Gottlob Frege

Review of Dr. E. Husserl’s Philosophy of Arithmetic

The author decides in the Introduction that for the time being he will consider (only) cardinal numbers (cardinalia), and thereupon launches into a discussion of multiplicity, plurality, totality, aggregate, collection, set. He uses these words as if they were essentially synonymous; the concept of a cardinal number1 is supposed to be different from this. However, the logical relationship between multiplicity and number (p. 9) remains somewhat obscure. If one were to go by the words “The concept of number includes the same concrete phenomena as the concept of multiplicity albeit only by way of the extensions of the concepts of its species, the numbers two, three, four, etc.…,” one might infer that they had the same extension. On the other hand, multiplicity is supposed to be more indeterminate and more general than number. The matter would probably be clearer if a sharper distinction were drawn between falling under a concept and subordination. Now the first thing he attempts to do is to give an analysis of the concept of multiplicity. Determinate numbers as well as the generic concept of number which presupposes them, are then supposed to emerge from it by means of determinations. Thus we are first led down from the general to the particular, and then up again.

On the law of inertia

It will undoubtedly seem strange to many that a law as long regarded as unquestionable as that of inertia should again receive exhaustive examination and that a new conception for it be sought. ‘In the absence of outside forces, a body at rest remains at rest and a body in motion retains its velocity’. This has been established in innumerable cases; and what is meant by ‘a body is in motion’ or ‘is at rest’ appears to be so clear that nothing remains to be explained. The purpose of the worthwhile work mentioned below [i.e. Lange’s book] is to upset this false security and inspire further speculation. It is well-known, and the author [Lange] brings it out in detail, that the philosophers of antiquity found it difficult to answer the question whether a given body move or not. I am thinking of the ship anchored in the current; and of the man moving toward the rear of a ship under sail, whose position with respect to objects on the shore does not change. In such cases our question may easily receive different answers, according to the weight one places on this or that relative position; a general criterion is wanting. All of these disputes would of course be quite simply settles if one recognized the incompleteness of an expression like ‘a moves’ and in its place wrote ‘a moves with respect to b’ The sentences, ‘a moves with respect to b’ and ‘a does not move with respect to c’ need not contradict one another. Indeed the physicists will admit that the motion of a body is never absolutely ascertainable, but rather is only so in relation to another. One thereby acknowledges the defectiveness of the above statement of the law of inertia; for the reference therein is to absolute motion and rest. And the bad part of this is that this defect cannot be corrected by including in the law a reference to another body; for which [reference body] should we choose? A given body would appear to be at rest, or to move in rectilinear or curved paths, uniformly or nonuniformly, depending on the choice of the reference body. The nature of the law of inertia prohibits reference to a particular body, since none exists which deserves this distinction; but at the same time absolute motion remains unrecognizable. This is the difficulty. How is it then that it is so little heeded by the physicists? The incomplete expression, ‘a moves’ is so convenient and sanctified by ordinary usage that it is used all too often even in physics. The theoretical impropriety of this expression is all the more readily forgotten as it easily helps to overcome many a difficulty. Whenever one cannot answer a question, one can at least allow it to disappear behind the cloud of imprecise speech - which in the present case is especially agreeable. For if one considered the matter quite openly, the entire foundations of physics would seem to totter. For this reason, one has unconsciously avoided throughout using the complete expression ‘a moves with respect to b’ Further, the law of inertia has become such unassailable common knowledge that we do not readily notice when we tacitly presuppose it in order to prove it. In this way we easily make use of laws of motion and of expressions like ‘mass’ and ‘force’ although the law of inertia is the foundation of all laws of motion and lends meaning to these expressions. How is it then that physics makes such sure progress despite this deficiency? Indeed, astronomy does introduce a coordinate system that suffices for practical purposes. When we express the law of inertia in terms of motion with respect to this system we find that all consequences are sufficiently in accord with experience. Theoretically, however, we gain nothing by this; for no one doubts that the fixed stars, which we need to set up our coordinate system, are only apparently at rest with respect to one another, and that this appearance is the consequence of our imprecise observations. Added to this is the fact that reference to specific bodies is contrary to the concept of a natural law, which requires generality. [2] On the other hand, no one would want to doubt that the completeness with which our coordinate system satisfies the requirements of scientific explanation indicates a regularity without which that satisfaction would be unexplainable.