% ----------------------------------------------------------------------

\chapter{Introduction}
\label{intro}
\index{derivation}
\index{proof}

\index{Pythagorean theorem}
\index{Pythagoras (c.~580--c.~500~B.C.)}
\index{diagonal}
\index{hypotenuse}
\index{leg}
The Pythagorean theorem holds that
\bq{alggeo:pythag}
  a^2 + b^2 = c^2,
\eq
where~$a$, $b$ and~$c$ are the lengths of the legs and diagonal of a
right triangle as in Fig.~\ref{alggeo:223:25}.
\begin{figure}
  \caption{A right triangle.}
  \label{alggeo:223:25}
  \bc
    {
      \nc\xax{-5}
      \nc\xbx{-1}
      \nc\xcx{ 4}
      \nc\xdx{ 3}
      \begin{pspicture}(\xax,\xbx)(\xcx,\xdx)
        \psset{dimen=middle}
        \small
        %\psframe[linewidth=0.5pt](\xax,\xbx)(\xcx,\xdx)
        \psset{linewidth=2.0pt}
        \psline{C-C}(0,0)(0,2.6)(-1.4,0)(0,0)
        \rput(-.7,-.3){$a$}
        \rput(.3,1.3){$b$}
        \rput(-1.0,1.45){$c$}
        \nc\tmpd{0.3}
        \psline[linewidth=0.5pt](-\tmpd,0)(-\tmpd,\tmpd)(0,\tmpd)
      \end{pspicture}
    }
  \ec
\end{figure}
Many proofs of the theorem are known.

\index{square!rotated}
\index{square!tilted}
\index{area}
One such proof posits a square of side length $a+b$ with a tilted square
of side length~$c$ inscribed as in Fig.~\ref{alggeo:223:35}.
\begin{figure}
  \caption{The Pythagorean theorem.}
  \label{alggeo:223:35}
  \bc
    {
      \nc\xax{-5}
      \nc\xbx{-3}
      \nc\xcx{ 5}
      \nc\xdx{ 3}
      \begin{pspicture}(\xax,\xbx)(\xcx,\xdx)
        \psset{dimen=middle}
        \small
        %\psframe[linewidth=0.5pt](\xax,\xbx)(\xcx,\xdx)
        \psset{linewidth=2.0pt}
        \nc\tmpa{0.6}
        \nc\tmpb{1.97}
        \nc\tmpc{2.0}
        \psline{C-C}( \tmpa,-\tmpb)( \tmpb, \tmpa)(-\tmpa, \tmpb)(-\tmpb,-\tmpa)( \tmpa,-\tmpb)
        \psline{C-C}( \tmpc,-\tmpc)( \tmpc, \tmpc)(-\tmpc, \tmpc)(-\tmpc,-\tmpc)( \tmpc,-\tmpc)
        \rput(1.3,-2.3){$a$}
        \rput(2.3,-.7){$b$}
        \rput(-.7,-2.3){$b$}
        \psline[linewidth=0.5pt](\tmpa,-2.0)(\tmpa,-2.3)
        \rput(1.0,-.55){$c$}
        \nc\tmpd{0.3}
        \rput(2,-2){
          \psline[linewidth=0.5pt](-\tmpd,0)(-\tmpd,\tmpd)(0,\tmpd)
        }
        \rput(\tmpa,-\tmpb)
          {
          \rput{-28.30}(0,0)
          {
            \psline[linewidth=0.5pt](-\tmpd,0)(-\tmpd,\tmpd)(0,\tmpd)
          }
        }
      \end{pspicture}
    }
  \ec
\end{figure}
The area of each of the four triangles in the figure is evidently
$ab/2$.  The area of the tilted inner square is~$c^2$.  The area of the
large outer square is $(a+b)^2$.  But the large outer square is
comprised of the tilted inner square plus the four triangles, so the
area of the large outer square equals the area of the tilted inner
square plus the areas of the four triangles.  In mathematical symbols,
this is that
\[
  (a+b)^2 = c^2 + 4\left(\frac{ab}{2}\right),
\]
which simplifies directly to~(\ref{alggeo:pythag}).

If you like the foregoing, then you might like the book.

This book is a book of applied mathematical proofs.  When you have seen
a mathematical result somewhere, if you want to know why the result is
so, then you can look for the proof here.

The book's purpose is to convey the essential ideas underlying the
derivations of a large number of mathematical results useful in the
modeling of physical systems.   To this end, the book emphasizes main
threads of mathematical argument and the motivation underlying the main
threads, de\"emphasizing formal mathematical rigor.  It derives
mathematical results from the applied perspective of the scientist and
the engineer.

The book's chapters are topical.  This first chapter explains the book's
philosophy and otherwise treats a few introductory matters of general
interest.

% ----------------------------------------------------------------------

\section{Applied mathematics}
\label{intro:220}
\index{applied mathematics}
\index{mathematics!applied}
\index{mathematician!applied}

What is applied mathematics?
%
\begin{quote}
  Applied mathematics is a branch of mathematics that concerns itself
  with the application of mathematical knowledge to other domains\mdots
  The question of what is applied mathematics does not answer to logical
  classification so much as to the sociology of professionals who use
  mathematics.~\cite{def-applied-math}
\end{quote}
%
That is about right, on both counts.  In this book we shall define
\emph{applied mathematics} to be correct mathematics useful to
scientists, engineers and the like; proceeding not from reduced,
well-defined sets of axioms but rather directly from a nebulous mass of
natural arithmetical, geometrical and classical-algebraic idealizations
of physical systems; demonstrable but generally lacking the detailed
rigor of the pure, professional mathematician.

