\documentclass[a4paper,notitlepage]{slides}
\usepackage{amsmath,amssymb,url}
\usepackage[mathscr]{euscript}
\usepackage{hhline}
\newcommand{\comp}{^{\mathrm{c}}} % set-theoretic complement
\newcommand{\pow}[1]{\mathscr{P}(#1)} % power set
\newcommand{\str}[1]{\mathfrak{#1}}
\newcommand{\lang}{\mathcal{L}} % a language or signature
\title{Mathematics as Philosophy}
\date{2003, October 27}
\author{David Pierce}
\newcommand{\hang}{\hspace{-1em}}
\newcommand{\gk}[1]{\mathsf{#1}}
\newcommand{\blakeskip}{\vspace{0.25\baselineskip}}
\newcommand{\Def}[2]{\mathscr D^{#1}(#2)}
\newcommand{\bDef}[1]{\mathscr D^{#1}}
\newcommand{\Fun}[1]{F(#1)}
\raggedright
\begin{document}
\begin{slide}
\maketitle
\begin{center}
\url{http://www.math.metu.edu.tr/~dpierce/talks/}
$\gk A\Gamma \gk E\Omega \gk{METPHTO}\Sigma\ \gk{MH\
EI}\Sigma\gk{IT}\Omega$ \\
(Geometri yapamayan girmesin):
Motto of the American Mathematical Society, presumably based on
Platonic tradition
Mathematics is justified by its service to philosophy!
\end{center}
\end{slide}
\begin{slide}
\begin{center}
FACIO LIBEROS EX LIBERIS LIBRIS LIBRAQUE\\
(Teraziyle ve kitaplarla \c cocuklardan \"ozg\"ur~insanlar yapar\i m):
Motto of St John's College, Annapolis, Maryland, and Santa Fe, New
Mexico, USA.
``The
following teachers will return to St John's College next year:
``Homer, \AE schylus, Herodotus, Plato, Sophocles, Aristophanes,
Thucydides, Aristotle, Euripides, Lucretius, the Bible, Plutarch,
Virgil, Tacitus, Epictetus, Plotinus, Augustine, Anselm, Aquinas,
Dante, Chaucer, Shakespeare, Machiavelli, Montaigne, Bacon, Descartes,
Cervantes, Pascal, Milton, Hobbes, Locke, Rousseau, Swift, Leibniz,
Berkeley, Hume, Kant, Wordsworth, Austen, Smith, Twain, Tolstoy,
Goethe, Hegel, Tocqueville, Kierkegaard, Dostoevski, Marx, Nietzsche,
Freud\dots''
\end{center}
\end{slide}
\begin{slide}
\begin{center}
\begin{tabular}{l||ll||l|l|l|l|l|l||l|l||l}
\multicolumn{12}{c}{\ }\\
\hhline{~|t:========:t|~~~}
&\multicolumn{8}{l||}{Science} & \multicolumn{3}{l}{}\\
\hhline{~||~~~~~~~~||~~~}
&\multicolumn{8}{l||}{} & \multicolumn{3}{l}{}\\
\hhline{~||~~|t:======#==:t|~}
&&&\multicolumn{6}{l||}{} & \multicolumn{2}{l||}{} &\\
\hhline{~||~~||~----~||~~||~}
&&&&\multicolumn{4}{l|}{Mathematics}&&\multicolumn{2}{l||}{}&\\
\hhline{~||~~||~|~-----~||~}
&&&&&\multicolumn{3}{l|}{}&&&&\\
\hhline{~||~~||~|~|~-~|~||~|~||~}
&&&&&&Model-theory&&&&&\\
\hhline{~||~~||~|~|~-~|~||~|~||~}
&&&&&\multicolumn{3}{l|}{}&&&&\\
\hhline{~||~~||~----~||~|~||~}
&&&\multicolumn{2}{l|}{}&\multicolumn{4}{l||}{}&&&\\
\hhline{~|b:==#==|====:b|~|~||~}
\multicolumn{3}{l||}{}&\multicolumn{2}{l|}{}&
\multicolumn{5}{l|}{Logic}&& \\
\hhline{~~~||~~-----~||~}
\multicolumn{3}{l||}{}&\multicolumn{8}{r||}{Philosophy}&\\
\hhline{~~~|b:========:b|~}
\multicolumn{12}{c}{}\\
\end{tabular}
Mathematical truths are:
like philosophical truths, \textbf{personal}: not needing verification by
experiment or by the agreement of a multitude; but,
like scientific truths, \textbf{universal}: one expects
them to be agreed on by all who take the trouble to understand
them---and the agreement generally happens.
\end{center}
Unwilling math-student's complaint: ``There's only one right answer!''
\end{slide}
\begin{slide}
\hang
In \textbf{model-theory}, a model is a \textbf{structure}, considered
with respect to some \textbf{theory} or \emph{theories} of which it is a
model.
\hang
(Likewise a person is a son or daughter with respect to his or her
parents.)
\hang
Model-theory is:
mathematics done with an awareness of the language with which one
does it;
the study of the construction and classification of structures within
specified classes of structures (Wilfred Hodges, \emph{Model Theory});
algebraic geometry without fields (Hodges, \emph{A Shorter Model
Theory});
the geography of \textbf{tame mathematics} (Lou van den Dries);
the study of structures \emph{qu\^a} models of theories.
