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Tin Lok Wong (uploaded by LianTze Lim)

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Abstract:

A thesis template for submission to the University of Birmingham, written by Tin Lok Wong (2010) and recommended by UoB Mathematics Postgraduate Society.

For those who would like a cover with the university crest, the optional `\makecrestcover`

command is provided.

Tags:

Author:

Tin Lok Wong (uploaded by LianTze Lim)

License:

Creative Commons CC BY 4.0
^{(?)}

Abstract:

A thesis template for submission to the University of Birmingham, written by Tin Lok Wong (2010) and recommended by UoB Mathematics Postgraduate Society.

` ````
\documentclass{bhamthesis}
\title{Annotated quotations:\\
an example using \clsname}
\author{Tin Lok Wong}
\date{December~2009} %% Version 2009/12/26
\usepackage{amsthm}
\usepackage{amssymb}
\usepackage{graphicx}
\newtheorem*{thm}{Theorem}
\theoremstyle{definition}
\newtheorem*{defn}{Definition}
\newcommand{\mar}[1]{\marginpar{\raggedright#1}}
\newcommand{\clsname}{\textsf{bhamthesis}}
\newcommand{\bktitle}[1]{\textit{#1}}
\newcommand{\ZF}{\mathrm{ZF}}
\newcommand{\IN}{\mathbb{N}}
\makeatletter
\newcommand{\makecrestcover}{%
\begin{titlepage}
\centering\singlespacing
\vspace*{1cm}
{\huge\bfseries University of Birmingham\par}
\vspace*{2cm}
\includegraphics[width=.5\textwidth]{crest}\par
\vspace*{\stretch{1}}
{\Huge\bfseries
\@author\par
\vspace{1cm}
\@title\par}
\vspace*{\stretch{1}}
{\Large\@date\par}
\end{titlepage}
}
\makeatother
\prefixappendix
\begin{document}
\frontmatter
%% Optional/alternative cover with crest
% \makecrestcover
\maketitle
\begin{abstract}
This document only aims to demonstrate how one can use \clsname,
a \LaTeX\ class written by me. Please see the source file for
more comments. Instead of making something up myself, I quote
several pieces that may be of interest to mathematicians, and
include some of my comments. \emph{The contents of this document
should not be treated seriously.} In particular, this document
is not meant to be a thesis intended for submission.
\end{abstract}
\tableofcontents
\mainmatter
\chapter{Thesis writing}
\section{Pictures}
It is not surprising that the English proverb `a picture is worth
a thousand words' has analogues in many languages, including
Chinese and Japanese. Pictures are a very useful tool in
explaining mathematics. The following is from Hardy's \bktitle{A
Mathematician's Apology}~\cite[\S23]{book:math-apol}.
\begin{quotation}
Let us suppose that I am giving a lecture on some system of
geometry, such as ordinary Euclidean geometry, and that I draw
figures on the blackboard to stimulate the imagination of my
audience, rough drawings of straight lines or circles or
ellipses. It is plain, first, that the truth of the theorems
which I prove is in no way affected by the quality of my
drawings. Their function is merely to bring home my meaning
to my hearers, and, if I can do that, there would be no gain in
having them redrawn by the most skillful draughtsman. They are
pedagogical illustrations, not part of the real subject-matter
of the lecture.
\end{quotation}
\TeX\ and \LaTeX\ are not very good at drawing pictures. However,
it may still worth the effort to include some more important
diagrams.
\section{The mathematical experience}
In the following excerpt from \bktitle{The Mathematical
Experience}~\cite[pp.~34--37]{book:math-exp}, Davis and Hersh
described how the ideal mathematician writes.
\begin{quotation}
To talk about the ideal mathematician at all, we must have a name
for his ``field,'' his subject. Let's call it, for instance,
``non-Riemannian hypersquares.'' [\ldots]
To his fellow experts, [the ideal mathematician] communicates
[his] results in a casual shorthand. ``If you apply a tangential
mollifier to the left quasi-martingale, you can get an estimate
better than quadratic, so the convergence in the Bergstein
theorem turns out to be of the same order as the degree of
approximation in the Steinberg theorem.''
This breezy style is not to be found in his published writings.
There he piles up formalism on top of formalism. Three pages of
definitions are followed by seven lemmas and, finally, a theorem
whose hypotheses take half a page to state, while its proof
reduces essentially to ``Apply Lemmas 1--7 to definitions
A--H.''
His writing follows an unbreakable convention: to conceal any
sign that the author or the intended reader is a human being. It
gives the impression that, from the stated definitions, the
desired results follow infallibly by a purely mechanical
