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
>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:
First a note on this.
I used to think the von neuman construction of the naturals was something super profound but its not really, its just an example of how we can construct the idea of the naturals in set theory.
The more important thing though is what exactly a number is, and this is the discussion it leads to.
The important idea of numbers is **numbers exist because we can count, more precisely they exist because there is always a “next” number.** Here we use the successor function, S, to define this counting, but the basic idea is that if we have something somewhere where for every element of something there exists a “next” element, we have defined the basics of counting and as a result there is some notion of number that exists with that. The von neumann construction is just an example of this, its not “the true hidden meaning behind the natural numbers.”
> But these things only tell us properties of addition. If we have no idea what it means to add
This leads to a philosophical discussion, unfortunately or fortunately something we have to get into when we get this low into math. Addition is a human invention, there is some notion of it in the real world but our conception of it at least is exactly that, our conception of it. We invented it, we define its properties, the only thing is these properties have to be logically consistent with the rest of our number system idea. We in fact have various notions of addition, natural addition, rational addition, finite field addition, even things like elliptic curve addition are all some sort of notions of addition, we call these all addition because they satisfy the certain addition properties, usually most if not all.
In the basic notion of the naturals that you have above, we can say that “addition a function where you apply succession to the first numeral for every time succession is applied to the second.” But what is important is that this definition can provably satisfy the properties of addition which we choose are the properties of addition. To create this we worked backwards, we start with the requirements and narrow ourselves onto a solution.
Unfortunately, what you’re asking for almost certainly does not exist.
Mathematics ultimately rests on a set of axioms that are not proven, but rather accepted as true. There is no universal set of axioms that everyone agrees on. Sometimes arithmetic is just assumed to be true axiomatically.
Nonetheless, people have attempted to “reduce” arithmetic to a more basic set of axioms, but the results are the antithesis of concise. Check out Principia Mathematicia if you’re interested.
Furthermore there are proven limits to the completeness and consistency of such systems that attempt to derive arithmetic (Gödel’s Incompleteness Theorems).
Or, for a great graphic novel treatment of the quest, check out Logicomics!
> 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.)
The everyday addition operation is pretty axiomatic. You might be interested in abstract algebra though, which takes a step back and handles said operation as one of a kind of “addition-flavored” operations that behave similarly. The concept of an algebraic *group*
is defined as a set and an operation that combines two of the elements of said set to result in a third, with the following features:
* The operation is associative
* One of the elements of the set is an identity element (the “zero”) – performing the operation with it as one of the operands just gives back the other operand.
* Every element has an inverse (if you perform the operation on an element and its inverse, you get back the “zero”).
The everyday set of integer numbers and addition form a group, but for example, you can also grab the numbers 0-19 and define
a special addition operation ⊕ to roll over back to 0 when you’d hit 20 and to 19 if you go under 0. You now have 20-modular arithmetic. The inverse works too: for every number n, its inverse is 20-n. 16 ⊕ 4 = 0. 16 ⊕ 8 = 4.
It doesn’t matter that S(S(S(S(1))) is clunky, the point is you can essentially do all of mathematics with this construction. You can define addition with just rules about sets. But it’s so clunky we often write “5” as a shorthand for S(S(S(S(1))).
>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.
>he ends up defining the properties of addition, not what addition is.
This is extremely common in mathematics. What is a vector space? Anything that has the properties that we require a vector space to have. That’s the whole purpose of axiomatic definitions.
The most abstract definition of addition I can think of is the following. Natural numbers are isomorphism classes in the category of finite sets. Addition is the binary operation on these classes induced by coproduct.
Not exactly ELI5, but you start getting into group and field theory.
A group has an operation, it can be considered a multiplication or addition operation. A field has two operations, ‘addition’ (+) and ‘multiplication’ (* = X)
For all m,n in set S, mXn or m+n is also in set S.
For multiplication operation, there is an Identity value I in set S. Then the multiplicative inverse exists v in set S where m X v = I. and m X I = m
For addition there is a ‘zero’ value Z, and an additive inverse v in set S where m + v = Z and m+Z = m
You then also have to follow distributive/commutative properties for each operation
(mXn)X o = mX(nXo) or mXn = nXm
(m+n)+o = m+(n+o) or m+n = n+m
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