Roger Penrose

  1. Mathematical objects are just concepts; they are the mental idealizations that mathematicians make, often stimulated by the appearance and seeming order of aspects of the world about us, but mental idealizations neverthless.
    (The emperor's new mind, Oxford University Press, p.94)
  2. The Mandelbrot set is not an invention of the human mind ; it was a discovery. Like Mount Everest, the Mandelbot set is just there!
    (The emperor's new mind, Oxford University Press, p.95)
  3. Is mathematics invention or discovery? When mathematicians come upon their results are they just producing elaborate mental constructions which have no actual reality, but whose power and elegance is sufficient simply to fool even their inventors into believing that these mere mental constructions are 'real' ? Or are mathematicians really uncovering truths which are, in fact, already 'there' - truths whose existence is quite independent of the mathematicians 'activities ? I think that, by now, it must be quite clear to the reader that I am an adherent of the second, rather than the first, view, at least with regard to such structures as complex numbers and the Mandelbrot set.
    (The emperor's new mind, Oxford University Press, p.96)