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teaching:topics:number:axioms-formal

a more formal way to define numbers

Mathematicians have used different collections of numbers for thousands of years, gradually making them more sophisticated and applying them to the world in new ways. We do some operations with them including

`quad a+b ,quad a-b ,quad a×b ,quad a/b quad` and `quad a^b quad`.

We make sure these operations that we define with numbers are consistent as we extend the kinds of numbers we use. Much later in mathematics (especially beyond school) you might study these ideas very closely. Some school students — those who like to think abstractly and have that particular kind of imagination and creativity — find these ‘Pure Maths’ ideas a very good way to understand the ideas in school math. Many (most!) students find it all a bit scary and prefer more practical examples. This means lots of practice is needed to replace thinking deeply and eventually it all clicks. We hope.

It is important to remember that historically these very formal definitions came a long time after the ideas they define were invented and you do not need them to do the routine maths, but after you get the general idea it can help you understand what is happening in a much deeper way — and so make much better use of that maths in the world (and in the more difficult problem questions you get in tests and exams). This is the way that those original ideas are extended and become a powerful tool, and a way of thinking and communicating.

using 'sets' to describe the properties of numbers

We can propose rules to construct a set which we will call `cc N` of number-like elements with the operations `+` and `×` between them. Next we can show that some, or all, of the 12 properties listed below are true whenever those rules are true. We need some other important properties including what equality and order mean — that is what we mean by `=`, `<` and `>.` But essentially we need to draw a line somewhere — it is actually logically impossible to give a complete set of self-contained rules like this, we must leave some things as ‘just understood intuitively’ and we can always pose unanswerable questions within any system like this. This conclusion was proved almost 100 years ago.

We can make interesting and useful sets of things where only some of these properties hold or we can make finite sets of things that otherwise have all the properties of, say, the integers or rationals. At school we are mostly looking at either whole numbers (dealing with discrete things) or fractions and real numbers (dealing with quantities we can measure like length or weight). The difference between these two kinds of number is profound, and starting to understand that difference is a crucial step in high school maths.

In this formal definition we see that by including reciprocals (rule (11)) along with the other rules we get the Rational Numbers (all the fractions) and that now we can always divide any number into smaller parts. With these numbers we can describe any measurement as accurately as we want by making the denominator of our number big enough.

Whole fields of Mathematics work on these ideas, and applying them in new situations. Much of Physics involves applying this mathematics to measurements in the world around us, usually the wildest and most exotic kinds of measurement and numbers we have invented at the time.

Often the behaviour of these abstract, mathematical number-sets seems to match patterns we observe in the world around us. Then we work with those observations assuming the measurements behave like that set, like one of our number systems — so we can use all the familiar methods and theorems that we know about that kind of number.

This means that we can use our familiar algebra techniques to help solve problems or make predictions about that aspect of the world. We can compare our calculated results with what we observe in experiments — perhaps straightforward measurements of a particular property, or perhaps something much more exotic. If those predictions are correct then the idea that the things we are observing really do behave like the mathematical set we are using as a model gets stronger.

Here is a very formal summary of some of the most basic rules, using notation that might be new to you …

some set notation, as used here ...

  • `\ b in ccN quad` means: we are giving the name `b` to “an element of” (that is something “in”) the set which we are calling `ccN`
    • this is exactly what we do when we use a pronumeral in algebra
  • `forall b in ccN quad` then means: for all `b` in `ccN`
    • in other words … “for every possible `b` that we could choose from `ccN` …”
  • `exists c in ccN quad` means: there exists something we will call `c` in `ccN`
    • in other words … “we can always find some suitable element `c` in this particular collection of numbers …” (usually with some property that will be given given next).
  • `quad : quad` means “such that”
    • it can often be read as “given” or “where”
    • in other words: '… with the following conditions …’
  • so rule `(5)` … `quad forall a, b in ccN qquad exists c in ccN : quad c = a × b`
    reads as:`quad`for every `a` and `b` in `ccN` there is some `c`, also in `ccN`, where `\ c = a × b`.
    • “whenever we multiply two of these numbers the result is also one of these numbers”
    • this is what we mean when we say: “`ccN` is closed under multiplication”

