Is there any such thing as totally new math?

[His Divine Shadow]

I am wondering, isn't all math, even the most complex and modern forms of math in existence today, still based on basic mathematical axioms? Or is there some form of totally revolutionary and new math that is well and truly "invented" instead of derived from existing mathematical rules?

I was given the example of Twistor Theory as a form of entierly new maths but suffice to say I don't have the education to even understand what twistor theory is.

[Wyrm]

I am wondering, isn't all math, even the most complex and modern forms of math in existence today, still based on basic mathematical axioms? Or is there some form of totally revolutionary and new math that is well and truly "invented" instead of derived from existing mathematical rules?

A new mathematical field is created when a set of axioms are chosen for the objects in the math, as the properties of those objects are characterized by those axioms. It's possible to create a totally new math by choosing a set of esoteric axioms, but usually math progresses by finding parallels between different fields; mathematical objects from one field will often have properties of objects in another field, so you can use the second field to make statements about objects in the first.

I was given the example of Twistor Theory as a form of entierly new maths but suffice to say I don't have the education to even understand what twistor theory is.

Twistors are a construction of more basic maths, as it founded upon an isomorphism between two previously known groups. Whoever told you it's 'totally new' is full of shit.

[The Grim Squeaker]

Depends what you call basic theorems.
0? Fractions? Negative numbers? Complex numbers (square root of -1)?

Negative numbers or even 0 would have been new up until a few centuries-millenia ago.

[Surlethe]

The OP's question also touches upon an interesting philosophical question: is math invented or discovered? After all, A -> B is always true whether we realize A -> B, but to show that A -> B, you have to construct a proof.

[The Grim Squeaker]

The OP's question also touches upon an interesting philosophical question: is math invented or discovered? After all, A -> B is always true whether we realize A -> B, but to show that A -> B, you have to construct a proof.

I would say invented. It belongs to the same intellectual plane as philosophy, humans invent it, rationalize and prove it (theorems, rules, etc') and use a moral (in philosophy's case) or numerical base (lookit my fingers!) which suits us.

Dammit, I don't know how to explain what I mean more clearly...

[Steel]

I'd lean towards discovery, as the existence of unknowable quantities (Incompleteness Th'm, etc) steers towards 'exploration' starting from a given axiom set, with some places being 'unreachable'.

Also, I can confidently say there is more totally new mathematics than there is new literature.

EDIT: Alas, DEATH, you dont know quite enough maths to appreciate the bigger picture

[Grog]

Is the distinction useful? Clearly it shares some properties with both but I wouldn't say it is either. When one knows how we reach new math one doesn't really need explanations like it is invented or discovered. And if you try to explain to someone who doesn't know it could possibly lead to confusion. Basically what other than a way to describe how math is done is the importance of the invention vs discovery argument?

[The Grim Squeaker]

EDIT: Alas, DEATH, you dont know quite enough maths to appreciate the bigger picture

Ask me again once I start the degree .

Also, what are you studying in Cambridge?

[Gigaliel]

I'm not one to speak for HDS, but I think he might be referring to calculus or set theory or something similar? They were both based on previous work, but have dominated notation in many fields.

I suppose the correct term might be "revolutionary" or "extremely useful", but I'd say that might be the closest criteria to what HDS laid out. Beyond some machine that invents arbitrary axioms until it generates something entirely disconnected from current research, anyway.

Depends what you call basic theorems.
0? Fractions? Negative numbers? Complex numbers (square root of -1)?

Negative numbers or even 0 would have been new up until a few centuries-millenia ago.

I'd say these would count, but I'm not really sure if these ideas could be considered any more of an intuitive leap than most mathematical research. Not that I'm sure you could even measure that, which would be the main problem with the term "revolutionary" other than defining it by popularity. And, needless to say, I do not consider that satisfactory.

[Themightytom]

The OP's question also touches upon an interesting philosophical question: is math invented or discovered? After all, A -> B is always true whether we realize A -> B, but to show that A -> B, you have to construct a proof.

I wouldn't use invented, or discovered, but rather conceptualized

Also this reminded me of a question someone my client was freaking out over last weak.

"I was just at work putting up a door frame, and it hit me: Does the door we put up define a space, or is that space there and we just put a frame around? Did we make a hole by building a wall? or did we build a walla round a hole!"

OCD and schizophrenia are a wierd combination...

[Darth Wong]

I am wondering, isn't all math, even the most complex and modern forms of math in existence today, still based on basic mathematical axioms? Or is there some form of totally revolutionary and new math that is well and truly "invented" instead of derived from existing mathematical rules?

Yes. You can find this new math on Wall Street. For example, Wall Street taught us that if you combine 100 objects, each with a 50% probability of decreasing in value, you get a giant meta-object with a 0% probability of decreasing in value!

[Kuroneko]

The last significant foundational development was due to category theory, which both generalizes set structure and shifts the fundamental relationship from membership (to a set) to functions between them. Or to be more precise, category-theoretic topoi generalize sets.

The OP's question also touches upon an interesting philosophical question: is math invented or discovered? After all, A -> B is always true whether we realize A -> B, but to show that A -> B, you have to construct a proof.

One should not equivocate the discovery of a result and the invention of a mathematical system. From a formalist point of view, the discovery a new theorem no more implies that mathematics is discovered than the discovery of a new strategy in a game implies that the game was not invented.

[Zadius]

One should not equivocate the discovery of a result and the invention of a mathematical system. From a formalist point of view, the discovery a new theorem no more implies that mathematics is discovered than the discovery of a new strategy in a game implies that the game was not invented.

That's basically what I wanted to say. It's a false dichotomy to say that mathematics is either invented or discovered. I would say that axioms are invented and their logical consequences are discovered.