_Origin of the hypothesis._--In our analysis of the process of judgment,
we attempted to show that the predicate arises in case of failure of
some line of activity going on in terms of an established habit. When
the old habit is checked through failure to deal with new conditions
(_i. e._, when the situation is such as to stimulate two habits with
distinct aims) the problem is to find a new method of response--that is,
to co-ordinate the conflicting tendencies by building up a single aim
which will function the existing situation. As we saw that, in case of
judgment, habit when checked became ideal, an idea, so the new habit is
first formalized as an ideal type of reaction and is the hypothesis by
which we attempt to construe new data. In our inquiry as to how this
formulation is effected, _i. e._, how the hypothesis is developed, it
will be convenient to take some of the currently accepted statements as
to their origin, and show how these statements stand in reference to the
analysis proposed.
_Enumerative induction and allied processes._--It is pointed out by
Welton[73] that the various ways in which hypotheses are suggested may
be reduced to three classes, viz., enumerative induction, conversion of
propositions, and analogy. Under the head of "enumeration" he reminds us
that "every observed regularity of connection between phenomena suggests
a question as to whether it is universal." There are numerous instances
of this in mathematics. For example, it is noticed that 1+3=2^2,
1+3+5=3^2, 1+3+5+7=4^2, etc.; and one is led to ask whether there is any
general principle involved, so that the sum of the first _n_ odd numbers
will be _n_^2, where _n_ is any number, however great. In this early
form of inductive inference there are two divergent tendencies. One is
the tendency to complete enumeration. This _tendency_ is clearly
ideal--it transcends the facts as given. To look for all the cases is
thus itself an experimental inquiry, based upon a hypothesis which it
endeavors to test. But in most cases enumeration can be only incomplete,
and we are able to reach nothing better than probability. Hence the
other tendency in the direction of an analysis of content in search for
a principle of connection in the elements in any _one_ case. For if a
characteristic belonging to a number of individuals suggests a class
where it belongs to all individuals, it must be that it is found in
every individual as such. The hypothesis of complete class involves a
hypothesis as to the character of each individual in the class. Thus a
hypothesis as to extension transforms itself into one as to intension.
But it is analogy which Welton considers "the chief source from which
new hypotheses are drawn." In the second tendency mentioned under
enumerative induction, that is, the tendency to analysis of content or
intension, we are naturally led to analogy, for in our search for the
characteristic feature which determines classification among the
concrete particulars our first step will be an inference by analogy. In
analogy attention is turned from the number of observed instances to
their character, and, because particulars have some feature in common,
they are supposed to be the same in still other respects. While the best
we can reach in analogy is probability, the arguments may be such as to
result in a high degree of certainty. The form of the argument is
valuable in so far as we are able to distinguish between essential and
nonessential characteristics on which to base our analogy. What is
essential and what nonessential depends upon the particular end we have
in view.
In addition to enumerative induction, which Welton has mentioned, it is
to be noted that there are a number of other processes which are very
similar to it in that a number of particulars appear to furnish a basis
for a general principle or method. Such instances are common in
induction, in instruction, and in methods of proof.
If one is to be instructed in some new kind of labor, he is supposed to
acquire a grasp of the method after having been shown in a few instances
how this particular work is to be done; and, if he performs the
manipulations himself, so much the better. It is not asked why the
experience of a few cases should be of any assistance, for it seems
self-evident that an experienced man, a man who has acquired the skill,
or knack, of doing things, should deal better with all other cases of
similar nature.
There is something very similar in inductive proofs, as they are called.
The inductive proof is common in algebra. Suppose we are concerned in
proving the law of expansion of the binomial theorem. We show by actual
calculation that, if the law holds good for the _n_th power, it is true
for the _n_+first power. That is, if it holds for any power, it holds
for the next also. But we can easily show that it does hold for, say,
the second power. Then it must be true for the third, and hence for the
fourth, and so on. Whether this law, though discovered by inductive
processes, depends on deduction for the conclusiveness of its proof, as
Jevons holds;[74] whether, as Erdmann[75] contends, the proof is
thoroughly deductive; or whether Wundt[76] is right in maintaining that
it is based on an exact analogy, while the fundamental axioms of
mathematics are inductive, it is clear that in such proofs a few
instances are employed to give the learner a start in the right
direction. Something suggests itself, and is found true in this case, in
the next, and again in the next, and so on. It may be questioned whether
there is usually a very clear notion of what is involved in the "so on."
To many it appears to mark the point where, after having been taken a
few steps, the learner is carried on by the acquired momentum somewhat
after the fashion of one of Newton's laws of motion. Whether the few
successive steps are an integral part of the proof or merely serve as
illustration, they are very generally resorted to. In fact, they are
often employed where there is no attempt to introduce a general term
such as _n_, or _k_, or _l_, but the few individual instances are deemed
quite sufficient. Such, for instance, is the custom in arithmetical
processes. We call attention to these facts in order to show that
successive cases are utilized in the course of explanation as an aid in
establishing the generality of a law.
