Lectio 11 Lecture 11 Solvuntur rationes Zenonis, quibus motum omnem excludere conatus est The arguments of Zeno, who tried to deny all motion, are answered Zeno autem male ratiocinatur et paralogizat. Si enim semper, dicit, quiescit omne aut movetur, cum est secundum aequale; est autem semper quod fertur in ipso nunc: immobilem eam esse sagittam quae fertur. Hoc autem falsum est. Non enim componitur tempus ex ipsis nunc indivisibilibus, sicut nec alia magnitudo ulla. Zeno’s reasoning, however, is fallacious when he says that, if everything is at rest when it occupies an equal space and if what is in locomotion is always occupying such a space at any moment, then the flying arrow is therefore motionless. This is false, for time is not composed of indivisible moments any more than any other magnitude is composed of indivisibles. Quatuor autem sunt rationes de motu Zenonis, ingerentes difficultatem solventibus. Zeno’s arguments about motion, which cause so much disquietude to those who try to solve the problems that they present, are four in number. Prima quidem de eo quod non movetur, propter hoc quod prius in medium oportet accedere quod fertur quam ad finem: de qua divisimus in prioribus rationibus. The first asserts the non-existence of motion on the ground that what is in locomotion must arrive at the half-way stage before it arrives at the goal. This we have discussed above. Secunda autem vocata Achilles. Est autem haec, quod tardius nequaquam iungetur currens a velocissimo: ante enim necesse est ire persequens unde movit fugiens; quare semper habere ante aliquid necesse est tardius. The second is the so-called “Achilles,” and it amounts to this: in a race, the quickest runner can never overtake the slowest, since the pursuer must first reach the point whence the pursued started, such that the slower must always hold a lead. Est autem haec eadem ratio in decidendo in duo: differt autem in dividendo non in duo acceptam magnitudinem. This argument is the same in principle as that which depends on bisection, though it differs from it in that the spaces with which we successively have to deal are not divided into halves. Non quidem igitur coniungi tardius accidit ea ratione: fit autem ad idem in duo decisioni. In utrisque enim accidit non attingere ad terminum, divisa quodammodo magnitudine. Sed apponitur in hac, quia neque velocissimum, quod cum tragoedia dictum est, in persequendo tardius. Quare necesse est eandem esse solutionem. The result of the argument is that the slower is not overtaken, but it proceeds along the same lines as the bisection-argument (for in both, a division of the space in a certain way leads to the result that the goal is not reached, though the “Achilles” goes further, in that it affirms that even the quickest runner in tragic tradition must fail in his pursuit of the slowest), such that the solution must be the same. Velle autem quod praecedens non iungatur, falsum est. Cum enim praecedit, non coniungetur: sed tamen coniungetur, si quidem dabitur transire finitam. Hae quidem igitur rationes sunt duae. And the axiom that that which holds a lead is never overtaken is false: it is not overtaken, it is true, while it holds a lead, but it is overtaken nevertheless if it is granted that it traverses the finite distance prescribed. These, then, are two of his arguments. Tertia autem, quae nunc dicta est, quoniam sagitta quae fertur, stat. Accidit autem quia accipit tempus componi ex ipsis nunc: non dato enim hoc, nullus erit syllogismus. The third is that already given above, to the effect that the flying arrow is at rest, which result follows from the assumption that time is composed of moments; but if this assumption is not granted, the conclusion will not follow. Quarta autem ex his quae moventur stadio, ex contrarietate aequalium magnitudinum iuxta aequalia, his quidem a fine stadii, illis vero a medio, aequali velocitate. The fourth argument is that concerning bodies moved in an equal manner on a race course with equal velocity in opposite directions—the one originally between the goal and the midpoint, and the other between the midpoint and the start. In quo accidere opinatur aequale tempus dimidium duplo. This, he thinks, involves the conclusion that half a given time is equal to double that time. Est autem deceptio in eo quod hoc quidem secus motum, illud autem secus quiescens, aequalem magnitudinem, velle aequali velocitate secundum aequale ferri tempus. Hoc autem falsum est. The fallacy of the reasoning lies in the assumption that a body occupies an equal time in passing with equal velocity a body that is in motion and a body of equal size that is at rest, which is false. Ut sint stantes aequales