One of the greatest problems people who are getting into hardcore maths face, one of the blocks, is how arbitrary the axioms look. Axioms are the fundamental truths of a mathematical system, the things we do not prove. Whenever people are introduced to axioms, though, the most common feeling is that we should be able to somehow prove all of them and end up with no unproved theorems, no fundamental truths.
Now here’s why this is wrong.
What’s a natural number? Depending on your level of mathematical sophistication, you might not even remember that there’s such a difference. There are numbers that are natural? What are those? And the answer is that the natural numbers are… well, the natural ones. The ones we first think of, as a species, when we think of numbers; the ones that are sort of already in our brain when we’re born.
They’re You know, the counting numbers.
Let’s add to the mix because is a Very Important Number.
Now, suppose you have a baby mathematician AI. It’s super intelligent, but it has no knowledge. You can communicate with it, to an extent, but it doesn’t know numbers. If you show it scribbles like , it will just stare at you blankly. If you try to point to apples on a table it will tilt its head and think you’re crazy. How do you explain to this super mathematician what numbers actually are, beyond your innate intuition?
Axioms are logical sentences. But they’re not just any logical sentences: they’re logical sentences that pinpoint certain concepts from conceptspace. What does this mean?
Picture some imaginary space where all possible ideas and concepts lie. It’s vastly huge, it’s really unimaginable, but you know that somewhere there you’ll find the natural numbers. They’re a concept, right? So they must be in there somewhere. Now, how do you find the natural numbers there? How do you pinpoint them?
And that’s what axioms do. Axioms are sentences that are true of the things you’re trying to find, and of nothing else. That is, if a thing is a natural number, then the Natural Number Axioms are true of it; if a thing is not a natural number, then at least one of those axioms isn’t. So what are the Natural Number Axioms? What do you tell a baby mathematician AI so that it can find them in conceptspace?
A0. is a natural number.
This is our first axiom. If a thing is a sequence with all natural numbers, then there has to be a in there somewhere. So right now, our natural number set is just .
Then, we define equality.
A1. Reflexivity: If is a natural number, then .
A2. Symmetry: If and are natural numbers and , then .
A3. Transitivity: If , , and are natural numbers, , and , then .
Not even equality is a logical concept we take for granted. The three axioms above are well-known equivalence relation axioms: every binary relation that obeys these three axioms is known as an equivalence relation. But there’s another equality axiom that’s exclusive of natural numbers:
A4. If is a natural number and then is also a natural number.
These four axioms don’t really change what our set of natural numbers is. Any set that contains a number called “zero” so far is a natural number set.
Next, we define the successorship operation, Sx.
A5. If is a natural number, is also a natural number.
A6. If and are natural numbers and , then .
A7. There is no natural number such that .
The fifth axiom defines the successorship operation and creates the first way of finding new natural numbers. Every other way – addition, multiplication, exponentiation – will be found via recursive application of that operation. To understand what it means: is just the number . is the number which is – you guessed! – . So this is an operation that takes a natural number and finds the next one. But this is not enough. Yes our regular natural numbers, obey this axiom, of course. But a set where, say, does too. This would be a set that has a loop: ‘s successor is , but ‘s successor is also , and there are only these two numbers.
So the sixth axiom is there to prevent loops: if two numbers have the same successor, then they’re the same number. But it’s still not enough. With it, we have that if (that is, if ‘s successor is also ), then it must be the case that (that is, ‘s successor is , which means that ), and nothing prevents this from being the case yet.
And finally the seventh axiom is one that makes special: is not the successor of any number. There are no numbers before it. (“What?! What about ?!” Chill, young padawan, chill. Everything in time. For now, does not exist.) Now this axiom, with the previous one, guarantees that there can be no loops, since every loop would imply at some point that has a predecessor, and this is impossible. So our set of natural numbers now is something like .
