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In a set theoretic point of view all mathematical objects are sets. We "call" some of them as numbers (e.g. sets in $\mathbb{N}$, $\mathbb{Q}$, $\mathbb{R}$, $\mathbb{C}$, $Ord$, $Card$) but what is rigorous definition of a number? In fact this question is not as simple as it seems. There is a long sequence of logical and philosophical papers related to this subject which unfolds the complexity of the notion of number.

Sometimes in my set theory courses when I refer to members of $Ord$ and $Card$ as ordinal and cardinal "numbers", some students ask me if they are really numbers or not? In such a situation I usually say:

What do you mean by a "number"?

They usually say:

We mean something like natural, integer, rational, real and imaginary numbers?

Then I ask:

Why do you think those objects are more natural numbers than infinitary numbers? Which properties of $\mathbb{N}$, $\mathbb{Q}$, $\mathbb{R}$, $\mathbb{C}$ makes them a set of numbers? Why do you believe that $i$ is a number but $\omega_{\omega+1}+2$ is not? Are you sure that you don't refer to members of $\mathbb{C}$ as "numbers" just because you "heard" this word from your teachers before and this is just a relative social common belief?!

When we reach this point in the discussion, I emphasize on complexity of the notion of number and refer my students to philosophical papers as a project.

Sometimes I think it is good if I know what a usual school/university math teacher who is not familiar with logic and philosophy of maths professionally, says when his/her students ask him/her about the notion of number and the reason which makes members of $\mathbb{N}$, $\mathbb{Q}$, $\mathbb{R}$, $\mathbb{C}$ numbers. Surely most of students ideas about numbers is inspired by ideas of their teachers who are not necessarily logicians and it is important to understand students (possibly false) common beliefs about numbers.

Question: What are the most common beliefs about notion of number amongst non-logician math teachers? What is a "number" in their point of view?

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    $\begingroup$ Why do you want a rigorous definition of "number"? We already have rigorous treatments of N, Q, R, Z, Ord, Card....what purpose would an all-encompassing definition of number serve, either mathematically or pedagogically? $\endgroup$ – user173 Jul 28 '14 at 2:28
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    $\begingroup$ Numbers for me also include such things as quaternions, hyperbolic numbers, dual numbers, finitely generated supernumbers, infinitely generated supernumbers, etc... any associative algbebra over $\mathbb{R}$ qualifies, but that is even too narrow. It is quite a nebulous term. See math.stackexchange.com/a/865624/36530 for one of a multitude of such questions fielded on the MSE. (I realize your question is more specific here, not dupicate) $\endgroup$ – James S. Cook Jul 28 '14 at 3:07
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    $\begingroup$ "it is important to understand students (possibly false) common beliefs about numbers." - Can you give some support or an illustrative example of an instance where understanding students' beliefs about numbers came into play in instruction? For example, I heard a teacher tell his students "infinity is a number" on the basis that you could do operations on it (e.g. multiply it by 2). I thought of it as an issue of his notion of infinity, but perhaps you might see it as stemming from a notion of number? Or both? $\endgroup$ – JPBurke Jul 28 '14 at 9:40
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    $\begingroup$ It's not necessary to define numbers in terms of sets. IIRC one of the classic presentations of the surreal numbers (Knuth's? Conway's?) explicitly avoids constructing them in terms of sets, partly as a matter of taste and partly for practical/technical reasons. More relevantly for this question, I don't think "constructed from sets" is in any way relevant to the question of "what is number." $\endgroup$ – Ben Crowell Jul 29 '14 at 15:01
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    $\begingroup$ @kjetilbhalvorsen: But then we do have number systems such as the extended reals, hyperreals, and surreals that include infinite quantities. In the context of the extended reals, $\infty$ certainly is a number. In the context of the reals, $\infty$ is not a number. In the context of the hyperreals $\infty$ is not a number because infinite hyperreals come in more than one size. $\endgroup$ – Ben Crowell Aug 1 '14 at 18:03
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I think the level of the student is very important to this question. If the student has never had an abstract math course (like my students), then the lack of a definition of "number" is a great way to introduce the idea of abstract algebra. They are very happy to initially believe a definition like

A number is anything that you can add and multiply, such that addition and multiplication work and follow the right rules.

