Mo.) Congress of Arts and Science (1904 : Saint Louis.

Congress of Arts and Science : universal exposition, St. Louis, 1904 online

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capable of rigid deduction from self-evident premises, so that, in
what regards arithmetic, we may say with Schroder that the famous
Kantian question "how are synthetic judgments a priori possible?"
is now known to be meaningless. As regards geometry, the case ap-


pears to a non-mathematician like myself more doubtful. Those
who hold with Schroder that geometry essentially involves, as Kant
thought it did, an appeal to principles not self-evident and depend-
ent upon an appeal to sensuous "intuition," are logically bound
to conclude with him that geometry is an " empirical," or as W. K.
Clifford called it, a "physical" science, different in no way from
mechanics except in the relative paucity of the empirical premises
presupposed, and to class it with the applied sciences. On the other
hand, if Mr. Bertrand Russell should be successful in his promised
demonstration that all the principles of geometry are deducible from
a few premises which include nothing of the nature of an appeal to
sensuous diagrams, geometry too would take its place among the
pure sciences, but only on condition of our recognizing that its
truths, like those of arithmetic, are one and all, as Leibniz held,
strictly analytical. Thus we obtain as a fir^t distinction between the
pure and the empirical sciences the principle that the propositions
of the former class are all analytical, those of the latter all synthetic.
It is not the least of the services which France is now rendering to
the study of philosophy that we are at last being placed by the
labors of M. Couturat in a position to appreciate at their full worth
the views of the first and greatest of German philosophers on this
distinction, and to understand how marvelously they have been
confirmed by the subsequent history of mathematics and of logic.

(2) A consequence of this distinction is that only the pure or
formal sciences can be matter of rigid logical demonstration. Since
the empirical or applied sciences one and all contain empirical pre-
mises, i. e., premises which we admit as true only because they have
always appeared to be confirmed by the appeal to " intuition,"
and not because the denial of them can be shown to lead to false-
hood, the conclusions to which they conduct us must one and all
depend, in part at least, upon induction from actual observation of
particular temporal sequences. This is as much as to say that all
propositions in the applied sciences involve somewhere in the course
of the reasoning by which they are established the appeal to the
calculus of Probabilities, which is our one method of eliciting general
results from the statistics supplied by observation or experiment.
That this is the case with the more concrete among such applied
sciences has long been universally acknowledged. That it is no less
true of sciences of such wide range as mechanics may be said, I
think, to have been definitely established in our own day by the
work of such eminent physicists as Kirchhoff and Mach. In fact,
the recent developments of the science of pure number, to which
reference has been made in a preceding paragraph, combined with
the creation of the "descriptive " theory of mechanics, may fairly
be said to have finally vindicated the distinction drawn by Leibniz


long ago between the truths of reason and the truths of e.
fact, a distinction which the Kantian trend of philosophical i
lation tended during the greater part of the nineteenth centur^
obscure, while it was absolutely ignored by the empiricist opponent
of metaphysics both in England and in Germany, The philosoph-
ical consequences of a revival of the distinction are, I conceive, of
far-reaching importance. On the one side, recognition of the em-
pirical and contingent character of all general propositions estab-
lished by induction appears absolutely fatal to the current mechan-
istic conception of the universe as a realm of purposeless sequences
unequivocally determined by unalterable "laws of nature," a result
which has in recent years been admirably illustrated for the Eng-
lish-speaking world by Professor Ward's well-known Gifford lectures
on "Naturalism and Agnosticism." Laws of physical nature, on the
empiristic view of applied science, can mean no more than observed
regularities, obtained by th£ application of the doctrine of chances,
— regularities which we are indeed justified in accepting with con-
fidence as the basis for calculation of the future course of temporal
sequence, but which we have no logical warrant for treating as ulti-
mate truths about the final constitution of things. Thus, for exam-
ple, take the common assumption that our physical environment
is composed of a multitude of particles each in every respect the
exact counterpart of every other. Reflection upon the nature of
the evidence by which this conclusion, if supported at all, has to
be supported, should convince us that at most all that the state-
ment ought to mean is that individual differences between the ele-
mentary constituents of the physical world need not be allowed
for in devising practical formulae for the intelligent anticipation of
events. When the proposition is put forward as an absolute truth
and treated as a reason for denying the ultimate spirituality of the
world, we are well within our rights in declining the consequence
on the logical ground that conclusions from an empirical premise
must in their own nature be themselves empirical and contingent.

