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Mathematical realists believe that mathematical entities exit independently of human minds. Mathematical objects have an objective independent existence, and they are discovered by mathematicians, not invented.

But does this apply to all mathematical entities?

It seems intuitive that concepts like the number 3, odd number, sqrt(2), triangle, sphere, cone, etc...have an independent existence.

On the other hand some of the more advanced mathematical concepts that one encounters when studying science and engineering seem definitely like artificial human constructs.

Concepts like the Fourier and the Laplace Transforms, Surface Integrals, Hermite Polynomials, Weil Groups, etc....seem closer to technical innovations, on par with computer algorithms or human languages, and seem definitely like objects that are invented by mathematicians, not discovered.

  • Do mathematical realists draw a line between discovered mathematical objects and invented mathematical objects? If so, how do they do so?
  • Or do they bite the bullet and claim that all mathematical entities are discovered? If so, doesn't this discredit the mathematical realist stance? Isn't something like the Discrete Short Time Fourier Transform obviously a human invention?
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Yes there is. Maddy takes this position in Perception and Mathematical Intuition, and so do recent mathematical Aristotelians. The alternative of believing it all real usually comes with full blown Platonism (forms in a separate realm), and is not very popular, see however Brown's Platonism, Naturalism, and Mathematical Knowledge for a modern defense. Let me add however that many modern realists would pragmatically admit fictional mathematical entities beyond merely constructible ones, like inaccessible cardinals or Lebesgue non-measurable sets, although Aristotle might have been more conservative.

Here is a long quote from Franklin's Aristotelian Realism, which I give in full because it seems to address the questions directly:

"The thesis defended has been that some necessary mathematical statements refer directly to reality. The stronger thesis that all mathematical truths refer to reality seems too strong... Statements about negative numbers can refer to reality in some way, since one can make true conclusions about debts by using negative numbers. But the reference is indirect, in the way that statements about the average wage-earner refer to reality, but not in the direct sense of asserting something about an entity, ‘the average wage-earner’. Indirect reference of this kind is not in principle mysterious, though it needs to be explained in each particular case.

So it can be conceded that many of the entities mentioned in mathematics are fictional, without any admission that this makes mathematics unique; minus 1 can be seen as like fictional entities elsewhere, such as the typical Londoner, holes, the national debt, the Zeitgeist and so on. What has been asserted is that there are properties, such as symmetry, continuity, divisibility, increase, order, part and whole which are possessed by real things and are studied directly by mathematics, resulting in necessary propositions about them."

What I described are variations on "genuine" mathematical realism, but there is also another kind, where this issue doesn't even arise. Here is Quine:"I see no way of meeting the needs of scientific theory... without admitting universals irreducibly into our ontology... Nominalism... is evidently inadequate to a modern scientific system of the world". This came to be known as the "indispensability argument" for the existence of universals, Putnam joined Quine in accepting it in 1970s, when he was a realist. However, one reading this should keep in mind that for Quine "to be is to be a value of a variable" in a theoretical ontology, and ontologies, scientific or otherwise, are "myths" and "bridge of our own making". So "universals exist" only means that our current scientific scheme needs them as variables, and whatever else it needs, exists too.

That was then. Since then nominalists, like Chihara and Field, showed that sets and numbers were dispensable in science after all. As Burgess, Maddy's mentor and fellow Aristotelian, put it in Why I am Not a Nominalist:"Quine and Putnam have been false friends of numbers in making the case, for their acceptance seems to depend on a claim of indispensability".

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This is an excellent question, because it gets right to the heart of the problem of mathematical realism which is essentially the same question as the problem of universals, which is essentially the same as the mind-body problem of Cartesian dualism.

At the heart of all of these questions is the issue of do we live a purely material existence, or is there a Ghost in the Machine.

So while I know nothing much about mathematical realists such as Roger Penrose, nor the reasons why they believe as they do, I can say how I think these issues impact what I think mathematical realism ought to be.

The issue in @Alexander’s question boils down to one of complexity. It seems simple enough to see the implied order of the integers, and then intuit that these are real for everyone therefore must have independent reality. But when it comes to complicated and inventive solutions to problems that maybe are not necessarily problems demanding resolution in the first place, then it intuitively seems that such creativity has to be original in all ways.

So the question is by implication about creativity, about Art, and where if anywhere we draw a line between Art and Science,

One way to address the issue is to consider poetry and the dictionary. The dictionary by its very existence implies all possible arrangements of the words it contains; therein lies its ghost, and among those possibilities are the complete works of Shakespeare, which we theoretically believe could be replicated by an army of chimps banging away on typewriters. So when Shakespeare writes “What's in a name? That which we call a rose by any other word would smell as sweet;” the phrase [and its words] gains meaning from its context in the play, just as the play gains meaning by being brought into the world by Shakespeare, and then more so from performances, its place in history, and the impact it has had on countless lives, and the spin offs in music and film.

All of this contextual meaning amplifies parts of the ghost that are in the potential in the dictionary, and therein lies the Art, the creative process of taking something of little meaning and imbuing it with greater meaning. This is what Marcel Duchamp did with Fountain, as did his audience.[Though not the exhibition panel!]

Creativity in mathematics is no different to creativity in art, literature, or any other act in life. It multiplies the sparse meaning in the ghostly relationships between universals, by giving life to them in ways that are to a greater or lesser extent original, which is borne out by the value we place on originality and creativity.

The bottom line is that far more complex and unpredictable solutions are added to the ghost by all of life, so there is no specific problem with mathematical complexity The problem is with the original premise of a ghostly mathematical realism in the first place. Is it really independently real?

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