Russell ‘Mathematics and the Metaphysicians’, summary [At-At theory of motion to solve Zeno’s Paradoxes]

 

by Corry Shores
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[Russell also gives the at-at account of motion in “Principles of Mathematics”]

Bertrand Russell’s “At-At” Theory of Motion,

His Critique of the Infinitesimal,

and His Solution to Zeno’s Paradoxes

 

 

in

 

Bertrand Russell

“Mathematics and the Metaphysicians”

Very brief summary:

Mathematics’ development was hindered from its dependence on notions of quantity. It advanced as soon as quantity was replaced with terms in sets whose properties can be deduced logically from axioms. One result of this is the rejection of the infinitesimal, which can only be understood quantitatively [according to Russell]. Without the infinitesimal, an object in motion can only be thought of as being in one place at one time. This is his ‘at-at’ account of motion. Between each temporal and spatial point there are still more, no matter how near they may be. This means no immediately consecutive point follows any other, and thus moving objects are never between points of space and time or spanning across more than one. [This is contrary to Leibniz’ conceptions: “One point of a moving body at the time of conatus, or in a time less than any assignable time, is in many places or points of space.” (Leibniz 140c)] But although there is no infinitesimal, there is still Cantor’s infinite set, whose subsets are no fewer than the whole. When Achilles reaches the Tortoise, the Tortoise is already advancing and creating an infinity more points for Achilles to cross. However, Achilles can go a greater length to overtake the Tortoise, but still only cross the same number of points, an infinite number of the them. Mathematics should continue using mathematical logic so to advance even further.

Brief summary:

Mathematics is now founded on logic. It’s propositions are deduced rigorously from simple axioms. This has cleared up some confusions and it has even helped solve Zeno’s paradoxes. Weierstrass banished the infinitesimal from calculus. So there are only cuts in a continuum but no infinitely small divisions. This means that for a continuum like time, there is no ‘next instant’. If there is no infinitesimal, then between any two points there are always an infinity more. This means you can never have consecutive points or moments. This also means that a moving object can never be between points or in two points at the same time. This is his ‘at-at’ account of motion: “When a body moves, all that can be said is that it is in one place at one time and in another at another. We must not say that it will be in a neighbouring place at the next instant, since there is no next instant. [...] a body in motion is just as truly where it is as a body at rest. Motion consists merely in the fact that bodies are sometimes in one place and sometimes in another, and that they are at intermediate places at intermediate times” [84] But while Russell rejects the concept of the infinitesimal, he still thinks there is a legitimate concept of the infinite, as it is defined by Cantor: “A collection of terms is infinite when it contains as parts other collections which have just as many terms as it has. If you can take away some of the terms of a collection, without diminishing the number of terms, then there are an infinite number of terms in the collection. For example, there are just as many even numbers as there are numbers altogether, since every number can be doubled. This may be seen by putting odd and even numbers together in one row, and even numbers alone in a row below.” [86] [It seems strange that Russell would accept the existence and conceptual viability of the infinite, but reject the infinitesimal, which is merely the inverse of the infinite. He later says “the smallest of the infinite integers is the limit of the finite integers, though all finite integers are at an infinite distance from it.” (92) Here already we might derive the concept of the infinitesimal. (It also contradicts his claim that infinity has nothing to do with magnitude or size, because here we have an infinite ‘distance’ between a finite and an infinite number). To define the infinitesimal, we would just borrow from the concepts already laid out in this quoted claim above. The set of infinitesimals are the set of all “infinite integers” divided by 1. (And since all infinite integers are at an infinite distance from finite ones, that means all infinitesimal figures are infinite reductions of finite ones, just like in Leibniz’ infinitesimal calculus.) So long as his mathematical philosophy includes the notion of infinite integers (whose ‘distance’ from the finite is immeasurably great), and also, so long as he allows for division and the unit 1 (which surely he does), then the concept of the infinitesimal is already inherent to his system, and it is easily deducible from the simplest of its concepts.] Mathematics has made great progress recently by rejecting the notions of quantity and space in favor of terms in sets and logical algebra. Mathematics should now focus even more on mathematical logic to continue its progress.

Summary

Pure mathematics was an invention of the 19th century.

It was discovered by Boole (Laws of Thought, 1854). Yet,

His book was in fact concerned with formal logic, and this is the same thing as mathematics.
[74]

Pure mathematics makes assertions about generalities that thereby apply to any particular case.

Pure mathematics consists entirely of assertions to the effect that, if such and such a proposition is true of anything, then such and such another proposition is true of that thing.
[75a]

Applied mathematics is more concerned with whether the assertions are true and what specific cases they might apply to.

We start, in pure mathematics, from certain rules of inference, by which we can infer that if one proposition is true, then so is some other proposition.
[75]

Formal logic does something similar. But whenever the hypotheses of logic are thought to apply to any possible case, then they are pure mathematics.

If our hypothesis is about anything, and not about some one or more particular things, then our deductions constitute mathematics. Thus mathematics may be defined as the subject in which we never know what we are talking about, nor whether what we are saying is true.
[75]

In pure mathematics, there are no indefinables and indemonstrables, except those belonging to general logic [on which the pure mathematics is expressed].

All pure mathematics Arithmetic, Analysis, | and Geometry is built up by combinations of the primitive ideas of logic, and its propositions are deduced from the general axioms of logic, such as the syllogism and the other rules of inference.
[75-76]

Formal logic is now understood as identical with mathematics. It was invented by Aristotle, and besides theology, it was the primary study of the Middle Ages. But there were not too great advances in these times, because Aristotle did go much beyond the syllogism, and the Middle Age scholars did not go too much beyond Aristotle. Yet from 1850 on, there have been remarkable advances in formal logic. It has been given a symbolic form which allows for deductions to be computed much like in Algebra. [76]

Before symbolization, many things seemed obvious and self-evident, and thus it was difficult to know what is being deduced from what else. Symbolization lays these logical relations bare.

Now, in the beginnings, everything is self-evident; and it is very hard to see whether one self-evident proposition follows from another or not. Obviousness is always the enemy to correctness. Hence we invent some new and difficult symbolism, in which nothing seems obvious. Then we set up certain rules for operating on the symbols, and the whole thing becomes mechanical. In this way we find out what must be taken as premiss and what can be demonstrated or defined. For instance, the whole of Arithmetic and Algebra has been shown to require three indefinable notions and five indemonstrable propositions. But without a symbolism it would have been very hard to find this out. It is so obvious that two and two are four, that we can hardly make ourselves sufficiently sceptical to doubt whether it can be proved.
[77]

One useful result of critically evaluating how obvious notions are deduced is that we may learn they are false. For example, we once thought that a number is increased by adding 1 to it. But infinite numbers do not increase with the addition of 1. [78]

Peano, for example, explains the basis of arithmetic symbolically using just the concepts on which all others are derived: 0, number, and next after [successor].

Leibniz dreamed of something similar, a universal characteristic [a universal logical calculus]. By means of it, any dispute could be resolved by taking to pen and paper and computing the truth. Mathematicians have accomplished something similar. This means that old debates involving mathematic notions (like infinity) can now be resolved more conclusively.

Hence many of the topics which used to be placed among the great mysteries for example, the natures of infinity, of continuity, of space, time and motion are now no longer in any degree open to doubt or discussion. Those who wish to know the nature of these things need only read the works of such men as Peano or Georg Cantor; they will there find exact and indubitable expositions of all these quondam mysteries. [80]

We can find examples of old disputes that are now solvable in the paradoxes of Zeno of Elea. These paradoxes supposedly prove that motion is impossible. Weierstrass eliminates the mathematically imprecise concept of infinitesimals from the calculus. If we apply his conceptions to the paradoxes, then we obtain a picture of motion as being made of a series of rests. The mistake is assuming that because everything is only ever at rest that the world would have to be unchanging.