On the other hand, the extreme empiricism which treats all know-
ledge whatsoever as merely relative to the total psychical state
of the knower, and therefore in the end problematic, must, I appre-
hend, go down before any serious investigation into the nature of
the analytic truths of arithmetic, a consequence which seems to be
of some relevance in connection with the philosophic view popularly
known as Pragmatism. Thus I should look to the coming regeneration
of metaphysics, of which there are so many signs at the moment, on
the one hand, for emphatic insistence on the right, e. g., of physics
and biology and psychology to be treated as purely empirical
sciences, and as such freed from the last vestiges of any domination
by metaphysical presuppositions and foregone conclusions, and on



jr, for an equally salutary purgation of formal studies like
and arithmetic from the taint of corruption by the irrelevant
. asion of considerations of empirical psychology.
We cannot too persistently bear in mind that there is, correspond-
ing to the logical distinction between the analytic and the synthetic
proposition, a deep and broad general difference between the wants
of our nature ministered to by the formal and the applied sciences
respectively. The formal sciences, incapable of adding anything to
our detailed knowledge of the course of events, as we have seen,
enlighten us solely as to the general laws of interconnection by which
all conceivable systems of true assertions are permeated and bound
together. In a different connection it would be interesting to de-
velop further the reflection that the necessity of appealing to such
formal principles in all reasoning about empirical matters of fact
contains the explanation of the famous Platonic assertion that the
" Idea of Good " or supreme principle of organization and order in
the universe, is itself not an existent, but something en eVeVetva r^s
ovcrtas, "transcending even existence," and the very similar declara-
tion of Hegel that the question whether "God" — in the sense of
such a supreme principle — exists is frivolous, inasmuch as existence
(Dasein) is a category entirely inadequate to express the Divine
nature. For my present purpose it is enough to remark that the
need to which the formal sciences minister is the demand for that
purely speculative satisfaction which arises from insight into the
order of interconnection between the various truths which compose
the totality of true knowledge. Hence it seems a mistake to say, as
some theorists have done, that were we born with a complete know-
ledge of the course of temporal sequences throughout the universe,
and a faultless memory, we should have no need of logic or meta-
physics, or in fact of inference. For even a mind already in possession
of all true propositions concerning the course of events, would still
lack one of the requisites for complete intellectual satisfaction
unless it were also aware, not only of the individual truths, but of
the order of their interdependence. What Aristotle said long ago
with reference to a particular instance may be equally said univers-
ally of all our empirical knowledge; "even if we stood on the
moon and saw the earth intercepting the light of the sun, we should
still have to ask for the reason why." The purposes ministered to
by the empirical sciences, on the other hand, always include some re-
ference to the actual manipulation in advance by human agency of
the stream of events. We study mechanics, for instance, not merely
that we may perceive the interdependence of truths, but that we
may learn how to maintain a system of bodies in equilibrium, or how
to move masses in a given direction with a given momentum. Hence
it is true of applied science, though untrue of science as a whole, that


it would become useless if the whole past and future course of events
were from the first familiar to us. And, incidentally it may be ob-
served, it is for the same reason untrue of inference, though true of
inductive inference, that it is essentially a passage from the known
to the unknown.

In dealing with the relation of metaphysics to the formal sciences
generally, the great difficulty which confronts us is that of determin-
ing exactly the boundaries which separate one from another. Among
such pure sciences we have by universal admission to include at
least two, pure formal logic and pure mathematics, as distinguished
from the special applications of logic and mathematics to an empiri-
cal material. Whether we ought also to recognize ethics and aesthet-
ics, in the sense of the general determination of the nature of the
good and the beautiful, as non-empirical sciences, seems to be a more
difficult question. It seems clear, for instance, that ethical discus-
sions, such as bulk so largely in our contemporary literature, as to what
is the right course of conduct under various conditions, are concerned
throughout with an empirical material, namely, the existing pecu-
liarities of human nature as we find it, and must therefore be regarded
as capable only of an empirical and therefore problematic solution.
Accordingly I was at one time myself tempted to regard ethics as
a purely empirical science, and even published a lengthy treatise
in defense of that point of view and in opposition to the whole
Kantian conception of the possibihty of a constructive Metaphysik
der Sitten. It seems, however, possible to hold that in the question
"What do we mean by good?" as distinguished from the question
" What in particular is it right to do? " there is no more of a reference
to the empirical facts of human psychology than in the question
"What do we mean by truth?" and that there must therefore be
a non-empirical answer to the problem. The same would of course
hold equally true of the question "What is beauty?" If there are,
however, such a pure science of ethics and again of sesthetics, it
must at least be allowed that for the most part these sciences are
still undiscovered, and that the ethical and sesthetical results hitherto
established are in the main of an empirical nature, and this must
be my excuse for confining the remarks of the next two paragraphs
to the two great pure sciences of which the general principles may
be taken to be now in large measure known.

That metaphysics and logic should sometimes have been absolutely
identified, as for instance by Hegel, will not surprise us when we
consider how hard it becomes on the view here defended to draw any
hard and fast boundary line between them. For metaphysics, accord-
ing to this conception of its scope, deals with the formulation of the
self-evident principles implied, in there being such a thing as truth
and the deductions which these principles warrant us in drawing.


