DAVIS 1931 Discovery Review

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REVIEWS 373

R. D. Carmichael.-The Logic of Discovery. IX+280 p. The Open

Court Publishing Co., Chicago, I930. (Price S 2.00).

Only at rare intervals does there appear a book devoted to the discussion

of problems in epistemology which is thoroughly lucid. At one placeor another undefinables are introduced into the discussion and the meaning

is then lost in a confusion of words. The book under review therefore

is especially noteworthy in the fact that it belongs to that limited class

of philosophical essays which defines its terms, states its postulates,

and then proceeds to the development of its theme with an inevitable

logic. A reader is thus privileged to disagree with the conclusions,

not from any feeling of insecurity derived from semantic uncertainties,

but only from unwillingness to grant the initial premises. One thusderives unusual satisfaction in the perusal of Professor CARMICHAEL'S

book since one is never on a single page uncertain as to the meaning

of the author.

The problem of the book is to delineate the character of a logic of

discovery, in contrast to a logic of demonstration, ain the sense of a logic

by which one infers from the known to that unknown which hitherto

has not been apprehended or suspected)). The questions for which

answers are sought are of the type: How does the mind behave in theprocess of investigation ? What constitute heuristic guides in the activities

of discovery? Are such guides essentially particular, depending upon

the nature of the material of the investigation, or do they possess general

attributes characteristic of every investigation?

The author, while admitting in the beginning that (( he mind cannot

comprehend its highest movements with clearness)) since this would

involve an explanation of explanation and so on with an infinite regression,

believes that a measure of delineation is still possible for the processesof discovery. In proof of this he cites two examples. The first of

these is the heuristic guide first employed by STURM nd CAUCHY hereby

known theorems about systems of algebraic equations are found to

suggest new results in the realm of transcendental equations through

limiting processes of doubtful rigor. This method has been amply justi-

fied as a tool of investigation in the beautiful results obtained in recent

times in the theory of integral equations by VOLTERRAand FREDHOLM.

The author sees in this a genuine logic of discovery by means of whichthe mind is led through essentially non-rigorous processes to conclusions

which must later be verified by the inevitable logic of demonstration.

The second example is EINSTEIN'Spostulate of relativity which may

be epitomized in the assertion that natural phenomena are independent

of mathematical axes. Once formulated the postulate is a certain

guide to investigations. That the discoveries so made may finally prove



374 ISIS, XV, 2

to be vacuous or not consonant with objective experiment has nothing

to do with the power of the postulateas the basis of a logic of discovery.

The answerin the case cited is of course too well known to requirefurther

comment. Other examples will readily occur. Nature acts in sucha way that the first variation of the integral of action is zero; matter

and energy are indestructible; the time-derivativeof entropy is always

positive. These postulateshave alreadyproducedvolumes of discoveries

and present tendencies of physical science lead one to suspect that these

logics of discovery have not yet been entirely exhausted.

It is to be regretted that Professor CARMICHAELt this point did not

explore more fully the nature of ((an insidious error)) to which he gives

the name of ((proof by ignorance )), wherein apparent indeterminationsin postulates are attributed to the insufficiency of our knowledge. Pos-

tulates which rest essentially upon the calculus of probabilities are apt

to have an element in them of this insufficient reason. The author

mentions the second law of thermodynamics as a postulate of this nature,

but it would have been of the greatest interest to have had also an appraise-

ment of perhaps the most celebrated example in science of this type,

namely, the rule of BAYESn the theory of inverse probability. In this

connection R. A. FISHERrecently made the following remarkable state-ment : ((I know only one case (BAYES ule) in mathematics of a doctrine

which has been accepted and developed by the most eminent men of

their time, and is now perhaps accepted by men now living, which at

the same time has appeared to a succession of sound writers to be funda-

mentally false and devoid of foundation )). (Proc. Cambridge Phil. Soc.,

vol. 26, 1930, p. 528). The difficulty in this theory is certainly to be

attributed to the very subtle introduction of a ((proof by ignorance )).

Having in the first chapter of the book exhibited the fruitfulnessof postulates as tools of investigation, Professor CARMICHAELroceeds

in succeeding pages to explore their nature. One is frankly surprised

to see at the beginning the statement: ((It seems clear that postulate

systems have been employed fully and consciously in only a small part

of that domain of thought in which their use will yield characteristic

values... Too much of what is known on the subject is now known to

a relatively small group of thinkers )). And yet the researches of GAUSS,

LOBACHEVSKI,BOLYAI, and RIEMANN in non-Euclidean geometry haveappeared only within the last hundred years and the postulational nature

of mathematics itself thus belongs to recent knowledge. The theory

of relativity and the new formulation of quantum mechanics begin

tardily to force a similar character ulpon the ((laws)) of physics. Less

fortunate fields have as yet been unable to achieve a postuilational basis.

In this connection the author says : ( ... the known fact of the existence



REVIEWS 375

of important and extensive bodies of doctrine in the ideal deduictive

form affordedby a postulationaltreatment should be kept in mind byevery scientific investigator as suggesting a fundamental criterion by

which he may estimate the degreeof advancementof his own science)).Having thus set up the postulateas the basis of any scientificdoctrine,

and having further defined the consistency of such a system as its iso-

morphism or one-to-one-ness with some set of objects, the authorturns to the concept of a doctrinalfunction as the primarybasis of exactthought. By the term thus so impressively elevated Professor CARMICHAEL

means ((any body of statements which is made up solely of a set of con-

sistent postulates about elements or objects not explicitly defined and

the logical consequences which flow from them)).The main contribution of the book is perhaps found in its development

of the thesis that the progress of any science is measured by the degree

to which it approximates the character of a doctrinal function. Omitting

hypotheses that never gain scientific currency, the author shows that

systems of postulates fall into one of two classes : first, those which

for a time are of great usefulness in suggesting the direction of empirical

investigation but are ultimately found to lead to contradiction, or, second,

those which attain the exalted position of forming the basic principlesof science. He concludes however, with a statement pessimistic for

those who labor in empirical fields that one may doubt whether any

system of the latter class will fail ultimately to recede into the former.

It is by means of this constant exploration of the consequences of postulates

and the ultimate comparison with experience that the doctrinal function

of any science is finally to be attained.

The book is full of illustrations mainly though not exclusively from

mathematical and physical sciences. The history of the atomic theoryis systematically developed as an indication of the growth of hypotheses

and the author concludes that ( it promises to hold this supreme place

during our generation )). It is quite interesting in this connection,

however, to note the opening statement in a recent treatise on atoms,

molecules, and quanta, where a certain sharp debate between the two

eminent physicists BOLTZMAANNnd MACH is recorded. The former had

categorically stated in a scientific meeting : ((I know that there are

molecules ,), to which MACH had immediately replied: ((You do not )).This modern treatise then makes the significant statement that the

present tendency is to side with MVACH.

The book concludes with a chapter somewhat unrelated to the main

theme in which the autor argues for the larger hluman worth of

mathematics. It is interesting, however, irz adding one more item to

the record of those who have felt the exaltation experienced by a survey

DAVIS 1931 Discovery Review

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