In geometry we find a class of proofs in which the successive steps seem
to have great significance. A common proof of the area of the circle
will serve as a fair example. A regular polygon is circumscribed about
the circle. Then as the number of its sides are increased its area will
approach that of the circle, as its perimeter approaches the
circumference of the circle. The area of the circle is thus inferred to
be [pi]_R_^2, since the area of the polygon is always (1/2)_R_×
perimeter, and in case of the circle the circumference = 2[pi]_R_.
Here again we get under such headway by means of the polygon that we
arrive at the circle with but little difficulty. Had we attempted the
transition at once, say, from a circumscribed square, we should
doubtless have experienced some uncertainty and might have recoiled from
what would seem a rash attempt; but as the number of the sides of our
polygon approach infinity--that mysterious realm where many paradoxical
things become possible--the transition becomes so easy that our polygon
is often said to have truly become a circle.
Similarly, some statements of the infinitesimal calculus rest on the
assumption that slight degrees of difference may be neglected. Though
the more modern theory of limits has largely displaced this attitude in
calculus and has also changed the method of proof in such geometrical
problems as the area of the circle, the underlying motive seems to have
been to make transitions easy, and thus to make possible a continued
application of some particular method or way of dealing with things.
But granted that this is all true, what has it to do with the origin of
the hypothesis? It seems likely that the hypothesis may be suggested by
a few successive instances; but are these to be classed with the
successive steps in proof to which we have referred? In the first place,
we attempt to prove our hypothesis because we are not sure it is true;
we are not satisfied that there are no other tenable hypotheses. But if
we do test it, is not such test enough? It depends upon how thorough a
grasp we have of the situation; but, in general, each test case adds to
its probability. The value of tests lies in the fact that they
strengthen and tend to confirm our hypothesis by checking the force of
alternatives. One instance is not sufficient because there are other
possible incipient hypotheses, or more properly tendencies, and the
enumeration serves to bring one of these tendencies into prominence in
that it diminishes other vague and perhaps subconscious tendencies and
strengthens the one which suddenly appears as the mysterious product of
genius.
The question might arise why the mere repetition of conflicting
tendencies would lead to a predominance of one of them. Why would they
not all remain in conflict and continue to check any positive result? It
is probably because there never is any absolute equilibrium. The
successive instances tend to intensify and bring into prominence some
tendency which is already taking a lead, so to speak. And it may be said
further in this connection that only as seen from the outside, only as a
mechanical view is taken, does there appear to be an excluding of
definitely made out alternatives.
In explanation of the part played by analogy in the origin of
hypotheses, Welton points out that a mere number of instances do not
take us very far, and that there must be some "_specification_ of the
instances as well as numbering of them," and goes on to show that the
argument by enumerative induction passes readily into one from analogy,
as soon as attention is turned from the number of the observed instances
to their character. It is not necessary, however, to pass to analogy
through enumerative induction. "When the instances presented to
observation offer immediately the characteristic marks on which we base
the inference to the connection of S and P, we can proceed at once to an
inference from analogy, without any preliminary enumeration of the
instances."[77]
Welton, and logicians generally, regard analogy as an inference on the
basis of partial identity. Because of certain common features we are led
to infer a still greater likeness.
Both enumerative induction and analogy are explicable in terms of habit.
We saw in our examination of enumerative induction that a form of
reaction gains strength through a series of successful applications.
Analogy marks the presence of an identical element together with the
tendency to extend this "partial identity" (as it is commonly called)
still farther. In other words, in analogy it is suggested that a type of
reaction which is the same in certain respects may be made similar in a
greater degree. In enumerative induction we lay stress on the number of
instances in which the habit is applied. In analogy we emphasize the
content side and take note of the partial identity. In fact, the
relation between enumerative induction and analogy is of the same sort
as that existing between association by contiguity and association by
similarity. In association by contiguity we think of the things
associated as merely standing in certain temporal or spatial relations,
and disregard the fact that they were elements in a larger experience.
In case of association by similarity we regard the like feature in the
things associated as a basis for further correction.
