topics: |  math

Gowers is very careful in positing ideas.  e.g. given two 5-D vectors x and
y, proposes that the distance between these may be taken as sqrt SUM_i
(xi-yi)², and shows how this may follow from pythagoras on a series of right
triangles - x1:x2, then diag(x1,x2):x3, and so on.

But then:

	Actually, however, what we have done is define a notion of
	distance.  No physical reality forces us to decide that
	five-dimensional distance should be calculated in this
	manner.  However, this approach is so c1early the natural
	generalization of 2D and 3D that it would be strange to adopt any
	other definition.  75

one of the main points that Gowers emphasizes is that there is no single
mathematical reality.  everything depends on how you define things.  one set
of definitions and assumptions leads to one kind of mathematics. there is no
convincing reason for stating that one set of assumptions are superior to
another.

Excerpts

from Preface

Early in the 20th century, the great mathematician David Hilbert noticed that
a number of important mathematical arguments were structurally similar. In
fact, he realized that at an appropriate level of generality they could be
regarded as the same. This observation, and others like it, gave rise to a
new branch of mathematics, and one of its central concepts was named after
Hilbert. 

The notion of a Hilbert space sheds light on so much of modern mathematics,
from number theory to quantum mechanics, that if you do not know at least
the rudiments of Hilbert space theory then you cannot claim to be a
well-educated mathematician.

What, then, is a Hilbert space? In a typical university mathematics course it
is defined as a complete inner-product space. Students attending such a
course are expected to know, from previous courses, that an inner-product
space is a vector space equipped with an inner product, and that a space is
complete if every Cauchy sequence in it converges. Of course, for those
definitions to make sense, the students also need to know the definitions of
vector space, inner product, Cauchy sequence and convergence. To give just
one of them (not the longest): a Cauchy sequence is a sequence x1, x2, x3,
. . . such that for every positive number ε exists an integer N such
that for any two integers p and q greater than N the distance from xp to xq
is at most ε.

[Gowers is going to try to give a more intuitive idea of some of the 
complexities of mathematical concepts.]

1 Models (and the relevance of mathematics)

How to throw a stone

Suppose you would like to throw [a stone] as far as possible. Given how
hard you can throw, the most important decision you must make is the
angle at which the stone leaves your hand. If this angle is too flat,
then although the stone will have a large horizontal speed it will land
quite soon and will therefore not have a chance to travel very far. If on
the other hand you throw the stone too high, then it will stay in the air
for a long time but without covering much ground in the process. Clearly
some sort of compromise is needed.

[Can be solved by a combination of science and mathematics] - can predict
the entire behaviour of the stone trom the moment it is launched until the
moment it lands. However, [this works] only under several simplifying
assumptions - does not consider air resistance, the rotation of the
earth, a small gravitational influence from the moon, the fact that the
earth's gravitational field is weaker the higher you are, and the gradually
changing direction of 'vertically downwards' as you move from one part of
the earth's surface to another.

Finally, the recommendation of 45 degrees is based on another implicit
assumption, namely that the speed of the stone as it leaves your hand does
not depend on its direction.  Again, this is untrue: one can throw a stone
harder when the angle is flatter.

Mathematics works on models

[In examining the solution to a physical problem, one can often]
draw a clear distinction between the contributions made by science and
those made by mathematics.  Scientists devise a theory, based partly on the
results of observations and experiments, and partly on more general
considerations such as simplicity and explanatory power.

Mathematicians, or scientists doing mathematics, then investigate the
purely logical consequences of the theory. Sometimes these are the results
of routine calculations that predict exactly the sorts of phenomena the
theory was designed to explain, but occasionally the predictions of a
theory can be quite unexpected. If these are later confirmed by experiment,
then one has impressive evidence in favour of the theory.

Mathematicians do not apply scientific theories directly to the world but
rather to models. A model in this sense can he thought of as an imaginary,
simplified version of the part of the world being studied, one in which exact
calculations are possible.

When choosing a model, one attempts to make its behaviour correspond
closely to the actual behaviour of the world, [while paying attention to]
simplicity and mathematical elegance. Indeed, there are very useful models
with aImost no resemblance to the world at aIl, as some of my examples will
illustrate.

nobody could be expected to tell me the outcome of a given roll in advance,
even if they had expensive technology at their disposal and the dice were to
be rolled by a machine.

If you want to get a feel for the difficulty of factoring primes, try finding
two prime numbers that multiply to give 6901 and another two that give
280123.