procedure. In fact, [to] read his proofs, one must be privy to a
whole subculture of motivations, standard arguments and examples,
habits of thought and agreed-upon modes of reasoning. The
intended readers (all twelve of them) can decode the formal
presentation, detect the new idea hidden in lemma~4, ignore the
routine and uninteresting calculations of lemmas~1, 2, 3, 5, 6,
7, and see what the author is doing and why he does it. But for
the uninitiate, this is a cipher that will never yield its
secret. If (heaven forbid) the fraternity of non-Riemannian
hypersquarers should ever die out, our hero's writings would
become less translatable than those of the Maya.
\end{quotation}
Perhaps mathematical writing, and especially thesis writing,
should not be this cold? Why should one take away \emph{all} the
human elements from his/her three years of mathematical
experience?
\section{Respectable mathematics}
How much is your thesis supervisor involved in your thesis
writing? Crilly and Johnson tell the story of Brouwer in their
chapter in \bktitle{History of
Topology}~\cite[Section~7]{incoll:emerg-topodim}.
\begin{quotation}
[Brouwer's] doctoral thesis of 1907, \textit{On the Foundations
of Mathematics} (\textit{Over de Grondslagen der Wiskunde}),
marked the real beginning of his mathematical career. The work
revealed the twin interests in mathematics that dominated his
entire career: his fundamental concern with critically assessing
the foundations of mathematics, which led to his creation of
Intuitionism, and his deep interest in geometry, which led to his
seminal work in topology [\ldots]. Brouwer quickly found that
his philosophical ideas sparked controversy. D.J.~Korteweg
(1848--1941), his thesis supervisor, had not been pleased with
the more philosophical aspects of the thesis and had even
demanded that several parts of the original draft be cut from the
final presentation [\ldots]. Korteweg urged Brouwer to
concentrate on more ``respectable'' mathematics, so that the
young man might enhance his mathematical reputation and thus
secure an academic career.
\end{quotation}
You may get the same `advice' from your supervisor, 90~years after
Brouwer's days, if you try to put some philosophy into your thesis
(and if your supervisor reads it). I do not think one should be
discouraged for putting his/her ideas into the thesis, as long as
the majority of the work is still mathematics. (This is only my
personal opinion.) Given Brouwer's strong personality, one may
expect him to have insisted on what he wanted to do. For some
reason, he did not. As Crilly and
Johnson~\cite{incoll:emerg-topodim} write:
\begin{quotation}
Brouwer was fiercely independent and did not follow in anybody's
footsteps, but he apparently took his teacher's advice and set
out to solve some really hard problems of mathematics. Brouwer
put in a prodigious effort in these early years and rapidly
produced a flood of papers on continuous group theory and
topology --- more than forty major papers in less than five
years [\ldots].
\end{quotation}
Perhaps I would only have a choice if I were as good as Brouwer.
By the way, the `rejected parts' of Brouwer's thesis is now
published~\cite{art:brouwer-rej}. You can even see the big
crosses his supervisor drew on his drafts.
\chapter{The nature of mathematics}
\section{Arithmetical splitting}
I was recently in a conversation with Andrey Bovykin, a former
graduate student of the University of Birmingham. He told me
about the idea of \emph{arithmetic splitting}, which essentially
says that there is no `absolute truth' about the natural numbers.
Pettigrew~\cite[pp.~19--20]{unpub:uniqueIN} explains it in a
better way.
\begin{quotation}
This list of facts [\ldots] gives a glimpse of the varied zoology
of natural number structures that it supports. This, I propose,
should be our foundation for arithmetic. [\ldots]
Of course, at first sight this proposal will seem extremely
radical. It will seem that it entails changing much
number-theoretic practice. Most importantly, if this foundation
were adopted, each of our number-theoretic statements would have
to be relativized to a particular collection of natural number
structures. For instance, if I were to say that there are
infinitely many primes, I would have to say in which natural
number structures I take this to holds. Does it hold only in
those closed under exponentiation? Or also in weaker structures?
Thus, on my proposal, arithmetic would come to resemble branches
of algebra such as group theory, in which we prove theorems that
