Is there anything currently that generates rejection from the mathematical community?

Question

Is there anything currently that generates rejection from the mathematical community, as happened with the complex roots of algebraic equations?

Answer 1

I don't think the assumptions of the question are correct. "Complex roots" were never rejected, they were not fully understood. I understand it is incredibly hard to place yourself in a world with less knowledge than what you currently possess; but when you "don't know something" you also typically "don't know that you don't know something". The tools of attack we have today, such as "graphing", "algebra", etc. were not always known; and it is typically foolish to think that mathematicians declare things unfit for study without reason.

A brief history of polynomials. Polynomials were typically understood mostly as geometric objects. They represented objects in geometry and their roots were typically specific solutions to pure geometric problems. The word "quadratic" stems from the concept of a "square" which has 4 sides. The notion of "completing the square" has an actual meaning in pure geometric terms. Of course the geometry makes no sense when the sides have negative length. Such geometry problems have no solution (and logically so). Unfortunately, quadratics are the base case of polynomials where the introduction of complex numbers does not add any insight into working with quadratics in geometry. Hence, there was never a need to go beyond the simple notions of "no solution" or "solutions".

However, this fails when we reach cubic equations. To "complete the cube" involves a fair number of steps, in particular one step requires the consideration of a square with "negative area". Such object is exceedingly complicated to think about. But the odd part is that in these weird cases, the full solution to the cubic might still come out real. This is a drastically different than quadratic. This is why "complex numbers" were not really understood as actual geometric solutions until the tedious work came about to fully understand the cubic formula. I highly recommend Veritasium's video. https://youtu.be/cUzklzVXJwo

Today, we enjoy thinking of polynomials graphically, algebraically, as field elements, and in geometric terms. Modern day understanding is exceptionally more complete than in the past. It is not correct to judge others in the past by what they didn't know or have access to. While on an individual level it may be possible to find instances of mathematicians with biases towards particular topics, as a whole it is hard to see a place in history where mathematicians as a community took a stance against a particular topic. Any perceived "rejection" is more likely a case of a lack of knowledge.