[i was able to do 6901.  i realized that 6901 mod 17 = -1.  so i tried
factors N s.t. N mod 17 = -1
[i am sure this is an incorrect reason, but this is what i did.]

and 67 did it.

later i realized that if p mod N = k and q mod N = k, then p and q are
congruent modulo N, and p-q is divisible by N, so
	6901-67 = 6834 is divisible by 17
but there is no reason why p (6901) should be divisible by q (67).
this is an example of a "justified true belief" also called the
Gettier's paradox in epistemology.


280123 i did with calculator:

400 * 700 -> lower by 123.  reduce 400 by 1, incr 700 x 2 -> +100
399 * 702 = 280098
383 * 731 = 279973
377 * 743 = 280111
373 * 761 = 283853  [trying for the last digit]
373 * 751 = 280123
]

2 Numbers and abstraction

Do numbers exist?

[ch 2 argues that mathematicians can, and even should, happily ignore this
seemingly fundamental question.]

there certainly are philosophers who take seriously the question of whether
numbers exist, and this distinguishes them from mathematicians, who either
find it obvious that numbers exist or do not understand what is being
asked. 17

[what matters about numbers (or other abstract objects] is not whether they
exist, or its intrinsic nature, but what it does.]

This attitude can be encapsulated in the following slogan:
		a mathematical object is what it does.

Similar slogans have appeared many times in the philosophy of language, and
can be highly controversial. Two examples are 'In language there are only
differences' and 'The meaning of a word is its use in the language', [from
Saussure's Course in General Linguistics and Wittgenstein's
Philosophical Investigations respectively.)]

What numbers do

[But then, what can] a number like 5 be said to do? It doesn't move around
like a chess piece. Instead, it seems to have an intrinsic nature, a sort
of pure fiveness that we immediately i grasp when we look at a picture [of
five cars, say].

[a similar sense of 7-ness can be there (clusters of 2 + 3 + 2) or even
12-ness (3 rows of 4 each), but it becomes harder for numbers like 47, say.]

then we have little choice but to count them, this time thinking of
47 as the number that comes after 46, which itself is the number
that comes after 45, and so on.

In other words, numbers do not have to be very large before we stop
thinking of them as isolated objects and start to understand them
through their properties, through how they relate to other numbers,
through their role in a number system. This is what I mean by what
a number 'does'.

[in doing operations such as multiplication, we implicitly assume that]
	if you multiply two numbers together, it doesn't matter which order
	you put them in, and that ifyou multiply more than two numbers
	tagether, then it makes no difference how you bracket them.
--> commutative, and distributive laws for multiplication.

    also introduces  associative law for addn and multiplicn,
    and multiplicative and additive identity [zero],
    also uses the successor function "2 is defined to be the number 1+1."  to
	derive that 2x0 = 0.

can we derive that 0 x 0 = 0?

Q. by Gowers' six year old son, John :
	how can nought times nought be nought, since nought
	times nought means that you have no noughts?

easier to handle such q's in the formal structure than by informal.

Negative numbers and fractions

As anybody with experience of teaching mathematics to small
children knows, there is something indirect about subtraction and
division that makes them harder to understand than addition and
multiplication.

e.g. subtraction questions: 'How many oranges will be left if you start
with five and eat two of them?'.

But not the best way to think about it - e.g. how to take 98 from 100?
better think about what one has to add to 98 to make 100. i.e. effectively
solving the equation 98 + x = 100.  p.25

can extend our system with two more rules related to negative numbers
(subtraction) and fractions (division):

* additive inverse:  for every num a, exists b s.t. a+b = 0
* multiplicative invrese: for every a not eq to zero, exists b s.t. ab=1

we can think of -a and 1/a for b above.

these axioms are mentioned as
      "rules we simply introduce to make a good game" [p.27]
and these differ from deductions we can make based on these.

[when my son zubin was 4 or so, he had asked me why -1 times -1 = +1.
This axiomatic system gives an answer, but not at a level understandable
by four year olds.

TG does not try to address why -1 x -1 = +1; but here is a proof:

Let -1 x 1 be x.
	-1 + 1 = 0.
	-1 x 0 = -1 x [1 + -1] = -1 + x = 0.  Hence x = +1
]

Large numbers

[large numbers are not really "counted"; while ]

	the natural number system, thought of as a model, is useful in certain
	circumstances whereas [very large] number systems are not.

