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.

Call for Nominations for the 2017 ICMI Felix Klein and Hans Freudenthal Awards

From IMCI Newsletter November 2016:

[All nominations must be sent by e-mail to the Chair of the Committee (annasd >>at<< edu.haifa.ac.il, sfard >>at<< netvision.net.il) no later than 15 April 2017.]

Since 2003, the International Commission on Mathematical Instruction (ICMI) awards biannually two awards to recognise outstanding accomplishments in mathematics education research: the Felix Klein Medal and the Hans Freudenthal Medal.

The Felix Klein medal is awarded for life-time achievement in mathematics education research. This award is aimed at acknowledging excellent senior scholars who have made a field-defining contribution over their professional life. Past candidates have been influential and have had an impact both at the national level within their own countries and at the international level. We have valued in the past those candidates who not only have made substantial research contributions, but also have introduced new issues, ideas, perspectives, and critical reflections. Additional considerations have included leadership roles, mentoring, and peer recognition, as well as the actual or potential relationship between the research done and improvement of mathematics education at large, through connections between research and practice.

The Hans Freudenthal medal is aimed at acknowledging the outstanding contributions of an individual’s theoretically robust and highly coherent research programme. It honours a scholar who has initiated a new research programme and has brought it to maturation over the past 10 years. The research programme is one that has had an impact on our community. Freudenthal awardees should also be researchers whose work is ongoing and who can be expected to continue contributing to the field. In brief, the criteria for this award are depth, novelty, sustainability, and impact of the research programme.
For further information about the awards and for the names of past awardees (seven Freudenthal Medals and seven Klein Medals, to date), see http://www.mathunion.org/icmi/activities/awards/the-klein-and-freudenthal-medals/

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Blind people use brain’s visual cortex to help do maths

From New Scientist, by Colin Barras, September 2016:

“It’s actually hard to think of a situation when you might process numbers through any modality other than vision,” says Shipra Kanjlia at Johns Hopkins University in Baltimore, Maryland.

But blind people can do maths too. To understand how they might compensate for their lack of visual experience, Kanjlia and her colleagues asked 36 volunteers – 17 of whom had been blind at birth – to do simple mental arithmetic inside an fMRI scanner. To level the playing field, the sighted participants wore blindfolds.

We know that a region of the brain called the intraparietal sulcus (IPS) is particularly active when sighted people process numbers, and brain scans revealed that the same area is similarly active in blind people too.

“It’s really surprising,” says Kanjlia. “It turns out brain activity is remarkably similar, at least in terms of classic number processing.”

Read the whole story. Journal reference: PNAS, DOI: 10.1073/pnas.1524982113

From the original research paper:

Significance

Human numerical reasoning relies on a cortical network that includes frontal and parietal regions. We asked how the neural basis of numerical reasoning is shaped by experience by comparing congenitally blind and sighted individuals. Participants performed auditory math and language tasks while undergoing fMRI. Both groups activated frontoparietal number regions during the math task, suggesting that some aspects of the neural basis of numerical cognition develop independently of visual experience. However, blind participants additionally recruited early visual cortices that, in sighted populations, perform visual processing. In blindness, these “visual” areas showed sensitivity to mathematical difficulty. These results suggest that experience can radically change the neural basis of numerical thinking. Hence, human cortex has a broad computational capacity early in development.

Abstract

In humans, the ability to reason about mathematical quantities depends on a frontoparietal network that includes the intraparietal sulcus (IPS). How do nature and nurture give rise to the neurobiology of numerical cognition? We asked how visual experience shapes the neural basis of numerical thinking by studying numerical cognition in congenitally blind individuals. Blind (n = 17) and blindfolded sighted (n = 19) participants solved math equations that varied in difficulty (e.g., 27 − 12 = x vs. 7 − 2 = x), and performed a control sentence comprehension task while undergoing fMRI. Whole-cortex analyses revealed that in both blind and sighted participants, the IPS and dorsolateral prefrontal cortices were more active during the math task than the language task, and activity in the IPS increased parametrically with equation difficulty. Thus, the classic frontoparietal number network is preserved in the total absence of visual experience. However, surprisingly, blind but not sighted individuals additionally recruited a subset of early visual areas during symbolic math calculation. The functional profile of these “visual” regions was identical to that of the IPS in blind but not sighted individuals. Furthermore, in blindness, number-responsive visual cortices exhibited increased functional connectivity with prefrontal and IPS regions that process numbers. We conclude that the frontoparietal number network develops independently of visual experience. In blindness, this number network colonizes parts of deafferented visual cortex. These results suggest that human cortex is highly functionally flexible early in life, and point to frontoparietal input as a mechanism of cross-modal plasticity in blindness.

Alexandra O Fradkin: Exploring Rectangles

Friday is a special day in our math classes at the Main Line Classical Academy.  We read and discuss mathematical stories and we do exploration projects.  Here is the project that we did with the 2nd-4th grades last Friday.

It began with one of my favorite questions to discuss with kids: What is a rectangle?  Some of the kids in each class had participated in previous discussions with me on this topic, but this was close to 2 years ago and so probably had very little effect on the outcome.