In conversion of propositions we try to reverse the direction of the
reaction, so to speak, and thereby to free the habit, to get a mode of
response so generalized as to act with a minimum cue. For instance, we
can deal with A in a way called B, or, in other words, in the same way
that we did with other things called B. If we say, "Man is an animal,"
then to a certain extent the term "animal" signifies the way in which we
regard "man." But the question arises whether we can regard all animals
as we do man. Evidently not, for the reaction which is fitting in case
of animals would be only partially applicable to man. With the animals
that are also men we have the beginning of a habit which, if unchecked,
would lead to a similar reaction toward all animals, _i. e._, we would
say: "All animals are men." Man may be said to be the richer concept, in
that only a part of the reaction which determines an object to be a man
is required to designate it as an animal. On the other hand, if we start
with animal, then (except in case of the animals which are men) there is
lacking the subject-matter which would permit the fuller concept to be
applied. By supplying the conditions under which animal=man we get a
reversible habit. The equation of technical science has just this
character. It represents the maximum freeing or abstraction of a
predicate _qua_ predicate, and thereby multiplies the possible
applications of it to subjects of future judgments, and lessens the
amount of shearing away of irrelevancies and of re-adaptation necessary
when so used in any particular case.
_Formation and test of the hypothesis._--The formation of the hypothesis
is commonly regarded as essentially different from the process of
testing, which it subsequently undergoes. We are said to observe facts,
invent hypotheses, and _then_ test them. The hypothesis is not required
for our preliminary observations; and some writers, regarding the
hypothesis as a formulation which requires a difficult and elaborate
test, decline to admit as hypotheses those more simple suppositions,
which are readily confirmed or rejected. A very good illustration of
this point of view is met with in Wundt's discussion of the hypothesis,
by an examination of which we hope to show that such distinctions are
rather artificial than real.
The subject-matter of science, says Wundt,[78] is constituted by that
which is actually given and that which is actually to be expected. The
whole content is not limited to this, however, for these facts must be
supplemented by certain presuppositions, which are not given in a
factual sense. Such presuppositions are called hypotheses and are
justified by our fundamental demand for unity. However valuable the
hypothesis may be when rightly used, there is constant danger of
illegitimately extending it by additions that spring from mere
inclinations of fancy. Furthermore, the hypothesis in this proper
scientific sense must be carefully distinguished from the various
inaccurate uses, which are prevalent. For instance, hypotheses must not
be confused with expectations of fact. As cases in point Wundt mentions
Galileo's suppositions that small vibrations of the pendulum are
isochronous, and that the space traversed by a falling body is
proportional to the square of the time it has been falling. It is true
that such anticipations play an important part in science, but so long
as they relate to the facts themselves or to their connections, and can
be confirmed or rejected any moment through observation, they should not
be classed with those added presuppositions which are used to
co-ordinate facts. Hence not all suppositions are hypotheses. On the
other hand, not every hypothesis can be actually experienced. For
example, one employs in physics the hypothesis of electric fluid, but
does not expect actually to meet with it. In many cases, however, the
hypothesis becomes proved as an experienced fact. Such was the course of
the Copernican theory, which was at first only a hypothesis, but was
transformed into fact through the evidence afforded by subsequent
astronomical observation.
Wundt defines a theory as a hypothesis taken together with the facts for
whose elucidation it was invented. In thus establishing a connection
between the facts which the hypothesis merely suggested, the theory
furnishes at the same time partly the foundation (_Begründung_) and
partly the confirmation (_Bestätigung_) of the hypothesis.[79] These
aspects, Wundt insists, must be sharply distinguished. Every hypothesis
must have its _Begründung_, but there can be _Bestätigung_ only in so
far as the hypothesis contains elements which are accessible to actual
processes of verification. In most cases verification is attainable in
only certain elements of the hypothesis. For example, Newton was obliged
to limit himself to one instance in the verification of his theory of
gravitation, viz., the movements of the moon. The other heavenly bodies
afforded nothing better than a foundation in that the supposition that
gravity decreases as the square of the distance increases enabled him to
deduce the movements of the planets. The main object of his theory,
however, lay in the deduction of these movements and not in the proof of
universal gravity. With the Darwinian theory, on the contrary, the main
interest is in seeking its verification through examination of actual
cases of development. Thus, while the Newtonian and the greater part of
the other physical theories lead to a deduction of the facts from the
hypotheses, which can be verified only in individual instances, the
Darwinian theory is concerned in evolving as far as possible the
hypothesis out of the facts.
Let us look more closely at Wundt's position. We will ask, first,
whether the distinction between hypotheses and expectations is as
pronounced as he maintains; and, second, whether the relation between
_Begründung_ and _Bestätigung_ may not be closer than Wundt would have
us believe.
As examples of the hypothesis Wundt mentions the Copernican hypothesis,
Newton's hypothesis of gravitation, and the predictions of the
astronomers which led to the discovery of Neptune. As examples of mere
expectations we are referred to Galileo's experiments with falling
bodies and pendulums. In case of Newton's hypothesis there was the
assumption of a general law, which was verified after much labor and
delay. The heliocentric hypothesis of Copernicus, which was invented for
the purpose of bringing system and unity into the movements of the
planets, has also been fairly well substantiated. In the discovery of
Neptune we have, apparently, not the proof of a general law or the
discovery of further peculiarities of previously known data, but rather
the discovery of a new object or agent by means of its observed effects.