[but -ve nums can be useful - e.g. in models of] temperatures, or dates, or
bank accounts.

There is nothing wrong with the numher
1394840275936498649234987 from a mathematical point of
view, but if we can't even count votes in Florida, it is inconceivable
that we might ever be sure that we had a collection of
1394840275936498649234987 objects. If you take two piles of
leaves and add them ta a third, the result is not three piles of leaves
but one large pile.

If you have just watched a rainstorm, then, as Wittgenstein said, 'The
proper answer to the question, "How many drops did you see?", is many, not
that there was a number but you don't know how many...

Real numbers

The real number system consists of ail numbers that can be
represented by infinite decimals. This concept is more sophisticated
tban it seems...

The most famous example of such an equation is : x^2 := 2. In the
sixth century it was discovered by the school of Pythagoras that
x is irrational, which means that it cannot be represented by a
fraction...

[proof: let s^2 be rational, then exists co-prime a,b, s.t. s^2=a/b.
	then b.s^2 = a, and b.b.2 = a.a
	but since b and a are co-prime, this is not possible, since there
	must be another 2 on the LHS or RHS]
ALT: a² is even hence a is even.  Let a = 2c.  but can also show that b.b =
2.c.c, hence b is also even, so cannot be co-prime.  ]

This discovery caused dismay when it was made, but now we cheerfully accept
that we must extend our number system if we want ta model things like the
length of the diagonal of a square. Once again, the abstract method makes our
task very easy. We introduce a new symbol, √2, and have one rule tbat tells
us what to do with it: it squares to 2.

it is useful as a solution to equations of the type x² = 2.

[problem: this definition does not distinguish between +√2 and -√2.

suggests one can handle this by distinguishing +ve from -ve.

but then half-backtracks and says do we really need to?  is this distinction
really needed? gives an example - √2-1 = 1/√2+1; here treating the √2 and
-√2 as same actually is informative. ]

Imaginary and Complex numbers

Historical suspicion of the abstract method has left its traces in the words
used to describe the new numbers that arose each time the number system was
enlarged, words like 'negative' and 'irrational'. But far harder to swallow
than these were the 'imaginary', or 'complex'... 29

[We are] simply continuing to extend our number system, by introducing a
solution to the equation x² = -1 and calling it i; Why should this be more
objectionable than our earlier introduction of √2?  e.g. we can now say
things like
		1/(i-1) = 1/2 (i+1)

one objection may be that we can think of √2 using a concrete representation
such as 1.4142...  but there is no such ossiblity for i.

To see why i is [so problematic], ask yourself: which of the two square roots
of -1 is i and which is -i?  The question does not make sense because the
only defining property of i is that it squares to -1. Since -i has the same
property, any true sentence about i remains true if one replaces it with the
corresponding sentence about -i. It is difficult, once one has grasped this,
to have any respect for the view that i might denote an independently existing
Platonic object.

Do these questions matter?

parallels with the theory of qualia

Might it be that when you perceive the colour red your
sensation is what I experience when I perceive green, and vice
versa? Somee philosophers take this question seriously and define
'qualia' ta be the absolute intrinsic experienccs we have when, for
example, we see colours. Others do not believe in qualia. For them, a
word like 'green' is defined more abstractly by its role in a linguistic
system, that 1s, by its relationships, with concepts like 'grass', 'red',
and so on.  It is impossible to deduce somebody's position on this
issue from the way they talk about colour, except during
philosophical debates. Similarly, all that matters in practice about
numbers and other mathematical objects is the rules they obey.

Remarkably, it turns out that any polynomial equation
can be solved within the complex number system. In
other words, we make the small investment of aceepting the
number i, and are then repaid many times over. 31

This fact, which has a complicated history but is usually attributed to
Gauss, is known as the fundamental theorem of algebra and it provides very
convincing evidence that there is something natural about i. It may he
impossible to imagine a basket of i apples, a car journey that lasts i hours,
or a bank account with an overdraft of i pounds, but the complex number
system has become indispensable to mathematicians, and to scientists and
engineers as weil - the theory of quantum mechanics, for example, depends
heavily on complex numbers.

It provides one of the best illustrations of a general principle: if an abstract
mathematical construction is sufficiently natural, then it will almost
certainly find a use as a model.

