Priest (11.5) In Contradiction, ‘The LCC and Contradiction’, summary

by Corry Shores
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[The following is summary. My own comments are in brackets, but please consult the original text, as I am not a logician. All boldface and underlining are my own.]

Graham Priest

In Contradiction:
A Study of the Transconsistent

Part III. Applications

Ch.11. The Metaphysics of Change I:
The Instant of Change

11.5 The LCC and Contradiction

Brief Summary:

By applying Liebniz’ Law of Continuity in states of change, we can say that during such transitions something is both in a state and not in that state.

Summary

[Now recall from section 11.2 the four types of changes we discussed.

Before a time t₀, a system s is in a state s₀, described by α. After t₀ it is in a state s₁, described by α. What state is it in at t₀? A priori, there are four possible answers:
(A) s is in s₀ and s₀ only.
(B) s is in s₁ and s₁ only.
(Γ) s is in neither s₀ nor s₁.
(Δ) s is in both s₀ and s₁.
(160)

The Δ sort of change involves something being both in a state and in its succeeding state at the same time.] Priest will now apply Leibniz’ Law of Continuity (LCC, for Leibniz’ Continuity Condition) to Δ type changes. [If something changes continuously from one state to another, there had to be a moment when both states held together.]

So much for the LCC itself. Let us now return to the question of the existence of type Δ changes, and apply the LCC. For the LCC implies that any change from a continuous state of p to a continuous state of ¬p is a type Δ change. More generally, suppose that φ and ψ are any distinct literals (propositional parameters or their negations). Suppose that prior to time t system s is in state s₀: φ is true. Posterior to time t, s is in state s₁: ψ is true. Since s₀ occurs arbitrarily close to t (and continuously), it occurs at t by the LCC. But s₁ occurs arbitrarily close to t (and continuously). Hence it too occurs at t. Thus, at t there is a nexus state at which both φ and ψ are realised. In particular, if φ is p and ψ is ¬p, p ∧ ¬p is realised at t. The LCC therefore implies that contradictions are realised at the nodal points of certain sorts of change.
(Priest 169)

Priest will now formulate this in tense logic semantics [see especially section 11.3. Recall that W is the set of moments. Propositions describe the states at certain moments. We can add the following modifiers to designate temporal locations.

P     “It has at some time been the case that …”
F     “It will at some time be the case that …”
H     “It has always been the case that …”
G     “It will always be the case that …”
(from Galton’s Stanford Encyclopedia article on tense logic.)

]

We can reproduce this argument in the tense logical semantics. Suppose that W is an appropriate stretch of time and that | g is the nodal point of a change from φ to ψ; i.e., Hφ ∧ Gψ holds at g. Then, assuming only that g is suitably distant from the ends of a < chain, it follows that FF(q ∨ ¬q) and PP(q ∨ ¬q) hold at g, whence, by (1) of the previous section [{PP(q ∨ ¬q), Hp} ⊨ p ], φ ∧ ψ holds at g. The dialetheia produced at a type Δ change need not be instantaneous (for all I have said so far, though this is a plausible additional constraint). For example, the interpretation with real time where v_x(p) is {1} if x < 0, {1, 0} if 0 ≤ x ≤ 1, and {0} if x > 1, is quite compatible with the LCC. Still, there must be at least an instantaneous dialetheia.
(Priest 169-170)

[So we think of moment g that is a transitional place between a prior period when φ held and following period when ψ will hold. This means that there is some specific prior moment or moments when (q ∨ ¬q) and some specific future moment when this holds as well too. (Again I do not know why it is important for it to be this particular formulation). But this means that whatever held in the prior period and whatever held in the forthcoming must both be held at g, in accordance with Priest’s formulation of LCC. Thus at g both φ and ψ hold, making it a type Δ change. Regarding the idea that the change need not be instantaneous, he seems to be saying that we can think of a range of moments when both φ and ψ together hold, but this is still compatible with LCC, since what holds just prior to that period also holds within it.]

Priest will now show that LCC can be applied not just to discrete changes like the above but also to continuous ones. He has us consider a body moving according to a particular equation. In it, x is the position and t is the time: x = kt (k≠0). So this formula will have a value at a particular time. It will have a different value before and after that time. But by LCC, that means it both will and will not have these different values.

It is not only for discrete changes that the LCC can be applied to show that contradictions arise. The LCC entails that contradictions arise in continuous change too. For example, consider a body that moves in accordance with the equation x = kt (k ≠ 0), where x is its position and t is the time, both with respect to some suitable coordinate system. Consider a point t₀. At t₀, x = kt₀. But for all points after (and before) t₀, x ≠ kt₀. Hence, by the LCC, at t₀, x ≠ kt₀. Thus at t₀, x = kt₀ and x ≠ kt₀. And since t₀ was arbitrary, we see that motion produces a continuous state of contradiction. What this might possibly mean I will return to in a moment; we can at least see it as vindicating dialecticians, such as Hegel, who claimed that change would be impossible without contradiction. As he put it, [the following is quotation]
. . . contradiction is the root of all movement and vitality; and it is only in so far as something contains a contradiction within it that it moves, has an urge and activity.
(Priest 170, quoting Hegel [1982] p.439 of the English translation)

People sometimes complain that they cannot conceive how it is for something to be in a state of true self-contradiction. But as we can see, it happens in moments of transition, which we commonly experience.

