I have recently posted a couple of comments relating to whether mathematics is real. These were blog comments or forum comments. And, to be clear, these posts were elsewhere (not on this blog). This post will expand on those.
Discovery or invention
A blogger asked whether mathematics is a matter of discovery or invention, and I commented. But I’m not sure where that was. So I am reconstructing and expanding my comment.
My first remark was that mathematics is an invention. I’m reminded that Kronecker famously said “God gave us the natural numbers; all else is the work of man”. But it has long seemed to me that Kronecker gave too much credit to God. It is all the work of man.
I did not want to dispirit the blogger, so I tried to connect invention with discovery.
Nobody doubts that Lego blocks were invented. But give children some Lego blocks, and they will quickly discover all kinds of interesting things that they can do with those blocks. You really cannot separate invention and discovery. If you invented something that was useless and boring, you would quickly forget it. Any important invention involves the discovery that what you invented is useful or interesting or both. We might say that “invention or discovery” is a false dichotomy. They go together hand in hand.
Getting back to mathematics, presumably at some time it was discovered that counting was a useful practice. But maybe the practice had to be invented before you could discover that it was useful. Again, discovery and invention are intertwined.
Numbers are not just counting. For counting, you need a sequence of names to use. But numbers themselves are abstract entities, presumably derived from the practice of counting. So the use of numbers in this way was a separate invention. But again, it was an invention that depended on discovering the usefulness of numbers.
You can build many interesting structures with Lego blocks as the basic components. Similarly, you can build many interesting abstract structures, with numbers as the basic building blocks. As a couple of simple examples, consider magic squares. These do use the arithmetic properties of numbers. Or consider Sudoku puzzles, which do not in any way depend on arithmetic properties of numbers. And, of course, arithmetic itself is a kind of structure built with numbers. And it, too, is both invention and discovery.
Is reality mathematical?
This question, on the relation of mathematics to reality, is what I thought I was responding to in a comment at the peaceful science forums. I will expand on that comment here.
As best I can tell, mathematics does not play any role at all in the cosmos. However, scientists find it useful to construct models of aspects of reality. And mathematics is very good as a modeling tool.
Mathematics works well in some sciences, because those mathematical models work very well. In fact, they work so well, that there is a tendency to conflate the model with the aspect of reality that it models. So we take a name from our mathematical model, and use that same name for the corresponding part of reality that is modeled. Because there is a mathematical relation between components of our model, we treat that as if it were a mathematical relation between the corresponding parts of reality. And this leaves the impression that those parts of reality are in a mathematical relation.
It is quite reasonable for us to think about those parts of reality as if they were in a mathematical relation. But it goes too far to jump to the conclusion that somehow that mathematical relation was part of the actual design of the cosmos. We simply cannot know that. We cannot even know whether the cosmos was designed.
The nature of mathematics
What is mathematics? Many people think of it as a branch of logic. However, mathematicians are more likely to see logic as a branch of mathematics. Logicism is the thesis that mathematics is just logic, or is reducible to logic. That has always seemed to me to be a mistaken view of mathematics.
Going back to the example of Lego blocks, mentioned early in this post, a logical analysis of Lego blocks would not be very interesting. You would, instead, need to analyze the creative inventions that people can build out of Lego. Similarly, with mathematics, it is not enough to study numbers. Mathematicians study the creative ways in which numbers and other mathematical objects have been used. As far as I know, we do not have a logical account of human creativity.
Perhaps we should consider mathematics to be an art form.
The Heretical Philosopher 