Weierstrass, by strictly | banishing from mathematics the use of infinitesimals, has at last shown that we live in an unchanging world, and that the arrow in its flight is truly at rest. Zeno s only error lay in inferring (if he did infer) that, because there is no such thing as a state of change, therefore the world is in the same state at any one time as at any other. This is a consequence which by no means follows; and in this respect, the German mathematician is more constructive than the ingenious Greek. Weierstrass has been able, by embodying his views in mathematics, where familiarity with truth eliminates the vulgar prejudices of common sense, to invest Zeno’s paradoxes with the respectable air of platitudes; and if the result is less delightful to the lover of reason than Zeno s bold defiance, it is at any rate more calculated to appease the mass of academic mankind.
[80|81]

Zeno was concerned with three problems: the problems of (1) the infinitesimal, (2) the infinite, and (3) continuity. In recent times, Weierstrass, Dedekind, and Cantor have solved these three problems. [81]

The Greeks had a notion of the infinitesimal, because the they thought of a circle being infinitesimally different from a polygon with infinite sides. [see for example Archimedes’ quadrature of the parabola, and discussion on in Katz and Sherry] Leibniz then used it for his infinitesimal calculus. Yet, the concept of the infinitesimal in mathematics is problematic, because it is not conceptually clear or precise. It is neither 0 nor an assignable fraction, yet it enjoys the privileges of both types of number in the calculations which use it. [see Leibniz’ explanation for this ambiguous dual status in his Cum Prodiisset, on the basis of the Law of Continuity.]

The Calculus required continuity, and continuity was supposed to require the infinitely little; but nobody could discover what the infinitely little might be. It was plainly not quite zero, because a sufficiently large number of infinitesimals, added together, were seen to make up a finite whole. But nobody could point out any fraction which was not zero, and yet not finite. Thus there was a deadlock.
[82]

But Weierstrass [by formalizing the concept of limits] did away with the infinitesimal and placed the calculus on rigorous sturdy foundations.

But at last Weierstrass discovered that the infinitesimal was not needed at all, and that everything could be accomplished without it. Thus there was no longer any need to suppose that there was such a thing.
[82]

Russell then notes a consequence of the banishment of infinitesimals from the calculus. It has to do with how time is composed. With the infinitesimal, you could have one moment and the next be cogiven because only an infinitesimally small amount of time would separate them. But without the infinitesimal, then no two moments can be consecutive, because between any two would always be another [and they would never terminate to an infinitely small difference, even though there are an infinity of such moments within any finite duration. This seems to be similar to the actual infinity/potential infinity distinction. With the concept of the infinitesimal, there is an actual infinity of moments, which means there really are already there an infinity of them, each infinitesimally apart from their neighbors. But without a smallest division, then infinity is potential. Each division produces more to be divided.] Adding this inexhaustible element would seem to create difficulties for solving Zeno’s paradoxes, but Russell will show it does not. [83]

The banishment of the infinitesimal has all sorts of odd consequences, to which one has to become gradually accustomed. For example, there is no such thing as the next moment. The interval between one moment and the next would have to be infinitesimal, since, if we take two moments with a finite interval between them, there are always other moments in the interval. Thus if there are to be no infinitesimals, no two moments are quite consecutive, but there are always other moments between any two. Hence there must be an infinite number of moments between any two; because if there were a finite number one would be nearest the first of the two moments, and therefore next to it. This might be thought to be a difficulty; but, as a matter of fact, it is here that the philosophy of the infinite comes in, and makes all straight.
[83]

Russell continues in the same vein:

The same sort of thing happens in space. If any piece of matter be cut in two, and then each part be halved, and so on, the bits will become smaller and smaller, and can theoretically be made as small as we please. However small they may be, they can still be cut up and made smaller still. But they will always have some finite size, however small they may be. We never reach the infinitesimal in this way, and no finite number of divisions will bring us to points. Nevertheless there are points, only these are not to be reached by successive divisions. Here again, the philosophy of the infinite shows us how this is possible, and why points are not infinitesimal lengths.
[83]

[Russell is not very clear here. He says that a finite number of divisions does not result in points. There are still infinitely many points between any two others. He does not clarify what happens when you make not a finite but an infinite amount of divisions. One view using the infinitesimal concept says that an infinity of cuts produces an infinity of points, and between each of which is not more points but rather infinitely small gaps. When Russell says that with each division there is always more to be divided, then he is dealing only with a finite number of divisions. Of course neither an infinitesimal nor a point can be obtained that way. So for one thing he does not explain why it is inconceivable for an infinity of divisions to produce points or infinitesimal intervals between points. Another problem is that if a continuum is inexhaustibly divisible, then he has not solved the Zeno sort of paradoxes. Before going from one point to another, the object needs to go half that distance. Before that, half that distance. But the continuum has inexhaustibly many divisions with no two points being consecutive. Russell’s account will say that the object will be in two locations in two moments. But he still does not explain how it is possible to cross through inexhaustibly many locations in a finite amount of time.]

Russell continues with this idea that because there is no  infinitesimal, that means there is no consecutively next moment. And for this reason, there is no state of motion, which requires a body traversing through consecutive locations. He then proposes what has come to be called Russell’s ‘at-at’ theory of motion. The moving object is at some position at some time. In another moment, it is at some position at some other time. [This would seem to be an account of objects being at rest in different locations, perhaps by moving, then stopping, moving, then stopping. Russell is using this definition to account for continuous motion. We might object and say that he never includes in his account that the object translocates from one position to another. But when taking his assumption of an inexhaustible division of time and space, this is somehow implied. For, if an object is found at time-point 1 (t1) at location-point 1 (p1), then at t2 at p2, it must have occupied all possible locations in between. Perhaps his view of time and space is that they are a dense continuity, which somehow at the basis squishes each location into its neighbors, and thus the object is statically at any point, but is already slipping into its neighbor which is perfectly continuous with itself. But this cannot be, because Russell says there are no neighbors to each point. It seems to be more of a description of movement rather than an explanatory account; for, it does not explain how a still object can translocate if it is not passing through neighboring points. But, it also seems to be fairly resistant to criticism, because if we say, how did the object get from A to B, the answer could be, by occupying every possible position in between. That would seem to be a sufficient account, and because there are no neighboring moments, there is no need to explain the movement from one to the other. However, it still makes it mysterious, as if by some magic the object changes location without moving from one point through its neighbors.] 

As regards motion and change, we get similarly curious results. People used to think that when a thing changes, it must be in a state of change, and that when a thing | moves, it is in a state of motion. This is now known to be a mistake. When a body moves, all that can be said is that it is in one place at one time and in another at another. We must not say that it will be in a neighbouring place at the next instant, since there is no next instant. Philosophers often tell us that when a body is in motion, it changes its position within the instant. To this view Zeno long ago made the fatal retort that every body always is where it is; but a retort so simple and brief was not of the kind to which philosophers are accustomed to give weight, and they have continued down to our own day to repeat the same phrases which roused the Eleatic’s destructive ardour. It was only recently that it became possible to explain motion in detail in accordance with Zeno’s platitude, and in opposition to the philosopher’s paradox. We may now at last indulge the comfortable belief that a body in motion is just as truly where it is as a body at rest. Motion consists merely in the fact that bodies are sometimes in one place and sometimes in another, and that they are at intermediate places at intermediate times. Only those who have waded through the quagmire of philosophic speculation on this subject can realise what a liberation from antique prejudices is involved in this simple and straightforward commonplace.
[83|84]

Russell does not think that there is such a thing as the infinitesimal. But he does think there is infinity. Only recently with Dedekind and Cantor do we have a rigorous definition of infinity. Cantor find that the proofs which are adverse to infinity also involved principles which were destructive to all of mathematics. However, the proofs more favorable to infinity did not have such destructive principles. [85]

Cantor’s definition of infinity does not have destructive consequences in the rest of mathematics:

A collection of terms is infinite when it contains as parts other collections which have just as many terms as it has. If you can take away some of the terms of a collection, without diminishing the number of terms, then there are an infinite number of terms in the collection. For example, there are just as many even numbers as there are numbers altogether, since every number can be doubled. This may be seen by putting odd and even numbers together in one row, and even numbers alone in a row below :–

1, 2, 3, 4, 5, ad infinitum.
2, 4, 6, 8, 10, ad infinitum.