The thesis that contradictions arise at the nodal points of certain transitions can also be used to free the mind of a certain mental cramp that often arises when people consider dialetheism. A commonly heard complaint is as follows (said with an air of puzzlement): ‘I just cannot see what it would be like for a contradiction to be true, what it would be like, for example, for something to be a cup and not a cup, or for a person to be in a room and not in a room.’ The answer to this (objection?) should now be obvious: something is a cup and not a cup the instant it breaks into pieces. Someone is in and out of the room the instant they leave. Contradictions occur at the nodal points of certain transitions and, as such, are perfectly familiar.
(170)

Regarding such moments of change when contrary states coincide, we can even think of it as a state in itself, a state of change. [In this respect is might be similar to Leibniz’ status transitus.] However, this might lead to infinite regress, for we then need to account for how we got to the state of change. Priest says there is not this problem, because in the nexus state getting into the state of change, the changed state both holds and does not hold [see footnote 17 below]:

We have seen that a certain kind of change from a holding to β holding, produces a nexus state where α ∧ β holds. We may, however, go a step further. We may take the nexus state produced to be the state of change itself. The state described by α ∧ ¬α just is the state described by a changing into the state described by ¬α. Thus, there is such a thing as a state of change, and it does take time, if only an instant. Notice how this relates to the discussion of the LCC in section 11.4. Not only is there a state of change that takes time, but it commences while the prior state obtains and terminates only after the posterior state has begun.¹⁷
[Foootnote 17: If we suppose there to be states of change, does this not start an infinite regress? For what of the change between, e.g. the prior state, described by a, and the state of change, described by α ∧ ¬α? | There is no infinite regress. The nexus state between these two states is described by a α ∧ (α ∧ ¬α), i.e. α ∧ ¬α, which is the original nexus state. Thus, to be changing into a state of change is already to be in that state of change, as one might expect.]
(170-171)

Motion is in a continual state of contradiction, because each instant it is both entering and leaving its location.

The notion that a contradictory state is a state of change also starts to make sense of the fact that motion is a continuous state of contradiction. For the contradictory state of the body at t₀ in the above example, x = kt₀ ∧ x ≠ kt₀, is then indicative of the fact that the body is not only occupying the spot kt₀, but, since its occupation is instantaneous, is at the same time both entering and leaving the spot. All this suggests that the thesis that certain kinds of contradictory state are states of change should be investigated further. To this I turn in the next chapter.
(171)

 

 

Most citations from:

Priest, Graham. In Contradiction: A Study of the Transconsistent. Oxford/New York: Clarendon/Oxford University, 2006 [first published 1987].

Or otherwise as noted from:

Hegel, G. W. F. (1812) Wissenschaft der Logik, published in English translation by A. V. Miller as Hegel’s Science of Logic, Allen & Unwin, 1969.

 

 

 

Priest (11.4) In Contradiction, ‘The Leibniz Continuity Condition’, summary

 

by Corry Shores
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Graham Priest

In Contradiction:
A Study of the Transconsistent

Part III. Applications

Ch.11. The Metaphysics of Change I:
The Instant of Change

11.4 The Leibniz Continuity Condition

Brief Summary:

One application of the dialetheic logic of change is Leibniz’ law of continuity. It says that what holds up to a limit holds at the limit too. Priest formulates this to mean that what holds at an intermediary moment holds as well at the surrounding moments it intermediates.

Summary

Priest previously formulated his dialectical conception of true contradiction in change in a formal semantical system using tense logic, which allows us to assign the truth value for instants of transition when it is both the case and not the case that something is in a particular state. The function that we use to assign truth values in that system was called v.  He ended that section by saying that “We will now look at a more sophisticated and important example of a condition that might naturally be placed on v.” (165)

He says now that this example incorporates a continuity principle into the semantical system. He begins with the historical background of this notion, going back to Leibniz’ Law of Continuity. [see excellent analysis here by Katz and Sherry. The Leibniz text Priest looks at is also examined here.] Quoting Leibniz

When the difference between two instances in a given series or that which is presupposed can be diminished until it becomes smaller than any given quantity whatever, the corresponding difference in what is sought or in their results must of necessity also be diminished or become less than any given quantity whatever. Or to put it more commonly, when two instances or data approach each other continuously, so that one at last passes over into the other, it is necessary for their consequences or results (or the unknown) to do so also.
[Priest 165, quoting Leibniz, (1687) ‘Letter of Mr. Leibniz on a General Principle Useful in Explaining the Laws of Nature through a Consideration of the Divine Wisdom; to Serve as a Reply to the Response of the Rev. Father Malebranche’. Published in English translation in Leibniz’ Philosophical Papers and Letters, ed. L. E. Loemker, Reidel, 1969, 351–4.]

[[Priest then gives a mathematical formulation for one interpretation of the principle. He seems to be working in a literal way with the first of Leibniz’ two explanations. If two values are brought infinitesimally close, then their difference will be 0 and they will be so close with so little difference between their values that they can be thought to be equal. In terms of the geometrical interpretation, this would be like two points brought so close together that no other points stand between them. In a sense, both points are inseparable and form a unit, because there is nothing extending between them to separate them. However, this unit can be thought of as a binary value, because there are still two points, two values. In terms of dialetheic logic, this allows for in the same instant (one infinitesimal interval) something to be in two states, or in the case of motion, two locations, all at once. In Russell’s at-at theory of motion, both moving objects and objects at rest are only ever at one location and never more. However, with the idea of the principle of continuity and the infinitesimal, we can think of there being moments when moving objects are between immediately neighboring locations and thus having one side facing the former and another side facing the latter.]] In the least Leibniz would be saying that:

(Priest 165)

But he is saying more as well. As his examples show, he intends it to apply to “limiting processes—not just arithmetic, but geometric, physical, temporal, and so on” (165).