Infinity

Once one accepts the power of abstract thinking,

	there seems to be nothing stopping us:
	infinity should mean something like 1 divided by 0, So why not let ∞
	be an abstract symbol and regard it as a solution to the equation
	Ox=l?

	The trouble with this idea emerges as soon as one tries to do
	arithmetic.  Here, for example, is a simple consequence of the
	associative law for multiplication, and the fact that 0 x 2 = 0.

	1 =  1= ∞ x O = ∞ x (Ox2) = (∞ xO) x2 = lx2 = 2

	What this shows is that the existence of a solution to the
	equation Ox = lleads to an inconsistency. 32

Does that mean that infinity does not exist? No, it simply means that no
natural notion of infinity is compatible with the laws of arithmetic. It is
sometimes useful to extend the number system to include the symbol ∞,
accepting that in the enlarged system these laws are not always
valid. UsuaIly, however, one prefers to keep the laws and do without
infinity. 32

Negative and Fractional powers

the abstract method allows us to extend familiar concepts to unfamiliar
situations.

e.g. we know that 2³ = 2 x 2 x 2.  But what is 2^{3/2}?

We define two rules for raising numbers to powers:

* a¹ = a for any real num a
* a^{m+n} = a^m x a^n - for any natural nums m,n.

[TG extends this second rule quietly to other than natural nums without much
discussion]

this lets us handle 2^{3/2}, say x.  Then we have
	x² = 2^{3/2} x 2^{3/2} = 2³ = 8.

This still leaves the q of whether x is -ve or +ve.

so it is customary to adopt the convention, that

* if a > 0 and b is a real number, then a^b is positive.

This is not a discovery of the 'true value' of 23/2. However, neither
is the interpretation we have given to the expression 2"/2 arbitrary it
is the only possibility if we want to preserve the three rules above.

can also derive a⁰ and negative powers of a from these axioms. 34

[also logarithms]

many such concepts seem puzzling when you try to understand them concretely,
but they lose their mystery when you relax, stop worrying about what they
are, and use the abstract method.

Ch3 : Proofs

opens with an impressive result that was new for me:


   chords on n points on a circle. for n = 1 to 5, the num of
   regions created goes as 1,2,4,8,16.

but will this trend hold for any n?  how do we prove this? p.35

the problem gets shelved for 11 pages while gowers analyzes the nature of
proof via sqrt(2) and area of a circle.

then we find that the number of elements added on the 5-circle when a new
point is introduced is 1+4+6+4+1 = 15.  Thus, the total regions for 6 is
31, not 32.



so what is going on? can we get the series right?

MY ATTEMPTS:
by induction:
	 f(3) = 1 + 1 (+2) = 4
	 f(4) : adds 1 + 2 + 1 -> 8
	 f(5) : adds 1 + 3 + 3 + 1 -> 16
	 f(6) : adds 1 + 4 + 5 + 4 + 1 -> 31
	 f(7) : adds 1 + 5 + 7 + 7 + 5 + 1 -> 57  [+7 is determined by observation]
	 f(8) : adds 1 + 6 + 9 + 10 + 9 + 6 + 1 -> 99

when adding lines from nth-point, each point has n-2 edges.  the
chord to the k-th neighbour misses out on k edges, so it cuts n-2-k
edges. Hence, for even n = 2 * SUM_to n/2 (n-k-2)*k - 1
[odd n, add the 2n+1/2-th term].

Simpler argument:
   num of regions added by Introducing a line = 1 + num of intersections.
   Num of lines = C(n,2)
   Num of intersections = C(n,4)
   also, 1 region to begin with.
   Hence num regions = 1 + C(n,2) + C(n,4)

the series goes: l, 2, 4, 8, 16, 31, 57, 99, 163, 256 (Sloane's A000127).

also see: http://spikedmath.com/449.html
     	    http://mathforum.org/library/drmath/view/55015.html
	    http://mathworld.wolfram.com/CircleDivisionbyChords.html


related: law of small numbers
loosely, this says that however many examples you take at the beginning,
they are not enough.

more alliteratively:

	Initial irregularities inhibit incisive intuition.
see http://www.math.sjsu.edu/~hsu/courses/126/Law-of-Small-Numbers.pdf

Proofs of rationality of square root of 2

Gives a series of proofs for the irrationality of √2.
The first proof (10 steps) has a step which says:
	if r² is even, then r is even.
The second subproof (5 steps)  establishes this.  But it has a step that says
	since r is odd, there is an integer t s.t. r = 2t+1
but
	Why should every whole number be either a multiple of two or one more
	than a multiple of two?
The third subproof has 4 steps that proves this.