There are obviously just as many numbers in the row below as in the row above, because there is one below for each one above. This property, which was formerly thought to be a contradiction, is now transformed into a harmless definition of infinity, and shows, in the above case, that the number of finite numbers is infinite.
[86]

It might seem strange to talk about a quantity which is too great to be counted. Yet, counting is not necessarily the best way to quantify a value. And also, counting only gives us the ordinal number; “it is impossible to count things without counting some first and others afterwards, so that counting always has to do with order.” [87] Counting can give us the quantity of finite figures, and we can count them in any order we want [front to back or back to front, or eenie meenie miney moe, perhaps: all ways would produce the same number]. But for infinite numbers [for some reason], “what corresponds to counting will give us quite different results according to the way in which we carry out the operation.” [87]

The quantities of infinite numbers, however, are ordinal. We quantify them by comparing their sizes, so to speak.

The fundamental infinite numbers are not ordinal, but are what is called cardinal. They are not obtained by putting our terms in order and counting them, but by a different method, which tells us, to begin with, whether two collections have the same number of terms, or, if not, which is the greater. It does not tell us, in the way in which counting does, what number of terms a collection has ; but if we define a number as the number of terms in such and such a collection, then this method enables us to discover whether some other collection that may be mentioned has more or fewer terms.
[87]

The way infinite sets sizes are compared is by mapping each member of one onto each of the other, which would tell you that both sets have the same size of infinity.

An illustration will show how this is done. If there existed some country in which, for one reason or another, it was impossible to take a census, but in which it was known that every man had a wife and every woman a husband, then (provided polygamy was not a national institution) we should know, without counting, that there were exactly as many men as there were women in that country, neither more nor | less. This method can be applied generally. If there is some relation which, like marriage, connects the things in one collection each with one of the things in another collection, and vice versa, then the two collections have the same number of terms. This was the way in which we found that there are as many even numbers as there are numbers. Every number can be doubled, and every even number can be halved, and each process gives just one number corresponding to the one that is doubled or halved. And in this way we can find any number of collections each of which has just as many terms as there are finite numbers. If every term of a collection can be hooked on to a number, and all the finite numbers are used once, and only once, in the process, then our collection must have just as many terms as there are finite numbers. This is the general method by which the numbers of infinite collections are defined.
[87-88]

But there are sets whose infinity is larger than the infinite set of finite numbers.

But it must not be supposed that all infinite numbers are equal. On the contrary, there are infinitely more infinite numbers than finite ones. There are more ways of arranging the finite numbers in different types of series than there are finite numbers. There are probably more points in space and more moments in time than there are finite numbers. There are exactly as many fractions as whole numbers, although there are an infinite number of fractions between any two whole numbers. But there are more irrational numbers than there are whole numbers or fractions. There are probably exactly as many points in space as there are irrational numbers, and exactly as many points on a line a millionth of an inch long as in the whole of infinite space, There is a greatest of all infinite numbers, which is the number of things altogether, of every sort and kind. It is obvious that there cannot be a greater number than this, because, | if everything has been taken, there is nothing left to add. [88-89]

With this in mind we can explain the Zeno’s paradox of Achilles and the Tortoise. [The editor notes that Russell is not giving the traditional account of this paradox. As it is in Aristotle, both the Tortoise and Achilles will be in motion. Just as soon as Achilles catches up with the Tortoise, the Tortoise, who is in motion, is already in the act of creating more distance for Achilles to catch up to. Achilles quickly closes that gap, only to find that the Tortoise has yet again created more distance for Achilles to cross. So long as the Tortoise is in motion, Achilles can never overtake him. What Russell says is not so clear. But Russell does state that neither Achilles nor the Tortoise is in the same place twice while in movement. So his reasoning might be similar. There might be moment that Achilles and the Tortoise are at the same distance. But this lasts only a moment. In the next time-point, both advance a space-point. And so long as both are in motion, neither one can overtake the other. This requires that one be at rest and thus be in the same space-point in two successive instants. … Another way to explain Russell’s reasoning is this. When Achilles reaches the Tortoise, the Tortoise is already advancing and creating an infinity more points for Achilles to cross. But if Achilles advancing past the Tortoise, he would be crossing the same number of points and not more, because no matter what the actual finite distance a moving body traverses, in every case the infinity of points through which they cross is the same, that is, the same size of infinity.] 

We can now understand why Zeno believed that Achilles cannot overtake the tortoise and why as a matter of fact he can overtake it. We shall see that all the people who disagreed with Zeno had no right to do so, because they all accepted premises from which his conclusion followed. The argument is this: Let Achilles and the tortoise start along a road at the same time, the tortoise (as is only fair) being allowed a handicap. Let Achilles go twice as fast as the tortoise, or ten times or a hundred times as fast. Then he will never reach the tortoise. For at every moment the tortoise is somewhere and Achilles is some where; and neither is ever twice in the same place while the race is going on. Thus the tortoise goes to just as many places as Achilles does, because each is in one place at one moment, and in another at any other moment. But if Achilles were to catch up with the tortoise, the places where the tortoise would have been would be only part of the places where Achilles would have been. Here, we must suppose, Zeno appealed to the maxim that the whole has more terms that the part.² [[ft 2: This must not be regarded as a historically correct account of what Zeno actually had in mind. It is a new argument for his conclusion, not the argument which influenced him. On this point, see e.g. C. D. Broad, “Note on Achilles and the Tortoise,” Mind, N.S., Vol. XXII, pp. 318-19. Much valuable work on the interpretation of Zeno has been done since this article was written. [Note added in 1917.] ]] Thus if Achilles were | to overtake the tortoise, he would have been in more places than the tortoise; but we saw that he must, in any period, be in exactly as many places as the tortoise. Hence we infer that he can never catch the tortoise. This argument is strictly correct, if we allow the axiom that the whole has more terms than the part. As the conclusion is absurd, the axiom must be rejected, and then all goes well. But there is no good word to be said for the philosophers of the past two thousand years and more, who have all allowed the axiom and denied the conclusion.
[89-90]

Russell diagnoses the problem as lying in the axiom that the whole has more terms than the part. He describes a similar paradox that he calls the Paradox of Tristram Shandy. He tried to chronicle the first two days of his life, but this took him two years. But this means that as time goes on, the current day of writing gets further and further way from the past day currently being chronicled at that time. But Russell, on account of his conception of infinity, thinks that eventually Tristam will finish his chronicle. Russell then has us line up chronicled days with days of writing, in the same way that he correlated every even number with every natural number. “Thus any day that may be mentioned will be written up sooner or later, and therefore no part of the biography will remain permanently unwritten.” [90d] [Russell continues by saying that the number of days is no greater than the number of years. This would be because they are the same size infinities. However, there might still be a problem. The gap between the chronicled day and the current one also rises to infinity, and an infinity of the same size as the total infinity of all time. For, each day between the chronicled and the chronicling days can be matched with each day of either other series, in the same way these two other series were matched to one another. This means the gap can never be closed, because it cannot be exhausted:] “This paradoxical but perfectly true proposition depends upon the fact| that the number of days in all time is no greater than the number of years.” [90-91]

On the surface, such conclusions seem paradoxical, and for that reason philosophers have criticized the notion of infinity for having inherent contradictions. But Cantor’s conception of infinity shows that these problems are oddities and not contradictions. [91]

The concept of quantity for long was thought to be the fundamental notion of mathematics. But now quantity has been largely put aside in favor of the concept of order.

