24 May 2014

Priest (11.2) In Contradiction, ‘The Instant of Change’, summary

 

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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


11.2 The Instant of Change



Brief Summary:

Things in the physical world change states through time. There are instants of such change when we have no more reason to say that it is in the prior or the following state. In such cases, it is true that the thing is both in a state and not in that state. This is a true contradiction, a dialetheia, that is a real and common situation in the physical world.



Summary


Priest will illustrate a problem with the instant of change.

As I write, my pen is touching the paper. As I come to the end of a word I lift it off. At one time it is on; at another it is off (that is, not on). Since the motion is continuous, there must be an instant at which the pen leaves the paper. At that instant, is it on the paper or off? [160]

[We have things and their interactions here operating in a describable way, so we will call it system s. There is a moment in time in question, the instant when the pen leaves the paper, which we will call t0. We also have different states the pen can be in, either on or off the paper, called s0 and s1. We will describe the state of being on the page in the statement called α (which reads “The pen is on the paper”). Otherwise, the pen would be off the paper, and then it will be in the next succeeding state, s1. After the pen leaves the paper, it is not the case that “the pen is on the paper”, which is a proposition we would then write as ¬α. Now we want to know, at the moment it leaves the paper, what state is it in? There are four possibilities. Either 1) it is only on the paper, 2) it is only off the paper, 3) it is neither on the paper nor off, or 4) or it is both on the paper and off the paper in the same instant.]

We may formulate the problem more generally. 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)

Also, there may not be an answer which holds for all cases, because “Different changes may be changes of different kinds.” (160) If we just assume classical logic, then all changes will have to be of types either A or B. But since we are testing the viability of classical logic for explaining change, we cannot assume from the beginning it is the only means to find the answer. Priest will argue that there are changes of type Δ, that the pen is both on the page and off it the instant it makes that transition.


Previously in section 4.7 Priest ruled out the possibility of Γ type changes, so at least one of α or ¬α must hold. [His argument there has to do with liar paradoxes, for example, ‘This sentence is false’. If it is false, then it is true, but if it is true, then it is false. Therefore it is both true and false. That is not a problem for dialetheians. However, those who reject this need to explain what is wrong with the argumentation that leads to the paradox. One solution is to say that there are truth gaps, that is, that there are sentences which are neither true nor false. Priest in that section showed how their arguments failed, and thus in this current section, he does not find the third option, that it is neither in one state nor the other, convincing.] Now we must argue that not all changes are of types A or B (being either on or off the paper exclusively). So recall that at  t0 the pen leaves the paper. Is it on or off? It seems we do not have a better reason to say one over the other. This means we might prefer a symmetrical answer like Γ or Δ (the pen is neither on nor off the paper, or the pen is both on and off the paper). We might break this asymmetry by identifying being on with being zero distance from. [It seems Priest might be saying something like the following here. Consider its motion coming off the paper. Because of infinite divisibility, there is no first point when it is off the paper. Thus maybe we might say that it cannot be both, because the final on point is determinable, but the first off point is not. However, this does not apply for objects falling to the ground for example. This is perhaps because there is a final terminating point for the motion without there being a continuation after it. Priest writes: “There is, however, a way of breaking the asymmetry in this case. Since the motion is continuous, there is, presumably, a last instant at which the distance between the point of my pen and the paper is zero, but no first point at which it is non-zero. (Perhaps more precisely, there is a last point at which the electrical repulsion between my pen and the paper is equal to the weight of the pen, but no first point at which this is not the case.) If we identify being on with being zero distance from, this makes the change of type A. But the identification is highly suspect. An arrow is fired into the ground. At the instant of impact, before the point of the arrow penetrates the ground, is the arrow on the ground?” (160)]


Priest says that even if we can preserve asymmetry in the above example, it will not work in all cases. If we discover some solution, then we have a symmetrical relation between the state before of not having it and the state after of having it.

A particularly striking example of this is a phenomenological one. For days I have been puzzling over a problem. Suddenly the solution strikes me. Now, at the instant the solution strikes me, do I or do I not know the answer? The situation is, again, symmetrical. Before, I did not know the answer; after, I did. Moreover, one cannot suppose that in this case there is some tie-breaking ulterior fact. My epistemological state is all there is, and that is symmetrical. It makes little sense to suppose that I either did or did not determinately know the answer at the instant of change, though I am unaware which. (161)

In the next example, Priest has us consider us walking into a room. There will be a point when we have no more reason to say we are in than we are out. Thus there are cases where we have enough reason to give answers of type A or B.

I am in a room. As I walk through the door, am I in the room or out of (not in) it? To emphasize that this is not a problem of vagueness, suppose we identify my position with that of my centre of gravity, and the door with the vertical plane passing through its centre of gravity. As I leave the room there must be an instant at which the point lies on the plane. At that instant am I in or out? Clearly, there is no reason for saying one rather than the other. It might be suggested that in this and similar cases we are free to stipulate that I was, say, in. Unfortunately this is not a solution, but simply underlines the problem. I am free to stipulate in this way only because neither being in nor not being in has a better claim than the other: I am neither determinately out rather than in, nor determinately in rather than out. Thus, intrinsically, the change is symmetrical, and therefore not of type A or type B. (161)


So we are arguing for type Δ changes (both on and off). Someone might argue against them by opening the possibility for Γ type changes (neither on nor off). They might do this by rejecting the exhaustion principle which says that if α is not true then ¬α is true. [Perhaps then in the case of the pen, it can both be that the pen is not on the table but also not-not on the table, or in other words, that the pen is neither on nor off the table.] This is possible if the arguments from section 4.7 can be met. Nonetheless, the above example still gives us ample reason to suppose that there are type Δ changes.


There is another issue to address. Instead of instants of time, there might only be intervals. If so, then there are no instants of change, and thus no contradictions [because it is never at the same time that the pen is on or off the paper]


There are some problems however with arguing that time is not composed of instants. Science operates as though time can be represented by the real line. Calculus’ application in physics presupposes this. Thus to say that they are wrong about instants is to say that much of their work [or at least methodology, maybe conclusions] are wrong. [[However, there is also a way to conceive of the instant as an infinitesimal interval. This would make it both a combination of states without any passage of duration between them.]]

a good part of science is based on the | assumption that physical continua have a structure that can be represented by the real line, and therefore that we can speak of instants of time. In particular, any science that uses the differential and integral calculus presupposes this. Therefore, this proposal, if adopted, would cause the demise of a good part of science. Or, to put it more tellingly, the proposal flies in the face of well corroborated scientific theories. Its correctness is, therefore, highly suspect. (161-162)


Another problem with the theory that time is composed of intervals and not units is that it fails to account for how (or when) change happens. If two successive intervals have different states, where does the change take place? It cannot be between them, because there are no instants between intervals. But it cannot be in either one, because there is only one state and not a change of states. Thus tie would just be a sequence of still moments. This is the cinematic account of change [we discuss it further in sect.12.2].

suppose that during a certain time a system, s, changes discretely from state s0 to states s1. Then there must be two abutting intervals, X and Y, X wholly preceding Y, such that s0 holds throughout X and s1 holds throughtout [sic] Y. Now, given that there is no instant dividing X and Y, we cannot ask what state s is in at it. However, just because there is no such instant, there is no time at which the system is changing. X is before the change. Y is after it. Thus, in a sense, there is no change in the world at all, just a series of states patched together. The universe would appear to be more like a sequence of photographic stills, shown consecutively, than something in a genuine state of flux or change. We might call this the cinematic account of change. As we will see, it has a habit of surfacing in consistent accounts of change. I will discuss it in more detail in section 12.2. For the present, let us just note that the cinematic account is highly counter-intuitive. (162)

Also, intervals would have to be divisible, which means at one line of subdivision there could be a case of α ∧ ¬α.

