Chapter X: The Definition and Division of PrinciplesEdit
- Such indemonstrable principles may be either peculiar to each science or common to several sciences, though common only by analogy. All demonstration involves three things:—the object demonstrated, common axioms or principles, and the special modifications or properties of the subject genus. The distinction between Hypothesis and Petition.
I mean by the elementary principles in each genus those whose existence it is not possible to prove. Now the meaning of the primary principles and that of their consequences are assumed; the existence of the elementary principles must also be assumed, that of everything else proved. For instance the meaning of Unit, or Straight, or Triangle must be assumed, that Unit and Magnitude exist must also be assumed, everything else must be proved.
Of the principles employed in demonstrative science some are peculiar to each science, others are common to all, i.e. common in the sense of analogous, since their use is confined to each genus as comprehended by a particular science. Principles peculiar to one science are such as the proposition ‘Line, or Straight, is of such and such a nature;’ common principles are such as, ‘If one take equals from equals the remainders are equal.’ Each of these principles is taken as applicable to all cases belonging to the particular genus; for its results will be the same whether it be treated universally or only particularly, e.g. in geometry to spatial magnitudes or in arithmetic to numbers.
Those principles too are peculiar whose existence is assumed not demonstrated, namely those whose essential attributes are investigated by the science; as arithmetic investigates units, geometry points and lines, for these sciences assume that the thing in question exists, and that it is identical with some particular object. They likewise assume the meaning of the essential attributes of the thing, as arithmetic assumes the meaning of Odd, Even, Square or Cube, and geometry that of Incommensurable, and Inclined or at an Angle, while the existence of these qualities is shewn by means of the common principles and the conclusions already demonstrated. The same thing applies to astronomy.
In short in every demonstrative science there are three elements: (1) the things whose existence it assumes, namely the subject or genus, the essential attributes of which are investigated by the science; (2) what are called ‘Common Axioms’ which the demonstration uses as its primary principles; and (3) Properties, the meaning of which is assumed.
However nothing prevents some sciences from overlooking one or other of these elements; e.g. a science may not expressly assume the existence of the subject genus if this be self-evident (for the existence of Number is more obvious than that of Cold or Heat), or it may not assume the meaning of the properties if it is obvious, just as in the case of their common principles the sciences do not assume the meaning of ‘taking equals from equals,’ because this is known. None the less however there are naturally these three elements in a science:—the subject of proof, the things proved and the grounds of proof.
That which must needs exist and must necessarily be supposed to exist is neither Hypothesis nor Petition but Axiom. Demonstration is not concerned with the outward expression of an idea but with its inner significance, for that is the case with syllogism in general, and one may always raise objections to the external expression but not always to the inner significance.
Everything which, being capable of proof, is assumed without being proved, if admitted by the learner is a Hypothesis, which hypothesis is not an absolute hypothesis but only one with reference to the person who accepts it.
If however something be assumed with regard to which the learner has no opinion or a contrary one it is a Petition. This then is the difference between hypothesis and petition; petition being that which is somewhat opposed to the learners opinion, or, in a wider sense, whatever, though capable of demonstration, is assumed and employed without any proof.
Definitions are not hypotheses, since it is not asserted that their subjects do or do not exist. Hypotheses are formulated as propositions, Definitions require only to be understood, and no Hypothesis consists in that alone, unless it be maintained that mere Hearing is a Hypothesis. Hypotheses are the premises from the existence of which the conclusion is inferred.
The hypotheses of the geometrician are not, as some assert, false, saying that, though one ought not to make use of false propositions, yet the geometrician calls a line a foot long which is not a foot long, or declares that he has drawn a straight line, though the line is really not straight. The geometrician in reality draws no conclusion from the fact of the particular line that he draws actually possessing the quality which he names, but from the existence of the things which that line represents.
Moreover all postulates and hypotheses are universal or particular, definitions are neither.