Geometry, while order more and more reigns supreme. The investigation of different kinds of series and their relations is now a very large part of mathematics, and it has been found that this investigation can be conducted without any reference to quantity, and, for the most part, without any reference to number. All types of series are capable of formal definition, and their properties can be deduced from the principles of symbolic logic by means of the Algebra of Relatives.
[91]

The concept of the limit [was once understood with regard to the infinitesimally small quantity and thus it] was defined by means of quantity “as a term to which the terms of some series approximate as nearly as we please.” [92]

Geometry is now built axiomatically, and thus it is not proven empirically. “Thus the geometer leaves to the man of science to decide, as best he may, what axioms are most nearly true in the actual world.” [93]

In geometry today, proofs no longer depend on figures. [93]

Peano conducts his geometry with points but without any reference to space. [94]

Euclid’s propositions are not entirely supported. [94]

Countless errors are involved in his first eight propositions. That is to say, not only is it doubtful whether his axioms are true, which is a comparatively trivial matter, but it is certain that his propositions do not follow from the axioms which he enunciates. A vastly greater number of axioms, which Euclid unconsciously employs, are required for the proof of his propositions. Even in the first proposition of all, where he constructs an equilateral triangle on a given base, he uses two circles which are assumed to intersect. But no explicit axiom assures us that they do so, and in some kinds of spaces they do not always intersect. It is quite doubtful whether our space belongs to one of these kinds or not. Thus Euclid fails entirely to prove his point in the very first proposition.
[94]

Weierstrass and his follows showed that many of the basic propositions in mathematics were false. [95]

Kant’s doctrine of a priori intuitions does not apply to today’s mathematics. [96] For math to progress, we need to devote attention to mathematical logic. [96]

 

 

 

Except for the Leibniz quote in the very brief summary, all citations and quotations from:

Russell, Bertrand. Mysticism and Logic, and Other Essays. London: George Allen & Unwin, 1917, second edition. [1st 1910, entitled “Philosophical Essays”]

https://archive.org/details/mysticism00russuoft

Otherwise:

Leibniz. Philosophical Papers and Letters. Ed. & Transl. Leroy E. Loemker. Dordrecht: D. Reidel Publishing Company, 1956.

See this entry.

Weichselgartner & Sperling (1985) ‘Continuous Measurement of Visible Persistence’, notes

by Corry Shores
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[The following is summary and quotation except for my bracketed commentary. Please consult the original text, as not all of this article was clear to me.]

Erich Weichselgartner; George Sperling

‘Continuous Measurement of Visible Persistence’

Brief Summary:

The authors conduct experiments meant to determine the duration of visual persistence, and they introduce the notion of ‘perceived brightness’, which might be similar to the concept of ‘presentedness’ in the phenomenological study of the specious present.

Notes:

The authors will examine the decay of persisting visual images.

A brief visual stimulus presented to a subject is not perceived to end abruptly but to fade out gradually.
[p.711]

There have been two primary ways that the duration of persisting images have been tested: 1) inferring from the accuracy of subjects’ reports, and 2) depending on subjects’ subjective reports of their experiences.

The time difference between the physical termination of the stimulus and its perceptual termination has been investigated in a variety of paradigms. These paradigms fall into two main classes: those that infer visual storage from the accuracy of subject's reports and those that depend on subjective reports (e.g., Brindley's [1960] Class A and Class B procedures).
[711]

The first type (accuracy procedures) are generally either partial report tasks or picture completion tasks. In partial report (like Sperling 1960), the subject views a flash of a matrix of letters and must report on a randomly selected portion. In picture completion, subjects must compare successive images.

In the partial report paradigm (Sperling, 1960), the observer views a brief flash of a matrix of letters. Afterwards, a tonal cue that can be precisely located in time is used to request report of a randomly selected row of the stimulus. The decline of response accuracy with cue delay indicates the duration of short-term visual storage (iconic memory— Neisser, 1967). Picture completion paradigms require the observer to integrate information from two successive flashes in order to identify a target letter (e.g., Eriksen & Collins, 1967) or to detect a missing dot in a regular dot matrix (Hogben & DiLollo, 1974).
[711]

In subjective procedures, the subject makes their own judgment to how long the stimulus remains in their sensory memory. They can do this by adjusting clicking sound [see section 2.3 of Sperling 1967] [711B]

For the most part, both these objective and subjective methods indicate a similar duration for sensory memory. However, there are some important discrepancies. [Please see the last full paragraph of p.711, as it is not clear to me how to summarize it. It suggests that there is a function to describe the decay curve, but “no one has come up yet with a measure for those decay functions.” (712Aa)]

WS then distinguish two classes of studies on visible persistence. (a) Studies that are “concerned with the the time difference between stimulus termination and perceptual termination”, and (b) “studies concerned with the total phenomenal duration of the stimulus that require reliable judgments of both onset and termination, the estimated stimulus duration being the time difference between judged onset and termination”. [712Aa.b] However, “There are no statements concerning the actual form of the rise and decay curves or of the complete representation of the stimulus as a function of time in the subject's visual system.” [712Ab]

Thus what is needed is “a method to measure the entire moment-to-moment time course of visible persistence.” [712A] But, the term ‘visible persistence’ already has various meanings. So WS will use another term to be more specific: temporal brightness response (TBR), which “describes how the perceived brightness of a brief visual stimulus changes as a function of time.” [712A] [note, this could be similar to the notion of presentedness in the phenomenology of the specious present.] “The purpose of this article is to prove the feasibility of measuring the TBR and to measure TBRs to brief flashes for 3 observers.” [712A]

Elaborated Synchrony Judgment Paradigm

The following will be an elaboration of Sperling’s (1967) method. It will be an “intermodal synchrony judgment paradigm.” [712A] In it, the subject views two adjacent stimuli: (a) a reference stimulus that is “presented at the beginning of the trial and remains on with constant luminance during the trial,” and (b) “a test stimulus of varying luminance, which is presented and terminated sometime during the middle of the trial. Luminances are are adjusted so that from the observer's point of view the test stimulus initially appears dimmer than the steady-state reference stimulus but increases in intensity until eventually it becomes as bright or brighter (Figure 1).” [712A]

[As we can see, the luminosity of the test image will become at least as bright as the test image, maybe even brighter.] The instant when the luminosity seems to match between test and reference stimuli is called the match time. This method wants to know when the match time occurs. “In a time interval around match time, the subject is presented with a click.” [712Ba]

If the subject perceives the click before they perceive the two stimuli matching, they indicate with a left-hand response key. But if the click occurs after they seem to match, then with a right-hand response key. [712B]

Or, the subject might be asked to press the right-hand key if at the time of the click the test was brighter than the reference stimulus. Otherwise [if it seemed the same or dimmer] then they press the left-hand key.  [If the test is dimming and they press the right-hand key, then this would suggest the brightness persisted beyond its actual luminance.]

Repeated judgments result in a psychometric function, the probability of a right response as a function of time. The point of subjective equality (pse) is the 50% point of the psychometric function; it corresponds to the time at which the brightness of the reference matches that of the test. A psychometric function obtained with a particular luminance of the reference stimulus determines only one match time. The entire TBR function is obtained by obtaining match times for a full range of luminances of the reference stimulus, and by determining match times near the onset and also near the termination of the test stimulus.
[712B]

Verification Procedure

[The actual stimulus change has a luminance function, its change over time.] They will for example have a light bulb increase linearly to its maximum at 300 ms, then dim to off in another 300 ms. While the stimulus is rising, WS expect the match times to increase with increasing luminance.

Our primary interest is in the TBR function of a very brief stimulus. However, it is important to first demonstrate that a subject can make reasonable match time judgments with a slowly varying stimulus for which the TBR can be assumed to approximately track the temporal luminance function of the physical stimulus. An example of such a physical stimulus is the gradual fading out of a light bulb turned off with a dimmer. Therefore, we first determine TBRs for a control condition in which test field luminance increases linearly during a 600-ms period, stays at its maximum for 300 ms, and then turns off linearly during another 600-ms period. The data from this control condition with real "physical persistence" can be used to evaluate the method. For example, when the rising part of the test stimulus is under investigation (onset trials), we expect the match times to increase with | increasing luminance of the reference stimulus, and when the decaying part of the test stimulus is under investigation (termination trials), we expect the match times to increase with decreasing luminance of the reference stimulus. After Experiment 1 (with ramped onsets and terminations) we proceed to Experiment 2, which determines TBRs for brief stimuli. [712-713]

General Method

Overview

There are two experiments, the verification and the main experiment. Both use the same apparatus and a similar method.

Apparatus and Stimuli

Spatial arrangement

Spatial Arrangement:

As depicted in figure 2, the stimuli were two square-wave gratings.

Stimulus intensity:

Reference took on 5 different luminances [see table 1].

Procedure

Individual trials:

(1) Reference stimulus first turned on. (It remains on for 3,000 ms)

(2) Secondly, the test stimulus is turned on. (It remains on for 1,500 ms in Experiment 1, and 31 ms in experiment 2.)

