Narrow Road

7/27/2019 Narrow Road

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The Narrow Roadto the InteriorA Mathematical Journey

Leland McInnes

Current as of July 2, 2007http://jedidiah.stuff.gen.nz/wp/


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Contents

First Steps 1

1.1 On Abstraction . . . . . . . . . . . . . . . . . . . . . . . . . . 21.2 The Slow Road . . . . . . . . . . . . . . . . . . . . . . . . . . 7

1.3 A Fraction of Algebra . . . . . . . . . . . . . . . . . . . . . . . 10

A Fork in the Road 17

2.1 The Paradoxes of the Continuum, Part I . . . . . . . . . . . . 18

2.2 Shifting Patterns . . . . . . . . . . . . . . . . . . . . . . . . . 24

2.3 Paradoxes of the Continuum, Part II . . . . . . . . . . . . . . 342.4 Permutations and Applications . . . . . . . . . . . . . . . . . 41

2.5 A Transfinite Landscape . . . . . . . . . . . . . . . . . . . . . 49

2.6 Grouping Symmetries . . . . . . . . . . . . . . . . . . . . . . . 58

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ontents iv


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First Steps 1

he Narrow Road draws its title from Oku no Hosomichi (The Narrow Roadthe Interior), the famous travel diary of Matsuo Basho as he journeyed into

orthern Japan. My aim is to follow a similar wandering journey, but insteadavelling into the abstract highlands of pure mathematics, pausing to admiree beauty and sights along the way, much as Basho did. That means we

ave a long way to travel: from the basics of abstract or pure mathematics,rough topology, manifolds, group theory and abstract algebra, category

eory, and more. There may well be some detours along the way as well.is going to take a long time to get to where we are going, but along the

ay well see plenty of things that make the trip worthwhile. Indeed, as isoften the case, the journey means more than the destination.

1


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irst Steps 2

.1 On Abstraction

ets begin with a short practical experiment. Pick up a pen, or whatever

milar sized object is handy, hold it a short distance above the ground, and

op it. The result that the pen falls to the ground is not a surprising one.he point of the experiment was not to note the result, however, but rather

note our lack of surprise at it. We expect the pen to fall to the ground;

ur expectation is based not on knowledge of the future however, but on

bstraction from past experience. Chambers Dictionary defines abstract,

e verb, to mean to generalize about something from particular instances,

nd it is precisely via this action that we come to expect the pen to fall to

e ground. By synthesis of many previous instances of objects falling whene drop them, we have generalized the rule that things will always fall when

e drop them1. We make this abstraction so instinctively, and take it so

mpletely for granted, that it is worth dwelling on it for a moment so we

n see how remarkable it actually is.

The circumstances surrounding each and every instance of you observing

n object falling to the ground are quite unique. Were it not for our brains

atural tendency to try to link together our experiences into some kind of

arrative we would be left contemplating each dropped object as an entirely

stinct instance and be in no position to have any expectation as to what

ill happen each time it would be an entirely new case. In our minds we

ave, rather than a vast array of disjoint and distinct instances, a single prin-

ple that knits together the common elements. This allows us to generalise

any new circumstances that share the same common elements as those of

ast experiences. This is something we do unconsciously; automatically: our

ains hunt for patterns in the world and try to generalise those patterns in

e expectectation that they will continue. Indeed, almost all our expecta-

ons are a result of such inductive knowledge and abstraction abstraction

fundamental to our experience.

Of course, not all abstractions are correct. Our minds are constantly on

e hunt for possible patterns, and dont always pick out valid ones. Theassic example is the Christmas goose, who, every day of the year has found

e arrival of the farmer results in the goose getting fed until Christmas day

hen the abstracted rule that Farmer implies food meets a painful end.

ven if we find abstractions that are ostensibly correct, that doesnt mean


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On Abstraction

and. Certainly that is true, but it leaves out what might be considered anmportant common property of things being dropped: the direction in which

e dropped object accelerates. Alternatively we could note that differentbjects, say a feather compared to a stone, behave very differently when

opped and arrive at a vast array of rules, one for each different kind ofbject.

This leads to the dilemma of finding the most effective or efficient abstrac-on the abstraction that most consistently produces the results you wantith the least effort. That is where science comes in: it is a systematic effort

refine our abstracted rules and principles, and continually check them fornsistency. We may, if we like, think of the different sciences as arising from

e results we want clause of our definition of effective abstraction. A bi-ogist tends to work with different abstractions (generally on very differentales) than a physicist because the sorts of results they are interested in

etermining are rather different.This is all very interesting, but you are probably starting to wonder what

ny of it has to do with mathematics. The answer is that mathematics reliesn precisely this sort of abstraction that is so integral to our experience of

e world. Mathematics simply attempts to take the abstraction as far as itossibly can. Part of what makes mathematics difficult is that it tends tole abstraction upon abstraction. That is to say, after developing a particu-r abstraction it is common for mathematics to then study that abstraction

nd, upon finding common properties when dealing with that abstraction,neralise that commonality into a new abstraction; mathematics develops

bstractions not only from a synthesis of experiences of the external world,ut also via synthesis of properties of existing abstractions. This layeringeans that unless youve gotten a good grasp of the preceding level of ab-raction, the current one can be extremely hard to follow. Once youveandered off the path, so to speak, it can be difficult to find your way back.

The other problem that people tend to face, when learning mathematics,that as you pile up abstractions and climb higher, and hence more distant

om everyday experience, intuition becomes less and less helpful, and ancreasing degree of pedantry is required. To give an example of what Iean by this, lets take a detailed look at a mathematical abstraction thatmost everyone takes for granted: numbers.

Natural numbers (also known as counting numbers) are one of those re-


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irst Steps 4

stance that possesses the property common to all the particular instancesom which the number 3 is generalised. Because we take natural numbers foranted we tend to make assumptions about how they work without thinkingrough the details. For instance we all know that 1 + 1 = 2 but does it?

onsider a raindrop running down a windowpane. Another raindrop can runown to meet it and the separate raindrops will merge into a single raindrop.ne raindrop, plus another raindrop, results in one raindrop: 1 + 1 = 1.he correct response to this challenge to common sense is to say but thatsot what I mean by 1 + 1 = 2 and going on to explain why this particular

ample doesnt qualify. This, however, raises the question of what exactlye do mean when we say that 1 + 1 = 2. To properly specify what weean, and rule out examples like putting 1 rabbit plus another rabbit in a

ox and (eventually) ending up with more than 2 rabbits, is rather harderan you might think, and requires a lot of pedantry. Ill quote the alwayscid Bertrand Russell2 to explain exactly what we mean when we say that+ 1 = 2:

Omitting some niceties, the proposition 1 + 1 = 2 can be inter-preted as follows.

We shall say that is a unit property if it has the two following

properties:

1. there is an object a having the property;

2. whatever property f may be, and whatever object x may be,

if a has property f and x does not, then x does not have the

property.

We shall say that is a dual property if there is an object c such that

there is an object d such that:

1. there is a propertyF belonging to c but not tod;

2. c has the property andd has the property;

3. whatever propertiesf andg may be, and whatever object x may

be, if c has the property f and d has the property g and x has

neither, then x does not have property.

We can now enunciate 1 + 1 = 2 as follows: If and are unitproperties, and there is an object which has property but not the


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On Abstraction

We define 1 as being the property of being a unit property and2 as being the property of being a dual property.

The point of this rigmarole is to show that 1 + 1 = 2 can beenunciated without mention of either 1 or 2. The point may become

clearer if we take an illustration. Suppose Mr A has one son and onedaughter. It is required to prove that he has two children. We intendto state the premise and the conclusion in a way not involving thewords one or two.

We translate the above general statement by putting:x. = .x is a son of Mr A,x. = .x is a daughter of Mr A.

Then there is an object having the property, namely Mr A junior;whateverx may be, if it has some property that Mr A junior does nothave, it is not Mr A junior, and therefore not a son of Mr A senior.This is what we mean by saying that being a son of Mr A is a unit

property. Similarly being a daughter of Mr A is a unit property.Now consider the property being a son or daughter of Mr A, whichwe will call. There are objects, the son and daughter, of which (1)the son has the property of being male, which the daughter has not;(2) the son has the property and the daughter has the property;(3) ifx is an object which lacks some property possessed by the sonand also some property possessed by the daughter, then x is not a sonor a daughter of Mr A. It follows that is a dual property. In shorta man who has one son and one daughter has two children.

