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

\addtocounter{chapter}{5}
\chapter{Remarks}
\label{conclu}
\index{remarks, concluding}
\index{concluding remarks}

A book could tell more about derivations of applied mathematics, maybe
without limit.  This book ends here.

If you have learned by reading, I too have learned by writing.  An
engineer, I have long observed advantages of the applied approach to
mathematics.  Writing however has let drawbacks of the applied approach
impress themselves as well.  Some drawbacks are nonobvious.

\section{Frege}

\index{Frege, Friedrich Ludwig Gottlob (1848--1925)}
\index{G\"odel, Kurt Friedrich (1906--1978)}
\index{Hilbert, David (1862--1943)}
\index{Kant, Immanuel (1724--1804)}
\index{Hume, David (1711--1776)}
\index{language, natural}
\index{natural language}
\index{arithmetic}
In~1879, the mathematician and philosopher Gottlob Frege explained,
\begin{quote}
  In apprehending a scientific truth we pass, as a rule, through various
  degrees of certitude.  Perhaps first conjectured on the basis of an
  insufficient number of particular cases, a general proposition comes
  to be more and more securely established by being connected with other
  truths through chains of inferences, whether consequences are derived
  from it that are confirmed in some other way or whether, conversely,
  it is seen to be a consequence of propositions already established.
  Hence we can inquire, on the one hand, how we have gradually arrived
  at a given proposition and, on the other, how we can finally provide
  it with the most secure foundation.  The first question may have to be
  answered differently for different persons; the second is more
  definite, and the answer to it is connected with the inner nature of
  the proposition considered.  The most reliable way of carrying out a
  proof, obviously, is to follow pure logic, a way that, disregarding
  the particular characteristics of objects, depends solely on those
  laws upon which all knowledge rests\mdots  \emph{To prevent anything
  intuitive from penetrating here unnoticed,}\footnote{Emphasis added.}
  I had to bend every effort to keep the chain of emphasis free of gaps.
  In attempting to comply with this requirement in the strictest
  possible way I found the inadequacy of language to be an obstacle; no
  matter how unwieldy the expressions I was ready to accept, I was less
  and less able, as the relations became more and more complex, to
  attain the precision that my purpose required.  This deficiency led me
  to the idea of [an] ideography [whose] first purpose \ldots\ is to
  provide us with the most reliable test of the validity of a chain of
  inferences and to point out every presupposition that tries to sneak
  in unnoticed, so that its origin can be investigated.~\cite[Preface]{Frege:1879}
\end{quote}
Frege's sentiments are poignant in that, like G\"odel and
unlike Hilbert, Frege can fairly be described as a Platonist.  (Witness
Frege's words in a later book: ``In arithmetic we are not concerned with
objects which we come to know as something alien from without through
the medium of the senses, but with objects given directly to our reason
and, as its nearest kin, utterly transparent to it.  And yet, or rather
for that very reason, these objects are not subjective fantasies. There
is nothing more objective than the laws of
arithmetic.''\footnote{\cite{Lotter}})  To Frege, unlike to
Hume or even to Kant, Platonic realism affords all the more cause to
distrust merely human processes of inference in mathematical matters.

The present book being long enough as it is, one can hardly see how to
have met a truly Fregean standard while covering sufficient applied
mathematical ground within the covers of a single volume; but yet does
Frege not still have a point?  I believe that he does.  In some Fregean sense,
the book you are reading has been a book of \emph{derivation sketches}
of various merit rather than of derivations proper.  I am reasonably
satisfied that the book does what it had set out to do, but also
observe that a close study of the book points out the supplementary
need too for more formulaic approaches.