(3) Click sounds. “On the first trial, the onset time of the click was randomly chosen within an interval of ± 200 ms around starting points (initial values), which were determined in preliminary experiments (see below).”

(4) Afterward, subject must decide whether click occurred before or after the instant of perceived brightness match between two visual stimuli. Either click before or click after choices available. [714B]

Blocks of trials:

Ran a number of trials varying brightness according to step procedure (below).

Staircase procedure:

WS used the stair-case procedure described by Lewitt (1971) [Please see the original article to be sure of its content. As far as I can summarize, it seems the problem that WS are contending with is that the subject cannot just press a button when it seems the stimuli match. For, this involves a motor response whose accuracy might be unreliable and variable. To narrow down to greater accuracy, WS suggest this step method. In it, the subject reports only after the stimuli have finished. The subject reports if the clicked seemed to come before or after the apparent match. If before, then the click is moved forward a step. If after, then moved back. It then seems by means of calculating this data, it can then be better approximated when it is that the subject perceives the match. The following diagram is from Levitt (1971) and it depicts this step ‘staircase’ procedure:

]

Experiment 1

Experiment 1 consisted of two phases. In Phase 1, stimulus parameters for Phase 2 were empirically determined. In Phase 2, the TBR for a ramped test stimulus was measured using the parameters from Phase 1.
[715A]

Method

Procedure

Phase 1:

The first phase determines appropriate luminances for the reference stimuli and also time for occurrence's of the click. They used the step method.

Together, the isolated brightness and temporal matches of Phase 1 pinpoint the brightness and time of the perceived brightness peak of the test stimulus.
[715]

Phase 2:

[Recall that “temporal brightness response (TBR)” “describes how the perceived brightness of a brief visual stimulus changes as a function of time.” (712A)]

In Phase 2 of Experiment 1, the TBR was determined for a ramped test stimulus by means of the psy -|- chophysical method described above in the Procedure section.
[715|716]

Results

The results of Experiment 1 can be found in table 1. Figure 3 shows the temporal brightness response times for the 3 subjects. The match times are open squares. “The five data points on the left side of the TBR are the match times for the onset judgment, and the five data points on the right side of the TBR are the match times for the termination judgment.” [716A, see for more detail]

The TBRs are comparable to the stimulus slopes. “In all cases of the onset judgment, the match times were ranked in an ascending order with increasing luminance, and in all cases of the termination judgment, the match times were ranked in an descending order with decreasing luminance.” [716B]

Discussion

[See text pp.176-178]

Conclusion:

From the results of Experiment 1, we conclude that the elaborated synchrony judgment method yields reasonable temporal brightness response functions, and there are pronounced individual differences.
[718B]

Experiment 2

Experiment 2 applies the paradigm of Ex1 to a 31ms flash to determine its TBR.

Method

Procedure

Phase 1:

A reference stimulus was found whose intensity matched the peak of the test stimulus.

Phase 2:

“Phase 2 of Experiment 2 determined the TBR for the 31-ms pulse with the elaborated synchrony judgment paradigm.” [718A]

Results:

Ex2 indicates match times. Results are shown in figure 4, which indicates the TBR for each subject.

General Discussion

Duration of Visible Persistence

Here the authors discuss “three fallacies in thinking of persistence as a concept that is adequately described by a single number, its duration.” [719A]

[skipping to]

Conclusion:

The elaborated synchrony judgment paradigm utilizes a highly refined form of introspection to trace out an entire temporal brightness response function to a test stimulus. The paradigm applies to a wide variety of possible temporal waveforms. For very brief test flashes, both the onset and the termination phases of the TBR differ widely across observers, with overall durations of the TBR varying from about 200 to about 500 ms.
[724B]

Weichselgartner, Erich, and George Sperling. "Continuous Measurement of Visible Persistence." Journal of Experimental Psychology 2.6 (1985): 711-725.

http://weichselgartner.de/erich/full_text/EW+GS_Visible_Persistence_JEP_1985.pdf

http://www.ncbi.nlm.nih.gov.sci-hub.org/pubmed/2934504

Also a reference to:

Levitt, H. "Transformed Up‐Down Methods in Psychoacoustics." The Journal of the Acoustical Society of America 49 (1971): 467-477.

http://bdml.stanford.edu/twiki/pub/Haptics/DetectionThreshold/psychoacoustics.pdf

http://scitation.aip.org/content/asa/journal/jasa/49/2B/10.1121/1.1912375

Levitt (1971) ‘Transformed up‐down methods in psychoacoustics’, notes

by Corry Shores
[Search Blog Here. Index-tags are found on the bottom of the left column.]

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H. Levitt

Transformed up‐down methods in psychoacoustics

Brief summary:

The authors discuss methods for doing psychoacoustic experiments in which the subject’s responses determine the course of the experiment. For our purposes, we focused on the step-procedure. Here, a positive response means reducing the stimuli by a step, and a negative response means stepping the stimuli up. Then, it seems a middle point can be determined mathematically. [In the case of Weichselgartner & Sperling (1985), the subject needs to indicate when the brightness of a steady stimuli matches with a stimuli whose brightness is varying. If the subject just pushes a button, there is the problem of response times and variations they might introduce (one time being quicker, another especially slow). So instead, after the stimuli finish, the subject says whether the click comes before or after the perceived match in visual stimuli. If says comes after, then move the click back a step. If before, then move forward. Then, with all this data, it is then calculated when in fact they do perceive it as matching, the X₅₀ value of the psychometric curve.]

Notes:

 

Introduction

A. Adaptive Procedures in Psychophysics

An adaptive procedure is one in which the stimulus level on any one trial is determined by the preceding stimuli and response. [467]

Up-down methods are a type of sequential experiment, which is one where the course of the experiment is determined by experimental data. There are two types: (a) ones where the number of observations is determined by the data and (b) ones where the choice of stimulus levels is determined by the data. [467A]

B. The Psychometric Function

Look at fig 1a. The abscissa is the stimulus level. The ordinate is the proportion of ‘positive’ responses. A positive response might for example be the report from the subject that ‘the signal is present’.

[Please see the article to obtain a correct interpretation of its contents. For our purposes, I think we can derive what we need from figure 4 below. It seems if the response is positive, then the next trial takes the stimulus down a step. If the response is negative, then it steps up. Perhaps over time this approximates where the match would be. In the case of the Weichselgartner & Sperling (1985) experiment, this would mean that they use this step procedure to estimate when the subject senses the match between the steady reference stimuli and the varying test stimuli. The problem with having the subject just press the button when they sense the match could be the response time and the inaccuracies and variances it can introduce. But in this step method, the subjects chose the option (‘before,’ ‘after’) after the stimuli are all done. So there is not a response delay involved. ]

Levitt, H. "Transformed Up‐Down Methods in Psychoacoustics." The Journal of the Acoustical Society of America 49 (1971): 467-477.

http://bdml.stanford.edu/twiki/pub/Haptics/DetectionThreshold/psychoacoustics.pdf

http://scitation.aip.org/content/asa/journal/jasa/49/2B/10.1121/1.1912375

Sperling (1967) ‘Successive approximations to a model for short term memory’, notes

by Corry Shores
[Search Blog Here. Index-tags are found on the bottom of the left column.]

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[The following is summary and quotation except for my bracketed commentary. All underlining and boldface are mine.]

George Sperling

‘Successive approximations to a model for short term memory’

Brief Summary:

By working through possible models for testing visual sensory memory, we learn that a sensory stimulus with information persists in our awareness even after it physically disappears. We read (‘scan’) parts of that image, and store them temporarily by repeating them vocally or subvocally (‘rehearsal’). But the parts that were never scanned disappear from awareness and seem to be lost forever to consciousness. [Whether or not they are registered subconsciously or unconsciousness would be another matter.]