As you can see, once we try to be specific about exactly what we mean,en simple facts that we assume to be self-evident become mired in technical

etail. For the most part people have a sufficiently solid intuitive grasp of

ncepts like number and addition that they can see that 1 + 1 = 2

ithout having to worry about the technical pedantry. The more abstractions

e pile atop one another, however, the less intuition people have about the

ncepts involved and it becomes increasingly important to spell things out

xplicitly. In fact much of modern mathematics has reached the point whereis sufficiently far divorced from everyday experience that common intuition

counterproductive, and leads to false conclusions. If you think that sounds

ly then consider that modern physics has also passed this threshold few

eople can claim quantum mechanics to be intuitive. Our experience of the


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irst Steps 6

mes, to avoid a sheer climb, the road is forced to take a tortuous, winding

ath. As you progress deeper into the mountains, however, you will find

aces where the road opens out to present you with a glorious vista looking

ut over where you have come from. Each new view allows an ever broader

ew of the landscape, allowing you to see further and more clearly, whileso seeing all the other different roads that all lead to this same peak. It is

ese unexpected moments, upon rounding a corner, of beauty, and clarity

nd insight, that, to me, make the study of mathematics worthwhile. I hope

at, in this journey into the interior of mathematics, I can impart to you

me of those moments of beauty and wonder.


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The Slow Road

.2 The Slow Road

natsukusa ya

tsuwamonodomo ga

yume no atoThe summer grasses:

The high bravery of men-at-arms,

The vestiges of dream.3

Matsuo Basho, on visiting Hiraizumi, once home to the great Fujiwara

clan whose splendid castles had been reduced to overgrown grass mounds.

A good haiku not only arrests our attention, it also demands reflection and

ntemplation of deeper themes. In Bashos Oku no Hosomichi, The Narrow

oad to the Interior, the haiku often serve as a point of pause amidst theavelogue, asking the reader to slow down and take in all that is being said.

he slow road to understanding is often the easiest way to get there. At

e same time the travelogue itself provides context for the haiku. Without

at context, both from the travelogue, and from our own experiences of the

orld upon which the haiku asks us to reflect, the poem becomes shallow:

ou can appreciate the sounds and the structure, but the deeper meaning

e real essence of the haiku is lost.Mathematics bears surprising similarities. A well crafted theorem or proof

emands reflection and contemplation of its deep and wide ranging implica-

ons. As with the haiku, however, this depth is something that can only

e provided by context. A traditional approach to advanced mathematics,

nd indeed the approach you will find in most textbooks, is the axiomatic

pproach: you lay down the rules you wish to play by, assuming the bare

inimum of required knowledge, and rapidly build a path straight up theountainside. This is certainly an efficient way to get to great heights, but

e view from the top is often not rewarding unless you have spent time

andering through the landscape you now look out upon. Simply put, you

k th t t t t l i t th l t d d i i ht th t th


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irst Steps 8

the previous entry, On Abstraction, I discussed the process of abstraction,nd how mathematics builds up layer upon layer of abstraction. The roade must take, the slow road, is the path that winds its way through theseyers. Each layer is, in a sense, a small plateau amidst the mountains; to be

plored before the next rise begins.The place to start, therefore, is with the area of mathematics that most

eople already have a fairly strong intuitive sense for: numbers. Many peo-e tend to assume that mathematics is all about numbers, something thatmply isnt the case. Numbers are just one of the more extreme abstractionsom the external world, amongst many different abstractions that make upodern mathematics. Even in antiquity mathematics was divided into arith-etic, which abstracted quantity, and geometry, which abstracted shape andrm. Numbers are, however, something that almost everyone has (or thinksey have) a solid intuitive grasp of and studying the nature of that ab-raction, and how it is made, will provide some context for other similar

bstractions, as well as providing a solid base from which to build furtheryers of abstraction.

The concept of number is both a greedy abstraction, and a remarkablene. It is greedy in that it tries to abstract away as much detail as possible.iven a collection of objects (for now well take collection as intuitive,ost peoples everyday experience is sufficient for elementary numbers and

oesnt run afoul of the pathological cases that require strict definitions tovoid) we forget absolutely everything about the collection, and about thebjects themselves, except for a single particular property. The abstraction is

markable because, by being so very greedy, it is applicable to everything ere is simply nothing in our experience of the world that doesnt fall undere umbrella of this particular generalisation: everything can be quantifiedsome sense, albeit trivially (as one) in many cases.This is the power of mathematics: by seeking greedy abstractions, by

neralising as much as possible, it finds properties or concepts that have nearniversal applicability. Mathematics allows you to speak about everything at

nce. The catch, of course, is that by abstracting too far you leave yourselfnable to say anything useful (you can say nothing about everything). Theick is to forget as much as possible about particular instances, while stillaving some property that can be worked with in a constructive mannerward some purpose or other Whether youve forgotten too much depends


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The Slow Road

e very abstract terms of addition of numbers we gain two things: first, bymoving the messy particularity of the world via abstraction we make theocess simple to deal with; second, by using the greedy abstraction of num-

ers we produce universally applicable results; i.e. 2 + 3 = 5 is a statement

bout any collection of 2 things and any collection of 3 things. Perform-g an addition is generalising across incredibly broad classes of real worldtuations. We are saying an enormous amount incredibly simply.

The beauty and complexity starts to unfold when, due to the greedinessthe abstraction, we find that numbers can reflect back on themselves.

or example, additions can form collections, and as noted, collections haveuantitative properties. Thus we can talk about a particular number of addi-ons, for example we might have 5 additions of 3, and arrive at multiplication

That is, 3 + 3 + 3 + 3 + 3 = 5 3). We can talk about a particular numbermultiplications and arrive at exponentiation, and so on. By folding the

bstraction back on itself we can build layers of structure structure thatay be far more complicated than we might first imagine. In introducingultiplication we raise the question of its inverse, division. That is, if we can

nd the quantity that results from some number of additions, we can ask too the other way and decompose a quantity into some number of additions.doing this, however, we introduce prime numbers (those which cannot be

ecomposed into any integer number of additions of integers) and fractions.Whole new expanses of complexity and structure open up before you there

apparently, a whole world to explore.In the next section well look into what kind of abstractions we can make

om the world of numbers, and dip our toes into the beginnings of algebra.the meantime, however, Ill leave you with a question about numbers toonder:

Can every even integer greater than 2 be written as the sum

of 2 primes?

his is commonly known as Goldbachs conjecture, and it remains an open

oblem to this day; no one knows the answer. It is worth taking a momentthink about the problem yourself, and wonder why it may, or may not, be

ue, and what it really means, and also how little we really know about therange world of numbers.


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.3 A Fraction of Algebra

s a mathematician there is a story I hear a lot. It tends to come up whenevertell someone what I do for the first time, and they admit that they dont

ally like, or arent very good at, mathematics. In almost every case, if Iother to ask (and these days I usually do), I find that the person, once upontime, was good at and liked mathematics, but somewhere along the wayey had a bad teacher, or struck a subject they couldnt grasp at first, andll a bit behind. From that point on their experiences of mathematics is ale of woe: because mathematics piles layer upon layer, if you fall behinden you find yourself in a never ending game of catch-up, chasing a horizon

at you never seem to reach; that can be very dispiriting and depressing. Ine previous entries we have dealt with subjects (abstraction in general, ande abstraction of numbers) that most people have a natural intuitive grasp, even if the details, once exposed, prove to be more complex than most

eople give them credit for. It is time to start looking at subjects that oftenove to be early stumbling blocks for some people: fractions and algebra.

There is a reason that these subjects give people pause when they firstncounter them, and that is, quite simply, that they are difficult. They arefficult in that they represent another order of abstraction. Both fractions

nd elementary algebra must be built from, or abstracted from, the basicncept of numbers. Because of the sheer prevalence of numbers and countingour lives from practically the moment we are born, people quickly develop

feel for this first, albeit dramatic, abstraction. It is when people encounter

e next step, the next layer of abstraction, in the form of fractions and/orgebra, that they have to actively stretch their minds to embrace a significantbstraction for the first time. Most of us, having won this battle long ago,ruggle to see the problem in hindsight we might recall that we had troubleith the subject when we were younger, but would have a hard time sayinghy. We have developed the same sort of intuitive feel for fractions andgebra as we have for numbers and have forgotten that this is hard won

nowledge.I want to begin with fractions because, ultimately, it is by far the easier

the two being only a semi-abstraction and will provide an example ofe process as background for stepping up to elementary algebra.

