Anticipation in Geometric Calculus: from Leibniz/Grassmann to E. Kähler José G. Vargas ...

44 Pages·2017·0.25 MB·English

Anticipation in Geometric Calculus: from
Leibniz/Grassmann to E. K¨ahler
Jos´e G. Vargas
July 26, 2017
josegvargas4 [at] sc [.] rr [.] com
PST Associates, LLC (USA)
ABSTRACT
We document the mathematical anticipation that started with Leibniz
under the name of geometric characteristic. It was Grassmann who, one
and a half centuries later, adopted the forward looking stance of Leibniz
and, under the name of geometric calculus, gave rise to an avalanche of
mathematical ideas not yet carried to their last consequences.
We proceed to explain how the main issues with Grassmann’s main sys-
tem, the progressive-regressive calculus, were solved. On the one hand, Clif-
ford extracted his namesake algebra from the web of entangled concepts
present in that system. On the other hand and independently of Clifford’s
´
work, E. Cartan enriched the calculus by basing it on another one of Grass-
mann’s great ideas, the exterior algebra. Years later, his profound knowledge
of the progressive-regressive calculus allowed Cartan to correct its limita-
tions. Thus, after reformulating the Erlangen program, he used his exterior
calculus to create modern differential geometry all the way to the famous
Cartan-Finsler connection (year 1934).
We show the naivet´e of what modernly goes by the name of geometric
calculus and proceed with advocation of the overlooked Ka¨hler calculus of
differential forms, where the underlying exterior algebra is replaced with a
superseding Clifford algebra. But K¨ahler’s work lacked sophistication when
post-scalar valued differential forms were concerned. We point out the exten-
sions of his calculus to one for Clifford-valued differential forms and conclude
with lessons of different types learned from Leibniz-Grassmann anticipation,
both for mathematics and, more importantly, for the formidable problems
that humanity, society and the culture face.
We wish to make potential readers aware of the fact that the term an-
ticipation is here used from the point of view of the history of ideas (or
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intellectual genealogy) and not from the acceptation of this term in papers
in this notebook. It is, however, fitting to mention the emergence from the
so extended Ka¨hler calculus of a program for physical unification.
1 Introduction
In this paper, we speak of the evolution of algebra, calculus and geometry
since the work by Grassmann in the mid eighteen hundreds as per the En-
glish translation by Kannenberg of his work [1]. This work was explicitly
acknowledged by Grassmann himself as the realization of the Leibnizian an-
ticipation. He is the one who was successful at launching the mathematical
developments that made of Leibniz vision the beginning of an anticipation.
Mutatis mutandis, the development of this anticipation takes us through
Clifford and specially E. Cartan to the Ka¨hler calculus (KC), developed in
1960-62[2], [3], [4]. Itissignificantthat(toourknowledge)henever returned
¨
to his 1933 paper: Uber einer bemerkenswerte Hermitesche Metric [5], which
arguably is the work for which he is best known, even more than for his
contribution to what is nowadays known as the Cartan/K¨ahler theory of
exterior systems [6] (His metric nevertheless appears now and then in his
work, but rarely if ever from 1958 on. And it is still more significant that in
1992, when he already was of very old age, he returned to the KC [7] with a
more ambitious goal than in the 1960s.
He wrote the KC in German, but it would have been difficult to under-
stand even if written in English. Several characteristics of his work go a long
way towards explaining why this is so, and why it is almost forgotten in spite
of its many merits. A main one is that virtually all of modern physics —-
whereitismostuseful—-isdonewithtechniquesthatwerefirmlyestablished
and widely used by the time when he made public the KC.
We advance that, as will become clear, any connection between the work
ofE.Cartanand/orKa¨hlerwithdifferentareasofmathematicswillbemeant
to refer to connections with the Leibniz/Grassmann anticipation. Cartan
and Ka¨hler constituting the maximum exponents of its development, their
names may occasionally be used to refer to the present end point of that
anticipation.
With regards to other authors that have contributed to this anticipation,
we confine ourselves to significant developments directly related to Grass-
mann’s work. Clifford falls in there, and less obviously Felix Klein, and less
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obviously but not less importantly Riemann. The latter’s geometry is con-
´
tained in the extensive work of E. Cartan on differential geometry, in turn
deeply connected with Grassmann’s contributions to algebra.