number axioms

\begin{align*}\label{numbers} \newcommand{\defeq}{\stackrel{\scriptsize{\mathsf{def}}}{=}} \text{We construct a set, }{\scr N}\text{, of numbers ...}\\ \\ \textbf{counting numbers}\\ \text{are built up starting from }1\text{ and }\,+ \\ \tag{1} ∃\, 1 ∈ {\scr N} :\ ∀\,a ∈ {\scr N} &&& a × 1 = 1 × a = a \\ \tag{2} \forall\, a, b \in {\scr N}\quad \exists\, c \in {\scr N} :\, &&& c = a + b \\ \text{then we find some important rules} \\ \tag{3} ∀\, a, b, c ∈ {\scr N} &&& a + (b + c) = (a + b) + c \\ \tag{4} ∀\, a, b ∈ {\scr N} &&& a + b = b + a \\ \tag{5} \forall\, a, b \in {\scr N}\quad \exists\, c \in {\scr N} :\, &&& c = a × b \\ \tag{6} ∀\, a, b, c ∈ {\scr N} &&& a × (b × c) = (a × b) × c \\ \tag{7} ∀\, a, b ∈ {\scr N} &&& a × b = b × a \\ \tag{8} ∀\, a, b, c ∈ {\scr N} &&& a × (b + c) = (a × b) + (a × c) \\ \ \\ \textbf{integers also have zero and negatives} \\ \tag{9} ∃\, 0 ∈ {\scr N} :\ 0≠1\ \text{ and }\ ∀\, a ∈ {\scr N} &&& a + 0 = a\ \text{ and }\ a × 0 = 0 \\ \tag{10} ∀\, a ∈ {\scr N} \quad ∃ (−a) ∈ {\scr N} : &&& a + (−a) = (−a) + a = 0 \\ \text{and now we can always }\textit{subtract} &&& a-b\defeq a+(-b) \\ \ \\ \textbf{rationals also have reciprocals}\\ \tag{11} ∀\, a ∈ {\scr N} :\ a≠0 \quad ∃(\tfrac{1}{a})∈{\scr N}: &&& a × \frac{1}{a} = \frac{1}{a} × a = 1 \\ \text{and now we can (almost!) always }\textit{divide} &&& \forall\ b\ne0\quad a\div b\defeq a\times\frac{1}{b}=\frac{a}{b} \\ \ \\ \textbf{reals also include the irrational numbers} \\ \text{these cannot be represented as a fraction} \\ \text{examples include }\pi, {\rm e}, \text{ and } \sqrt{2} \\ \text{they can be defined by geometric constructions} \\ \text{or as the limits of series of fractions} \\ \text{or as the solution to algebraic equations} \\ \text{for example} \quad \sqrt{2}\defeq\ x \ : \ x^2=2 \text{ and } x\ge 0\\ \text{real numbers have all the above properties} \\ \ \\ \text{positive rationals are very ancient knowledge} \\ \text{irrationals were understood long before integers} \\ \text{zero was invented last ... at least 1400 years ago} \\ \ \\ \textbf{complex numbers also have } \mathrm{i} \\ \tag{12} \exists\, \mathrm{i} \in {\scr N} : &&& \mathrm{i} \times \mathrm{i} = -1 \\ \text{then we find that ...} \\ \forall\, a, b ∈ {\scr N} :\ a≠0 \quad ∃\, c ∈ {\scr N} : &&& c = a^b \\ \ \\ \text{complex numbers were developed 500 years ago} \\ \text{they are }\textit{algebraically closed} \\ \text{now we have solutions for } x^2 + 1 = 0 \\ \text{and lots more!}\quad \text{we find that} &&& {\rm e}^{{\rm i}x}=\cos x + {\rm i}\sin x \\ \text{this is the way we understand complex powers} \\ \text{and link algebra, geometry and trigonometry!} \\ \text{... complex numbers have many applications} \\ \ \\ \text{this formal description of numbers is more recent} \\ \ \\ \text{there is a }\textbf{lot}\text{ more if you dig just a little bit deeper}\\ \text{mathematics is }\textbf{always} \text{ adding to these ideas! } \\ \ \text{we next ask for formal axioms for equality} \\ \ \text{and sets} \\ \ \text{but be careful ... that could go on forever} \\ \ \text{... and }\textit{incompleteness}\text{ has been proved ...} \\ \text{that is: we can never answer}\textbf{ everything }\text{this way} \\ \ \\ \textbf{rather than starting by counting things} \\ \text{we might try another suitable collection} \\ \text{we must define }=,\ +\ \text{ and }\times\text{ for our elements} \\ \text{then we show which of the rules apply} \\ \text{we could start with some basic entity} \\ \text{... perhaps take the empty set } \emptyset \text{ as the number } 0 \\ \text{... or use a function, like in }\textit{lambda calculus} \\ \ \\ \textbf{we have developed many collections like this} \\ \text{if rules (1) to (10) apply we call them a }\textit{Ring} \\ \text{if rule (11) also applies we call them a }\textit{Field} \\ \textit{Ring}\text{ applies to discrete kinds of entities} \\ \textit{Field}\text{ often applies to continuous things} \\ \text{but it is much more interesting than that!} \\ \ \\ \text{we consider cases where (7) does not apply} \\ \text{and many other possibilities} \end{align*}

logic, mathematics, the world and philosophical considerations

Here we have been considering the properties of numbers and the structure of collections of number-like things. This helps us build and work with mathematical models of systems and patterns we see around us. Numbers, and mathematical representations of certain ideas, are part of language. They have been part of human communication, reasoning and decision making since well before any recorded history.

The description of numbers presented here is quite distinct from any attempt to formalise mathematics within some form of abstract, symbolic logic. It is also not especially concerned with the details of implementing calculating, reasoning or decision-making machines. It does present a structure which any symbolic representation or machine implementation must provide if it is to be able to work with this idea we call ‘number’, an abstraction we have worked with and developed over many millenia.

Strategies to make a symbolic logical base for mathematics were pursued especially vigorously around 1900. Taking the idea of a ‘Set’ as the starting point then building everything including numbers from there was an influential one. Another powerful strategy is to start with a ‘Function’ and construct everything from there. A more recent path has developed the idea of ‘Categories’ as a way of representing mathematical structures.

The philosophical questions “What is the relationship between truth and mathematics?” then “What is mathematics?” and “What is truth?” have been driving some of this quest. It is a fascinating topic, but not the one addressed here.

teaching/topics/number/axioms-formal.txt · Last modified: 2020/10/25 23:50 by simon