hold of all Abelian groups, for instance, or all cyclic groups;
or field theory, in which sometimes we prove theorems that hold
of all finite fields. [\ldots]
But perhaps [this] feature is [not] as revolutionary as it seems
at first. [\ldots] Much current research in mathematical logic
concerns the strength of the number-theoretic assumptions
required to prove certain propositions. For instance, in the
case of Euclid's theorem that there are infinitely many primes,
it has recently been shown that this holds in more natural number
structures than previously thought.[\ldots] This result, along
with many, many similar to it, belongs to the research project
known as \emph{Weak} or \emph{Bounded Arithmetic}, which studies
the deductive power of a certain sort of fragment of first-order
arithmetic.[\ldots] In a similar vein, the research project of
\emph{Reverse Mathematics}, inaugurated by Harvey Friedman and
carried on by Stephen~G. Simpson, aims to identify, for a given
proposition, the weakest fragment of second-order arithmetic in
which that proposition may be proved.[\ldots] Both research
projects occupy central positions in contemporary research in
arithmetic (more usually called \emph{number theory} by
mathematicians). So, it seems that a foundation for arithmetic
in which \textsc{Uniqueness} fails might be more appropriate to
the concerns of contemporary arithmetic than the traditional
foundation, which guarantees \textsc{Uniqueness}.
Furthermore, arithmetic belongs to that part of mathematics that
is used to model phenomena in other disciplines. And, in some of
these disciplines, it may well help to be able to choose between
natural number structures with different properties. For
instance, consider the notion of feasibility in computer science.
We might well wish to say that the class of natural numbers that
measure the feasible computations is closed under successor, but
not under exponentiation [\ldots]. And, if so, it will be
difficult to model the notion of feasibility in arithmetic with
the traditional foundation. But it is straightforward on the
foundation proposed in this paper since there are natural number
structures closed under successor but not under exponent[i]ation.
Such a proposal requires more work than I have space to carry out
here, but it suggests that there may be advantages to the
foundation described and advocated in the previous section.
[\ldots]
I conclude that mathematics took a wrong turning when it accepted
the uniqueness of natural number structures and built into its
foundations presuppositions that guarantee that uniqueness. Had
it not taken that wrong turning, the orthodox foundation for
arithmetic might have been $\mathsf{BST}^{\sf Bnd}_2$, which
permits many different sorts of natural number structure:
structures closed under addition and structures closed under
multiplication, structures shorter than a structure that is
closed under exponentiation, and so on. Had this been the
foundation for arithmetic, we might have begun the study of
so-called Weak Arithmetic and Reverse Mathematics much earlier
than we did. I propose we rectify our error now.
\end{quotation}
From my education and experience, I am convinced that the set of
natural numbers exists objectively. Therefore, in Pettigrew
terminology, there is only one true `natural number structure' for
me. Set theory is different. We thought we knew a lot about
sets, and we thought we could `visualize' the universe of sets.
Then it came Russell's Paradox, and everyone, including me, gave
up (some of) our intuitions and work with formal axioms instead.
Therefore, I do not believe there is an objectively existing
universe of sets, just as I do not believe set theory is
consistent.\footnote{I suppose it is now common consensus amongst
set theorists that there is a variety of set theoretic universes,
none of which is superior to the others. There are (only) one or
two exceptions. For example, Woodin thinks
$2^{\aleph_0}=\aleph_2$ with strong reasons.}\mar{This is a widow,
probably caused by the footnote.} %% This is a marginal note.
I tried to think as Bovykin and Pettigrew suggest, but I still
cannot convince myself. Here is one of the arguments Bovykin told
me: you were convinced that natural numbers exist in exactly the
same way as you were convinced `the' universe of sets exists; why
should you believe in the first argument but not the second? There
is also some objective evidence supporting arithmetical splitting,
e.g., G\"odel's Incompleteness Theorems (depending on how you view
it), but \emph{there has to be one and only one $\IN$!} Perhaps I