Abstract [quoting]:

Experimental data are considered from a simple task in which an observer looks at letters and then writes them down. Three models are proposed. Model 1 consists of only two components: a visual memory for the letters and a motor translation component to enable copying a visual memory onto paper. Model 1 is inadequate because the visual image is shown not to persist until the time of reproduction. Model 2 corrects this deficiency by incorporating the possibility of subvocal rehearsal of the stimulus letters and an auditory memory for the rehearsal. However, Model 2 cannot account for performance with extremely short duration images because of the limit on the maximum rehearsal rate. The critical improvement in Model 3 is a more detailed specification of scanning, recognition and rehearsal, including a form of memory which is inherent in the process of recognition itself. Model 3 accounts for these data and incidently gives rise to some interesting inferences about the nature of consciousness. [285]

Summary

1. Introduction

Sperling proposes some simple memory tasks involving seeing random letters and writing them down from memory. On the basis of these simple methods, more complex ones might be derived.

2. Models

2.1 Model 1

The subject briefly sees a series of letters, and then from memory must immediately transcribe them from what they see in their ‘mind’s eye’. “While the subject is writing, the contents of his visual memory  are decaying, so that when he finally comes to write the fifth or sixth letter his visual memory of the stimulus no longer is legible.” [286]

The problem with this model is that just as soon as the subject begins writing the letters, the persisting image has already faded away.

2.2 Model 2

In various such experiments, Sperling recorded the voice of the subject, and found that often they speak the letters while writing. If the delay before reporting is extended to about 20 seconds or so, often times the subject will repeat the series and then when time comes to report them, speaks the letters while writing them. This could be a memory mechanism which refreshes the fading image.

Occasionally a subject, when he is writing down letters, can be heard to mumble the letters as he is writing them. His tendency to say the letters aloud can be emphasized by playing loud noise into his ears. Noise itself not seem to alter performance in any other significant way. We have this technique, together with a microphone placed near the subject’s mouth, to record the actual letters the subject is saying. We also recorded automatically whenever the subject was writing. The most interesting results with this technique are obtained when the subject is required to wait (e.g., | for 20 sec) after the stimulus exposure before writing the letters. He repeats (rehearses) the entire letter sequence several times with a pause between each repetition during the interval. Then, at the time of writing each letter, he also may speak it simultaneously.

Rehearsal suggests an obvious memory mechanism. The subject says a letter, hears himself saying it, and then remembers the auditory image. As the auditory image fades, he repeats it to refresh it. Most of our subjects do not vocalize during recall, but they all concur in stating that they rehearse subvocally. Therefore, we assume that the sound-image of a letter lean enter auditory memory directly from subvocal rehearsal without the necessity of actually being converted into sound and passing into the external world. These relations are illustrated in fig. 2.
[286|287]

According to Model 2, stimulus letters first are retained in visual storage. They are rehearsed, one at a time (i.e., converted from a visual to an auditory form), and then remembered in auditory storage. Subsequently they may be rehearsed again and again as required until they are written down. The limits on performance may arise either from the limited duration visual storage (so that some letters decay before they can be rehearsed) or from the limited capacity of the rehearsal-auditory storage loop, depending on the stimulating conditions. [288]

The problem with this method is the following. [Someone can recollect three letters from a retained image (having been given or lasting in the mind) for 0.1 sec. This means that in order to have pulled those letters out by reading them (vocally or subvocally), they had to do so at a speed of 30 letters a second.]

Attractive as Model 2 seems, it is inadequate for the following reason: it is possible to generate an image in visual storage which has a duration of definitely less than .1 sec and from which 3 letters can be reported. This would require a rehearsal rate of over 30 letters per second, which clearly is completely beyond the capabilities of the rehearsal processes described for Model 2.  [288]

2.3 Short duration visual images

[When testing recollection after many letters, the subject may only store four or five items in their memory. But it is also possible that the image of more of these items was there, but faded before they could report them. The solution to this is partial reporting. The idea is that the subject does not know until after which portion of the whole array they need to report. In

Sperling 1960, for example, there were grid arrays of letters, and subjects were asked after a flash of them to report on just one row. The idea here seems to be that if they can remember 4 or 5 items from one row, which was requested after the image faded, then they must have remembered that many in each of the other rows. For, were any of those other rows requested instead, they would have likewise reported just as many items. That would mean, that if a subject could recall all of one row a second or so after the image disappeared, then probably the whole image remained in sensory memory for that amount of time.] Partial reporting has shown that for a visual image of 18 letters given at 1/20 of a second, up to 10 items remain for as long as 2 seconds after the exposure.

There were also ‘letter-noise’ stimulus sequences, where visual noise was given after the stimulus. Then, by using clicks, the subject subjectively determined the onset of the visual image with its disappearance. This experiment found that

The apparent image duration of the letters in a letter-noise sequence is zero for extremely brief exposures (e.g., less than 10 msec) and then increases linearly with increasing exposure duration for durations exceeding about 20 msec.
[290a]

The results also showed that each subject had a particular order where they were most correct. In one case it was left-to-right, but more scrambled in others. However, the fact that all positions are reported better than chance indicates that the retentional images are not placed in the mind serially but rather in parallel [and then only afterward ‘read’ in some idiosyncratic order. Please see the first full paragraph on p.290 to be sure this is a correct interpretation.] [290]

Here is quotation for the above summarized parts:

In a letter-noise stimulus sequence, a second, interfering, stimulus (visual 'noise') is exposed immediately on termination of the letter stimulus. The duration of the letter images can be estimated by comparing them to an | auditory signal. Two different methods were used. In the first method two clicks were produced at the ears of the subject. He then adjusted the interval between the clicks until the auditory interval was judged equal to the visual duration. In the second method, the subject heard only one click at a time. We adjusted this click to occur so that it coincided subjectively with the onset of the visual image. After this judgment was complete, he made | another adjustment of the click to coincide with the termination of the visual image. The measured interval between clicks – taken to be the duration of the visual image – was the same by both methods. The apparent image duration of the letters in a letter-noise sequence is zero for extremely brief exposures (e.g., less than 10 msec) and then increases linearly with increasing exposure duration for durations exceeding about 20 msec (fig. 3a).

When stimuli of 5 letters, followed by noise, are exposed for various durations, the accuracy of report increases with exposure duration as shown in fig. 3b. The most interesting aspect of these data is revealed by analyzing separately the accuracy of report at each of the 5 locations (fig. 3c). The accuracy of report at each location reported increases continuously as a function of exposure duration. For this subject, the order of the successive locations which are reported correctly is generally left-to-right (I to V), except that location V is reported correctly at shorter exposures than location IV. Other subjects have different idiosyncratic orders, e.g., I, V, III, II, IV. By definition, in a purely serial process the nth location is not reported better than chance until the exposure duration at which the n-1th location is reported with maximum accuracy is exceeded. The observation that all locations begin to be reported at better than chance levels even at the briefest exposures, may be interpreted as evidence of an essentially parallel process for letter-recognition. This process gives the illusion of being serial because the different locations mature at different rates (cf. GLEZER and NEVSKAIA, 1964; SPERLING 1963). These findings are taken into account in Model 3 (fig 4).

2.4 Model 3

Model three has a scan-rehearsal component like Model 2.
But this part is subdivided into three other components. The first is the scan component. It “determines – within a limited range – the sequence of locations from which information is entered into subsequent components.” [291] [It seems the scan is reading from the persisting sensory image, and then the order of that scan is carried over into the rehearsal.]

The second different sub component is “recognition buffer-memory,” which “ converts the visual image of a letter provided by the scanner into a ‘program of motor-instructions’, and stores these instructions.” [191] And this program is then executed by the rehearsal component. [So the ordering of the scan is then carried into a program for repeating that order.]  What is important is that creating this program can be carried out in relatively short time (about 50 msec for 3 letters, for example) compared to the time it will take to execute that program of rehearsal (500 msec for 3 letters, for example.) [290]

The third new sub-component then is the rehearsal, in which the visual information is now stored in auditory cycles governed by motor-instructions. This then translates back to the visual image when reporting the letters on paper. [290]

3. Consciousness in the Memory Models

We can infer that there is an act of consciousness involved in the scan component of the process. Contents of visual memory which are not scanned fade away. But also, parts that were never scanned we never conscious. This is strange because it means neither subjectively nor objectively can these unscanned elements be observed.