it is not even clear that dialetheism can be avoided by eschewing instants of time in favour of intervals; for, unless there are atomic intervals, a possibility that raises the shades of Zeno and exacerbates both the previous problems, intervals must be indefinitely subdivisible. Now, note that the fact that a holds at an interval, X, does not necessarily imply that it holds at every subinterval of X (or else the sun’s shining on a certain day would imply that it shone during every part of the day). There is therefore nothing, in principle, to rule out the possibility of an interval such that every subinterval where a holds has a subinterval where :a holds and vice versa. What holds at this interval? What could it be but α ∧ ¬α?
(162)



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

8 May 2014

Graham Priest, entry directory


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Corry Shores
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An Introduction to Non-Classical Logic: From If to Is

 

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In Contradiction:
A Study of the Transconsistent


Graham Priest, In Contradiction, entry directory


 

 

Beyond the Limits of Thought

 

Priest, Beyond the Limits of Thought, entry directory

 

 


One:
Being an Investigation into the Unity of Reality and of its Parts, including the Singular Object which is Nothingness


Graham Priest’s One, entry directory

 


Doubt Truth To Be a Liar

 

Priest. Doubt Truth To Be a Liar, entry directory

 

 

 

“Dialectic and Dialetheic”


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Logic: A Very Short Introduction

 

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Graham Priest In Contradiction, entry directory

 

by Corry Shores
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In Contradiction:
A Study of the Transconsistent


Part I. The Logical Paradoxes


Ch.1. Semantic Paradoxes


1.1 Logical Paradoxes


1.2 The Semantic Paradoxes


1.3 Truth Value Gaps 


Part II. Dialetheic Logical Theory

Ch.4. Truth or Falsity


4.7 Truth or Falsity: Truth Value Gaps



Part III. Applications

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


11.1 Contradictions in the World


11.2 The Instant of Change


11.3 Dialectical Tense Logic


11.4 The Leibniz Continuity Condition


 11.5 The LCC and Contradiction


Ch.12. The Metaphysics of Change II: 
Motion


12.1 Change and Motion


12.2 The Orthodox Account of Change


12.3 The Hegelean Account of Motion


12.4 … And Its Consequences





Priest (11.1) In Contradiction, ‘Contradictions in the World’, summary

 

by Corry Shores
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[The following is summary. All boldface, underlying and bracketed commentary 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


11.1 Contradictions in the World


Brief Summary:

Priest will discuss true contradictions in the physical world.



Summary


Previously Priest examined dialetheias [true contradictions] belonging to the “abstract realm of logic (set theory and semantics)” [159]. Priest turns now to true contradictions found concretely in the empirical world. Perhaps the world is not something which can be consistent or inconsistent like propositions can be. [But we can make statements about the world, and perhaps there are physical situations whose descriptions produce dialetheias, true contradictions.]




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



 

5 May 2014

Augustine on Time, Confessions Book 11, summary



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[The following is summary and quotation. All boldface, underlining, and bracketed commentary are mine. Proofreading is incomplete, so mistakes are still present.]





Augustine


Confessions


Book XI
 
Brief summary:
Time is mysterious. We know there is a present. But it passes. The future and past do not exist, but we remember and anticipate them. Can the present have a duration? But whatever its duration, there will always be within it a segment that is more present. Thus the present is momentary and the past and future do not exist. However, because we remember and anticipate them right now in the present, there is “a time present of things past; a time present of things present; and a time present of things future.” Another mystery is when God created the world and thus when time began. God could not have existed in a moment before it, because there was no time before it. But God is eternal, and eternity does not have successive moments. All its parts are simultaneous. God did not exist successively before creation, but creation is the product of him. These mysteries of time tell us of the greatness of God whose ways we can scarcely comprehend.
 
Summary

Quoting Outler’s summarization at the beginning:
The eternal Creator and the Creation in time. Augustine ties together his memory of his past life, his present experience, and his ardent desire to comprehend the mystery of creation. This leads him to the questions of the mode and time of creation. He ponders the mode of creation and shows that it was de nihilo and involved no alteration in the being of God. He then considers the question of the beginning of the world and time and shows that time and creation are cotemporal. But what is time? To this Augustine devotes a brilliant analysis of the subjectivity of time and the relation of all temporal process to the abiding eternity of God. From this, he prepares to turn to a detailed interpretation of Gen. 1:1, 2. [215]
 
Chapter I
 
§1

Augustine notes that God is eternal but his current appeals are temporal, and so he wonders if God can even hear them. [For, he might be somewhere else in time or even nowhere in time.]
Is it possible, O Lord, that, since thou art in eternity, thou art ignorant of what I am saying to thee? Or, dost thou see in time an event at the time it occurs? If not, then why am I recounting such a tale of things to thee? Certainly not in order to acquaint thee with them through me; but, instead, that through them I may stir up my own love and the love of my readers toward thee, so that all may say, “Great is the Lord and greatly to be praised.” I have said this before406 and will say it again: “For love of thy love I do it.”
[155]


Chapter II

§2

Augustine confesses his ignorance of God’s divine law and how much he wants to meditate on it.

§3

He asks for mercy for his longing.
“Thine is the day and the night is thine as well.” At thy bidding the moments fly by. Grant me in them, then, an interval for my meditations on the hidden things of thy law, nor close the door of thy law against us who knock.
[...]
Let me drink from thee and “consider the wondrous things out of thy law”416--from the very beginning, when thou madest heaven and earth, and thenceforward to the everlasting reign of thy Holy City with thee.
[156]

§4

Augustine’s desire  for this knowledge comes from his love for God.
 
Chapter III

§5

Augustine wants to know how God created heaven and earth. Moses wrote of it, but he is dead now.
since I cannot inquire of Moses, I beseech thee, O Truth, from whose fullness he spoke truth; I beseech thee, my God, forgive my sins, and as thou gavest thy servant the gift to speak these things, grant me also the gift to understand them.
[157]
 


Chapter IV

§6

Because heaven and earth are changing, they must have been something different before. From this Augustine concludes they were created. [If something was always the same, then maybe it was always there without origin.]
Look around; there are the heaven and the earth. They cry aloud that they were made, for they change and vary. Whatever there is that has not been made, and yet has being, has nothing in it that was not there before. This having something not already existent is what it means to be changed and varied. Heaven and earth thus speak plainly that they did not make themselves: “We are, because we have been made; we did not exist before we came to be so that we could have made ourselves!” And the voice with which they speak is simply their visible presence.
[157]


Chapter V

§7

Augustine wonders how God created the world. Man creates by changing the shapes of things already given to him. But God creates from nothing. And from what place did he create them? God must have created solely by his word.
But how didst thou make them? How, O God, didst thou make the heaven and earth? For truly, neither in heaven nor on earth didst thou make heaven and earth--nor in the air nor in the waters, since all of these also belong to the heaven and the earth. Nowhere in the whole world didst thou make the whole world, because there was no place where it could be made before it was made. And thou didst not hold anything in thy hand from which to fashion the heaven and the earth, for where couldst thou have gotten what thou hadst not made in order to make something with it? Is there, indeed, anything at all except because thou art? Thus thou didst speak and they were made, and by thy Word thou didst make them all.
[158]


Chapter VI

§8

Augustine now wonders how God’s voice sounded out in creation. When announcing his son of God, it was a voice in time in the created world. But the eternal word that created the world could not be corporeal, because then something created would have preceded it. Augustine wonders if the word itself decreed that there be some source from which that word word be spoken?
And what these words were which were formed at that time the outer ear conveyed to the conscious mind, whose inner ear lay attentively open to thy eternal Word. But it compared those words which sounded in time with thy eternal word sounding in silence and said: “This is different; quite different! These words are far below me; they are not even real, for they fly away and pass, but the Word of my God remains above me forever.” If, then, in words that sound and fade away thou didst say that heaven and earth should be made, and thus madest heaven and earth, then there was already some kind of corporeal creature before heaven and earth by whose motions in time that voice might have had its occurrence in time. But there was nothing corporeal before the heaven and the earth; or if there was, then it is certain that already, without a time-bound voice, thou hadst created whatever it was out of which thou didst make the time-bound voice by which thou didst say, “Let the heaven and the earth be made!” For whatever it was out of which such a voice was made simply did not exist at all until it was made by thee. Was it decreed by thy Word that a body might be made from which such words might come?
[158]