Of course I haven't defined "number" at all -- I've defined something else entirely -- but this is a great opening that allows you to discuss new objects. $\mathbf{Z}/12$ is a pretty easy starting point. You can explore $\mathbf{Z}/12$ and find all kinds of strange properties, but still see that addition and multiplication play nicely together, so it should still count as a set of "numbers."

Then you get to the point: you can use the definition of a "number" as something pliable that is open to opinion. For example:

  • $\mathbf{Z}/12$ has zero-divisors (don't say this or define it or even give the name "zero-divisor," just look around and find some). Does that feel illegal enough that these should not count as numbers? Or are they just strange numbers?
  • $\mathbf{Z}/12$ does not permit division. Does that feel illegal enough that these should not count as numbers?
  • $\mathbf{Z}/12$ does not allow you to add things like $\frac34$ to a number. Does that feel illegal enough that these should not count as numbers?

The resolution, which is that in abstract algebra all these things have names ("ring," "field," "$\mathbf{R}-algebra$," etc) is a stupid punchline that I don't even give them. The idea of abstract algebra is cooler than the names. Leaving the punchline as "the definition of number is open to opinion" is way, way better.

The best students will be really bothered by this open-ended piece of mathematics, and come back the next week and want to know if we can make $\infty$ is a number, or if we can make $i^i$ a number, of if 3:00pm $\cdot \frac34$ could be a number, and then we get to talk about ordinals, or surreals, or extensions, or parallel parking, or whatever else is in that direction.

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  • $\begingroup$ This answer feels misleading to me, since it suggests that "3/4" and "2 mod 12" are both numbers of the same sort. A more honest answer would use the word "number system", to distinguish those as numbers in different number systems, and would not say "the right rules" (as if there is only one set of rules) but rather "reasonable rules". Or, if that answer would be too abstract for the students you describe, I'd rather deflect the question. $\endgroup$ – user173 Jul 28 '14 at 12:13
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    $\begingroup$ @Matt F. Many things about my answer are misleading. Of course the eventual goal is to show that there are many different "right rules" and that as a result we should totally avoid trying to claim there is one set of "right rules." $\endgroup$ – Chris Cunningham Jul 28 '14 at 13:16
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    $\begingroup$ I'd second the notion of communicating to students that there is no single collection of "right rules". Context matters. Not everything that's "legal" is useful or sensible, while some arguably-good things are "illegal" in some locales, in math as in life... $\endgroup$ – paul garrett Jul 29 '14 at 16:44
  • $\begingroup$ You mentioned a good point on importance of having a nice "arithmetic" in definition of "numbers". $\endgroup$ – user230 Aug 2 '14 at 0:47
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    $\begingroup$ Worth adding the point that if numbers are numbers because they obey the right rules, then technically no single number can be a number by itself. They are only numbers because they come in a collection of objects that all follow the rules together. $\endgroup$ – DavidButlerUofA Aug 4 '14 at 20:12
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Whatever teachers may think about the nature of numbers, the foundations of "arithmetic" and the nature and concept of number in particular are very subtle. For a recent and sophisticated look at the issues, see: John Horton Conway, On Numbers and Games, second edition, A.K. Peters, 2001. Conway gives his approach to the surreal numbers, relates these to what has come before, and puts what he did a broader setting.

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I think this question is important. I'd love to see an actual answer to it and cannot upvote it enough. I don't have an answer, but would like to share some intuitions/speculation. I think the following is what teachers have absorbed as the notion of number. I do not think most teachers could articulate this.

Our early exposure to numbers comes about usually by naming groups of objects. The action of "counting" as pointing to objects in succession and saying number words allows us to move from the objects counted to the process of counting itself. The number words are then "reified" into objects themselves. One may regard this as a primitive "language game" in the sense of Wittgenstein. From this point of view a (natural) number encodes an action...something we do when faced with a set of objects where we can carry out a physical/linguistic process. It seems important that the stability or consistency of the linguistic pattern with the physical process be abstracted into language. When we abstract the process into language it becomes number.

Other simple physical/linguistic processes like grouping objects yield "addition" and "subtraction" language games. Again, working with a few stones is good enough to "understand" addition and subtraction, but bootstrapping using the brain's language capability to form a "pattern language" that encodes and extrapolates the observed regularity in the physical objects yields a first concept of number.

