing on the shoulders of giants.” Einstein built on Newton’s discoveries in classic physics and, in turn, stood on Newton’s shoulders with the formulation of relativity. Modern physicists stand on Einstein’s shoulders. The advancement in technology can likewise be considered standing on an ever-increasing stack of shoulders. This is certainly the case in artificial intelligence. Rosenblatt and Widrow’s early work on AI led to discovery of error backpropagation neural network training that led to deep convolution neural networks, deep learning, and the generative AI we use today.

Inventions can be discovered. An example of an invention being discovered by two men is the telephone. Alexander Graham Bell is credited with inventing the telephone. But according to the Library of Congress:

Elisha Gray, a professor at Oberlin College, applied for a caveat of the telephone on the same day Bell applied for his patent of the telephone … Bell’s lawyer got to the patent office first. The date was February 14, 1876. He was the fifth entry of that day, while Gray’s lawyer was 39th. Therefore, the U.S. Patent Office awarded Bell with the first patent for a telephone, US Patent Number 174,465 rather than honor Gray’s caveat.

If true, both Gray and Bell were standing on the shoulders of those who proposed the telegraph and glimpsed the possibility of the telephone.

Philosophers might contemplate the similarity of the discovery of invention with the debate between predestination and free will. If inventions and advancements in mathematics are discovered, the future is, in a sense, predestined by our discoveries. The pros and cons of the debate will continue well beyond the arguments presented here.

Cross-posted at Mind Matters News.


  1. A. Papoulis. A new method of image restoration. Joint Services Technical Activity Report, 39, 1973–74
  2. R.W. Gerchberg. Super-resolution through error energy reduction. Optica Acta, Vol. 21, pp. 709–720, 1974.
  3. Kari Karhunen ‘Zur Spektraltheorie Stochastischer Prozesse’, Ann. Acad. Sci. Fennicae, (1946), 37
  4. Michel Loève ‘Probability Theory’, Princeton, N.J.: VanNostrand, 1955
  5. László Jenkovszky, Matthew J. Lake, and Vladimir Soloviev. “János Bolyai, Carl Friedrich Gauss, Nikolai Lobachevsky and the New Geometry: Foreword.” Symmetry 15, no. 3 (2023): 707.
  6. R.J. Marks II, J.F. Walkup, M.O. Hagler and T.F. Krile “Space-variant processing of one-dimensional signals,” Applied Optics, vol. 16, pp.739-745 (1977).
  7. Joseph W. Goodman, Peter Kellman, and E. W. Hansen. “Linear space-variant optical processing of 1-D signals.” Applied Optics 16, no. 3 (1977): 733-738.