Chapter VII

§9

It seems Augustine is saying that the eternal word never finished. It also seems Augustine is saying that although the eternal word is not temporally located, God does say other things which are.
Thou dost call us, then, to understand the Word--the God who is God with thee--which is spoken eternally and by which all things are spoken eternally. For what was first spoken was not finished, and then something else spoken until the whole series was spoken; but all things, at the same time and forever. For, otherwise, we should have time and change and not a true eternity, nor a true immortality. [...] But there is nothing in thy Word that passes away or returns to its place; for it is truly immortal and eternal. And, therefore, unto the Word coeternal with thee, at the same time and always thou sayest all that thou sayest. And whatever thou sayest shall be made is made, and thou makest nothing otherwise than by speaking. Still, not all the things that thou dost make by speaking are made at the same time and always.
[159]


Chapter VIII

§10

It seems here that Augustine is saying that because we see change, it tells us there must be an origin of change.

Why is this, I ask of thee, O Lord my God? I see it after a fashion, but I do not know how to express it, unless I say that everything that begins to be and then ceases to be begins and ceases when it is known in thy eternal Reason that it ought to begin or cease--in thy eternal Reason where nothing begins or ceases. And this is thy Word, which is also “the Beginning,” because it also speaks to us.424 Thus, in the gospel, he spoke through the flesh; and this sounded in the outward ears of men so that it might be believed and sought for within, and so that it might be found in the eternal Truth, in which the good and only Master teacheth all his disciples. There, O Lord, I hear thy voice, the voice of one speaking to me, since he who teacheth us speaketh to us. But he that doth not teach us doth not really speak to us even when he speaketh. Yet who is it that teacheth us unless it be the Truth immutable? For even when we are instructed by means of the mutable creation, we are thereby led to the Truth immutable. There we learn truly as we stand and hear him, and we rejoice greatly “because of the bridegroom’s voice,” restoring us to the source whence our being comes. And therefore, unless the Beginning remained immutable, there would then not be a place to which we might return when we had wandered away. But when we return from error, it is through our gaining knowledge that we return. In order for us to gain knowledge he teacheth us, since he is the Beginning, and speaketh to us.
[159]


Chapter IX

§11.

Augustine praises God.
“How wonderful are thy works, O Lord; in wisdom thou hast made them all.”428 And this Wisdom is the Beginning, and in that Beginning thou hast made heaven and earth.
[160]


Chapter X

§12

In this section Augustine raises the issue of what comes before creation, but more importantly, if the creation was the will of God, and that will is as eternal as he is, then why is the world not as eternal as he is? Why did he have to create it?
 
Chapter XI

§13

Augustine says that those who asks such questions do not understand how God creates things. They also do not understand the nature of eternity. It is not progressive time going on forever. To think that is to then be confused about beginnings and ends. In a long progressive time, what makes it long is that all the parts of that time are happen one after each other. But in eternity, the whole of time is simultaneous with itself.
They endeavor to comprehend eternal things, but their heart still flies about in the past and future motions of created things, and is still unstable. Who shall hold it and fix it so that it may come to rest for a little; and then, by degrees, glimpse the glory of that eternity which abides forever; and then, comparing eternity with the temporal process in which nothing abides, they may see that they are incommensurable? They would see that a long time does not become long, except from the many separate events that occur in its passage, which cannot be simultaneous. In the Eternal, on the other hand, nothing passes away, but the whole is simultaneously present. But no temporal process is wholly simultaneous. Therefore, let it see that all time past is forced to move on by the incoming future; | that all the future follows from the past; and that all, past and future, is created and issues out of that which is forever present. Who will hold the heart of man that it may stand still and see how the eternity which always stands still is itself neither future nor past but expresses itself in the times that are future and past? Can my hand do this, or can the hand of my mouth bring about so difficult a thing even by persuasion?
[161]
 
Chapter XII

§14

[God is eternal. This means ‘before’ and ‘after’ does not have the normal meaning in terms of succession. The whole of eternity is simultaneous with itself. So we cannot designate in eternity points where one thing comes before or after another thing.] We would be misunderstanding the nature of eternity if we asked what God was doing before he made heaven and earth.
 
Chapter XIII

§15

When God created the world, he created successive progressive time with it. There could not be a ‘before’ to is since no successivity existed until the world was created.
For in what temporal medium could the unnumbered ages that thou didst not make pass by, since thou art the Author and Creator of all the ages? Or what periods of time would those be that were not made by thee? Or how could they have already passed away if they had not already been? Since, therefore, thou art the Creator of all times, if there was any time before thou madest heaven and earth, why is it said that thou wast abstaining from working? For thou madest that very time itself, and periods could not pass by before thou madest the whole temporal procession. But if there was no time before heaven and earth, how, then, can it be asked, “What wast thou doing then?” For there was no “then” when there was no time. [161]
 

§16

For eternal God, all of time is altogether at once. His years do not pass but rather are always abiding. All of time for Him is an eternal today.
Nor dost thou precede any given period of time by another period of time. Else thou wouldst not precede all periods of time. In the eminence of thy everpresent eternity, thou precedest all times past, and extendest beyond all future times, for they are still to come--and when they have come, they will be past. But “Thou art always the Selfsame and thy years shall have no end.” Thy years neither go nor come; but ours both go and come in order that all separate moments may come to pass. All thy years stand together as one, since they are abiding. Nor do thy years past exclude the years to come because thy years do not pass away. All these years of ours shall be with thee, when all of them shall have ceased to be. Thy years are but a day, and thy day is not recurrent, but always today. Thy “today” | yields not to tomorrow and does not follow yesterday. Thy “today” is eternity. Therefore, thou didst generate the Coeternal, to whom thou didst say, “This day I have begotten thee.” Thou madest all time and before all times thou art, and there was never a time when there was no time.
 
Chapter XIV

§17

Augustine acknowledges that we have implicit knowledge of time, but we cannot say what it is.
For what is time? Who can easily and briefly explain it? Who can even comprehend it in thought or put the answer into words? Yet is it not true that in conversation we refer to nothing more familiarly or knowingly than time? And surely we understand it when we speak of it; we understand it also when we hear another speak of it.
What, then, is time? If no one asks me, I know what it is. If I wish to explain it to him who asks me, I do not know.
[162]
He then wonders about the present. The present must pass, or else it would be eternal. But what defines time is that it is passing away. But then its cause of being seems like a contradiction. The present comes to exist because it ceasing to exist.
But, then, how is it that there are the two times, past and future, when even the past is now no longer and the future is now not yet? But if the present were always present, and did not pass into past time, it obviously would not be time but eternity. If, then, time present--if it be time--comes into existence only because it passes into time past, how can we say that even this is, since the cause of its being is that it will cease to be? Thus, can we not truly say that time is only as it tends toward nonbeing?
[162]
 
Chapter XV

§18

We say that some time ago was long ago. [It seems Augustine is then saying that a period of time can only be long if it is present, perhaps because when it is past it no longer exists. Perhaps Augustine is saying that the present of which it was a part carried on for a long time. Please see for yourself:]
And yet we speak of a long time and a short time; but never speak this way except of time past and future. We call a hundred years ago, for example, a long time past. In like manner, we should call a hundred years hence a long time to come. But we call ten days ago a short time past; and ten days hence a short time to come. But in what sense is something long or short that is nonexistent? For the past is not now, and the future is not yet. Therefore, let us not say, “It is long”; instead, let us say of the past, “It was long,” and of the future, “It will be long.” And yet, O Lord, my Light, shall not thy truth make mockery of man even here? For that long time past: was it long when it was already past, or when it was still present? For it might have been long when there was a period that could be long, but when it was past, it no longer was. In that case, that which was not at all could not be long. Let us not, therefore, say, “Time past was long,” for we shall not discover what it was that was long because, since it is past, it no longer exists. Rather, let us say that “time present was long, because when it was present it was long.” For then it had not yet passed on so as not to be, and therefore it still was in a state that could be called long. But after it passed, it ceased to be long simply because it ceased to be.
[162]