magnitudines in quibus sunt AAA; aliae autem in quibus ipsa BBB, incipientes a medio ipsorum A, quae aequales secundum numerum et magnitudinem sunt; aliae autem, in quibus ipsa CCC, ab ultimo, aequales numero his et magnitudine, et aeque veloces ipsis B. For instance (so runs the argument), let AAA be the bodies of equal size which are standing still, and let BBB the bodies equal in number and in size to AAA and originally occupying the half of the course from the starting post to the middle of AAA, and let CCC be those originally occupying the other half from the goal to the middle of AAA, equal in number, size, and velocity to BBB. Contingit igitur primum B simul cum ultimo A esse, et primum C secus invicem motorum. Then three consequences follow: first, as BBB and CCC pass one another, the first B reaches the last C at the same moment as the first C reaches the last B. Accidit autem ipsum C iuxta omnia A transire, et ipsum B secus media A. Quare medium esse tempus: aequale enim utrumque est secus unumquodque. Second, at this moment, the first C has passed all of AAA, whereas the first B has passed only half of AAA, and has consequently occupied only half the time occupied by the first C, since each of the two occupies an equal time in passing each A. Simul autem accidit ipsum B secus omnia C transactum esse: simul enim erit primum B et primum C in contrariis ultimis; in aequali tempore ad unumquodque factum ipsorum B, quantum quidem ipsorum A, ut ait, propter ambo aequali tempore secus ipsa A fieri. Third, at the same moment, all of BBB has passed all of CCC, for the first C and the first B will simultaneously reach the opposite ends of the course, since (so says Zeno) the time occupied by the first C in passing each B is equal to that occupied by it in passing each A, because an equal time is occupied by both the first B and the first C in passing all of AAA. Ratio igitur haec est. Incidit autem ad dictam falsitatem. This is the argument, but it presupposed the previously stated fallacious assumption. Neque igitur secundum mutationem in contradictione, nihil nobis erit impossibile: ut si ex non albo in album mutetur, et in neutro est, tanquam ergo neque album erit neque non album. Non enim si non totum in quolibet est, non dicetur album aut non album: Nor, in reference to contradictory change, shall we find anything unanswerable in the argument that, if a thing is changing, say, from not-white to white and is in neither condition, then it will be neither white nor not-white. For the fact that it is not wholly in either condition will not preclude us from calling it white or not-white. album enim dicitur aut non album, non quod totum sit huiusmodi, sed quod plures aut magis propriae partes; non idem autem est non esse in hoc, et non esse in hoc totum. We call a thing white or not-white not necessarily because it is wholly either in one or the other, but because most of its parts or the most essential parts of it are so: not being in a certain condition is different from not being wholly in that condition. Similiter autem est et in esse et in non esse, et in aliis quae secundum contradictionem sunt: erit enim ex necessitate in altero oppositorum, in neutro autem non semper totum. It is the same in the case of being and non-being and all other conditions that stand in a contradictory relation: while the changing thing must of necessity be in one of the two opposites, it is never wholly in either. Iterum autem in circulo et in sphaera, et omnino in his quae in ipsis moventur, quia accidet ipsa quiescere. In eodem enim loco secundum tempus quoddam sunt et ipsa et partes: quare quiescent simul et movebuntur. Again, in the case of circles and spheres and everything whose motion is confined within the space that it occupies, it is not true to say the motion can be nothing but rest on the ground that such things in motion—themselves and their parts—will occupy the same position for a period of time, and that therefore they will be at once at rest and in motion. Primum namque partes non sunt in eodem nullo tempore. Postea et totum mutatur semper in alterum. For in the first place, the parts do not occupy the same position for any period of time; in the second place, the whole also is always changing to a different position; Non enim eadem est ab ipso A accepta circulatio, et quae est ab ipso B et C, et ad aliorum unumquodque signorum, nisi sicut musicus homo et homo, quia accidit. for if we take the orbit as described from a point A on a circumference, it will not be the same as the orbit as described from B or C or any other point on the same circumference, except in an accidental sense—the sense, that is to say, in which a musical man is the same as a man. Quare mutatur semper altera in alteram, et nequaquam quiescet. Eodem autem modo est et in sphaera et in aliis quae in seipsis moventur. Thus, one orbit is always changing into another, and the thing will never be at rest. And it is the same with the sphere and all other things that are moved within themselves. 