In virtue of this, we might state the principle thus: given any limiting process, whatever holds up to the limit holds at the limit; or, as L’Huilier, who, like most eighteenth-century mathematicians, endorsed the principle, put it: if a variable quantity at all stages enjoys a certain property, its limit will enjoy the same property.
(166)

The continuity principle could not apply to all possible cases, because then we would say that “every real number is rational (since it is the limit of a sequence of rationals).” Also “it is not the case that every parabola is a closed and bounded figure, even though every ellipse is closed and bounded.” [165, see also Katz and Sherry, Leibniz, Leibniz again, and an explanation here for that example.] Priest assumes that Leibniz thought there was some limitation to the application of the principle, but it is not clear what that is. 

Priest will now focus on changes in physical states of affairs. If we think of these changes as events, we could formulate it as: “any event that is occurring at a continuous set of times is occurring at any limit of those times”. If we do away with the “limit jargon” we could formulate it thus:

anything going on arbitrarily close to a certain time is going on at that time too. Let us call this, in honour of Leibniz, the Leibniz Continuity Condition, LCC for short.
(166)

Leibniz’ justification for the principle is this. If at the limit the principle did not hold, then for some reason, that would mean the behavior at the limit would be capricious. [This seems to assume that it would not hold because of a law of non-continuity which would say that whatever holds up to the limit does not hold at the limit, in which case it would be law governed.] But since God, the designer of the world, is not capricious, that means the world cannot behave in this capricious way. Later Priest will reexplain this justification in a non-anthropomorphic way.

Priest now looks for a way to establish the LCC. We cannot verify it empirically, since no clock is precise enough [to measure the infinitely brief]. In mathematics, the value of a function at some argument is logically independent of the value at others, so capricious behavior is not mathematically impossible. But in nature, neighboring moments are not independent.

How might one establish the LCC? Clearly there is no possibility of verifying the principle by experiment. No measuring instrument, particularly no clock, is accurate to more than a finite number of decimal places. There is therefore no way in which we might hope to observe the situation at a certain time to the exclusion of states at arbitrarily close times. Neither is there any question of proving the principle by pure mathematics. There is nothing mathematically impossible in such capricious behaviour. This is because the mathematical representation of states of affairs is quite atomistic. The value at some argument of a mathematical function in extension is logically independent of its value at all others. But it is precisely here that nature may plausibly be thought to differ from such a representation. For succeeding states of affairs in nature are not atomistic: there are connections. This would be denied by a Humean. For her, if the principle held it could only be by a global accident. I therefore see no possibility of convincing a Humean of the plausibility of the principle. But of course, for a Humean, every sequence of events is a global accident; hence there is no possibility of convincing her of anything. Let us therefore leave this scepticism aside. For the non-sceptic there are nexuses that serve to make the state of affairs at a certain time dependent on those at other times.
(167)

Priest will now discuss why a change that violates LCC would be unintelligible. Unless we say that there is a moment when a changing thing is in its former and following states at the same time, then we would have to say the change happened during no time at all. If we say that something is in one state up to a particular time, with not transitional state, then when does the transition happen? We cannot say the change happened before the limit, because it has not yet happened. We cannot say it happened at the limit, because then it already happened. But since there is no time in between or no transitional status where both states were together at the same time, then the change must have happened without happening in time.

I suspect that a change which violates the LCC is capricious in the sense that it is incompatible with the existence of some of these nexuses. How does this work exactly? There is, I think, a good deal to be said about this, and the following is at least part of it. A change that violated the LCC would be unintelligible because of the following sorts of considerations. Let us suppose that a state of affairs, s, holds before, and all the way up to, a limit time, t, but fails at t. Then, clearly, a change has occurred. But when did this change occur? It cannot occur before t, since at any time before t there are later times at which s held; but it cannot occur at t (or at any subsequent time), because at this time the change is all over: s is already terminated! We can reason similarly if the state holds after, and at all times down to, a prior limit time, t, but not at t. When did the change occur? It cannot happen after t—that is too late: at any time after t there are prior times at which s already holds; but it cannot happen at t (or at any prior time), because at that time the change has not yet started: the old state is still in place. It therefore seems, in either case, that something, namely a change, has occurred, but that it took place at no time. But this is very strange. We may countenance things that happen very quickly, but if something happens it must take some time, if only an instant. (For just this reason, theories of action at a distance, which require something to happen in no time, namely the transmission of an effect, have always been felt philosophically puzzling.) A possible response to this train of | thought is simply to deny that there is any such thing as change itself. A change occurs when one state is replaced by another, and that’s that. This response just endorses the cinematic account of change, which we met in section 11.2. As we noted there, it, too, is highly counter-intuitive.
(167-168)

Priest will now incorporate LCC into the semantics of tense logic. [On the real number line, there are intervals with limits that lie outside the interval. Recall Russell’s description of intervals whose limits lie outside them: “a limit may or may not belong to the class u of which it is a limit, but it always belongs to some series in which u is contained, and if it is a term of u, it is still a limit of the class consisting of all terms of u except itself.” (p. 279 of Bertrand Russell. Principles of Mathematics. London/New York: Routledge, 2010 [1st published 1903].)]