Thus it might be fairly said to be the supreme science of order, and
it would not be hard to show that all the special questions commonly-
included in its range, as to the nature of space, time, causation, con-
tinuity, and so forth, are all branches of the general question, how
many types of order among concepts are there, and what is their
nature. A completed metaphysics would thus appear as the realiza-
tion of Plato's splendid conception of dialectic as the ultimate reduc-
tion of the contents of knowledge to order by their continuous de-
duction from a sujjreme principle (or, we may add, principles). Now
such a view seems to make it almost impossible to draw any ulti-
mate distinction between logic and metaphysics. For logic is strictly
the science of the mutual implication of propositions, as we see as
soon as we carefully exclude from it all psychological accretions. In
the question what are the conditions under which one proposition
or group of propositions imply another, we exhaust the whole scope
of logic pure and proper, as distinguished from its various empirical
applications. This is the important point which is so commonly
forgotten when logic is defined as being in some way a study of " psy-
chical processes," or when the reference to the presence of "minds"
in which propositions exist, is intended into logical science. We can-
not too strongly insist that for logic the question so constantly raised
in a multitude of text-books, what processes actually take place when
we pass from the assertion of the premises to the assertion of the
conclusion, is an irrelevant one, and that the only logical problem
raised by inference is whether the assertion of the premises as true
warrants the further assertion of the conclusion, supposing it to be
made. (At the risk of a little digression I cannot help pointing out that
the confusion between a logical and a psychological problem is com-
mitted whenever we attempt, as is so often done, to make the self-
evidence of a principle identical with our psychological inability to
believe the contradictory. From the strictly logical point of view,
all that is to be said about the two sides of such an ultimate contra-
diction is that the one is true and the other is false. Whether it is
or is not possible, as a matter of psychical fact for me to afhrm with
equal conviction, both sides of a contradiction, knowing that I am
doing so, is a question of empirical psychology which is possibly
insoluble, and at any rate seems not to have received from the
psychologists the attention it deserves. But the logician, so far as
I can see, has no interest as a logician in its solution. For him it
would still be the case even though all mankind should actually and
consciously affirm both sides of a given contradiction, that one of the
affirmations would be true, and the other untrue.) Logic thus seems
to become either the whole or an integral part of the science of order,
and there remain only two possible ways of distinguishing it from
metaphysics. It might be suggested that logical order, the order of


implication between truths, is only one species of a wider genus,
order in general by the side, for example, of spatial, temporal, and
numerical order, and thus that logic is one subordinate branch of
the wider science of metaphysics. Such a view, of course, implies
that there are a plurality of ultimately independent forms of order
irreducible to a single type. Whether this is the case, I must confess
myself at present incompetent to decide, though the signal success
with which the principles of number have already been deduced
from the fundamental definitions and axioms of symbolic logic, and
number itself defined, as by Mr. Russell, in terms of the purely logical
concept of class-relation, seems to afford some presumption to the
contrary. Or it may be held that the difference is purely one of the
degree of completeness with which the inquiry into order is pursued.
Thus the ordinary symbolic logic of what Schroder has called the
"identical calculus," or "calculus of domains," consists of a series
of deductions from the fundamental concepts of class and number,
identical equality, totality or the "logical 1," zero or the null-class,
and the three principles of identity, subsumption, and negation. The
moment you cease to accept these data in their totality as the given
material for your science, and to inquire into their mutual coherence,
by asking for instance whether any one of them could be denied,
and yet a body of consistent results deduced from the rest, your
inquiry, it might be said, becomes metaphysics. So, again, the dis-
cussion of the well-known contradictions which arise when we try to
apply these principles in their entirety and without modification to
classes of classes instead of classes of individuals, or of the problem
raised by Peano and Russell, whether the assertions "Socrates is
a man" and "the Greeks are men" affirm the same or a different
relation between their subject and predicate (which seems indeed to
be the same question differently stated), would generally be allowed
to be metaphysical. And the same thing seems to be equally true
of the introduction of time-relations into the interpretation of our
symbols for predication employed by Boole in his treatment of
hypotheticals, and subsequently adopted by his successors as the
foundation of the "calculus of equivalent statements."