% ----------------------------------------------------------------------

\section{Rigor}
\label{intro:284}
\index{rigor}
\index{formal mathematical rigor}
\index{professional mathematics}
\index{pure mathematics}
\index{mathematics!professional or pure}
\index{mathematician!professional}
\index{philosophy}

\index{definition!abstract}
\index{subimposition}
\index{algebra!formal}
\index{formal algebra}
Applied and pure mathematics differ principally and essentially in the
layer of abstract definitions the latter subimposes beneath the physical
ideas the former seeks to model.  That subimposed layer, the disciplined
use of it, and the formal algebra associated with it may together be
said to institute pure mathematical \emph{rigor.}

Such pure mathematical rigor tends to dwell more comfortably in lone
reaches of the professional mathematician's realm than among the hills
and plats of applications, where it does not always travel so
gracefully.  If this book will be a book of mathematical derivations,
then it might speak a little of rigor here at the start.

\subsection{Axiom and definition}
\label{intro:284.2}
\index{axiom}
\index{definition}

\index{irreducibility}
\index{pedantry}
\index{infinity}
\index{axiomatic method}
Whether explicitly or implicitly, the professional mathematician usually
founds his rigor upon what he calls the \emph{axiomatic method}---an
\emph{axiom}, according to Webster, being ``a self-evident and necessary
truth, or a proposition whose truth is so evident at first sight that no
reasoning or demonstration can make it plainer; a proposition which it
is necessary to take for granted.''%
\footnote{\cite{Webster1913}}

\index{set}
For example, the following could be an axiom: ``For every set~$A$ and
every set~$B$, $A=B$ if and only if for every set~$x$, $x$ is a member
of~$A$ if and only if~$x$ is a member of~$B$.''\footnote{%
  The source quoted is \cite[\S~2.1]{Wilson}, which however uses the
  symbol~$\in$ for ``is a member of.''%
}

Axioms undergird the work of the professional mathematician.
Indeed, so fundamental are axioms to the professional mathematician's
work that---\emph{ideally and at least in principle}---it may be that
the professional
will derive nothing until he has first declared the axioms upon which his
derivations will rely; that is, until he has stated the least premises
upon which he will argue.  Conversely, aiming deeper---promoting the
latter of Webster's two readings---the professional can
illuminate the wisdom latent in his very axioms by their use in a
suitable derivation.%
\footnote{\cite[Einleitung]{Hilbert:geometrie}}
Irreducibility is a prime aesthetic on either
level: at best, no axiom should overlap the others or be specifiable in
terms of the others.  Nonaxiomatic geometrical argument---proof by
sketch if you like, as the Pythagorean with its figures at the head
of this chapter---is distrusted.\footnote{\cite[chapters~1
and~2]{Weierstrass:werke}}
The professional mathematical literature discourages undue pedantry
indeed, but its readers do implicitly demand a convincing assurance that
its writers \emph{could} derive results in pedantic detail if called
upon to do so.  Precise definition here is critically important, which
is why the professional mathematician tends not to accept blithe
statements such as that
\[
  \frac{1}{0} = \infty,
\]
among others, without first inquiring as to exactly what is meant by
symbols like~$0$ and~$\infty$.

\index{model}
\index{geometrical argument}
\index{proof!by sketch}
\index{sketch, proof by}
\index{bridge}
\index{soil}
\index{wind}
The applied mathematician begins from a different base.  His ideal lies
not in precise definition or irreducible axiom, but rather in the
elegant modeling of the essential features of some physical system.
Here, mathematical definitions tend to be made up \emph{ad hoc} along
the way, based on previous experience solving similar problems, adapted
implicitly to suit the model at hand.  If you ask the applied
mathematician exactly what his axioms are, which symbolic algebra he is
using, he usually does not know; what he knows is that the physical
system he is analyzing, describing or planning---say, a bridge---is to
be founded in certain soils with observed tolerances, is to suffer
such-and-such a wind load, and so on.  To avoid error, the applied
mathematician relies not on abstract formalism but rather on a thorough
mental grasp of the essential physical features of the phenomenon he is
trying to model.  An equation like
\[
  \frac{1}{0} = \infty
\]
may make perfect sense without further explanation to an applied
mathematical readership, depending on the physical context in which the
equation is introduced.  Nonaxiomatic geometrical argument---proof by
sketch---is not only trusted but treasured.  Abstract definitions are
wanted only insofar as they smooth the analysis of the particular
physical problem at hand; such definitions are seldom promoted for their
own sakes.