we notice that in the real line (which is the paradigm representation of time), with the usual ordering and topology, the (open) continuous intervals are just sets of the form {x | r < x < s} for real numbers r and s; and r and s are the only limits of the interval that are not already in it.
(168)

[To understand the following formulation, first note that a ‘parameter’ is “A variable to which arbitrary values may be assigned for a specific purpose” (p. 159 of Greenstein Dictionary of Logical Terms and Symbols). A propositional parameter would be a variable which would take one proposition or another. “basic propositions will be represented in PL [propositional logic] by simple capitals letters (called “ sentence letters,” “propositional constants,” or “propositional parameters”): A, B, C, …, P, Q, R, …” (p.31-32 of Smith Logic: The Laws of Truth). Priest emphasizes that his formulation works with propositional parameters, because he means that the propositions involved do not have a tense, so they refer only to their own moment and not to one coming before or after. We can say that ‘I am alive’ applies to all moments we are alive and as well to the transitional moment into our death, during which we as well are not alive. But ‘there is a later moment of my life’ does not apply at that final transition. The formulation then seems to say that if a proposition holds (or does not hold) at some moment, then it holds (or does not hold) during its immediate predecessor and successor as well. In this formulation, it seems to imply that the moments are in immediate succession, and the brackets seem to mean that the formulation can be read as having all trues or all falses.]

For every propositional parameter, p, and every x, y ∈ W, if 1[0] ∈ v_z(p) for every z such that x < z < y, then 1[0] ∈ v_x(p) and ∈ v_y(p).
(168)

under no circumstances should it be extended to tensed formulas. For suppose the LCC did apply to tensed formulas, and consider the moments of someone’s life. Being alive is certainly a continuous state of affairs, and so we can apply the LCC to conclude that this set contains all its limit points. In particular, it has a last moment, assuming, of course, that it does not go on for ever. Call this z. At any point prior to z, ‘There will be a (later) time of life’ is true. If we could apply the LCC to tensed sentences we could apply it to this one to conclude that it is true at z, which, manifestly, it is not. The point, of course, is that, though ‘There will be a later time of life’ may be true at time t, it does not describe a state of affairs that holds at time t in the pertinent sense.
(168)

[To understand the next formulation, we should establish a few things. The first is that if the is no tense modifier, we can perhaps assume the proposition refers to this current moment or the moment it is spoken.

(6') T'(It is sunny and warm today)
[...]
a certain redundancy is present in (6'); not only does the statement operator T signify present tense but the interior statement in is the present tense as well. Thus the T operator is superfluous and can be dispensed with.
(McArthur 3)

[…]

'It did rain in Boston but isn't now' being Pp & ~p
(McArthur 4)

The next thing to note is what happens when we combine two P tense markers.

Below are statements in two common perfect tenses (past and future) with symbolizations.
(8) It had rained in Boston = PPp
(9) It will have rained in Boston = FPp.
(McArthur 4)

Note that the past perfect implies two temporal positions in the past, with one preceding the other.

Past Perfect
FORM
[had + past participle]
Examples: You had studied English before you moved to New York.

(Text and image above from englishpage.com)

So in the following proposition, it seems we are referring to q or not-q holding a moment before another past moment, p always having held in the past, and p holding now in the present.]

it is clear that the LCC will validate the following inference:
{PP(q ∨ ¬q), Hp} ⊨ p
(169)

[This seems to mean that if q or not-q hold a moment before a past moment, and if p always held in the past, then right now p holds. Putting Priest aside for a moment, if we say that p holds all instants in the past (understood as infinitesimal intervals of time), including the infinitesimal interval leading into the present, then p holds for the present interval too. But as we saw before, Priest uses three moments to define LCC:

For every propositional parameter, p, and every x, y ∈ W, if 1[0] ∈ v_z(p) for every z such that x < z < y, then 1[0] ∈ v_x(p) and ∈ v_y(p).
(168)

There he seemed to be saying that whatever proposition holds for an (immediately) intermediary moment holds for those (immediately) surrounding moments neighboring it. So perhaps that is why it is not enough for Priest to just say Hp ⊨ p. So let’s think of the present moment as t₃. We know that p holds for all prior moments t₂, t₁, etc. In our evaluation so far, we do not yet know if p holds for present moment t₃. But we do know that at least two moments in the past q or not-q held, while at the same time p held. And this this one precedes another past moment that is the intermediary moment, t₂, during which p held. Since p holds in an intermediary moment t₂, standing between two other moments, then what was true of t₂ must hold for them as well, and thus p must hold for the present moment. In the following explanation, Priest seems to have us suppose that PP(q ∨ ¬q) and p hold at (present) moment g. To clarify what the PP means, let’s look at this formulation from Galton’s Stanford Encyclopedia article on tense logic.

Fp→FFp
“If it will be the case that p, it will be — in between — that it will be”
(Galton)

Here the FFp refers to a moment nearer to the present than Fp.]

For suppose the premises hold at g. Then there is some x < g such that p and P(q ∨ ¬q) hold at x. Hence there is some y < x. By the LCC, p holds at y and g. The inference holds if we replace p by its negation. The future-symmetric analogues of these inferences also hold. However, all of these inferences may break down if the LCC does not hold.
(169)

[Let me offer an experimental explanation (meaning likely wrong). Let’s assume as he says that PP(q ∨ ¬q) holds at moment g. This means literally ‘It has  at some time been the case that it has at some time been the case that q or not-q.’ So ‘It has at some time been the case [intermediary moment x]’ ‘that it has at some time been the case [most prior moment y]’ ‘that q or not-q’. This means that at one moment prior, P(q ∨ ¬q), because at that intermediary moment there was a prior where q or not-q held. I am assuming then that at most prior moment (q ∨ ¬q) held, but we cannot say at middle moment x what held other than p, which always had held. So we seem to have this sequence:

moment y, at which p and (q ∨ ¬q) held;
moment x, at which p and P(q ∨ ¬q) held; and
moment g, at which Hp and PP(q ∨ ¬q) holds.

