Famous Math Problems/A

19. Irrationals
How do we model the continuum? This is a problem with a long and storied history. It is difficult to unravel it from physics and studies of the material world. People today believe this problem has been solved, that the Real number line models the continuum well, which is false. Also related, the question of homogeniety (or, sameness at any level of magnification) arises; it is by no means certain that what works on one scale works at another.

Rather than defining points on the continuum arithmetically (or using analytic geometry), we might begin with geometry, in terms of lengths (and areas).

22. Infinitesimals
Like the dihedral group for the three kinds of 2D or imaginary numbers, there is a 2x2 matrix which best encodes the concept of an infintesimal. It can be thought of as nilpotent, or "squareing to zero":

$$\epsilon = \begin{bmatrix} 0 & 0 \\ 1 & 0 \end{bmatrix}$$