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5. Lie's concept of a continuous group of transformations without the
assumption of the differentiability of the functions defining the group
It
is well known that Lie, with the aid of the concept of continuous groups
of transformations, has set up a system of geometrical axioms and, from
the standpoint of his theory of groups, has proved that this system of
axioms suffices for geometry. But since Lie assumes, in the very
foundation of his theory, that the functions defining his group can be
differentiated, it remains undecided in Lie's development, whether the
assumption of the differentiability in connection with the question as to
the axioms of geometry is actually unavoidable, or whether it may not
appear rather as a consequence of the group concept and the other
geometrical axioms. This consideration, as well as certain other problems
in connection with the arithmetical axioms, brings before us the more
general question: How far Lie's concept of continuous groups of
transformations is approachable in our investigations without the
assumption of the differentiability of the functions. Lie
defines a finite continuous group of transformations as a system of
transformations xi'
= fi(x1, ... , xn; a1,
... , ar) (i = 1, ..., n) having
the property that any two arbitrarily chosen transformations of the
system, as xi'
= fi(x1, ... , xn; a1,
... , ar) applied
successively result in a transformation which also belongs to the system,
and which is therefore expressible in the form xi"
= fi{f1(x, a), ... , fn(x,
a); b1, ... , br} = fi(x1,
... , xn; c1, ... , cr)
where
c1, ... , cr are certain functions of
a1, ... , ar and b1,
... , br. The group property thus finds its full
expression in a system of functional equations and of itself imposes no
additional restrictions upon the functions f1, ... , fn;
c1, ... , cr. Yet Lie's further
treatment of these functional equations, viz., the derivation of the well-known
fundamental differential equations, assumes necessarily the continuity
and differentiability of the functions defining the group. As
regards continuity: this postulate will certainly be retained for the
present--if only with a view to the geometrical and arithmetical
applications, in which the continuity of the functions in question
appears as a consequence of the axiom of continuity. On the other hand
the differentiability of the functions defining the group contains a
postulate which, in the geometrical axioms, can be expressed only in a
rather forced and complicated manner. Hence there arises the question
whether, through the introduction of suitable new variables and
parameters, the group can always be transformed into one whose defining
functions are differentiable; or whether, at least with the help of
certain simple assumptions, a transformation is possible into groups
admitting Lie's methods. A reduction to analytic groups is, according to
a theorem announced by Lie10
but first proved by Schur,11
always possible when the group is transitive and the existence of the
first and certain second derivatives of the functions defining the group
is assumed. For
infinite groups the investigation of the corresponding question is, I
believe, also of interest. Moreover we are thus led to the wide and
interesting field of functional equations which have been heretofore
investigated usually only under the assumption of the differentiability
of the functions involved. In particular the functional equations treated
by Abel12
with so much ingenuity, the difference equations, and other equations
occurring in the literature of mathematics, do not directly involve
anything which necessitates the requirement of the differentiability of
the accompanying functions. In the search for certain existence proofs in
the calculus of variations I came directly upon the problem: To prove the
differentiability of the function under consideration from the existence
of a difference equation. In all these cases, then, the problem arises: In
how far are the assertions which we can make in the case of
differentiable functions true under proper modifications without this
assumption? It
may be further remarked that H. Minkowski in his above-mentioned Geometrie
der Zahlen starts with the functional equation f(x1 + y1,
... , xn + yn)
and
from this actually succeeds in proving the existence of certain
differential quotients for the function in question. On
the other hand I wish to emphasize the fact that there certainly exist
analytical functional equations whose sole solutions are non-differentiable
functions. For example a uniform continuous non-differentiable function
where
6.
Mathematical treatment of the axioms of physics
The
investigations on the foundations of geometry suggest the problem: To
treat in the same manner, by means of axioms, those physical sciences in
which mathematics plays an important part; in the first rank are the
theory of probabilities and mechanics. As
to the axioms of the theory of probabilities,14 it seems to me desirable that their logical investigation should be
accompanied by a rigorous and satisfactory development of the method of
mean values in mathematical physics, and in particular in the kinetic
theory of gases. Important
investigations by physicists on the foundations of mechanics are at hand;
I refer to the writings of Mach,15
Hertz,16
Boltzmann17
and Volkmann. 18
It is therefore very desirable that the discussion of the foundations of
mechanics be taken up by mathematicians also. Thus Boltzmann's work on
the principles of mechanics suggests the problem of developing
mathematically the limiting processes, there merely indicated, which lead
from the atomistic view to the laws of motion of continua. Conversely one
might try to derive the laws of the motion of rigid bodies by a limiting
process from a system of axioms depending upon the idea of continuously
varying conditions of a material filling all space continuously, these
conditions being defined by parameters. For the question as to the
equivalence of different systems of axioms is always of great theoretical
interest. If
geometry is to serve as a model for the treatment of physical axioms, we
shall try first by a small number of axioms to include as large a class
as possible of physical phenomena, and then by adjoining new axioms to
arrive gradually at the more special theories. At the same time Lie's a
principle of subdivision can perhaps be derived from profound theory of
infinite transformation groups. The mathematician will have also to take
account not only of those theories coming near to reality, but also, as
in geometry, of all logically possible theories. He must be always alert
to obtain a complete survey of all conclusions derivable from the system
of axioms assumed. Further,
the mathematician has the duty to test exactly in each instance whether
the new axioms are compatible with the previous ones. The physicist, as
his theories develop, often finds himself forced by the results of his
experiments to make new hypotheses, while he depends, with respect to the
compatibility of the new hypotheses with the old axioms, solely upon
these experiments or upon a certain physical intuition, a practice which
in the rigorously logical building up of a theory is not admissible. The
desired proof of the compatibility of all assumptions seems to me also of
importance, because the effort to obtain such proof always forces us most
effectually to an exact formulation of the axioms. So far we have considered only questions concerning the foundations of the mathematical sciences. Indeed, the study of the foundations of a science is always particularly attractive, and the testing of these foundations will always be among the foremost problems of the investigator. Weierstrass once said, "The final object always to be kept in mind is to arrive at a correct understanding of the foundations of the science. ... But to make any progress in the sciences the study of particular problems is, of course, indispensable." In fact, a thorough understanding of its special theories is necessary to the successful treatment of the foundations of the science. Only that architect is in the position to lay a sure foundation for a structure who knows its purpose thoroughly and in detail. So we turn now to the special problems of the separate branches of mathematics and consider first arithmetic and algebra. |
f(x1,
... , xn) + f(y1, ... , yn)
