100 decimal places. But the following example may possibly give us
some conception of it. Conceive a sphere constructed with the earth
as centre, and imagine its surface to pass through Sirius, which is
134-1/2 million million kilometres distant from us. Then imagine
this enormous sphere to be so packed with microbes that in every
cubic millimetre millions of millions of these diminutive animalcula
are present. Now conceive these microbes to be all unpacked and so
distributed singly along a straight line, that every two microbes are
as far distant from each other as Sirius from us, that is 134-1/2
million million kilometres. Conceive the long line thus fixed by
all the microbes, as the diameter of a circle, and imagine the
circumference of it to be calculated by multiplying its diameter with π
to 100 decimal places. Then, in the case of a circle of this enormous
magnitude even, the circumference thus calculated would not vary from
the real circumference by a millionth of a millimetre.
This example will suffice to show that the calculation of π to 100 or
500 decimal places is wholly useless.
#Professor Wolff's curious method.#
Before we close this chapter upon the evaluation of π, we must mention
the method, less fruitful than curious, which Professor Wolff of Zurich
employed some decades ago to compute the value of π to 3 places. The
floor of a room is divided up into equal squares, so as to resemble
a huge chess-board, and a needle exactly equal in length to the side
of each of these squares, is cast haphazard upon the floor. If we
calculate, now, the probabilities of the needle so falling as to lie
wholly within one of the squares, that is so that it does not cross
any of the parallel lines forming the squares, the result of the
calculation for this probability will be found to be exactly equal to π
- 3. Consequently, a sufficient number of casts of the needle according
to the law of large numbers must give the value of π approximately. As
a matter of fact, Professor Wolff, after 10000 trials, obtained the
value of π correctly to 3 decimal places.
#Mathematicians now seek to prove the insolvability of the problem.#
Fruitful as the calculus of Newton and Leibnitz was for the evaluation
of π, the problem of converting a circle into a square having exactly
the same area was in no wise advanced thereby. Wallis, Newton,
Leibnitz, and their immediate followers distinctly recognised this.
The quadrature of the circle could not be solved; but it also
could not be proved that the problem was insolvable with ruler and
compasses, although everybody was convinced of its insolvability. In
mathematics, however, a conviction is only justified when supported
by incontrovertible proof; and in the place of endeavors to solve
the quadrature there accordingly now come endeavors to prove the
impossibility of solving the celebrated problem.
#Lambert's contribution.#
The first step in this direction, small as it was, was made by the
French mathematician Lambert, who proved in the year 1761 that π was
neither a rational number nor even the square root of a rational
number; that is, that neither π nor the square of π can be exactly
represented by a fraction the denominator and numerator of which are
whole numbers, however great the numbers be taken. Lambert's proof
showed, indeed, that the rectification and the quadrature of the
circle could not be possibly accomplished in the particular way in
which its impossibility was demonstrated, but it still did not exclude
the possibility of the problem being solvable in some other more
complicated way, and without requiring further aids than ruler and
compasses.
#The conditions of the demonstration.#
Proceeding slowly but surely it was next sought to discover the
essential distinguishing properties that separate problems solvable
with ruler and compasses, from problems the construction of which is
elementarily impossible, that is by solely employing the postulates.
Slight reflection showed, that a problem elementarily solvable, must
always possess the property of having the unknown lines in the figure
relating to it connected with the known lines of the figure by an
equation for the solution of which equations of the first and second
degree alone are requisite, and which may be so disposed that the
common measures of the known lines will appear only as integers. The
conclusion was to be drawn from this, that if the quadrature of the
circle and consequently its rectification were elementarily solvable,
the number π, which represents the ratio of the unknown circumference
to the known diameter, must be the root of a certain equation, of a
very high degree perhaps, but in which all the numbers that appear are
whole numbers; that is, there would have to exist an equation, made
up entirely of whole numbers, which would be correct if its unknown
quantity were made equal to π.