§19

Augustine then points out that a long time cannot all be present. Only its most current part can be. [[This is a useful point for the discussion of Barry Dainton’s notion of the specious present. He claims that the present can have a length, and to say what Augustine is saying here that there must always be a privileged part of any duration that is present is called ‘presentism’]].
Is a hundred years when present a long time? But, first, see whether a hundred years can be present at once. For if the first year in the century is current, then it is present time, and the other ninety and nine are still future. Therefore, they are not yet. But, then, if the second year is current, one year is already past, the second present, and all the rest are future. And thus, if we fix on any middle year of this century as present, those before it are past, those after it are future. Therefore, a hundred years cannot be present all at once.
Let us see, then, whether the year that is now current can be present. For if its first month is current, then the rest are future; if the second, the first is already past, and the remainder are not yet. Therefore, the current year is not present all at once. And if it is not present as a whole, then the year is not present. For it takes twelve months to make the year, from which each individual month which is current is itself present one at a time, but the rest are either past or future.
[163]

§20

But the present cannot have any length, because that length would always have a smaller more present point within it. The present must then be a durationless moment.
Thus it comes out that time present, which we found was the only time that could be called “long,” has been cut down to the space of scarcely a single day. But let us examine even that, for one day is never present as a whole. For it is made up of twenty-four hours, divided between night and day. The first of these hours has the rest of them as future, and the last of them has the rest as past; but any of those between has those that preceded it as past and those that succeed it as future. And that one hour itself passes away in fleeting fractions. The part of it that has fled is past; what remains is still future. If any fraction of time be conceived that cannot now be divided even into the most minute momentary point, this alone is what we may call time present. But this flies so rapidly from future to past that it cannot be extended by any delay. For if it is extended, it is then divided into past and future. But the present has no extension whatever.
Where, therefore, is that time which we may call “long”? Is it future? Actually we do not say of the future, “It is long,” for it has not yet come to be, so as to be long. Instead, we say, “It will be long.” When will it be? For since it is future, it will not be long, for what may be long is not yet. It will be long only when it passes from the future which is not as yet, and will have begun to be present, so that there can be something that may be long. But in that case, time present cries aloud, in the words we have already heard, that it cannot be “long.”
[163]
 
Chapter XVI

§21

We measure time, even after it has passed. But Augustine also says that we cannot measure passed time, since it no longer exists. [I am not sure how it is we measure the present if it has no duration, unless we compare it with the past or future. So I am not sure what Augustine is saying here:]
And yet, O Lord, we do perceive intervals of time, and we compare them with each other, and we say that some are longer and others are shorter. We even measure how much longer or shorter this time may be than that time. And we say that this time is twice as long, or three times as long, while this other time is only just as long as that other. But we measure the passage of time when we measure the intervals of perception. But who can measure times past which now are no longer, or times future which are not yet--unless perhaps someone will dare to say that what does not exist can be measured? Therefore, while time is passing, it can be perceived and measured; but when it is past, it cannot, since it is not.
[163]
 
Chapter XVII

§22

But we have much reason to think that there does exist a past and future.
Who can say that there is only time present because the other two do not exist? Or do they also exist; but when, from the future, time becomes present, it proceeds from some secret place; and when, from times present, it becomes past, it recedes into some secret place? For where have those men who have foretold the future seen the things foretold, if then they were not yet existing? For what does not exist cannot be seen. And those who tell of things past could not speak of them as if they were true, if they did not see them in their minds. These things could in no way be discerned if they did not exist. There are therefore times present and times past.
[164]
 
Chapter XVIII

§23

So there is a past in future in some sense. But if they exist, they must be there somehow as present. When we remember something, that image is there presently to our mind. And it seems when we think about the future, it is also being conceived or imagined presently.
Give me leave, O Lord, to seek still further. O my Hope, let not my purpose be confounded. For if there are times past and future, I wish to know where they are. But if I have not yet succeeded in this, I still know that wherever they are, they are not there as future or past, but as present. For if they are there as future, they are there as “not yet”; if they are there as past, they are there as “no longer.” Wherever they are and whatever they are they exist therefore only as present. Although we tell of past things as true, they are drawn out of the memory--not the things themselves, which have already passed, but words constructed from the images of the perceptions which were formed in the mind, like footprints in their passage through the senses. My childhood, for instance, which is no longer, still exists in time past, which does not now exist. But when I call to mind its image and speak of it, I see it in the present because it is still in my memory. Whether there is a similar explanation for the foretelling of future events--that is, of the images of things which are not yet seen as if they were already existing--I confess, O my God, I do not know. But this I certainly do know: that we generally think ahead about our future actions, and this premeditation is in time present; but that the action which we premeditate is not yet, because it is still future. When we shall have started the action and have begun to do what we were premeditating, then that action will be in time present, because then it is no longer in time future.
[164]

§24

So we do not foresee the things themselves, rather we see their signs or causes existing presently.
Whatever may be the manner of this secret foreseeing of future things, nothing can be seen except what exists. But what exists now is not future, but present. When, therefore, they say that future events are seen, it is not the events themselves, for they do not exist as yet (that is, they are still in time future), but perhaps, instead, their causes and their signs are seen, which already do exist. Therefore, to those already beholding these causes and signs, they are not future, but present, and from them future things are predicted because they are conceived in the mind. These conceptions, however, exist now, and those who predict those things see these conceptions before them in time present.
[164]
 
Chapter XIX

§25

Augustine now asks how it is that God teaches the prophets of things in the future. [165]


Chapter XX

§26

So there are three times, past, present, and future, but only insofar as they are given in the present. It is more accurate to speak of a time present of things past; a time present of things present; and a time present of things future.
But even now it is manifest and clear that there are neither times future nor times past. Thus it is not properly said that there are three times, past, present, and future. Perhaps it might be said rightly that there are three times: a time present of things past; a time present of things present; and a time present of things future. For these three do coexist somehow in the soul, for otherwise I could not see them. The time present of things past is memory; the time present of things present is direct experience; the time present of things future is expectation. If we are allowed to speak of these things so, I see three times, and I grant that there are three. Let it still be said, then, as our misapplied custom has it: “There are three times, past, present, and future.” I shall not be troubled by it, nor argue, nor object-- always provided that what is said is understood, so that neither the future nor the past is said to exist now. There are but few things about which we speak properly-- and many more about which we speak improperly--though we understand one another’s meaning.
[165]
 
Chapter XXI

§27

We measure time, like from the past to know, as though it had extension. But we just said it does not have extension.
I have said, then, that we measure periods of time as they pass so that we can say that this time is twice as long as that one or that this is just as long as that, and so on for the other fractions of time which we can count by measuring.
So, then, as I was saying, we measure periods of time as they pass. And if anyone asks me, “How do you know this?”, I can answer: “I know because we measure. We could not measure things that do not exist, and things past and future do not exist.” But how do we measure present time since it has no extension? It is measured while it passes, but when it has passed it is not measured; for then there is nothing that could be measured. But whence, and how, and whither does it pass while it is being measured? Whence, but from the future? Which way, save through the present? Whither, but into the past? Therefore, from what is not yet, through what has no length, it passes into what is now no longer. But what do we measure, unless it is a time of some length? For we cannot speak of single, and double, and triple, and equal, and all the other ways in which we speak of time, except in terms of the length of the periods of time. But in what “length,” then, do we measure passing time? Is it in the future, from which it passes over? But what does not yet exist cannot be measured. Or, is it in the present, through which it passes? But | what has no length we cannot measure. Or is it in the past into which it passes? But what is no longer we cannot measure. [165-166]
 
Chapter XXII

§28

Our assessments of measured time have meaning, but how so given that time has no extension? Augustine pleads with God for this knowledge.
 
