Mathematical Platonism

Mathematical Platonism is the form of realism that suggests that mathematical entities are abstract, have no spatiotemporal or causal properties, and are eternal and unchanging. This is often claimed to be the view most people have of numbers.

• I don’t call myself a “Platonist,” since I don’t even know what it even means to say that mathematical truths “exist in a Platonic realm.” Indeed, that concrete way of putting things even strikes me as a step backwards—as if astronomers could one day send probes to Plato’s realm to find out what’s in it, or prove that it never existed. The concepts of “existence” and “reality” that we use when discussing physical objects are just flat-out irrelevant here.
  I instead call myself an “anti-anti-Platonist”: someone who’s not literally committed to a “Platonic realm” of mathematical truth, but who given the choice, would much rather that people believe in such a realm than that they spout the self-evident garbage that goes with denying math’s intellectual autonomy.
  • Scott Aaronson, comment (March 10th, 2016)

• What’s the greatest discovery in the history of thought? Of course, it’s a silly question – but it won’t stop me from suggesting an answer. It’s Plato’s discovery of abstract objects. Most scientists, and indeed most philosophers, would scoff at this. Philosophers admire Plato as one of the greats, but think of his doctrine of the heavenly forms as belonging in a museum. Mathematicians, on the other hand, are at least slightly sympathetic. Working day-in and day-out with primes, polynomials and principal fibre bundles, they have come to think of these entities as having a life of their own. Could this be only a visceral reaction to an illusion? Perhaps, but I doubt it.
  • James Robert Brown, Philosophy of Mathematics (2008), Chap. 2 : Platonism

• Perhaps nobody should be particularly surprised by this, as, after all, the laws of nature physicists acknowledge also seem to be timelessly true independently of whether anyone takes them to be true. Where do they come from? Since there are somewhat mundane interpretations of what laws of nature are – including the possibility that they are accidental generalities valid in this particular universe and/or within a certain time-span – the case posed by mathematical constructs seems to be even more clear and powerful. Math, like diamonds, truly seems to be forever.
  If one ‘goes Platonic’ with math, one has to face several important philosophical consequences, perhaps the major one being that the notion of physicalism goes out the window. Physicalism is the position that the only things that exist are those that have physical extension [ie, take up space] – and last time I checked, the idea of circle, or Fermat’s theorem, did not have physical extension. It is true that physicalism is now a sophisticated doctrine that includes not just material objects and energy, but also, for instance, physical forces and information. But it isn’t immediately obvious to me that mathematical objects neatly fall into even an extended physicalist ontology. And that definitely gives me pause to ponder.
  • Massimo Pigliucci, "Mathematical Platonism" (2011)

• One cannot escape the feeling that these mathematical formulas have an independent existence and intelligence of their own, that they are wiser than we are, wiser even than their discoverers, that we get more out of them than was originally put into them.
  • Heinrich Hertz, as quoted in Morris Kline (1980). Mathematics: The Loss of Certainty. Oxford University Press, p. 338.

• How is it that mathematical ideas can be communicated in this way? I imagine that whenever the mind perceives a mathematical idea, it makes contact with Plato's world of mathematical concepts. ... When one 'sees' a mathematical truth, one's consciousness breaks through into this world of ideas, and makes direct contact with it ('accessible via the intellect'). I have described this 'seeing' in relation to Gödel's theorem, but it is the essence of mathematical understanding. When mathematicians communicate, this is made possible by each one having a direct route to truth, the consciousness of each being in a position to perceive mathematical truths directly, through this process of 'seeing'. (Indeed, often this act of perception is accompanied by words like 'Oh, I see'!) Since each can make contact with Plato's world directly, they can more readily communicate with each other than one might have expected. The mental images that each one has, when making this Platonic contact, might be rather different in each case, but communication is possible because each is directly in contact with the same externally existing Platonic world!
  • Roger Penrose. The Emperor's New Mind, 1989 Penguin edition, ISBN 0140145346. Ch. 10, Where lies the physics of mind?, p. 428.