Process and Reality — the Gifford Lectures, 1929

It is at once evident that all these tests dependent on a direct intuition of permanence. This ‘permanence’ means ‘permanence in respect to congruence,’ for the various instruments employed, namely, the yard measure, or the optical instruments, or analogous instruments. For example, the yard measure is assumed to remain congruent to its previous self, as it is transferred from one setting to another setting. It is not sufficient to intuit that it remains the same body. Substances that are very deformable preserve that sort of self-identity. The required property is that of self-congruence. Minute variations of physical conditions will make the rod vary slightly; also sense-perception is never absolutely exact.

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But unless there be a meaning to ‘exactitude,’ the notions of a ‘slight variation’ and of a ‘slight defect from exactitude’ are nonsense. Apart from such a meaning the two occasions of the rod’s existence are incomparable, except by another experiment depending upon the same principles. There can only be a finite number of such experiments; so ultimately we are reduced to these direct judgments.

However far the testing of instruments and the corrections for changes of physical factors, such as temperature, are carried, there is always a final dependence upon direct intuitions that relevant circumstances are unchanged. Instruments are used from minute to minute, from hour to hour, and from day to day, with the sole guarantee of antecedent tests and of the appearance of invariability of relevant circumstances.

This ‘appearance’ is always a perception in the mode of presentational immediacy. If such perception be in any sense ‘private’ in contradistinction to a correlative meaning for the term ‘public,’ then the perceptions, on which scientific measurement depend, merely throw light upon the private psychology of the particular observer, and have no ‘public’ import.

Such a conclusion is so obviously inconsistent with our beliefs as to the intercommunication of real actualities in a public world, that it may be dismissed as a reductio ad absurdum, having regard to the groundwork of common experience which is the final test of all science and philosophy. A great deal of modern scientific philosophy consists in recurrence to the theory of ‘privacy’ when such statements seem to afford a short cut to simplicity of statement, and—on the other hand—of employment of the notion of observing a public world when that concept is essential for expressing the status of science in common experience. Science is either an important statement of systematic theory correlating observations of a common world, or is the daydream of a solitary intelligence with a taste for the daydream of publication. Butp. 503 it is not philosophy to vacillate from one point of view to the other.

SECTION V

Finally, meaning of ‘congruence’ as a relation between two geometrical elements in a strain-locus must be considered. It will be sufficient to consider this meaning in reference to two segments of straight lines, and to treat all other meanings as derivative from this.

A strain-locus is defined by the ‘projectors’ which penetrate any one finite region within it. Such a locus is a systematic whole, independently of the actualities which may atomize it. In this it is to be distinguished from a ‘duration’ which does depend on its physical content. A strain-locus depends merely upon its geometrical content. This geometrical content is expressed by any adequate set of ‘axioms’ from which the systematic inter-connections of its included straight lines and points can be deduced. This conclusion requires the systematic uniformity of the geometry of a strain-locus, but refers to further empirical observation for the discovery of the particular character of this uniform system. For example, the question as to whether a complete straight line be a ‘closed’ serial locus of points or an ‘open’ serial locus, is entirely a question for such discovery. The only decision is to be found by comparing the rival theories in respect to their power of elucidating observed facts.

The only relevant properties of straight lines are (i) their completeness, (ii) their inclusion of points, (111) their unique definition by: any pair of included points, (iv) their possibility of mutual intersection in a single point. The additional axioms which express the systematic geometrical theory must not have reference to length or to congruence. For these notions are to be _ derived from the theory. Thus the axioms must have exclusive reference to the intersection of straight lines, and to their inclusion or exclusion of points indicated by the intersections of other lines. Such sets of axioms arep. 504 well known to mathematicians. There are many such sets which respectively constitute alternative geometrical theories. Also given one set of axioms constituting a definite geometrical theory, different sets of axioms can easily be obtained which are equivalent to each other in the sense that all the other sets can be deduced from any one of them. All such equivalent sets produce the same geometrical theory. Equivalent sets have their importance, but not for the present investigation. We can therefore neglect them, and different sets of axioms will mean sets of axioms which constitute incompatible geometrical theories.

There are many such sets, with a great variety of peculiar properties. There are, however, three such sets which combine a peculiar simplicity with a very general conformation to the observed facts. These sets give the non-metrical properties of the three geometrical theories respectively known to mathematicians as the theory of Elliptic Geometry, of Euclidean Geometry, and of Hyperbolic Geometry. It will serve no purpose to give the three sets of axioms. But it is very easy to explain the main point of difference between the theories, without being led too far from the philosophical discussion.

In the first place, a definition of a ‘plane’ can be given which is common to all the three theories. The definition already given in Chapter III of this Part will suffice. But an alternative definition can be stated thus: If A, B, C be any three non-collinear points, and AB, BC, CA denote the three complete straight lines containing respectively, A and B, B and C, C and A, then the straight lines which respectively intersect both members of any pair of these three lines, not both lines at one of the corners A or B or C, pass through all the points constituting one plane, and all their incident points are incident in the plane.

Thus a plane is defined to be the locus of all the points incident in at least one of such a group of straight lines. The axioms are such that this definition is equivalent top. 505 the definition in Chapter III. Also the axioms secure that any straight line passing through two points in a plane, itself is wholly incident in that plane. Also it follows from the definition of a plane that a line / and a point P, not incident in J, are coplanar.