#Final success of Prof. Lindemann.#
Since the beginning of this century, consequently, the efforts of a
number of mathematicians have been bent upon proving that π generally
is not algebraical, that is, that it cannot be the root of any equation
having whole numbers for coefficients. But mathematics had to make
tremendous strides forward before the means were at hand to accomplish
this demonstration. After the French Academician, Professor Hermite,
had furnished important preparatory assistance in his treatise "Sur la
Fonction Exponentielle," published in the seventy-seventh volume of
the "Comptes Rendus," Professor Lindemann, at that time of Freiburg,
now of Königsberg, finally succeeded, in June 1882, in rigorously
demonstrating that the number π is not algebraical,[52] thus supplying
the first proof that the problems of the rectification and the squaring
of the circle, with the help only of algebraical instruments like ruler
and compasses are insolvable. Lindemann's proof appeared successively
in the Reports of the Berlin Academy (June, 1882), in the "Comptes
Rendus" of the French Academy (Vol. 115. pp. 72 to 74), and in the
"Mathematischen Annalen" (Vol. 20. pp. 213 to 225).
[52] For the benefit of my mathematical readers I shall
present here the most important steps of Lindemann's
demonstration, M. Hermite in order to prove the transcendental
character of
_e_ = 1 + 1/1 + 1/1.2 + 1/1.2.3 + 1/1.2.3.4 + ....
developed relations between certain definite integrals
(_Comptes Rendus_ of the Paris Academy, Vol. 77, 1873).
Proceeding from the relations thus established, Professor
Lindemann first demonstrates the following proposition: If
the coefficients of an equation of _n_th degree are all real
or complex whole numbers and the n roots of this equation
_z_{1}, _z_{2}, ..., _z_{_n_} are different from zero and
from each other it is impossible for
_e_^_z_{1} + _e_^_z_{2} + _e_^_z_{3} ... + _e_^_z_{_n_}
to be equal to _a_/_b_, where _a_ and _b_ are real or complex
whole numbers. It is then shown that also between the functions
_e_^{_rz_{1}} + _e_^{_rz_{2}} + _e_^{_rz_{3}}
+ ... _e_^{_rz_{_n_}},
where _r_ denotes an integer, no linear equation can exist
with rational coefficients variant from zero. Finally
the beautiful theorem results: If _z_ is the root of an
irreducible algebraic equation the coefficients of which are
real or complex whole numbers, then _e_^_z_ cannot be equal
to a rational number. Now in reality _e_^{t√-1} is equal to a
rational number, namely,-1. Consequently, π√-1, and therefore
π itself, cannot be the root of an equation of _n_th degree
having whole numbers for coefficients, and therefore also not
of such an equation having rational coefficients. The property
last mentioned, however, π would have if the squaring of the
circle with ruler and compasses were possible.
#The verdict of mathematics.#
"It is impossible with ruler and compasses to construct a square
equal in area to a given circle." These are the words of the final
determination of a controversy which is as old as the history of the
human mind. But the race of circle-squarers, unmindful of the verdict
of mathematics, that most infallible of arbiters, will never die out so
long as ignorance and the thirst for glory shall be united.
HERMANN SCHUBERT.
THE CRITERION OF TRUTH.
A DISSERTATION ON THE METHOD OF VERIFICATION.
Modern science rests upon the recognition of the truth that all
knowledge is a statement of facts. The formulation of natural laws is
nothing but a comprehensive description of certain kinds of natural
processes. Natural laws are generalisations of facts. Similarly,
any philosophical theory is, or from the modern standpoint ought to
be, simply a systematised representation of facts. Facts are the
bottom-rock to which, everywhere, we have to go down.
The recognition of this maxim is called, most appropriately,
positivism; and I take it that as a matter of principle all modern
thinkers can and perhaps do agree to it. A Roman Catholic philosopher
may consider some things as facts which a scientist of heretic England,
for instance, does not; yet it is from facts, or what is thought to be
facts, that every one derives his conception of the world.