\section{Temperament}

\index{teacher}
\index{instructor}
\index{course}
\index{undergraduate}
\index{circuit theory}
\index{continuous and discrete systems}
\index{discrete and continuous systems}
\index{system!continuous or discrete}
\index{electronics}
\index{industrial electronics}
\index{programming}
\index{C++}
\index{electromagnetics}
\index{freshman}
\index{sophomore}
\index{junior}
\index{United States}
\index{Fourier, Jean Baptiste Joseph\\(1768--1830)}
\index{set theory}
\index{foundations of mathematics}
\index{screwdriver}
The matter in question is, naturally, not one you and I are likely to
settle in a few paragraphs, so for the present purpose let me turn
attention to a facet of the matter which has proved significant at least
to me.  Though I am a practicing engineer rather than a teacher, it so
happens that (besides being the father of six, which has made me a teacher
of another kind) I have over the years, on a part-time basis, taught
several state-university courses in electrical engineering, having
instructed in sum about~1000 engineering undergraduates in subjects like
circuit theory, industrial electronics, continuous and discrete
systems,~C++ programming, and electromagnetics.  My undergraduates have
been U.S.  freshmen, sophomores and juniors mostly aged~18 to~21, so none of
the teaching has been very advanced; and indeed as measured in U.S.\
academia such occasional instructional experience as mine, sans
academic research, counts for so little that I should hardly mention it
here except for one point: an instructor cannot instruct so many
engineering undergraduates without coming to understand somewhat of how
young future engineers think and learn.  When an engineering undergraduate is
finding, say, Fourier's concepts hard to grasp, his engineering
instructor will not extrude the topic into formulations congenial to set
theory.  Rather, the instructor will sketch some diagrams, assign some
pencil-and-paper exercises, require the hands-on construction/testing of
a suitable mechanical/chemical/electrical apparatus, and then field
questions by engaging the undergraduate's physical intuition as directly
as the instructor can.  Of course, professional mathematicians likewise
brandish partly analogous intuitional techniques from
their own instructional arsenals; but the professional carries the
additional burden of preparing \emph{his} students, by gradual stages,
to join mathematics' grand investigation into foundations---or at least
he carries the burden of teaching his students, in the spirit of Frege,
how to deploy formal methods to preclude error.  The engineering student
lacks the time and, usually, the temperament for that.  He has a
screwdriver in his hand.

\index{bridge}
\index{logic}
And this state of affairs is right and proper, is it not?  Give the
professional his due.  While we engineers are off designing bridges or
whatever, the professional mathematician will make it his business to be
better at logic than we.

\section{Foundations athwart intuition}
\label{conclu:athwart}

\index{Cantor, Georg (1845--1918)}
\index{philosophy!Greek}
\index{Greek philosophy}
\index{Reginald of Piperno, Father (c.~1230--c.~1290)}
\index{Aquinas, St.~Thomas (1225-1274)}
\index{physical-intuitional methods}
The experience of writing this book has strengthened my own conviction
that the search for the ultimate foundations of mathematics is probably
futile.  If we have not unearthed the ultimate foundations by now,~2500
years on from when Greek philosophy started excavating, then the
coming~2500 years seem unlikely to reveal them.  Some may think, ``We
have the computer now.  We have Cantor's set theory.  It is different.''
But I am not convinced.
Not a professional mathematician nor a philosopher, I
neither expect nor ask a reader to lend my conviction in such a matter
much weight---nor does \emph{this} book seriously attempt to support
the conviction\footnote{Try~\cite{Feser}
and~\cite{Feser:StTh}, rather.}---but as for myself, I doubt that it is
in the nature of mortal human
intellect to discover or grasp ultimate foundations of such a kind.
Like Father Reginald, I credit St.~Thomas' last report.\footnote{%
  For reasons that have little to do with mathematics, it has been
  fashionable to impute such credit to fanaticism.  Fashion, however, is
  as fleeting as it is shallow.  You and I must aim deeper than that.
  Refer to~\cite[``St.~Thomas Aquinas'']{CatholicEncyc}.%
}
% I did not wish to strike the following footnote.  However, the
% footnote drags the innocent reader into a dispute for which he did not
% ask.  Unfortunately, I'll just have to take the bigotry.  Fortunately,
% I am not important enough for the bigotry, because of a sentence in
% the last chapter of a long book, to have much practical effect.  At
% any rate, the footnote is struck.
%\footnote{%
%  Though I am aware of the careless bigotry (how could one be unaware?)\
%  that affects to sneer at such remarks in the regrettably unstable
%  country and era in which I write, the notice of St.~Thomas is not a
%  social signal.  The book is not about signaling.  It should go without
%  saying (but somehow it doesn't) that you simply cannot be serious
%  about~2500 years of Western philosophy while summarily writing off the
%  schoolmen.
%
%  That this footnote should be necessary shames reason and embarrasses
%  posterity.  As Gibbon might have added---no friend to the schoolmen
%  he---that this footnote should be necessary condemns the taste of our
%  age.%
%  %What a waste of good ink.%
%}

Even so, unexpectedly, the experience of writing the book has
illuminated in my sight a certain mathematical inadequacy of
physical-intuitional methods.  Wherever large enough a mathematical
structure is built by mostly physical-intuitional methods, a fact of
experience seems to emerge: the structure begins to creak.

