[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
“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.
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”  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.”  [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.
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.
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.
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.
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.
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.
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. 
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.
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. 
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. 
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.
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. 
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.
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.
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. 
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.
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.
[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.
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. 
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.
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.”  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.” 
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.
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.
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.
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. 
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.
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.” 
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.” 
In geometry today, proofs no longer depend on figures. 
Peano conducts his geometry with points but without any reference to space. 
Euclid’s propositions are not entirely supported. 
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.
Weierstrass and his follows showed that many of the basic propositions in mathematics were false. 
Kant’s doctrine of a priori intuitions does not apply to today’s mathematics.  For math to progress, we need to devote attention to mathematical logic. 
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”]
Leibniz. Philosophical Papers and Letters. Ed. & Transl. Leroy E. Loemker. Dordrecht: D. Reidel Publishing Company, 1956.