I have been doing mathematics for 57 years. What is my experience? Do I discover or invent? Am I a James Cook, finding what was already there, or a Thomas Edison, bringing something new into being?… My answer is unequivocal: for me, the experience is one of invention.
Edward Nelson, (2002) Syntax and Semantics
Is mathematics invention or discovery? When mathematicians come upon their results are they just producing elaborate mental constructions which have no actual reality… 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…
Roger Penrose, (1989) The Emperor’s New Mind. Oxford University Press, p126
The mathematician is an inventor, not a discoverer.
Ludwig Wittgenstein, (1938) Remarks on the Foundations of Mathematics, I-168