As intimated in the abstract, this paper started being a historical narra-
tive of a particular line of mathematical development. But this mathemat-
ics becomes physics through the judicious choice of equations of formidable
structural simplicity. That is an amazing feature of Ka¨hler’s mathematical
and physical work, if one judges by the results that he himself showed. As
for classical physics, which he did not deal with, the statement of zero cur-
vature plays a similar structural role for classical physics, though less clearly
so. Let us here refer to zero (affine curvature) as teleparallelism. Einstein
proposed it for physical unification, and Cartan instructed him on how to
extract physics out of it, but the former did not really listen [8]. The present
author has taken Cartan’s proposals to Einstein further, but we shall not
deal with this for fear of overextending ourselves.
The issue then arises of the absence in this anticipation of what most
wouldconsiderthemodernmathematicsusedinstandardtheoreticalphysics.
The K¨ahler calculus, which presently is the best implementation of Leibniz
anticipation, is modern, but not main stream. As for mathematics that are
viewed as modern, there are cases like category theory, which is virtually
unrelated to physics at this time, and string theory, which is related.
Category theory can be viewed as a sector of algebra. It is so presented
in the authoritative book on algebra by MacLane and Birkhoff [9]. It there
constitutes the fifteenth of sixteen chapters, thus not a foundation of algebra.
It may, however, be viewed as being intimately related to the foundations of
other fields of other branches of mathematics.
The motivation for category theory is largely the attempt to avoid the
concept of set as a corner stone in the foundations of arithmetic. The reason
for such an attempt is the existence of paradoxes, the most famous of which
is known as the Russell’s paradox, or the paradox of the set of all sets. For
a clear sophisticated description yet easily understandable statement of the
paradox see for instance the book “G¨odel’s proof” [10]. Their sophistication
is precisely what makes the authors use the term class where others would
use the term set. Their definition of class has the flavor of the concept of
category. In one case as in the other, the salient common feature of the
definitions of those terms is the emergence and preeminence of the role of
the relations that implicitly define the class or the category. The concept
of category, or of class for that matter, may thus be relevant for a rigorous
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approach to analysis and differentiable manifold theory [11] or to differential
forms in algebraic topology [12]. By the time that physics gets involved with,
say, differential forms, the rigor in relating these objects to mathematical
foundations has become irrelevant, provide that the use of these differential
forms (meaning how we operate with them) is correct. So, category theory
is presently irrelevant in the development of the Leibnizian anticipation.
There is then the modern mathematics inspired by physics, string the-
ory being the most significant example. Half a century of work by many
string theory groups has produced nothing but more string theory, without
approaching its original goal of applicability to the foundations of physics,
though producing other applications. Contrast that type of mathematics
with the Ka¨hler calculus and its immediate generation of new foundations
for quantum mechanics. Leibniz anticipation is very old, but the application
to physics of the Cartan/Ka¨hler phase of its development beats easily any-
thing that alternative mathematical developments have to offer. The proof
is in the pudding.
The paper is organized as follows. In section two, we cite extensively from
Grassmann’s analysis [1] of what Leibniz did or failed to do in connection
with his geometric characteristic [13]. In section three, we deal with the lines
of evolution, respectively by Clifford and Cartan, of the work of Grassmann
on that characteristic. In section four, we discuss the misappropriation by
some practitioners of Clifford algebra of the term geometric calculus to refer
to their work. In section five, we explain how the integration by Ka¨hler of the
Clifford and Cartan lines of evolution of Grassmann’s ideas come together
in his calculus. This helps to understand where the Leibniz/Grassmann
anticipation appears to be going, which we explain in section six.
In section seven, we deal with considerations on anticipatory processes as
judged from the Leibniz/Grassmann anticipation. We mean issues like the
fog enveloping research in general and the development of an anticipation in
particular, bifurcations in the process to get there, impact on paradigms (or
the status quo) and the latter’s’ response.
In section eight, we deal with lessons on societal and historical issues that
the Leibniz/Grassmann anticipation teaches us.
Section nine is a digression that points out at how much mathematics the
Leibniz/Grassmann anticipation brings together.
