What is the point in pure maths?

What is called "pure" maths at A-level, isn't pure maths

Pure Maths is generally university topics such as Analysis and Group/Number Theory.
These topics are mainly based off proofs and generally are more rigorous than there applied counterparts. (Most applied Maths has theorems based off and proven rigorously from Pure Maths)

Applied Maths is generally seen as Calculus, Statistics, Mechanics, etc...(All of A level Maths generally falls in here). (Not to sure where Linear Algebra falls in this one as it can be a bit of both depending on how it's taught)

A level Maths contains little/no actual Pure Maths in it apart from maybe Proof by Induction in FP1 which begins to follow the logical thought which is used in topics like Analysis.

The A level course really should cover more Analysis based material as A level Maths barely scratches the surface on what studying Maths is truly about.

Finding the dot product of two vectors isn't pure maths

As for the OP, if pure maths is no longer allowed to be classed as such when it starts being useful, obviously no pure maths will be 'useful' by your definition.

No, I'm asking what the point of it is. It seems selfish (that probably isn't the right word though) to study something just because you enjoy it.
I am well aware that eventually much pure maths has applications, but can't we just 'come up' with that mathematics when the real-world problem arises?




I only know about, like differentiation and integration but I think that makes more sense in an applied context (where the variables actually mean something i.e. displacement, velocity, acceleration, rate of change of acceleration with respect to time) than in a core context.
My point is, it didn't need to be discovered until people realised the real-world problems.
A lot of maths was learnt in a real-world context was it not? 'Pure' maths came after 'science' or 'physics' or 'applied maths' whatever you want to call it. I know this doesn't mean much, but still...

The A level course really should cover more Analysis based material as A level Maths barely scratches the surface on what studying Maths is truly about.

I'm fairly sure if analysis was part of the A level maths syllabus I would've been scarred for life and done something else instead.
I think it's probably best to trick people into the subject first with some easy ideas about integrals and then slap them in the face with analysis only after they have paid their first year of University fees.

Off topic but I think people would much prefer it, people want to be challenged. Maths is only interesting when it's hard.

I'm fairly sure if analysis was part of the A level maths syllabus I would've been scarred for life and done something else instead.
I think it's probably best to trick people into the subject first with some easy ideas about integrals and then slap them in the face with analysis only after they have paid their first year of University fees.

Off topic but I think most people would much prefer it, people want to be challenged. Maths is only interesting when it's hard.

While I agree with many sentiments from your post, this one is a bit dubious. We shouldn't forget Gödel's lesson that every set of proofs always has to have some axioma's that cannot be proven, but are rather assumed. Of course, they are often very plausible and agreeable, but it's an illusion to consider it some form of "pure truth" that is completely without assumption. Of course it's silly to suggest that proofs as such are 'silly' though.

Perhaps you are referring to Godel's incompleteness theorem? It sounds a bit like someone described it to you over the phone.

Perhaps I should clarify, Godel's (first) incompleteness theorem states for any consistent axiomatic system there are "simple" arithmetical statements about integers that can be neither proved nor disposed of, using only the axioms of that system.

The second incompleteness theorem states that for any "reasonably powerful" axiomatic systems A and B, the following is impossible: A proves the consistency of B and B proves the consistency of A. So there you have it.

Perhaps this has something useful to say about "mathematical truth", whatever that is. Although I can't see exactly what that would be that you couldn't say anyway without Godel's theorems. Most "ordinary" mathematical theorems don't have anything to do with any kind of Godelian phenomena.

Godel's theorems lead to a slew of developements in computer science. That, apart from mathematical logic, is where they made the biggest bang, when all is said and done. Sometimes in pure maths the non-algorithmicity of deciding whether a certain type of mathematical object has a certain property, is used to deduce that there exists at least one object of that sort which has that property (otherwise there would be an algorithm: one that just says "NO").

Does number theory have any applications? Apparently Paul Erdos, who studied number theory, said that he is proud that number theory has no applications. Has any of his research into number theory become useful yet?

Don't know about Erdos specifically, but number theory finds application in online security where encryption/decryption algorithms usually depend on aspects of primality and factorisation.

I would just like to point out that number theory is itself a huge branch of maths with several subfields and the applications (that have been stated so far) form a very small fraction of all the knowledge in this branch (mostly drawn from elementary number theory). I am starting to feel a kind of a confirmation bias when people start talking about applications of number theory in this way.

I would just like to point out that number theory is itself a huge branch of maths with several subfields and the applications (that have been stated so far) form a very small fraction of all the knowledge in this branch (mostly drawn from elementary number theory). I am starting to feel a kind of a confirmation bias when people start talking about applications of number theory in this way.

Thank you for added clarification. I am by no means a mathematician and I shall not pretend to know the exact details. Although it isn't the case that "someone only described it to me on the phone" , it is true that my understanding lacks mathematic foundation. Even so, I guess it got the point accross

Pure Maths is generally university topics such as Analysis and Group/Number Theory.
These topics are mainly based off proofs and generally are more rigorous than there applied counterparts. (Most applied Maths has theorems based off and proven rigorously from Pure Maths)

Applied Maths is generally seen as Calculus, Statistics, Mechanics, etc...(All of A level Maths generally falls in here). (Not to sure where Linear Algebra falls in this one as it can be a bit of both depending on how it's taught)

A level Maths contains little/no actual Pure Maths in it apart from maybe Proof by Induction in FP1 which begins to follow the logical thought which is used in topics like Analysis.

The A level course really should cover more Analysis based material as A level Maths barely scratches the surface on what studying Maths is truly about.