Here is what the boards looked like after the 2nd grade and the 3rd/4th grade discussions respectively:

img_2963

img_2978-1

The kids used a lot of hand motions in their initial descriptions, but I told them to pretend that we were talking on the phone and I couldn’t see them.  They would also sometimes come up with very long and convoluted explanations, which I also refused to write on the board.  After each initial set of properties, I’d try to draw a shape on the board that fit them all but was not a rectangle or did not fit some of them and was a rectangle (some of the shapes unfortunately did not make it into the pictures).  The kids had a lot of laughs when I would draw a silly shape and ask them “is this a rectangle?”  In the end though, I believe that we settled on a set of properties that succinctly characterized rectangles.

The second part of the class consisted of making all possible rectangles out of a given number of squares.  The kids had to make them out of snap cubes and then draw them on graph paper.  The second graders all got 12 snap cubes while the 3rd/4th graders initially got 12 and then each their own different number between 18 and 32.

I was very surprised that no one tried to draw the same rectangle in different orientations.  Some kids did, however, try to make and draw rectangles with holes in them.  A few of the second graders initially had trouble because the squares on the graph paper were smaller than the snap cubes, so tracing the structure did not work.  However, after a brief discussion, they were all able to make the one-to-one correspondence between the cubes and the squares.

Here are some pictures of the process:

img_2979

In the end, we discussed with both groups how to make sure that we have made all the possible rectangles.  One of the older kids pointed out the connection with factors/divisors of a number.  None of the kids had formally studied area or multiplication (although most know what those are to various degrees), but those will both be big topics in the 3rd/4th grade class this year.  I think that this served as a good indirect introduction to them.

Olivier Gerard: Learning mathematics as a Russian interpreter

You might be interested in reading How I Rewired My Brain to Become Fluent in Math, by Barbara Oakley, in Nautilus, October 2, 2014.

A quote:

“Time after time, professors in mathematics and the sciences have told me that building well-ingrained chunks of expertise through practice and repetition was absolutely vital to their success. Understanding doesn’t build fluency; instead, fluency builds understanding. In fact, I believe that true understanding of a complex subject comes only from fluency.

In other words, in science and math education in particular, it’s easy to slip into teaching methods that emphasize understanding and that avoid the sometimes painful repetition and practice that underlie fluency. “

How to raise a genius: lessons from a 45-year study of super-smart children

How to raise a genius: lessons from a 45-year study of super-smart children, by Tom Clynes, 07 September 2016, in Nature | News Feature.

On a summer day in 1968, professor Julian Stanley met a brilliant but bored 12-year-old named Joseph Bates. The Baltimore student was so far ahead of his classmates in mathematics that his parents had arranged for him to take a computer-science course at Johns Hopkins University, where Stanley taught. Even that wasn’t enough. Having leapfrogged ahead of the adults in the class, the child kept himself busy by teaching the FORTRAN programming language to graduate students.

Unsure of what to do with Bates, his computer instructor introduced him to Stanley, a researcher well known for his work in psychometrics — the study of cognitive performance. To discover more about the young prodigy’s talent, Stanley gave Bates a battery of tests that included the SAT college-admissions exam, normally taken by university-bound 16- to 18-year-olds in the United States.

Read the rest of the story

Tim Gowers: In case you haven’t heard what’s going on in Leicester …

[Reposted from Tim Gowers’ Blog, 15 Sept 2016]

Strangely, this is my second post about Leicester in just a few months, but it’s about something a lot more depressing than the football team’s fairytale winning of the Premier League (but let me quickly offer my congratulations to them for winning their first Champions League match — I won’t offer advice about whether they are worth betting on to win that competition too). News has just filtered through to me that the mathematics department is facing compulsory redundancies.

The structure of the story is wearily familiar after what happened with USS pensions. The authorities declare that there is a financial crisis, and that painful changes are necessary. They offer a consultation. In the consultation their arguments appear to be thoroughly refuted. The refutation is then ignored and the changes go ahead.

Here is a brief summary of the painful changes that are proposed for the Leicester mathematics department. The department has 21 permanent research-active staff. Six of those are to be made redundant. There are also two members of staff who concentrate on teaching. Their number will be increased to three. How will the six be chosen? Basically, almost everyone will be sacked and then invited to reapply for their jobs in a competitive process, and the plan is to get rid of “the lowest performers” at each level of seniority. Those lowest performers will be considered for “redeployment” — which means that the university will make efforts to find them a job of a broadly comparable nature, but doesn’t guarantee to succeed. It’s not clear to me what would count as broadly comparable to doing pure mathematical research.

How is performance defined? It’s based on things like research grants, research outputs, teaching feedback, good citizenship, and “the ongoing and potential for continued career development and trajectory”, whatever that means. In other words, on the typical flawed metrics so beloved of university administrators, together with some subjective opinions that will presumably have to come from the department itself — good luck with offering those without creating enemies for life.

Oh, and another detail is that they want to reduce the number of straight maths courses and promote actuarial science and service teaching in other departments.

There is a consultation period that started in late August and ends on the 30th of September. So the lucky members of the Leicester mathematics faculty have had a whole month to marshall their to-be-ignored arguments against the changes.

It’s important to note that mathematics is not the only department that is facing cuts. But it’s equally important to note that it is being singled out: the university is aiming for cuts of 4.5% on average, and mathematics is being asked to make a cut of more like 20%. One reason for this seems to be that the department didn’t score all that highly in the last REF. It’s a sorry state of affairs for a university that used to boast Sir Michael Atiyah as its chancellor.

I don’t know what can be done to stop this, but at the very least there is a petition you can sign. It would be good to see a lot of signatures, so that Leicester can see how damaging a move like this will be to its reputation.