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2 Leibniz and Grassmann on the former’s
geometric characteristic
In general Leibnizian circles of study, in contradistinction to more math-
ematical circles, one usually speaks of universal characteristic rather than
geometric characteristic. Comparison of the terms geometric and universal
suffices to intimate that the latter was meant to be far more comprehensive
than the former. It is worth being aware of the fact that Grassmann used the
term geometric analysis to refer to the Leibnizian geometric characteristic.
Nothing much has come out from the anticipatory stance of universal
characteristic. A cursory look at Wikipedia will be more than enough to get
a bird’s eye idea of what Leibniz might have had in mind when speaking
of the geometric characteristic, which is the one in which we are interested
here. It should be thought of as the restriction to geometry of his anticipated
universal characteristic.
The cited material that follows has been extracted from the book [1]. The
page numbers given below refer to this book.
In page 404, we find that Leibniz stated the following:
I was no longer content with algebra, which provides neither ... nor...
This was because I believe we need another analysis properly geometric or
linear, by which we express place directly, just as algebra expresses magni-
tude. And I believe I have seen the means, and how to represent figures
and even machines and movements in characters, just as algebra represents
numbers of quantities, ...” Emphasis in original.
Later in the same page, we read: “I have found some elements of a new
characteristic, entirely different from algebra, ...”.
And further down: “Algebra is nothing but the characteristic of inde-
terminate numbers or quantities. But it does not directly express situation,
angles and motion.” Emphasis in original.
Finally: “Algebra is obliged to assume the elements of geometry, whereas
this characteristic carries analysis right through to the end. If it is achieved
in the manner I conceive it, one would be able to cast into symbols, that
would be only letters of the alphabet, the description of ...”
Mathematical details beyond furthering this description can be found in
the aforementioned essay.
We now let Grassmann speak, first of what Leibniz intended and then of
his failure to achieve it. From here on, all emphasis is added.
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GrassmannillustratedtheexceptionalityoftheLeibnizianideasbyspeak-
ing of a class to which they belong (p. 316): “... all the more does such an
intellect stand out in those ideas which are ahead of the times and even
anticipate ...” Further down:
Those ideas —Grassmann goes on— “are of a type that... only a priv-
ileged few contemporaries are permitted to enter and discern the wealth of
developments that will be manifold a future time”. In the same paragraph,
he stated: “Often only after centuries when time has pushed development
to the point anticipated by that intellectual force, will those thoughts sow a
rich harvest”.
He starts a new paragraph as follows: “That Leibniz’s sublime idea, ...,
namely the idea of a genuine geometric analysis, belongs among these leading
and, as it were, prophetic ideas cannot be doubted ...”.
From his next paragraph, we extract the following: “For on the one hand
the ideal of this geometric analysis, as conceived of by Leibniz for future
development, has by no means yet been completely achieved, ...”. Retrospec-
tively, the use of “completely” speaks of how unaware Grassmann was of the
fact that his own ideas would have to evolve (and the process may not yet
be complete!) before it is justified to speak of any degree of completeness.
At the top of the next page (p. 317), Grassmann starts to speak of what
amounts to be the two elements of anticipation. Indeed we read “Leibniz
himself distinguished most clearly on the one hand his idea of a pure geo-
metric analysis, whose formulation and completion floated before his eyes as
a distant goal, whose importance however he completely understood, and on
the other his attempt at such a new characteristic, ...”. He concludes the
paragraph with the statement: “The two must therefore be sharply distin-
guished if one is properly to appreciate Leibniz’s contribution.”
Grassmann then writes about Leibniz’s failure at the use of the antici-
pated outcome to guide his efforts. He thus said: “In fact, one easily con-
vinces oneself that the characteristic attempted by Leibniz does not in the
leastrealizewhathegenerallypromisedofgeometric analysis,onthecontrary
... ” And regarding Leibniz’s awareness of his own limitations, Grassmann
wrote: “... he by no means ascribed to his attempt that merit which he
would ascribe in rich measure to geometric analysis in general”.
Grassmann proceeded to rhetorically ask why Leibniz had such strong,
emphatic, esteemed convictions about a subject “whose consideration he did
not know”. He then announced that his paper would show that Leibniz
perceived correctly, and continued: “This I will do by formulating at least in
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outline an analysis which in general actually accomplishes what he regarded
as the goal of geometric analysis ...” While further elaborating on how he
intended to go about his proof, he mentioned that the essential merits of the
analysis were already enumerated by Leibniz with a degree of completeness.
