[sleepyams]

Good points, a large chunk of math is different formalizations of the notion of size, for example (in order of increasing absurdity):

https://en.wikipedia.org/wiki/Cardinality

https://en.wikipedia.org/wiki/Measure_(mathematics)

https://en.wikipedia.org/wiki/Ultrafilter_(set_theory)

https://en.wikipedia.org/wiki/Euler_characteristic

https://golem.ph.utexas.edu/category/2008/02/metric_spaces.h...

https://golem.ph.utexas.edu/category/2006/10/euler_character...

(and of course there are certainly many that I'm missing)

For more fun, see these slides from John Baez: https://math.ucr.edu/home/baez/counting/counting.pdf