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Transcript
```ANALYSIS
History &
Philosophy of
Calculus
WHAT IS ANALYSIS?
 Over the past few weeks we’ve begun to see some problems
with the calculus as it was developed by Newton and Leibniz.
 Though it seemed to give good results, it also seemed to use dubious
arguments to arrive at them.
 Analysis is the attempt to fix these problems and put calculus
on a firm footing.
 This means the ideas used in its arguments must be made precise
and clear, and shown to be logically consistent.
 Analysis is, in a sense, more philosophical than calculus.
 Note that in older texts (C17/18), “Analysis” is often used to
mean what we mean by “Calculus”. The distinction between
the two only arises in the C19/20.
CALCULUS VS ANALYSIS
Calculus
 Practical
 Methods(“how?”)
 Science & engineering
 Easy to learn
Analysis
 Theoretical
 Proofs (“why?”)
 “Pure” maths
 Hard to learn
AUGUSTIN-LOIS CAUCHY
 Like the Calculus itself, Analysis was arrived at by a number of
mathematicians over a long period.
 Cauchy was one of the key early figures in this process.
 He did a great deal of work in both physics and pure maths.
 In 1821 his book, the Cours d’Analyse, was published. It is the
textbook of a (reportedly very unpopular) course he taught at
that time at the École Polytechnique.
 In this book he sought to overturn what he called the assumption of the
“generality of algebra”, which had led C18 mathematicians to
manipulate infinite sequences using ordinary algebra, sometimes getting
absurd results.
 He replaced this with a rigorous notion called a “limit”, an idea already
used by Newton and others but never made explicitly clear.
 Limits allowed some later Analysts to reconstruct calculus without
referring to infinitesimal quantities.
 In the modern version (though not quite in Cauchy’s), infinitesimals are
completely expunged.
LIMITS
JOHN WALLIS’S INFINITE FRACTIONS
“Since, moreover, as the number of terms increases, that excess over one third
is continually decreased, in such a way that at length it becomes less than any
assignable quantity (as is clear); if one continues to infinity, it will vanish completely.”
-- John Wallis, The Arithmetic of Infinitesimals, Prop 20.
PARABOLA
AN INFINITE SUM…
∞
𝐴𝑟𝑒𝑎 = 𝐴 +
𝑖=1
1
4
𝑖
4
𝐴= 𝐴
3
Though the series of
powers of ¼ goes on
them we get another
series that seems to
converge towards the
total area of all the
white squares.
LIMITS
 Both of these arguments use the same general idea.
 They create an infinitely long sequence of values that gradually
gets closer and closer to some particular value.
 Since the sequence is always getting closer to that
value, we say that “in the limit” it becomes equal to
it.
 This is the definition of the limit – we don’t have to “go to
infinity” to complete the sequence in order to get there.
 Historically, the notion of limits took a long time to
crystallize and in C18 texts is often vague, like the
description above. Joseph-Lois Lagrange (17361813), among others, famously lamented the lack of
a clear definition of the term.
LIMITS
 We define a limit to be the value that a sequence tends
towards. For example:
1
lim = 0
𝑛→∞ 𝑛
 Our intuition here is that 1/n gets smaller and smaller – closer
and closer to 0 – as n gets bigger. We can make 1/n as close to
0 as we like just by making n very big.
 This does NOT commit us to saying something mathematically
silly like
1
=0
∞
 We can countenance a process that goes on as long as we like
(the potential infinite) but not an actual infinite quantity.
ε-δ DEFINITION OF A LIMIT
ε-δ DEFINITION OF A LIMIT
 The definition works like a game:
 I give you a sequence, s(n), and claim that its limit is L.
 You give me an “error”, ε, as a challenge.
 I can meet your challenge if I can give you a number, δ, such that for
every n > δ, |s(n) – L| < ε.
 |x| is the “absolute value” or “size” of x – when x is negative, it flips the
sign to positive.
 In other words, after a certain point in the sequence it is
always within the margin of error defined by ε. (That is, it
never leaves that small neighbourhood around L again).
 BUT you could give me any value of ε, no matter how small.
 We can keep playing this game with ε=0.1, ε=0.01, ε=0.001,
ε=0.0001 and so on, and I will always have to find a point in the
sequence where it never leaves that neighbourhood of L.
 If I can prove I can always do this, then L is indeed the limit of the
sequence s(n).
THE DERIVATIVE
 We can actually use the “converging secant lines” idea of the
tangent to define the derivative using limits.
 Define the following sequence:
𝑠 𝑛 =
𝑓 𝑥 + 1/𝑛 − 𝑓(𝑥)
1/𝑛
 Then we have:
𝑑𝑓
𝑓 𝑥 + 1/𝑛 − 𝑓 (𝑥)
= lim
𝑑𝑥 𝑛→∞
1/𝑛
 Notice that 1/n is always a “proper” number, with a size –
never an infinitesimal.
 The derivative is defined to be the value this sequence
converges towards.
The only way to “see” that this works is by calculating a
derivative, so with apologies…
Suppose f(x) = x 2 :
𝑑𝑓
𝑓 𝑥 + 1/𝑛 − 𝑓 (𝑥)
= lim
𝑑𝑥 𝑛→∞
1/𝑛
= lim
𝑛→∞
(𝒙 +
Definition of the derivative.
Replace f with what it actually
does.
𝟏/𝒏) 𝟐 −𝒙 𝟐
1/𝑛
Multiply out the bracket.
