Richard T. W. Arthur

On thought experiments as a priori science

Abstract Against Nortons claim that all thought experiments can be reduced to explicit arguments, I defend Browns position that certain thought experiments yield a priori knowledge. They do this, I argue, not by allowing us to perceive “Platonic universals” (Brown), even though they may contain non‐propositional components that are epistemically indispensable, but by helping to identify certain tacit presuppositions or “natural interpretations” (Feyerabends term) that lead to a contradiction when the phenomenon is described in terms of them, and by suggesting a new natural interpretation in terms of which the phenomenon can be redescribed free of contradiction.

Leibniz’s Theory of Time

On one level, there is little disagreement concerning the main features of Leibniz’s philosophy of time. No one would dispute, for instance, that Leibniz maintained the following three theses: (i) Time is relational. That is, time is not itself a physical entity, but is rather a relation or ordering of such entities as are not coexistent. (ii) Time is ideal. Being relational, time has no existence apart form the things it relates; it is therefore an ideal entity. What exactly Leibniz meant by this is, as we shall see, a matter of dispute: Russell’s view that it follows from an ontology which denies the reality of relational facts (such as “a is before b”) has recently been forcefully challenged by Ishiguro and others.1 But whatever its exact meaning, the ideality of time is clearly consonant with Leibniz’s belief that continuity is a concept that strictly applies only to ideal entities, and that (iii) Time is a continuous quantity.

Space and Relativity in Newton and Leibniz

In this paper I challenge the usual interpretations of Newtons and Leibnizs views on the nature of space and the relativity of motion. Newtons ‘relative space’ is not a reference frame; and Leibniz did not regard space as defined with respect to actual enduring bodies. Newton did not subscribe to the relativity of intertial motions; whereas Leibniz believed no body to be at rest, and Newtons absolute motion to be a useful fiction. A more accurate rendering of the opposition between them, I argue, leads to a wholly different understanding of Leibnizs theory of space, one which is not susceptible to the objections Newton had raised against Descartes regarding the representation of motion. This in turn suggests a new approach for contemporary theory of space, one which neither hypostatizes space nor tries to reduce it to relations among actual things.

Minkowski’s Proper Time and the Status of the Clock Hypothesis

In this chapter I argue that the concept of proper time must be regarded as one of Minkowski’s enduring contributions to physics. I examine some confusions that still interfere with an appreciation of this, including a conflation of proper time with the co-ordinate time of the inertial frame of a system at rest, and the related mistaken notion that Special Relativity cannot be applied to accelerating systems. This sets the stage for a treatment of the so-called clock hypothesis, according to which the instantaneous rate of a clock depends only on its instantaneous speed. I argue that this does not have the status of an independent hypothesis, but is simply a description of the behaviour of an ideal clock as predicted by (classical, special and general) relativity theory. The question whether this hypothesis holds, moreover, must be distinguished from the question of whether the restorative acceleration of the mechanism within any real system acting as a clock is sufficiently great (relative to the acceleration undergone by the system) that the system will be able to approximate such an ideal clock. The failure of the clock hypothesis would entail the falsity of relativity theory in the form proposed by Einstein, as Weyl had sought to demonstrate with his unified theory of gravity and electromagnetism in 1918. I argue that it is the Strong Equivalence Principle in General Relativity that preserves the chronometric significance that the metric had in Special Relativity, and thereby preserves the relation of inertia to time assumed classically.

Chapter 7: Minkowski Spacetime and the Dimensions of the Present

Abstract In Minkowski spacetime, because of the relativity of simultaneity to the inertial frame chosen, there is no unique world-at-an-instant. Thus the classical view that there is a unique set of events existing now in a three-dimensional space cannot be sustained. The two solutions most often advanced are (i) that the four-dimensional structure of events and processes is alone real, and that becoming present is not an objective part of reality; and (ii) that present existence is not an absolute notion, but is relative to inertial frame; the world-at-an-instant is a three-dimensional, but relative, reality. According to a third view, advanced by Robb, Capek and Stein, (iii) what is present at a given spacetime point is, strictly speaking, constituted by that point alone. I argue here against the first of these views that the four-dimensional universe cannot be said to exist now, already, or indeed at any time at all; so that talk of its existence or reality as if that precludes the existence or reality of the present is a non-sequitur. The second view assumes that in relativistic physics, time lapse is measured by the time co-ordinate function; against this I maintain that it is in fact measured by the proper time, as I argue by reference to the Twin Paradox. The third view, although formally correct, is tarnished by its unrealistic assumption of point-events. This makes it susceptible to paradox, and also sets it at variance with our normal intuitions of the present. I argue that a defensible concept of the present is nonetheless obtainable when account is taken of the non-instantaneity of events, including that of conscious awareness, as (iv) that region of spacetime comprised between the forward light cone of the beginning of a small interval of proper time τ (e.g. that during which conscious experience is laid down) and the backward light cone of the end of that interval. This gives a serviceable notion of what is present to a given event of short duration, as well as saving our intuition of the “reality” or robustness of present events.

