Most mathematicians operate on the assumption that they are studying something that already exists: their theorems are discoveries and not creations in contrast with poems and plays. But how is that possible? It sounds like they are saying that the theorems exist before they are contemplated: if so, where are they?
I’m going to make 2 broad arguments: the first is adapted from the physicist Roger Penrose (2007). The second comes from the philosopher Bertrand Russell’s The Problems of Philosophy (1912), chapters 8 - 9. The ideas and good parts are their’s, the slapdash and inaccurate aspects of the presentation are mine.
Consider fractals, in particular the Mandelbrot set. It’s created by a fairly simple rule. Yet it is incredibly elaborate: in fact its detail is basically infinite, or so I understand. Regardless, it’s quite a sizable thing, too big for a single person to hold in their head in its entirety. The set itself (as opposed to the algorithm behind it) simply can’t be an invention. So it must have been discovered.
Buh wah? If the Mandelbrot set was discovered what the hell is it? It has no mass, no energy. How can we say that it exists? [sup]1[/sup]
It’s real. It just has different properties than physical objects, energy and even thoughts. In such cases, it’s useful to introduce new terminology, new definitions, so that we can wrap our heads around unfamiliar ideas. We’ll say that the Mandelbrot set (or mathematics in general) are examples of Universals.
Let’s think a little about mind and matter: we will see that universals are neither:
So again: that something can be inside something else: that’s real. It certainly isn’t merely a thing that is only in our minds. Rocks and sticks are real as well. So is the relation 2+2=4 -if you put together 2 objects with another 2 objects there must be 4 objects-, although that’s harder to see than something being inside something else. The notion of being closer to one object vs another - say a given bird is closer to New York than to Chicago - that universal is real as well. As is the bird.
To pull out some more terminology, we can say that rocks, sticks and thoughts exist while universals like math and being inside something else subsist. Things that exist have a location in time and space. Universals though subsist across time. All time, AFAIK. The property of being in something else isn’t dependent upon someone thinking it and it isn’t dependent upon any particular physical example. It’s not thought and it’s not matter: Russell labels such things as universals.
Now at this point Plato voyages into mysticism[sup]2[/sup], which is why I couldn’t take him too seriously. But we don’t have to do that. Material objects are real, but so are universals. Plato implied that universals were more real than physical things, which were mere examples. But (really!) both entities are real and which category you focus on will vary with personal preference. Personally, I’m an empiricist. But from time to time idealists will post on this message board.
[sup]1[/sup] For those keeping score, the material before this point came from Penrose, chapter 1. The material after it comes from Bertrand Russell (1912). Russell adapted it straight from Plato. I studied Plato in college, but this stuff -core concepts really- sailed over my head. Thanks Professor Russell!
[sup]2[/sup]See eg the allegory of the cave.
2000: Is math truly objective?
2001: Mathematics: Invented or Discovered?
2001: Is mathematics made up?
2004: Epistemologically is mathematics considered more of an invention, or a discovery?