**
The Letter “G”**
By H. L. Haywood
*
The New York Masonic Outlook *
– December 1927
**A**mong
the features of The Regius Poem, the oldest existing
MS. of Freemasonry, written in 1390 or thereabouts
to set forth the Legend of the Craft as then
understood, none is more remarkable than the great
importance attributed to Euclid and his famous
“Elements of Geometry.” The fact that its author
knew little about Euclid is of small importance; the
fact that he boldly placed EucLETTERGlid among the
founders of “thys craft”—if indeed he may not be
said to have made Euclid the founder—is of great
importance; and the fact that he made Masonry and
Geometry almost synonymous is of greater importance
still. This emphasis on the central place of
Geometry in the system of Masonry reappears again
and again in the documents that followed the Regius,
each of them a variation of the old Legend. They one
and all laid primary stress on Geometry, of which
the Dowland MS. (A.D. 1500) said, for example, “It
is called throughout all this land Masonrye.” What
led our ancient brethren thus to exalt Euclid? The
same thing obviously that led them to hold Geometry
as a sacred secret to be conveyed to their initiates
under solemn oath-bound forms. That led them to hold
in reverence “our worthy Brother Pythagoras,” who
had made a religion of Geometry. That had led them
to transform numbers (3, 5, 7, for example), lines,
angles (the square), circles (the compasses), points
(point with a circle), triangles (the Forty-fifth
Proposition), cubes, and many other mathematical and
geometric figures into symbols, whereby to teach
their young men how to build and how to live, many
of which we have inherited. That led their
successors, the Speculative Masons, to hang the
Letter “G,” initial of the word Geometry, above the
Master’s station as a perpetual reminder that the
art of Masonry owed its existence to the science of
Geometry. As students and devotees of present-day
Masonry we cannot appreciate the full force and
significance of this, or win from it that full
comprehension of the philosophy of Masonry which all
of us desire, unless we recall the fact—frequently
overlooked — that in the period when our oldest
records were written Geometry and Mathematics meant
the same thing, so that if the authors of those
records were now living they would give the primacy
to Mathematics, of which Geometry is only one among
many branches. The Operative Masons strove—in their
own way, under their own limitations, and suffering
from their heavy handicaps—to develop what we in
this day would describe as a Mathematical Philosophy
of Life. It is not difficult for us to imagine what
led them to such a position. They lived in a period
when many of the arts and most of the sciences were
lost; when the majority of men and women were living
like pigs in a sty under a system of brutish serfdom
or slavery; when ignorance and superstition lay like
darkness over Europe and England; when whole
populations were swept away by famine, or
pestilence, or by numberless petty wars that meant
little to them except an opportunity to escape from
the miseries of life; when any discoverer, inventor,
or thinker (Roger Bacon, for example) stood in
danger of being accused of black magic; when most of
the values of life lay at a subhuman level. In the
midst of all this the Craft of Masons was able to
produce the cathedrals and all that went with them,
an achievement that stood as far above the average
of human production in that day as the Alps stand
above their valleys. It is little wonder that the
Masons felt themselves in possession of secrets
almost supernatural; or that they exalted the
science by means of which they had perfected their
art; or that they reasoned that the whole system of
society might be redeemed by that science, if only
it could be broadened out to cover all human
interests and activities. I am not implying that
they succeeded in thus broadening it out, or that
any such thing could have been accomplished at that
stage of development by anybody else; neither was
possible; I am only saying that it was an ideal, a
reasonable ideal, and a dream. It was a dream that
had haunted others many centuries before the Masons’
Craft of the cathedral builders had come into
existence. The Egyptians had dreamed it, for it was
they who first discovered the rudiments of Geometry
while learning to measure the waters of the Nile and
to survey their fluviatile lands, and they held it
to be the foundation of wisdom; Pythagoras had
dreamed it, fondly hoping to find a universal
philosophy in the science of numbers and, as already
stated, making a religion out of mathematics; and
Plato had dreamed it also, and Aristotle, the former
becoming thereby the chief of philosophers, the
latter the first of scientists, with equal reverence
for mathematics. Indeed, it is said of Plato that
when a disciple inquired, “What does God do all the
time,” Plato replied, “God geometrizes.” These men,
and many like them in their day, had a double
feeling about mathematics. From one point of view
they saw in it a revelation of order in the
universe, suggesting that back of the chaos and
welter of the world which so troubled the minds of
men, there is an inner system and symmetry. From
another point of view they saw in the method of
mathematics a marvellous instrumentality of thought
that might be used in all the fields of life.
