Math is critical in most science related subjects, such as physics and astronomy. People would agree that math has helped and even influenced science. Yet no one mentioned the other way around: science influencing and helping math.

My question is that has science influenced and/or helped math in any way? All help is appreciated!

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    $\begingroup$ This question is much too broad to properly address in a 10,000 page book, let alone on this site. The mutual influences of mathematics and physics (to name one of the sciences) on each other are for all practical purposes innumerable. $\endgroup$ – Danu Mar 9 '15 at 18:24

On sciences other than physics and astronomy, there was already such question: What new mathematics was inspired by biology and chemistry?

Concerning physics and astronomy, examples are so abundant that no list of reasonable length can be made.

Here are just some random examples. In fact more than half of the whole mathematics comes from physics.

Trigonometry was invented for the needs of astronomy. (First spherical, then flat). Calculus was developed with one of the main motivation coming from physics. Almost all theory of differential equations, ordinary and partial was invented for applications to physics (in 18 century the main motivation came from celestial mechanics, later also from other parts of physics). Fourier transform was invented by Fourier when he studied heat conduction in solids. Ergodic theory and "chaos theory" originate in celestial mechanics. Tensor calculus originates from Riemann's work on continuous mechanics.

Quantum mechanics still makes an increasing influence on pure mathematics. First it stimulated the development of functional analysis, operator theory, then quantum algebra. Statistical mechanics gave rise to many new mathematical theories. Of the modern physical theories, let me mention string theory and quantum gravity which make very large influence on the modern mathematics.

So it is not an exaggeration to say that more than 1/2 of the whole body of mathematics comes from physics and would not be invented otherwise. Perhaps much more than 1/2. When I write "comes from" I mean it: not only problems are suggested by physics but also methods of their solution, and in many cases physics suggests solution of the problems which DO NOT come from physics.

An interesting case study is contained in the paper of Ruelle, Is our mathematics natural? in Bull. Amer. math. Soc. (available free online). He discusses the following question: if an extraterrestrial civilization will be ever discovered, will its mathematics be similar to ours? The answer is probably "yes", exactly for the reason that mathematics comes from physics.


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