\index{Heaviside, Oliver (1850--1925)}
\index{Hilbert, David (1862--1943)}
\index{Courant, Richard (1888--1972)}
\index{Pinter, Charles~C. (1938--)}
\index{Shilov, Georgi~E. (1917--75)}
\index{perspective, Courant-Hilbert-Shilov}
\index{Courant-Hilbert-Shilov perspective, the}
\index{physicist}
The irascible Oliver Heaviside (1850--1925), responsible for the
important applied mathematical technique of phasor analysis, once said,
\begin{quote}
  It is shocking that young people should be addling their brains over
  mere logical subtleties, trying to understand the proof of one obvious
  fact in terms of something equally \ldots\ obvious.~\cite{mathbios}
\end{quote}
Exaggeration, perhaps, but from the applied mathematical perspective
\linebreak % bad break
Heaviside nevertheless had a point.  The professional mathematicians
\linebreak % bad break
Richard Courant (1888--1972) and David Hilbert (1862--1943) put it more
soberly in~1924 when they wrote,
\begin{quote}
  Since the seventeenth century, physical intuition has served as a
  vital source for mathematical problems and methods.  Recent trends and
  fashions have, however, weakened the connection between mathematics
  and physics; mathematicians, turning away from the roots of
  mathematics in intuition, have concentrated on refinement and
  emphasized the postulational side of mathematics, and at times have
  overlooked the unity of their science with physics and other fields.
  In many cases, physicists have ceased to appreciate the attitudes of
  mathematicians.~\cite[Preface]{Courant/Hilbert}
\end{quote}
And what are these ``attitudes'' of which Courant and Hilbert speak?
To the mathematician Charles~C.\ Pinter, they are not attitudes, but
principles:
\begin{quote}
  Since the middle of the nineteenth century, the axiomatic method has
  been accepted as the only correct way of organizing mathematical
  knowledge.~\cite[chapter~1]{Pinter}
\end{quote}
But accepted by whom?  The mathematician Georgi~E.\ Shilov, less
enthusiastic than Pinter for the axiomatic method, is not so sure:
\begin{quote}
  There are other approaches to the theory \ldots\ where things I take
  as axioms are proved\mdots  Both treatments have a key deficiency,
  namely the absence of a proof of the compatibility of the axioms\mdots
  The whole question, far from being a mere technicality, involves the
  very foundations of mathematical thought.  In any event, this being
  the case, it is not very important where one starts a general
  treatment\mdotx~\cite[Preface]{Shilov}
\end{quote}
Although the present book responds to ``the attitudes of mathematicians''
with greater deference than some of Courant's and Hilbert's unnamed~1924
physicists might have done, though Shilov himself admittedly is more
rigorous than his own, seemingly casual words let on, still, Courant and
Hilbert could have been speaking for the engineers and other applied
mathematicians of our own day as well as for the physicists of theirs;
and still, Shilov like Heaviside has a point.  To the applied
mathematician, the mathematics is not principally meant to be developed
and appreciated for its own sake; it is meant to be \emph{used.}  This
book adopts the Courant-Hilbert-Shilov perspective.%
\footnote{\label{intro:284.2:fn1}%
  It is acknowledged that Hilbert at other times took what seems to be
  the opposite perspective; and that there remains the historically
  important matter of what the early twentieth century knew as
  ``Hilbert's program,'' a subject this book will not address.  Hilbert
  however, perhaps the greatest of the mathematical
  formalists~\cite[chapter~1]{Feferman}, was a broad thinker, able to survey
  philosophical questions seriously from each of multiple points of
  view.  What Hilbert's ultimate opinion might have been, and whether
  the words quoted more nearly represent Hilbert's own conviction or his
  student Courant's, and how the views of either had evolved before or
  would evolve after, are biographical questions this book will not try
  to treat.  The book would accept the particular passage recited rather
  on its face.

  Regarding Shilov, his formal mathematical rigor is easy and fluent,
  and his book~\cite{Shilov} makes a good read even for an engineer.
  The book you now hold however adopts not Shilov's methods---for one
  can read Shilov's book for those---but only his perspective, as
  expressed in the passage recited.

  For a taste of Shilov's actual methods, try this, the very first proof
  in his book:  ``Theorem.  The system [of real numbers] contains a
  unique zero element.  Proof.  Suppose [that the system] contains two
  zero elements~$0_1$ and~$0_2$.  Then it follows from [the axioms of
  commutation and identity] that $0_2=0_2+0_1=0_1+0_2=0_1$\@.
  Q.E.D.''~\cite[\S~1.31]{Shilov}.
}

But why?  Is the Courant-Hilbert-Shilov perspective really necessary, after
all?  If unnecessary, is it desirable?  Indeed, since the book you are
reading is a book of derivations, would it not be a more elegant book if
it began from the most primitive, pure mathematical fundamentals, and
proceeded to applications thence?

If Heaviside was so irascible, then wasn't he just plain wrong?

\subsection{Mathematical Platonism}
\label{intro:284.4}
\index{Platonism}
\index{mathematical Platonism}
\index{formalism}
\index{mathematical formalism}

\index{Bell, John~L.\ (1945--)}
\index{Kort\'e, Herbert}
\index{Weyl, Hermann (1885--1955)}
\index{knee}
To appreciate the depth of the trouble in which the applied
mathematician may soon find himself mired, should he too casually reject
the Courant-Hilbert-Shilov perspective, consider John~L.\ Bell's and Herbert
Kort\'e's difficult anecdote regarding Hilbert's brilliant student and
later cordial rival, Hermann Weyl (1885--1955):
\begin{quote}
  \index{pencil}
  \index{Sirius}
  Weyl \ldots\ considers the experience of seeing a pencil lying on a
  table before him throughout a certain time interval. The position of
  the pencil during this interval may be taken as a function of the
  time, and Weyl takes it as a fact of observation that during the time
  interval in question this function is continuous and that its values
  fall within a definite range. And so, he says, ``This observation
  entitles me to assert that during a certain period this pencil was on
  the table; and even if my right to do so is not absolute, it is
  nevertheless reasonable and well-grounded. It is obviously absurd to
  suppose that this right can be undermined by `an expansion of our
  principles of definition'---as if new moments of time, overlooked by
  my intuition, could be added to this interval; moments in which the
  pencil was, perhaps, in the vicinity of Sirius or who knows
  where\mdotx''~\cite{Bell/Korte}
\end{quote}
In Weyl's gentle irony lies a significant point, maybe, yet how should
applied mathematics advocate such a point, or dispute it?  Is applied
mathematics even suited to such debates?  What of engineering questions
of a more mundane cast---such as, for instance, how likely Weyl's pencil
might be to roll off if his knee bumped the table?  After all, there is
a pencil, and there is a table, and Sirius seems to have little
immediately to do with either; and whether the pencil rolls off might
concern us, irrespective of any particular substruction pure mathematics
sought to build to support the technique we had used to model and
analyze the case.  Indeed, Weyl himself---a great mathematician, a
consistent friend of the engineer and of the scientist, and a wise
man---warns,
\begin{quote}
  \index{foundations of mathematics}
  The ultimate foundations and the ultimate meaning of mathematics
  remain an open problem; we do not know in what direction it will find
  its solution, nor even whether a final objective answer can be
  expected at all.~\cite{Weyl:1950}
\end{quote}
Just so.  Fascinating as it is, we shall not try to answer Weyl's deep
question of mathematical philosophy here.