As was noted in the last entry the complexity of mathematics begins to


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1 A Fraction of Algebra

e same way that subtraction provides an inverse to addition. That is, whileddition asks if I add a collection of size 3 to a collection of size 2, what size

the resulting collection?, subtraction asks the inverse question if I got asulting collection of size 5 by adding some collection to a collection of size

how much must I have added?; parallelling that we have multiplicationking if I add together 5 collections of size 3, what size is the resultingllection, and division reversing the question: if I have a resulting collec-

on of size 15, how many collections of size 3 must I have added together?.verything seems fine so far, but there is some subtlety here that complicatese issue.

If we are still thinking in terms of collections then dividing a collection2 objects into 4 parts doesnt make sense. If we are viewing numbers and

perations on them as entities in their own right then we can at least forme construction 2/4, and ask if it might have a practical use. It turns outat it does, since it allows a change of units. What do I mean by this?

We can say a given collection has the property of having 2 objects in it,ut to do so is make a decision about what constitutes a discrete object.

eciding what counts as an object, however, is not always clear there areten several possible ways to do it, depending on what you wish to considerwhole object (that is, the base unit which you use to count objects in thellection). A simple example: in the World Cup soccer finals, do you counte number of teams, or the number of individual players? Both make sense

epending on the kind of result you want to obtain, so considering a team, orch individual player, as a discrete object is a choice. The problem is even

ore common when dealing with measurement: a distance is measured as artain number of basic lengths, but what you use as your basic length (thenit of measure) is quite arbitrary. We tend to measure highway distances

miles or kilometres and peoples heights in feet or metres, but we couldst as easily switch to different units and measure highway distances in feetmetres and still be talking about the same distance.Most importantly we can change our minds, or re-interpret, what con-

itutes a distinct object after the fact. Using this re-interpretation of whatnstitutes and discrete object we can make sense of 2/4. If we re-interpret astinct object such that what we had previously considered a single objectnow considered two objects then we will have 4 objects in the collection,

nd we need 4 of these new objects to arrive at a collection that would be


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me new object as 2 old objects re-interpreted as 4.Indeed, we can go on like this, with 3/6, 4/8, 5/10, and so on, all ex-

essing the same relationship of new object to old - all different ways torive at the same size of new object. And so we have a catch - what are

n inspection quite different expressions will, in practice, behave the same.e-interpreting 1 object as 2, or 2 objects as 4 results in the same new

bjects, so counting, and hence addition, subtraction, and multiplication ofese new objects will give the same result, whichever re-interpretation wee. Perhaps it doesnt seem like much of revelation that 2/4 is the same as2, but that is simply because we have learned, through practice, to auto-atically associate them. The reality is that 2/4 and 1/2 are quite distinct,

nd it is only because they behave identically with regard to arithmetic thate regard them as the same. In identifying them as the same we are ab-racting over such expressions, forgetting the particularities of what size ofitial collection we were dividing, caring only about the common behaviourith regard to arithmetic. Making sense of fractions involves abstracting

ver numbers - they are another level of abstraction, and this, I suspect, is

hy people find them difficult when they first encounter them.There is an important idea in this particular abstraction that is worthaying attention to - it leads the way to algebra. We have an infinite num-er of different objects: 1/2, 2/4, 3/6, /4/8, . . . but because they all behaveentically with respect to a given set of rules (in this case basic arithmetic)e pick a single symbol to denote the entire class of possible objects. Alge-a can be thought of as extending that idea to its logical conclusion. The

sight we need to make the step to algebra is that there is a subset of theles of arithmetic for which all numbers behave identically. For exampleversing the order of addition makes no difference to the result, no matterhat numbers you are adding: 1 + 2 = 2 + 1, and 371 + 27 = 27 + 371.you can identify which rules have the property that the specific numbers

ont matter, then you can pick a single symbol to denote the entire class ofumbers for any manipulations within that set of rules. This is algebra.

This is important because it is a layer of abstraction over and above thebstraction of numbers. With numbers we considered many different collec-ons and abstracted away everything about them except a certain propertythe number of objects they contain. This proved to be useful because

ith regard to a certain set of rules the rules of arithmetic that was the


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ature of the collections beyond the number of objects. Now, with algebra,e can perform calculations and have the result be true regardless of thearticular numbers involved. This is an exceptionally powerful abstraction:essentially does for numbers what numbers do for collections. This is why

e rules of algebra, that subset of arithmetic rules under which all numbersehave identically, are so important.

In particular we can say that, no matter what numbers x, y and z are,e following are always true:

1. x + y = y + x and x y = y x. These are referred to as commutativeproperties.

2. x + (y + z) = (x + y) + z and x (y z) = (x y) z. These arereferred to as associative properties.

3. x(y+z) = xy+xz. This is referred to as a distributive property.

4. x+0 = x and x1 = x. This property of 0 and 1 is referred to as being

anidentity element

for addition and multiplication (respectively).5. There is a number, denoted x such that x + x = 0. This refers to

the existence of inverses for addition.

We also have one odd one out the existence of inverses for multiplication.he catch here is that it does matter what number x is; inverses exist formost every number, but if x = 0 there is no multiplicative inverse of x.

hus we have:

6. Ifx is any number other than zero then there is a number, denoted 1x

,such that 1

x x = 1.

you have any curiousity you will be wondering why this special case oc-urred, breaking the pattern. Remember that we are talking about abstract

operties common to all numbers, so the fact that this is a special case saysmething quite deep about both multiplication, fractions, and the numberro. Indeed, because we are two layers of abstraction up, referring to all

umbers, which in turn each refer to all collections with a given property,e fact that this is a special case has significance with regard to almost


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tuitive (that is, most people have a firm enough grasp on numbers) thatwont get into details here; just be forewarned that numbers as order andumbers as size are actually distinct concepts that, at some point, we willave to carefully tease apart.

7. Either x < y, y < x, or x = y.

8. Ifx < y and y < z then x < z.

9. Ifx < y then x + z < y + z.

10. Ifx < y and 0 < z then x z < y z.

ote that, again, 0 and multiplication have a significant interaction andovide another special case.

Note that I gave names to properties 1 through 5 because these propertiesill keep cropping up again and again later; some will prove to be important,hers less so. Which ones are important and which are not may be somewhata surprise, but Ill leave that surprise till later.At this point it is worth taking stock of how far weve come. Not only

ave we built up two layers of abstraction, each of which can be used toeat practical effect (just witness how much of modern technology and engi-

eering is built upon arithmetic and elementary algebra!), in doing so weveegun to uncover an even deeper principle the principle that will form theundation for much of the modern mathematics that is to follow. What do I

ean? There is a common thread to how these successive abstractions haveeen built: we discerned a set of rules for which an entire class of objectsotentially even completely abstract objects) behave identically, and this al-wed us to abstract over the entire class. The broader the class the broadere results we can draw; the higher the abstraction (in terms of successive

yers) the deeper the results we can draw. The approach now will be toek out rules, and classes that they allow us to abstract over; the broader

nd more layered, the better. In so doing we will part ways with numbersntirely. Fractions, ordering, and the difficulties of 0, will lead us towards and of generalised geometry, while consideration of properties 1 through 6ill lead us to a language of symmetry.

We have come to the first truly significant incline on our road Behind us


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mportance of this cannot be overstated! We are abstracting over the process

abstraction itself! This is the path to high places from which, when we

nally arrive, we can look out, over all the plains we now leave behind, with

esh eyes, and deeper understanding.


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A Fork in the Road 2

Alice came to a fork in the road. Which road do I take? she asked.Where do you want to go? responded the Cheshire cat.I dont know, Alice answered.Then, said the cat, it doesnt matter.

Lewis Carroll, Alices Adventures in Wonderland[2]

In the later years of his life, after his journey to the interior, Basho lived insmall abandoned thatched hut near lake Biwa that he described as being ate crossroads of unreality1. Now, still early in our journey, we have comeour own crossroads of unreality. We are caught between dichotomies of

nreal, abstract, objects. One road leads to consideration of finite collections,nd properties of composition (the algebraic properties 1 through 5 from the

evious section); the other road leads to the continuum and questions of

dering and inter-relationship (properties 7 through 10 from the previousction). The first road will lead to a new fundamental abstraction from

nite collections, different from, and yet as important as, the abstraction thate call numbers; this way lies group theory and the language of symmetryat has come to underlie so much of modern mathematics and physics.

he second road will lead to deep questions about the nature of reality,nd, brushing past calculus along the way, lead to a new and minimalist

terpretation of a continuous space through the concept of topology.Which road do we take? As the cat said to Alice, It doesnt matter. We

e at the crossroads of unreality, and the usual rules need not apply. Whichad do we take? Both.

