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
> 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)?
If you want to be really formal, you can, for example, define a set whose members are things like {1, 1, {2}} and {4, 11, {15}}, and then define the notation “a + b = c” as meaning that {a, b, {c}} is a member of this set. There are many other ways of doing essentially the same thing.
I think set theory texts tend to focus on defining slightly more general things like ordered n-tuples, relations, and functions. The basic arithmetic operations are essentially just special cases of functions, so they may not bother to define them specifically. It’s important to bear in mind that none of these definitions are really canonical – there are lots of different ways of defining, say, numbers or functions as sets, and none of them are really better than the others. Plus there are plenty of people who disagree that it’s philosophically useful to define everything as ZFC-style sets.
> 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
But people don’t learn basic concepts from formal definitions. People learn how to add numbers by being shown examples and developing an intuitive understanding of how it works (or it’s possible that humans have some instinctive ability to do arithmetic and they’re just learning to tap into it, or learning to relate shapes and symbols to knowledge that is already in their brain).
We only reach the level of wanting a formal treatment of arithmetic once we have experience with much more complicated concepts that need formal definitions to be understandable. It’s at this point that we start asking things like “well, if I need this whole proof to know that sqrt(2) is irrational, then how do I really know that 2+2=4”? If you try and teach an 8-year-old a formal definition of addition, they’ll think you’re bonkers.
> So if a tight, concise definition of addition can’t be reached, then the whole foundation is shaky.
There isn’t really anything special about addition, and I would say that the foundation is shaky. At some level you have to work with things that you can’t justify from scratch. If you define addition in terms of sets, then you end up asking the same questions about sets. How do we really define what a set *is*? How do we know that our definition makes sense? There are numerous philosophical approaches to answering these kinds of questions, but none of them are really completely satsifying, and none of them have ever been universally regarded as the correct answer.
What foundations like set theory do is help to simplify and focus the philosophical questions. Prior to set theory, you could ask all these philosophical questions about arithmetic (what is addition, exactly? how do we know that you can’t prove 1=2?), but you could ask corresponding questions about geometry and graph theory and probability theory and so on (what is probability, exactly? how do we know that you can’t prove a square is a triangle?). In set theory, you can define all these diverse concepts purely as sets, and you’re left with one focused set (sorry) of philosophical questions about sets. (Though it isn’t universally accepted that this really helps. The reason why we can do all this stuff with set theory is because it’s so rich and general, so you could argue that it’s much harder to justify set theory than it is to justify arithmetic or geometry.)
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