Professional mathematicians would have told us as much.

And yet---the na\"ive, merely plausible extension of mathematical
methods carries greater impact, and retains more power, than pure
mathematics may like to admit, as in \S~\ref{prob:200.90} for example.
Anyway, such na\"ive extension is more
than \emph{merely} plausible.  Such na\"ive extension yields correct
results in the main, too, as for example in this book's applied-level
development of complex exponentials and analytic continuation.
Sometimes, bending the arc of reason to follow the trail of intuition
is the right thing to do.

Substantial value subsists in the applied approach.

\section{Convergence}
\label{conclu:convergence}

\index{convergence}
\index{professor}
One of the more debatable choices I have made during the writing of this
book has been to skip explicit justification of the convergence of
various sums and integrals---or, if you prefer, has been to leave the
justification in most instances as an exercise.  A pure mathematician
would not have done so, not at any rate in the same way.  I still recall
an undergraduate engineering lecture, though, decades ago, during which
the lean, silver-haired engineering professor---pausing the second time
to justify an interchange of summing operators---rather turned to the
class and confided, ``Instead of justifying the interchange again, let's
just \emph{do it,} okay?''  That professor had his priorities straight.

%\index{Weierstrass, Karl Wilhelm Theodor (1815--1897)}
Admittedly, to train the intuition, a mathematical education probably
ought at some stage to expose the student to formal, Weierstrassian
tests of convergence.  However, except at that stage, the repetitive
justification of convergence soon grows tiresome.  If the reader cannot
readily tell for himself, \emph{in a given concrete case,} whether a sum
converges, is this not probably because the reader fails to understand
the term being summed?  If the reader indeed fails to understand, then
Weierstrass can hardly help.

Though the proposition remains debatable, I believe---at least insofar
as the book you are reading is an applied work---that the book's
approach to convergence has been the right one.

%Weierstrassian tests have their place in mathematics.
%Weierstrassian tests have, however, had less of a place in this book.

\section{Klein}
\label{conclu:klein}

\index{Klein, C.~Felix (1849--1925)}
\index{G\"ottingen}
November 2,~1895, at G\"ottingen, the mathematician Felix Klein (like
Frege a German) masterfully summarized both sides of these matters.  His
lecture and its English translation having passed into the public
domain, we are free to quote Klein at length as follows.
\begin{quotation}
  \index{Newton, Sir Isaac (1642--1727)}
  \index{Leibnitz, Gottfried Wilhelm (1646--1716)}
  \index{Gauss, Carl Friedrich (1777--1855)}
  \index{Abel, Niels Henrik (1802--1829)}
  \index{Cauchy, Augustin Louis (1789--1857)}
  \index{Dirichlet, J.~Peter Gustav Lejeune (1805-1859)}
  \noindent
  \ldots\ With the contemplation of nature as its starting point, and its
  interpretation as object, a philosophical principle, the principle of
  continuity, was made fundamental; and the use of this principle
  characterizes the work of the great pioneers, Newton and Leibnitz, and
  the mathematicians of the whole of the eighteenth century---a century of
  discoveries in the evolution of mathematics.  Gradually, however, a more
  critical spirit asserted itself and demanded a logical justification for
  the innovations made with such assurance, the establishment, as it were,
  of law and order after the long and victorious campaign.  This was the
  time of Gauss and Abel, of Cauchy and Dirichlet.  But this was not the
  end of the matter.  Gauss, taking for granted the continuity of space,
  unhesitatingly used space intuition as a basis for his proofs; but
  closer investigation showed not only that many special points still
  needed proof, but also that space intuition had led to the too hasty
  assumption of the generality of certain theorems which are by no means
  general.  Hence arose the demand for exclusively arithmetical means of
  proof; nothing shall be accepted as a part of the science unless its
  rigorous truth can be clearly demonstrated by the ordinary operations of
  analysis\mdots [W]here formerly a diagram served as proof, we now find
  continual discussions of quantities which become smaller than, or which
  can be taken smaller than, any given small quantity.  The continuity of
  a variable, and what it implies, are discussed\mdots

