Talons Philosophy

An Open Online Highschool Philosophy Course


Abstract Objects and Metaphysics


Abstract objects and metaphysics

Q1: Do numbers exist?

Q2: What are abstract objects?

Q3: Why would ideal forms not exist in space-time? Why would they have to be abstract?


I stumbled upon the topic of abstract objects while I was planning to do my research on what numbers are and whether these numbers existed (first question). I began to realize – from my reading – that numbers are a part of the concept of abstract objects (second question). And abstract objects are ideal forms that are mentioned in the third question. So, as I read more and had a deeper understanding of abstract objects, I was able to escalate the complexity and connect each question to one another.

Concrete object: an identifiable collection of matter, which may be more or less constrained by an identifiable boundary, to move together by translation or rotation, in 3-dimensional spaces

Abstract object: an object that does not exist at any particular time or place, but rather exists as a type of thing (such as an idea or abstraction)


Q1: Do numbers exist?

Numbers are obviously not something that can be physically manipulated – you can’t just simply pick up a number and throw it around. You can pick up numerals (which are the concrete versions of numbers) and throw them – the printed numerals on receipts, price tags, books etc. – but it isn’t like you’re actually throwing the numbers. Numbers exist as what philosophers call abstract objects.

But you don’t, by virtue of tearing out page three of a book and tossing it out a window, throw the number 3 out the window, any more than you throw me out of a window by drawing a picture of me and throwing that out the window.

Q2: What are abstract objects?

Mathematical objects, chess moves, games, pieces of music, and propositions are all examples of abstract objects. Every chess move we make, circle we draw, or number we write are all ideas/replications of the original move/object. There is only one real idea of the move/object – and it is what we call an abstract object. Every physical version of that move we make or circle we draw or number we write is a concrete imitation. Just like Plato’s theory of forms: when we think of a circle, every drawing we make is an imitation of this circle. This perfect circle we picture and this chess move we have an idea of are abstract objects because they are the basis of what we try to recreate.

We generally think of a chess move as something that exists by virtue of a concrete chess player actually moving a concrete chess piece in accordance with the rules of the game. But that seemingly concrete move can be instantiated in so many concrete ways — you could be replicating someone else’s game on your own chess board, you could make the move on a hundred different boards all at (nearly) the same time, you could make the move in your head before you make it on the board,… and all of these concrete possibilities point to the metaphysical problem here: If you believe there is only one move, and it’s concrete, then which move is the one move? And then what are the other moves? Copies of the move? Or instantiations of the same move?

Q3: Why would ideal forms not exist in space-time? Why would they have to be abstract?

Objects in the real world (space-time) are all imperfect copies of something. They have to be abstract because you could draw a circle or number or make a chess move the same way for a numerous amount of times and then you would have a difficulty picking out the ideal/perfect version from all the copies. So, if we use an abstract version of the move/object – something perfect and outside space-time – we won’t have to choose from all the similar instantiations.

Thinking about geometric objects is perhaps the clearest way to think about abstract objects. A line segment (a true, geometric line segment) is a perfectly straight, one-dimensional object with a determinate length. There are no such objects in space-time. So if there does, somehow, exist a true line segment, it certainly isn’t in the concrete world, and therefore it must be in some sort of abstract realm.

Personal Interest:

When reading the metaphysics package, I found the topic of Plato’s theory of forms (his idea of the perfect circle and never being able to recreate it) to be the most interesting and something I was drawn to. So I thought doing a project related to that would encourage me to do more research and be more interested in the metaphysical side of philosophy.


I found an article (where I got all my quotes from) that broke down abstract objects to a level that was easier to understand: https://welovephilosophy.com/2012/12/17/do-numbers-exist/

Another article that goes into depth about Plato’s theory of the forms and abstract objects is: http://www.iep.utm.edu/plato/#SH6b

Where to next?: 

From here I plan to continue on this topic and I hope to research the side of people who don’t believe in the theory of abstract objects (the nominalist point of view).



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