David Pierce: The geometry of numbers in Euclid

Reposted from David Pierce’s blog Polytropy

This is about how the Elements of Euclid shed light, even on the most basic mathematical activity, which is counting. I have tried to assume no more in the reader than elementary-school knowledge of how whole numbers are added and multiplied.

How come 7 ⋅ 13 = 13 ⋅ 7? We can understand the product 7 ⋅ 13 as the number of objects that can be arranged into seven rows of thirteen each.

Seven times thirteen

Seven times thirteen

If we turn the rows into columns, then we end up with thirteen rows of seven each; now the number of objects is 13 ⋅ 7.

Thirteen times seven

Thirteen times seven

In the end, it doesn’t matter whether we have arranged the objects into rows or columns of thirteen. Either way, when we gather up the objects and count them, we must always get the same result.

A heap of a hundred bottle-caps and five bottle-caps.

A heap of a hundred five bottle-caps.

Must we really? We believe from childhood that we must. As children, we learn to say certain words in a certain order: one, two, three, four, and so on. We learn to say these words as we move objects, one by one, from one pile to another. As we move the last object, the last word we say is supposed to be the number of objects in the original pile. We have now counted that pile. In the process, we have removed the pile; but if we count the new pile, we get the same number.

At least we think we do. Does anybody ever question this? If we do question it, and if we are familiar with some mathematical terminology, we may decide that, in technical terms, what we are asking is whether all linear orderings of the same finite set are isomorphic, or whether all one-to-one functions from the set to itself are also onto the set. We can prove that they are, in either case, by the method of mathematical induction. However, I suppose it takes some mathematical sophistication, not only to understand the terminology, but to believe that anything is accomplished by its use. If one is being asked to learn the method of mathematical induction for the first time, I doubt one will be impressed by its usefulness in establishing that, no matter how many times you count a bag of bottle-caps, you will always get the same number.

Meanwhile, there is a more fundamental question: what is a number in the first place? As it happens, for me, the best theoretical answer is that a number, a counting number, is a nonempty ordinal that neither contains a limit nor is itself a limit. An ordinal is a set with two properties: (1) it contains every member of each of its members, and (2) among the members of each of its nonempty subsets, there is one that has no other as a member. The empty set is an ordinal, and if a set called alpha is an ordinal, then so is the set that contains every member of alpha, along with alpha itself. This new set is the successor of alpha, and every ordinal that is neither the empty set nor a successor is a limit. Now, using the method given by von Neumann in 1923, I have defined counting numbers in simple terms, but in a complicated way that cannot be made sense of without some work. I am not going to do that work here, but I shall instead suggest that Euclid’s Elements offers an understanding of numbers that is unmatched, as far as I know, until the work of Dedekind in 1888. It some ways it may remain unmatched in the twenty-first century.

For Euclid, a number is a magnitude. A pile of bottle-caps might be called a magnitude; at least it has a weight, to which may be assigned a number. No matter how the bottle-caps are piled into the pan of a scale, we expect the same weight to be found; but it is hard to see how this observation can be made into a mathematical principle.

Euclid’s typical magnitudes—the ones seen in his diagrams—are bounded straight lines, or what we call line segments. What makes one of these a number is that some specified segment measures it, or goes into it evenly. This is the fundamental notion. The measuring segment is a unit, as is any other segment that is equal to it—equal in the sense of being congruent.

A number then is a magnitude that can be divided into units. Unless it is prime, it can be divided into numbers as well. Thus a number consisting of fifteen units can be divided into those fifteen units, or into five numbers of three units each, or three numbers of five units each. In the last case, we might refer to each of those three numbers as five; but then, strictly speaking, we are using the adjective five as a noun meaning five units. The units may vary. All fives are equal—equal in number— but they are not all the same.

