\documentclass[a4paper,notitlepage]{slides}
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\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}}

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\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 $j<n$.  Then $n=\{j:j<n\}$.

\hang
Let $I$ be a finite subset of $\omega$.  The \textbf{Cartesian power}
$M^I$ is 
the set of functions from $I$ to $M$; a typical element $a$ of $M^I$
can be written $(a_j:j\in I)$.

\hang
Since $0$ is empty, $M^0$ consists of the empty function, $0$; so
$M^0=\{0\}=1$. 

\hang
$M^n$ consists of $(a_0,\dots,a_{n-1})$, where $a_j\in M$.

\hang
An \textbf{$I$-ary operation} on $M$ is a function from $M^I$ to $M$.

\hang
Atn
\textbf{$I$-ary relation} is a subset of $M^I$.  

\hang
A \textbf{structure}
on $M$ is a set of operations and relations on $M$, each of them
$n$-ary for some $n$ in $\omega$.
  \end{slide}

  \begin{slide}
\hang
    In the background are three ``primordial'' structures:

$\bullet$ the structure of the \textbf{natural numbers:}
$$(\omega,{}',0),$$
 where ${}'$ is the unary operation $x\mapsto x+1$; 

$\bullet$ the \textbf{Boolean algebra} of subsets of a ``universal'' set
      $U$:
$$(\pow U,\cap,{}^{\mathrm
      c},\cup,\varnothing,U,\subseteq);$$

$\bullet$ \textbf{propositional logic:} 
$$(\mathbb B,\land,\lnot,\lor,0,1,\Rightarrow),$$
where $\mathbb B=\{0,1\}=\pow{\{\varnothing\}}$ and can be understood as
$\{\text{false},\text{true}\}$. 
  \end{slide}

  \begin{slide}
\hang
    A structure on $M$ may be denoted $\str M$.  This has a
    \textbf{signature}: a set $\lang$ of symbols for the specified
    operations and relations.  

\hang
The symbols in $\lang$ are primary; say they are
    $f$ and $R$, symbolizing $n(f)$-ary operations $f^{\str M}$ and
    $n(R)$-ary relations $R^{\str M}$ on $M$.

\hang
Any element $a$ of $M$ can be understood as a nullary
operation-symbol: a \textbf{constant-symbol}.
For any subset $A$ of $M$, for the signature $\lang\cup A$ there is a
first-order language, $\lang_{\omega\omega}(A)$.

\hang
For an $I$-ary formula $\phi$ of $\lang_{\omega\omega}(A)$, there is an
$I$-ary relation
$\phi^{\str M}$ on $M$.  

\hang
Such relations are the \textbf{$A$-definable sets}
of $\str M$; they compose Boolean sub-algebras $\Def IA$
of the $\pow{M^I}$. 
  \end{slide}


  \begin{slide}    

\hspace{-10mm}
\begin{tabular}{|r|l|}\hline
SYMBOL & INTERPRETATION\\
SYNTAX & SEMANTICS\\ 
\hline\hline\hline
{signature} $\lang$ & $\str M$, an {$\lang$-structure}\\
\hline\hline  
& OPERATIONS ON $M$: \\ \hline
%& \textsc{basic operations:}\\ 
\textbf{variable} $x_j$ & $ a\mapsto a_j:M^I\to M$\\ 
\textbf{constant-symb.} $c$ & $c^{\str M}$, an element of $M$ \\ 
\textbf{function-symb.} $f$ & $f^{\str M}:M^{n(f)}\to M$\\ %\hline
{term} $t$ & $t^{\str M}$, a composition \\
\hline\hline
LOGICAL: & FUNCTIONS ON  $\pow{M^I}$\\ \hline
\textbf{connectives:} & {operations}: \\
$\land$ & $\cap$ \\ 
$\lnot$ & $A\mapsto A\comp$ \\
$\lor$ & $\cup$ \\
$\to$ & $(A,B)\mapsto A\comp\cup B$ \\
%$\iff$ & $(A,B)\mapsto (A\comp\cup B)\cap (A\cup B\comp)$\\ 
\hline
\textbf{quantifiers:} & {projections:} \\ 
$\exists x_j$ & $\pi_j^I$: $a\mapsto(a_i:i\in
I\setminus\{j\})$ 
\\ 
$\forall x_j$ & $A\mapsto(\pi_j^I(A\comp))\comp$ \\ \hline\hline
& RELATIONS ON $M$:\\ \hline
%& \textsc{basic relations:}\\
\textbf{equals-sign} $=$ & equality\\
\textbf{relation-symb.} $R$ & $R^{\str M}$, a subset of $M^{n(R)}$\\
%\hline
%\textsc{formulas:} & \textsc{definable relations:} \\ 
{$I$-ary formula} $\phi$ & $\phi^{\str M}$, a subset of $M^I$ \\
nullary formula \phantom{$\phi$} & \\
or \emph{sentence} $\sigma$ & $1$ (true) or $0$ (false)\\ \hline
%\hline
%theory $T$ & $\Mod T$ \\
%$\str M\models\sigma$ & $\sigma^{\str M}=1$ \\
%$T\models \sigma$ & $\Mod{T}\included\Mod{\sigma}$\\ \hline
\end{tabular}




  \end{slide}


  \begin{slide}
\hang
One may care only about the definable sets:

\hang
Let also $J$ be a finite subset of $\omega$.  If 
\begin{equation*}
  \sigma:I\to J,
\end{equation*}
then
$\sigma^*:M^J\to M^I$,
whence
\begin{gather*}
  \sigma^*:\pow{M^J}\to \pow{M^I},\\
\sigma_*:\pow{M^I}\to \pow{M^J},
\end{gather*}
where $\sigma^*(c)=(c_{\sigma(i)}:i\in J)$ if $c\in M^J$, and
\begin{gather*}
  \sigma^*A=\{\sigma^*(c):c\in A\},\\
\sigma_*B=\{c:\sigma^*(c)\in B\}.
\end{gather*}

\hang
We can re-define a \textbf{structure} on $M$ to be $\bDef{}$, which comprises,
for each $I$, a Boolean sub-algebra $\bDef I$ of
$\pow{M^I}$ such that 
\begin{gather*}
\{(c,c)\in M^2\}\in\bDef 2;\\
  A\in\bDef J\Rightarrow\sigma^*A\in\bDef I;\\
  B\in\bDef I\Rightarrow\sigma_*B\in\bDef J.
\end{gather*}
From $\bDef{}$ we can recover a structure on $M$ in the original sense
whose $0$-definable sets are just the sets in the $\bDef I$.

  \end{slide}


\end{document}

