Arithmetic reasoning captures deeply important properties of the world. ~ American psychologist C.R. Gallistel & Canadian psychologist Rochel Gelman
Are the maxims of mathematics invented or discovered? The answer to this question can be found in a grander inquiry, more readily answered – is there a natural order?
Nature commonly exhibits self-organization and hidden order within apparent disorder. Chaotically tumultuous fluids spontaneously create stripes of coherent flow alternating with turbulent regions. Liquids self-organize into crystalline structures: a phenomenon known as disordered hyperuniformity. Photons in laser light self-organize into fractal patterns. Viewed as particles in a system (instead of linearly), prime numbers exhibit an ordered structure.
The studies of cosmology, physics, chemistry, and biology all aver a natural order. Hence, the universal means for understanding Nature’s ways – through mathematics – cannot be an invention.
During the Edo Period (1603–1867), Japan had limited contact with foreign cultures. (The Japanese had an active interest in Chinese science from 1600 to 1675. The bit of math imported from China during this time was significantly advanced by the Japanese. This was at a time when China was abandoning its own native mathematics under the influence of the Jesuits.) A Japanese school of mathematics – wasan – flourished, without knowledge of European mathematics. Mathematical discoveries in Japan often contemporaneously mirrored those in Europe.
Proceeding from a different basis, calculus was independently discovered by the Japanese and Europeans at roughly the same time. Whereas Europeans had in mind the numerics of atomic infinitesimals, the Japanese approached calculus geometrically – an unlikely convergence if math were mere invention.
When Japan became infested with Western culture during the Meiji era (1868–1912), Japanese scholars lost interest in wasan, as the Chinese had in their own mathematics almost 3 centuries earlier.
One cannot escape the feeling that these mathematical formulae have an independent existence and an intelligence of their own, that they are wiser than we are, wiser even than their discoverers, that we get more out of them than was originally put into them. ~ German physicist Heinrich Hertz
The realm of reason is entangled: structured by Nature, and in a way that that the order may be discerned, using the abstract means by which the order is expressed.
Understanding is a matter of discovery. We invent only contrivances of misunderstanding.
Every new body of discovery is mathematical in form. ~ Charles Darwin