In each of these instances we admit that the hypothesis was not readily
suggested or easily and directly tested.
If we turn to Galileo's pendulum and falling bodies, it is clear first
of all that he did not have in mind the discovery of some object, as was
the case in the discovery of Neptune. Did he, then, either contribute to
the proof of a general law or discover further characteristics of things
already known in a more general way? Wundt tells us that Galileo only
determined a little more exactly what he already knew, and that he did
this with but little labor or delay.
What, then, is the real difference between hypothesis and expectation?
If we compare Galileo's determination of the law of falling bodies with
Newton's test of his hypothesis of gravitation, we see that both
expectation and hypothesis were founded on observation and took the form
of mathematical formulæ. Each tended to confirm the general law
expressed in its formula, though there was, of course, much difference
in the time and labor required. If we compare the Copernican hypothesis
with Galileo's supposition concerning the pendulum, we find again that
they agree in regard to general purpose and method, and differ in the
difficulty of verification. If the experiment with the pendulum only
substituted exactness for inexactness, did the Copernican theory do
anything different in _kind_? It is true that the more exact statement
of the swing of the pendulum was expressed in quantitative form, but
quantitative statement is no criterion of either the presence or the
absence of the hypothesis.
Again, we may compare the pendulum with Kepler's laws. What was Kepler's
hypothesis, that the square of the periodic times of the several planets
are proportional to the cubes of their mean distances from the sun,
except a more exact formulation of facts which were already known in a
more general way? Wundt's position seems to be this: whenever a
supposition or suggestion can be tested readily, it should not be
classed as a hypothesis. This would make the distinction one of degree
rather than kind, and it does not appear how much labor we must expend,
or how long our supposition must evade our efforts to test it, before it
can win the title of hypothesis.
In the second place, we have seen that Wundt draws a sharp line between
_Begründung_ and _Bestätigung_. It is doubtless true that every
hypothesis requires a certain justification, for unless other facts can
be found which agree with deductions made in accordance with it, its
only support would be the data from which it is drawn. Such support as
this would be obtained through a process too clearly circular to be
seriously entertained. The distinction which Wundt draws between
_Begründung_ and _Bestätigung_ is evidently due to the presence of the
experimental element in the latter. For descriptive purposes this
distinction is useful, but is misleading if it is understood to mean
that there is mere experience in one case and mere inference in the
other. The difference is rather due to the relative parts played by
inference and by accepted experience in each. In _Begründung_ the
inferential feature is the more prominent, while in _Bestätigung_ the
main emphasis is on the experiential aspect. It must not be supposed,
however, that either of these aspects can be wholly absent. It is
difficult to understand how any hypothesis can be entertained at all
unless it meets in some measure the demand with reference to which it
was invented, viz., a unification of conflicts in experience. And, _in
so far_, it is confirmed. The motive which casts doubt upon its adequacy
is the same that leads to its re-forming as a hypothesis, as a mental
concept.
The difficulties in Wundt's position are thus due to a failure to take
account of the reconstructive nature of the judgment. The predicate,
supposition, or hypothesis, whatever we may choose to call it, is formed
because of the check of a former habit. The judgment is an ideal
application of a new habit, and its test is the attempt to act in
accordance with this ideal reconstruction. It must not be thought,
however, that our supposition is first fully developed and then tried
and accepted or rejected without modification. On the contrary, its
growth is the result of successive minor tests and corresponding minor
modifications in its form. Formation and test are merely convenient
distinctions in a larger process in which forming, testing, and
_re_-forming go on together. The activity of experimental verification
is not only a testing, a confirming or weakening of the validity of a
hypothesis, but it is equally well an evolution of the _meaning_ of the
hypothesis through bringing it into closer relations with specific data
not previously included in defining its import. _Per contra_, a purely
reflective and deductive consideration which develops the idea as
hypothesis, _in so far_ as it introduces the determinateness of
previously accepted facts within the scope, comprehension, or intension
of the idea, is in so far forth, a verification.
If the view which we have maintained is correct, the hypothesis is not
to be limited to those elaborate formulations of the scientist which he
seeks to confirm by crucial tests. The hypothesis of the investigator
differs from the comparatively rough conjecture of the plain man only in
its greater precision. Indeed, as we have attempted to show, the
hypothesis is not a method which we may employ or not as we choose; on
the contrary, as predicate of the judgment it is present in a more or
less explicit form if we judge at all. Whether the time and labor
required for its confirmation or rejection is a matter of a lifetime or
a moment, its nature remains the same. Its function is identical with
that of the predicate. In short, the hypothesis is the predicate so
brought to consciousness and defined that those features which are not
noticed in the ordinary judgment are brought into prominence. We then
recognize the hypothesis to be what in fact the predicate always is,
viz., a method of organization and control.