𝒙 𝟐 + 𝟐 𝟏/𝒏 𝒙 + (𝟏/𝒏) 𝟐 −𝑥 2
= lim
𝑛→∞
1/𝑛
2 1/𝑛 𝑥 + (1/𝑛)
= lim
𝑛→∞
1/𝑛
= lim 2𝑥 + 1/𝑛 = 2𝑥
𝑛→∞
2
The x2 and –x2 cancel each other
out.
The 1/n on top and bottom cancel
out. Now the limit is easy to find.
CAUCHY SEQUENCES
 Intuitively speaking, sequences that converge to a limit must
“bunch up together” around that limit in the long term. We
can use this to make a definition of convergence that doesn’t
need the idea of a limit.
 The definition is a bit unintuitive but it’s very useful in
practice.
A sequence is Cauchy if for
every error term e, we can find
an N such that, whenever n, m
> N, |s(n) – s(m)| < e.
In other words, “past N,
everything stays bunched up
in an e-sized neighbourhood”.
COMPLETENESS
 In any continuum, every sequence that converges to
a limit is Cauchy.
 In a “nice” continuum (technically, a “complete
metric space”), every Cauchy sequence converges to
a limit, and the limit is a point in the continuum.
So “convergent sequences” and “Cauchy sequences”
are the same objects, if your continuum is “nice” (i.e.,
complete).
HOW TO MAKE A NICE
CONTINUUM
DIFFERENT CONSTRUCTIONS OF THE
REAL NUMBERS
 Cauchy’s Cours still used infinitesimals, but tried to put them on
a more rigorous footing.
 In the C19 quite a few attempts made to use Cauchy’s ideas to
either get rid of infinitesimals or make them logically sound.
 Richard Dedekind saw Real numbers as arising from all the ways you
could “cut” the continuum (under suitable definitions, of course).
 Karl Weierstrass saw Real numbers as infinite decimals.
 All these start from the assumption that the continuous
geometric line is a coherent idea, and that it’s made up of points
that can be represented by numbers.
 All give rise to essentially the same version of the continuum by a variety
of methods.
 But some rest on more “fancy” arguments than others.
 The version we will use is the one first described by Charles
Méray in his Nouveau Précis d’Analyse Infinitesimale (1872).
 This is the standard one used in most (though not all) introductory
Analysis courses.
 It’s also easy to generalise beyond the Real numbers, unlike some of the
others.
 It’s also probably the “fanciest” of all.
 Usually we just use numbers as if they were all given to
us in a lump. But it helps to sort them out and be clear
about which ones we need for dif ferent jobs.
 The natural numbers, ℕ, are the basic counting
numbers.
 This is fine if you just want to add and multiply.
 The integers, ℤ, are the negative and positive whole
numbers, plus zero.
 This is handy if you want to be able to subtract.
 The rationals, ℚ, are the positive and negative
fractions, plus zero.
 Here we can also divide.
 The reals, ℝ, are the numbers you need to identify the
points in a continuous line. Question: are the reals and
the rationals the same numbers?
THE RATIONAL NUMBERS AREN’T “NICE”
 Our only model of a continuum so far has been the
rational numbers – the fractions a/b, where a and b
are whole numbers and b isn’t 0.
 But Cauchy sequences of rational numbers don’t
always converge to a rational number limit.
 It’s not hard to prove that 2 is not a rational number; it’s also
quite easy to produce a sequence whose limit ought to be 2.
 This can be repeated many times with square roots, cube roots,
numbers like pi and so on.
 This means the rational numbers are “incomplete” – there are
lots of “gaps” where there ought to be limits of Cauchy
sequences.
 We will now, by magic, construct a number system that is
complete.
THE ALGEBRA OF LIMITS
 It turns out that the following likely -seeming things are true of
Cauchy sequences:
lim 𝑠 𝑛 + 𝑡 𝑛
= lim 𝑠 𝑛 + lim 𝑡 𝑛
lim 𝑠 𝑛 − 𝑡 𝑛
= lim 𝑠 𝑛 − lim 𝑡 𝑛
lim 𝑠 𝑛 × 𝑡 𝑛
= lim 𝑠 𝑛 × lim 𝑡 𝑛
lim 𝑠 𝑛 ÷ 𝑡 𝑛
= lim 𝑠 𝑛 ÷ lim 𝑡 𝑛
𝑛→∞
𝑛→∞
𝑛→∞
𝑛→∞
lim
𝑛→∞
𝑛→∞
𝑛→∞
𝑛→∞
𝑛→∞
𝑠(𝑛) =
𝑛→∞
𝑛→∞
𝑛→∞
𝑛→∞
lim 𝑠(𝑛)
𝑛→∞
 The things on the left are ordinary numbers – limits of Cauchy
sequences. The things on the right are arithmetic operations
on the limits of other Cauchy sequences.
 This means we can “do arithmetic with Cauchy sequences” –
they “look like a number system”.
THE REAL NUMBERS
 We now define a real number to be a Cauchy sequence of
rational numbers.
 (Actually, this is a bit like defining a rational number to be a/b. We want
1/4 to be the same number as 2/8, so we have to do a bit more work to
“group equivalent fractions together”. In the same way we want two
Cauchy sequences that converge to the same limit to represent the same
real number. But this is a detail.)
 This new model of the continuum has no “gaps” because every
Cauchy sequence converges to the number that it represents !
 Is this cheating?
 Number systems (except maybe the “natural numbers”) don’t drop out of
the sky; we must construct them.
 Any continuum worth the name should contain all the limits of its Cauchy
sequences. We just constructed it in a way that guaranteed that.
 And remember, the limits of Cauchy sequences obey the normal laws of
arithmetic, so although its elements look a bit weird, this new continuum
behaves just the way we expect it to.
```