In the scientific definition of force given above force appears as the result of a multiplication of two other magnitudes. Now as is well known, it is essential for the operation of multiplication that of the two factors forming the product at least one should exhibit the properties of a pure number. For two pure numbers may be multiplied together - e.g. 2 and 4 - and a number of concrete things can be multiplied by a pure number - e. g. 3 apples and the number 4 - but no sense can be attached to the multiplication of 3 apples by 4 apples, let alone by 4 pears! The result of multiplication is therefore always either itself a pure number, when both factors have this property; or when one of the two factors is of the nature of a concrete object, the result is of the same quality as the latter. An apple will always remain an apple after multiplication, and what distinguishes the final product (apples) from the original factor (apples) is only a pure number.
If we take seriously what this simple consideration tells us of the nature of multiplication, and if we do not allow ourselves to deviate from it for whatever purpose we make use of this algebraic operation, then the various concepts we connect with the basic measurements in physics undergo a considerable change of meaning.
Let us test, in this respect, the well-known formula which, in the conceptual language of physics, connects 'distance' (s), 'time' (t), and 'velocity' (c). It is written
c = s / t, or s = ct.
In this formula, s has most definitely the meaning of a 'thing', for it represents measured spatial distance. Of the two factors on the other side of the second equation, one must needs have the same quality as s: this is c. Thus for the other factor, t, there remains the property of a pure number. We are, therefore, under an illusion if we assume the factor c to represent anything of what velocity implies in outer cosmic reality. The truth is that c represents a spatial distance just as s does, with the difference only that it is a certain unit-distance. Just as little does real time enter into this formula - nor does it into any other formula of mathematical physics. 'Time', in physics, is always a pure number without any cosmic quality. Indeed, how could it be otherwise for a purely kinematic world-observation?
We now submit the formula F=ma to the same scrutiny. If we attach to the factor a on the right side of the equation a definite quality, namely an observable acceleration, the other factor in the product is permitted to have only the properties of a pure number; F, therefore, can be only of the same nature as a and must itself be an acceleration. Were it otherwise, then the equation F=ma could certainly not serve as a logical link between the Velocity and Force parallelograms.
Our present investigation has done no more than grant us an insight into the process of thought whereby the consciousness limited to a purely kinematic experience has deprived the concept of force of any real content. Let us look at the equation F=ma as a means of splitting of the magnitude F into two components m and a. The equation then tells us that F is reduced to the nature of pure acceleration, for that which resides in the force as a factor not observable by kinematic vision has been split away from it as the factor m. For this factor, however, as we have seen, nothing remains over but the property of a pure number.
Let us note here that the first thinker to concern himself with a comprehensive world-picture in which the non-existence of a real concept of force is taken in earnest-namely, Albert Einstein - was also the first to consider mass as a form of energy and even to predict correctly, as was proved later, the amount of energy represented by the unit of mass, thereby encouraging decisively the new branch of experimental research which has led to the freeing of the so-called atomic energy. Is it then possible that pure numbers can effect what took place above and within Nagasaki, Hiroshima, etc.? Here we are standing once again before one of the paradoxes of modern science which we have found to play so considerable a part in its development.
To find an interpretation of the formula F=ma, which is free from illusion, we must turn our attention first of all to the concepts 'force' and 'mass' themselves. The fact that men have these two words in their languages shows that the concepts expressed by them must be based on some experience that has been man's long before he was capable of any scientific reflexion. Let us ask what kind of experience this is and by what part of his being he gathers it.
The answer is, as simple self-observation will show, that we know of the existence of force through the fact that we ourselves must exert it in order to move our own body. Thus it is the resistance of our body against any alteration of its state of motion, as a result of its being composed of inert matter, which gives us the experience of force both as a possession of our own and as a property of the outer world. All other references to force, in places where it cannot be immediately experienced, arise by way of analogy based on the similarity of the content of our observation to that which springs from the exertion of force in our own bodies.
As we see, in this experience of force that of mass is at once implied. Still, we can strengthen the latter by experimenting with some outer physical object. Take a fairly heavy object in your hand, stretch out your arm lightly and move it slowly up and down, watching intently the sensation this operation rouses in you.2 Evidently the experience of mass outside ourselves, as with that of our own body, comes to us through the experience of the force which we ourselves must exert in order to overcome some resisting force occasioned by the mass. Already this simple observation - as such made by means of the sense of movement and therefore outside the frontiers of the onlooker-consciousness - tells us that mass is nothing but a particular manifestation of force.