It is natural that the range of individual experience should be
very limited in comparison with the knowledge indispensably needed
for acquiring an adequate conception of the world in which we live.
We have, to a great extent, to rely on statements of facts which
we ourselves have not observed. To enrich and to enlarge our own
experience we have to imbibe the experience of others. Sometimes we
can, but sometimes we cannot, verify what we have been told. For
instance, that stones fall through empty space with a velocity of
32·18 English feet at the end of the first second can be verified
by experiment, i. e., the experiment can be repeated under the same
circumstances. But historical data such as whether Buddha died under
a fig-tree, or whether Christ was crucified under Pontius Pilate,
cannot be verified by experiment. Historical data are statements, not
of general truths, but of single facts, which, if they are accepted
at all, have to be taken on authority. The authority may be weak or
strong; it may be strong enough to be equivalent practically to a
certainty, which latter case occurs, for instance, when the fact in
question in its direct consequences perceptibly affects our life, and
its causal connection can thus be directly and indubitably traced.
It is not intended here to emphasise the difference between facts
verifiable by experiment, and historical facts; yet it is desirable
with reference to all kinds of facts stated on authority, to understand
the importance of a criterion of truth. We do accept and we have to
accept, every one of us, without any exception, the most discriminate
scientist even and most of all the philosopher, innumerable statements
of facts as they have been observed by others. We all have to rely on
the authority of others. The time of the longest human life would be
too short to repeat all the experiments made by others, with a view
to verifying them in detail. On the other hand, it is obvious that no
statement of facts should be accepted on pure authority. We must have
a means, a sieve as it were, by which the wheat can easily be winnowed
from the chaff; a sieve that will enable us to discard at once those
statements that are positively erroneous. In this way our attention
can be confined to statements of things that are possible, those that
need not, but _may_ be true. "Possible" in German is very appropriately
called _möglich_, i. e. _mayable_.
The criterion of that which 'may be' true is the first step towards
ascertaining truth; and although it does not exhaust the methods of
arriving at truth it is of greatest consequence, for if properly
understood and applied, it would save from the start many useless
efforts in the investigation of truth.
* * * * *
The question arises then, What is the criterion of the possible? We
reject statements, sometimes, as _prima facie_ untrue. Have we a right
to do so? And if we have, by what standard do we determine this?
Let us first take into consideration how people really behave when a
statement of new facts is made. Take, for instance, the following case.
Two strangers meet; A. and B. Mr. A. relates to Mr. B. some incident
of his life. He is apparently a very trustworthy person and during
the conversation remains perfectly serious. He tells a ghost story in
detail, how a departed friend of his appeared to him in distinctly
visible form; he says that the spirit spoke to him and told him many
strange things, and that he pointed out to him an imminent danger.
We suppose that on the one hand A. makes his statement in good faith
and that on the other hand B. is a spiritualist. Will B. consider
A.'s story as possible? B., being a spiritualist, most probably will
consider A.'s story as possible, and, if he is convinced of A.'s
honesty, he will believe the story the same as if he had experienced
it himself; no less than a scientist will rely on the statement of an
experiment made by one of his colleagues whose scientific veracity he
has no reason to doubt.
Suppose A. tells the same story to C. Mr. C. is an infidel and a
materialist. As characteristic features of his personality we might
mention that he considers religion as pure superstition originated by
the fraud of cunning priests. This man will, we may fairly suppose,
laugh at A.'s story, because it appears to him an out and out lie. Mr.
A. as well as Mr. B., he who tells and he who believes the story, C.
will declare, are either insane or they are both impostors.
The difference of opinion in B. and C. indicates that the criterion
of truth is different with different persons and that it depends upon
their conception of the world. Men who have the same world-conception
will also have the same criterion of truth.
