One of the amazing things about mathematics in general is the way we can continuously discover and prove new results on the basis of simple definitions and axioms. But in a way, this concept is similar to those ancient myths that our planet is sitting on the back of a giant turtle. It sits on another turtle, which sits on another, and it’s turtles all the way down. Where’s the bottom? In order to prove anything, we need to start from somewhere, right?
For millennia after the dawn of mathematics, people generally assumed that the starting point had to be the whole numbers and their basic operations: addition, multiplication, etc. How much simpler could you get than that? But in the 19th century, philosophers and mathematicians began serious efforts at trying to improve the overall rigor of their endeavor, by defining simpler notions from which you could derive whole numbers and prove their basic properties. The average person may not need proofs that whole numbers exist, but mathematicians can be a bit picky sometimes. One of the most successful of these efforts was by Giuseppe Peano from Italy, who published his set of axioms in 1889.
The Peano axioms can be stated in several equivalent forms, but for the moment we’ll use the version in Bertrand Russell’s nice “Introduction to Mathematical Philosophy”. As Russell states it, they are based on three primitive notions and 5 axioms. The notions are the concepts of zero, number, and successor. Note that he’s not saying we start by knowing what all the numbers are, just that we are assuming some set exists which we are calling “numbers”. Based on these, the 5 axioms are:
- Zero is a number.
- The successor of any number is a number.
- No two numbers have the same successor.
- Zero is not the successor of any number. (Remember that we’re just defining whole numbers here; negative numbers will potentially be a later extension to the system.)
- Induction works: if some property P belongs to 0, and we can prove that if P is true for some number, it’s true for its successor, then P is true for all numbers.
With these primitive notions, we can derive the existence of all the whole numbers, without having them known at the start. For example, we know Zero has a successor which is a number, so let’s label that 1. Then 1 has a successor number as well, so let’s call that 2, and so on. We can also define the basic operations we’re used to, simply building on these axioms: for example, let’s define addition. We’ll create a plus operation, and define “a + 0” as equal to a for any number a. Then we can define “a + the successor of b” as “the successor of (a+b)”. So, for example, a + 1 equals a + “the successor of 0”, which becomes “the successor of a + 0”; by our original definition, this boils down to “the successor of a”. Thus we have shown that the operation a+1 always leads to a’s successor, a good sign that we are using a reasonable definition of addition. We can similarly define other operations such as multiplication and inequalities.
The Peano axioms were quite successful and useful, and a great influence on the progression of the foundations of mathematics. Yet Russell points out that they had a few key flaws. Think again about the ideas we started with: zero, successors, and induction. They certainly apply to the natural numbers… but could they apply to other things, that are not what we would think of as the set of natural numbers? The answer is yes— there are numerous other sets that can satisfy the axioms.
- One example: let’s just define our “zero” for these axioms as the conventional whole number 100. Then what is being described is the set of whole numbers above 100. If you think about it for a minute, this won’t violate any of Peano’s axioms— we are still defining a set of unique numbers with successors, none of which is below our “zero”, and to which induction applies.
- As another example, let’s keep zero as our conventional zero, but define the “successor” operation as adding 2. Now our axioms describe the set of even whole numbers. A very useful set, indeed, but not the true set of whole numbers we were aiming to describe.
- As an even more absurd case, let’s define our “zero” as the conventional number 1, and the successor operation as division by 2. Then we are describing the infinite progression 1, 1/2, 1/4, 1/8, and so on. Another very useful series, but not at all matching our intention of describing the whole numbers.
Choosing alternate interpretations of this set, naturally, can lead to very weird interpretations of our derived operations such as addition and multiplication. But Russell’s point is that despite the power and utility of Peano’s axioms, there is clearly something lacking. This situation actually reminds me a bit of some challenges we encounter in my day job, proving that computer chip designs work correctly: it’s nice if you can prove that your axioms lead to desired conclusions for your design, but you also need evidence that there aren’t also BAD designs that would satisfy those axioms equally well. If such bad designs do exist, your job isn’t quite done.
Russell’s answer to this issue was to seek improved approaches to defining whole numbers based on set theory, which would more precisely correspond to our notion of what these numbers really are. We’ll discuss this topic in a future podcast.