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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.

Mark
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    You already have the ability to rediscover mathematical formulas or theorems that have previously been discovered. So just keep learning, and one day when you know enough stuff some of the theorems you discover will be new. For most mathematicians this starts to happen a few years into grad school. – littleO Aug 24 '19 at 23:34
  • That is a very interesting question. It brings one back to this question: Is mathematics discovered or invented? – Sigma Aug 24 '19 at 23:34
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    The first time I created a theorem was in my first year of grad school. I thought that something was true, then I decided to try to prove it. It was very much like a homework assignment that I made up. Slowly but surely, I repeated this act more and more often. Think of something that seems like it should be true, then prove it. The only difference between a homework proof and new math is that you invent the questions yourself. BTW, at my school we did not do many proofs until the third year of undergraduate studies. – irchans Aug 25 '19 at 00:18

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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.

Sigma
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    "Maybe this object has already been defined and I just don't know about it, because I never studied Number Theory." -- It seems like it's based on the notion of twin primes for what it's worth. In that sense a twin prime pair is a pair $p,p+2$ which are both prime numbers. Also, $3$ would not be "super prime" since, generally, $1$ is not considered to be prime (nor composite). $5$ on the other hand would be since $3,5,7$ are all prime. ... I wonder if there are any other super primes actually. – PrincessEev Aug 24 '19 at 23:51
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    "Maybe "super prime numbers" are considered useless and have never been studied." -- Mmm I feel that the notion of "usefulness" is sort of fleeting for mathematical discoveries. Of course sometimes the sort of usefulness is obvious, but at other times it might only serve to give new insights into math. Or maybe what you're trying to solve itself isn't useful, but leads to the development of new techniques to verify your results which are useful. Or perhaps it's just interesting and fun for its own sake, which is useful in its own right if just on a personal level. What's "useful" is relative. – PrincessEev Aug 24 '19 at 23:55
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    Note that, for any integer $n$, one of the three numbers ${n-2,n, n+2}$ is divisible by $3$, which implies that $5$ is the only super prime. – lulu Aug 24 '19 at 23:57
  • Thanks @lulu. So since 5 is the only super prime number, you might think super prime numbers are not so special or useful for anything. – Sigma Aug 24 '19 at 23:58
  • Exactly @EeveeTrainer. Some mathematical research being done today may prove to be useful 100 years from now. Who knows. – Sigma Aug 25 '19 at 00:01
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    Well, maybe not that precise definition. But suppose you look at primes for which $p\pm 6$ are also prime. Those primes include $11, 13, 23, 37,47$. Are there infinitely many? I'll bet that is not known. – lulu Aug 25 '19 at 00:02
  • only if $$6ab+a+b\6cd+c-d\6ef-e-f$$ have gaps of three that none can represent @lulu –  Aug 25 '19 at 00:10
  • @RoddyMacPhee Not sure I follow. Are you suggesting that there's a simple way to show the list is finite? Or are you rephrasing the definition? – lulu Aug 25 '19 at 00:13
  • Very interesting @lulu. Now you got yourself a conjecture! :) And why 6? Define super prime numbers to be numbers such that $p \pm k$ are also prime numbers, given that $k \in \mathbb{Z}$. I would like to know if the set of super prime numbers, defined in this manner, is infinite. – Sigma Aug 25 '19 at 00:14
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    @VictorS. Well, $6$ is the smallest number that makes sense for that. Can't be an odd number (else some of the terms are even), and it has to be divisible by $3$ else my earlier argument would work. So $6$ is the smallest interesting case. In general though, if these sort of problems don't have simple congruence reasons why the list must be finite, they tend to be hopelessly difficult. – lulu Aug 25 '19 at 00:16
  • it's the sieve of sundaram on multiplcative group mod 6 @lulu. If a number $n$ is of one of these 3 forms, then one of $6n\pm1$ is composite. –  Aug 25 '19 at 00:17
  • @RoddyMacPhee I see. So that sounds like another way to phrase the "conjecture". There are so many examples of primes such that $p\pm 6$ are also prime that I'd be very surprised to learn that there was any simple way to prove the list finite, though. – lulu Aug 25 '19 at 00:19
  • yeah both that conjecture, and twin prime conjecture might need to be proved in unison. As twin prime conjecture comes down to that there are infinitely many natural numbers not of those forms. –  Aug 25 '19 at 00:23
  • @RoddyMacPhee Indeed, I see no reason why this should be any easier than the twin prime conjecture. – lulu Aug 25 '19 at 00:24
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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.

Cristian Dumitrescu
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People tend to find new results for the first time when they are grad students. This is because they work under the supervision of a mathematician who is working on many projects that are likely going to yield new publishable results within several months of research work.

Count Iblis
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Almost anytime. I have a thread on mersenneforum that taught me set theory. Here are a random thing you can find yourself:

Goldbach Collatz interaction

Claim: The primes $p$ that have $\lceil {3p\over 2}\rceil$ prime interlink the two conjectures.

Show: Take any number of form $x=4n+2=r+q$ that are a sum of these primes r,q , and follow it through 2 iterations of the Collatz map. We find, that we get $$6n+4={3\over 2}(4n+2)+1=({3r\over 2}+{1\over2})+({3q\over 2}+{1\over 2})$$ Since the primes are odd, the first part of each term is a half-integer . Equally, we have the adding half to a half integer, is equivalent to the ceiling function output. Showing the last iterate, to be the sum of two primes, by the definition of the sequence we got the previous two primes from.

You can also link twin prime conjecture to goldbach partitions, or to a statement on the natural numbers as a whole, etc.

Most new math looks at things from different viewpoints, not necessarily with entirely new fields of mathematics.