\index{Sunday}
\index{weekday}
\index{Hersh, Reuben (1927--)}
To the extent to which a professional mathematician classified the book
at all, he might properly call the book you are reading a
\emph{Platonist} work.  Now, that is a fine adjective, is it not?  It is
most subtle, most lofty; and perhaps the author had better not be too
eager to adorn his own work with it; yet let us listen to what the
professional mathematician Reuben Hersh has to say:
\begin{quote}
  Most writers on the subject seem to agree that the typical ``working
  mathematician'' is a Platonist on weekdays and a formalist on Sundays.
  That is, when he is doing mathematics, he is convinced that he is
  dealing with an objective reality whose properties he is attempting to
  determine.  But then, when challenged to give a philosophical account
  of this reality, he finds it easiest to pretend that he does not
  believe in it after all\mdots

  The basis for Platonism is the awareness we all have that the problems
  and concepts of mathematics exist independently of us as individuals.
  The zeroes of the zeta function%
  \footnote{%
    This book stops short of treating Riemann's zeta function, a rather
    interesting special function that however seems to be of even
    greater interest in pure than in applied mathematics.  If you want
    to know, the zeta function is $\zeta(z) \equiv \sum_{k=1}^\infty
    1/k^z.$~\cite[chapter~10]{Spiegel}
  }
  are where they are, regardless of what I may think or know on the
  % bad break
  sub\-ject\mdotx%
  \footnote{%
    The generous Hersh, who has thus so empathetically sketched
    mathematical Platonism, goes on tactfully to confess that he
    believes mathematical Platonism a myth, and to report (admittedly
    probably correctly) that most professional mathematicians also
    believe as he does on this point.  The present writer however
    accepts the sketch, appreciates the tact, and believes \emph{in} the
    myth, for the reasons outlined in this introduction among others.
  }%
  ~\cite{Hersh}
\end{quote}
Your author inclines to Platonism%
\footnote{\cite[chapter~2]{Feser}}
on Sundays, too, yet even readers who do not so incline
should find the book nonetheless edifying the other six days of the
week at least.

\index{Tymoczko, Thomas (1943--1996)}
\index{Frege, Friedrich Ludwig Gottlob (1848--1925)}
\index{Wittgenstein, Ludwig (1889--1951)}
\index{No\"e, Alva (1964--)}
\index{Bertrand Russell (1872--1970)}
\index{mathematician!applied, chief interest of}
\index{ontology}
\index{epistemology}
Hersh goes on with tact and feeling at some length in the article from
which his words are quoted, and it is fair to say that he probably would
not endorse the present writer's approach in every respect.
Notwithstanding, the philosopher Thomas Tymoczko---who unlike Hersh but
like the present writer might fairly be described as a Platonist%
\footnote{%
  Tymoczko's preferred term is not ``Platonist'' but
  ``quasi\"empiricist,'' a word Tymoczko lends a subtly different
  emphasis.~\cite{Tymoczko:vat}
}%
---later writes of Hersh's article,
\begin{quote}
  \mbox{\ldots\ }In so far as [the working philosophy of the professional
  mathematician] is restricted to the usual mix of foundational ideas,
  Hersh charges, [this philosophy] is generally inconsistent, always
  irrelevant and sometimes harmful in practice and teaching.