Dividing is not the same as measuring, but complementary. Dividing fifteen apples among five children is a different activity from measuring how many five-apple collections can be formed from fifteen apples. In the first case, each child gets three apples, in the second, three collections of apples are formed. A number of three things arises in each case, because multiplying three by five has the same result as multiplying five by three.

This is only a special case of Euclid’s general result, which is Proposition 16 of Book VII of the Elements:

Ἐὰν δύο ἀριθμοὶ πολλαπλασιάσαντες ἀλλήλους ποιῶσί τινας,
οἱ γενόμενοι ἐξ αὐτῶν ἴσοι ἀλλήλοις ἔσονται.

If two numbers multiply one another,
their products will be equal to one another.

Actually the Greek is a bit wordier: If two numbers, multiplying one another, make some things, the products of them [that is, the original numbers] will be equal to one another. This multiplication is defined as follows:

Ἀριθμὸς ἀριθμὸν πολλαπλασιάζειν λέγεται,
ὅταν,
ὅσαι εἰσὶν ἐν αὐτῷ μονάδες,
τοσαυτάκις συντεθῇ ὁ πολλαπλασιαζόμενος,
καὶ γένηταί τις.

A number is said to multiply a number
whenever,
however many units are in it,
so many times is the the number being multiplied laid down,
and something is produced.

Jonathan Crabtree argues strenuously that multiplying A by B does not mean adding A to itself B times, since this would result in a sum of B + 1 copies of A.

A product is the multiple of a multiplicand by a multiplier. Euclid proves that the roles of multiplicand and multiplier are interchangeable: in modern terms, multiplication is commutative. The proof uses a theory of proportion. In this theory, there are several ways to say the same thing:

  1. the numbers A, B, C, and D are in proportion;
  2. the ratio of A to B is the same as the ratio of C to D;
  3. A is to B as C is to D.

I shall abbreviate these by writing

A : B :: C : D.

The meaning of this for Euclid may not be crystal clear to modern readers; but I think it can only mean that when the so-called Euclidean algorithm is applied to C and D, the algorithm has the same steps as when applied to A and B.

Applied to any two magnitudes, each step of the Euclidean algorithm has the following two parts:

  1. Judge whether one of the magnitudes is greater than the other.
  2. If it is, then subtract from it a piece that is equal to the less magnitude.

Repeat as long as you can. There are three possibilities. In the simplest case, you will keep subtracting pieces equal to the same magnitude, until what is left of the other magnitude is equal to it. In this case, the one magnitude measures the other: alternatively, it is a part of the other. In case the algorithm never ends, then the original magnitudes must not have been numbers, but they were incommensurable. In the third case, you end up with a greatest common measure of the original two numbers, and each of these numbers is said to be parts of the other.

That is Euclid’s terminology. By the definition at the head of Book VII of the Elements, A : B :: C : D means A is the same part, or parts, or multiple of B that C is of D. Again, if we assume Euclid knows what he’s doing, this can only mean that, at each step of the Euclidean algorithm, the same magnitude (first or second, left or right) is greater, whether we start with A and B or C and D. Thus 8 : 6 :: 12 : 9, because

8 > 6, 8 – 6 = 2;
2 < 6, 6 – 2 = 4;
2 < 4, 4 – 2 = 2;

while in the same way

12 > 9, 12 – 9 = 3;
3 < 9, 9 – 3 = 6;
3 < 6, 6 – 3 = 3;

the pattern >, <, < is the same in each case. We discover incidentally that the greatest common measure of 8 and 6 is 2; and of 12 and 9, 3. In fact

8 = 2 ⋅ 4,  6 = 2 ⋅ 3,  12 = 3 ⋅ 4,  9 = 3 ⋅ 3.

The repetition of the multipliers 4 and 3 here also ensures the proportion 8 : 6 :: 12 : 9, but only because 3 and 4 are prime to one another: they have no common measure, other than a unit. If we did not impose this condition on the multipliers, then the definition of proportion alone would not ensure the transitivity of sameness of ratios: the definition alone would not guarantee that ratios that were the same as the same ratio were the same as one another. But every kind of sameness has this property. Therefore, although Euclid does not quite spell it out, I contend that his definition of proportion of numbers has the meaning that I have given.

