Imagining the Real or Realizing the Imaginary: Platonism Versus Imaginism

Undoubtedly, a vast majority of mathematicians are Platonists. They believe that mathematical objects exist “out there” independently of the human mind, and mathematicians merely discover them. The Platonists believe that a mathematical statement, such as AC, is objectively either true or false – we simply do not know which it is, although in a poll of mathematicians, “AC is true” would win hands down. Likewise, a question, what is the chromatic number of the first Shelah–Soifer graph G, surely, must have a definitive answer; it cannot be “2 or uncountable infinity.” Therefore, for the Platonists either ZFC or ZF+DC+LM is true, we just do not know which. Platonists imagine the real.