just have a psychological barrier?
On the other hand, I would never call what is commonly known as
arithmetic `number theory', nor would I call number theory
`arithmetic'. They are different.
Logicians may be biased. What do other mathematicians think?
\section{Topological spaces}
\begin{defn}
A \emph{topological space} is a pair $(X,\mathcal{T})$ where $X$
is a nonempty set and $\mathcal{T}$ is a collection of subsets of
that contains the empty set and $X$, and is closed under finite
intersections and arbitrary unions.
\end{defn}
As mathematicians, we are familiar with the idea of a topological
space. The notion of topology abstracts the idea of open sets in
a metric space, and the axioms for a topological space come from
the properties of open sets. I remember having some difficulties
accepting the definition of a topological space in my second year.
Why are the open sets a good candidate for abstracting the notion
of continuity? What do they model? If they model open sets in a
metric space, then why should we allow topological spaces that are
not Hausdorff? As Vickers said~\cite{unpub:vickers/dptsem},
mathematicians sometimes choose to work with a notion just because
it works. Perhaps mathematicians should look for better
motivations. In the particular case of topological spaces, you
may find the answers to my questions in Vicker's
book~\cite{book:topologic}.\mar{This is another awkward looking
widow.}
\section{Set theory as \emph{the} foundation}
I recently participated in a conversation between several
mathematicians, most of which are, strictly speaking, not
logicians. One of them, who happens to know quite a lot of logic,
insisted that \emph{all} mathematicians should know enough set
theory to distinguish whether they are working with sets, (proper)
classes, or families of class, etc.\ to avoid running into set
theoretic paradoxes.
I disagree with him. As a logician, I like set theory being
popularized too, but mathematics \emph{is} not set theory. Conway
devoted a whole appendix in his \bktitle{On Numbers and
Games}~\cite{book:numgames} to this topic. Here is an excerpt on
pages~65--67.
\begin{quotation}
In this simpler formalisation, a number is still a pretty
complicated thing, namely a certain function in $\ZF$, which is
of course a certain set of Kuratowskian ordered pairs. The first
members of these ordered pairs will be ordinals in the sense of
von Neumann, and the second members chosen from the particular
two-element set we take to represent $\{{+},{-}\}$.
The curiously complicated nature of these constructions tells us
more about the nature of formalisations within $\ZF$ than about
our system of numbers, and it is partly for this reason that we
did not present any such formalised theory in this book. But the
main reason was that we regard it as almost self-evident that our
theory is as consistent as $\ZF$, and that formalisation in $\ZF$
destroys a lot of its symmetry. [\ldots]
It seems to us, however, that mathematics has now reached the
stage where formalisation within some particular axiomatic set
theory is irrelevant, even for foundational studies. It should be
possible to specify conditions on a mathematical theory which
would suffice for embeddability within $\ZF$ (supplemented by
additional axioms of infinity if necessary), but which do not
otherwise restrict the possible constructions in that theory. Of
course the conditions would apply to $\ZF$ itself, and to other
possible theories that have been proposed as suitable foundations
for mathematics (certain theories of categories, etc.), but would
not restrict us to any particular theory. This appendix is in
fact a cry for a Mathematicians' Liberation Movement!
Among the permissible kinds of construction we should have:
\begin{enumerate}
\renewcommand{\theenumi}{\roman{enumi}} %% Changing the way things look.
\renewcommand{\labelenumi}{(\theenumi)}
\item Objects may be created from earlier objects in any
reasonably constructive fashion.
\item Equality among the created objects can be any desired
equivalence relation.
\end{enumerate}
In particular, set theory would be such a theory, sets being
constructed from earlier ones by processes corresponding to the
usual axioms, and the equality relation being that of having the
same members. But we could also, for instance, freely create a
new object $(x,y)$ and call it the ordered pair of $x$ and $y$.
We could also create an ordered pair $[x,y]$ different from
$(x,y)$ but co-existing with it, and neither of these need have
any relation to the set $\{\{x\},\{x,y\}\}$. If instead we