We can now describe Euclid’s proof of the commutativity of multiplication as follows. We accept that addition is commutative:

A + B = B + A.

This means, if you pick up a rod, turn it end to end, and put it back down, it will still occupy the same distance. One might try to imagine a geometry in which this is not true; but we assume it is true. It follows then that, for any multiplier x,

Ax + Bx = (A + B)x,

that is,

A + … + A + B + … + B = A + B + … + A + B,

where the ellipses represent the same number of missing terms in each case.

Given A : B :: C : D, we now show A : B :: (A + C) : (B + D). Assuming, as we may, that A is less than B, we have two possibilities:

  1. A is part of B, and so, for some x, we have
    B = Ax, D = Cx.
  2. A is parts of B, and so, for some x and y that are prime to one another, for some E and F, we have
    B = Ex, A = Ey, D = Fx, C = Fy.

In the first case, B + D = Ax + Cx = (A + C)x. Similarly, in the second case, B + D = (E + F)x, while A + C = (E + F)y. In either case, we have the desired conclusion, A : B :: (A + C) : (B + D). As special cases, we have

A : B :: A + A : B + B :: A + A + A : B + B + B

and so on; in general, A : B :: Ax : Bx.

Given again A : B :: C : D, we now show A : C :: B : D. We consider the same two cases as before. In case (1), we have A : C :: Ax : Cx :: B : D. In the same way, in case (2), we have A : C :: E : F and B : D :: E : F, so again A : C :: B : D.

Finally, denoting a unit by 1, since by definition we have 1 : A :: B : BA and 1 : B :: A : AB, and the latter gives us now 1 : A :: B : AB, we conclude BA = AB. This is Proposition 16 of Book VII of Euclid’s Elements.

Thus, if we lay out seven sticks end to end, each thirteen units long, we reach the same length as if we lay out thirteen sticks, each seven units long. This is not obvious, even though, if you follow the rules of computation learned in school, you will find that 7 ⋅ 13 and 13 ⋅ 7 are equal. Euclid proves that this will be so, without any need for computation—which anyway will apply only to the particular example in question.

Thales and the Nine-point Conic

New paper in the De Morgan Gazette:

David Pierce, Thales and the Nine-point Conic, The De Morgan Gazette 8 no. 4 (2016)  27-78. bit.ly/2hlyHzZ. ISSN 2053-1451

Abstract: The nine-point circle is established by Euclidean means; the nine-point conic, Cartesian.Cartesian geometry is developed from Euclidean by means of Thales’ s Theorem. A theory of proportion is given, and Thales’s Theorem proved, on the basis of Book I of Euclid’s Elements, without the Archimedean assumption of Book V. Euclid’s theory of areas is used, although this is obviated by Hilbert’s theory of lengths. It is observed how Apollonius relies on Euclid’s theory of areas. The historical foundations of the name of Thales’s Theorem are considered. Thales is thought to have identified water as a universal substrate; his recognition of mathematical theorems as such represents a similar unification of things.

A. Borovik: Sublime Symmetry: Mathematics and Art

A new paper in The De Morgan Gazette:

Form the Introduction:

This paper is a text of a talk at the opening of the Exhibition Sublime Symmetry: The Mathematics behind De Morgan’s Ceramic Designs  in the delighful Towneley Hall  Burnley, on 5 March 2016. The Exhibition is the first one in Sublime Symmetry Tour  organised by The De Morgan Foundation.

I use this opportunity to bring Sublime Symmetry Tour to the attention of the British mathematics community, and list Tour venues:

06 March to 05 June 2016 at Towneley Hall, Burnley
11 June to 04 September 2016 at Cannon Hall, Barnsley
10 September to 04 December 2016 at Torre Abbey, Torbay
10 December 2016 to 04 March 2017 at the New Walk Gallery, Leicester
12 March to 03 September 2017 at the William Morris Gallery, Walthamstow

William De Morgan, Peacock Dish. The De Morgan Foundation

William De Morgan, Peacock Dish. The De Morgan Foundation.

