Can you provide me historical examples of pure mathematics becoming "useful"?

I am trying to think/know about something, but I don't know if my base premise is plausible. Here we go.

Sometimes when I'm talking with people about pure mathematics, they usually dismiss it because it has no practical utility, but I guess that according to the history of mathematics, the math that is useful today was once pure mathematics (I'm not so sure but I guess that when the calculus was invented, it hadn't a practical application).

Also, I guess that the development of pure mathematics is important because it allows us to think about non-intuitive objects before encountering some phenomena that is similar to these mathematical non-intuitive objects, with this in mind can you provide me historical examples of pure mathematics becoming "useful"?

Solution 1:

Here are few such examples

• Public-key cryptosystems based on elliptic curves, factorization, trapdoor functions, lattices and hyperelliptic curves.
• Use of algebraic topology in distributed computing and sensor networks. Topology has uses in various other branches engineering as well.
• Use of differential geometry for computer graphics, computer vision algorithms, robotics and general relativity.
• Error-correcting codes based on algebraic geometry. Algebraic geometry has also applications in robotics.
• Lattice theory is used in program analysis and verfication.
• Group theory is used in chemistry as well as physics
• Tropical geometry, a branch of algebraic geometry, has applications in mathematical biology.
• Digital electronics is impossible without Boolean algebra.
• Topos theory has been applied to music theory

Solution 2:

Negative numbers and complex numbers were regarded as absurd and useless by many mathematicians prior to $15^{th}$ century. For instance, Chuquet referred negative numbers as "absurd numbers." Michael Stifel has a chapter on negative numbers in his book "Arithmetica integra" titled "numeri absurdi". And so too were complex/imaginary numbers. Gerolamo Cardano in his book "Ars Magna" calls the square root of negative numbers as a completely useless object.

I guess the same attitude towards Quaternions and Octonions would have been prevalent, when they were initially discovered.

This is from my answer to a similar question here.

Below are some uses of negative and complex numbers.

Solution 3:

Here are some examples of pure mathematics that has shown to have real applications - however I am not sure of the origins.

• Radon transform applied to tomography e.g ultrasound detection of babies. ;)
• Partial differential equations applied to heat, waves, weather,finance etc.
• Graph theory applied to logistics of transportation.
• Stochastic analysis applied to finance e.g. option pricing leaded by Black, Scholes and Merton.
• Discrete Fourier transform (or rather discrete cosine transform) applied to image analysis e.g. jpeg.
• Control theory is used in order to strengthen signals in the telecom industry, as well as calibrating cd-drives. Control theory is pretty much based on Fourier analysis and on the theory of $H^\infty(\mathbb{D})$ (i.e. the space of bounded analytic functions on the unit disc).

Solution 4:

The discussion of conic sections by the ancient Greeks, see the wikipedia article, gave the basic definitions required by Kepler to formulate his law of planetary orbits. Of course the Greeks did not have term "pure mathematics".

An example from pure mathematics of the 20th century is the applications of category theory to computer science.

People also forget that the notion of the graph of a function was invented by Descartes and of course is now ubiquitous in our daily papers, to show clearly how bad things are getting! For more information on the invention of Cartesian coordinates, see the wikipedia entry on Descartes.