Abstract
None of the four options for describing the moment of change has revealed itself as satisfactory in all cases:
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(a)
Application of the either/or-option suggests itself for C-changes (as one naturally wants to keep LNC and LEM). It also suggests itself for the beginnings and ends of processes, because if only one state may obtain at the limiting instant there are, in this case, good reasons as to which state it is. However, such reasons could not be found for s-changes between comparative properties, and so the impression that applying the either/oroption to them would involve an arbitrary decision prevails. Discontinuous changes remain unexplained. (Cf. ch.II,1)
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(b)
Application of the either-way-option has proved incomprehensible for C-changes. For s-changes between rest and motion the adherents of this option have to rely on concepts of motion and rest at an instant which are counter-intuitively weak. (Cf.ch.II,2)
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(c)
Application of the both-states-option necessitates the postulation of objective contradiction in nature and, thus, the denial of the LNC. (Cf.ch.II,3)
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(d)
If one permits states at instants (for which there is ample reason), then the use of the neither/nor-option for C-changes inescapably violates the LEM. (Cf. ch.II,4)
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Notes
To my knowledge, the term is from Kretzmann who uses it in ‘Incipit/Desinit’. Sorabji uses this name as well when criticizing Kretzmann (‘Time...’, ch. 26).
The relation of rest and motion in the context of the NIA can, thus, also be described in terms familiar from the chapter on Plato: rest and motion together are incompatible but dispensable for instants; in relation to periods, however, they are both incompatible and indispensable. Cf.ch. I,1.3. §3f.
Cf. Kretzmann: Incipit/Desinit, p.114.
The fact that Kretzmann not only discovers the NIA as implicitly contained in the medieval approach (which is plausible), but also in Aristotle (which is not plausible at all), does not matter in this chapter, where it is dealt with purely as a contemporary systematic solution.
In Bostock: Plato on Change...
Although, as was shown in ch.I,1.2.1., he does not attribute the use of the neither/noroption to Plato.
Bostock: Plato on Change..., p.238.
Ibid.
If it sounds implausible that a when changing from being at rest to not being at rest, for example, is already not at rest, this demonstrates some plausibility of Plato’s premiss.
So, for the time being I am following a proposal by Sorabji (Time..., p.404). In part III, however, the position of an object is to be defined as an extended part of space.
Perhaps it was conscious Anti-Hegelianism which provoked Russell to state his intuition so very clearly here. Cf.ch.II,3.
The reason for this is that it becomes very easy to obtain a parallel definition of motion. If only rest at an instant were to be defined, it would not be necessary to require the two instants to be different. So in the definition of rest this requirement is, strictly speaking, redundant.
It is assumed here that there are no gaps in the existence of a physical object, and that at any time of its existence it is somewhere. If this is assumed it is true that if there are any positions of a at all both before and after t, then there are positions of a both arbitrarily shortly before t and arbitrarily shortly after t.
Cf.ch.II,1.1.2.
So Sorabji is absolutely right to remark (Time..., p.416): “Bertrand Russell gives a definition of motion at a moment in §446 of the Principles of Mathematics and denies [...] that the instant of motion can be an instant of rest. He does not, however, make it so clear whether or not it can be an instant of motion.”
This is clearly noticed by Priest (‘Inconsistencies..’ p.339): “Russell is actually slightly inconsistent, since after giving his definition he permits that something may be momentarily at rest if its position derivative with respect to time is zero. This is quite compatible with its being in motion in the official sense.”
Cf. ch.II,1.1.
Kretzmann: Incipit/Desinit, p.114.
Kretzmann: Incipit/Desinit, p.115:“The instant t c is the extrinsic limit of the state of rest as well as the extrinsic limit of the state of motion. But how can it be both at once? [...] Considered as belonging to the interval during which the ball is in its initial state of rest, t c is the t NBZ [= last instant of non-being] of the interval during which the ball is in motion; considered as belonging to the interval of motion, t c is the tNBa [= first instant of non-being] of the interval of rest.” Fortunately, the wording is unambiguous, but nevertheless this much has to be said against it: (1) It is not quite clear what ‘belonging to’ is supposed to mean here. t cannot be an element of an interval all instants of which are instants of rest of a, and at the same time be the first instant at which a is not at rest. Therefore ‘belonging to’ cannot refer to the relation ‘is an element of’, as one would expect at first sight;but it means something like ‘being an extrinsic boundary of the interval’. (2) Of course, there is no “first/last instant of non-being” of an interval but, at most, of a state obtaining during an interval. An open interval (and this is what Kretzmann is concerned with here) simply has no first or last instant. Kretzmann originally introduces his index labels ‘NBa’ and ‘NBz’ with respect to medieval ‘res permanentes’ and ‘res successivae’, which he regards as states. That is where they make sense.
As for the classification of the moment of change, Kretzmann is not quite as clear here as he is in his description of it. On the one hand, he seems to deny an instantaneous event for the s-change between rest and motion.(Kretzmann: Incipit/Desinit, p.115: “There is no change of rest to motion or of motion to rest. [...] The instant is simply the the extrinsic limit of the state of rest as well as the extrinsic limit of the state of motion.”) On the other hand, he states that the verbs ‘leave’, ‘arrive’ and ‘begin’ may refer to what occurs at an instant. In connection with this, his opinion is that the last instant at which a is in position A is the very instant for which it is true to say “[a] is leaving from [position] A”. (Ibid., p.116.)
Wolfgang Büchner has pointed out to me that the second conjunct is strictly speaking redundant. With his permission I will keep this redundant expression because in this way the intuition behind the definition is easier to recognize.
According to these definitions turning-point instants are not neutral. If they are to be, one has to add the requirement that is an instant of motion of a only if it falls within a period of uninterrupted motion of a and if its M-value at t differs from zero.
Cf. Introduction2.
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Strobach, N. (1998). The ‘Neutral Instant Analysis’ (NIA). In: The Moment of Change. The New Synthese Historical Library, vol 45. Springer, Dordrecht. https://doi.org/10.1007/978-94-015-9127-0_10
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