Alas,
this latter idea, so fruitful in the thought of
Plato and Aristotle, did not take root, but
slumbered for some two thousand years, waiting for a
generation of men capable of conceiving and
developing a conception so profound. Save for the
dream of the Freemasons, mathematics continued to be
considered a merely technical affair, useful to
engineers and carpenters, but nobody guessed that it
might possess any usefulness outside that narrow
technical field. Mathematics a possible philosophy
of human life, useful everywhere, to everybody, in
all possible fields of human activity! It was
impossible for men during those two thousand years
even to formulate such an idea. It has remained for
our own generation, building on the work of two or
three generations immediately proceeding, to make
the old dream come true. That it has at last come
true everybody knows who knows anything at all about
present day mathematics, more especially
mathematical logic and mathematical philosophy; the
story of how it has come true is a romantic story of
great achievement—the greatest achievement, many
believe, in all the long annals of mankind; and it
is a story of grave moment to every Mason who in any
degree cherishes that old ideal of the Craft that
led our forbears to hang above the Master’s station
the letter “G.” If only space permitted, and if this
were the medium proper to it, it would be a pleasure
to repeat in condensed form the history of those
astonishing discoveries which have made of
mathematics a new thing. Such a history would begin,
perhaps, with the great Leibnitz, a contemporary of
Sir Isaac Newton, who said, “Mathematics is my
philosophy.” It would touch upon Spinoza, the mighty
Dutch philosopher, to whom “belongs the credit of
having been the first important thinker to see
clearly that the method of Euclid’s ‘Elements’ was
far more general than its matter,” and who tried to
erect a system of ethics on the basis of Geometry,
failing because the time was not ripe. It would tell
of the discovery of the non-Euclidean Geometries by
Bolyai, Lobachevski, Riemann, Klein, Lie, etc., and
of the epoch-making definition of infinity made by
Dedekind and Cantor. It would touch upon the work of
scores of other thinkers to whom the world is
infinitely more indebted than to most of its popular
heroes. It would reach its climax with the crowning
discovery of all, the discovery that at bottom
mathematics and logic are one and the same
thing—“one,” as Keyser has put it, “in the sense in
which the roots the trunk and the branches of a tree
are physically one.” And it would then only remain
to show that on the basis of that system of thought
which is at once pure mathematics and pure logic the
best thinkers of our own day are even now
painstakingly building a new philosophy of life,
destined as surely to replace the older
philosophies, as mathematics itself replaced the
crude rule-of-thumb methods of primitive men. This,
however, as I have already said, is not the place
for such a history (though it is one of a kind of
things that should be more discussed in Masonic
journals); the studious reader must be referred to
the works of Keyser, Russell and Whitehead,
especially the “Mathematical Philosophy” of Keyser
and “The Principles of Mathematics” of Russell. But
how, you may well be asking, has it been possible to
make a philosophy out of mathematics? The answer
lies in the fact—at first of seeming small
importance—that mathematics is a system of thought
without content; the system holding true whatever
kind of content you may be able to pour into it. If
you pour into it the problems that bother you as a
human being, problems of thought, or of conduct, or
of the nature of the world, the result will be a
philosophy; and just because the system you will be
using is the system of mathematics you will find
yourself in possession of a mathematical philosophy.
One
example will suffice. Euclid believed—like everybody
else until less than a century ago, and like the
vast majority of people even now—that Geometry has a
subject matter peculiar to itself, consisting of
space, points, lines, surfaces, angles, planes,
circles, volumes, spheres, spirals, and similar
entities. These entities were supposed to be the
content of Geometry, and there was supposed to be no
Geometry apart from such contents. This is known to
be false. Geometry may have to do with such
entities, but there are countless other kinds of
entities that will serve equally well. Geometry is a
self-consistent system of axioms or postulates,
postulation functions, and doctrinal functions—in
other words, a system of logical forms. Fit anything
you can into those forms and the result will be
Geometry of that thing. It is a mere accident that
Euclid selected the set of entities he did; he might
have selected any one of an infinite number of other
sets, and the result would have been the same. What
is true of pure Geometry is true of all pure
mathematics. As written in the usual text-books it
appears to deal with numbers, figures, letters, and
a great variety of strange diagrams and bewildering
formulate. But all that symbolic material is merely
the technical machinery of the science; you may
substitute for the numbers, figures, x’s, y’s, z’s,
etc.—any suitable things you wish— and the result
will be a mathematical treatment of those things.
This means that many of the experiences, problems,
ideas, and ideals of what we call our everyday life
are all mathematizable. It was mathematics that
enabled the physicists, chemists, biologists,
astronomers and workers in other applied sciences to
give us the steam-engine, the dynamo, the aeroplane,
telephone, radio, modern surgery and all the other
wonders; there is no reason why it should not lead
to results equally marvellous if ever we learn to
use it in dealing with all those momentously serious
matters we call our problems of human life.
As a
matter of fact, mathematics is closer to everyday
life than we are accustomed to think. Each of us is
a mathematician unaware. A man may not be able to
divide one fraction by another; he may be unable to
count to ten; nevertheless he understands somewhat
of mathematics if he understands anything at all,
because mathematics is involved in the act of
understanding itself. Without those ideas which
mathematics makes the object of its special study,
and to which its name is therefore given, no ideas
at all would be possible; thinking itself would fall
to pieces. Consider such ideas as these: counting,
comparing, grouping, relating, serializing, adding,
dividing, separating, order, dimensions, variation,
dating, function, relations, infinity, etc., etc.
Mathematics is nothing other than a rigorously
precise definition and use of these and similar
ideas and of the organization of them in various
combinations. If we struck from our familiar English
language every word used to express such ideas in
all their multitudinous forms and ramifications our
noble speech would be instantly reduced to an
unintelligible chaos of meaningless noises.
Mathematics is intelligence in its purest form. In
it our most useful ideas are comprehended with
complete thoroughness and defined with ultimate
precision. Nay, one may justly say even more! One
may say that it is in mathematics and in mathematics
alone, that some of these ideas receive the only
meaning they have, so that if a man is to use them
at all he must know them as the mathematician knows
them. When therefore our modern mathematicians
released their science from its forbidding technical
apparatus, setting mathematical ideas free, so that
they, in all their precision, purity,
self-consistency, and luminousness became available
to all men, and for all purposes, they achieved for
us all a service of unlimited usefulness, and they
made possible a new future, a new development, a new
hope for the world. In a newer manner, and in a way
that only their rich modern equipment made possible,
they have hung the letter “G” within reach of all
who would become masters of the art of life, thereby
fulfilling for us the old dream of our Masonic
forefathers who dared to hope that their beloved art
of Masonry might become as universal as the light. |