process of coming up with a definition or axiom?

Question

I am not a maths student but am very interested in the mathematical processes. A major part of mathematics is proofs. The steps carried out on the proofs are all based on the initial axioms or definitions. So my question is, are the initial axioms based on common sense and real world observation? is this normally how the first initial axiom or definition of any mathematical concept is created? Secondly, can it be the case where the definition can be wrong(since it is kind of an assumption but a very good one) when we apply the proof to tell something about the world? I hope to get some insights possibly from the mathematics majors or professors who happened to glance at it. Thanks!

Answer 1

There are different collections of initial axioms to choose from.

Today, the standard is Zermelo–Fraenkel set theory with axiom of choice. There is also some interest in Zermelo-Fraenkel set theory without the axiom of choice, but it's not the mainstream.

One thing that should be made clear is that there is a big difference between axioms and definitions. Axioms are extremely simple, super super super essential and basic things, like "two sets are the same if they contain the same things" or "there is a such thing as an infinite set". These are things that you really can't argue with if you want to have a system of mathematics that makes any sense. There are very few options when it comes to what kind of axioms are reasonable.

Definitions, on the other hand, are much more specific, and you have lots of options there. You ask whether a definition can be wrong-- well, it can't really be wrong, no, because it's something you're assuming. But a definition can fail to be useful, or it can fail to describe something that you think it ought to. For example, there is always some discussion as to why $1$ isn't considered a prime number. What you're really asking there is whether the definition of a prime can include $1$. Whether you decide that $1$ is a prime number or not doesn't change the actual truth behind mathematics, but it will mean you have to phrase things differently. If you were to say $1$ is prime, then most theorems about prime numbers would have to be about "prime numbers except for 1", so ultimately it makes more sense to not call $1$ a prime number. This is the closest one gets to a definition being "wrong." Definitions are ultimately just a linguistic issue.

Answer 2

I'll approach this topic from something like a Platonist perspective.

We all know that there is such a thing as a "counting number". If we have a discrete, finite collection of apples, we can describe the collection by saying there are 2, 3, or even zero of them. We also know that when we put three apples with seven, we always get ten apples. And similarly, when we have a collection of three pairs added to one of seven, we get a collection of ten pairs. This leads us to the intuition that there are some inherent properties about things like "seven" and "three" which are independent of apples and pears. That is, this leads us to the notion that numbers (and the addition operation) exist in their own right.

Since we've decided that there is a "collection" of "counting numbers", the next step is to describe its properties in the hopes of making deductions about these "counting numbers" from things we know to be true about them. Axioms are therefore viewed as a way of specifying logically what we're trying to describe.

Suppose I told you of a mathematical collection known as $\mathbb{N}$ which is the collection of "counting numbers". How could I mathematically convey to you what it is that I'm describing?

As I describe adding each axiom, I encourage you to visualise the "bad $\mathbb{N}$" examples I'm providing by drawing out $\mathbb{N}$ as a collection of dots with an arrow between $n$ and $S(n)$. The imagery will be clearer if you can visualise it.

I would probably begin by telling you that something called $0$ is an element of $\mathbb{N}$ and that for every $n$ which is an element of $\mathbb{N}$, we have some element known as $S(n)$ which is also in $\mathbb{N}$.

This doesn't tell you much of anything about $\mathbb{N}$. The collection could have as few as $1$ element (perhaps it contains only $0$, and $S(0) = 0$), or an arbitrarily large collection of elements.

So we add in the principle that for every $n$, $S(n) \neq 0$. What does this tell us about $\mathbb{N}$?

It tells us that $\mathbb{N}$ has at least two elements, $0$ and $S(0)$. But it could be that $S(S(0)) = S(0)$ - that is, $\mathbb{N}$ could have exactly 2 elements. Obviously, a set with two elements isn't what we mean by the collection of "counting numbers", so I will have to be more specific.

The next principle I would articulate is that for every $n$ and every $m$, if $S(n) = S(m)$ then $n = m$. This tells us that $\mathbb{N}$ must have infinitely many elements. But it does not necessarily describe what we're looking for. For example, we could have $\mathbb{N}$ consisting of two chains $0, S(0), S(S(0)), ...$ and $w, S(w), S(S(w)), ...$ where $w$ is not the successor of any $n$. This is obviously not what we're looking for.