860. Postquam Philosophus determinavit de divisione motus et quietis, hic excludit quaedam, ex quibus errabant aliqui circa motum. 860. After finishing with the division of motion and of rest, the Philosopher now refutes certain opinions that have been the source of error in regard to motion. Et circa hoc tria facit: About this, he does three things: primo solvit rationes Zenonis, negantis totaliter motum esse; first, he answers the arguments of Zeno, who denies that motion exists at all; secundo ostendit quod indivisibile non movetur, contra Democritum, qui ponebat indivisibilia moveri semper, ibi: ostensis autem his etc.; second, he shows that an indivisible is not moved, against Democritus, who said that they are always in motion, at our next point is that what is (240b8; [872]). tertio ostendit mutationem omnem esse finitam, contra Heraclitum, qui ponebat omnia moveri semper, ibi: mutatio autem etc. third, he shows that all change is finite, against Heraclitus, who said that all things are eternally moved, at our next point is that no process (241a26; [879]). Circa primum duo facit: About the first, he does two things: primo ponit quandam rationem Zenonis et solvit eam, quae pertinet ad id quod immediate de motu praemiserat; first, he gives and rejects one of Zeno’s arguments, which pertains to what Zeno had accepted about motion; secundo explicat omnes rationes eius per ordinem, ibi: quatuor autem sunt rationes etc. second, he explains all his arguments in order, at Zeno’s arguments (239b9; [862]). 861. Dicit ergo primo quod Zeno male ratiocinabatur, et apparenti syllogismo utebatur ad ostendendum quod nihil movetur, etiam illud quod videtur velocissime moveri, sicut sagitta quae fertur. Et erat ratio sua talis. Omne quod est in loco sibi aequali, aut movetur aut quiescit: sed omne quod fertur, in quolibet nunc est in aliquo loco sibi aequali: ergo et in quolibet nunc aut movetur aut quiescit. Sed non movetur: ergo quiescit. Si autem in nullo nunc movetur, sed magis videtur quiescere, sequitur quod in toto tempore non moveatur, sed magis quiescat. 861. He says, therefore (239b5), that Zeno reasoned badly and used what had only the appearance of a syllogism to show that nothing is being moved—even what seems to be in rapid motion, as an arrow in flight. And this was his argument. Anything that is in a place equal to itself is either being moved or is at rest. But whatever is being moved is at each instant in a place equal to itself. Therefore, even at each instant, it is either in motion or at rest. But it is not in motion; therefore, it is at rest. But if it is not in motion at any instant but at rest, as it seems, then it is at rest throughout the entire time and not in motion. Posset autem haec ratio solvi per id quod supra ostensum est, quod in nunc neque movetur neque quiescit. Sed haec solutio intentionem Zenonis non excluderet: sufficit enim Zenoni, si ostendere possit quod in toto tempore non movetur; quod videtur sequi ex hoc quod in nullo nunc eius movetur. Et ideo Aristoteles aliter solvit, et dicit falsum esse quod ratio concludit, et non sequi ex praemissis. Now, this argument could be answered by appealing to something already proved: in an instant, there is neither motion nor rest. But such a solution would not cripple Zeno’s intention, for he is satisfied to show that, through the entire time, there is no motion. This seems to follow if there is no motion at any instant of the time. Therefore, Aristotle answers in a different manner and says that the conclusion both is false and does not follow from the premises. Ad hoc enim quod aliquid moveatur in tempore aliquo, oportet quod moveatur in qualibet parte illius temporis: ipsa autem nunc non sunt partes temporis; non enim componitur tempus ex ‘nunc’ indivisibilibus, sicut neque aliqua magnitudo componitur ex indivisibilibus, ut supra probatum est: unde non sequitur quod in tempore non moveatur aliquid, ex hoc quod in nullo nunc movetur. For in order that something be moved in a given period of time, it has to be moved in each part of the time. But instants are not parts of time, for time is not made up of instants any more than a magnitude is made of points, as we have already proved. Hence, it does not follow that a thing is not in motion in a given time just because it is not in motion in any instant of that time.