  \mbox{\ldots\ }Hersh suggests [that] the best explanation of foundational
  concerns is in terms of the historical development of
  mathemat\-ics\mdots  [H]e isolates some of the basic presuppositions of
  foundation studies:  ``that mathematics must be provided with an
  absolutely reliable foundation'' and ``that mathematics must be a
  source of indubitable truth.''  Hersh's point is that it is one thing
  to accept the assumption when, like Frege, Russell or Hilbert, we feel
  that the foundation is nearly attained.  But it is quite another to go
  on accepting it, to go on letting it shape our philosophy, \emph{long
  after}%
  \footnote{%
    Emphasis in the original.
  }
  we've abandoned any hope of attaining that goal\mdotx~\cite{Tymoczko}
\end{quote}
The applied mathematician who rejects the Courant-Hilbert-Shilov perspective
and inserts himself into \emph{this} debate%
\footnote{%
  See also, in no particular order,
  \cite{Carson/Huber}\linebreak[0]%
  \cite{Courant/Hilbert}\linebreak[0]%
  \cite{Feferman}\linebreak[0]%
  \cite{Frege}\linebreak[0]%
  \cite{Hamming}\linebreak[0]%
  \cite{Hardy}\linebreak[0]%
  \cite{Hersh}\linebreak[0]%
  \cite{Knopp}\linebreak[0]%
  \cite{Krader}\linebreak[0]%
  \cite{Lebedev}\linebreak[0]%
  \cite{Pinter}\linebreak[0]%
  \cite{Shilov}\linebreak[0]%
  \cite{Sieg/Schlimm}\linebreak[0]%
  \cite{Toader}\linebreak[0]%
  \cite{Tymoczko}\linebreak[0]%
  \cite{Tymoczko:vat}\linebreak[0]%
  \cite{Watson}\linebreak[0]%
  \cite{Weierstrass:werke}\linebreak[0]%
  \cite{Weiner}\linebreak[0]%
  \cite{Weyl:1950}.
}
may live to regret it.  As the mathematician Ludwig Wittgenstein
illustrates,
\begin{quote}
  [Bertrand] Russell [co\"author of \emph{Principia
  Mathematica} and archexponent of one of the chief schools of pure
  mathematical
  \linebreak % bad break
  thought]\cite{Whitehead/Russell} gives us a calculus
  here.  How this calculus of Russell's is to be \emph{extended} you
  wouldn't know for your life, unless you had ordinary arithmetic in
  your bones.  Russell doesn't even prove $10\times 100 = 1000.$

  What you're doing is constantly taking for granted a particular
  interpretation.  You have mathematics and you have Russell; you think
  mathematics is all right, and Russell is all right---more so; but
  isn't this a put-up job?  That you can correlate them in a way, is
  clear---not that one throws light on the
  other.~\cite[lecture~XVI]{Wittgenstein-lectures}
\end{quote}
The book you hold will not correlate them but will (except for some
nonessential side commentary) confine its attention to the applied
mathematician's honorable, chief interest---which is to describe,
quantify, model, plan and analyze particular physical phenomena of
concern; and to understand topically why the specific mathematical
techniques brought to bear on such phenomena should prosper; but not to
place these techniques in the context of a larger ontological or
epistemological dispute---a dispute that, though important in itself,
does not directly move the applied mathematician's interest one way or
the other.

Indeed, as the philosopher Alva No\"e observes,
\begin{quote}
  [T]here is no stable or deeply understood account of how these
  autonomous domains fit together. The fact that we are getting along
  with business as if there were such an account is, well, a political
  or sociological fact about us that should do little to
  reassure.~\cite{Noe}
\end{quote}
No\"e is writing here about the nature of consciousness but could as
well, with equal justice and for similar reasons, be writing about our
problem of mathematical foundations.

To conclude this subsection's glance upon mathematical Platonism
we may well quote Plato himself:
\begin{quote}
  \index{arithmetic}
  \index{geometry}
  Then this is a kind of knowledge which legislation may fitly
  prescribe; and we must endeavour to persuade those who are to be the
  principal men of our State to go and learn arithmetic, not as
  amateurs, but they must carry on the study until they see the nature
  of numbers with the mind only; nor again, like merchants or
  retail-traders, with a view to buying or selling, but for the sake of
  their military use, and of the soul herself; and because this will be
  the easiest way for her to pass from becoming to truth and being\mdots
  I must add how charming the science is!\,\ldots\quad
  [A]{r}{i}{t}{h}{m}{e}{t}{i}{c} has a very great and elevating effect,
  compelling the soul to reason about abstract number, and rebelling
  against the introduction of visible or tangible objects into the
  argument\mdots [T]his knowledge may be truly called necessary,
  necessitating as it clearly does the use of the pure intelligence in
  the attainment of pure truth\mdots

  And next, shall we enquire whether the kindred science [of geometry]
  also concerns us?\,\ldots\  [T]he question relates \ldots\ to the
  greater and more advanced part of geometry---whether that
  % bad break
  \linebreak
  tends in any degree to make more easy the vision of the idea of good;
  and thither, as I was saying, all things tend which compel the soul to
  turn her gaze towards that place, where is the full perfection of
  being, which she ought, by all means, to behold\mdots [T]he knowledge
  at which geometry aims is knowledge of the eternal, and not of aught
  perishing and transient.  [G]{e}{o}{m}{e}{t}{r}{y} will draw the soul
  towards truth, and create the spirit of philosophy, and raise up that
  which is now unhappily allowed to fall down.~\cite[book~VII]{Plato}.
\end{quote}
\index{indulgence}
\index{indictment}
\index{vanity}
The vanity of modern man (I do not say, modern mathematician) may affect
to smile upon the ancient; but his vanity less indulges the
ancient, who hardly needs indulgence, than indicts the modern.

\index{4th century~B.C.}
Plato is not less right today than he was in the fourth century~B.C.