1From the translation of Genjan no fu by Donald Keene, in Anthology of Japaneseterature[3]

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Fork in the Road 18

.1 The Paradoxes of the Continuum, Part I

finity is a slippery concept. Most people tend to find their metaphoricalaze just slides off it, leaving it as something that can only ever be glimpsed,

urry and unfocused, out of the corner of their eye. The problem is that,r the most part, infinity is defined negatively; that is, rather than sayinghat infinity is, we say what it is not. This, in turn, is due to the nature ofe abstraction that leads to the concept on infinity in the first place.

The ideas of succession and repetition are fairly fundamental, and arepparent in nature in myriad ways. For example, the cycle of day and nightpeats, leading to a succession of different days. Every such series of suc-

ssive events is, in our experience, bounded it only extends so far; up toe present moment. Of course such a series of events can extend back to

ur earliest memories. Via the collective memory of a society, passed downrough written or oral records, it can even extend back to well before we

ere born. Thus, looking back into the past, we come to be aware of seriessuccessive events of vastly varying, though always bounded, length. We

n then, at least by suitable juxtaposition of a negation, form the concepta sequence of succession that does not have a bound. And thus arises thencept of infinity. Is the concept coherent? Does succession without boundake any sense? With this conception of infinity it is hard to say, for we have

nly really said it is a thing without a bound. We have said what propertyfinity does not have, but we have said little about what properties it does

ave.

Indeed, despite the basic concept of infinity extending back at least as farancient Greece, whether infinity is a coherent concept has been a pointbitter debate, with no significant progress made until as recently as the

nd of the 19th century. Even now, despite having a fairly well groundedefinition and theory for transfinite numbers, there is room for contentionnd differing conceptions of infinity, and in particular of the continuum.uch modern debate divides over subtle issues which we will come to in due

urse. First, however, it will be educational to look at some of the moreraightforward reasons that people have difficulty contemplating infinity:e apparent paradoxes and contradictions that arise.

Some of the earliest apparent paradoxes that involve the infinite are fromncient Greece Among the more well known are the paradoxes proposed


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9 The Paradoxes of the Continuum, Part I

at motion, and change, are actually just illusions. The paradoxes have,owever, come to be associated with the paradoxical nature of the infinite.

The first of Zenos paradoxes, the Dichotomy, essentially runs as follows:efore a moving body can reach a given point it must traverse half the

stance to that point, and before it can reach that halfway point it mustaverse half ofthat distance (or one quarter of the distance to the end point),

nd so on. Such division of distance can occur indefinitely, however, so tot from a starting point to anywhere else the body must traverse an infinite

umber of smaller distances and surely an infinite number of tasks cannote completed in a finite period of time?

The second paradox, the most well known of the three, is about a race

etween Achilles and a tortoise, in which the tortoise is granted a head start.eno points out that, by the time Achilles reaches the point where the tortoisearted, the tortoise will have moved ahead a small distance. By the timechilles catches up to that point, the tortoise will again have moved ahead.his process, with the tortoise moving ahead smaller and smaller distances,n obviously occur an infinite number of times. Again we are faced withe difficulty of completing an infinite number of tasks. Thus Achilles will

ever overtake the tortoise!The third paradox, the Arrow, raises more subtle questions regarding the

ntinuum, so I will delay discussion of it until later. Taken together thearadoxes were supposed to show that motion is paradoxical and impossible.ew people are actually convinced, however: everyday experience contradictse results that the paradoxes claim. The common reaction is more along the

nes of Okay, sure. Whats the trick?. The trick is actually relativelybtle, and while rough and ready explanations can be given by talking aboutonvergent series, it is worth actually parsing out the fine details here (aseve seen in the past, the devil is often in the details), as it will go a longay toward informing our ideas about infinity and continuity.

Let us tackle the Dichotomy first. To ease the arithmetic, let us assumeat the moving body in question is traversing an interval of unit length

which we can always do, since we are at liberty to choose what distance wensider to be our base unit), and that it is travelling at a constant speed.

We can show that, contrary to Zenos claim, the object can traverse thisstance in some unit length of time (again, a matter of simply choosing

n appropriate base unit) despite having to traverse an infinite number of


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Fork in the Road 20

deed, be completed in finite time. This tends to be the point where mostxplanations stop, possibly with a little hand-waving and vague geometric

gument about progressively cutting up a unit length. It is at this point,owever, that our discussion really begins. You can make intuitive arguments

to why the sum turns out to be 1, but, given that we werent even thatear about what 1 + 1 = 2 means, a little more caution may be in order articularly given that infinity is something completely outside our practical

perience, so our intuitions about it are hardly trustworthy.

Since we cant trust our intuitions about infinite sums yet, it seems sen-ble that we should look at finite sums instead. Certainly we can calculate

e sum 1/2 + 1/4 = 3/4,and1/2 + 1/4 + 1/8 = 7/8, and so on. Each of thesems will, in turn, give a slightly better approximation of the infinite sum weish to calculate; the more terms we add, the better the approximation. The

bvious thing to do, then, is to consider this sequence of ever more accuratepproximations and see if we can say anything sensible about it. To save my-lf some writing I will use S

nto denote the sum 1/2 + 1/4 + 1/8 + ... + 1/2n

hus S2 = 1/2 + 1/4 and S4 = 1/2 + 1/4 + 1/8 + 1/16, and so on), and talk

bout the sequence of partial sums S1, S2, S3, . . .It may not seem that weve made much improvement, having shifted from

mming up an infinite number of terms to considering an infinite sequence ofms, but surprisingly infinite sequences are easier to deal with than infinitems and we at least only have finite sums to deal with now. The trick

om here is to deal with the nth term of the sequence for values ofn thate finite, but arbitrarily large. That means we get to work with finite sumsince for any finite n, S

nis a finite sum) which we can understand, but at

e same time have no bound on how large n can be, which brings us intontact with the infinite. In a sense we are building a bridge from the finitethe infinite: any given case is finite, but which term the case deals with is

ithout bound. Before we can get to the arbitrarily large, however, we mustst deal with the arbitrarily small.

In some ways it was the arbitrarily small that lead to this problem thearadox is founded on the presumption that the process of dividing in half can

on indefinitely, resulting in arbitrarily small distances to be traversed. It isecisely this property of infinite divisibility that is a necessary feature of theea of a continuum: something without breaks or jumps The opposite of the


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1 The Paradoxes of the Continuum, Part I

ay talk about the arbitrarily small (a result of arbitrarily many divisionsnote the relationship between the infinite and the continuous). What wee really after is a concept of convergence; the idea that as we move furtherong the sequence we get closer and closer, and eventually converge to,

me particular value. That is, we want to be able to say that, by lookingr enough along the sequence we can end up an arbitrarily small distance

way from some particular value that the sequence is converging to. This,turn, leads us to the next concept: distance.

We need to be careful here because while the original problem was aboutmoving object covering a certain distance in the real world, we have ab-racted away these details so as to have a problem solely about sequences

numbers. That means we are no longer dealing with practical physicalstance, but an abstract concept ofdistance between numbers. So what doesmean for one number to be close to another? We need a concrete def-ition rather than vague intuition if we are to proceed. Since numbers are

urely abstract objects we could, in theory, have close mean whatever weoose. There is a catch, however: when talking about numbers we gener-

ly assume that they are ordered in a particular way. For example, whenriving at rules for algebra we included rules for ordering numbers. Thismplicit ordering defines closeness in the sense that we would like to think

at x < y < z means that y is closer to z than x is. Looking back ate rules regarding ordering we find that this means that the closer z yto 0, the closer y is to z. Thats really just saying that the smaller the

fference between y and z, the smaller the distance between them, and so

e definition of distance we need is the difference between y and z! Thenal catch is that we would like to be able to consider the distance from z toto be the same as the distance from y to z, but z y = (< y z). Thelution is simply to say that the direction of measurement, and hence the

gn of the result, is irrelevant and take the absolute value to get:

The distance between y and z is |y z|.