Chapter XXIII

§29

Some say that the motion of the heavenly bodies constitute time. But we can have ongoing cycles without them. And while they might measure time, we cannot thereby equate the two.

§30

There is an absolute sort of time, such that if the sun stopped moving, we could still count the ‘days’ that it has not moved, since it measures time and does not constitute its passage. So it seems time in a way is an extension, but how?
I see, then, that time is a certain kind of extension. But do I see it, or do I only seem to? Thou, O Light and Truth, wilt show me.
[167]
 
Chapter XXIV

§31

Some say that motion is time. But we measure also how long something is at rest.
 
Chapter XXV

§32

Augustine confesses his ignorance of how we measure time when time has no extension, pleading for God to enlighten him on the matter.
 
Chapter XXVI

§33

Augustine restates the question. How can we say that one duration is twice as long as another if neither extends and if both no longer exist?
 
Chapter XXVII

§34

Augustine again restate the problem. We measure an event after it happened. But it no longer exists.

§35

Augustine illustrates this problem with a phrase. It has short and long syllables. How is it that we measured them?

§36

The future is constantly moving into the past, making the past larger and the future shorter. “The past increases by the diminution of the future until by the consumption of all the future all is past.” [170]
 
Chapter XXVIII

§37

future time, which is nonexistent, is not long; but “a long future” is “a long expectation of the future.” Nor is time past, which is now no longer, long; a “long past” is “a long memory of the past.”
[170]

§38

Augustine now describes what happens when he repeats a Psalm. It is in his memory. And he anticipates saying it in the future. The future then carries over into the past, and yet his attention remained in the present the whole time. [171]
 
Chapter XXIX

§39

Augustine finds inspiration in God. [Perhaps the point here is that because God is eternal, he is always before us.]
Thus through him I may lay hold upon him in whom I am also laid hold upon; and I may be gathered up from my old way of life to follow that One and to forget that which is behind, no longer stretched out but now pulled together again--stretching forth not to what shall be and shall pass away but to those things that are before me.
[171]
 
Chapter XXX

§40

[It seems Augustine is no longer trying to get answers for his questions but is instead interesting in focusing on the divine reality of eternal time, which does not present these problems.]
 
Chapter XXXI

§41

[It seems again that the idea is that for eternal God, all times are one. Perhaps the overall message is that time as we experience presents us with mysteries whose inexplicability results from the fact that time was caused by something eternal, and perhaps for that reason it is an odd mixture of eternal impermanence and momentary change. What is most important is that we take these mysteries of the time we experience as pointing to the majesty of the eternal creator. By meditating on the unsolvable mysteries of time, our attention and appreciation are directed to God.]
For whatever is past and whatever is yet to come would be no more concealed from him than the past and future of that psalm were hidden from me when I was chanting it: how much of it had been sung from the beginning and what and how much still remained till the end. But far be it from thee, O Creator of the universe, and Creator of our souls and bodies--far be it from thee that thou shouldst merely know all things past and future. Far, far more wonderfully, and far more mysteriously thou knowest them. For it is not as the feelings of one singing familiar | songs, or hearing a familiar song in which, because of his expectation of words still to come and his remembrance of those that are past, his feelings are varied and his senses are divided. This is not the way that anything happens to thee, who art unchangeably eternal, that is, the truly eternal Creator of minds. As in the beginning thou knewest both the heaven and the earth without any change in thy knowledge, so thou didst make heaven and earth in their beginnings without any division in thy action. Let him who understands this confess to thee; and let him who does not understand also confess to thee! Oh, exalted as thou art, still the humble in heart are thy dwelling place! For thou liftest them who are cast down and they fall not for whom thou art the Most High.
[172]
 
 
Augustine. Confessions. Ed. and Trans. Albert C. Outler. First published MCMLV Library of Congress Catalog Card Number: 55-5021 This book is in the public domain. It was scanned from an uncopyrighted edition. Available online here:
http://www9.georgetown.edu/faculty/jod/augustine/conf.pdf
http://www.ling.upenn.edu/courses/hum100/augustinconf.pdf
 
 
 
 
 

Augustine, entry directory


by Corry Shores
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Entry Directory for Augustine



Augustine



Confessions

Augustine on Time, Confessions Book 11, summary




Heywood & Zihl (1999) Case Study of L.M.’s Inability to Perceive Motion, in their book chapter “Motion Blindness”, summary notes


by
Corry Shores
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Heywood and Zihl


“Motion Blindness”




Brief Summary:

L.M. is a woman who suffered a rare case of brain damage that resulted in her being unable to detect motion. She could see the different positions of moving objects but not the motion between them. Motion in place, like the stream of pouring water, seemed “frozen like a glacier”. I appeal to this case for the phenomenological argument that motion, change, and time itself are differential phenomenon: the phenomenal content is not just the different positions of a moving object, for example, but as well, the difference between them is itself phenomenal content. In other words, time and change phenomenologically speaking are differences and not objects or processes.


Summary


Neurophysiological studies indicate that our vision system has highly specialized parts.

The discovery of an impressive patchwork of cortical visual areas that lies in the extrastriate cortex of the monkey has led to the suggestion that each is relatively specialised for the processing of a particular visual attribute.
[1]

Thus, “the destruction of a single area will result in the disturbance of a single function.” [2a] But there are problems with this view. For example, already we have discovered 30 or so areas of  the macaque monkey brain used for vision, but there are not that many visual attributes which each one might specialize in. Also, there is no good way to make such a correlation [as the are regional variations which complicate such an analysis]. The second problem is that we have only identified a small  number of selective disorders [and so we cannot identify all the possible regions], and thirdly, “surgical removal of a single visual area in the monkey has rarely, if ever, resulted in a deficit that parallels any of the clinical findings.” [2]


Nonetheless, we still have cases of selective disorders that are useful to our scientific investigation into the brain’s visual areas.


There is a particularly important part of the brain for vision (found in the monkey brain). It is the cortical area V5, also known as MT. Research on this area indicated that the visual cortex of monkey brains has highly specialized parts. Regarding this V5 region of the visual cortex, “Neurons in this region are finely tuned to the direction of visual motion and it was promptly referred to as ‘the motion area.’” [2d] And shortly after this discovery there was a case study of a patient (L.M.) “with a relatively selective and profound deficit in the perception of visual motion […] (Zihl, von Cramon, & Mai, 1983)”. [3a]



The Case of L.M.


Patient L.M. had a brain injury that seemed to have disabled her ability to perceive motion (the injury occurred 1978, studies on her case began 1980, and Zihl et al. publish their report 1983). She could see the positions of mobile things but never the movements between those positions.