Now recall Priest’s formulation for LCC:

For every propositional parameter, p, and every x, y ∈ W, if 1[0] ∈ v_z(p) for every z such that x < z < y, then 1[0] ∈ v_x(p) and ∈ v_y(p).
(168)

So since p holds in this case in the middle moment x, it must also hold at neighboring moments y and g. But without this condition, we might not have any basis to conclude p. The role of (q ∨ ¬q) is not clear to me.]

[Now Priest will say that it is not interesting if LCC holds for an order of moments that is linear and discrete, because it would mean that what is said of one moment holds for all (through transitivity). He says it does become interesting when the sequence is continuous or dense. Perhaps what he means is that the density makes us lose the total transitivity. If between x and z there is y where p holds, then it holds for x and z. But between x and y is another moment d, which maybe cancels the application to z (since it is now two steps away); and, between x and d there is e, and so on. Perhaps what he is saying is that the transitivity breaks down or is called into question when we cannot assign an immediate neighbor, and thus we cannot say that a proposition holds for all possible intermediary values.]

In general, the effects of the LCC are not very interesting if the order is both linear and discrete. For, given any three consecutive points x, y, and z, the LCC ensures that whatever propositional parameters or their negations hold at y hold at x and z. If this does not render the evaluations at all indices identical, it does so near enough to make the situation rather uninteresting. The LCC assumes real interest mainly when the ordering is continuous, or at least dense.
(169)

 

Most citations from:
Priest, Graham. In Contradiction: A Study of the Transconsistent. Oxford/New York: Oxford, 2006 [first published 1987]

Or otherwise as indicated from:

Greenstein, Carol Horn. Dictionary of Logical Terms and Symbols. New York / Cincinnati / Toronto / London / Melbourne: Van Nostrand Reinhold, 1978.

Smith, Nicholas J.J. Logic: The Laws of Truth. Princeton / Oxford: Princeton University Press, 2012.
 

Katz and Sherry’s [Pt.4.6] “Leibniz’s Infinitesimals: Their Fictionality, Their Modern Implementations, And Their Foes From Berkeley To Russell And Beyond,” 4.6 ‘Assignable Versus Unassignable’, summary

summary by Corry Shores
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[The following is summary. My own comments and citations are placed in double brackets. All boldface and underlying are mine.]

 

Mikhail G. Katz  and David Sherry

“Leibniz’s Infinitesimals: Their Fictionality, Their Modern Implementations, And Their Foes From Berkeley To Russell And Beyond”

4. Cum Prodiisset

 

4.6 Assignable Versus Unassignable

Brief Summary:

In Cum Prodiisset Leibniz works with assignable (finite) and unassignable (infinitesimal / infinite) values. On account of his law of continuity, we can use the same mathematical operations even when we substitute one for the others.

Summary

 

Leibniz’ infinitesimal values dx and dy are ‘unassignable,’ and Leibniz writes that

the unassignables dx and dy may be substituted for them by a method of supposition even in the case when they are evanescent (Child [Leibniz] 1920, p. 153).
[KS 583]

KS will examine Leibniz’ multiplicative example:

ay = xv

[We will look briefly at Leibniz’ procedures to follow Katz and Sherry. Leibniz writes: “Let ay = xv, then a(d)y = x(d)v + v(d)x.” Leibniz then adds to their corresponding values dy, dx, and dv:

Proof. ay + ady = (x + dx) (v+ dv) “

then expanding

= xv + xdv + vdx +dxdv.

And so together:

ay + ady = (x + dx) (v+ dv) = xv + xdv + vdx +dxdv

or

ay + ady = xv + xdv + vdx +dxdv

(note: there is a transcription error in the translation. In the following, the original has ‘xv’ but the translation has instead ‘xy’.)

We then remove ay and xy (perhaps because they are repeated now in their infinitesimal renditions)

ady = xdv + vdx + dxdv

(then dividing both sides by dx)

This is the formulation KS move next to.]

“Simplifying the differential quotient, Leibniz obtains”

[It seems then that Leibniz will replace the infinitesimals dy and dx with finite differences (d)y and d(x), but he will not do the same for the second d(v) for some reason, leaving it as infinitesimal. Returning to KS:]

At this point Leibniz proposes to transfer ‘‘the matter, as we may, to straight lines that never become evanescent’’, obtaining

KS explain that (d)y and (d)x are assignable, while dv is ‘superfluous’ as ‘it alone can be evanescent’ [KS 583, quoting Leibniz].

[This means that dv can be treated as zero and removed from the equation. Then, we multiple both sides by (d)x to obtain:

Then Leibniz divides both sides by a and by (d)x to obtain:

Then Leibniz notes that (d)y / (d)x “always” equals dy/dx. Thus we can substitute them. This is an application of the principle of continuity again:

Also, since (dy) : (d)x always = dy : dx, it will be allowable to suppose this is true in the case when dy, dx become evanescent, and to say that dy : dx = x + v : a, or ady = xdv + vdx.
(Leibniz 154)

(above by multiplying both sides by a and dx).]

