Mathemathics in expanding brain memes

Tiny_brain.jpg: Infinity is a number
Normal_brain.jpg: 0.999...=1
Illuminated_brain.jpg: 1+2+3+4+5+...=-1/12
Enlightened_Mind.jpg: Real numbers aren't real
Master_of_All_Cosmos.jpg: Ultrafinitism

Pardon my pea brain but by what grounds are finitism and its case made? Is it posited axiomatically or observed in the sense of mathematical realism?