ForamodernprintoutonLeibniz’sgeometriccharacteristic,seetheFrench
translation of the original work under the title ”La Caracteristique Ge-
ometrique” [13].
3 Of Grassmann, Clifford and E. Cartan
Leibniz’s forward looking stance was adopted by H. Grassmann, who used it
to produce a formidable amount of mathematical work. The most impressive
part of this work was the so called Grassmann’s system. This is not the
exterior algebra also discovered by him and that has received his name. It
is far more than that, involving multiplications that go by the names of
progressive, regressive and interior products (For a superb description of
Grassmann’s mathematical work and of this system in particular, see the 17
pages dealing with it in Cartan’s Complex Numbers paper [14]).
Grassmann intended his work to be essentially geometric, not purely or
predominately algebraic. This transpires clearly from his system. An impor-
tant question then is: what is the connection between algebra and geometry?
We are referring here to geometries where both points and lines, or both
points and vectors, are involved.
Retrospectively, this question is not precise as formulated. Algebra con-
nects directly only with so called elementary or Klein geometries [15], [16],
not with the generalized ones. This distinction was not present in Klein’s
original Erlangen program [17]. For him, the geometry of the sphere was
like the geometry of the plane, the difference lying just in the group at work
in each case (Nowadays the geometry of the sphere is viewed, in spite of its
great symmetry. as a generalized geometry not as a Klein geometry which
the geometry of the plane is. The difference was not understood in those
times). It should then not be surprising that the existence of such different
fields of mathematics as algebra, elementary geometry, calculus and differ-
ential geometry was virtually impossible to fathom at mid century. This did
not impede, however, that in the foreword to A New Branch of Mathemat-
ics, Albert C. Lewis would state: “One outcome of Grassmann’s effort to
clarify the foundations of mathematics, for example, is a degree of generality
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and rigor that evidently could not be fully appreciated until the twentieth
century” [1].
In order to make the question precise, we concern ourselves at this point
withthedifferencebetweenaffineandvectorspaces, or, ifyouprefer, between
Euclidean spaces and Euclidean vector spaces. The difference is that affine
and Euclidean spaces do not have a zero, or a special point for that matter.
Vector spaces do. There is a proper way to deal with this difference, certainly
present in Cartan’s work, but also present in disguised form in the work of
several French mathematicians who worked with so called moving frames
before him. Grassmann made significant progress in his dealing with that
difference, but failed in a most important respect, as we now explain.
In applying his system to 3-D Euclidean space, Grassmann would distin-
guish four classes of elements, as Cartan explained [14]. They bear resem-
blance to the grades of a graded algebra. The first one, called the primitive
class, is constituted by the points of the space and the vectors. But points
do not have dimension; vectors do. This unfortunate feature of his system
carries on to other classes, which is why, in dimension n, he ended up iden-
tifying n-vectors and scalars. This is only part of the larger mess that his
work looks like, given the lack of ordering and structure in his myriad of new
ideas.
In his paper The Tragedy of Grassmann [18], the notable French math-
ematician Dieudonn´e speaks eloquently of what is best and worst in Grass-
mann’s works. In particular, he starts a paragraph with the statement “It
is almost unbelievable that such an original work should have been entirely
ignored by contemporary mathematicians, ...” And finishes the paragraph
as follows: “... the triumph and the failure of Grassmann, namely a lot of
wonderful ideas rendered ineffective by lack of the necessary technique, and
ending up in a frightful mess”.
Howthencomesthat, inspiteofthebadconsequencesofputtingtogether
points and vectors in the same class, Cartan would refer to Grassmann’s
calculus as being very fruitful. The reason is that he apparently understood
where the flaw was and that its correction resided in connecting it with the
theory of moving frames. This highly plausible influence of Grassmann on
Cartan is overlooked in the literature.
We have intimated that a formal concept of geometry did not exist in
Grassmann’s time, except during the very last years of his life with the
advent of the Erlangen program [17]. He, however, anticipated the eventual
formulation of geometries in terms of groups when he said that “all concepts
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and laws of the new analysis can be developed completely independently
of spatial intuitions, since they can be tied to the abstract concept of a
continuous transformation...” (page 384 in Kannenberg’s translation).
