Reading Notes: March 10th, 2022

“A “constant” magnitude, in any given process, is one which does not change its value. A magnitude to which, in the course of any given process, different values are assigned, is said to be “variable.” The earlier letters a, b, c,… of the alphabet are generally used to denote constant, and the later letters …u, v, w, x, y, z to denote variable magnitudes.” (Lamb, Infinitesimal Calculus, 1) 

“One variable quantity is said to be a “function” of another when, other things remaining the same, if the value of the latter be fixed that of the former becomes determinate. The two quantities thus related are distinguished as the “dependent” and the “independent” variable respectively.” (Lamb, Infinitesimal Calculus, 12) 

“The following investigations proceed on the assumption that the matter with which we deal may be treated as practically continuous and homogeneous in structure; i.e. we assume that the properties of the smallest portions into which we can conceive it to be divided are the same as those of the substance in bulk. The fundamental property of a fluid is that it cannot be in equilibrium in a state of stress such that the mutual action between two adjacent parts is oblique to the common surface. (Lamb, Hydrodynamics, 1) 

“For some good observations on the fallacy of assuming that mathematical symbolism must always be interpretable, see B. Russell, Foundations of Geometry, p. 45-46…” (Taylor, Elements of Metaphysics, 36) 

“[W]e may set up…the following definitions: a priori applies to any pieces of knowledge which, though perhaps elicited by experience, is logically presupposed in experience: subjective applies to any mental state whose immediate cause lies, not in the external world, but within the limits of the subject….Now the only mental states whose immediate cause lie in the external world are sensations. A pure sensation is, of course, an impossible abstraction—we are never wholly passive under the action of an external stimulus.” (Russell, An Essay on the Foundations of Geometry, 2) 

“But modern logic has shown that necessary propositions are always, in one aspect at least, hypothetical. There may be, and usually is, an implication that the connection, of which necessity is predicated, has some existence, but still, necessity always points beyond itself to a ground of necessity, and asserts this ground rather than the actual connection.” (Russell, An Essay on the Foundations of Geometry, 4) 

“I use “experience” here in the widest possible sense, the sense in which the word is used by Bradley.” (Russell, An Essay on the Foundations of Geometry, 4) 

“Conceptions of magnitude, according to Riemann, are possible there only, where we have a general conception, capable of various determinations (Bestimmungsweisen). The various determinations of such a conception together form a manifold, which is continuous or discrete, according as the passage from one determination to another is continuous or discrete. Particular bits of a manifold, or quanta, can be compared by counting when discrete, and by measurement when continuous….We thus reach the general conception of a manifold of several dimensions, of which space and colours are mentioned as special cases….But by the assumption, from the start, that space can be regarded as a quantity, he has been led to state the problem as: What sort of magnitude is space? Rather than: What must space be in order that we may be able to regard it as a magnitude at all? He does not realize, either…that an elaborate Geometry is possible which does not deal with space as a quantity at all. His definition of space as a species of manifold, therefore…leaves obscure the true ground for this nature, which lies in the nature of space as a given system of relations, and is anterior to the possibility of regarding it as a system of magnitudes at all.” (Russell, An Essay on the Foundations of Geometry, 4) 

“We may define a continuous manifold as any continuum of elements, such that a single element is defined by n continuously variable magnitudes. This definition does not really include space, for coordinates in space do not define a point, but its relations to the origin, which is itself arbitrary. It includes, however, the analytical conception of space with which Riemann deals, and may, therefore, be allowed to stand for the moment. Riemann then assumes that the difference—or distance, as it may be loosely called—between any two elements is comparable, as regards magnitude, to the difference between any other two.” (Russell, An Essay on the Foundations of Geometry, 20) 

