1) Some people enjoy researching it. Therefore it adds to their lives.

2) Some people enjoy knowing about it. The researchers therefore add to these people's lives too.

3) Pure research frequently turns out to have an application later - numerous branches of pure mathematics later turn out to have applications. New knowledge is added to the toolkit we can use to understand the world.

entirely in agreement with 3)

I know you were answering the question as written, but IMO 1) and 2) are out of place in a discussion about 'value', with the implied context of 'and why are we spending public money on this useless rubbish?'.

you may as well replace 'learning about number theory' with 'windsurfing'. sure, it adds to their lives, but it's only the public good that has an impact on the discussion.

Why do we build concert halls and stadiums? There's this thing called "culture"... It's hard to define, but you know when it's missing.

(Note, the original question was deleted, so I can't see context.)

The public good most certainly includes the good of the people doing it, though. The notion that those who act on behalf of the public good must commit a sacrifice of their own, individual good is simply wrong.

The most famous quote for this is by G. H. Hardy, a famous number theorist. He said:

"No one has yet discovered any warlike purpose to be served by the theory of numbers or relativity, and it seems unlikely that anyone will do so for many years."

Yet here we are a few decades later with the fundamentals of modern cryptography based on number theory results. Specifically, the difficult in factorizing the product of two primes into the original prime factors (RSA).

So while this specific might seem abstract and without purpose, it often goes in mathematics that we'll find the purpose for it later.

In Sid Meier's Civilization you can only build catapults after you research mathematics.

http://hwhelp.s3.amazonaws.com/hwhdocs/The%20Math%20Behind%2... [PDF]

Pythagoras: why we throw giant rocks at walls.

And those fancy JDAMs and other GPS-guided toys wouldn't be nearly as accurate if GPS did not account for the effects of both the special and the general relativity.

That's physics, not mathematics (let alone number theory).

Mathematics purists would disagree http://xkcd.com/435/

I referred to the fact that Prof. Hardy specifically mentioned relativity in the quote.

But sometimes we discover the math to support we discover the physics. Knowing the physics alone is less useful (but not useless) than knowing the underlying mathematical relationship of the physics.

One of my favorite short science fiction stories is about inter-dimensional warfare in which numbers are the weapons.

Dark Integers by Greg Egan

http://www.asimovs.com/_issue_0710/Dark.shtml

The material on which RSA is based was once thought completely useless, studied for interest alone. You never know when things like this may turn out to be useful.

I'd recommend A Mathematician's Apology for a defense of number theory by someone who was quite prominent in the field. It's slightly dated ("No one has yet discovered any warlike purpose to be served by the theory of numbers or relativity, and it seems unlikely that anyone will do so for many years."), but a lot of it is still relevant, or at very least, interesting.

Whether or not it's relevant, I still second the recommendation to read this book - it's a great read :)

First of all you have to understand that when something is not solved, it's in human nature to try to find a way to solve it, that said, it is also hard to know the implications of a new discovery/demonstration.

That said, prime numbers are a very important and discussed topic in math, they also have high implications on computer systems since prime numbers are at the core of many secure systems' implementations.

So, when we know something new about primer numbers, new doors may open concerning the generation of prime numbers or the detection of prime numbers, basically it's just that a small step in knowledge, let's see where it takes us.

of itself, probably nothing.

asking this question is somewhat like driving to your mother's house, looking out the window on the way and asking, "Why the hell am I on a freeway onramp? I didn't want to be here, I want to be at my mother's house!"

academic discovery is a journey, building on top of the progress made by others along the way. not all points along that route have intrinsic value.

public-key crypto has incredibly high real-world value. most of the things learnt by number theory researchers building up to RSA's discovery - not so much.

What does this electron circling an atomic nucleus adds to our lives? It's purely theoretical

What does this adding numbers that only can be zero or one add to our lives? It's purely theoretical

What does this imaginary number things and the square root of negative numbers adds to our lives? It's purely theoretical

It would be nice to take a common gadget, say an iPhone or an Android phone, and split it up into small bits and pieces (including software) and then list all the pieces which are based upon a theory or research which decades ago was considered irrelevant and useless. I.e. all the findings which make up an iPhone, but where decades ago people would have said 'what does this add to our life'. What I can think of, right now, is

- Cryptography (iCloud, iMessage, Online Banking): Is all based on prime numbers. For hundreds of years, prime number theory in math was considered a purely academic exercise, something with no real world usage at all. And suddenly it is the basis of many things we consider granted. [1]

[1] http://www.claymath.org/posters/primes/

One example of the applicability of number theory is cryptography.

Pure mathematics has a habit of trickling down to real world applications.

Sometimes that influence is direct, as seen with cryptography. In fact much of the worlds security is based on prime numbers. eg HTTPS (SSL / TLS) uses RSA.

Sometimes that influence is indirect. Such as math theorems that are applied to physics. Which in turn lead to real world products (eg faster processors, GPS, etc).

You ask: 1. Is X good? How is it good? 2. Is it entirely separate from reality?

While you might think of it as a simple question, you asked it in the general sense, without any context. In the general sense, these are two central questions in 2 separate domains of philosophy. 1. Ethics/Moral Philosophy 2. Dualism (philosophy of mind). Neither has a simple answer, although you could start form Wikipedia or the Stanford Encyclopedia of Philosophy.

Furthermore, the idea and enterprise of science and math is complex. It's not familiar to the layman. Instead, try approaching the question with more familiar ideas:

Think of a person who doesn't use Facebook at all. 1. What does Facebook add to his life? 2. As far as he can see, Facebook is a purely virtual social network, isn't it? 3. Why is it worth everyone's time to check/use Facebook?

Think about it from the perspective of a non-Facebook user. Perhaps one of the tribesmen in Africa should you know any.

Because any new discovery could open up possibilities that we don't yet understand?

Because it's there :)

In addition to what other people said, some of the methods used to approach the problem may be reused on other problems.

This question prompts the discussion - which can be a good thing, please don't just downvote because you disagree.

On the other hand, it's a discussion I've seen over and over and over again. And it's not actually specific to this particular link at all, so it would be reasonable to downvote if you feel the discussion is out of place.

And you'll note that currently, ~90% of the comments in this thread are in response to the question -- it's something of an attractive hazard.