Clifford replaced progressive and regressive products with Clifford prod-
ucts [19]. In the case of product of two vectors, or of product of vector and
arbitrary element of the algebra, this reduces to the sum of exterior and in-
terior, which are largely similar to progressive and regressive (The exterior
product was so obviously present in Grassmann’s work that exterior algebra
is nowadays known as Grassmann’s algebra). Clifford created his algebra
by cutting and pasting from Grassmann’s own work, but without including
points in his graded system. He treated the dot product as clearly as the ex-
terior one, and put both of them together without being distracted by other
products. Grassmann had gone through all of them since the 1840’s. It was
only in the 1870s that he started to put the exterior and interior products
together in that special case. He had already done that in 1846, but without
doing much about it in one case as in the other. Clifford formalized an al-
gebra based on these two products, and did some work with it. The algebra
got his name.
In the aforementioned paper, “Complex Numbers” [14], Cartan devotes
three pages to Clifford and Lipschitz jointly. The former was luckier in that
their common algebra was named after him, even if, in certain respects,
Lipschitz appears to have gone further, specially with regards to rotations
[20]. In footnote #280, Cartan calls attention to the fact that both of those
authors, and another two that he mentions in the same footnote, only dealt
with units of square minus one. This lack of generality betrays the influence
of Hamilton. All these authors were thinking of the generalization of the
complex numbers without a real concern for a new view of geometry. Clifford
nevertheless emphatically recognized that Grassmann had done the heavy
lifting in the process of reaching Clifford algebra. And, like Grassmann, his
work was essentially algebraic, not geometric. If any or both of them was
thinking of doing geometry, they had not understood the difference between
algebra and geometry.
It was Cartan who decisively contributed to the further development of
Grassmann’s work, and thus to the Leibnizian anticipation. In 1899, he
formulated the exterior calculus [21]. It is based on Grassmann’s exterior
algebra, but with elements that are differential forms, not multivectors. In
additionheunderstoodthatdifferentialgeometryhadtodofirstwithgroups,
not with algebras. In 1910, just two years after the 140 page “Complex
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Numbers” paper, Cartan was already dealing with groups of transformations
within bundles of frames. But that was only the beginning.
It is easy to appreciate the relation of exterior and Clifford algebra to sev-
eral of the existing calculi. It is also easy to appreciate the relation between
calculus and differential geometry, as the main branch of this geometry is un-
derstood to be the calculus of vector-valued differential forms. Not as easy to
appreciate is the relation between the Erlangen program and what became
of Grassmann program before Cartan, not what it first intended. Geometry,
and thus the Erlangen program, is first about Lie groups of transformations.
Grassmann’s program became one about algebras, not about groups. Alge-
bras in geometry come in a second phase of the latter’s development.
The Erlangen program was right in aim, but wrong in details. Cartan
modified the Erlangen program in such a way that it entered the realm of
Leibniz anticipation. He also put the original Riemannian geometry in the
context of the theory of connections, enriching both in the process.
In 1908, Cartan published a second major paper (The Subgroups of Con-
tinuous Groups, 131 pages) of relevance for the present discussion [22] . Re-
garding the impact of that paper, Dieudonn´e [23] said that Felix Klein
”... had envisaged groups of isometries of Riemannian spaces
as a field of study of this program, but in general a Rieman-
nian space does not admit any isometries except for the identity
´
transformation. By an extremely original generalization, E. Car-
tan was able to show that there as well the idea of “operation”
still plays a fundamental role; but it is necessary to replace the
group with a more complex object, “called the principal fiber
space”; one can roughly represent it as a family of isomorphic
groups parametrized by the different points under consideration;
...”.
The “more complex objects” replace groups. These thus cease to be
the cornerstone of geometry, though they remain very close to it in a more
general scenario. Those statements by Dieudonn´e are not totally fortunate.
When he refers to the principal fiber space he has in mind frame bundles, but
without stating the relation of bundles to groups. That is what a connection
does, if one integrates it along a curve between two points on a manifold.
The result is a transformation that takes us from one frame to another frame
(i.e. another basis at, in general, another point). It is thus a path-dependent
member of the group.
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