The particular topic is the general notion of whether mathematics, and similar sorts of activities, are describing something that is real. I have touched upon this topic in previous posts. In some ways, this argument is at the very heart of the worldviews that assert there can be some absolute truth, moral principles, authority, etc. You can't hang an absolute morality or absolute knowledge on a logical foundation that is arbitrary, at least not without some cognitive discomfort.
So, n Anderson's post on mathematics, we find Anderson arguing for how mathematicians feel mathematics is a feature of reality. All quotes following are from his post, unless otherwise noted.
Mathematics is also a very interesting field if you have an interest in philosophy and questions about design in nature. Almost all mathematicians are in practice realists - they believe that as they make progress in their field they are involved in discovering and not in inventing. (See here for more on this distinction). That is, they act and research as if there is already a transcendent, pre-existing mathematical universe "out there" that is waiting for us to find and explore it. The opposite of that is behaving as if mathematics is our arbitrary toy, to be played with, deconstructed and rebuilt as we please. Shall we adopt the convention that 2+2 = 5 from now on and see where that takes us?
Yes, the good old 2+2 = 5 argument (it never occurs to such people that it is perfectly reasonable in some circumstances to say 2 + 2 = 1). Now, I am sure there are mathematicians who think there is some big design they are uncovering, and that belief does not prevent them from being fine mathematicians. It's probably even a positive when you are working closely with computer scientists, engineers, or physicist in pushing the boundaries of mathematics for direct applications. However, having enjoyed some 66 semester hours of undergraduate mathematics and another 36 at the graduate level, I can say that I ran into more than a few instructors and fellow students who treated mathematics exactly like an arbitrary toy, something you could play with, take apart, and build to order. Most of them, to my knowledge, would not have cared if there were a philosophical position that reflected this attitude, but I find fictionalism to be a decent approximation. We were playing in a universe we created for our benefit, and occasionally something useful would pop out.
In my view, the atheist materialists who have tried to explain their view of reality are in an exceptionally weak position when they seek to explain mathematics in non-transcendent terms. Mathematics resists, at multiple levels, any attempt to treat it as an arbitrary invention of the human mind. Almost at every turn it cries out "I was here before you, and I am bigger than you!". Maths is a very theistic subject!
Actually, mathematics is highly receptive to humans treating it like an arbitrary invention. It's why we have Euclidean, Lobachevkian, and Riemannian geometries to describe different sorts of space. It's we we have intuitionist, constructivist, and para-consistent logics and their mathematical descendants. We add or remove axioms at our pleasure, look at the results, and call it fun (and occasionally useful).
I can of course always add two apples to two apples and will always get four apples (an inconvenient truth for the atheists who want to argue that mathematical truths are not transcendent!) - but as I do so I'm conscious that there is a notion of "two-ness" or "four-ness" that goes far beyond the tasty bits of fruit and is independent of them.
Now, this is a great example of fuzzy-headed thinking, an absolute truth that is right except where it is wrong. If I smash two apples against two apples, I will quite possibly have over 100 bits of apple. If I pour two liters of water into two liters of alcohol, I will have under 3.9 liters of fluid. Except in very limited circumstances, numbers are not conserved!
The more complicated the mathematics gets, the more obvious this becomes. I can move from the simple adding of objects to a dimension up and do calculus to work out the area under a graph. I can then accelerate to five or six dimensional spaces and work out their corresponding concept of volume. I can work out the properties of completely theoretical objects. you get the idea. Mathematics speaks to us of an ideal reality which depends on the mind.
Here's an interesting question: how the fundamental nature of an ideal reality depend upon a human mind? Personally, I don't think is can or does. Perhaps Anderson mistyped, and meant that the reality is revealed to the mind, or perhaps it was a Freudian slip. I do think that the reality of any particular five-dimensional construction we create depends on our mind. That makes it not a fundamental property of nature.
Whilst it depends on the mind, mathematics also seems to have an unbreakable link to the physical world. In the most simple example, there's something about those two oranges that has the notion of two-ness. The notion of two-ness is contained, but not exhausted, by them.
The notion of twoness, and similar notions, is the pattern that we humans impose upon our world to make it simpler. Since we are identifying a pattern, it is unsurprising that no one instance of the pattern will form a complete rendition.
I can create a two-dimensional shape that is approximately (but never exactly - because we live in a world of discrete atoms and molecules) equal to the one in the equation of the graph I was using. This is all simple enough.
Anderson references quantum mechanics later on. Perhaps it's my ignorance, but wouldn't it be much more correct to say we live in a worlds of fuzzy atoms and molecules behaving in probabilistic ways? Even here, the need to have absolutes alters the mindset.
What is more breath-taking, though, is to understand that correspondences between abstract mathematics and the physical world have also been discovered in far more complicated cases. In some areas, mathematicians discovered new theorems in highly abstract areas that nobody thought would ever turn out to have a practical application - but in fact they actually perfectly described physical phenomena observed decades later. Do you get that? Away in his dusty study somewhere, the mathematician was working on a problem that was thought to be far too abstract to have any real application. Some time later, a physicist realised that this bit of mathematics was the key to something that he was observing. Quantum physics provides a number of illustrations of this.
I'm not sure why this is particularly breathtaking. While we play with mathematics as a toy, we also do like to make it useful. So, a mathematician, playing around with equations that are useful descriptions of quantum mechanics, comes across a new feature which also has utility, based upon deductively playing with something already utilized. I don't see why that would be surprising. I don't think it would be particularly shocking to Anderson if an engineer predicted that a steel beam would collapse under a certain weight load, and was correct.
Observation one: Mathematics has its seat in minds. Observation two: We also now know that mathematics is also embedded at a fundamental and essential in physical reality. Inescapable conclusion: Physical reality is the product of a mind.
Anderson very obligingly illustrated my point for me. This notion of real versus useful fictions is not just semantics and playing with words. It's a part of how and why believers believe, one of the pillars they use to prop up their world view.
I'll offer one final link to a rant by a mathematician, offered because he views mathematics as an art. I don't endorse everything he says about education, but his view of mathematics fits much more closely with the mathematicians I learned from and with.