Number is a fundamental concept of math and of life. Bones have been found for the stone age with notches in them – used for counting.
There is more than one kind of number. We start from Natural numbers for counting from 0, 1, 2, and on through 3. From these we create numbers that can be less than zero, or a fractional part of a number.
It might surprise you then, that the formal basis for these foundational numbers, only goes back to the later part of the 19th century.
The different branches of math like numbers, geometry and trigonometry …… are based on what we consider to be simple assumed facts, which we can just accept. We call these simple assumed facts axioms.
The trick is, that from these axioms, we can continually derive new facts. This is how the branches of math have grown, and continue to grow.
Axioms of Natural numbers
The axioms of the Natural numbers are known as the Dedekind–Peano axioms, and are described here. Let’s take a quick look at these axioms, and I will try to provide some commentary.
We have to start somewhere, so we assume 0 is a natural number:
The axioms then go on to define the equals or equality relation. Relations are used in math to compare things.
- For every natural number x, x = x
So something is equal to itself, e.g. 2 = 2
- For all natural numbers x and y, if x = y, then y = x.
So the order does not matter, say x = my age, which say is 25
Let’s also say y = Alice’s age which is also 25
Then x = y, and y = x
- For all natural numbers x, y and z, if x = y and y = z, then x = z.
We have my age x, and Alice’s age y, but let me tell you that Alice’s age is the same as Diana’s age z.
So we have now y = z, we know x = y …… so we must have x = z
- For all a and b, if a is a natural number and a = b, then b is also a natural number.
This means that every Natural number, has another Natural number it is equal to – the same number. It is not as if say 35 is not equal to anything. The bottom line is that we can use = without worrying it will suddenly stop working and leave us in the lurch.
Operations are the actions of math. The Dedekind–Peano axioms now go on to define a successor operation which we will call S(n), where n is some Natural number.
- For every Natural number n, S(n) is a natural number.
So we are saying that every Natural number has a successor.
S(0) = 1, S(1) = 2, S(2) = 3 ……
But wait, there is a chain to these numbers, we could also say
S(0) = 1, S(S(0)) = 2, S(S(S(0))) = 3 ……
- For every natural number n, S(n) = 0 is false.
So this is just saying there is no Natural number who has a successor of 0. So 0 is the start of the Natural numbers.
- For all natural numbers m and n, if S(m) = S(n), then m = n.
This axiom is telling us that the Natural numbers are ordered. No two different Natural numbers can have the same successor.
- We need to show that the successor operation can produce all the Natural numbers, or put another way, that every Natural number is included in (or is a subset of) – the set of 0 and its successors:
Let’s say K is a set of such that:
If 0 is in K, and
for every natural number n, if n is in K, then its successor S(n) is also in K,
then K contains every natural number.
Finally defining addition
Using the above, we can define the arithmetic operation of addition.
We define addition recursively, that is if terms of a repeating definition of itself. Let’s say that
- a + 0 = a
- a + S(b) = S(a+ b)
Now we can define a + 1 as:
a + 1 = a + S(0) = S(a + 0)
S(a + 0) = S(a).
So if a was 5, then we have S(5) = 6
We can continue this and define a + 2 as:
a + 2 = a + S(1) = S(a + 1)
S(a + 1) = S(a + S(0)) = S(S(a + 0))
S(S(a + 0)) = S(S(a)))
So if a was 5 again, then S(S(5)) = S(6) = 7
We build the branches of math from axioms.
The Natural numbers are based on axioms know as the Dedekind–Peano axioms.
Axioms provide a foundation to continuously derive the rest of the math facts from. However as you can see, even for something as simple as the Natural numbers, it takes quite a bit of work and it is low-level detail
So for most people, learning math involves just learning the higher level derived math facts, or rules. They can take for granted that these were derived from the axioms by others and not worry about the axioms themselves.
However, I hope now you have seen how math is created in layers, with axioms being the lowest or most fundamental layer.