From this point of view, numbers are abstracted from processes found in language games. One might say that numbers, since tied with processes, are connected with the movement of time in some way. A good text on abstract algebra defined that subject as concerned with "general notions of number" as another answer to this question does.

Of course there is a significant leap from the above naive experience with number and the axiomatic approach to number, which employs quantifiers to axiomatize the abstracted properties even further. Of course we know what happens if we try to completely logicize basic number: Whitehead and Russel's 200 page proof of 1+1=2.

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    $\begingroup$ 379 pages actually. $\endgroup$ – skullpatrol Jul 30 '14 at 14:48
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    $\begingroup$ @skullpatrol:I was hoping someone would check that! $\endgroup$ – Jon Bannon Jul 30 '14 at 15:12
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I did a search on this question as well as the three existing answers for the term "applications" and did not find any occurrences. The natural numbers $\mathbb N$ do not cause students any particular difficulties of motivation. Arguably the main point to emphasize in introducing the successive enlargements $\mathbb{N}\subset\mathbb{Z}\subset\mathbb{Q}\subset\mathbb{R}\subset\mathbb{R^\ast}\ldots{}$ is their usefulness in applications.

The first two extensions are helpful in solving the simplest linear equations. If we wish to solve harder problems in algebra and geometry (finding diagonal of a square, area of a circle) it is convenient to extend further to the reals. To do calculus, it is convenient to include infinite numbers and infinitesimals, so we extend to the hyperreals. Complex numbers are of course easily motivated in terms of roots of polynomials, quaternions in terms of spacial rotations, etc.

Without applications it is difficult to motivate today's students. Specifically I wonder how useful it would be to tell them about transfinite ordinals which are after all not necessarily the most useful way of modeling infinity as far as concrete applications are concerned.

Responding to the comment by Matt, one of my favorite applications of the extenstion $\mathbb{R}\subset\mathbb{R^\ast}$ is Cauchy's definition of continuity of $f$, namely an infinitesimal increment $\alpha$ always leads to infinitesimal change $f(x+\alpha)-f(x)$. In my teaching experience students have an easier time relating to this than to the four-quantifier definition used in the epsilon-delta approach.

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  • $\begingroup$ Here is an argument from applications that I like: Applications of $\mathbf{N}$ and $\mathbf{Z}$ are made easier by using $\mathbf{Q}$. E.g.: Consider independent linear equations with coefficients in $\mathbf{N}$; the easiest way to check whether they have a solution in $\mathbf{N}$ is to find the solution in $\mathbf{Q}$ and see if that solution is in $\mathbf{N}$. It'd be harder to find a good argument from applications for going from $\mathbf{Q}$ to $\mathbf{R}$, and even harder for going from $\mathbf{R}$ to $\mathbf{R}^*$. $\endgroup$ – user173 Aug 5 '14 at 1:58
  • $\begingroup$ @MattF., I will spell out one such application in my answer. $\endgroup$ – Mikhail Katz Aug 5 '14 at 11:41
  • $\begingroup$ Thanks! I like continuity as an example for $\mathbf{R}^*$. But 1) Do you also have an argument from $\mathbf{Q}^*$ to $\mathbf{R}$? 2) I realize I don't know how to take limits uniquely in $\mathbf{R}^*$. Do you have an online reference to suggest? $\endgroup$ – user173 Aug 5 '14 at 22:25
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    $\begingroup$ @Matt, (1) do you mean from Q to R or from Q* to R? (2) This seems like a good candidate for an SE math question in nonstandard-analysis. $\endgroup$ – Mikhail Katz Aug 6 '14 at 7:57
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'Number' is just a word. It's meaning strongly depends on the audience and the information you want to give. For a small child a number is something she can show on her fingers. Growing up with number, it is something used to count things (N), measure dimensions (R), describe positions (C) and so on. You decide where to stop. For me, the number is the smallest piece of information you can reason about in the given context.

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How about:

The Book of Numbers by John H. Conway, Richard Guy

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    $\begingroup$ Why might this be a good choice? $\endgroup$ – Benjamin Dickman Aug 7 '14 at 16:42

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