\subsection{Methods, notations, propositions and premises}
\label{intro:284.45}
\index{method}
\index{notation}
\index{proposition}
\index{premise}

% bad break in the next index entry
\index{Zermelo-Fraenkel and Choice set theory (ZFC)}
\index{ZFC}
\index{set theory}
\index{Zermelo, Ernst (1871--1953)}
\index{Fraenkel, Abraham (1891--1965)}
\index{Cantor, Georg (1845--1918)}
\index{Weyl, Hermann (1885--1955)}
The book purposely overlooks, and thus omits, several of the mathematics
profession's pure methods and some of its more recondite notations,
unsuited to (or at any rate unintended for) applied use.  Most notably,
the book overlooks and omits the profound methods and notations of the
Zermelo-Fraenkel and Choice set
theory~(ZFC)\footnote{%
  \index{Bertrand Russell (1872--1970)}%
  See \cite{Wilson}\cite{Dasgupta}.  The~ZFC is a near descendant of
  the work of the aforementioned Bertrand Russell and, before Russell,
  of Georg Cantor.%
}
and its consequents.  The book inclines rather toward Weyl's view:
\begin{quote}
  [A set-theoretic approach] contradicts the essence of the continuum,
  which by its very nature cannot be battered into a set of separated
  elements.  Not the relationship of an element to a set, but that of a
  part to a whole should serve as the basis\mdots~\cite{Scholz}
\end{quote}
The years have brought many mathematical developments since Weyl wrote
these words in~1925 but the present author still tends to think as
Weyl does.

\index{intuitive proposition}
\index{unsupported proposition}
\index{premise}
\index{square!tilted}
\index{square!rotated}
\index{continuum}
\index{mean-value theorem}
\index{identifying property}
\index{discontinuity}
This does not of course mean that the author or this book intended to peddle
nonintuitive mathematical propositions, unsupported, as fact.  The book
could hardly call itself a book of derivations if it did.  What it does
mean is that the book can assume without further foundation or
explication---and without extremes of formal definition---for example,
that a rotated square remains square; that a number like $\sqrt{3}/2$
occupies a definite spot in a comprehensible
continuum;\footnote{\cite{Weyl:1918}}
that no numbers in the continuum other than~0 and~1 enjoy these two
numbers' respective identifying properties;%
\footnote{See footnote~\ref{intro:284.2:fn1}.}
that one can impute an unobserved population against an observed
sample;\footnote{\cite[chapter~11]{Bulmer}}
that a continuous, differentiable,\footnote{%
  Are burdensome adjectives like ``continuous, differentiable''
  necessary?  Are they helpful?  Do they sufficiently illuminate
  one's understanding that they should be suffered to clutter the text
  so?

  Maybe they are indeed necessary.  Maybe they are even helpful but,
  even if so, does the discerning reader want them?  Does the
  \emph{nondiscerning} reader want them?  If neither, then whom do
  they serve?  If the only answer is that they serve the investigator
  of foundational concerns, then what does this tell us about the
  wisdom of injecting foundational concerns into applications?

  Regarding continuity and differentiability: the applicationist is
  inclined to wait until a specific problem arises in which a
  particular, concrete discontinuity or undifferentiability looms, when
  he will \emph{work around} the discontinuity or undifferentiability as
  needed---whether by handling it as a parameterized limit or by
  addressing it in some other convenient way.  None of this has much to
  do with foundations.

  To treat every such point as a fundamental challenge to one's
  principles of definition just takes too long and anyway does not much
  help.  The scientist or engineer wants to save that time to wrestle
  with physical materialities.%
}
real function's average slope over a real interval equals the
function's instantaneous slope at at least one
point;\footnote{\cite[\S~7.4]{Shilov}}
and so on (and if you did not understand all of that, that is
all right, for to explain such things is what the rest of the
book's chapters are for).  This is what the book's approach means.

\index{premise!implied, unstated}
The Pythagorean theorem at the chapter's head examples the approach.
The theorem is proved briefly, without excessive abstraction, working
upon the \emph{implied, unstated premise} that a rotated square remains
square.

If you can accept the premise, then you can accept the proof.  If you
can accept the \emph{kind} of premise, then you can accept the book.

\subsection{Rigor to forestall error}
\label{intro:284.5}
\index{rigor!to forestall error}
\index{error!forestalled by rigor}

\index{ontology}
\index{epistemology}
Aside from foundational concerns---whatever the ontological or
epistemological merits of formal mathematical rigor may be---some will
laud such rigor too for forestalling error, even in applications.%
\footnote{%
  To cite a specific source here might be more captious than helpful,
  so the reader is free to disregard the assertion as unsourced.
  However, if you have heard the argument---perhaps in conjunction
  with the example of a conditionally convergent sum or the
  like---then the writer has heard it, too.
}
Does the rigor deserve this praise?  Well, perhaps it does.  Still,
though the writer would not deny the praise's decorum in every instance,
he does judge such praise to have been oversold by a few.

\index{Hamming, Richard~W.\ (1915--1998)}
Notwithstanding, formal mathematical rigor serves two, distinct
programs.  On the one hand, it embodies the pure mathematician's noble
attempt to establish, illuminate or discover the means and meaning of
\emph{truth.}\footnote{%
  \index{Aristotle (384--322~B.C.)}%
  \index{Aquinas, St.~Thomas (1225-1274)}%
  \index{G\"odel, Kurt Friedrich (1906--1978)}%
  As courtesy to the reader, I should confess my own opinion in the
  matter, which is that it is probably, fundamentally not given to
  mortal man to lay bare the ultimate foundations of truth, as it is not
  given to the beasts, say, to grasp mathematics.  Like the beasts, we
  too operate within the ontological constraints of our nature.