We might try to add in an axiom that every $n$ is either $0$ or a successor. However, this still leaves the possibility in play that $\mathbb{N}$ could consist of sequence $0, S(0), S(S(0)), ... $ together with some $w$ such that $w = S(w)$. This is again obviously not what we mean.

We could try to add in another axiom that no $n$ can equal $S(n)$. But this still leaves open yet another possibility - a "single-ended chain" consisting of $0, S(0), S(S(0)), ...$ and a "double-ended chain" $..., S^{-1}(S^{-1}(w)), S^{-1}(w), w, S(w), S(w), S(S(w)), ...$. Again, this is not what we know "counting numbers" should look like!

In order to fully characterise the natural numbers, I would therefore need the Axiom of Induction. (For simplicity's sake, we'll gloss over differences in 1st and 2nd order logic). This says that for every property $P$ that a number might have (where $n$ having property $P$ is written as $P(0)$), if $P(0)$ and if for every $n$, $P(n)$ implies $P(S(n))$, then it must be the case that for all $n$, $P(n)$.

This axiom is enough to specify exactly what we mean by the "counting numbers".

In the same way, the axioms of Set Theory are designed to describe some abstract notion of "set". Generally, the notion of a set in ZFC is some kind of well-founded tree. However, we can prove interesting things about the logical system of ZFC by finding other collections which seemingly have little to do with the intuitive notion of a set but which nevertheless satisfy the axioms of ZFC. This is how one can prove, for example, the independence of the continuum hypothesis (though obviously, a lot of mathematical sophistication is required to understand this proof).

Can an axiom be "wrong"? According to the school of thought that axioms are there to describe mathematical things which exist in their own right, an axiom is "wrong" if it doesn't describe the objects you think it describes. This is obviously very subjective in practice - to some, the Axiom of Choice is a self-evident property of sets, while to others, it self-evidently cannot be.

There is one way to prove beyond a shadow of a doubt that an axiom is wrong. You could prove that the axiom is inconsistent - that is, that it's possible to derive a contradiction from it. Bertrand Russell famously gave set theory a frightful scare by using the widely accepted axioms of set theory to prove Russell's Paradox. After he did this, everyone agreed that since contradictions cannot exist, there's no way that the axioms leading to the contradiction could possibly all be true. This led to the revision of the axioms of set theory.

Interestingly, a similar paradox was found during the development of Martin-Lof type theory, a theory with very different characteristics from set theory, by considering a "type of all types" just as set theory once incorrectly considered a "set of all sets". This suggests that there are some deep underlying constraints on how "big" collections can be that are in some sense independent of foundation.

I myself lean more towards the Formalist view of axioms, which goes something like this:

When mathematicians prove something, they are just following a set of rules in a game to reach an outcome. There is not necessarily any philosophical significance at all in the fact that some theorem holds in ZFC other than the fact that, following the axioms of ZFC and the rules of first-order logic, it is possible to prove that theorem. ZFC doesn't necessarily "describe" a "real collection of sets" or indeed any real collection at all; rather, it is a set of axioms which can apply to a wide range of mathematical constructions, many of which look nothing alike on the surface.

I do, however, tend to believe that there "really is" such a thing as $\mathbb{N}$. In that sense, I don't take the extreme positions of the finitists, who say that (in some sense, some kinds of) infinities do not exist and therefore, any mathematical theory dealling with infinite objects has value only insufar as it can tell us truths about finite objects.

As far as proofs being applied to things in the "real world", this ventures more into the realm of physics and the sciences than pure mathematics. Most scientists, like Feynman famously did, would say that the only test of a scientific fact is experiment. On this view, the fact that mathematics can help us make sense of the world is, at worst, a happy coincidence and at best a result of the fact that humans specifically developed math to try to understand the real world.

After all, the theory of "counting numbers" originally came about to understand such phenomena as putting three apples with seven, just as the theory of plane geometry came about to understand real physical surfaces and the theory of calculus came about to describe physics. It's no surprise that our mathematical tools do exactly what we designed them to.