The problem consequently is, whether this criterion of truth (i. e. the
criterion of what is possible) is necessarily wholly subjective, or
whether we can arrive at an objective criterion. It is apparent that
this question is intimately connected with another problem, namely,
Is every world-conception necessarily subjective, or, Is it possible
to arrive at an objective world-conception? It appears to me that we
can; and the ideal of philosophy to-day is just such an objective
representation of facts.
The difficulty that presents itself lies mainly in the confusion
between facts and our interpretation of facts. If A. declares that he
saw a ghost, he does not relate a fact, but his interpretation of a
fact. Let us suppose that he tells his story again to a third person
D., who is a psychologist. D. most likely will not think him a liar.
D. will accept the statement _bona fide_ as a mere interpretation of a
fact and will inquire after the causes that produced the hallucination.
He may be able, possibly, to lay bare the facts disfigured by the wrong
interpretation of A. And having clearly stated the objective state
of things he may with the assistance of his experience explain the
origin of the whole process, partly from the mental condition and the
physiological constitution of A., partly from individual circumstances
that gave rise to the hallucination. He will not doubt that something
extraordinary has happened to Mr. A. The latter's mind has been, and
perhaps still is in an abnormal state. And as to B.'s believing the
ghost story, Mr. D. will not think that he is insane; though we may
presume that he will regard B.'s views of the world as resting upon
unfirm grounds; and he will not believe him to be a man of critical
ability.
The notion is very common among idealists that we can never go beyond
our subjective states of consciousness. This would be tantamount to
saying that there is no difference between dreams and real life, except
that a dream is cut off by awaking while life lasts comparatively much
longer and ceases with death, which may also be an awakening from
a dream. In that case hallucinations would be of the same value as
sensations. Both would be interpretations of facts for which we do not
have an objective criterion of truth. Interpretations of facts would
be the sole facts, and it would be quite indifferent whether they were
misinterpretations or correct interpretations.
Take a simple instance. We see a tree. The perception of a tree is an
interpretation of a set of facts. Interpretations of facts, whether
correct or not, are of course also facts. Thus the perception of a tree
is a fact which, if all matter were transparent, would, physiologically
considered, appear to the eye of an observer as special vibrations in
the brain. But the peculiarity of this fact is that it represents
other facts. The question is no longer whether there is a perception of
a tree taking place in a brain, but whether this perception is true,
i. e., whether it agrees with the facts represented. Every perception
has a meaning beyond itself; every perception is a fact representing
other facts, and the question of truth or untruth has reference to the
agreement between representations and facts represented.
Professor Mach says in his essay "The Analysis of Sensations" (_The
Monist_, Vol. I. No. I, p. 65):
"Bodies do not produce sensations, but complexes of sensations
(complexes of elements) form bodies."[53]
[53] Professor Mach in thus speaking of bodies uses the
word in the sense of representations and not in the sense
of objects represented. He calls them in the sentence next
following "thought-symbols for complexes of sensations
(complexes of elements)."
And, certainly, we do not deny that upon a closer analysis the
perception of a tree appears as a bundle, or a complex of sensations;
there is the green of the leaves, the color of the bark, the different
shades of the color indicating its bodily form, the shape of the
branches, and their slight motions in the breeze that gently shakes the
tree. Yet the perception of a tree does not consist of these sensations
alone. All these sensations might be so many isolated sensations; and
if they remained isolated, they would not produce the percept of a
tree. These sensations are interpreted; they have acquired a meaning
and are combined into a unity. It is this unity which constitutes the
perception of a tree. This unity has grown from sensations; and that
process which develops and, as we have learned, naturally must develop
sensations from sense-impressions, and from sensations perceptions
that are representative of a group of facts outside of the perceptions
themselves,—that process we define as mind-activity.
What does the 'perception of a tree' mean? It means that if the person
perceiving it moves in a certain direction and over a certain distance,
he will have certain sensations which upon the whole can be correctly
anticipated. Every perception and also every sensation contains a
number of anticipations. The perception of a tree is in so far to be
considered correct, as the anticipations which it contains, and of
which it actually consists, can be realised. If and in so far as these
anticipations when realised tally with the perception, if and in so far
as they justify it, or can justify it, if and in so far as they fulfil
the expectations produced by the perception, if and in so far as they
make no alteration of the perception necessary, but being in agreement
with it confirm the representation it conveys: the perception is said
to be true. Moreover, we can predict similar results with regard to
beings of a similar constitution.