“In the first place, externality is an essentially relative conception—nothing can be external to itself. To be external to something, is to be an “Other” with some relation to that thing. Hence, when we abstract a form of externality from all material content, and study it in isolation, position will appear, of necessity, as purely relative—it can have no intrinsic quality, for our form consists of pure externality, and externality contains no shadow or trace of an intrinsic quality….The same argument may also be stated as follows: If we abstract the conception of externality, and endeavour to deal with it per se, it is evident that we must obtain an object alike destitute of elements and of totality. For we have abstracted from the diverse matter which filled our form, while any element, or any whole, would retain some of the qualities of a matter. Either an element or a whole, in fact, would have to be a thing not external to itself, and would thus contain something not pure externality. Hence arise infinite divisibility, with the self-contradictory notion of the point, in the search for elements, and unbounded extension, with the contradiction of an infinite regress or a vicious circle, in the search for a completed whole. Thus again, our form contains neither elements nor totality, but only endless relations—the terms of these relations being excluded by our abstraction from the matter which fills our form. In like manner we can deduce the homogeneity of our form. The diversity of content, which was possible only within the form of externality, has been abstracted from, leaving nothing but the bare possibility of diversity, the bare principle of differentiation, itself uniform and undifferentiated. For if diversity presupposes such a form, the form cannot, unless it were contained in a fresh form, be itself diverse or differentiated.” (Russell, An Essay on the Foundations of Geometry, 136-138) 

“Finally, it is interesting to observe that…the relativity of position involves…the most glaring contradictions. For, as we have seen, it is only possible to define points by their relations, i.e., by lines, while lines can only be defined by the points they relate. This involves either an infinite regress or a vicious circle, the penalty of our attempt to give thinghood to a mere complex of relations.” (Russell, The À Priori in Geometry, 112) 

“What our intellect can accept as finally real, we saw, must be indissolubly one with actual experience, and it must be an internally coherent system….When we say that a thing “is” or “has Being,” we seem primarily to mean that it is an object for the knowing consciousness, that it has its place in the system of objects which coherent thought recognizes. When we call the same object “real” or a “reality,” we lay the emphasis rather on the consideration that it is something of which we categorically must take account, whether we like it or not, if some purpose of our own is to get it fulfilment. Thus again the “non-existent” primarily means that which finds no place in the scheme of objects contemplated by consistent scientific thought; the “unreal,” that with which we have not, for any human purpose, to reckon.” (Taylor, Elements of Metaphysics, 50-51) 

“Produce any instance you please, we said to the realist, whom we had not then learned to know by that name, of what you personally regard as reality, and we will undertake to show that it derives its reality for you from the very fact that it is not ultimately separable from the experience of a subject. A thing is real for you, and not merely imaginary, precisely because in some aspect of its character it enters into and affects your own experience. Or, what is the same thing in other words, it is real for you because it affects favourably or otherwise some subjective interest of your own. To be sure, the thing as it enters into your experience, as it affects your own subjective interests, is not the thing as it is in its fullness; it only touches your life through some one of its many sides. And this may lead you to argue that the real thing is the unexperienced “condition” of a modification of your experience. But then we had again to ask what you mean by saying that facts which you do not experience are real as “conditions” of what you do experience. And we saw that the only meaning we could attach to the reality of the “condition” was presence to an experience which transcends your own.” (Taylor, Elements of Metaphysics, 69-70) 

“To this general argument we may add two corollaries or supplementary considerations, which, without introducing anything fresh, may help to make its full force more apparent. (1) The argument, as originally presented, was concerned directly with the that of reality, the mere fact of its existence. But we may also state it, if we please, from the side of the what, the nature possessed by the real. You cannot affirm any doctrine about the real existence of anything without at the same time implying a doctrine about its nature. Even if you say “Reality is unknowable,” you are attributing something beyond mere independence of experience to your reality; you are asserting that what is thus independent possesses the further positive quality of transcending cognition. Now what, in logic, must be your ground for attributing this rather than any other quality to your independent reality? It can only be the fact or supposed fact…that our experiences themselves are all found to be self-contradictory. There is no ground for taking the unknowable of Reality to be true unless you mean by it a character which belongs not to something which stands outside all experience, but to experience itself. The same contention applies to any other predicate which the realist affirms as true of his ultimate reality. (2) Again, we may with effect present our argument in a negative form. Try, we may say, to think of the utterly unreal, and see how you will have to conceive it. Can you think of sheer unreality otherwise than as that of which no mind is ever aware, of which no purpose has ever to take account as a condition of its fulfilment? But to think of it thus is to attribute to it, as its definition, precisely that independence in which the realist finds the mark of ultimate reality. And if “independence” constitutes unreality, presence to and union with experience must be what constitutes reality.” (Taylor, Elements of Metaphysics, 70)