Demonstrates the idea of mathematical induction - given as an axiom - which
can be used to avoid the term "and so on" in a step in the third subproof.

A fact of fundamental importance to mathematics is that this process
eventually comes to an end.  In principle, if you go on and on splitting
steps into smaller ones, you will end up with a very long argument that
starts with axioms that are universally accepted and proceeds to the desired
conclusion by means of only the most elementary logical rules (such as 'if A
is true and A implies B then B is true').

What l have just said in the last paragraph is far from obvious: in
fact it was one of the great discoveries of the early 20th century,
largely due to Frege, Russell, and Whitehead...

This discovery has had a profound impact on mathematics, because it means
that any dispute about the validity of a mathematical proof can always be
resolved. In the 19th century, by contrast, there were genuine disagreements
about matters of mathematical substance. For example, Georg Cantor, the
father of modern set theory, invented arguments that relied on the idea that
one infinite set can be 'bigger' than another. These arguments are accepted
now, but caused great suspicion at the time.

Actually, this does not mean that disagreements never occur.  For example,
it quite often happens that somebody produces a very long proof that is
unclear in places and contains many small mistakes, but which is not obviously
incorrect in a fundamental way. Establishing conclusively whether such an
argument can be made watertight is usually extremely laborious, and there is
not much reward for the labour. Even the author may prefer not to risk
finding that the argument is wrong.

Nevertheless, the fact that disputes can in principle be resolved does make
mathematics unique. There is no mathematical equivalent of astronomers who
still believe in the steady-state theory of the universe, or of biologists
who hold, with great conviction, very different views about how much is
explained by natural selection, or of philosophers who disagree fundamentally
about the relationship between consciousness and the physical world, or of
economists who follow opposing schools of thought such as monetarism and
neo-Keynesianism.

[discusses issues of human nature in mathematical proofs:]

	mathematical papers are written for highly trained readers who do not
	need everything spelled out. However, if somebody makes an important
	claim and other mathematicians find it hard to follow the proof, they
	will ask for clarification, and the process will then begin of
	dividing steps of the proof into smaller, more easily understood
	substeps. Usually, again because the audience is highly trained, ihis
	process does not need to go on for long until either the necessary
	clarification has been provided or a mistake comes to light. Thus, a
	purported proof of a result that other mathematicians care about is
	almost always accepted as correct only if it is correct. 41

We usually want more from a proof than a mere guarantee of correctness. We
feel after reading a good proof that it provides an explanation of the
theorem, that we understand something we did not understand before. 42

[proof that golden ratio is irrational - geometric construction - if
ratio = p/q then process of subdivision must come to an end (at most pq
squares].

elegance of proofs

take a 4x4 square. remove two corners along either diagonal.  try covering
the remaining  with domino-like pieces that cover two squares.

can show  that this is impossible - with any start, leads to a dead end.
but cannot generalize this method to a 8x8 board.

On the other hand, the chessboard analogy - alternating colours, both corners
removed are same colour so now we have 32 blacks and 30 whites say.  After 31
dominoes, we have two squares of the same colour left, which can never be
covered by a domino.

the second proof is not only more extensible (e.g. it holds for a
thousand-by-thousand board), but it also reveals something like a reason
why this theorem holds.

	It often puzzles people when mathematicians use words like 'elegant',
	'beautiful', or even 'witty' to describe proofs, but an example such
	as this gives an idea of what they mean. Music provides a useful anal
	ogy: we may be entranced when a piece moves in an unexpected harmonic
	direction that later comes to seem wonderfully appropriate, or when
	an orchestral texture appears to be more than the sum of its parts in
	a way that we do not fully understand. Mathematical proofs can
	provide a similar pleasure with sudden revelations, unexpected yet
	natural ideas, and intriguing hints that there is more to be
	discovered. Of course, beauty in mathematics is not the same as
	beauty in music, but then neither is musical beauty the same as the
	beauty of a painting, or a poem, or a human face. 51

Obvious-seeming statements that need proofs

An aspect of advanced mathematics that many find puzzling is that
sorne of its theorems seem too obvious to need proving. Faced with
such a theorem, people will often ask, 'If that doesn't count as
obvious, then what does?' A former colleague of mine had a good
answer to this question, which is that a statement is obvious if a
proof instantly springs to mind.