You might be thinking that this is enough to pin down the natural numbers uniquely. This is true of every natural number, yes? So let’s put this to the test. Take a set that goes like this: . The numbers that have a after them are greater than any regular natural number. They’re also infinite in both directions. What axioms is this set violating?
is in it, so A0 is ok. We can just assume that Axioms 1 through 4 are also obeyed by the set. Axiom 5 is also obeyed: every number’s successor is a natural number, yes? So are axioms 6 and 7, there are no loops there, has no predecessor ( is not the same as , it’s not special, and it’s in fact infinite, so it can have predecessors). So, this weird set with those bizarre starry numbers obeys all axioms so far! We need the final and most important axiom: the axiom of mathematical induction.
A8. Induction: If is a set that contains the number and has the property “If is in then is also in ,” then contains all natural numbers.
Now we’re done. Let’s say the set is just , the set of all natural numbers. is certainly in it, and Axiom 5 means every number that’s reachable via successorship from is also in it. Therefore, we have finally pinned down uniquely the set using those axioms.
Now, if you followed this construction closely, you’ll have seen that without the axioms, lots of other concepts could also fit the concept of natural number! That’s what axioms do, they slowly constrain conceptspace until they’ve squeezed every other concept out.
A historical example of the importance of axioms is Euclidean Geometry. Euclid (and everyone else) had an intuitive understanding of what geometry is and how it should behave. So he made four postulates that he thought were necessary, and a fifth, the Parallel Postulate, which he and many other people throughout the ages felt should be derivable from the first four postulates. The first four postulates defined the mathematical entities like “point” and “line,” and the fifth said that two parallel lines never intercept. From the wordless, intuitive concept of point and line, that sounds pretty obvious, so Euclid always felt that we should be able to somehow prove that fifth postulate without having it as an axiom. That is, he wanted to prove its theoremhood, instead of assuming it.
Turns out he couldn’t. Why? Because the wordless intuitive entities he called “point” and “line” had, imbued in them, the Parallel Postulate. But they weren’t fully defined by the other four postulates. That is, to talk about points and lines in the intuitive fashion, you need all five postulates, and if you discard the fifth, you get the so-called Non-Euclidean Geometries. Everything that obeys the first four is a geometry, and there you bend and twist the meaning of point and line beyond recognition, unless you also have the fifth one.
As a final consideration, I’ll comment on what exactly I mean when I say I’m a mathematical realist/platonist. I think that mathematical entities have some reality external to the humans who think them. And what is that reality? It’s embodied in the axioms, and in model theory: everything that obeys certain axioms has to obey all of their consequences. No matter what any human or other sapient being has ever thought, regardless of any person’s beliefs and theorems and ideas, it’s simply true that if two apples fall from a tree, and then another two, there will have fallen four apples.
In fact, “quantity of apples that fell from a tree” (henceforth called “quapp”) obeys all 9 of the above axioms! To see that, let’s say our successorship operation is the physical act of “an apple falling from the tree,” and equality is “comparing the number of apples that fell from two trees.” Then:
A0. Naturally, there can be apples on the ground, because no apples fell.
A1. If you compare a tree with itself, the number of apples that fell from it is the same.
A2. If you see that as many apples fell from tree A as from tree B, then as many apples fell from tree B as from tree A.
A3. If you see that the same number of apples fell from tree A and tree B, and also tree B and tree C, then the same number of apples fell from tree A and tree C.
A4. This is playing with words, but if apples fell from tree A, apples fell from tree B, and both are the same number, then the same number of apples fell from tree B.
A5. If apples fell from a tree and a further apple falls, then that new number of apples that fell is a quapp.
A6. If the quapp from tree A is the same as that from tree B, and they’re the successor of some other quapp, then before the last apple fell from each tree they also had the same quapp.
A7. There is no quapp that comes before no apples having fallen from a tree.
A8. The same argument we used to define the set can be used to define the set of all apples that fell from a tree.
Well, what do you know, the quantity of apples that fell from a tree is, in fact, a number! And so, everything you prove about natural numbers can also be said of quapps. And it’s in that sense that mathematics is external to humans, and has a separate reality.
It’s in that sense that I’m a platonist.