L.M. reported that looking at objects in motion made her feel quite unwell. The explanation she gave sounded rather odd. She claimed that she no longer saw movement; objects which should move, as she well remembered, now appeared as "restless" or “jumping around". Although she could see objects at different locations and distances, she was unable to find out what happened to them between these locations. She was sure that objects did not move, but appeared as "jumping from one position to the next, but nothing is in between''. Because of' these difficulties she avoided streets, busy places, supermarkets and cafés. Traffic had become very frightening; she could still identify cars without any difficulty but could not tell whether they were moving or stationary. The only w:ay for her to establish this was to wait until the car became either conspicuously bigger or small. However, this turned out to be very complicated, especially when there were other cars in the vicinity. As a consequence, she no longer risked crossing the street except at pedestrian crossings. When people walked nearby, she usually waited until they passed, because the ''restlessness"' they produced by their walking irritated her so much that she had to interrupt her walking to find a "resting point for my eyes". Furthermore, she reported substantial difficulty in pouring fluids into a cup or glass, because the tea, coffee or orange juice appeared "frozen like a glacier'. She could not see the fluid rising, and therefore, couldn't establish when to stop pouring. In addition, she felt very irritated when looking at people while they were speaking: their lips appear to ''hop up and down", so she had to look away so as not to become confused. "'To my friends, this behavior appears very strange if not unkind; they believe that I am no longer interested in their conversation because I am always looking absent-minded. But it is the only way to listen to them without being disturbed". For this reason she had decided no longer to meet her friends.
[3]



Behavior Consequences of L.M.’ Movement Vision Disorder


In this section L.M. is quoted as saying:

“Sometimes I do not even know whether a person is approaching me or is receding.”
[6a]



Neuropsychological Assessment


In this section we learn that since L.M.’s case, there have been a number of other reported cases of deficits in motion perception. However, L.M. is still the case most extensively studied. Her condition has been coined akinetopsia by Zeki (1991)



Cerebral Akinetopsia


There is an effective test for the brain’s motion systems called ‘random dot cinematograms.’

one very effective way of establishing the capacity of the motion system is by testing with a class of visual stimuli known as random dot cinematograms. These are composed of a random display of elements which lack an overall conspicuous form. When some of the dots are spatially displaced during sequential frames of the display, the normal observer effortlessly perceives smooth visual motion. By varying the parameters of the display, such as the density, distributions of direction and distance of the displaced elements, exposure duration and interstimulus interval, the limits of motion vision can be characterised. The displays have the particular advantage that the observer is unable to determine which element in the second exposure corresponds to which element in the second [sic?]. It is therefore impossible to infer the motion from the change in location of individual elements. Processes that extract such motion have been termed “short-range” (Braddick, 1974), in contrast to longer-range processes that extract motion information from displays that contain small numbers of clearly defined elements.

wiki.Random_Dot_Kinematogram_(Elliptical)

[“Random Dot Kinematogram” animated gif from wikimedia commons]

Using this test, it was determined that at certain levels of variation, L.M. could not perceive motion.


Besides being unable to detect motion at a certain level, she was also unable to discriminate other properties of motion such as direction or velocity also at a certain level. “Even at low velocities, L.M. required a twenty-fold increase in contrast, compared with the normal observer, to correctly judge the direction of motion.” [9a] However with other testing it was determined that her deficit was not “in the direction of motion but in making judgments of the attributes of stimulus motion.” [9a]





Heywood, C. A., & Zihl, J. (1999). Motion blindness. In G. W. Humphreys  (Ed.), Case Studies in the Neuropsychology of Vision (pp. 1-16). Hove: Psychology Press.
Summary based on limited preview at google books:
http://books.google.com.tr/books?id=YyahncBD_FMC&printsec=frontcover


Random Dot Kinematogram animated gif from:
Wikipedia commons
Random Dot Kinematogram (Elliptical).gif
http://commons.wikimedia.org/wiki/File:Random_Dot_Kinematogram_%28Elliptical%29.gif



4 May 2014

Russell, Ch.40 of Principles of Mathematics, ‘The Infinitesimal and the Improper Infinite’, summary notes

 

by Corry Shores
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[The following is summary and quotation. All boldface, underlining, and bracketed commentary are mine. Please see the original text, as I did not follow it closely. Proofreading is incomplete, so mistakes are still present.]

 


 

Bertrand Russell


Principles of Mathematics


Part 5: Infinity and Continuity


Ch.40: The Infinitesimal and the Improper Infinite





Brief Summary:

The infinitesimal was once an important concept in mathematics, especially for understanding continuity. Now that we have Cantor’s more precise definition of infinity, we find that the infinitesimal is found only in very special cases and it has not usefulness in mathematics anymore. Something can be infinitesimal with regard to something much greater than it. For example the side of a square is infinitesimal in relation to its area. However, mathematics considers these two sorts of magnitude as of different kinds and as being incomparable. This is the only actual instance of infinitesimals, and it has no mathematical importance. Infinitesimals were traditional understood however as absolute and not as relative as in this case. Russell shows that an absolute infinitesimal cannot exist. For example, if we divide a segment more and more, we keep getting finite valued parts, which can be summed to obtain the value of the whole. But if the parts get below the finite, then they can no longer be added to obtain a finite value. If we add an infinitely long segment to another, we do not increase its cardinal value. It will be infinite. Likewise, if we add one infinitesimal to another, it will also not become finite. Thus, a finite segment cannot be made of infinitesimals. Hence a magnitude could not be absolutely infinitesimal.

 



Summary

 

§309


Until recently (ca. 1900), continuity was understood by means of the concept of the infinitesimal. But now that concept has been abandoned. [336]


The infinitesimal has been given certain senses, but none have been mathematically precise. It is for example the distance between a point and its immediate neighbor. But we now know that there is no such thing.

The infinitesimal has, in general, been very vaguely defined. It has been regarded as a number or magnitude which, though not zero, is less than any finite number or magnitude. It has been the dx or dy of the Calculus, the time during which a ball thrown vertically upwards is at rest at the highest point of its course, the distance between a point on a line and the next point, etc., etc. But none of these notions are at all precise. The dx and dy, as we saw in the last chapter, are nothing at all: dy/dx is the limit of a fraction whose numerator and denominator are finite, but is not itself a fraction at all. The time during which a ball is at rest at its highest point is a very complex notion, involving the whole philosophic theory of motion; in Part VII we shall find, when this theory has been developed, that there is no such time. The distance between consecutive points presupposes that there are consecutive points—a view which there is every reason to deny. And so with most instances—they afford no precise definition of what is meant by the infinitesimal.
[336]



§310


[We should first examine the axiom of Archimedes. We want to know if two values are finite in relation to one another or infinite in relation to one another. Consider values 4 and 6. We can multiply 4 by 2 and get 8, which is larger than 6. This means they are finite in relation to one another, or their difference in value is finite. Now consider 4 and the cardinal value for the natural numbers. Or let’s just say, consider 4 and infinity. There is no finite number that we can multiply 4 by in order to obtain a number greater than infinity. That means they are infinite in relation to one another, or their difference is infinite. So the first example illustrates relative finitude. Absolute finitude would require some anchoring points you say, 0 and 1, and as well a principle of composing finite numbers, namely mathematical induction, the successor function. As we can see, the notion of relative finitude applies to any kind of magnitude, but absolute infinity has more limited application to numbers, classes and divisibilities. And also note that an inch and a foot both are magnitudes consisting of an infinity of terms (leading up to their total value, all the sizes smaller than an inch that are implicitly contained within it). So both an inch and a foot are absolute infinities. However, they are finite in relation to one another and are thus relative finitudes. So “any two numbers, classes, or divisibilities, which are both absolutely finite are also relatively finite; but the converse does not hold”.]

There is, so far as I know, only one precise definition, which renders the infinitesimal a purely relative notion, correlative to something arbitrarily assumed to be finite. When, instead, we regard what had been taken to be infinitesimal as finite, the correlative notion is what Cantor calls the improper infinite (Uneigentlich-Unendliches). The definition of the relation in question is obtained by denying the axiom of Archimedes, just as the transfinite was obtained by denying mathematical induction. If P, Q be any two numbers, or any two measurable magnitudes, they are said to be finite with respect to each other when, if P be the lesser, there exists a finite integer n such that nP is greater than Q. The existence of such an integer constitutes the axiom of Archimedes and the definition of relative finitude. It will be observed that it presupposes the definition of absolute finitude among numbers—a definition which, as we have seen, depends upon two points, (1) the connection of 1 with the logical notion of simplicity, or of 0 with the logical notion of the null-class; (2) the principle of mathematical induction. The notion of relative finitude is plainly distinct from that of absolute finitude. The latter applies only to numbers, classes and divisibilities, whereas the former applies to any kind of measurable magnitude. Any two numbers, classes, or divisibilities, which are both absolutely finite are also relatively finite; but the converse does not hold. For example, ω and ω.2, an inch and a foot, a day and a year, are relatively finite pairs, though all three consist of terms which are absolutely infinite.
[337]


[Russell will now definite the infinitesimal and improper infinite. Consider 2 values. If no matter what finite value we multiply one by that it can in no case be greater than the other, then this term is infinitesimal or improperly infinite. This can only apply to numbers and not magnitudes.]