But the law of homogeneity is not mentioned here in Cum Prodiisset, so we do not here have sufficient rational for that step in the operation.

The authors then show a case in the Leibniz text of division by second differentials, which they say is incompatible with the nilsquare approach. [The author’s mention the nilsquare approach in section 4.1]

Bibliography:

Katz, M.; Sherry, D. Leibniz's Infinitesimals: Their Fictionality, Their Modern Implementations, And Their Foes From Berkeley To Russell And Beyond. Erkenntnis 78 (2013), no. 3, 571-625. See http://dx.doi.org/10.1007/s10670-012-9370-y, http://www.ams.org/mathscinet-getitem?mr=3053644, and http://arxiv.org/abs/1205.0174

The above bibliography material taken from the following source, a page by Mikhail Katz, which links to many other recent publications on infinitesimals.

http://u.cs.biu.ac.il/~katzmik/infinitesimals.html

Regarding the Leibniz text:

English references from:

Leibniz. The Early Mathematical Manuscripts of Leibniz. Trans. J.M. Child. Mineola, NY: Dover, 2005 [1920 Open Court].
1920 Edition available at archive.org:
https://archive.org/details/earlymathematic01gerhgoog

Latin references from:

Leibniz. Historia et origo calculi differentialis. Ed. C.I. Gerhardt. Hannover: Im Verlage der Hahn'schen Hofbuchhandlung, 1846]
Available at archive.org:
https://archive.org/details/historiaetorigo00gerhgoog

Katz and Sherry’s [Pt.4.5] “Leibniz’ Infinitesimals: Their Fictionality, Their Modern Implementations, And Their Foes From Berkeley To Russell And Beyond,” 4.5 ‘Mathematical Implementation of Status Transitus,’ summary

summary by Corry Shores
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Mikhail G. Katz  and David Sherry

“Leibniz’s Infinitesimals: Their Fictionality, Their Modern Implementations, And Their Foes From Berkeley To Russell And Beyond”

4. Cum Prodiisset

 

4.5 Mathematical Implementation of Status Transitus

Brief Summary:

In Cum Prodiisset Leibniz discusses his Law of Continuity, and he provides some examples. Katz and Sherry will examine some of the mathematical procedures involved in these examples. What we find is that the same formulations and procedures are used regardless of whether we are dealing with finite or infinite values. This is because the law of continuity postulates that what holds for the finite holds for the infinite (and infinitesimal), [because there is a continuous transition from one to the other.]

Summary

 

In section 4.2 and 4.4, Katz and Sherry (KS) discuss three applications of Leibniz’ law of continuity and infinitesimals in his Cum Prodiisset. Now they will examine them more mathematically.

Leibniz distinguishes assignable finite quantity d(x) from infinitesimal value dx.

The assignable quantity (d)x passes via infinitesimal dx on its way to absolute 0. Then the infinitesimal dx is the terminus, or the status transitus. Zero is merely the shadow of the infinitesimal. This particular status transitus is the foundation rock of the Leibnizian definition of the differential quotient.
[KS 581]

[Recall Leibniz’ example of finding the tangent to a parabola. He writes:

“let ₁X₂X, the difference between A₁X and A₂X, be called dx;”

“and similarly, let D₂Y, the difference between ₁X₁Y and ₂X₂Y, be called dy.” (Leibniz 151)

Then, we shrink dx and dy down to find the tangent.

(Animated diagram by Corry Shores, using OpenOffice Draw and Unfreez)

Here Leibniz does not notate a difference between dx and d(x), but we would think of the finitely large line as being d(x) and the infinitesimally small one as dx.

KS also refer to Z as being the shadow of the infinitesimal. In the appendix they explain the shadow as being a real number assigned to hyperreal infinitesimals. It is the next closest value, only infinitesimally different. In these cases below, the shadow is like rounding the infinitesimal value to the nearest real number.]]

Let’s recall the first example Katz and Sherry examine:

(1) In the context of a discussion of parallel lines, he writes: when the straight line BP ultimately becomes parallel to the straight line VA, even then it converges toward it or makes an angle with it, only that the angle is then infinitely small (Child 1920, p. 148).
[[KS579]]

[Image from Leibniz/Child 148]

[We animated it thus:

(Animated diagram by Corry Shores, using OpenOffice Draw and Unfreez)

]

In KS’s rendition, they have us think of this diagram on its side with the moving line intercepting the x axis. In their formulation, H is the x-intercept value.

Example (1) of parallel lines can be elaborated as follows. Let’s follow Leibniz in building the line through the point (0,1) parallel to the x-axis in the plane. Line L_H with y-intercept 1 and x-intercept H is given by y = x / H .
[KS 581]

[Referring back to Leibniz’ diagram, the y-intercept here is like point P, which remains the same, all while the x-intercept, equivalent to point C, moves to greater distances.]

We will move the x-intercept to infinity. [KS later refer to the ‘finite portion of L_H.’ Perhaps one way to picture this is to think that the lines meet way off into infinity, but they ‘drag’ the finite part of the line all the way up such that it is only infinitesimally away from being y = 1. Yet, somewhere the lines must meet, which means it cannot stay at y = 1 everywhere. So does that mean that in the infinite part, it is no longer infinitely close to being parallel? I am not sure, but I think it is always infinitely close to being parallel, but just not always being infinitely close to y = 1. So far off into infinity, it will maybe be infinitely close to being parallel to y = 0.5, and so on diminishing. For every y position of the line in its infinite part, there would be an infinite stretch both ways, making every position be nearly parallel to the line.]

