At what point of mathematical education can you start inventing new math?

I am a 2nd year student doing an honors program in math and statistics.

Everything that I have been learning has been formulas, theorems, and mathematical concepts that other people have discovered/invented/created.

Some very simplistic formulas I am able to modify to meet the needs of what I am trying to accomplish, but still I am using someone else discoveries as a basis for what I am modifying.

Friends and family say I am becoming a mathematician, but I dont feel that way as I do not have the ability to invent/create/discover new math, I am simply regurgitating what others have discovered.

At what point in your mathematical education are you able to invent new math?

For example, the linear regression formula. How did Francis Galton know that the formula he created would accomplish what he wanted to accomplish?

Note: Sorry to the editors as I could not find a relevant tag for this question.

Solution 1:

You can start inventing new math at almost any level, by defining new mathematical objects. For example, I can define a "super prime number" to be a prime number $p$ such that $(p + 2)$ and $(p - 2)$ are also prime numbers. By my weird new definition, $5$ is a "super prime number".

Maybe this object has already been defined and I just don't know about it, because I never studied Number Theory. Maybe "super prime numbers" are considered useless and have never been studied.

The point is, you can define some mathematical objects however you like. The bigger point is: is your newly created mathematical object useful?

For example, Joseph Fourier literally invented/discovered the Fourier Transform, which is a fantastically useful mathematical object that has tens (if not hundreds) of applications in many different areas, and he just wanted to find solutions to the Heat Equation. And Fourier became one of the "immortals" in mathematics because of his Transform.

So, as you can imagine by now, new mathematics is discovered/created by attempting to solve important problems for which there are currently no solutions.

You can also create/invent new math by attempting to create objects that do something you want them to do, or have properties you want them to have. Hamilton, for example, was trying to create 3-dimensional numbers, because he liked complex numbers, which are 2-dimensional numbers. After much effort, he discovered/created what are now called quaternions, which are 4-dimensional numbers.

Hamilton was bashed by other famous mathematicians of his time (such as Heaviside) when he created quarternions. However, nowadays quaternions are very, very useful in Robotics because they are very useful for describing motion in 3D space.

Solution 2:

I assume that new math means new mathematical theories, not new results within an established mathematical framework (axiomatic system ).

Mathematical evolutionary theory applied for mathematics itself, does it sound familiar (self-referential technique,  from ancient logical paradoxes to Godel's results, as a methodology) ?

I would suggest reading this reference (I confess I didn't yet, just some reviews).

Note that you can encode a mathematical theory in a multitude of ways (as sets of strings of fundamental symbols, and inference laws  ). You need to define the "objects" you're working with. 

As for selection principles, I would consider consistency, completeness,  soundness (defined within a mathematical framework),  but many others are worth considering (like usefulness in our real physical universe, sometimes apparent only hundreds of years later ).

Crossover, the role of  randomness  in all this, that's  an interesting problem all by itself.

I suspect Turing machines and AIT (algorithmic information theory ) are essential in this meta-theoretical framework. 

So at what point in evolutionary theory new species emerge? Can you construct a mathematical theory that can deal with this problem (too hard for me)?

Note that if at most only a finite number of inference steps N are allowed, this problem can effectively be approached using computer simulations (the selection principles become effective). The real problem appears when N tends to infinity.