As a momentary aside, it is worth noting that we have defined a distanceetween numbers to be another number, but that the number that definese distance is, in some sense, not the same type of number. The number

efining the distance is a higher level of abstraction since it is a number


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Fork in the Road 22

It is time to put the power of algebra the ability to work with a numberithout having to specify exactly which number it is to use. Let epsilone some non-zero positive number, without specifying exactly what numberm using because it is the traditional choice among mathematicians to

enote a number that we would like to presume is very small that is, veryose to zero). Then I can choose N to be a number large enough that 2N isgger than 1/, and hence 1/2N is less than . Exactly how big N will havebe will depend on how small is, but since there is no bound on how bigcan be, we can always find a big enough N no matter how small turns

ut to be. Now, if we note that, for any n,

Sn = 2n

12n

which you can verify for yourself fairly easily) then, if we assume that n isgger than N, we find that the distance between 1 and Sn is:

|1 Sn| =

2n

2n

2n 1

2n

=

1

2n

0 there exists an integer N 1ch that, for all m, n N,

|Sn Sm| <

ecalling that |x y| gives the distance between numbers x and y). Suchsequence is called a Cauchy sequence. Now, since any Cauchy sequencenverges to something, we can identify (consider equivalent) the sequence

nd the point at its limit. Furthermore, since we know that using fractionse can get arbitrarily close to any point on the continuum, there must beme sequence of fractions that converges to that point, and so if we considerl the possible infinite Cauchy sequences of fractions, we can cover all the

oints on the continuum we are assured that no holes or gaps can slip inis time. Weve caught all the holes without even having to find them!

It is worth looking at an example: can we find a sequence of fractionsnverges to

2? Consider the decimal expansion of

2 which starts out

41421... and continues on without any discernible pattern; clearly the se-uence 1, 1.4, 1.41, 1.414, 1.4142, 1.41421, . . . (where the nth term agrees with2 for the first n1 decimal places) converges to 2. More importantly eachrm can be rewritten as a fraction since each term has only finitely many

on-zero decimal places; for example 1.4 = 14/10 and 1.4142 = 14142/10000c. Finally it is not hard to see that this sequence is a Cauchy sequence. Wen do the same trick for any other decimal expansion, arriving at a Cauchy

quence that converges to the point in question. Of course there are manyher Cauchy sequences of fractions that will converge to these values: we are

ealing with something similar to our dilemma with fractions when we foundat there were an infinite number ofdifferent pairs of natural numbers thatl described the same fraction In that case we simply selected a particular


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Fork in the Road 38

e sequence constructed via the decimal expansion will do nicely in somense you can think of the Cauchy sequence as an infinite decimal expansion.

Now that we at least have some idea of what these sequences might lookke, it is time to take a step back and consider what is actually going on

ere. Back in The Slow Road we constructed natural numbers as a propertycollections of objects. Then, in A Fraction of Algebra, we created fractionsallow us to re-interpret an object within a collection. This was another

yer of abstraction fractions were not really numbers in the same way thatatural numbers were fractions were a way of re-interpreting collections,nd we could describe those re-interpretations by pairs of natural numbers.erhaps rather providentially it turned out that the rules of algebra, theles of arithmetic that were true no matter what natural numbers we chose,so happened to be true no matter what fractions we chose. It is thisroke of good fortune, combined with the fact that certain fractions canke the role of the natural numbers, that allows us to treat what are really

uite different things in principle (fractions and natural numbers) as theme thing in practice: for practical purposes we usually simply consider

atural numbers and fractions as numbers and dont notice that, at heart,ey are fundamentally different concepts. Now we are about to add a newyer of abstraction, built atop fractions, to allow us to describe points incontinuum. While all that was required to describe the re-interpretationobjects that constituted a fraction was a pair of numbers, points in a

ntinuum can only7 be described by an infinite Cauchy sequence of fractions.hus, in the same way that natural numbers and fractions are actually very

fferent object, so fractions and points in a continuum are quite different.gain, however, we find that when we define arithmetic on sequences (whichcurs in the obvious natural way) they all behave appropriately under ourgebraic rules. en we consider that it is easy enough to find sequences that

ehave as fractions (any constant sequence for instance) it is clear that, again,r practical purposes, we can call these things numbers and assume werelking about the same thing regardless of whether we are actually dealing

ith natural numbers, fractions, or points in a continuum.It should be pointed out that sometimes these distinctions are actually

mportant. A simple example is computer programming, which does botherdistinguish floating point numbers (ultimately fractions) from integers.

ou can usually convert or cast from one to the other via a function (and


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9 Paradoxes of the Continuum, Part II

times that function can be implicit), but the distinction is important.ater we will start getting into mathematics where the distinction becomes

mportant.So, now that we have this construction, several layers of abstraction deep,

at allows us to describe the continuum, does it resolve the problem theythagorean Brotherhood struggled with? Certainly within the continuumere is a point corresponding to

2, but even with our construction it is the

mit of an infinite sequence we still require a completed infinity. Of coursecepting the idea of a completed infinite would get us out of this conundrum;

hat we require is a coherent theory of the completed infinite were we toave that, then we neednt fear the idea as the Pythagorean Brotherhoodd. The next time we venture along this particular road we will discuss justch a theory, and explore the remarkable transfinite landscape that it leads. We would be remiss to conclude here, however, without noting that theresome dissent on this topic. While the theory of the continuum based onmpleted infinites we will cover is remarkably widely accepted and used,ere are still those who do not wish to have to deal with the completed

finite. So what is the alternative?The idea is to construct a continuum using infinitesimals: a number suchat we have 2 = 0, yet = 0. Using such a value we can create a continuum

ithout holes as desired. If adding a seemingly arbitrary new element to theumber system seems like cheating, remember that both fractions, and thefinite decimals via Cauchy sequences, are just as much artificial additionsthe number sequence they just happen to be ones were familiar with

nd now take for granted. The real dilemma is that, assuming the requiredoperties of infinitesimals, we can deduce contradictions. As we noted ate start of this post, when a mathematical argument leads you somewhere

ou dont wish to go you are left having to challenge the very foundationslogic itself. Surprisingly, that turns out to be the resolution: smooth

finitesimal analysis rejects the law of the excluded middle. The logic usedr this alternative conception of the continuum rejects the idea that, given a

oposition, either the proposition is true, or its negation is true. That meansat saying that x = y is not true, does not mean that x = y. This sounds

ke nonsense at first, because we generally take the law of excluded middle foranted, and it is ingrained in our thinking. We have to remember, however,at this is a theory dealing in potential but not completed infinites and it


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Fork in the Road 40

llion decimal places, and they might still agree; that does not mean theye equal, since it might be the billion and first decimal place at which theysagree; and yet we still cant conclude they are unequal theyve agreed sor and could continue to do so. We even check the first 1020 decimal places,

nd still we cant conclude either way whether x and y are equal or unequal.ecause we can never complete the infinity and check all the decimal places,nless we have more information (such as that both number are integers8), it

not possible conclude either way we have an in between state where theumbers are neither equal, nor unequal, and it is this in-between possibilityat causes the law of excluded middle to fall apart. To say that x = y is

ot true simply means we have not yet concluded that x = y, but that doesot require that we must have concluded x = y since we may still be torn inetween, unable to reach a conclusion. As strange as this sounds at first, ittually provides a surprisingly natural and intuitive model of the continuum

and a remarkably different one from the classical one we will be developing.nough sidetracks, however; it is time to return to the path.

We have rounded the bend, and can make out the rough expanse of the

ndscape below, but the land itself remains unexplored, and potentiallyuite alien. The next time we return to this road well try and understande implications of a continuum of completed infinites, including a varietyinitially unsettling results. In the meantime, however, we will return the

udy of patterns and symmetry, and try and build a robust theory from ourmple examples.


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1 Permutations and Applications

.4 Permutations and Applications

umbers are remarkably tricky. We tend not to notice because we live in aorld that is immersed in a sea of numbers. We see and deal with numbers

l the time, to the point where most basic manipulations seem simple andbvious. It was not always this way of course. In times past anything mucheyond counting on fingers was the domain of the educated few. If I ask youhat half of 60 is, youll tell me 30 straight away; if I ask you to stop and think

bout how you know that to be true youll have to think a little more, andart to realise that there is a significant amount of learning there; learningat you now take for granted. Almost everyone uses numbers regularly every

ay in our current society, be it through money, weights and measures, timesday, or in the course of their work. Through this constant exposure and

e weve come to instinctively manipulate numbers without having to evenink about it anymore (in much the same way that you no longer have tound out words letter by letter to read). That means that when we meet a

ew abstraction, like the symmetries discussed in Shifting Patterns, it seemsmparatively complex and unnatural. In reality the algebra of symmetriesin many ways just as natural as the algebra of numbers, we just lack

xperience. Thus, the only way forward is to look at more examples, and seeow they might apply to the world around us.

