 
                        20080527.00

                  History and Mathematics

     Between history and mathematics there are differences
with far-reaching implications.  History is my
recapitulation of the past.  Fundamentally, history is my
story of my memory of what I have seen and heard and felt,
in short, what I have experienced.(erlebt) My memory is
notoriously feeble, fragile and unreliable.  Mathematics is
a skill that I acquire and cultivate by practice. It
entails no memory of the past.

     I can have no memory of what I have not experienced, -
but I can hear and read the accounts of the experiences of
others, and vicariously make these experiences my own.  To
be able to tell the story of other persons, I must share
their experiences, if only vicariously.  To purport to tell
the stories of other persons is to claim the vicarious
sharing of their experiences.  That is the function of the
historian.

     All history is of the past. Even "current events" are
past events. Their currency is always only relative.  When
I report what I am "presently" experiencing, I am, strictly
speaking, reporting about the past.  The present, if it
exists at all, is left behind in the processes of
consciousness and communication.  The present is an
illusion: it is a consequence (an expression, a reflection)
of the vividness and force of recent memory.  All memory is
of the past, all memory is fleeting, fallible and
incomplete; so is all knowledge of the past.

     Memory is an echo, a recapitulation of a prior
original mental state.  That original mental state is but
an inference, for in and of itself the original mental
state is inaccessible.  Even the initial awareness of that
original mental state is already a memory. My knowledge of
myself, my story of myself, my history of myself, is a
chain of progressively attenuated memories.

     I tell my story in words, by means of language. There
is no other method or mechanism of substantive
communication. The words themselves, the story itself,
becomes a subject of memory. I remember not only the
original experience. I remember also my account, my story,
my history of it. The memory of the account merges with and
often largely masks the memory of the original experience.
The elements that enter into my present memory fuse and
become indistinguishable.

     I tell my (hi)story to others and they tell their
(hi)stories to me.  The histories fuse and the elements of
the combined histories become indistinguishable. There
develops then a very large and complex body of accounts of
the past, i.e. of memories of uncounted individuals,
documented in thousands of books, massive and impermeable
to analysis.  This accretion and concatenation appears to
take on a reality of its own, independent of the memory of
any given individual.  We henceforth think and speak as if
the cumulative histories made the past accessible to us.
But this accessibility is an illusion.  History, the story
that we tell each other, remains rooted only in fragile and
evanescent memory.

     Discipline, especially academic discipline, will
nurture and enforce a certain uniformity of account and
interpretation; and the consistency of accounts and stories
will seem to vouch for their validity and truth. Yet it can
easily be demonstrated that the objective validity and
truth seemingly vouched for by consistency are only
relative, subject to revision, rescission and correction.
Dilthey's claim of validity for the historical "sciences"
notwithstanding, The bounds of history are undefinable in
practice as well as in theory.  dependent as they are on
individual memory and individual experience (Erleben).
Their objective truth is always only an approximation.
Their real truth is subjective; it inheres in the imagery
and drama that they evoke in the mind of the historian,
imagery and drama that is peculiar to each re-teller of the
history,

     Nonetheless, the "objective" truth of historical
statements is of great practical importance, as exemplified
by the sworn testimony of the witness in a court of law.
The objective truth or falsehood of historical statements
requires to be reconciled with the insight that history is
always deficient and incomplete.  The crux here is that
subjective truth derives from its passionate reflection and
expression of experience, whereas objective truth reflects
the consistency of statements, is therefore a logical,
hence a mathematical truth.  Another way of considering
this is to note that if subjective truth is the
correspondence of the account with the experience there can
also be subjective falsehood when the account does not
correspond to the experience; and while experience is
admittedly and assertedly subjective, the subjective
accounts of (other) witnesses to the same (given) event
will also shed light on the subjective and objective
veracity of any given account.

     The "memory" on which mathematics relies is
fundamentally and radically different from historical
memory.  Mathematics knows no past; it is practiced in an
eternal present.  It knows no past.

     Although mathematics can, and does rise (develop) to
extraordinary complexity which is far beyond me, I content
myself with reflecting on the simplest of mathematical
procedures, counting, addition, subtraction, multiplcation
division of natural numbers, the simple operations of
algebra and analytic geometry.  Implicitly assuming that
there is no fundamental qualitative difference between the
epistemological significance of the simplest mathematical
operations and the most complex.  If I am wrong, an author
more sophisticated than I will have to discover and define
these differences.

     All the mathematics that I know (understand) entails
the assimilation into intellect and the expression
thereafter as spontaneous intellectual (mental) function of
what is intuitively obvious or becomes intuitively obvious
with suitable demonstration. The difference between those
of us mathematically talented and those of us not
mathematically talented, as myself, is the facility of
intuition. And indeed it is arguable whether there might
not be some mathematicians with intuition overly open to
novelty, to whom seems obvious what no one else is able to
understand.  But this is an issue which does not require to
be broached in this context.

     What does matter is that neither the mathematical
intuition nor its formal integration into (mathematical)
thought, although like all other experience, it occurs in
time, is an historical event. I.e. the historical
circumstance of its discovery and assimilation, the name of
the teacher or the textbook, the date and circumstance of
the lecture, are all irrelevant to its meaning. The
correctness of our counting, addition, subtraction, etc, is
independent of the occsions on which we do so, and
independent also of the occasions on which we learned. The
skill that we have acquired has become an integral
component of our mental function, no less compelling, and
ultimately qualitatively indistinguishable from a language
that we have learned to speak or a musical instrument that
we have learned to play.

