CONTENTS

1   INTRODUCTION

2   TECHNICAL DEVELOPMENTS

3   B-THEORY: EXPOSITION

4   A-THEORY: EXPOSITION AND PARADOX

5   ARGUMENTS FOR A-THEORY

6    A-THEORY: REVISED EXPOSITION

7    TIME AND ASSERTION

8    THE PAST-FUTURE ASYMMETRY

9    CONCLUSIONS

3   B-THEORY: EXPOSITION

After the technical developments and associated digressions, let us return to the main line of the argument. Because B-theory is conceptually simpler, we develop it first. The reduction of the A-series to the B-series was due to Russell (1903), before McTaggart described his paradox. In the notation developed above the reduction can be stated as follows:

Tense-xs (x)  =  Later-xs (x, this utterance)  .

The tense of an event is the B-relation of that event to the utterance in which the tense occurs, treated as an event. All tensed sentences are therefore seen as being token reflexive. Different utterances of the same sentence-type will contain different reference, because they refer to the utterance, that is, the sentence-token, itself. In the case of tensed statements, interpreted in this way, different utterances of the same sentence-type will contain a different reference unless they are simultaneous.

In regimented Fregean language, token reflexivity is allowed, but it is untidy. Quine complains,

“Our ordinary language shows a tiresome bias in its treatment of time. Relations of date are exalted grammatically as relations of position, weight and colour are not. This bias is of itself an inelegance, or breach of theoretical simplicity.”

We shall see that the core of the A-theory counter-argument is that the treatment of time in natural language is no mere “tiresome bias”, but something forced on language by the nature of time. To avoid tensed language is simply to fail to refer adequately to temporal phenomena.

In the ontology of B-theory, the enduring objects which are the basis of our common sense ontology (and that of A-theory) are composite objects, made out of events strung out along the time axis. These special events, corresponding to an enduring object simply being at a particular time, can be called instances of the enduring object. Although for purposes of exposition, the notion of instance has been introduced here by reference to enduring objects, in B-theory the instances are primary, and enduring objects emerge as composites whose constituents are instances. For example, while I may think of myself as a fundamental, unitary object, I am according to B-theory a composite object made up of all the instances of the form:

myself-at-d          for all dates  d  Π [B, D)

 where B and D are the dates of my birth and death respectively. Determining the criteria for an event to be an instance, and for instances to constitute jointly an enduring object is then an important theoretical problem for B-theory. It is however outside the scope of this essay.

B-theory produces a picture of events placed in a one-dimensional continuum like geometric object in space. The A-theory criticism of this picture is that it is too spatial, too static. It lacks what A-theorists would see as the essential kinetic ingredient of change. Not so, argue B-theorists: the B-theory ingredients are exactly what we mean by time and change. There is no need to invoke this additional mystical notion of the “passage of time”, which is just as well because attempts to explicate this notion lead directly to McTaggart’s paradox, which shows that it was incoherent to start with. An enduring object is said to change when later instances have different properties to the earlier instances. Russell defines change as follows.

“Change is the difference, in respect of truth or falsehood, between a proposition concerning an entity and a time T and a proposition concerning the same entity and another time T ¢, provided that the two propositions differ only by the fact that T occurs in the one where T ¢ occurs in the other.”

This, according to B-theorists is all that change could, non-paradoxically, mean.

This said, for B-theory the totality of objective reality is the whole of the time continuum and the events it contains. This timeless viewpoint of the whole of time is the one suitable for the science of the objective, namely physics. In a famous passage, Weyl expressed this thought as follows.

“The objective world simply is, it does not happen. Only to the gaze of my consciousness, crawling upward along the life line of my body, does a section of this world come to life as a fleeting image in space which continuously changes in time.”

Grünbaum argues for the subjectivity of temporal passage based on a total absence of reference to it in physical theory.

“It seems to me of decisive significance that no cognizance is taken of nowness (in the sense associated with becoming) in any of the extant theories of physics. If nowness were a fundamental property of physical events themselves, then it would be very strange indeed that it could go unrecognized in all extant physical theories without detriment to their explanatory success. And I hold with Reichenbach that ‘if there is Becoming [independently of awareness] the physicist must know it.’ ”

(Grünbaum’s italics and insertion into the quotation from Reichenbach.)

Another major motivation for adopting B-theory was the advent of Special Relativity (SR). A-theory is founded on the objectivity of the Now, which in turn is dependent on the objectivity of the simultaneity relation. The Now is one of the equivalence classes defined by simultaneity. At the heart of SR, of the transition from the Galilean to the Lorentzian transformation, is the axiom of the relativity of simultaneity. (Newtonian mechanics is relativistic, in the Galilean sense. What makes SR “more relativistic” is the relativity of simultaneity.) In the absence of a unique simultaneity relation, there can not be a unique Now. The Now, like simultaneity, becomes relativised to a frame of reference. From frame-dependence it is a short (though philosophically dubious) step to subjectivity.

Within SR, as Minkowski showed, to talk about the frame-independent B-relations between events, it becomes necessary to talk jointly of spatial and temporal relations. Space and time become merged in a single space-time continuum. To be sure, time retains some distinction from space because the continuum is a (1,3)-dimensional Minkowski space rather than a plain 4-dimensional Euclidean space. This means that the relations between events divide, in a frame-independent way, into space-like and time-like. Nevertheless Minkowski’s picture of the world is a natural extension of the B-theory view of time as a continuum containing events.

Historically SR was a major influence in propagating B-theory, especially among physicists and philosophers strongly influenced by developments in physics. But logically the arguments of Russell and McTaggart take priority. As Quine says:

“Einstein’s discover and Minkowski’s interpretation of it provided and essential impetus, certainly, to spatiotemporal thinking, which came afterward to dominate philosophical constructions in Whitehead and others. But the idea of paraphrasing tensed sentences in terms of eternal relations of things to times was clear enough before Einstein.”

Conversely it makes sense to divide a defence of A-theory into two steps:

This essay is concerned mostly with the first step, giving only some hints as to how the second step might be undertaken.

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