Russell’s Conundrum: On the Relation of Leibniz’s Monads to the Continuum

In his influential book on Leibniz (1900), Bertrand Russell often let his new-found anti-Hegelianism come between him and his subject, attacking doctrines held by the likes of Bradley and MacTaggart in the belief that he was attacking one of their main sources. This is particularly true of his chapter on Leibniz’s treatment of the continuum, where after rather wildly accusing Leibniz of courting “the essentially Hegelian view that abstraction is falsification”,1 he proceeds to the charge that “his whole deduction of Monadism from the difficulties of the continuum, seems to bear a close analogy to a dialectical argument” (110). This is intended as a hostile criticism, as Russell makes clear, explaining that in calling the argument “dialectical” he means that Leibniz infers the reality of monads “from premisses admittedly false, and inconsistent with each other”. He charges that Leibniz’s argument, “in obtaining many reals, assumes that these are parts of matter — a premiss which it is compelled to deny in order to show that the reals are not material”.2

Leibniz’s Actual Infinite in Relation to His Analysis of Matter

In this paper I examine the relationship between Leibniz’s thinking on the infinite and his analysis of matter. After contrasting his views on these subjects with those of Georg Cantor, I outline Leibniz’s doctrine of the fictionality of infinite wholes and numbers by reference to his 1674 quadrature of the hyperbola, and defend its consistency against criticisms. In the third section I show how this same conception of the infinite informs Leibniz’s thesis of the actually infinite division of matter. I defend his views on aggregation from Russell’s criticism that they would make plurality a merely mental phenomenon, and expound Leibniz’s argument that body is aggregated from unities that are not themselves parts of matter, although they are presupposed by them. I then argue that these unities of substance make actual the parts of matter, according to Leibniz, by being the foundation of the motions that individuate the actual parts of matter from one instant to another.

Chapter 11 Time Lapse and the Degeneracy of Time: Gödel, Proper Time and Becoming in Relativity Theory

Abstract In Special Relativity time bifurcates into coordinate time and proper time. I argue that confusion between these two concepts is largely responsible for the (fallacious) argument that becoming is incompatible with relativity theory.

Leibniz’s Syncategorematic Actual Infinite

It is well known that Leibniz advocated the actual infinite, but that he did not admit infinite collections or infinite numbers. But his assimilation of this account to the scholastic notion of the syncategorematic infinite (more accurately, the infinite syncategorematically understood) has given rise to controversy. A common interpretation is that in mathematics Leibniz’s syncategorematic infinite is identical with the Aristotelian potential infinite, so that it applies only to ideal entities, and is therefore distinct from the actual infinite that applies to the actual world. Against this, I argue in this paper that Leibniz’s actual infinite, understood syncategorematically, applies to any entities that are actually infinite in multitude, whether numbers, actual parts of matter, or monads. It signifies that there are more of them than can be assigned a number, but that there is no infinite number or collection of them (the categorematic infinite), which notion involves a contradiction. Similarly, to say that a magnitude is actually infinitely small in the syncategorematic sense is to say that no matter how small a magnitude one takes, there is a smaller, but there are no actual infinitesimals. In geometry one may calculate with expressions apparently denoting such entities, on the understanding that they are fictions, standing for variable magnitudes that can be made arbitrarily small, so as to produce demonstrations that there is no error in the resulting expressions.

Leibniz, Organic Matter and Astrobiology

In this chapter I discuss some of the leading ideas in Leibniz’s philosophy of biology with an eye to their relevance to modern astrobiology. Leibniz’s views make for an interesting contrast with the modern synthesis in biology, since although he posited the encoding of genetic information in each individual as a programme for its structure and development, his biological philosophy was developmental rather than evolutionary. But he had interesting and still topical views on (among other things) the compatibility of teleological explanation with mechanism, on what constitutes a living organism, on what a biological individual is, and on organic matter being more fundamental than inorganic. In what follows we will see that many of his insights and distinctions have been rediscovered by contemporary biologists working in the modern synthesis of evolution, natural selection, mutation, molecular biology and speciation, anatomy, physiology and homeostasis, developmental biology and reproduction, ecology and geochemistry.