  That this should be my opinion will not perhaps surprise readers who
  have read the preface and the chapter to this point.  As far as I
  know, Aristotle, Aquinas and G\"odel were right.
  %(Some sophomore will fell compelled to interject that Aristotle was
  %misinformed regarding astronomy, but that obviously isn't my point.)
  However, be that as it may, my opinion in the matter is not very
  relevant to this book's purpose.  I do not peddle it but mention it
  only to preclude misunderstanding regarding the sympathies and biases,
  such as they are, of the book's author.  ---THB---%
}
On the other hand, it cross-checks intuitive \emph{logic} in a
nonintuitive way.  Neither hand holds absolute metaphysical guarantees;
yet, even if the sole use of formal mathematical rigor were to draw the
mathematician's attention systematically to certain species of
questionable reasoning for further examination, such rigor would merit
the applicationist's respect.  As the mathematician Richard~W.\ Hamming
writes,
\begin{quote}
  When you yourself are responsible for some new application of
  mathematics in your chosen field, then your reputation, possibly
  millions of dollars and long delays in the work, and possibly even
  human lives, may depend on the results you predict.  It is then that
  the \emph{need} for mathematical rigor will become painfully obvious
  to you.  Before this time, mathematical rigor will often seem to be
  needless pedantry\mdots~\cite[\S~1.6]{Hamming}
\end{quote}
Sobering words.  Nevertheless, Hamming's point is not a point this book
will pursue.

\index{style}
The introduction you are reading is not the right venue for a full essay
on why both kinds of mathematics, applied and pure, are needed at any
rate.  Each kind has its place; and though it is a stylistic error to
mix the two indiscriminately, clearly the two have much to do with one
another.  However that may be, this book is a book of derivations of
applied mathematics.  The derivations here proceed by an applied
approach.

\section{Mathematical extension}
\label{intro:284.1}
\index{extension}

Profound results in mathematics occasionally are achieved simply by
extending results already known.  For example, the negative integers and
their properties can be discovered by counting backward---3, 2, 1,
0---and then asking what follows (precedes?)\ 0 in the countdown and what
properties this new, negative integer must have to interact smoothly
with the already known positives.  The astonishing Euler's formula
(\S~\ref{cexp:230}) is discovered by a similar but more sophisticated
mathematical extension.

\index{European Renaissance}
\index{Renaissance}
More often however, the results achieved by extension are unsurprising
and not very interesting in themselves.  Such extended results are the
faithful servants of mathematical rigor.  Consider for example the
triangle on the left of Fig.~\ref{intro:284:fig}.
\begin{figure}
  \caption{Two triangles.}
  \label{intro:284:fig}
  \bc
    \begin{pspicture}(-4.2,-1.1)(5.8,3)
      \small
      \nc\xa{3.0}
      \nc\xb{2.0}
      \nc\xc{0.6}
      \nc\xd{3.6}
      \nc\xe{2.0}
      \nc\xee{0.9}
      \nc\xq{0.3}
      \nc\xf{0.15}
      \nc\xk{0.3}
      \nc\xkk{0.2}
      \nc\xkkk{0.25}
      \nc\xg{0.2}
      \nc\xj[2]{
        \psline[linewidth=0.5pt](-#1,\xf)(-#1,-\xf)
        \psline[linewidth=0.5pt]( #1,\xf)( #1,-\xf)
        \psline[linewidth=0.5pt]{->}(-\xg,0)(-#1,0)
        \psline[linewidth=0.5pt]{->}( \xg,0)( #1,0)
        \rput(0,0){#2}
      }
      \rput(-\xa,0){
        \pspolygon[linewidth=2.0pt](-\xb,0)(\xb,0)(\xc,\xe)
        \psline[linewidth=0.5pt,linestyle=dashed](\xc,0)(\xc,\xe)
        \rput(\xc,0){\psline[linewidth=0.5pt](-\xq,0)(-\xq,\xq)(0,\xq)}
        \rput(0,-\xk){\xj{\xb}{$b$}}
        \rput(\xc,\xee){\rput(\xkk,0){$h$}}
        \rput(-0.6,0){\rput(0,\xkkk){$b_1$}}
        \rput( 1.2,0){\rput(0,\xkkk){$b_2$}}
      }
      \rput( \xa,0){
        \psline[linewidth=2.0pt]{c-c}(\xd,\xe)(-\xb,0)(\xb,0)(\xd,\xe)
        \psline[linewidth=0.5pt,linestyle=dashed](0,0)(\xd,0)(\xd,\xe)
        \rput(\xd,0){\psline[linewidth=0.5pt](-\xq,0)(-\xq,\xq)(0,\xq)}
        \rput(0,-\xk){\xj{\xb}{$b$}}
        \rput(\xd,\xee){\rput(\xkk,0){$h$}}
        \rput( 2.9,0){\rput(0,\xkkk){$-b_2$}}
      }
    \end{pspicture}
  \ec
\end{figure}
This triangle is evidently composed of two right triangles of areas
\bqb
  A_1&=&\frac{b_1h}{2}, \\
  A_2&=&\frac{b_2h}{2}
\eqb
(each right triangle is exactly half a rectangle).  Hence, the main
triangle's area is
\[
  A = A_1 + A_2 = \frac{(b_1+b_2)h}{2} = \frac{bh}{2}.
\]
Very well.  What about the triangle on the right?  Its~$b_1$ is not
shown on the figure, and what is that~$-b_2$, anyway?  Answer: the
triangle is composed of the \emph{difference} of two right triangles,
with~$b_1$ the base of the larger, overall one: $b_1=b+(-b_2)$.
The~$b_2$ is negative (whereby~$-b_2$ is positive) because the sense of
the small right triangle's area in the proof is negative: the small area
is subtracted from the large rather than added.  By extension on this
basis, the main triangle's area is seen again to be $A = bh/2$.  The
proof is the same.  In fact, once the central idea of adding two right
triangles is grasped, the extension is really rather obvious---too
obvious to be allowed to burden such a book as this.
%\footnote{%
%  One is given to understand that ``[n]{e}{g}{a}{t}{i}{v}{e} numbers
%  were a fruit of the European Renaissance''~\cite{Derbyshire}\@.
%  Mathematical life before negative numbers is not easy to imagine.
%  If one were unacquainted with the negative, then would the triangles'
%  extension be nonobvious?  That it truly would be seems hard to
%  credit.%
%}

\index{edge case}
Excepting the uncommon cases in which extension reveals something
interesting or new, this book generally leaves the mere extension of
proofs---including the validation of edge cases and over-the-edge
cases---as an exercise to the interested reader.