\end{slide}
\begin{slide}
\hang
\textbf{Bourbaki} on the role of logic:
``In other words, logic, so far as we mathematicians are concerned, is
no more and no less than the grammar of the language which we use,
a language which had to exist before the grammar could be
constructed\dots
\hang
``The primary task of the logician is thus the analysis of the body of
existing mathematical texts, particularly of those which by common
agreement are regarded as the most correct ones, or, as one
formerly used to say, the most `rigorous.'{}'' (``Foundations of
Mathematics for the Working Mathematician'',
\emph{Journal of Symbolic Logic} \textbf{14}, 1949)
\begin{center}
\textbf{Is mathematics one?}
Some possible divisions:
pure/applied
geometry/algebra/arithmetic/analysis/\dots
\end{center}
\end{slide}
\begin{slide}
\begin{center}
\textbf{Euclid} gives
all mathematical facts in geometric terms, even those facts
that we should call arithmetic or algebraic
\textbf{Descartes:} Euclid must have used non-geometrical means to
discover some of those facts
Euclid, \emph{Elements} II.4: ``If a straight line be cut at random,
the square on the whole is equal to the squares on the segments and
twice the rectangle contained by the segments.''
Descartes, \emph{Geometry:} \emph{By geometric means,} the squares
and rectangles can be
replaced by straight lines, once we have chosen a straight line to
represent unity.
Thus, all arithmetic operations can be referred to
geometry---though there may be no practical need to do so.
Old-style geometry is not abandoned: see Lobachevski.
\end{center}
\end{slide}
\begin{slide}
\begin{center}
A possible analogy:
Algebra, geometry \&c. are to the mathematical world as the senses
are to the physical world.
We have a \textbf{common sense} whereby the things that we see, hear and touch
are known to be part of one world.
The visible, audible and tangible worlds are the same in principle,
but it would be foolish to treat the world as \emph{simply} one of these.
We might even hope to develop additional senses, the better to
understand the world:
\end{center}
\end{slide}
\begin{slide}
\begin{center}
THERE is NO NATURAL RELIGION (William Blake, c.~1788):
\end{center}
\hang
The Argument: Man has no notion of moral fitness but from
Education. Naturally he is only a natural organ subject to
Sense.\\
\blakeskip
\hang
I Man cannot naturally Percieve, but through his natural or
bodily organs\\
\blakeskip
\hang
II Man by his reasoning power. can only compare \&\ judge of
what he has already perciev'd.\\
\blakeskip
\hang
III \emph{From a perception of only 3 senses or 3 elements none
could deduce a fourth or fifth}\\
\blakeskip
\hang
IV None could have other than natural or organic thoughts if
he had none but organic perceptions\\
\blakeskip
\hang
V Mans desires are limited by his perceptions. none can desire
what he has not perciev'd\\
\blakeskip
\hang
VI The desires \&\ perceptions of man untaught by any thing but
organs of sense, must be limited to objects of sense.
\end{slide}
\begin{slide}
\hang
``The essential business of \textbf{language} is to assert or deny facts.''
(Bertrand Russell, introduction to Wittgenstein's
\emph{Tractatus}, 1922)
\hang
``\emph{Bodily actions expressing certain emotions,} in so far as they come
under our control and are conceived by us, in our awareness of
controlling them, as our way of expressing these emotions, \emph{are
language}\dots The grammatical and logical articulations of
intellectualized language are no more fundamental to language as such
than the articulations of bone and limb are fundamental to living
tissue'' (R.G. Collingwood, \emph{The Principles of Art}, 1937).
\end{slide}
\begin{slide}
\hang
What \emph{is} an assertion of a fact?
\hang
Say it is an utterance, in a certain manner, of a \emph{statement} (or
\emph{proposition}).
\hang
What is a statement?
\hang
Some authors say it is a sentence that is either true or false.
\hang
I think a statement is a sentence distinguished from other sentences
by its \emph{form}: it is not a question, but its verb is declarative,
not subjunctive or imperative, \dots
\hang
Then a statement
\emph{becomes} true or false when placed in an appropriate
\emph{context}.
\hang
A true statement (in context) need not be a \emph{correct} answer to
the question it is intended to answer.
\end{slide}
\begin{slide}
\hang
Bourbaki: What are the \emph{structures} at the heart of mathematics?
They can be:
\hang
\emph{algebraic,} like:\\
the \emph{fields} $(\mathbb R,+,\times)$ and $(\mathbb C,+,\times)$,\\
the \emph{vector-spaces} $\mathbb R^n$ and $\mathbb C^n$,\\
the \emph{group} of rigid motions of the Euclidean plane \dots;
\hang
\emph{ordered:---}\\
totally, like $(\mathbb Z,\leqslant)$ and $(\mathbb R,\leqslant)$, or\\
partially, like $(\pow{\Omega},\subseteq)$, \dots;
\hang
\emph{topological:} the real number $x$ is \emph{in the interior of}
the set $A$ if: for some positive $\varepsilon$,\\ if
$\left|y-x\right|<\varepsilon$, then $y$ is in $A$; symbolically,\\
$$\exists\varepsilon>0\;\forall
y\;(\left|y-x\right|<\varepsilon\Rightarrow y\in A)$$
\hang
Bourbaki aims to characterize structures by means of \emph{axioms.}
\hang
\textbf{Model-theory} treats directly of the first two kinds of structures and
examines how axioms \emph{fail} to characterize any one structure.
\end{slide}
\begin{slide}
\hang
Let
$\omega$ be the set of \textbf{natural numbers:} the smallest of sets
$\Omega$ of sets such that $\varnothing\in\Omega$ and,
for all sets $A$, if $A\in\Omega$, then $A\cup\{A\}\in\Omega$. Let
$j$ and $n$ range over $\omega$.
\hang
We write $0$ for $\emptyset$, and $1$ for $\{0\}$, and so forth;
also $n+1$ for $n\cup\{n\}$. If $j\subset n$,
we write $j