I have written (references below) about the way we seem to think about math objects using our mind’s mechanisms for thinking about physical objects. What I want to do in this post is to establish a vocabulary for talking about these ideas that is carefully enough defined that what I say presupposes as little as possible about how our mind behaves. (But it does presuppose some things.) This is roughly like Gregor Mendel’s formulation of the laws of inheritance, which gave precise descriptions of how characteristics were inherited while saying nothing at all about the mechanism.

I will use module as a name for the systems in the mind that perform various tasks.

Examples of modules

a) We have an “I’ve seen this before module” that I talked about here.

b) When we see a table, our mind has a module that recognizes it as a table, a module that notes that it is nearby, and in particular a module that notes that it is a physical object. The physical-object module is connected to many other modules, including for example expectations of what we would feel if we touched it, and in particular connections to our language-producing module that has us talk about it in a certain way (a table, the table, my table, and so on.)

c) We also have a module for abstract objects. Abstract objects are discussed in detail in the math objects chapter of abstractmath.org. A schedule is an abstract object, and so is the month of November. They are not mathematical objects because they affect people and change over time. (More about this here.) For example, the statement “it is now November” is true sometimes and false sometimes. Abstract objects are also not abstractions, like “beauty” and “love” which are not thought of as objects.

d) We talk about numbers in some ways like we talk about physical objects. We say “3 is a number”. We say “I am thinking of the only even prime”. But if we point and say, “Look, there is a 3”, we know that we have shifted ground and are talking about, not the number 3, but about a physical representation of the number 3. That’s because numbers trigger our abstract object module and our math object module, but not our physical object module. (Back and fill time: if you are not a mathematician, your mind may not have a math object module. People are not all the same.)

More about modules

My first choice for a name for these systems would have been object, as in object-oriented programming, but this discussion has too many things called objects already. Now let’s clear up some possible misconceptions:

e) I am talking about a module of the mind. My best guess would be that the mind is a function of the brain and its relationship with the world, but I am not presuppposing that. Whatever the mind is, it obviously has a system for recognizing that something is a physical object or a color or a thought or whatever. (Not all the modules are recognizers; some of them initiate actions or feelings.)

f) It seems likely that each module is a neuron together with its connections to other neurons, with some connections stronger than others (our concepts are fuzzy, not Boolean). But maybe a module is many neurons working together. Or maybe it is like a module in a computer program, that is instantiated anew each time it is called, so that a module does not have a fixed place in the brain. But it doesn’t matter. A module is whatever it is that carries out a particular function. Something has to carry out such functions.

Math objects

The modules in a mathematician’s mind that deal with math objects use some of the same machinery that the mind uses for physical objects.

g) You can do things to them. You can add two numbers. You can evaluate a function at an input. You can take the derivative of some functions.

h) You can discover properties of some kinds of math objects. (Every differentiable function is continuous.)

i) Names of some math objects are treated as proper nouns (such as “42”) and others as common nouns (such as “a prime”.)

I maintain that these phenomena are evidence that the systems in your mind for thinking about physical objects are sometimes useful for thinking about math objects.

Different ways of thinking about math objects.

j) You can construct a mathematical object that is new to you. You may feel that you invented it, that it didn’t exist before you created it. That’s your I just created this module acting. If you feel this way, you may think math is constantly evolving.

k) Many mathematicians feel that math objects are all already there. That’s a module that recognizes that math objects don't come into or go out of existence.

l) When you are trying to understand math objects you use all sorts of physical representations (graphs, diagrams) and mental representations (metaphors, images). You say things like, “This cubic curve goes up to positive infinity in the negative direction” and “This function vanishes at 2” and “Think of a Möbius strip as the unit square with two parallel sides identified in the reverse direction.”

m) When you are trying to prove something about math objects mathematicians generally think of math objects as eternal and inert (not affecting anything else). For example, you replace “the slope of the secant gets closer and closer to the slope of the tangent” by an epsilon-delta argument in which everything you talk about is treated as if it is unchanging and permanent. (See my discussion of the rigorous view.)

Consequences

When you have a feeling of déjà vu, it is because something has triggered your “I have seen this before” module (see (a)). It does not mean you have seen it before.

When you say “the number 3” is odd, that is a convenient way of talking about it (see (d) above), but it doesn’t mean that there is really only one number three.

If you say the function x^2 takes 3 to 9 it doesn’t have physical consequences like “Take me to the bank” might have. You are using your transport module but in a pretend way (you are using the pretend module!).

When you think you have constructed a new math object (see (j)), your mental modules leave you feeling that the object didn’t exist before. When you think you have discovered a new math object (see (k)), your modules leave you feeling that it did exist before. Neither of those feelings say anything about reality, and you can even have both feelings at the same time.

When you think about math objects as eternal and inert (see (m)) you are using your eternal and inert modules in a pretend way. This does not constitute an assertion that they are eternal and inert.

Is this philosophy?

My descriptions of how we think about math are testable claims about the behavior of our mind, expressed in terms of modules whose behavior I (partially) specify but whose nature I don’t specify. Just as Mendel’s Laws turned out to be explained by the real behavior of chromosomes under meiosis, the phenomena I describe may someday turn out to be explained by whatever instantiation the modules actually have – except for those phenomena that I have described wrongly, of course – that is what “testable” means!

So what I am doing is science, not philosophy, right?

Now my metaphor-producing module presents the familiar picture of philosophy and science as being adjacent countries, with science intermittently taking over pieces of philosophy’s territory…

Links to my other articles in this thread

Math objects in abstractmath.org
Mathematical objects are “out there”?
Neurons and math
A scientific view of mathematics (has many references to what other people have said about math objects)
Constructivism and Platonism