* * * * *
Now let us suppose that an apple falls from a considerable height to
the ground. Knowing, from former experiences, the hardness of the soil
as well as the density of the apple, we can anticipate the effect of
the fall. The soil will not show any considerable impression, yet
one side of the apple will be crushed. In predicting this result
we anticipate sensations that we shall have under a certain set of
circumstances. In so far as we shall necessarily have these sensations
we have to deal with facts. Not as if our sensations constitute the
entire existence of facts; our sensations, being the effects of
so-called objective processes upon our senses, are only one end of a
relation, which as a matter of course never exists without the other
end. Sensations are the one end; they depend upon and vary with the
other end. Showing within certain limits as many varieties here as
occur there, they represent the other end.
We can, and for certain purposes we must, entirely eliminate the
subjective and sensory part of our sensations, in order to represent
in our minds not how two objects affect our senses of sight or touch
but how two or more objects affect each other. Thus we arrive at an
objective statement of facts, how the falling apple affects the soil,
and the soil the apple; while the relation of both to our senses is
to be eliminated. This objective statement of facts is the ideal of
all natural sciences. The physicist states the interaction between the
falling apple and the soil. He does not care how many sentient beings
witness the fall; he does not care about the psychological element in
their observations. He abstracts from the subjective elements in their
observations as well as in his own, and confines his attention to the
objective facts represented in their minds.
The objection to this conception of things is made by a consistent
idealist, that these observations must always exist in some mind, they
do not exist outside of a mind, and mind can as little go beyond itself
as a person can walk outside of his skin. Certainly, observations
always exist in some mind; they have always a subjective element. But
they have also an objective element. No sensation, no perception, no
observation is without an objective feature. This objective feature
in a sensation or a perception, and also in an abstract idea, is the
element that if true has to agree with other facts outside of the
sentient being of whose mind the perception is a part. An idealist who
is pleased to deny this would either have to identify hallucinations
with sensations, or he would be obliged to consider the objective
elements of his mind merely and solely as subjective states, having
no representative value. In that case he would necessarily be obliged
to consider the facts represented, i. e. the things outside the body,
as parts of his mind. This being granted, every mind would appear
as congruent and coextensive with the universe. We should have as
many universes as there are minds, and yet all universes would be
only one and the same universe, their sole difference being that of
a difference of centres. With the death of every living creature a
universe would die; but notwithstanding the chain of consciousness were
broken forever in death, the existence of his mind, being that which
is commonly considered as the objective universe, would not cease;
merely a view-centre would be lost. That which we have characterised
as representations in feeling-substance (which according to our
terminology constitutes mind) would be a transient and unessential
feature of mind only; and if it should cease to be, mind would
still exist in what we have defined as the outside facts, the facts
represented in mental symbols. In short, mind would be the All, it
would be a synonym of God. And not only all mental beings actually
existing or having existed would each, one and all, constitute the
universe, but also all potential minds, every atom and all possible
combinations of atoms that possibly might play a part in the mental
activity of a sentient being, would constitute it.
The views of an idealist who accepts these consequences are undeniably
correct, although we may quarrel about the propriety of his
terminology. Yet an idealist of this type, we may fairly assume, will
have little difficulty in adapting himself to our terminology, and in
that case we might easily agree about the possibility of arriving at a
criterion of truth; for his world-conception (aside from a difference
in terms) might, or rather would be practically the same as ours.
If truth is the agreement of certain mental facts with other
facts outside of the mind—if it is the agreement of subjective
representations with objective things or states of things represented,
the problem is whether we have any means of revising or examining this
agreement.
* * * * *