Stewart Shapiro on Ontological Realism in Mathematics

I've decided to start posting interesting quotations from the books I read here on this blog, and even if the content may be more relevant to my other blogs, I will keep the brief quotations here and keep the articles and other content in the topic blogs.

I am reading Stewart Shapiro's book Thinking About Mathematics (2000 Oxford University Press) and in the second chapter he discusses the ontological theories of mathematical objects.  That is, are mathematical objects such as numbers, points, lines, and sets real in the sense that they exist independently of any mathematicians?  This is certainly my worldview, the alternative being that mathematical objects are not really real but are made up by mathematicians-- that if there were no mathematicians, there would be no mathematical objects.  The first view is called realism, while the second is called idealism.  This is not the space to discourse about the merits of either view, but I think it really depends on your worldview and your belief structure.  The realist worldview is what comes out of Plato's teachings, and that is what excited me the most.

Plato the realist, point up to the World of mathematical Forms.
Aristotle the idealist, pointing outward to the phyiscal world of Matter.

Here is what Shapiro says on p. 27:

The scientific literature contains no reference to the location of numbers or to their causal efficacy in natural phenomenon or to how one could go about creating or destroying a number.  There is no mention of experiments to detect the presence of numbers or determine their mathematical properties.  Such talk would be patently absurd.  Realism in ontology is sometimes called 'Platonism', because Plato's Forms are acausal, eternal, indestructible, and not part of space-time.

This of course opens the door to many questions, such as how do science and mathematics differ, and what can be said about how theoretical physics differs from experimental physics?  Is theoretical physics just a branch of mathematics?  If so, how does its application lead to experimental results?  What is the relation between the Platonic world of Forms and the physical world of matter?

Books mentioned in this post:

Video - Ancient Philosophy of Mathematics 07 - The One, Limit, and Unlimited in Geometry

I have added another video to the video series Ancient Philosophy of Mathematics.  This is part seven, titled The One, Limit and Unlimited in Geometry.  We will explore the metaphysics of geometry through The One, Limit and Unlimited.  We will show how geometric constructions correspond to these first principles, with the point symbolizing The One, the circle symbolizing the Limit, and the line symbolizing the Unlimited. We give a basic description of the practice of geometric theurgy, a form of meditation that requires using body movement, thereby grounding the highest metaphysical principles down into the lowest level.  We read from Plato's Philebus and Proclus' Elements of Theology and Commentary on Euclid's Elements.  This video is the foundation for the next series of videos on geometric theurgy.