The definition of the infinitesimal and the improper infinite is then as follows. If P, Q be two numbers, or two measurable magnitudes of the same kind, and if, n being any finite integer whatever, nP is always less than Q, then P is infinitesimal with respect to Q, and Q is infinite with respect to P. With regard to numbers, these relative terms are not required; for if, in the case supposed, P is absolutely finite, then Q is absolutely infinite; while if it were possible for Q to be absolutely finite, P would be absolutely infinitesimal—a case, however, which we shall see reason to regard as impossible. Hence I shall assume in future that P and Q are not numbers, but are magnitudes of a kind of which some, at least, are numerically measurable. It should be observed that, as regards magnitudes, the axiom of Archimedes is the only way of defining, not only the infinitesimal, but the infinite also. Of a magnitude not numerically measurable, there is nothing to be said except that it is greater than some of its kind, and less than others; but from such propositions infinity cannot be obtained. Even if there be a magnitude greater than all others of its kind, there is no reason for regarding it as infinite. Finitude and infinity are essentially numerical notions, and it is only by relation to numbers that these terms can be applied to other entities.
[337]



§311


[Russell will now consider instances of infinitesimal values. We first consider divisible magnitudes. If we compare something with a finite number of parts to one with an infinite number, than the first is infinitesimal in relation to it. But we cannot compare such magnitudes on the basis of placing into a ration the cardinal numbers of their parts. Russell gives two reasons. The first is that we cannot place transfinite values into ratios (his explanation begins with saying we cannot place two transfinite cardinals into ratios. His example is of a finite and a transfinite. So for some reason it still applies in this other case). He second reason is equally unclear, but it seems he is saying that in order to make our original comparison, the divisibilities of each magnitude must be equal, but that is not the case for the transfinite value for some reason. Here is the text:]

The next question to be discussed is: What instances of infinitesimals are to be found? Although there are far fewer instances than was formerly | supposed, there are yet some that are important. To begin with, if we have been right in regarding divisibility as a magnitude, it is plain that the divisibility of any whole containing a finite number of simple parts is infinitesimal as compared with one containing an infinite number. The number of parts being taken as the measure, every infinite whole will be greater than n times every finite whole, whatever finite number n may be. This is therefore a perfectly clear instance. But it must not be supposed that the ratio of the divisibilities of two wholes, of which one at least is transfinite, can be measured by the ratio of the cardinal numbers of their simple parts. There are two reasons why this cannot be done. The first is, that two transfinite cardinals do not have any relation strictly analogous to ratio; indeed, the definition of ratio is effected by means of mathematical induction. The relation of two transfinite cardinals α, γ expressed by the equation αβ = γ bears a certain resemblance to integral ratios, and αβ =γδ may be used to define other ratios. But ratios so defined are not very similar to finite ratios. The other reason why infinite divisibilities must not be measured by transfinite numbers is, that the whole must always have more divisibility than the part (provided the remaining part is not relatively infinitesimal), though it may have the same transfinite number. In short, divisibilities, like ordinals, are equal, so long as the wholes are finite, when and only when the cardinal numbers of the wholes are the same; but the notion of magnitude of divisibility is distinct from that of cardinal number, and separates itself visibly as soon as we come to infinite wholes.
[337-338]


We can even have examples where one thing is infinitely less divisible than another, as for example a line compared to a square. [This is an example of an infinitesimal. But it seems Russell is saying that they are just relative infinitesimals and not the kind we are more concerned with, like in the infinitesimal calculus.]

Two infinite wholes may be such that one is infinitely less divisible than the other. Consider, for example, the length of a finite straight line and the area of the square upon that straight line; or the length of a finite straight line and the length of the whole straight line of which it forms part (except in finite spaces); or an area and a volume; or the rational numbers and the real numbers; or the collection of points on a finite part of a line obtainable by von Staudt’s quadrilateral construction, and the total collection of points on the said finite part.* All these are magnitudes of one and the same kind, namely divisibilities, and all are infinite divisibilities; but they are of many different orders. The points on a limited portion of a line obtainable by the quadrilateral construction form a collection which is infinitesimal with respect to the said portion; this portion is ordinally infinitesimal† with respect to any bounded area; any bounded area is ordinally infinitesimal with respect to any bounded volume; and any bounded volume (except in finite spaces) is ordinally infinitesimal with respect to all space. In all these cases, the word infinitesimal is used strictly according to the above definition, obtained from the axiom of Archimedes. What makes these various | infinitesimals somewhat unimportant, from a mathematical standpoint, is, that measurement essentially depends upon the axiom of Archimedes, and cannot, in general, be extended by means of transfinite numbers, for the reasons which have just been explained. Hence two divisibilities, of which one is infinitesimal with respect to the other, are regarded usually as different kinds of magnitude; and to regard them as of the same kind gives no advantage save philosophic correctness. All of them, however, are strictly instances of infinitesimals, and the series of them well illustrates the relativity of the term infinitesimal.
[338-339]


[Russell examines another example of comparing magnitudes divided infinitely. It is not clear to me, but it seems to be saying that if the divisions of a magnitude get smaller than the finite, then if we add up all their values, it will be 0. But please read it for yourself to decide what it means.]

An interesting method of comparing certain magnitudes, analogous to the divisibilities of any infinite collections of points, with those of continuous stretches is given by Stolz,* and a very similar but more general method is given by Cantor.† These methods are too mathematical to be fully explained here, but the gist of Stolz’s method may be briefly explained. Let a collection of points x' be contained in some finite interval a to b. Divide the interval into any number n of parts, and divide each of these parts again into any number of parts, and so on; and let the successive divisions be so effected that all parts become in time less than any assigned number δ. At each stage, add together all the parts that contain points of x' . At the mth stage, let the resulting sum be Sm. Then subsequent divisions may diminish this sum, but cannot increase it. Hence as the number of divisions increases, Sm must approach a limit L. If x' is compact throughout the interval, we shall have L = b − a; if any finite derivative of x' vanishes, L = 0. L obviously bears an analogy to a definite integral; but no conditions are required for the existence of L. But L cannot be identified with the divisibility; for some compact series, e.g. that of rationals, are less divisible than others, e.g. the continuum, but give the same value of L.
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§312


[Normally we think of the infinitesimal as composing a dense or ‘compact’ series. For, if all its parts were finite and there are infinitely many, than the whole segment would be infinite. If the parts were 0, then it would have 0 value. But if there were infinitely many infinitely small part, then those infinitive values would cancel one another generating a finite value. Russell will show that either it is impossible for the parts to be infinitesimal or at least that if they were, they would be indefinable. First he establishes that any segment is infinitely divisible, because between any two values is another. Next, he explains that segments can be added by placing one at the end of the other, which increases the total magnitude. If the added segments are equal, the new total will be double. Segments without terminal endings included in them (where they tend toward limits without attaining them), we can add them by adding such terminal segments. So we can define any finite multiple of segments (by adding them). For some reason, it seems we will draw these conclusions: if a smaller segment obeys the axiom of Archimedes with respect to the larger (if no matter how many times we multiply it, it will not be greater than the larger), then the larger will contain all the terms coming after the smaller. However, if the smaller is infinitesimal with respect to larger ones, then the larger one will not contain points of the first segment. (This is too unclear for me to understand). (It seems now we are working with the idea that an infinite segment cannot be increased by doubling it. Only terminating segments can.) Thus the larger segment is not terminating. On account of this, for some reason Peano concludes that the larger segment cannot be an element in finite magnitudes. Russell draws a stronger conclusion. an infinitesimal cannot have determinate bounds. So it cannot be added so to produce larger segments. Consult the original text:]