Sperling, G. "Successive approximations to a model for short term memory." Acta psychologica 27 (1967): 285-292.

http://psych.stanford.edu/~jlm/pdfs/Sperling67.pdf

http://www.ncbi.nlm.nih.gov.sci-hub.org/pubmed/6062221

Sperling (1960) ‘The Information Available in Brief Visual Presentation’, notes

by Corry Shores
[Search Blog Here. Index-tags are found on the bottom of the left column.]

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George Sperling

‘The Information Available in Brief Visual Presentations’

Brief Summary:

One way to know how long images stay in our mind after they disappear is to have people see a flash of objects then recall them after the image disappears. If they try to recall as many as possible regardless of position or type, then it is a whole report. One theory is that the image persists in the mind after the stimulus physically disappears, and when reporting, we select from the whole group, but that selection is arbitrary for the most part. One problem with the whole report method is that it alone does not tell us if the unselected parts persisted or not. If the whole image remains to our awareness, then this would be a persistence of the image, and if not, then either it does not persist or it does only partially. To find out which possibilities is true, Sperling employs a ‘partial report’ method. Here, the subject is to remember only a part of the set of flashed objects, but which part they might not know until shortly after the image has disappeared. If it can be shown that it does not matter from which part they are to report on, that suggests all of it remains in their mind’s eye, and they select from the whole. The partial report experiments here described show that not only does it not matter which part, and thus that the image probably persists, but also that subjects do better when they only need to recall part. One explanation for this again supports the image persistence theory: in whole reports, the subject must consume time deciding which figures to report, but in partial reports, that selection is made for them, giving them more time to make reports while the image persists in their minds.

Summary notes:

We often remember less than we original saw.

So we see a lot, but we can only remember part of it. [In the following, it seems our concern is not just with the quantity of content recalled in immediate sensory memory, but also with the duration of it. After the visual impression fades, it might become encoded as memory in a form which is not the sustained visual impression that remains in our vision. So we do not what to ask the subject to recall more than memory allows them too. They only need to report a portion of what they saw, and the question is, for how long does the impression stay in their minds so that they can still look at in it in their minds and withdraw information from it.?]

In order to circumvent the memory limitation in determining the information that becomes available following a brief exposure, it is obvious that the observer must not be required to give a report which exceeds his memory span. If the number of letters in the stimulus exceeds his memory span, then he cannot give a whole report of all the letters. Therefore, the observer must be required to give only a partial report of the stimulus contents. Partial reporting of available information is, of course, just what is required by ordinary schoolroom examinations and by other methods of sampling .available information.
[p.1]

After the stimulus terminates, the subject can be instructed to recall a certain part of the stimulus.

The experiments described in the following were conducted “to study quantitatively the information that becomes available to an observer following a brief exposure.” [2]

Letters were arranged in following sorts.

The stimulus array was flashed for 50 msec [see p.3]. Subjects were to report what they saw [depending on instruction for whole or partial reporting]. They were scored both for number correct and positions correct. [p.4]

Experiment 1: Immediate memory

Sperling writes:

When an S is required to give a complete (whole) report of all the letters on a briefly exposed stimulus, he will generally not re- | port all the letters correctly. The average number of letters which he does report correctly is usually called his immediate memory span or span of apprehension for that particular stimulus material under the stated observation conditions. An expression such as immediate-memory span ( Miller, 1956a) implies that the number of items reported by S remains invariant with changes in stimulating conditions. [p.4-5]

All 12 types of arrays [‘of stimulus materials’] were used.

Results:

the average number of correct letters contained in an S's whole report of the stimulus is approximately | equal to the smaller of (a) the number of letters in the stimulus or (b) a numerical constant-the span of immediate-memory which is different for each S. The use of the term immediate-memory span is therefore justified within the range of materials studied.
[5-6]

Experiment 1 showed that, regardless of material, Ss could not report more than an average of about 4.5 items per stimulus exposure.
[p6]

Experiment 2: Exposure duration

In the previous experiment, the exposure duration to the stimulus was short, 0.05s. In order to determine whether the 4.5 item limitation results from the shortness of the duration, we need to vary that duration.

Procedure:

Subjects were given tasks from prior experiment, except at varying durations of exposure to the stimulus: 0.015, 0.050, 0.150, 0.200, and 0.5oo second durations.

Results:

The main result is that exposure duration, even over a wide range, is not an important parameter in · determining the number of letters an S can recall correctly. Both individually and as a group, Ss show no systematic changes in the number of letters correctly reported as the exposure duration was varied from 0.015 to 0.500 sec. The invariance of the number of letters reported as a function of exposure durations up to about 0.25 sec. for the kind of presentation used ( dark pre- and postexposure fields) has long been known ( Schumann, 1904) .
[p6]

Experiment 3: Partial Report

[The controls experiments above tell us the full total quantity of the amount of information available to immediate recollection. This experiment will determine if the subject has more information than she can indicate in the immediate memory report. What this would suggest is that in the full reports, the subjects could have recalled different parts of the arrays, because all of it is available, but it decays, and the subject needs to report as much as possible, choosing arbitrarily from the retained image. If the subjects are able to recall the same number of figures but at varying spatial locations, then this suggests there is such a retained image from which a partial report can be selected.]

Experiments 1 and 2 have demonstrated the span of immediate-memory as an invariant characteristic of each S. In Experiment 3 the principles of testing in a perceptual situation that were advanced in the introduction are applied in order to determine whether S has more information available than he can indicate in his limited immediate-memory report.

The S is presented with the stimulus as before, but he is required only to make a partial report. The length of this report is four letters or less, so as to lie within S's immediate-memory span. The instruction that indicates which row of the stimulus is to be reported is coded in the form of a tone. The instruction tone is given after the visual presentation. The S does not know until he hears the tone which row is called for. This is therefore a procedure which samples the information that S has available after the termination of the visual stimulus.
[p.6]

Procedure:

Arrays with only two lines were used. Right after they disappear, the subject heard either a high or low tone, indicating whether to report the upper or lower row. [It then seems cards with more rows were shown and cued with  more tones. see p.6Bd]

Results:

The subjects improved, at first averaging 4.5 and later 5.6 letters [compare to the 4.5 limit of the control whole report sessions.] The diagram below shows in fact that partial reporting yields higher recollection [the lower curve is average number of figures recalled in whole reporting, the middle in partial reporting. Perhaps it is for this reason. In both cases, the image decays rapidly. In whole reporting, it might take more time to decide where in the retained image to select the figures to report. But in partial reporting, those briefly time-consuming choices are made already for the subject, allowing for them to draw out more information in the short time that the image remains in their vision.]

In Fig. 3 the number of letters available as a function of the number of letters in the stimulus are graphed as the upper curves. For all stimuli and for all Ss, the available information calculated from the partial report is greater than that contained in the immediate-memory report. Moreover, from the divergence of the two curves it seems certain that, if still more complex stimuli were available, the amount of available information would continue to increase. [p.6]

Experiment 4: Decay of Available Information

Part 1: Development of Strategies of Observing

We will now explore the decay of available information by delaying the cue telling which parts to recall. They used these variations on the cue time (even placing it before and during the stimulus): “0.05 sec. before stimulus onset (-0.10 sec.), ±0.0-, +0.15-, +0.30-, +0.50-, + 1.0-sec. delays after stimulus off-go”. [p.8]

Each subject went through all of these variations in delay, going either in ascending or descending order.

Results:

Look first in fig.5 to the leftmost panel (5a). These are the results of one subject giving partial reports at varying delay times. (Arrows tell what order the sequence went in on a whole, thus here began with ascending). We see that the lines go down, indicating that there is a decay, and much of the visual information is lost after 0.25s. The second session (5b) began with descending, and the results were not so orderly. The third session (5c) had more trials and also had a part where the the cue came before the stimulus.

The variability of 5b may have resulted from the subject choosing a strategy where she first guesses which row will be requested, and is sometimes right and sometimes wrong. Another subject described changing strategies after 0.15s  from trying to remember all equally and guessing the row, and his diagram [see p.9]

Part 2: Final Level of Performance

The experimenters attempted to eliminate this factor of choosing a strategy of guessing the row. Three proposed ways are: increasing the number of figures, making selection less useful; have tones come slightly before the stimulus; have the experimenter say something that would change the subject’s approach to the experiment, for example “not testing memory but reading, don’t read the card until hear the tone”. [It is not clear to me if the following results come from an experiment implementing some or all or none of these, but it seems at least some. Given that the results show a variation of times, it is not clear that the second modification was used any differently than before.]