     These considerations open yet another dimension the
analysis of which we need not pursue.  While it is the
dominant symbolism of both mathematics and language which
makes them unhistorical, there are important differences
between mathematics and language in respect to the
relationship of each to experience.  The reality which
mathematics reflects and to which mathematics makes its
appeal is an element of our mental constitution as
intellectual individuals.  The reality which language
reflects and to which language makes its appeal is an
element of our social constitution as intellectual
community.

     I propose that the validity (truth) of mathematics is
its objectivity, while the validity (truth) of history is
its subjectivity.  One does not have to accept
Kierkegaard's assertion that subjectivity is the only
truth, to acknowledge that subjectivity entails elements of
truth peculiar to it.  As I cannot conceive of telling a
story independent of memory, so I cannot conceive of the
history of anything whatsoever independent of the memory of
the individual who purports to tell - or to write an
historical account.  All history shares the qualities of
memory, its evanescence and its incompleteness.  Every
story is dependent on the memory and hence on the
experience of the individual who tells it.  Academic
history, no matter how abstract and defined, cannot escape
the dependence on memory and on experience.

     The memory entailed in mathematics is so different
that it requires to be distinguished by a different name.
Training in mathematics is training to think by rote,
independent of memory, to recapitulate without reflection,
patterns of symbols by which thought is henceforth defined
and controlled.  The truth of mathematics derives from the
power which such symbols exercise in defining and
controlling thought.  The truth of mathematics is the
predictable uniformity which in turn makes it indispensable
in constructing the network of unambiguous statements that
constitutes the ideal, if not the actual achievement of
natural science.  As mathematical sciences are in theory
invariably definable, so historical "sciences",
notwithstanding Dilthey, are in theory invariably
undefinable, dependent as they are on individual memory and
individual experience (Erleben).  Their "truth" is the
imagery and drama, peculiar to each re-teller of the
history, that they evoke in the mind of the historian.

     Admittedly, both terms, history and mathematics, serve
as acronyms, as summarizing definitions for much broader
fields; for fields so broad that I can only sketch their
extent.

     It is persuasive to reduce (zurueckfuehren) history to
the story which I tell from memory; but as I have pointed
out, the telling, the story, die Erzaehlung, then itself
becomes an experience, becomes something which is in turn
remembered, which re-enforces, overlaps, and may even
replace or mask the original.  And beyond the story there
are the concurring reports from other tellers of stories
(historians), relating their experience of the signal event
from their different perspective.  There will be mementos,
artifacts, pictures, paintings, drawings, photographs,
which themselves become subjects of history, which add to
the complexity and richness of the historical picture.

     There are, come to think of it, many species of
"history".  There are the accounts of purported glaciers,
ice ages and volcanoes 500 million years ago; there are
accounts of dinosaurs and other exotic animals, now long
extinct, who have never been the subject of any
individual's experience but whose existence is (merely) the
inference of paleological studies.  There is folklore which
merges into history and there are myths too numerous to
relate. There are the histories and gospels on which
religious beliefs and practices are based. There are
cosmological extrapolations of what "happened" in the
universe uncounted light years ago.  There are scientific
"observations", past present and contemplated.  There are
reports of organizational, especially judicial proceedings,
and there is the sworn judicial testimony of witnesses
purporting to be the truth, the whole truth and nothing but
the truth.  All of this history makes claims to objective
truth, claims which require to be reconciled with the
hypothesis that all history, in essence, addresses memory,
and that therefore the truth of all history is subjective.

     It is not possible to find, or to define, an algorithm
that will do justice to the claims of each of these diverse
histories to objective truth.  One must proceed
empirically, and for each history examine the evidence, the
rationale, the conclusions and the claims of truth, to
understand what they are about and how they should be
interpreted. The one unifying factor which may provide a
common denominator to all these claims of objective truth,
is that in all cases, the story which is told, is told in
words, words which are common to the hearers, words which
are therefore elements of objectivity, subject to social
assessment and comparison.  The objective truth of the
story which is told, cannot therefore relate to the
experience, which is unique to each individual historian:
It must relate to the (words of the) story itself, which by
virtue of having been told has become public. That makes
sense.

     It appears then that history, as the telling of
experience, logicizes - if I may coin a word - that
experience, makes it socially accessible and makes it, by
that token, the subject of objective truth.  Logic is a
subset of mathematics and exhibits the same qualities of
objective truth.  The objective truth, to be sure
(wohlbemerkt) is a quality of the account; it is not a
quality of the experience which the account relates.

     As a corollary and in an appendix, one must consider
the phenomena of fantasy, invention, lying and pathological
lying. It seems that for various reasons, or none at all,
one may invent or fabricate a story simulating experience,
and leading the hearer, and perhaps oneself to be persuaded
of the truth of ones invention.

     The conviction of subjective truth is appropriately
and properly called faith; and it may be that it was the
subjectivity of the truth of history which explains the
predomenant role of faith in Luther's theology.

     Just as by virtue of its being expressed in language,
history become susceptible to a degree of objective truth,
so the natural sciences, as soon as they deviate or detach
themselves from mathematics, become susceptible to a degree
of subjective truth.  In order to assess this propensity,
one must closely scrutinize each science and interpret it
accordung to its specific and peculiar characteristics.

                     * * * * *
Copyright 2008, Ernst Jochen Meyer