  Of course even this assigns no absolute standard of exactness; we can
  introduce further refinements if still stricter limitations are placed
  on the association of the quantities.  This is exemplified \ldots\ in
  the efforts to introduce symbols for the different logical processes, in
  order to get rid of the association of ideas, and the lack of accuracy
  which creeps in unnoticed, and therefore not allowed for, when
  ordinary language is used\mdots

  Summing up all these developments in the phrase, \emph{the arithmetizing
  of mathematics,} I pass on to consider the influence of the tendency
  here described on parts of the science outside the range of analysis
  proper.  Thus, as you see, while voluntarily acknowledging the
  exceptional influence of the tendency, I do not grant that the
  arithmetized science is the essence of mathematics; and my remarks have
  therefore the two-fold character of positive approbation, and negative
  disapproval.  For since I consider that the essential point is not the
  mere putting of the argument into the arithmetical form, but the more
  rigid logic obtained by means of this form, it seems to me
  desirable---and this is the positive point of my thesis---to subject the
  remaining divisions of mathematics to a fresh investigation based on the
  arithmetical foundation of analysis.  On the other hand I have to point
  out most emphatically---and this is the negative part of my task---that
  it is not possible to treat mathematics exhaustively by the method of
  logical deduction alone\mdots

  In the short time at my disposal I must content myself with presenting
  the most important points; I begin therefore by tracing the relation of
  the positive part of my thesis to the domain of geometry.  The
  arithmetizing of mathematics began originally, as I pointed out, by
  ousting space intuition; the first problem that confronts us as we turn
  to geometry is therefore that of reconciling the results obtained by
  arithmetical methods with our conception of space\mdots  The net result
  is, on the one hand, a refinement of the process of space intuition; and
  on the other, an advantage due to the clearer view that is hereby
  obtained of the analytical results considered, with the consequent
  elimination of the paradoxical character that is otherwise apt to attach
  itself to them\mdots  [T]here still remains the more important question:
  What justification have we for regarding the totality of points in space
  as a number-manifoldness in which we interpolate the irrational numbers
  in the usual manner between the rational numbers arranged in three
  dimensions?  We ultimately perceive that space intuition is an inexact
  conception, and that in order that we may subject it to mathematical
  treatment, we idealize it by means of the so-called axioms\mdots

  \index{Jacobi, Carl Gustav Jacob (1804--1851)}
  Another question is this: Practical physics provides us plentifully with
  experimental results, which we unconsciously generalize and adopt as
  theorems about the idealized objects\mdots  [T]he theorem that every
  finite elastic body is capable of an infinite series of harmonic
  oscillations [belongs to this category]\mdots  [Is such a theorem],
  taken in the abstract, [an] exact mathematical [theorem], or how must
  [it] be limited and defined in order that [it] may become so?\,\ldots\
  You see here what is the precise object of \ldots\ renewed
  investigations; not any new physical insight, but abstract mathematical
  argument in itself, on account of the clearness and precision which will
  thereby be added to our view of experimental facts.  If I may use an
  expression of Jacobi's in a somewhat modified sense, it is merely a
  question of intellectual integrity, ``die Ehre des menschlichen
  Geistes.''

  After expressing myself thus it is not easy, without running counter to
  the foregoing conclusions, to secure to intuition her due share in our
  science;%
  \footnote{Klein characterizes mathematics as a ``science'' so
  often that, insofar as my book is quoting Klein with approbation, I
  should note that I have never been persuaded that \emph{science} is
  the right word for it.  This is a minor quibble, and my witness
  may not weigh much in comparison with that of the eminent Klein,
  but I would nevertheless prefer rather to regard mathematics as a
  branch of \emph{philosophy.}} and yet it is exactly on this antithesis
  that the point of my present statements depends.  I am now thinking
  not so much of the cultivated intuition just discussed, which has been
  developed under the influence of logical deduction and might almost be
  called a form of memory; but rather of the na\"ive intuition, largely
  a natural gift, which is unconsciously increased by minute study of
  one branch or other of the science.  The word \emph{intuition} is
  perhaps not well chosen; I mean it to include that instinctive feeling
  for the proportion of the moving parts with which the engineer
  {c}{r}{i}{t}{i}{c}{i}{s}{e}{s} the distribution of power in any piece
  of mechanism he has constructed; and even the indefinite conviction
  the practiced calculator possesses as to the convergence of any
  infinite process that lies before him.  I maintain that mathematical
  intuition---so understood---is always far in advance of logical
  reasoning and covers a wider field.

