$2 + 2 = 4$.
That’s another way of saying that $••$ near $••$ is $••••$, no matter what $•$ is standing in for. This seems almost trivially obvious.
But it’s not always true. If you have two apples in a bowl, and then you go out and pick two more apples and put them in your bowl, you don’t always have four apples in the bowl.
Sometimes a kid will have eaten an apple while you were gone, and you’ll only have three apples. Sometimes your friend picked one too, and you’ll have five. Sometimes it takes you a hundred years to pick the apples, and when you get back, your original apples have rotted away. Sometimes they’ve grown into trees, and now you have a hundred apples.
In these cases we don’t say that $2 + 2 \ne 4$. We say that the apples weren’t acting like numbers. In order for the rules of numbers to apply, your apples can’t disappear into the mouths of children or appear in the hands of your friend. If they do, they aren’t acting like numbers. Numbers can’t be eaten. Numbers can’t rot away. Numbers can’t sprout trees.
There are very specific rules that specify when numbers apply. These special rules are rules that you have to satisfy yourself before you can even use the model. They’re not laws of the system – they’re barriers to entry.
We call these axioms. Axioms don’t have to be true, but when they are the rest of the model is also true.
Whenever the axioms of addition apply, $2 + 2$ is always $4$. If you find a real-world scenario where $2 + 2 \ne 4$ that’s not because addition is wrong, it’s because the axioms don’t apply in your situation.
In this way, statements about addition are always true. $2 + 2 = 4$ is a true statement. If you find a violation, it’s your fault, not addition’s fault.
$2 + 2 = 4$ is a true statement, but it’s not necessarily helpful. This is also a true statement:
If all elephants are purple, and all purple elephants have penguin riders, then all elephants have penguin riders.
Here the axioms (if we may abuse the term) are
- All elephants are purple
- All purple elephants have penguin riders
and the statement is
- All elephants have penguin riders
Given the axioms, this statement is true. If you believe the axioms, the statement follows. Hopefully you don’t believe the axioms.
Just because something is true doesn’t mean it is useful.
We can easily prove that if the axioms of addition apply to reality then two real things plus two real things is four real things. The hard part is proving that reality follows the axioms of addition.
It seems intuitive to us that apples follow the rules, and thus we feel like we can take the deductions of addition and apply them to apples without hesitation. But human intuition has been wrong before. How do you know that apples actually act like numbers?
Two apples plus two apples is four apples. You can go test that if you don’t believe me. But what about $107,179,482,832,000$ apples plus $3$ apples?
You’ve been taught since childhood that this yields $107,179,482,832,003$ apples. But are you sure? You haven’t tried it. You haven’t put one hundred and seven trillion, one hundred and seventy nine billion, four hundred eighty two million, eight hundred and thirty two thousand apples in one place, added three more apples, and then counted them up. You’ve never seen it done. You’re just trusting that apples work like numbers and then manipulating the numbers.
What if real things don’t actually follow the axioms of addition?
What if in the real world, $107,179,482,832,000$ apples added to $3$ apples is actually $107,179,482,832,001$ apples? What if apples don’t quite follow the axioms of addition? What if there’s some tiny subtlety where our axioms aren’t consistent with reality, a tiny difference that we don’t notice for small numbers but which shows up when you try to add big amounts of real things?
It wouldn’t be the first time that seemingly simple laws were demolished upon the discovery that reality doesn’t quite follow the ‘obvious’ axioms: 30 meters per second plus 30 meters per second yields a speed slightly less than 60 meters per second.
With formal notation and rational thought we can prove how things work given a set of axioms – but you can never be certain that reality follows your rules.
Fortunately, all is not lost. Even when reality doesn’t completely obey our axioms, our axioms are not useless. Our models don’t need to be perfect in order to be useful.
As it turns out, if you try to add a trillion apples to a trillion apples, you won’t end up with two trillion apples. You’ll end up with a tiny planet made from the crushed remains of two trillion apples.
Addition doesn’t apply to apples in the long term, because apples don’t follow all the axioms. Apples can get crushed into a tiny planet. Numbers can’t.
As far as the propositions of mathematics refer to reality, they are not certain; and as far as they are certain, they do not refer to reality.
But we don’t need an “if” for the model to work. We need a “when”.
When apples follow the axioms of addition – when your kid isn’t eating them, and your friend isn’t gathering more, and they aren’t crushing themselves into a tiny planet – then the discoveries of addition apply to apples.
We can never prove that reality follows our axioms all the time, in all possible scenarios. But when it does, the models that flow from our axioms are like windows into the workings of the world.