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REQUISITES OF HYPOTHESES

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cation are not already at hand they must at least be conceivable and their discovery must be within the bounds of possibility. Herodotus, in discussing the various theories of the rise of the Nile, says of the one which connected it with the mythical stream of Ocean: "The person who speaks about the Ocean, since he has transported the question to the dominion of the inscrutable, does not admit of refutation." 12

4. Other things being equal, choose the simplest hypothesis.

Making hypotheses involves mental activities which go beyond perception and memory. They are often discussed under the heading of "Imagination.” But imagination in this sense is not simply imagination in the limited sense of making mental pictures. It involves constructive activities often of a highly complicated sort. As "creative" imagination it differs from that of the poet in that it does not have to do necessarily nor primarily with concrete experiences. 13

EXERCISES.

State the ground of the hypothesis in each of the following examples and estimate the value of the hypothesis:

1. Geologists, watching at what rate changes are occurring in the earth's surface at the present time,—e. g., making of valleys, glacier movements, etc., determine the length of time it must have taken to produce the corresponding changes during the so-called geological periods.

2. In looking at the pictures in an art gallery, our attention is specially attracted by one picture whose characteristics impress themselves in our mind. Years afterward, in another country, we again see those characteristics in another picture and we feel certain that both pictures are the work of the same artist.

12 Gomperz, Greek Thinkers, Bk. III, chap. p. 6. 13 Minto, Logic, pp. 335, 336.

3. Cutting tools have edges and places for handles. These flints have edges and places for handles; they are therefore, cutting tools.

4. Some Northwest Coast Indians after seeing and hearing a phonograph for the first time, were asked what they thought it was. Their answer was that it was a very powerful echo which the white man controlled by means of a "strong medicine" or magic.

5. The theory that many philologists hold, that many of the languages of the world may be traced back to a common stock, known as the Aryan, is based on analogy. In Persian, Greek, Sanscrit, etc., several very simple words, usually verbs, such as to give and to be, are found to have almost identically the same root, from which resemblances the common descent is argued.

6. Certain mountains, which have large deposits of basalt, contain gold. When large deposits of basalt are found in other mountains, we may suppose that they also contain gold. If gold is not found, tin is. There seems to be a relation between deposits of basalt and deposits of gold.

7. Noting that certain substances expand when they crystallize, and noting also that certain other substances expand when heated, I might infer that heat causes the latter substance to crystallize and hence to expand.

8. Since ether has been offered as the medium of transmission of light-waves, and since some forms of electricity are forms of wave-motion, we might say that ether is the medium of transmission of electricity.

9. The U. S. is a republic and its citizens are prosperous and contented; we may therefore infer that if Cuba were a republic, her citizens would be prosperous and happy too.

10. Hydrochloric acid turns blue litmus paper red; sulphuric acid has similar properties, and we may infer that it, too, will turn blue litmus paper red.

11. Bones resembling those of an elephant were found in a given locality. We conclude that, at some time or other, elephants lived in this locality.

12. Cotton is grown in the U. S. in a moist, warm climate and a sandy soil; we may infer that Egypt, which has these characteristics, will also grow cotton.

CHAPTER III

TYPICAL SYSTEMS OF KNOWLEDGE

An examination of the methods employed in establishing certain typical varieties of systems of knowledge may help to make clearer the complexity of knowledge and the relations of the processes involved in getting it. Every system, as we have seen, contains laws. Some of them are systems of laws and general concepts; others include also concrete facts. Let us take as an example of the first, the sort of system which is to be found in mathematics or mechanics; and as examples of the second, the system of related facts which the historian or the criminal lawyer aims to establish. The other sciences lie between.

We cannot, in our present discussion, begin at the very beginning. We must grant to the historian and the lawyer the generally accepted laws of human behavior, the accepted principles of science, in short, the working materials of his science. To the mathematician, we must grant his concepts and axioms and postulates, and to both the general principles of scientific method. We wish merely to see how each employs these principles, what his method is. All these concepts and principles have been brought to light in the course of human experience. The mathematician employs chiefly the processes of analysis and deductive reasoning. Observation, testimony and, in general, the means

for knowing the concrete are, for the most part, left aside in his work.

1

The Geometric System. Let us take an example of scientific method as it appears in the science of geometry. Other fields of mathematics differ from this in important respects, but for the purposes of illustration geometry will be sufficiently representative. What is the starting point in geometry and what sort of system does it attempt to build? Geometry starts, not with perceived objects as the natural sciences do, but with a set of concepts and propositions. Among its concepts are those of point, line, magnitude, equality, and so on. Some of these are definable in terms of the others, as point is that which has no magnitude." There remain, however, certain concepts which are indefinable, viz., those by means of which all the others are defined. Of these concepts there are two kinds: concepts of elements, and concepts of relations. Besides these concepts geometry has among its data certain propositions which express the relations which hold among its elements. These propositions are known as axioms and postulates.2

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1 See Oswald Veblen, Popular Science Monthly, Vol. LXVIII, Art. The Foundations of Geometry"; and Transactions of the American Mathematical Society, Vol. 5, No. 3, Art. "A System of Axioms for Geometry."

2 No clear line of distinction was drawn by Euclid between axioms and postulates. Both were regarded as unproved and unprovable propositions which must be admitted as true by every one who understood them, as a priori truths. At the present day their a priori character is very widely questioned, but they are unprovable in that they can not be deduced from any simpler propositions. One way of distinguishing them was to define the axioms as common notions and the postulates as geometrical premises which must be taken for granted. But the line was not clearly drawn and propositions which sometimes appeared as postulates were at other times put among the axioms.

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MATHEMATICAL SYSTEMS

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As an axiom we may cite the first in the list: Things which are equal to the same thing are equal to one another "; and as a postulate: “ A straight line may be drawn from any one point to any other point." "All right angles are equal to one another " has sometimes been classed as an axiom and sometimes as a postulate. Euclidian geometry may be defined as a system of propositions codifying in a definite way our spatial judgments." Every one of its propositions can be deduced from its axioms and postulates, excepting, of course, the axioms and postulates themselves. To prove any proposition we have simply to combine certain concepts and propositions into a coherent whole.

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Let us examine Proposition XV, Book I. two straight lines (AB, CD) intersect each other, the vertically opposite angles made by them, are equal.”

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The demonstration is as follows: "For the angle CFA + the angle AFD = two right angles (prop. 13), and also the angle AFD + the angle DFB = two right angles; therefore the angle CFA + the angle AFD = the angle AFD + the angle DFB (axiom 1); and the common angle AFD being taken away from both, there remains the angle CFA the angle DFB (axiom 3); but these are vertically opposite angles. In like manner it may be proved that the vertically opposite angles

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