Here is an elegant description of the “discovered” or “Platonic” point of view, taken from Mario Livio’s excellent book on this question: “Platonism in its broadest sense espouses a belief in some abstract eternal and immutable realities that are entirely independent of the transient world perceived by our senses. According to Platonism, the real existence of mathematical objects is as much an objective fact as is the existence of the universe itself….Platonism views mathematicians as explorers of foreign lands; they can only discover mathematical truths, not invent them.”
Livio also includes this quote from the famous British mathematician Sir Michael Atiyah, espousing the “invented” point of view: “Had the universe been one dimensional or even discrete it is difficult to see how geometry would have evolved. It might seem that with the integers we are on firmer ground, and that counting is a primordial motion. But let us imagine that intelligence had resided, not in mankind, but in some vast solitary and isolated jell-fish, buried deep in the depths of the Pacific Ocean. It would have no experience of individual objects, only with the surrounding water. Motion, temperature and pressure would provide its basic sensory data. In such a pure continuum the discrete would not arise and there would be nothing to count.”
As far as numbers go, the question boils down to this: Are they purely a construct of our brains? Had the universe been one uninhabited amorphous mass, would they not have existed? Certainly, Professor Ray’s construction suggests numbers can be invented – in fact, invented from nothing. But couldn’t one also argue that the blueprint for such invention is always there, whether or not there is human intelligence to actually implement it? In other words, Professor Ray’s construction, unrealized as it might be, is an algorithm that would always exist?
Discussion: Consider the example of intelligence residing in a jellyfish, with the surrounding Pacific Ocean providing its only sensory data. Can you argue that there would, indeed, still be things for the jellyfish to count?
Let’s now turn to a very practical consequence of the “discovered/invented” question: patent laws. In just about every country where such laws exist, only inventions, not discoveries, can be patented. The reason is that something discovered is assumed to come from Nature or otherwise pre-exist, so that a human being cannot claim to have produced it. An invention, on the other hand, is a product of the inventor’s ingenuity, and it is the right to be the beneficiary of this work that a patent protects. As a result, a new bacterium that one has discovered, say, cannot be patented, but if one develops a new medical technique that uses the bacterium, this procedure can.
What about mathematical innovation? If one considers mathematics to be discovered, then it shouldn’t be patentable, but if it is invented, it should. You might wonder why anyone might want to patent something like a theorem, but as it turns out, some mathematical results can lead to enormously profitable applications. For instance, a technique discovered in 1984 by Narendra Karmarkar drastically reduced the amount of computer time taken to solve some ubiquitous, practically essential optimization problems. When Karmarkar’s employer AT&T Bell Laboratories realized the potential commercial benefits, it promptly applied for a patent. After much controversy and lengthy court proceedings, a patent was granted, not on the method itself, but software based on it.
Such controversies continue. In general, courts have taken the Platonic view of mathematics being a discovery, not an invention, which means the odds are stacked against mathematics researchers ever getting rich from their work. This article by David Edwards argues why such patents should be granted.
Discussion: Do you agree or disagree with the arguments Edwards makes?
The Big Bang of Numbers (video)