The case in which infinitesimals were formerly supposed to be peculiarly evident is that of compact series. In this case, however, it is possible to prove that there can be no infinitesimal segments,‡ provided numerical measurement be possible at all—and if it be not possible, the infinitesimal, as we have seen, is not definable. In the first place, it is evident that the segment contained between two different terms is always infinitely divisible; for since there is a term c between any two a and b, there is another d between a and c, and so on. Thus no terminated segment can contain a finite number of terms. But segments defined by a class of terms may (as we saw in Chapter 34) have no limiting term. In this case, however, provided the segment does not consist of a single term a, it will contain some other term b, and therefore an infinite number of terms. Thus all segments are infinitely divisible. The next | point is to define multiples of segments. Two terminated segments can be added by placing a segment equal to the one at the end of the other to form a new segment; and if the two were equal, the new one is said to be double of each of them. But if the two segments are not terminated, this process cannot be employed. Their sum, in this case, is defined by Professor Peano as the logical sum of all the segments obtained by adding two terminated segments contained respectively in the two segments to be added.* Having defined this sum, we can define any finite multiple of a segment. Hence we can define the class of terms contained in some finite multiple of our segment, i.e. the logical sum of all its finite multiples. If, with respect to all greater segments, our segment obeys the axiom of Archimedes, then this new class will contain all terms that come after the origin of our segment. But if our segment be infinitesimal with respect to any other segment, then the class in question will fail to contain some points of this other segment. In this case, it is shown that all transfinite multiples of our segment are equal to each other. Hence it follows that the class formed by the logical sum of all finite multiples of our segment, which may be called the infinite multiple of our segment, must be a non-terminated segment, for a terminated segment is always increased by being doubled. “Each of these results”, so Professor Peano concludes, “is in contradiction with the usual notion of a segment. And from the fact that the infinitesimal segment cannot be rendered finite by means of any actually infinite multiplication, I conclude, with Cantor, that it cannot be an element in finite magnitudes” (p. 62). But I think an even stronger conclusion is warranted. For we have seen that, in compact series, there is, corresponding to every segment, a segment of segments, and that this is always terminated by its defining segment; further that the numerical measurement of segments of segments is exactly the same as that of simple segments; whence, by applying the above result to segments of segments, we obtain a definite contradiction, since none of them can be unterminated, and an infinitesimal one cannot be terminated.
[339-340]


[Next Russell will argue that rational and real numbers cannot be made of infinitesimals. He seems to be saying that the real numbers are made of rational numbers. The real numbers are a class of real numbers. So any member of them will as well contain rational numbers, no matter how small. But an infinitesimally small term does not contain with in any rational numbers, because it is too small. Hence the real numbers cannot be made of infinitesimals. He might very well be saying something else, so please consider the original:]

In the case of the rational or the real numbers, the complete knowledge which we possess concerning them renders the non-existence of infinitesimals demonstrable. A rational number is the ratio of two finite integers, and any such ratio is finite. A real number other than zero is a segment of the series of rationals; hence if x be a real number other than zero, there is a class u, not null, of rationals such that, if y is a u, and z is less than y, z is an x, i.e. belongs to the segment which is x. Hence every real number other than zero is a class containing rationals, and all rationals are finite; consequently every real number is finite. Consequently if it were possible, in any sense, to speak of infinitesimal numbers, it would have to be in some radically new sense.
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§313


[Russell now examines an interesting question regarding orders of infinity and infinitesimality of functions. Russell does not draw any conclusions (although he seems to want at the end to say that this material supports the notion that infinitesimals are mathematical fictions), so I will just place the very technical material below:]

I come now to a very difficult question, on which I would gladly say nothing—I mean, the question of the orders of infinity and infinitesimality of functions. On this question the greatest authorities are divided: Du Bois Reymond, Stolz, and many others, maintaining that these form a special class of magnitudes, in which actual infinitesimals occur, while Cantor holds strongly that the whole theory is erroneous. To put the matter as simply as possible, consider a function f(x) whose limit, as x approaches zero, is zero. It may happen that, for some finite real number α, the ratio f(x)/xα has a finite limit as x approaches zero. There can be only one such number, but there may be none. Then α, if there is such a number, may be called the order to which f(x) becomes infinitesimal, or the order of smallness of f(x) as x approaches zero. But for some functions, e.g. 1/log x, there is no such number α. If α be any finite real number, the limit of 1/xα logx, as x approaches zero, is infinite. That is, when x is sufficiently small, 1/xα log x is very large, and may be made larger than any assigned number by making x sufficiently small—and this whatever finite number α may be. Hence, to express the order of smallness of 1/log x, it is necessary to invent a new infinitesimal number, which may be denoted by 1/g. Similarly we shall need infinitely great numbers to express the order of smallness of (say) e−1/x as x approaches zero. And there is no end to the succession of these orders of smallness: that of 1/log (log x), for example, is infinitely smaller than that of 1/log x, and so on. Thus we have a whole hierarchy of magnitudes, of which all in any one class are infinitesimal with respect to all in any higher class, and of which one class only is formed of all the finite real numbers.

In this development, Cantor finds a vicious circle; and though the question is difficult, it would seem that Cantor is in the right. He objects (loc. cit.) that such magnitudes cannot be introduced unless we have reason to think that there are such magnitudes. The point is similar to that concerning limits; and Cantor maintains that, in the present case, definite contradictions may be proved concerning the supposed infinitesimals. If there were infinitesimal numbers j, then even for them we should have

Limx = 0 1/ (log x. xj) = 0

since xj must ultimately exceed ½. And he shows that even continuous, differentiable and uniformly growing functions may have an entirely ambiguous order of smallness or infinity: that, in fact, for some such functions, this order oscillates between infinite and infinitesimal values, according to the manner in which the limit is approached. Hence we may, I think, conclude that these | infinitesimals are mathematical fictions. And this may be reinforced by the consideration that, if there were infinitesimal numbers, there would be infinitesimal segments of the number-continuum, which we have just seen to be impossible.
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§314


[Russell now summarizes. He has shown that the infinitesimal can never be anything but a relative term. When it does have an absolute meaning, it is indistinguishable from finitude (perhaps this is from the idea that an infinitely small segment cannot be increased by doubling it, so were a segment made of infinitesimals, they would have to have the properties of finite magnitudes.) There are cases of infinitesimals, for example the side of a square is infinitesimal compared with its area. But mathematicians consider each magnitude as different in kind and thus incomparable. We also saw that compact (dense) series cannot be made of infinitesimals. Thus the infinitesimal has not many important manifestations and it is not important mathematically.]

Thus to sum up what has been said concerning the infinitesimal, we see, to begin with, that it is a relative term, and that, as regards magnitudes other than divisibilities, or divisibilities of wholes which are infinite in the absolute sense, it is not capable of being other than a relative term. But where it has an absolute meaning, there this meaning is indistinguishable from finitude. We saw that the infinitesimal, though completely useless in mathematics, does occur in certain instances—for example, lengths of bounded straight lines are infinitesimal as compared to areas of polygons, and these again as compared to volumes of polyhedra. But such genuine cases of infinitesimals, as we saw, are always regarded by mathematics as magnitudes of another kind, because no numerical comparison is possible, even by means of transfinite numbers, between an area and a length, or a volume and an area. Numerical measurement, in fact, is wholly dependent upon the axiom of Archimedes, and cannot be extended as Cantor has extended numbers. And finally we saw that there are no infinitesimal segments in compact series, and—what is closely connected—that orders of smallness of functions are not to be regarded as genuine infinitesimals. The infinitesimal, therefore—so we may conclude—is a very restricted and mathematically very unimportant conception, of which infinity and continuity are alike independent.
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Sources [unless otherwise noted, all bracket page citations are from]:

Bertrand Russell. Principles of Mathematics. London/New York: Routledge, 2010 [1st published 1903].