Operational thinking will at first prove to be an unsocial virtue; one will find oneself perpetually unable to understand the simplest conversation of one's friends, and will make oneself universally unpopular by demanding the meaning of apparently the simplest terms of every argument. Possibly after every one has schooled himself to this better way, there will remain a permanent unsocial tendency, because doubtless much of our present conversation will then become unnecessary. The socially optimistic may venture to hope, however, that the ultimate effect will be to release one's energies for more stimulating and interesting interchange of ideas. Not only will operational thinking reform the social art of conversation, but all our social relations will be liable to reform. Let any one examine in operational terms any popular present-day discussion of religious or moral questions to realize the magnitude of the reformation awaiting us. Wherever we temporize or compromise in applying our theories of conduct to practical life we may suspect a failure of operational thinking. CHAPTER II OTHER GENERAL CONSIDERATIONS THE APPROXIMATE CHARACTER OF EMPIRICAL KNOWLEDGE ALTHOUGH many aspects of the processes by which we obtain knowledge of the external physical world are much beyond the scope of our present inquiry, one matter must be mentioned in detail because it tacitly underlies all our discussion, the fact, namely, that all results of measurement are only approximate. That such is true is evident after the most superficial examination of any measuring process; any statement about numerical relations between measured quantities must always be subject to the qualification that the relation is valid only within limits. Furthermore, all experience seems to be of this character; we never have perfectly clean-cut knowledge of anything, but all our experience is surrounded by a twilight zone, a penumbra of uncertainty, into which we have not yet penetrated. This penumbra is as truly an unexplored region as any other region beyond experiment, such as the region of high velocities, for example, and we must hold no preconceived notions as to what will be found within the region. The penumbra is to be penetrated by improving the accuracy of measurement. Within what was at one time penumbra has been found the displacement of angular position of the stars near the edge of the solar disc, and within the penumbra as yet unpenetrated we look for such effects as the equivalence of mass and energy. Many of the great discoveries of the future will probably be made within the penumbra: we have already mentioned that increased knowledge of phenomena of a cosmic scale is to be obtained by increasing the accuracy of measurement of the very small.