Examples:

1.  The fundamental theorem of arithmetic states that every natural
    number can be written in one and only one way as a product of
    prime numbers...

is it so obvious that 7 x 13 x 19 does not equal 37 x 47?

(one, as any mathematician will tell you, is more interesting than the other)
[7x13x19 is 1729, Ramanujan's instant answer to Hardy]

In fact, there is no easy proof of the theorem...
if a proof instantly springs to mind, then you have a very unusual mind.

2.  trefoil knot



can this knot be untied without cutting the string?
though the answer seems obviously "no", a proof is far
from obvious.

3. A curve in the plane means anything that you can draw without lifting your
pen off the paper. It is called simple if it never erosses itself, and closed
if it ends up where it began.  Clearly, every simple closed curve splits the
plane into two parts, the inside and the outside (three parts if one includes
the curve itself as a part).

[Jordan's curve theorem.  all known proofs are well beyond this book]

4: Limits and infinity

In one way or another, the concept of infinity is indispensable to
mathematics, and yet it is a very hard idea to make rigorous.

	The square root of 2 is about 1.41421356

Where is infinity involved in a simple statement like the above,
which says merely that one number is roughly equal to
another?

The phrase "square root of 2 is" -> assumes square root of 2 exists - if so,
what sort of object is it?  --> infinite decimal.

the closely related statement:

	1.41421356 squared is close to 2

[draws attention to the fact that this statement is entirely finite, though
it seems to say roughly the same thing.

will show how this îs important]

how do we add two infinite decimals?  cannot start from right as with normal
(finite) decimals.  adding from the left creates instability.
A long series of 9's can all become zeros -
however, no digit will need to be corrected more than once.

[what if adding more than 10 fractions? ]

multiplying infinite fractions is more complicated.  but given  k digits of
√2, gives a procedure for identifying (k+1)-th (largest value after which
square crosses 2).

But what makes us confident that x² = 2?
We might argue as follows:
                                    1² = 1
                                  1.4² = 1.96
                                 1.41² = 1.9881
                                1.414² = 1.999396
                               1.4142² = 1.99996164
                              1.41421² = 1.9999899241
                             1.414213² = 1.999998409469
                            1.4142135² = 1.99999982368225
                           1.41421356² = 1.9999999933878736

so the more digits we keep, the more nines we get - hence the entire infinite
expansion of ~2, we should get infinitely
many nines, and 1.99999999 - which is equal to 2.

Two Difficulties:
1. why does one point nine recurring equal two?
2. what does it mean to 'use the entire infinite expansion'? That is what wc were
   trying to understand in the first place.

[2nd is more serious]  <--

[AM: what does it mean to be "more serious" - how does this fit into  the
notion of a t/f logical world? ]

To dispose of the first objection, we must once again set aside any Platonic
instincts.

[what is a Platonic instinct?]

It is an accepted truth of mathematics that one point nine
recurring equals two, but this truth was not discovered by sorne process of
metaphysical reasoning. Rather, it is a convention. However, it is by no
means an arbitrary convention, because not adopting it forces one either to
invent strange new objects or to abandon sorne of the familiar rules of
arithmetic.  [e.g. will need to invent a new object for 2 - 1.999999.... - as
"a decimal point, followed by infinitely many zeroes, then something
non-zero"]

but doing this only makes for more difficulties.   what happens when you
multiply this new object with itself?  What of 1/3 = 0.333333... - what if
this number is now multiplied by 3, etc.
such tricky questions continue to arise.

(Tricky but not impossible: a coherent notion of 'infinitesimal'
numbers was discovered by Abraham Robinson in the 1960s, but
non-standard analysis, as his theory is called, has not become part
of the mathematical mainstream.)

The second difficulty is a more genuine one, but it can be
circumvented.
One may interpret x² = 2 as meaning that the more digits you take in x, the
closer the square is to 2.
-->   desire to generate a number with a given (small) error - i.e. a large
      number of nines.

given we want k nines,  this can be achieved by  having no more
than k+1 digits in the decimal.
[can be proved in terms of the error in x being bounded by 10^{-k}.

The fact that this can be done for any k is what i8 meant by saying that the
infinite decimal x, when multiplied by itself, equals 2.

with this definition, once we check that it does not conflict with the
commutative and associative axioms, we are free to think abstractly again.