Now let H be infinite. The resulting line LH has negative infinitesimal slope, meets the x-axis at an infinite point, and forms an infinitesimal angle with the x-axis at the point where they meet. We will denote by st(x) the assignable (i.e., real) shadow of a finite x. [KS 581]

[They now will formulate for finite values for the x and y locations included in Line H.]

Then every finite point (x,y) ∈ L_H satisfies

[by substituting the formula for y. Now, because we are supposing H to be infinite, that means you have a finite value divided by an infinite one, which means it is nearly zero, which means that the y value is very close to 1, and thus:]

Hence the finite portion of L_H is infinitely close to the line y = 1. The line y = 1 is parallel to the x-axis, and is merely the shadow of the inassignable L_H. Thus, the parallel line is constructed by varying the oblique line depending on a parameter. Such variation comprises the status transitus L_H defined by an infinite value of H.
[KS 582]

Now recall example 3:

(3) Finally, a conception of a parabola expressed by means of an ellipse with an infinitely removed focal point is articulated in the following terms: a parabola is the ultimate form of an ellipse, in which the second focus is at an infinite distance from the given focus nearest to the given vertex (Child 1920, p. 148).
[KS 579]

[[Recall from our prior discussion that by moving one focus of the ellipse infinitely away from the other, the ellipse transforms into a parabola.

(Animated diagram by Corry Shores, using OpenOffice Draw and Unfreez)

]

KS will provide a formulation to illustrate this better.

To implement example (3), let’s follow Leibniz in deforming an ellipse, via a status transitus, into a parabola. The ellipse with vertex (apex) at (0,-1) and with foci at the origin and at (0,H) is given by

[They then square both sides and expand the values:]

Then they kept the radical on the left side, moving the rest over:

Then they square both sides, and reduce to get:

[We notice that even though H is an infinite number, they still apply operations on the figures that normally apply to finite numbers. But whether or not that is mathematically admissible is uncertain, hence it the ‘postulate’ of the law of continuity that we can treat them in like manner.]

The calculation (4.1) through (4.4) depends on the following habits of general reasoning (to echo Child’s translation) with assignable quantities, which are generalized to apply to inassignable quantities (such as the terminus/status transitus) in accordance with the law of continuity:

• squaring undoes a radical;

• the binomial formula;

• terms in an equation can be transferred to the other side; etc.

General reasoning of this type is familiar in the realm of ordinary finite real numbers, but why does it remain valid when applied to the realm of infinite or infinitesimal numbers? The validity of transfering such general reasoning originally instituted in the finite realm, to the realm of the infinite is postulated by Leibniz’s law of continuity.
[KS 582]

[Because the same general reasoning is used for finite and infinite, that might give a ‘between’ status or ambiguous status to the status transitus. Notice that we are still using a formulation for an ellipse. And yet, the resulting figure is in transit to being a parabola. Off in infinity, it might be still elliptical. But it draws out the finite part such that only parabolic parts remain within the realm of assignable values.]

We therefore apply Leibniz’s law of continuity to Eq. (4.4) for an infinite H. The resulting entity is still an ellipse of sorts, to the extent that it satisfies all of the Eqs. (4.1) to (4.4). However, this entity is no longer finite. It represents a Leibnizian status transitus between ellipse and parabola. This status transitus has foci at the | origin and at an infinitely distant point (0, H).
[KS 582|583, boldface mine]

[[Recall that we left off before with:

It seems in the following that KS will assign real numbers so to find the “real shadow” of this figure, that is, the finite assignable values it is infinitely close to.]]

Assuming x and y are finite, we set x₀ = st(x) and y₀ = st(y), to obtain a real shadow of this entity:

then simplifying:

Thus, the finite portion of the status transitus (4.4) is infinitely close to its shadow (4.5), namely the real parabola y ¼ x2 4 1 (in Leibniz’s terminology as translated by Child, ‘‘it is really true’’ that this parabola has no focus at infinity—see Sect. 4.4). This is the kind of payoff Leibniz is seeking with his law of continuity.
[KS 583]

Bibliography:

Katz, M.; Sherry, D. Leibniz's Infinitesimals: Their Fictionality, Their Modern Implementations, And Their Foes From Berkeley To Russell And Beyond. Erkenntnis 78 (2013), no. 3, 571-625. See http://dx.doi.org/10.1007/s10670-012-9370-y, http://www.ams.org/mathscinet-getitem?mr=3053644, and http://arxiv.org/abs/1205.0174

The above bibliography material taken from the following source, a page by Mikhail Katz, which links to many other recent publications on infinitesimals.

http://u.cs.biu.ac.il/~katzmik/infinitesimals.html

Katz and Sherry’s [Pt.4.4] “Leibniz’s Infinitesimals: Their Fictionality, Their Modern Implementations, And Their Foes From Berkeley To Russell And Beyond,” 4.4 ‘Status Transitus,’ summary

summary by Corry Shores [Search Blog Here. Index-tags are found on the bottom of the left column.]
[Central Entry Directory]
[Calculus Entry Directory]
[Leibniz Entry Directory]
[Entry Directory for this Article]

[The following is summary. My own comments and citations are placed in double brackets. All boldface and underlying are mine.]