Yagmur Denizhan: Performance-based control of learning agents and self-fulfilling reductionism.

Yagmur Denizhan: Performance-based control of learning agents and self-fulfilling reductionism. Systema 2 no. 2 (2014) 61-70. ISSN 2305-6991. The article licensed under the Attribution-NonCommercial-NoDerivatives 4.0 International License. A PDF file is here.

Abstract: This paper presents a systemic analysis made in an attempt to explain why half a century after the prime years of cybernetics students started behaving as the reductionist cybernetic model of the mind would predict. It reveals that self-adaptation of human agents can constitute a longer-term feedback effect that vitiates the efficiency and operability of the performance-based control approach.

From the Introduction:

What led me to the line of thought underlying this article  was a strange situation I encountered sometime in 2007 or 2008. It was a new attitude in my sophomore class that I never observed before during my (by then) 18 years’ career. During the lectures whenever I asked some conceptual question in order to check the state of comprehension of the class, many students were returning rather incomprehensible bulks of concepts, not even in the form of a proper sentence; a behaviour one could expect from an inattentive school child who is all of a sudden asked to summarise what the teacher was talking about, but with the important difference that –as I could clearly see– my students were listening to me and I was not even forcing them to answer. After observing several examples of such responses I deciphered the underlying algorithm. Instead of trying to understand the meaning of my question, searching for a proper answer within their newly acquired body of knowledge and then expressing the outcome in a grammatically correct sentence, they were identifying some concepts in my question as keywords, scanning my sentences within the last few minutes for other concepts with high statistical correlation with these keywords, and then throwing the outcome back at me in a rather unordered form: a rather poorly packaged piece of Artificial Intelligence.
It was a strange experience to witness my students as the embodied proof of the hypothesis of cognitive reductionism that “thinking is a form of computation”. Stranger, though, was the question why all of a sudden half a century after the prime years of cybernetic reductionism we were seemingly having its central thesis1 actualised.

Rebecca Hanson: National Assessment Reform – Where are we now?

R. Hanson, National Assessment Reform – Where are we now? The De Morgan Gazette 5 no. 5 (2014), 33-39.

This short report summarises the pending changes to national assessment at 4/5, 6/7, 10/11, 15/16 and 17/18.  It attempts to list the key concerns about the reforms and to describe the likely imminent calls for modifications.

It can also be downloaded as a word document here:
National Assessment Reform Where are we now 1 Sept 2014

If you have any questions you can contact the author.

Misha Gavrilovich: Point-set topology as diagram chasing computations

M. Gavrilovich, Point-set topology as diagram chasing computations, The De Morgan Gazette 5 no. 4 (2014), 23-32.

Abstract:

We observe that some natural mathematical definitions are lifting properties relative to simplest counterexamples, namely the definitions of surjectivity and injectivity of maps, as well as of being connected, separation axioms \(T_0\) and \(T_1\) in topology, having dense image, induced (pullback) topology, and every real-valued function being bounded (on a connected domain).

We also offer a couple of brief speculations on cognitive and AI aspects of this observation, particularly that in point-set topology some arguments read as diagram chasing computations with finite preorders.

Tony Gardiner: Teaching mathematics at secondary level

A. D. Gardiner, Teaching mathematics at secondary level. The De Morgan Gazette 6 no. 1 (2014), 1-215.

From the Introduction:

This extended essay started out as a modest attempt to offer some supporting structure for teachers struggling to implement a rather unhelpful National Curriculum.  It then grew into a Mathematical manifesto that offers a broad view of secondary mathematics, which should interest both seasoned practitioners and those at the start of their teaching careers.  This is not a DIY manual on How to teach.  Instead we use the official requirements of the new National Curriculum in England as an opportunity:

  • to clarify certain crucial features of elementary mathematics and how it is learned — features which all teachers need to consider before deciding `How to teach’.