Instantaneous speeds

while observing the speedometer in a car, one may observe it go from 30 to 50
mph, and one might say:

	"We reached a speed of 40 m.p.h. just as we passed that lamp-post"

--> one way to model the instantaneous speed of an accelerating object is to
    measure the distance travelled during an infinitely small period of
    time.

This ridiculous fantasy leads to problems very similar to those with the idea
that one point nine recurring might not equal two.

Is t zero? If so, then quite clearly dist must be zero ...  but one cannat
divide zero by zero and obtain an unambiguous answer. On the other hand, if t
is not zero, then the car accelerates during those t hours and the
measurement is invalid.

Resolution: if t is "small" then acceleration within t is very small. as t
becomes smaller, our estimate of the present speed will become increasingly
more accurate.

Once again, we are regarding a statement that involves intinity as
a convenient way of expressing a more complicated statement
concerning approximations. Another word, which can be more
suggestive, is 'limit'.

Mathematicians often talk about what happens 'in the limit', or 'at
infinity', but if you press them, they will start to talk about
approximations instead.

Area of a circle

The realization that the infinite can be understood in terms of the
tinite was one of the great triumphs of 19th-century mathematics,
[or 16th-c kerala mathematics?]
although its roots go back much earlier [area of circle  - archimedes'
argument from 3d c BC]

But what is area? is it something like the amount of stuff in the shape
(two-dimensional stuff, that is)?  but can one one make this
more precise?

area of curved shape - cannot measure precisely - e.g. will not match a grid,
however fine.   but if we say the area is 12, it is suff to show that one can
show it to be greater
than 11.999... (and less than 12.000..1 ) using a suitably fine grid.

Alternately: instead of trying to prove that it is 12, we can disprove that
it is anything else, however close, with a suitably fine grid.  This way we
don't need infinity.

archimedes' argument: cut into thin pie slices and lay out alternately to
form rectangle  with base ~= 1/2 circumference, and height ~= r.

as slices become smaller - rectangle becomes more precise -> area becomes pi.r²

5: Dimension

much of advanced mathematics is concerned with geometry in more than three
dimensions. e.g.  twenty-six-dimensional or even infinite-dimensional
geometry can be mathematically important.  .

High-dimensional geometry - another example of a concept that is best
understood from an abstract point of view.

Rather than worrying about the existence, or otherwise,
of twenty-six-dimensional space, let us think about its properties.

examples of properties:

1. it should take twenty-six numbers to specify a point, just as it takes
    two numbers in 2D and three in 3D
2. doubling the size of an shape in every dim --> multiplying it by 226

there is more to space than just a collection of single points. For example,
we talk about the distance between a pair of points, and about straight
lines, circles, and other geometrical shapes.

what are the counterparts for these in higher dimensions?

e.g. distances between points x and y in any dimensions - can be computed as
sqrt SUM_i (xi-yi)² - can relate this to 2D by repeatedly applying
pythagoras to a series of "rt-angled" triangles - x1:x2, then diag(x1,x2):x3,
and so on.

but this is really a convention -

	Actually, however, what we have done is define a notion of
	distance. No physical reality forces us to decide that
	five-dimensional distance should be calculated in this
	manner. However, this approach is so c1early the natural
	generalization of 2D and 3D that it would be strange to adopt any
	other definition.  75

now extends the idea of a circle to higher-D spheres (based on distance);
and the idea of a high-D cube --> num of edges in a 5-D cube?  each of the
2⁵ vertices has 5 edges --> hence 160/2 = 80.

	One way of summarizing what we are doing is to say that we are
	converting geometry into algebra, using coordinates to translate
	geometrical concepts into equivalent concepts that involve only
	relationships between numbers. Though we can't generalize the
	geometry, we can generalize the algebra, and it seems reasonable to
	call this higher-dimensional geometry.  [though not related] to our
	immediate experience, [can still be] useful as a model.

at the same time it is not quite impossible to "visualize" higher-D cubes.
E.g. though one has no clear picture of 4-D space, one can visualize a 4-D
cube as two 3-D ones, with edges joining all the vertices.  So one can
still 'see' that there 12 edges for each 3-D cube, and eight edges linking
their vertices  -> total 32. Further, one can then 'see' that a
5-D cube has two of these w vertices joined: total of 32 + 32 + 16 = 80
edges.