wanted to make $(x,y)$ into an unordered pair, we could define
equality by means of the equivalence relation $(x,y) = (z,t)$ if
and only if $x = z$, $y = t$ \textit{or} $x = t$, $y = z$. I
hope it is clear that this proposal is not of any particular
theory as an alternative to $\ZF$ (such as a theory of
categories, or of the numbers or games considered in this book).
What is proposed is instead that we give ourselves the freedom to
create arbitrary mathematical theories of these kinds, but prove
a metatheorem which ensures once and for all that any such theory
could be formalised in terms of any of the standard foundational
theories. The situation is analogous to the theory of vector
spaces. Once upon a time these were collections of $n$-tuples of
numbers, and the interesting theorems were those that remained
invariant under linear transformations of these numbers. Now
even the initial definitions are invariant, and vector spaces are
defined by axioms rather than as particular objects. However, it
is proved that every vector space has a base, so that the new
theory is much the same as the old. But now no particular base
is distinguished, and usually arguments which use particular
bases are cumbrous and inelegant compared to arguments directly
in terms of the axioms.
We believe that mathematics itself can be founded in an invariant
way, which would be equivalent to, but would not involve,
formalisation within some theory like $\ZF$. No particular
axiomatic theory like $\ZF$ would be needed, and indeed attempts
to force arbitrary theories into a single formal strait-jacket
will probably continue to produce unnecessarily cumbrous and
inelegant contortions.
For those who doubt the possibility of such a programme, it might
be worthwhile to note that certainly principles (i) and (ii) of
our Mathematicians' Lib movement can be expressed directly in
terms of the predicate calculus without any mention of sets (for
instance), and it can be shown that any theory satisfying the
corresponding restrictions can be formalised in $\ZF$ together
with sufficiently many axioms of infinity.
Finally, we note that we have adopted the modern habit of
identifying $\ZF$ (which properly has only sets) with the
equiconsistent theory $\rm NBG$ (which has proper Classes as
well) in this appendix and elsewhere. The classification of
objects as Big and small is not peculiar to this theory, but
appears in many foundational theories, and also in our formalised
versions of principles (i) and (ii).
\end{quotation}
Formalization is probably still quite important to logicians
though.
\appendix
\chapter*{Two}
The following is a famous quote from
Knuth~\cite[Section~3.1]{book:ArtProgram}.
\begin{quotation}
In a sense there is no such thing as a random number; for
example, is 2 a random number?
\end{quotation}
I strongly believe that \emph{two is not a random number}. There
are many ways to see this. For example, two is the first number
that allows \emph{branching}. So while unary trees are not so
interesting and ternary trees are too complicated, binary trees
are just good.
More importantly, if one is interested in the relationships
between objects, then one naturally considers \emph{binary}
relations to start with. We can draw the diagram
\[ {\circ} \stackrel{R}{\longrightarrow} {\circ} \]
to mean the left object is in relation $R$ with the right object.
This leads us to the study of \emph{arrows}, or in technical
terms, \emph{categories}. Arrows have two ends. So they have a
special kind of symmetry: one can flip an arrow by changing its
head to its tail, and its tail to its head. When you do this
\emph{twice}, you get back to where you started. Such kinds of
symmetries are called \emph{involutions}, and this phenomenon is
known as \emph{duality}. In conclusion, if mathematics is about
relationships between objects, then \emph{two} must be a very
important number.
If the previous paragraph does not look convincing to you, then
you may find the following well-known theorem easier to take in.
\begin{thm}[Folklore]
All natural numbers are interesting.
\end{thm}
\begin{proof}
Suppose not. Then there is a natural number that is not
interesting. Since the natural numbers are well-ordered, we can
find a smallest such number $n$. Nothing can be more interesting
than being the smallest uninteresting natural number. Therefore,
$n$ is both interesting and uninteresting, which is a
contradiction.
\end{proof}
\backmatter
\bibliographystyle{plain}
\bibliography{bibeg}
\end{document}
```

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