In terms of examples I would like to take a step into the more abstract ther than dealing with a physical example and determining an abstractionom it, well start with a slightly abstract example and explore from there.

he example I have in mind is that of permutations. By a permutation Imply mean a rearrangement of unspecified objects, mapping one positionanother. We can view a permutation as a kind of wiring diagram, suchthe one depicted in figure 2.11. meaning that we shift whatever is in po-

tion 1 to position 3, whatever is in position 2 to position 1, and whateverin position 3 to position 2. Hopefully you can see how such rearrange-

ents are essentially what we were doing in Shifting Patterns, but here we

ent starting out with a specific pattern in mind, but considering all sucharrangements in general.

As before we can combine two rearrangements together to get another. this case we simply connect one wiring diagram to the next and followe paths from the top right the way to the bottom We can then simplify


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Fork in the Road 42

Figure 2.11: A permutation of three objects

Figure 2.12: Combining two permutations to obtain a new permutation

ull permutation where we do nothing (the first item is connected to the firstem, and the second item is connected to the second item), and a simple

wap where we reverse the items (connect the first item to the second, ande second item to the first). Using the algebraic terms we established pre-

ously we end up a two element algebra: let s be the permutation where wewap, and then we have the rule:

ss =


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On the other hand, as soon as we consider permutations of three objectse find things get more complicated. There are a total of 6 (thats 3 1) permutations of three objects. If we select two basic permutations

ppropriately we can generate them all as various combinations of the two.

here are, in fact, several different pairs we could select (though the resultinggebra will turn out to be the same, no matter how we do it the namesight change, but the underlying rules will be the same), and Ive opted fore two depicted in figure ??, where a swaps the first two elements and leaves

Figure 2.13: Two permutations, labelled a and b

e third alone, and b swaps the second two elements, and leaves the firstone. Now as with the permutations of two elements, if we swap a pair, and

en swap them again, we end up back where we started, so we can see thate have the following two rules:

aa =

bb =

ow, however, we have the possibility of combining together a and b. We

ready saw thatab

results in the first item going to the third place, thecond item to the first position, and the third item to the second positions shown in figure 2.12); but if we swap things around to find ba we get thether different situation shown in figure 2.14, which reverses the situation,ith the third item moving the first place, while the first and second items


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Fork in the Road 44

Figure 2.14: The result of combining b and then a

Figure 2.15: Showing how aba is the same as bab

We can see that this gives us all the permutations by counting up the

mbinations of a and b that havent been ruled out as being reducible to

mething simpler. We have

1. The null permutation:

2. a

3. b


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nd anything with four or more as and bs will be reducible. Why is that?nce aa = and bb = any sequence will have to alternate as and bs,herwise we can just cancel down consecutive pairs. On the other hand, ife have a sequence of more than three alternating as and bs then well have

sequence aba or bab that we can convert using the fact that aba = bab, andnd up with a pair of consecutive as or bs that we can then cancel down. Forxample, if we tried to have a sequence of four as and bs like abab, then we

n say

abab = (aba)b

= ba(bb)

= ba

= ba

With a little thought you can see that this sort of procedure can reduce anyquence of four or more as and bs down to one of three or less. So for

ermutations of three objects we get an algebra that is described by three

les:aa =

bb =

aba = bab

If we were to consider permutations of four items we would have 24 per-

utations (thats 4

3

2

1) to deal with, and things would be moremplicated yet again. Permutations of five items provide a total of 1204321) permutations, and an even more complicated algebra witht more subtle and interesting properties.

There are two things that you should take notice of here. The first is thaten simple changes to a pattern as simple as changing the number of itemsvolved can give rise to very different dynamics. The character of the al-

bra that arises from permutations of two objects is very different from thatthe three object permutation algebra, and four objects is different again.

o reiterate the point: different patterns can have surprisingly different andmarkable dynamics. The second thing that you should be noticing is thathile we can work with permutations as wiring diagrams and connect them


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Fork in the Road 46

e algebra is that we can reduce the whole problem of patterns to the simplesk of manipulating algebraic expressions according to particular rules. By

bstracting up to the algebra, weve made the problem much easier to thinkbout and manipulate.

Hopefully by this stage youre developing a feel for how this abstractionocess works. With numbers we start with a collection and abstract awayl the details save a single property: the quantity. Here we have somethinglittle more complex; we start with a pattern and abstract away as much ofe detail as possible, while still retaining some information about the naturethe pattern. That information can be efficiently encoded into a sort of

gebra, in the same way that we encode information about quantity intombols (numbers). The exact nature and rules of the algebra we generatethe information about the pattern that we have kept. Now, numbers allow to reason about quantity in general via arithmetic, which we can reducea game of manipulating symbols. Our abstraction of pattern allows us to

ason about patterns via their associated algebra, which we can also reducea game of manipulating symbols. We have turned thinking about patterns

to a kind of arithmetic; and doing so allows us to be systematic in studyingnd analysing such patterns.This, of course, raises the question of why we should be interested in

udying and analysing patterns at all. The same question can be asked aswhy we should be interested in studying and analysing quantity. The

fference is that our culture is steeped in analysis of and use of quantity; weke its usefulness for granted. So lets step back, and ask why using numbers

useful. As was pointed out in The Slow Road, numbers and quantity areeful because they are everywhere we can apply quantitative analysis tomost everything (and often do, sometimes even where it isnt appropriate).is worth pointing out that patterns and symmetry are every bit as prevalentthe world. All around us things can be described in terms of their patterns.ck any collection of objects you care to set your eyes upon, and they willrm some manner of pattern; perhaps they will only have a trivial symmetry,

perhaps they will have more complex symmetries. The point is that,st like numbers, symmetries are all around us. The study of pattern andmmetry in the manner weve been describing is very new however, and thiseans it hasnt entered the mainstream consciousness, nor the language, ine same way that numbers have We dont describe the world around us in


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those already using advanced mathematical methods. Right now thatnds to mean fields such as physics and chemistry. To give examples of

pplying our abstraction of pattern to physics and/or chemistry runs thesk of delving into the technical details of those subjects, as well as requiring

ath that is currently beyond the scope of our discussion. For that reason,oull have to forgive me if I gloss over things quite liberally in what follows.

Everything has a pattern, and symmetries associated with that pattern,en if it is just the trivial symmetry. In the case of chemistry the obviousing to start looking at is molecules. Unsurprisingly, the structure of aolecule has a pattern that depends, to a large extent, on its constituentements. More interestingly, molecules often have interesting symmetries.

We can, using a naive view, picture a molecule as a pattern of colouredalls, not dissimilar to our patterns of coloured marbles discussed in Shiftingatterns. Of course the patterns are now in 3 dimensions rather than 2, andnnections between the balls/marbles are important, but the fundamentalea is there. Consider, for instance, the picture of the ammonium molecule

NH4) shown in figure 2.16. We can, with little trouble, consider the various

gure 2.16: An ammonium molecule; nitrogen is coloured blue, while hy-ogen atoms are depicted as white.


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Fork in the Road 48

r any other molecule we care to consider. The exact algebra that resultsll differ from molecule to molecule, with the individual idiosyncrasies of thefferent algebras describing the the individual idiosyncrasies of the differentolecules. To understand the particular nature of the associated algebra is

understand a great deal about the particular nature of the molecule.It is, of course, possible to do this sort of analysis just by staring at the

atterns and never resorting to the sort of abstraction weve been discussing.his approach runs the risk of being both haphazard, and superficial. Byntrast, working in terms of pattern abstraction algebras affords us the

bility to be both comprehensive and systematic in our analysis. Ratheran trying to divine properties out of thin air via visual inspection, weke the resulting algebra and, by merely pushing symbols around on a piecepaper, pick apart every last nuance of its behaviour. Indeed, this sortanalysis (which extends to a level of detail in characterising the algebraat we wont touch on for some time) is now fundamental to understandinguch in chemistry, from spectroscopy to crystallography. Similar approachespatterns and symmetry of particles lead to a variety of important results

quantum physics.Our world is filled with patterns that are worth analysing with a system-ic approach: understanding the peculiarities of the algebras associated toose patterns can tell us a great deal about our world. New applicationsthis theory to new fields are still regularly occurring. There is a quiet

volution underway that is changing how we see and describe the world,nd the abstraction of pattern is at its heart.