Continue reading

Tony Gardiner: National curriculum – Comments and suggested necessary changes

Published today:

A. D. Gardiner, National curriculum (England), September 2013; Attainment targets and programmes of study (key stages 1–3). Comments and suggested necessary changes. The De Morgan Gazette 4 , no. 3 (2013), 13-57

From the Introduction:

The Education Order 2013 was “made” on 5 September 2013. The relevant details were “laid before parliament” on 11 September 2013, and will come into effect on 1 September 2014. Some of the details for GCSE were published on 1 November 2013. Further elaboration of GCSE assessment structure, and curriculum guidance for Key Stage 4 (Years 10–11, ages 14–16) are awaited.

It is generally agreed that the curriculum review process adopted over the last 3–4 years has been seriously flawed. Those involved worked hard, often under very difficult conditions. But the overall approach (of relying on civil servants and drafters whose responsibilities and constraints remained inscrutable) has merely demonstrated that drafting and maintaining curricula is a specialist task, requiring dedicated professionals with specialist experience.

Whatever flaws there may have been in the process, we will all have to live with the new curriculum for some years. So it is important to have an open discussion of the likely difficulties. This article is an attempt to indicate aspects of the National curriculum in England: mathematics programmes of study that will need to be handled with considerable care, and revised in the light of experience.

After three years of widespread unease about the process of the curriculum review and its apparent direction, it is remarkable that there has been almost no media coverage, and no clear professional response to the final mathematics programmes of study for ages 5–14. There is therefore a real danger that insights that emerged along the way will simply be forgotten, and that the same mistakes may then be made next time. […]

The details laid before parliament are `statutory’; but they incorporate basic flaws, and significant contradictions between the statutory list of content (which could all-too-easily be imposed uncritically) and the declared over-arching “aims” (which could get forgotten, or ignored). Given these flaws, the fate of the new programmes of study will depend on how sensitively their implementation is handled—whether slavishly, or intelligently. Teachers—and Ofsted, senior management, etc.—need to be alert to those aspects of the stated programmes of study that incorporate predictable pitfalls.

We summarise here what seem to be the two most important flaws.

Some material in Key Stage 1 and 2 is very poorly specified (especially from Year 4 onwards).

Some items are listed unnecessarily and unrealistically early, and so may be introduced at a stage:

  • where they are not yet needed,
  • where they will not be understood,
  • where they will be badly taught, and
  • where – if the relevant requirements were relaxed – the premature material could easily be delayed without causing any subsequent problems.

The listing of content for Key Stage 3 is in some ways reasonable, but too many things are left implicit. The programme of study is less structured than, and contains less detail than, that for Key Stages 1 and 2. Hence the details of the Key Stage 3 programme need interpretation. At present:

  • the words of each bullet point are rarely elaborated;
  • the connections between themes are mostly suppressed; and
  • there is no mention of essential preliminaries.

In addition

  • the Key Stage 3 programme has no accompanying `Notes and guidance’.

In summary, if the declared goals for Key Stage 4 are to be realised,

  • we need some way of clarifying the specified content and relaxing the unnecessary and potentially damaging pressures built in to the Key Stage 1–2 curriculum as it stands; and
  • the centrally prescribed curriculum for Key Stage 3 needs to be much more clearly structured to help schools understand what it is that is currently missing at this level—initially by providing suitable non-statutory `Notes and guidance’.

The De Morgan Journal: change of the name

By a decision of the LMS Education Committee, The De Morgan Journal changes its name to The De Morgan Gazette (ISSN 2053-1451).

The last paper of the old Journal and the first paper of the new Gazette are two parts of Tony Gardiner’s analysis of changes in Mathematics GCSE: