What are numbers?

# What are numbers? ([home](https://github.com/alexhkurz/introduction-to-numbers/blob/master/README.md) ... [next](https://hackmd.io/@alexhkurz/HykB3VPhU)) Every mathematical investigation starts with a question. Sometimes, at the beginning there is a very general question, more philosophical than mathematical. For example: What are the numbers? Could numbers be different? Can we calculate only with numbers or also with other objects? What is special about the numbers we learn in school? You may want to add your own questions now. In mathematics we can investigate such philosophical questions using the **axiomatic method**. For example, instead of debating the difficult question of what *are* the numbers, we can take a more pragmatic view: What if we **define** the numbers by the computation rules we all know from school? From a philosophical point of view, we turn the question of what *are* the numbers into the question of what we can *do* with numbers. And then investigate the consequences of such a definition. Such a line of investigation will teach us a lot about numbers without a commitment to answering the ontological question what the numbers "are". We can always revise the definition by changing the rules of computation and see what happens then. Maybe disappointingly to the philosopher, we do not obtain one answer but many answers. But one of the beauties of mathematics is that, eventually, all the different answers, even if they looked far fetched at the beginning, will turn out to be useful and important (disclaimer: I love mathematics). We will see many examples of this. To summarize, motivated by the big philosophical questions, we will start by investigating the following fairly narrow and special technical question. **Question:** Are there are other systems of numbers that satisfy the well-known laws of addition, subtraction, multiplication and division? I collected a quick recap of the laws of addition, subtraction, multiplication and division, which I will call [the axioms](https://hackmd.io/@alexhkurz/BJw5gSMi8) from now on. Just have a brief glance, I hope they look familiar. We will study them later in detail. To come back to the question: We are trying to find systems of numbers other than the ones we already know, satsifying the same rules for the four basic operations of high school algebra $+,-,\times, /$. Actually, we should ask whether there are any such "non-standard" numbers at all. And if there are not, why not? And if there are, how many and can we classify them? But before we start on this we should take away from this discussion a broader outlook on the history of mathematics: ## Takeaway The axiomatic method plays an important role in mathematics. Axiomatising a certain mathematical notion, or even a whole area of mathematics, can be a major step forward. Before looking at some examples, I want to emphasise that we should not expect the axiomatic method to lay to rest all philophical questions such as what is infinity, probablity, computability, causality, time, etc. But mathematices does build powerful theories on simple answers to such difficult questions and these theories are much more consequential than the philosophical questions on their own. ### Geometry The most famous example is probably Euclid. Is there any other science that can teach from a book that is over 2000 years old? If you don't believe it, check out this beautiful version of Euclid, it is still as fresh as it was in ancient Greece. Euclid's method was taken up in modern times again by mathematicians such as Felix Klein, David Hilbert and Emmy Noether who laid the foundations of modern mathematics and physics. While this might not have been even a question in Euclid's time, it is an important question in modern physics: What is space? ### Caclulus/Analysis What is a continuous function? What does it mean for an infinite sequence to approximate a limit? Calculus suffered for centuries from the problem that it seemed to built on a contradiction: The existence of elements that are larger than 0 but smaller than all positive numbers. Finding the proper axiomatisation answering to the questions above was a crucial step in the history of mathematics. ### Logic What are the laws of thought? Can we compute with thoughts in much the same way as we compute with numbers? How do we have to modify the laws of algebra in order to compute with propositions instead of numbers? In 1854, George Bool published a book with this title in which he proposed an axiomatisation of the laws of thought inspired by calculus. Booles axiomatisation, known today as Boolean algebra, provides the foundation of many areas of science including computer hardware. ### Set Theory ... ### Computability ... ### Probablity ... ### Information ... ### Causality ... ### Process algebra ... Items I cannot add to this list are data, money, time, consciousness, ... feel free to ask your own ...