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9 A Transfinite Landscape

.5 A Transfinite Landscape

roblems that involve infinity have a tendency to read a little like Zen koans.ake, for example, this problem: Suppose we have three bins (labelled bin

, bin B and bin C) and an infinite number of tennis balls. We starty numbering the tennis balls 1,2,3,... and so on, and put them all in bin

Then we take the two lowest numbered balls in bin C (thats ball 1, andall 2 to start) and put them in bin A, and then move the lowest numberedall in bin A from bin A to bin B (that would be ball 1 in the first round).

We repeat this process, moving two balls from bin C to bin A, and one ballom bin A to bin B, an infinite number of times. The question is, how many

alls are in bin A and how many balls are in bin B when were done? Thinkrefully!

The difficulty is in being sure of your answer10. We require a way to thinknsistently and coherently about such matters. So what is the answer? binhas no tennis balls in it, while bin B has an infinite number! Does thatund wrong? It certainly seems confusing: we are consistently putting two

alls into bin A and only taking one out at each step, so how can we end upith no balls in bin A? The key is to think in terms of a finished state, whene infinite process is somehow complete. Every ball is eventually movedbin B, thus after an infinite number of steps all the balls must have been

oved to bin B. The counterintuitive aspect is that we dont expect movingne ball at a time to ever catch up with moving two balls at a time, yet oddly

is happens.

Another tale that highlights this point is that of the hotel with an infiniteumber of rooms11. The story usually begins with the hotel finding itself fullne evening. A lone traveller then arrives, very weary, and asks the hotelanager if there is any chance at all that he can get a room. The hotelanager ponders this for a moment, and then has an idea. He asks each

uest to move to the room numbered one higher than their current room.nce every number has a number one greater, and there are an infinite

umber of rooms, everyone is housed; and yet room number 1 is now empty,nd the traveller has somewhere to stay. It doesnt end there though. After

e lone traveller, an infinite tour bus arrives, carrying an infinite numberpassengers all looking for rooms. After having solved the first problem,


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Fork in the Road 50

owever, the hotel manager isnt phased. He asks each guest to move intoe room with number twice that of their current room. Again, each number

as a number twice as large, and there are an infinite number of rooms,again everyone is housed; this time, however, everyone is housed in even

umbered rooms, which leaves infinitely many odd numbered rooms in whiche bus-load of tourists can be put up for the night.

This brings us a little closer to the sticky point where our intuitionsart to go astray. For any finite number n we expect there to be (roughly,depends on whether n is odd or even) half as many even numbers lessan n than there are natural numbers less than n; when we have infinitelyany numbers, however, there seems to be exactly as many even numbers as

atural numbers. Its this sort of unexpected equality with a set we initiallytuitively think should be half as big that allows the tennis ball problem tool us. In a sense, looking back from a completed infinity, 1 and 2 look prettyuch the same. What it really comes down to, however, is the very simple

uestion of what we mean by how many. As weve often seen before, theevil is usually in the details, and even simple things that we think we know

nd understand bear some thinking about if we want to be sure we actuallynow what we mean.What happens when we count things? Because counting is a fairly innate

ill for most adults it is helpful to consider what children, for whom countingstill somewhat new, do. Usually they count on their fingers (or other

milar things), making a correlation between objects counted and fingerseld up. At the more advanced adult level we do much the same thing, but

e correlate with abstract objects (numbers, which by that time weve solidlyeaten into instinctual memory). The point Im trying to get at here is thatunting is a matter of correlation; more importantly it is a very particularnd of correlation: in mathematics it is known as a one-to-one correlation.his means that each object corresponds with exactly one other object, andce versa in practice each object corresponds to exactly one number in ourunt, and each number in the count corresponds to exactly one object. If you

n accept that, at the heart of it, it is that one-to-one correspondence thatatters in counting, that it is the correspondence that ultimately determineshat we mean by quantity, then we can pull out the mathematicians handyol of abstraction and forget the other unimportant trivialities we mightsociate with counting and use the idea of one-to-one correspondence to


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So how do we count infinite sets with one-to-one correspondences? Ratheran actually counting infinitely many things, we provide an explicit process

y which the count could (in theory) be done. Thus, we simply try to setp a one-to-one correspondence just as before, the difference being that it

ill be given as a rule we can apply element by element as needed, ratheran having every single element to element correspondence laid out aheadtime. If such a correspondence exists then the sets have the same infinite

uantity. And that is exactly what we are doing, for example, with thefinite hotel story. First we are comparing the sizes of the sets {1, 2, 3, . . .}

nd {2, 3, 4, . . .} by noting that we have a correspondence

1 2 3 4 n 2 3 4 5 n + 1

nd that since both sets are infinite we will have exactly one element ine second set for every element in the first set, and vice versa; a one-to-

ne correspondence. The sets have the same quantity thus we can shuffleeryone down one room and still house them all. When the tour bus shows

p we end up comparing the sizes of the sets {1, 2, 3, . . .} and {2, 4, 6, . . .} byaking the correspondence

1 2 3 4 n

2 4 6 8 2n

here again the infinite sets ensure that each element corresponds to exactlyne element in either direction; another one-to-one correspondence, demon-rating the sets have the same quantity we can move everyone to even

umbered rooms and still house them all.

At this point most people are happy enough to accept that there are the

me number of even numbers as there are natural numbers. Their argumentns roughly well there is an infinite number of both, and infinity is as big a

umber as there can be, so of course theyre the same. There is, possibly, atle squirming under the fact that the what is clearly only a part apparently

as the same size as the whole, but that tends to get swept under the rug of


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The classic example of something infinite and larger in size than the natu-l numbers is the continuum (at least as classically conceived; the construc-

vist/intuitionist continuum is a little more tricky on this front) as discussedParadoxes of the Continuum, Part II. In that post we determined that

oints on the continuum were able to be identified with Cauchy sequences,hich were akin to (though a little more technical than) infinite decimal ex-ansions. Well stick with infinite decimal expansions here as most peopleave a better intuitive grasp of decimals than they do of Cauchy sequences oredekind cuts. To make things simple well consider the continuum rangingetween 0 and 1; that is, all the possible infinite decimals between 0 and

such as 0.123123123... We do have to be a little bit careful here since,you should recall from Paradoxes of the Continuum, Part II, in the same

ay that there are many fractions that represent the same ratio, there wereany Cauchy sequences that represent the same point in the continuum, andparticular there are different decimal expansions that represent the same

oint, such as 0.49999999... and 0.50000000...12. To be careful we have toake sure we always pick and deal with just one representative in all such

ses; to do this we can simply only consider representations that have in-nitely many non-zero places. Showing this is sufficient, and still covers allal numbers between 0 and 1 isnt that hard, but amounts to some technical

oop jumping that is necessary for formal proofs, but not terribly elucidatingr discussions such as this. Suffice to say that it all works out.

The catch now is that we need to show not just that we are incapable ofnding a one-to-one correspondence between these points on the continuum

nd the natural numbers, but that no such correspondence can exist. We dois by the somewhat backwards approach of assuming there is such a corre-ondence, and then showing that a logical contradiction would result. Fromat we can conclude that any such correspondence would be contradictory,

nd thus cant actually exist (at least not in any system that doesnt haventradictions). So, to begin, lets presume we have a correspondence like13

12People have a tendency to object to this, and other similar claims, such as that99999... is equal to 1. The easiest way to see this simple but slightly unintuitive fact isnote that we should be able to take 0.99999..., move the decimal place right one place,

btract 9, and arrive back at the same value (this is akin to shifting the hotel guests downe room to make a spare room at the front but in reverse) That is if x = 0 99999


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1 2 3 4

0.a1a2a3a4 . . . 0.b1b2b3b4 . . . 0.c1c2c3c4 . . . 0.d1d2d3d4 . . .

here the a1, a2, ldots etc. are just digits in the decimal expansion. The trickto show that despite our best efforts to set up a one-to-one correspondence,e list of points in the continuum given by this correspondence (and since we

avent specified what the correspondence actually is, any such one-to-onerrespondence) is actually incomplete: weve missed some. We do this bynstructing a decimal as follows: for the first decimal place, choose a digitfferent from a1, for the second decimal place choose a digit different from

, for the third choose a digit different from c3, and so on. Now clearly thisecimal is between 0 and 1, and hence ought to be in our list somewhere,ut by its very manner of construction it will differ from the nth decimalumber on the list at the nth decimal place that is, we are guaranteed that

is different from every decimal weve listed! Thus despite our assumptionat we had a one-to-one correspondence, it isnt, since weve found a pointthe continuum for which there is no corresponding natural number. Given

ch a contradiction, the only conclusion we can draw is that we cannoteate a one-to-one correspondence between natural numbers and points ine continuum no matter how we try, well always end up with extra pointsthe continuum for which there is no corresponding natural number; that there are more points in the continuum than there are natural numbers.