% ----------------------------------------------------------------------

\section{Deduction, adduction and the complex variable}
\label{intro:310}
\index{complex variable}
\index{variable!complex}

\index{adduction}
\index{deduction}
The English language derives from the Latin a nice counterpart to the
transitive verb \emph{to deduce,} a verb whose classical root means ``to
lead away.''  The counterpart is \emph{to adduce,} ``to lead toward.''
Adduction, as the word is here used, subtly reverses the sense of
deduction: it establishes premises from necessary conclusions rather
than the other way around.%
\footnote{%
  As \S~\ref{intro:284.2} has discussed, pure mathematics can
  occasionally, implicitly illuminate its \emph{axioms} in the light of
  necessary conclusions.  Since axioms are by definition a restricted
  kind of premise, one might arguably regard the illumination named as
  an elevated species of adduction.  However, that is not what this
  section is about.
}

Applied mathematics sometimes prefers adduction to deduction.
Attention is drawn to this preference because the preference governs the
book's approach to the \emph{complex number} and the \emph{complex
variable.}  The book will speak much soon of the complex number and the
complex variable, but we mention them now for the following reason.

An overall view of relevant analytical methods---including complex
% bad break
meth\-ods---and
of their use in the modeling of physical systems, marks the applied
mathematician more than does the abstract mastery of any received
program of pure logic.%
\footnote{%
  Pure logic is a worthy endeavor, though whether such
  logic is more properly \emph{received} or rather \emph{illuminated} is
  a matter of long dispute.  Besides Hilbert, of whom we have already
  spoken, see also Frege~\cite{Frege} and the commentary~\cite{Weiner}.
}
A \emph{feel} for the math is the great thing.
Formal definitions, axioms, symbolic algebras and the like, though often
useful, are esteemed less as primary objects than as secondary supports.
The book's adductive, rapidly staged development of the complex number
and the complex variable is planned on this sensibility.

Sections~\ref{alggeo:225}, \ref{trig:278}, \ref{trig:280},
\ref{drvtv:230.35}, \ref{drvtv:240}, \ref{noth:320}, \ref{inttx:250}
and~\ref{inttx:260}, plus all of chapters~\ref{cexp} and~\ref{taylor},
constitute the book's principal stages of complex development.  In these
sections and throughout the book, the reader comes to appreciate that
most mathematical properties which apply for real numbers apply equally
for complex, that few properties concern real numbers alone.

% diagn:  The next paragraph is partially quoted in purec.tex.

Pure mathematics develops its own, beautiful, abstract theory of the
complex variable,%
\footnote{\cite{Arnold:1997}\cite{Shilov}\cite{Fisher}\cite{Spiegel}\cite{Hildebrand}}
a theory whose
arc regrettably takes off too late and flies too far from
applications for such a book as this.
Less beautiful, less abstract, more practical, nonaxiomatic paths to
the topic exist,%
\footnote{%
  See chapter~\ref{taylor}'s footnote~\ref{taylor:320:fn10}.
}
and this book leads the reader along one of these.

For supplemental reference, a bare sketch of the abstract theory of the
complex variable is found in appendix~\ref{purec}\@.

\section{On the text}
\label{intro:290}

The book gives numerals in hexadecimal.  It denotes variables in Greek
letters as well as Roman.  Readers unfamiliar with the hexadecimal
notation will find a brief orientation thereto in appendix~\ref{hex}\@.
Readers unfamiliar with the Greek alphabet will find it in
appendix~\ref{greek}\@.

\index{GNU General Public License}
\index{General Public License, GNU}
\index{GNU GPL}
\index{GPL}
\index{Debian}
\index{Debian Free Software Guidelines}
\index{DFSG (Debian Free Software Guidelines)}
Licensed to the public under the GNU General Public Licence~\cite{GPL},
version~2, this book meets the Debian Free Software
Guidelines~\cite{DFSG}.

% Well, this paragraph is a flight of rhetoric, is it not?  Does
% it work?  I am not sure.  However, I like it enough that I will keep
% it for the indefinite (not unlikely permanent) time being.  --THB--
By its nature, a book of mathematical derivations can make strait,
colorless reading.  To delineate logic as it were in black and white is
the book's duty.  What then to tint?  Is naught to be warm nor cool,
naught ruddy nor blue?  Though mathematics at its best should serve the
demands not only of deduction but equally of insight, by the latter of
which alone mathematics derives either feeling or use; though
this book does occasionally
try (at some risk to strict consistency of tone) to add color in
suitable shades, to strike an appropriately lively balance between the
opposite demands of logical progress and literary relief; nonetheless,
neither every sequence of equations nor every conjunction of figures is
susceptible to an apparent hue the writer can openly paint upon it---but
only to that abeyant hue, that luster which reveals or reflects the fire
of the reader's own mathematical imagination, which color remains
otherwise unobserved.

The book begins by developing the calculus of a single variable.

