What is a number? Discuss this and see that it is some sort of abstraction of a set of objects.
Ask what "there are as many chairs as there are people here" means? Most people will answer something like "if you count the number of chairs and the number of people you get the same number". But this means they need to know how to count. Ask if this could be done without counting.
It can be done by saying that two sets are equal in size if there's a perfect matching between the two. Explain that you can use this to teach this to children, even if they don't know how to count. This is the definition of equality between natural numbers.
What do < and > mean? Discover a similar definition.
Now comes addition and multiplication. Addition is just the combination of two (disjoint?) groups. What is multiplication? It can be defined as repeated addition, or the number of dots in a rectangle, or the number of ways to connect an object from one set to an object from another set.
Whichever definition you use, prove that:
a+b = b+a
(a+b)+c = a+(b+c)
a*b = b*a
a*(b*c) = (a*b)*c
a*(b+c) = a*b + a*c