However we may decide such questions, we seem at least driven
by their existence to the recognition of two important conclusions.
(1) The relation between logical and metaphysical problems is so close
that you cannot in consistency deny the possibility of a science of
metaphysics unless you are prepared mth the absolute skeptic to
go the length of denying the possibility of logic also, and reducing
the first principles of inference to the level of formulae which have
happened hitherto to prove useful but are, for all we know, just as
likely to fail us in future application as not. (Any appeal to the
doctrine of chances would be out of place here, as that doctrine is


itself based on the very principles at stake.) (2) The existence of
fundamental problems of this kind which remained almost or wholly
unsuspected until revealed in our own time by the creation of a science
of symbolic logic should console us if ever we are tempted to suspect
that metaphysics is at any rate a science in which all the main con-
structive work has already been accomplished by the great thinkers
of the past. To me it appears, on the contrary, that the recent enor-
mous developments in the purely formal sciences of logic and mathe-
matics, with the host of fundamental problems they open up, give
promise of an approaching era of fresh speculative construction
which bids fair to be no less rich in results than any of the great
''golden" periods in the past history of our science. Indeed, but
that I would avoid the slightest suspicion of a desire to advertise
personal friends, I fancy I might even venture to name some of those
to whom we may reasonably look for the work to be done.

Of the relation of metaphysics to pure mathematics it would be
impertinent for any but a trained mathematician to say very much.
I must therefore be content to point out that the same difficulty
in drawing boundary lines meets us here as in the case of logic. Not
so long ago this difficulty might have been ignored, as it still is by too
many writers on the philosophy of science. Until recently mathematics
would have been thought to be adequately defined as the science of
numerical and quantitative relations, and adequately distinguished
from metaphysics by the non-quantitative and non-numerical char-
acter of the latter, though it would probably have been admitted that
the problem of the definition of quantity and number themselves is
a metaphysical one. But in the present state of our knowledge such
an account seems doubly unsatisfactory. On the one hand, we have
to recognize the existence of branches of mathematics, such as the
so-called descriptive geometry, which are neither quantitative nor
numerical, and, on the other, quantity as distinct from number appears
to play no part in mathematical science, while number itself, thanks
to the labors of such men as Cantor and Dedekind, seems, as I have
said before, to be known now to be only a special type of order in
a series. Thus there appears to be ground for regarding serial order
as the fundamental category of mathematics, and we are thrown back
once more upon the difficult task of deciding how many ultimately
irreducible types of order there may be before we can undertake any
precise discrimination between mathematical and metaphysical
science. However we may regard the problem, it is at least certain
that the recent researches of mathematicians into the meaning of
such concepts as continuity and infinity have, besides opening up new
metaphysical problems, done much to transfigure the familiar ones,
as all readers of Professor Royce must be aware. For instance I
imagine all of us here present, even the youngest, were brought up on


the Aristotelian doctrine that there is and can be no such thing as an
actually existing infinite collection, but which of us would care to
defend that time-honored position to-daj^? Similarly with continuity
all of us were probably once on a time instructed that whereas " quan-
tity" is continuous, number is essentially "discrete," and is indeed
the typical instance of what we mean by the non-continuous. To-day
we know that it is in the number-series that we have our one certain
and familiar instance of a perfect continuum. Still a third illustration
of the transforming light which is thrown upon old standing meta-
physical puzzles by the increasing formal development of mathe-
matics may be found in the difficulties attendant upon the conception
of the "infinitely little," once regarded as the logical foundation of
the so-called Differential Calculus. With the demonstration, which
may be found in Mr. Russell's important work, that "infinitesimal,"
unlike "infinite," is a purely relative term, and that there are no
infinitesimal real numbers, the supposed logical significance of the
concept seems simply to disappear. Instances of this kind could easily
be multiplied almost indefinitely, but those already cited shovild be
sufficient to show how important are the metaphysical results which
may be anticipated from contemporary mathematical research, and
how grave a mistake it would be to regard existing metaphysical con-
struction, e. g., that of the Hegelian system, as adequate in principle
to the present state of our organized knowledge. In fact, all the mate-
rials for a new Kategorienlehre, which may be to the knowledge of our
day what Hegel's Logic was to that of eighty years ago, appear to lie
ready to hand when it may please Providence to send us the meta-
physician who knows how to avail himself of them. The proof, given
since this address was delivered, by E. Zermelo, that every assem-
blage can be well ordered, is an even more startling illustration of
the remarks in the text.

It remains to say something of the relation of metaphysical specu-
lation to the various sciences which make use of empirical premises.
On this topic I may be allowed to be all the more brief, as I have quite
recently expressed my views at fair length in an extended treatise
{Elements of Metaphysics, Bks. 3 and 4), and have nothing of conse-
quence to add to what has been there said. The empirical sciences,
as previously defined, appear to fall into two main classes, distin-
guished by a difference which corresponds to that often taken in the
past as the criterion by which science is to be separated from philo-
sophy. We may study the facts of temporal sequence either with a
view to the actual control of future sequences or with a view to
detecting under the sequence some coherent purpose. It is in the
former way that we deal with facts in mechanics, for instance, or in
chemistry, in the latter that we treat them when we study history for
the purpose of gaining insight into national aims and character. We

Online LibraryMo.) Congress of Arts and Science (1904 : Saint LouisCongress of Arts and Science : universal exposition, St. Louis, 1904 → online text (page 26 of 68)