What all of this means is that we have to come to terms with the factat some infinities are bigger than others. In fact, it gets even worse: some

finities are entirely incompatible with others. This particular catch hidesa slightly over-zealous abstraction of numbers. For the most part we do

ot differentiate between numbers describing order (2nd position, as opposed5th position, etc.) and numbers describing quantity (which is generally

e notion of number with which weve been dealing). There is a perfectlyood reason for this: when it comes to using and manipulating numbers

ing standard arithmetic, the numbers describing order (so called ordinalumbers) behave completely identically to those describing quantity (whiche call cardinal numbers). As so often happens with abstraction (indeed, itsentially is the core idea of abstraction) if there are no practical differences

ch that there is a first element) This question quickly wades into very deep waters


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t least as far as the practical purposes we care about are concerned) betweenbjects, we simply forget that there are any differences at all. And, indeed,r finite numbers this is a perfectly reasonable thing to do. The catch is that,

nce we start dealing with infinities, ordinals and cardinals start behaving

ther differently it is no longer safe to consider them the same, or even,r that matter, comparable to one another.

I wont go into the rather technical theory of transfinite ordinals here,nd instead just give you a precis of where the difficulties lie. To start,ts introduce some standard mathematical notation, and let 0 denote thest infinite cardinal (that is, the quantity of natural numbers), and let

enote the first infinite ordinal (that is, the first position reached after wevehausted all finite positions). Now, as weve already seen, if we have infiniteoms, we can house an extra guest even if theyre all full; that is,0+1 = 0.n the other hand, if we tack on an extra position after ; and all the finite

nes, i.e. we have 1st, 2nd, 3rd,...,,ath, then the ath position turns out to beppreciably different to all the positions before it. In other words + 1 = .ow, whereas with finite numbers where are adding one produces the same

sult for both ordinals and cardinals, for infinite numbers it makes a hugefference (in one case we simply end up with what we started with, and ine other we end up with something entirely new). It shouldnt be too hardsee that, from that simple difference, whether you have are dealing with

cardinal or ordinal transfinite number is going to matter for arithmeticperations; we can no longer ignore the difference; cardinals and ordinalsave to be considered as quite separate and distinct kinds of objects!

At this point you might be trying to reconcile the fact that 0 + 1 = 0ith the previously observed fact that there are bigger cardinal infinities.ow can we get to a bigger infinity if adding to 0 ends up going nowhere?o answer that Im going to need to discuss power sets. Given a set And for now well keep things informal, the finer technicalities of what ac-ally is and is not a set will come further along our road), the power setA is the set of all possible subsets of A. An example will help clar-

y. If we have a set A = {a,b,c}, then the power set of A is P(A) =}, {a}, {b}, {c}, {a, b}, {a, c}, {b, c}, {a,b,c}}. Thus each element of the set

(A) is itself a set, and in particular, a subset ofa; note that we consideroth the empty set and A itself to be subsets ofA. With a little combina-rics you can see that if a set has n elements (where n is finite) then its


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n show that even if a set has infinite cardinality its power set will have arger cardinality. Thus, borrowing notation from the finite case, 0 < 2

0.sing this trick, which applies to any infinite set, we can develop an entireerarchy of different orders of infinity:

0 < 20 = 1 < 2

1 = 2 < 22 = 3 <

similar, but different, hierarchy of infinite ordinals also exists (above andeyond the obvious option of simply adding one to an existing infinite ordi-al to get a larger one), spiralling ever higher, this time using the concept

tetration14 rather than exponentiation. Contrary to initial expectation,

finities exist in infinite variety. How many infinities are there? We cannoty, on pain of paradox, since such a statement would only reflect back on

self in a vicious circle of contradiction.While we began with just a hazy view of the infinite, the mists have

eared to reveal a strange and remarkable valley; a transfinite landscapeith a veritable zoo of infinities of different kinds and sizes; it is, indeed, ahole new landscape of numbers and possibilities to explore; a hidden valleythe infinite. Running through the middle of the valley is a large river, andwe wade in we will find very deep waters. It is all a matter of asking the

ght questions. The question we can start with seems innocently simple:

Is the cardinality of points on the continuum bigger, smaller,

or the same as 1 = 20?

he answer is deceptively complex. It can be established, with a little work,at the number of points in the continuum is not bigger than 1, which

aves us with either smaller, or the same size as 1. From there thingst complicated quickly, and mired in a certain degree of technicality, butsentially the result is that the answer doesnt matter. What I mean byat is we may assume that the number of points in the continuum is 1 and

o problems or contradictions will arise, yet at the same time we can equallyell assume that the number of points in the continuum is strictly less than

and still no problems or contradictions arise. Indeed, whether there is anyrdinal number between 0 and 1 falls into this category. There is, in anse, no truth here, merely preference.


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This highlights deep facts about mathematics. When our journey begane considered numbers, and fractions, and algebra. Relatively speaking thesee fairly simple abstractions, and, more importantly, they are abstractionsat we tend to use each and every day (particularly in the case of numbers

nd fractions). Through a mix of immediate concrete associations due to thelatively low level of abstraction, and the sense reality imbued by constante and exposure, we tend to think of numbers, fractions, algebra, and evenathematics itself, as something real, fixed, and concrete. That is, we thinkmathematics as describing some platonic reality, that the objects it de-

ribes, while abstract, have some real existence. It is natural, then, to thinkat a number between 0 and 1 either exists, or doesnt exist, in some real

nd concrete way yet that is not how things have worked out. Imagineeing told that whether the number five existed or not was quite optional,ithmetic would work just fine either way! We have, in essence, been toldat existence is merely a preference, not a reality; truth is up for grabs,

n option rather than a cold hard absolute.Going all the way back to the first section, On Abstraction, things start

get a little clearer however. As long as we view mathematics as a matter ofaking effective and powerful abstractions from the real world, rather thanescribing some platonic universe, having a choice of abstraction doesnt seem

bad. We can choose how to interpret the continuum to suit our needs deed, we can even reject transfinite arithmetic and opt for the intuitionistnception of the continuum if we wish; we choose the abstraction that bestits our purposes for the moment. You could view it as little different than

hoosing to work at the genetic level as a molecular biologist instead of thensidering subatomic particles as a physicist would: the level and mannerabstraction matters only with regard to the level and manner of detailu wish to obtain in the way of results. The more layers of abstraction

e apply, the greater the chances of running into quandaries and choices;y abstracting away more and more detail, and by piling abstractions uponbstractions, we push further and further into the realm of pure possibility.

his has the potential to lead us to strange and confusing trails, but it alsoves us the power to see beyond our own limited horizons. In broadening ourinds to embrace worlds of possibility we conceive of realities that transcend

ur conceptions, and probe our own reality in ways far beyond the limitsolution has shackled our perceptions with


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the dance between logic and mathematics that will follow. We passed by

e crossroads of unreality some time ago, yet there is still a very long way

go.


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.6 Grouping Symmetries

oming soon...


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List of Figures

2.1 Pattern of coloured marbles . . . . . . . . . . . . . . . . . . . 25

2.2 Labelled pattern of marbles . . . . . . . . . . . . . . . . . . . 262.3 A different arrangement of marbles that preserves the pattern 26

2.4 Showing how the rearrangement was made . . . . . . . . . . . 26

2.5 The same marbles in a rectangular arrangement . . . . . . . . 27

2.6 A square with labelled corners . . . . . . . . . . . . . . . . . . 28

2.7 The three different rotations of a square . . . . . . . . . . . . 28

2.8 The four different flips of a square . . . . . . . . . . . . . . . . 29

2.9 A distorted square

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