 

Mikhail G. Katz  and David Sherry

“Leibniz’s Infinitesimals: Their Fictionality, Their Modern Implementations, And Their Foes From Berkeley To Russell And Beyond”

4. Cum Prodiisset

 

4.4 Status Transitus

Brief Summary:

A state of transition (status transitus) is an infinitely small variation that explains how things can change from one state to a contrary one. For example, parallel lines might converge, so long as the angle of that convergence is infinitely small. That angle would then be the ‘between’ transitional state between its states of being parallel and non-parallel. Such an angle is infinitesimally more than completely superposing the other line, but infinitesimally less than diverging sharply from it, at the vertex of that angle. Looking away from the intersection, we would say it is not yet parallel. Looking toward it, we would say it is not yet superposed. Transition can be explained by means of a law of continuity which says that continuous changes occur by infinitely small bridging variations, allowing us to go from opposing states, so long as that change is so small that we cannot even assign it a magnitude (and thus make an ‘actual’ distinction between the states).

 

Summary

Previously we noted three of Leibniz’ examples for the Principle of Continuity: (1) lines can both be parallel yet also converge at an infinitely small angle [[because there is so little difference between an infinitely small angle and none at all]] (2) two lines can be equal even if they differ in length by an infinitely small amount, and (3) a parabola is an ellipse with one side extended to infinity. Yet we find him also stating that:

Leibniz introduces his next observation by the clause ‘‘of course it is really true that’’, and notes that ‘‘straight lines which are parallel never meet’’ (Child 1920, p. 148); that ‘‘things which are absolutely equal have a difference which is absolutely nothing’’ (Child 1920, p. 148); and that ‘‘a parabola is not an ellipse at all’’ (Child 1920, p. 149).
[[KS 580]]

To explain then the original three examples we observed, he proposes his notion of status transitus, or ‘state of transition’. [[This concept will introduce the idea of event, change, motion, time, process, and the like, because]] in one such a state of transition, ‘there has not yet arisen exact eqauality’.

a state of transition may be imagined, or one of evanescence, in which indeed there has not yet arisen exact equality … or parallelism, but in which it is passing into such a state, that the difference is less than any assignable quantity; also that in this state there will still remain some difference, … some angle, but in each case one that is infinitely small; and the distance of the point of intersection, or the variable focus, from the fixed focus will be infinitely great, and the parabola may be included under the heading of an ellipse (Child 1920, p. 149).
[[KS 580]]

Recall also Leibniz’s notion of terminus: in a continuous transition, the final ending (the terminus) of the transition may be included with that transition. [[Consider something in motion slowing to a stop. That final ending, rest, can be included with the motion preceding it, as a part of that motion, even if it lies at the end of the motion. The speed while it is moving could be assigned a value, and its speed at rest can be assigned the value of 0. But because of continuity, perhaps we might say that between its states of motion and rest, it is going an inassignable, infinitely slow speed. By ‘between’ we do not mean during some duration of time it is moving some extent of distance. Say we come to the end of the object’s motion. It first comes to complete rest at time point 2 (t2) at location point 2 (p2). Immediately prior it was at p1, t1. There is no p or t between them, if we are dealing with an actual infinity (one that is already divided infinitely and not perpetually divided potentially and thus interminably). But, at p1,t1, its state of motion was immediately in relation with its state of rest at p2,t2. And at p2,t2, it is still in relation to its prior state of motion at p1,t1. This is because motion is always a difference between times and locations. If we divide motion, we do not divide it ultimately into p’s and t’s. Rather, we divide it into the smallest possible differences between p’s and t’s. There is a difference between p1,t1 and p2,t2, because one lies at a state of motion, and the other lies at a state of rest. When we concern ourselves with that tiny moment when the object goes from p1,t1 to p2,t2, then we have the terminus (rest state at p2,t2) included in the motion (as it is part of the interval with the motion state at p1,t1). We also have this state of transition, status transitus.]] “Thus, status transitus is subsumed under terminus, passing into an assignable entity, but is as yet inassignable.” [KS 580] We would not say that the status transitus is a ‘limit’ (as some translators have rendered it), because a limit is an assignable entity, where that status transitus is not. [580]

Yet for Leibniz, the metaphysical reality of the infinitesimal is open to question.

whether such a state of instantaneous transition from inequality to equality, … from convergence [i.e., lines meeting—the authors] to parallelism, or anything of the sort, can be sustained in a rigorous or metaphysical sense, or whether infinite extensions successively greater and greater, or infinitely small ones successively less and less, are legitimate considerations, is a matter that I own to be possibly open to question (Child 1920, p. 149).
[KS 580]

Yet this uncertain to the ontological reality of infinitesimals should not stop mathematicians from using them effectively in their calculations. Leibniz asserts

the possibility of the mathematical infinite: ‘‘it can be done’’, without ontological commitments as to the reality of infinite and infinitesimal objects.
[KS 581]

 

Bibliography:

Katz, M.; Sherry, D. Leibniz's Infinitesimals: Their Fictionality, Their Modern Implementations, And Their Foes From Berkeley To Russell And Beyond. Erkenntnis 78 (2013), no. 3, 571-625. See http://dx.doi.org/10.1007/s10670-012-9370-y, http://www.ams.org/mathscinet-getitem?mr=3053644, and http://arxiv.org/abs/1205.0174

The above bibliography material taken from the following source, a page by Mikhail Katz, which links to many other recent publications on infinitesimals.

http://u.cs.biu.ac.il/~katzmik/infinitesimals.html