[TG uses the term "just see" for the number of edges in both 3-D and 5-D; i
am sure for most of us, this won't be the same "just see", though I agree,
that for the 3-D cube also, one has to mentally count of the 4 + 4 + 4 = 12
edges. ]

Sorne mathematicians specialize in four-dimensional geometry, and their
powers of four-dimensional visualization are highly developed.

This psychological point has an importance to mathematics that goes weIl
beyond geometry. One of the pleasures of devoting one's life to mathematical
research is that, as one gains in expertise, one finds that one can 'just
see' answers to more and more questions that might once have required an hour
or two of hard thought, and the questions do not have to be geometrical. 80

Fractional dimension

What properties might we expect to  hold in a fractional dimensional space?
When we multiply the extent of a shape by t in every direction, we expect its
volume to go up by t raised to the power of the dimension.

Q.  is there a shape for which we can reason as above
and obtain an answer that is not an integer? The answer is yes.

e.g. Koch snowflake - every segment is a copy of itself - each part has four
copies, each  shrinking the wholeshape by 1/3d.
if we triple it, then the "size" goes up by 4.  hence 3^d = 4; since 3¹ = 3,
the dim d is abt 1.262.

As with our other uses of the abstract method, this does not mean that we have
discovered the 'true dimension' of the Koch snowflake and similar exotic
shapes - but merely that we have found the only possible definition
consistent with certain properties.

In fact, there are other ways of defining
dimension that give different answers. For example, the Koch
snowflake has a 'topological dimension' of 1. Roughly speaking, this
is because, like a line, it can be broken into two disconnected parts
by removing  any interior segment.

abstraction and generalization:

to generalize a concept one should find sorne properties associated with it
and generalize those.  Sometimes there  may be a only one natural way to do
this, but sometimes different sets of properties lead to different
generalizations, of which more than one may be fruitful [interesting?]

6 Geometry

Euclid proposed five axioms for geometry:

1. Any two points can be joined by exactly one line segment.
2. Any line segment can be extended to exactly one Hne.
3. Given any point P and any length r, there is a circle of radius r with
   P as its centre.
4. Any two right angles are congruent.  [can be slided into the other]

5. [the famous 5th axiom]:

	If a straight line N intersects two straight lines L and M, and if
	the interior angles on one side of N add up to less than two right
	then the lines L and M intersect on that side of N.

-> equiv to the parallel postulate - only one parallel line to L can be drawn
   through point  x not on  L.



so convinced was Kant about the correctness of Euclid that he devoted a
signïficant part of his Critique of Pure Reasan to the question of how
one could be absolutely certain that Euclidean geometry was true.

thirty years later the great mathematician Carl Friedrich Gauss could
conceive of such a triangle, from three points on a sphere (the mountain
peaks of Hohenhagen, Inselberg, and Brocken in the kingdom of Hanover).
[but they would have been too close for his purposes]

the parallel postulate

has aroused the most suspiciion (or uneasiness) - many mathematicians before
tried to prove it from the other four - until the 19th c.  at that point,
other geometries started coming up.

if we have a geometry in which the first four axioms hold but not the fifth,
then we can say that the first four do not logically entail the fifth.

spherical geometry

clearly, ideas such as "line" will need to be interpreted differently on the
sphere.

	The idea, which is a profound example of the abstract method at work,
	is to reinterpret what is meant by a straight line, so that the
	surface of a sphere does contain 'straight lines after aIl.

a natural definition: a line segment from x to y is the shortest path from x
   to y that lies entirely within the surface of the sphere.  93

if we now take the equator as one line, then any great circle through any
other point x will cut this line at two (antipodal) points.   in fact, all
lines through x will cut the equator.

but this seems like cheating - if I define 'straight Hne' in a new way,
then it is not particularly surprising if the parallel postulate ceases
to hold.

but spherical geometry is not a good example, since the first four axioms
also do not hold:
axiom 3 fails: a sphere does not contain circles of arbitrarily large radius
axiom 1 fails: there are infinitely many shortest routes from the North to
	the South Pole 98

hyperbolic geometry

hyperbolic tiling: escher fish by sylvio levi
these are defined as a set of regular p-gons on
the poincare disk.
source: http://www.geom.uiuc.edu/graphics/pix/Special_Topics/Hyperbolic_Geometry/escher.html





Contents
Preface ix
List of diagrams xiii
1 Models 1
2 {#num|Numbers and abstraction] 17
3 Proofs 35
4  86
7 Estimates and approximations 112
8 Some frequently asked questions 126
Further reading 139
Index 141