  I might now introduce an historical excursus, showing that in the
  development of most of the branches of our science, intuition was the
  starting point, while logical treatment followed\mdots The question in
  all such cases, to use the language of analysis, is one of
  interpolation, in which less stress is laid on exactness in particular
  details than on a consideration of the general conditions\mdots  Logical
  investigation is not in place until intuition has completed the task of
  idealization\mdots

  \index{Gymnasium}
  \index{instructor}
  \index{university}
  \index{professor}
  I must add a few words on mathematics from the point of view of
  pedagogy.  We observe in Germany at the present day a very remarkable
  condition of affairs in this respect; two opposing currents run side
  by side without affecting one another appreciably.  Among instructors
  in our Gymnasia [that is, roughly as understood in North American
  terms, in Germany's elite, preparatory high schools] the need of
  mathematical instruction based on intuitive methods has now been so
  strongly and universally emphasized that one is compelled to enter a
  protest, and vigorously insist on the necessity for strict logical
  treatment\mdots Among the university professors of our subject exactly
  the reverse is the case; intuition is frequently not only undervalued,
  but as much as possible ignored.  This is doubtless a consequence of
  the intrinsic importance of the arithmetizing tendency in modern
  mathematics.  But the result reaches far beyond the mark.  It is high
  time to assert openly once for all that this implies, not only a false
  pedagogy, but also a distorted view of the science.  I \ldots\ have
  always discouraged the laying-down of general rules for higher
  mathematical teaching, but this shall not prevent me from saying that
  two classes at least of mathematical lectures must be based on
  intuition; the elementary lectures which actually introduce the
  beginner to higher mathematics---for the scholar must naturally follow
  the same course of development on a smaller scale, that the science
  itself has taken on a larger---and the lectures which are intended for
  those whose work is largely done by intuitive methods, namely, natural
  scientists and engineers.  Through this one-sided adherence to logical
  form we have lost among these classes of men much of the prestige
  properly belonging to mathematics, and it is a pressing and urgent
  duty to regain this prestige by judicious treatment.

  \index{botanist}
  \index{tree}
  \index{branch}
  \index{root}
  To return to theoretical considerations, the general views which I
  uphold in regard to the present problems of mathematical
  science need scarcely be specially formulated.  While I desire in
  every case the fullest logical working out of the material, yet I
  demand at the same time an intuitive grasp and investigation of the
  subject from all sides.  Mathematical developments originating in
  intuition must not be considered actual constituents of the science
  till they have been brought into a strictly logical form.  Conversely,
  the mere abstract statements of logical relations cannot satisfy us
  until the extent of their application to every branch of intuition is
  vividly set forth, and we recognize the manifold connections of the
  logical scheme, depending on the branch which we have chosen, to the
  other divisions of our knowledge.  The science of mathematics may be
  compared to a tree thrusting its roots deeper into the earth and
  freely spreading out its shady branches into the air.  Are we to
  consider the roots or the branches as the essential part?  Botanists
  tell us that the question is badly framed, and that the life of the
  organism depends on the mutual interaction of its different
  parts.~\cite{Klein}
\end{quotation}
I entertain no illusions that the book whose end you have reached
measures up to Klein's high standard---nor is the book a product of the
mathematical profession in any case, so Klein's standard was never one
the book precisely sought to meet---yet more than a century after Klein
spoke, I can still think of no more fitting way to end the book than
with Klein's robust reflections.  During quiet moments, when the applied
mathematician is not out throwing bridges across chasms and such, he may
well ponder that which Klein has taught.

\nopagebreak

\noindent\\
THB

