Why is pure mathematics important? More generally, why do some scientists deal with inaplicable notions?

Question

I am a freshman, mathematics. I have a presentation assignment for a class. I am expected to talk about the necessity of abstract sciences. The thesis I need to argue is the following: ''Some people are against some researches like space explorations because they find those just a waste of time and money.'' Indeed, even in movies we can see this, for instance ''Interstellar''. I have already prepared a speech in which I mention ''A Mathematician's Apology'' by Hardy. There is also some information from sources like NASA about why we should explore space.

Personally, I believe that humankind has much more to do than just surviving, and so we need science that is not necessarily "applicable" in the narrow sense. But I feel like I need some historical sources discussing the role of "pure" mathematics and science, like the Hardy's book. Are there other such sources? Were there historical episodes when society or a subset of society turned against scientists because their science was "inapplicable"? How did scientists respond? Any help would be appreciated.

Answer 1

Historical experience shows that pure mathematics is one of the most useful parts of science. Pure mathematicians discover things which find applications later. Without pure mathematics, most of the "applied mathematics" and other sciences would be simply impossible. Let me give just one example. Ancient Greeks intensively studied curves called conic sections. There are various speculations of historians, about what motivated them, and it seems that nothing but pure curiosity, and other related pure mathematics problems.

1000 years later it was discovered by Kepler that planets and satellites move on conic sections. This was an application to the "real world", astronomy, but still astronomy is a "pure science". But discovery of Kepler triggered discoveries of Newton: to explain this motion of planets the whole science of mechanics was developed. Which is a basis of all modern engineering.

Discoveries of Kepler and Newton would be impossible without that work of the Greeks 1000 years ago. And this pattern is essentially omnipresent, if one traces the development of science. In the modern times, the period between a discovery in pure mathematics and its influence on applied sciences is usually shorter.

I am not saying that EVERY mathematical discovery found or will find applications, but Mathematics is like a living organism: you cannot remove parts of it, all parts are necessary. And you never know in advance which part will play a role in applications.

Answer 2

Basic cryptography used everyday by millions of people (security of email/text communications or e-commerce) depends on ideas from pure mathematics that were developed from pure intellectual interest rather than practical applications:

1. Fermat's little theorem is the backbone to RSA and is attributed to Fermat in the 1600s. Its standard formulation in terms of modular arithmetic goes back to Gauss in the early 1800s.

2. Elliptic curve cryptography depends on elliptic curves over finite fields. Elliptic curves were first studied over the complex numbers in the 19th century, developing from the study of elliptic functions (more pure math). Finite fields, in general, were created by Galois in the early 19th century as a topic in pure math. The consideration of elliptic curves over finite fields first came about in pure math and dates back to the first decades of the 20th century.