Some thoughts toward revising my article on mathematical objects.
Mathematical objects are a kind of abstract object. There are lots of abstract objects that are not mathematical objects, For example, if you keep a calendar or schedule for appointments, that calendar is an abstract object. (This example comes from ).
It may be represented as a physical object or you may keep it entirely in your head. I am not going to talk about the latter possibility, because I don't know what to say.
- If it is a paper calendar, that physical object represents the information that is contained in your calendar.
- Same for a calendar on a computer, but that is stored as magnetic bits on a disk or in flash memory. A computer program (part of the operating system) is required to present it on the screen in such a way that you can read it. Each time you open it, you get a new physical representation of the calendar.
Your brain contains a module (see , ) that interprets the representation in (1) or (2) and which has connections with other modules in your brain for dates, times, locations and whether the appointment is for a committee, a medical exam, or whatever.
The calendar-interpreter module in your brain is necessary for the physical object to be a calendar. The physical object is not in itself your calendar. The calendar in this sense does not exist in the physical world. It is abstract. Since we think of it as a thing, it is an abstract object.
The abstract object "my calendar" affects the physical world (it causes you to go to the dentist next Tuesday). The relation of the abstract object to the physical world is mediated by whatever physical object you call your calendar along with the modules in the brain that relate to it. The modules in the brain are actions by physical objects, so this point of view does not involve Cartesian style dualism.
Note: A module is a meme. Are all memes modules? This needs to be investigated. Whatever they are, they exist as physical objects in people's brains.
A rigorous proof of a theorem about a mathematical object tends to refer to the object as if it were absolutely static and did not affect anything in the physical world. I talked about this in , where I called it the dry bones representation of a mathematical object. Mathematical objects don't have to be thought of this way, but (I suggest) what makes them mathematical objects is that they can be thought of in dry bones mode.
If you use calculus to figure out how much fuel to use in a rocket to make it go a mile high, then actually use that amount in the rocket and send it off, your calculations have affected your physical actions, so you were thinking of the calculations as an abstract object. But if you sit down to check your calculations, you concentrate on the steps one by one with the rules of algebra and calculus in mind. You are looking at them as inert objects, like you would look at a bone of a dinosaur to see what species it belongs to. From that point of view your calculations form a mathematical object, because you are using the dry-bones approach.
All this blather is about how you should think about mathematical objects. It can be read as philosophy, but I have no intention of defending it as philosophy. People learning abstract math at college level have a lot of trouble thinking about mathematical objects as objects, and my intention is to start clarifying some aspects of how you think about them in different circumstances. (The operative word is "start" — there is a lot more to be said.)
About the exposition of this post (a commercial)
You will notice that I gave examples of abstract objects but did not define the word "abstract object". I did the same with mathematical objects. In both cases, I put the word "abstract object" or "mathematical object" in boldface at a suitable place in the exposition.
That is not the way it is done in math, where you usually make the definition of a word in a formal way, marking it as Definition, putting the word in bold or italics, and listing the attributes it must have. I want to point out two things:
- For the most part, that behavior is peculiar to mathematics.
- This post is not a presentation of mathematical ideas.
This gives me an opportunity for a commercial: Read what we have written about definitions in References ,  and .
- Atish Bagchi and Charles Wells, Varieties of Mathematical Prose, 1998.
- Reuben Hersh, What is mathematics, really? Oxford University Press, 1997.
- Charles Wells, Handbook of Mathematical Discourse.
- Charles Wells, Mathematical objects in abstractmath.org
- Math and modules of the mind (previous post)
- Mathematical Concepts (previous post)
- Thinking about abstract math (previous post)
- Terrence W. Deacon, Incomplete Nature. W. W. Norton, 2012. [I have read only a little of this book so far, but I think he is talking about abstract objects in the sense I have described above.]
- Gideon Rosen, Abstract Objects. Stanford Encyclopedia of Philosophy.
- Representations II: Dry Bones (previous post)