I recently saw a [great video](https://www.youtube.com/watch?v=dKtsjQtigag) by Another Roof on youtube using set theory to explain what numbers are. He explains that when we try to define a number, it’s hard because we tend to describe the adjective form of the number, e.g., “there are three people.” but what IS three? How do we define it with as few assumptions about mathematics as possible? He goes on to explain and show how the numbers can be derived using set theory, and that got me thinking:
Can we apply the same rigorous definitions to the arithmetic operators?
I saw another one of his videos which I thought would help, literally called [How to Add](https://www.youtube.com/watch?v=rhhhUAAAh-g), but he ends up defining the properties of addition, not what addition is. For example, when defining addition, he lists off a few defining factors:
*n + 0 := n*
*n + 1 := S(n)* (where S(n) means the successor of n, i.e. the next natural number)
But these things only tell us properties of addition. If we have no idea what it means to add, then how can we know that n + 0 is n, or that n + 1 is S(n)? These defining factors assume that we know what it is to add a posteriori. If we are to assume as little as possible about math in order to create it from the ground up, then we can’t assume to know what addition is.
He then gives the more general case:
*n + S(k) = S(n + k)*
which recursively just breaks into *S ( S ( S… S (1) ) )*, so that 2 + 3 is just *S ( S ( S ( S (1) ) )*
However, this is quite clunky, and if someone were to know literally nothing about addition except for this, they would have a terrible time calculating the sum of large numbers. Remember, we don’t want to assume anything, and I want to extend that to the reader of the definition. If we assume that the reader can remember large sums, then this definition is easy to carry out. if we remember what 2 + 3 is, then it’ll be much easier to figure out 2 + 4 (though this in itself poses more problems if we are to assume little to nothing). However, that is still an assumption, and I’d like to eliminate outside variables completely from the definition, and focus on the required logic alone.
I also wish to find a completely abstract definition, because many definitions on the internet describe addition by showing the combination of two groups of items. But physical things are an assumption. It assumes that in the mathematical universe, there are things. It’s a bit pedantic, but I believe it’s important. It’s best to have a definition that relies on logic alone.
So my question is:
Is there a way to define the addition operation in such a way that:
1. it captures what it means to “add”
2. it is rigorously provable
3. It assumes little to nothing about mathematics and little to nothing about the observer
4. it is concise as possible, so that any possible reader of the definition can know how to add by knowing only the definition and the natural numbers, and
5. it is completely abstract
I’ve tried to answer this question by myself but all the sources i could find defined properties of addition, not a definition for the operation itself.It’s an important question, because addition is one of, if not the, most basic operations you can perform on a set of numbers. And as we know, math builds on itself. So if a tight, concise definition of addition can’t be reached, then the whole foundation is shaky.
This is probably a noob question for set theorists, and I may be flawed in some of my reasoning, but I’m not a mathematician so sue me.
TL;DR : read the list
In: 3
>Is there a way to define the addition operation in such a way that:
it captures what it means to “add”
it is rigorously provable
It assumes little to nothing about mathematics and little to nothing about the observer
it is concise as possible, so that any possible reader of the definition can know how to add by knowing only the definition and the natural numbers, and
it is completely abstract
This list is impossible for a really boring reason not having anything to do with addition: you can’t prove a definition. A definition associate a previously-unknown symbol/word with some concept. There are no proofs that it is true or false, the same way you can’t prove or disprove the fact that “cat” refers to a cat. Some people choose to use that convention, and some do not.
But let me analyze line by line:
#4. It is very concise. It uses exactly the minimum needed. Natural numbers (with the successor operator) are about induction: induction captures what it means to assign a natural number. So every definitions involving natural numbers must – directly or indirectly – use induction.
#1. This definition captures what it means to add 2 natural numbers. Natural numbers are for counting, so when you add them, it means you keep on counting the amount you counted.
#3. This definition only assume that there are natural numbers (with the successor operation), and nothing else. There are no observers in the picture. The definition define what it *means* to add 2 numbers, not *how* to do so, which is a completely different thing.
#2. As mentioned, you can’t prove a definition. The most you can do is the prove that the definition satisfies a few properties, but this just move the goalpost: why are these properties important?
#5. This one is abstract. It only uses natural numbers. You don’t need physical objects to define them. In fact, you can count sets of abstract objects as well.
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