Results:

The data indicate that, for all Ss, the period of about one sec. is a critical one for the presentation of the instruction to report. If Ss receive the instruction 0.05 sec. before the exposure, then they give accurate reports : 9 1 % and 82% of the letters given in the report are correct for the 9- and 12-letter materials, respectively. These partial reports may be interpreted to indicate that the Ss have, on the average, 8.2 of 9 and 9.8 of 12 letters available. However, i f the instruction is delayed until one sec. after the exposure, then the accuracy of the report drops 32% (to 69% ) for the 9-letter stimuli, and 44% ( to 38%) for the 12-letter stimuli. This substantial decline in accuracy brings the number of | letters available very near to the number of letters that Ss give in immediate-memory ( whole) reports. [p.11-12]

[Experiment 5, skipped. It experiments with different pre- and post-exposure fields.]

Experiment 6: Letters and Numbers

[It could be that what is being remembered is a result of remembering locations. The following experiment reduces the importance of location.]

Procedure:

Subjects saw arrays mixed with both letters and numbers, and afterward were asked to recall only one or the other. [It seems this experiment also had trials asking for top and bottom rows too, regardless of symbol/number. see page 14Ad]

The results show that reporting either letters or numbers only is little better than immediate memory. [But it is better with locations. So asking for locations is a better method for finding how long images stay in our minds. Presumably there is not an unwanted advantage with positions, because in both cases the cue can come after the fact (and not be from guessing location ahead of time, see experiment 4). So presumably whether the subject is asked for types or for locations, in both cases, the image would have been equally available to their sensory memory.]

The failure in Experiment 6 to detect a substantial difference in accuracy between partial reports of only letters (or only numbers) and whole reports clearly illustrates that partial reports by position are more effective for studying the capacity of short-term information storage than partial reports by category.
[p.16]

[Experiment 7: Order of Report, skipped. From the results: “The results obtained in this experiment support the conclusions that both a position preference and the order of report ordinarily correlate with the accuracy of response, but that probably neither are necessary conditions for response accuracy.” p.19]

Discussion:

Sperling has two questions regarding why subjects remember more in partial reports than in whole reports.

(a) Why is the partial report more accurate than the whole report ? (b) Why does the partial report retain this added accuracy only for a fraction of a second after the exposure?
[20]

One way to answer [question b]  is with the subjective accounts of the subjects. They say that the image stays in their vision even when the tone sounds 150 msec after the image has physically disappeared.

The answers proposed are a systematic elaboration of an observation that is readily made by most viewers of the actual tachistoscopic presentation. They report that the stimulus field appears to be still readable at the time a tone is heard which follows the termination of the stimulus by 150 msec. In other words, the subjective image or sensation induced by the light flash outlasts the physical stimulus at least until the tone is heard. The stimulus information is thus "stored" for a fraction of a second as a persisting image of the objective stimulus. As the visual image fades, its legibility (information content) decreases, and consequently the accuracy of reports based upon it decreases. [p20AB]

Sperling then notes that sensation is not instantaneous. So the fact that the subject still sees the image after its extinction partly results from the fact that it takes time for the sensation to register in the subject’s awareness. But this is not necessarily related to the persistence of the image. It only refers to the delay. So what matters is not the delay after the extinction of the stimulus but rather the duration it remains before decaying.

There is other evidence, besides such phenomenological accounts, that suggests that information is available in the form of an image for a short time after extinction of the physical stimulus. In the first place, it is inconceivable that the observers should stop seeing the stimulus at exactly the moment the light is turned off. The rise and fall of sensation may be rapid, but they are not instantaneous. The question is not whether the observer continues to see the stimulus after the illumination is turned off, but for how long he continues to see the stimulus.
[p. 20]

Sperling then cites other research which would estimate the persistence of vision to be from 0.05 to 1.0 sec, and most probably around 1/6 of a sec.

These estimates of the persistence of the visual sensation vary from a minimum of 0.05 sec. (Wundt, 1899) to almost one sec. (McDougall, 1904). The most representative estimates are in the neighborhood of l /6 sec. (cf. Pieron, 1934), a figure that is in good agreement with the results.¹³ [ft 13: Measurements of the persistence of sensation have almost invariably used techniques which have at most questionable validity. Wundt's method depends upon masking, the effect of the persisting stimulus upon another stimulus. The masking power of a stimulus may be quite different from its visibility. McDougall's measurements, as well as those cited by Pieron, depend upon motion of a stimulus across the retina. Such measurements are undoubtedly influenced by the strong temporal and spatial interactions of the eye (Alpern, 1953). Schumann's ingenious application of the method of Baxt to the determination of persistence is probably the only experiment that utilizes pattern stimulation. The other methods have not been tried with pattern stimuli although there .is, a priori, no good reason why they have not been. The possibility that the persistence of pattern information is quite different from persistence of "brightness" has not been investigated.] [Citing these sources: Wundt, W. An introduction to psychology. London: Allen & Unwin, 1925 (reprinted.) Transl. from 2nd German ed, by R Pinter. Edinburgh, Ballentyre Press, 1912. ((see pdf, where much of the bib info is overwritten by hand.)). McDougall, W. The sensations excited by a single momentary stimulation of the eye. Brit. J. Psychol., 1904, 1, 78-113. Pieron, H. L'evanouissement de la sensation lumineuse: Persistance indifferenciable et persistance totale. Ann. psychol., 1934, 35, 1-49. Alpern, M. Metacontrast. J. Opt. Soc. Amer., 1953, 43, 648-657.]
[p.20]

Sperling again on the persistence of vision:

This then is the evidence-phenomenological reports, the effects of the postexposure fields, the known facts of the persistence of sensation, and the detailed characteristics of the responses-that is consistent with the hypothesis that.information is initially stored as a visual image and that the Ss can effectively utilize this information in their partial reports. In the present context, the term, visual image, is taken to mean that (a) the observer behaves as though the physical stimulus were still present when it is not (that is, after it has been removed ) and that (b) his behavior in the absence of the stimulus remains a function of the same variables of visual stimulation as it is in its presence, The units of a visual image so defined are always those of an equivalent "objective image," the physical stimulus. It is as logical or illogical to compute the information contained in a visual image (as was done in Experiments 3 and 4) as it is to compute the information in a visual stimulus.
[p.21]

But as indicated above, the subject behaves ‘as though’ it were present. Perhaps in fact it is not visibly present but only seems be in their behavior.

"Visual image" and "persistence of sensation" are terms suggested by the asynchrony between the time during which a stimulus is present and the time during which the observer behaves as though it were present. Although asynchrony is inevitable for short exposure durations, there is, of course, no need to use the term "visual image" in a description of this situation. One might, for example, refer simply to an "information storage" with the characteristics that were experimentally observed. This form of psychological isolationism does injustice to the vast amount of relevant researches.
[p.22]

Persistence of Vision and Afterimages

[It would seem the difference so far used with the terminology is this: persistence of vision is the very immediate lingering impression, and the afterimage remains shortly after, less than longer-term memory.]

Between the short persistence of vision and the remembrance of a long-passed event, there is an intermediate situation, the afterimage, which requires consideration. In discussing afterimages, it will be useful to distinguish some phases of vision that normally follow an intense or prolonged stimulus. First, there is the "initial" (or primary, or original) "image" (or sensation, or impression, or perception, or response). Any combination of a term from the first and from the second of these groups may be used. The initial image is followed by a latent period during which nothing is seen and which may in turn be followed by a complex sequence of afterimages.
[p.22]

George Sperling. The information available in brief visual presentations. Psychological monographs: General and applied, Vol. 74, No. 11. (1960), pp. 1-29.

http://www.citeulike.org/group/1696/article/995840

http://aris.ss.uci.edu/HIPLab/staff/sperling/PDFs/Sperling_PsychMonogr_1960.pdf