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Sextus Empiricus - Against the Professors

Sextus Empiricus' main surviving work, Against the Professors is also known as Against the Mathematicians (Πρὸς μαθηματικούς in Greek, Adversus Mathematicos in Latin), based on the older meaning of mathematician (máthēma simply meaning "knowledge"), or Against Those in the Disciplines. It is a work mainly dedicated to providing counterarguments to many common positions on the various topics it broaches during that era (the second century CE), in line with his own philosophical position of Pyrrhonism, or skepticism, which rejects dogma and advocates the suspension of judgement over the truth of all beliefs.

For the purpose of this project, I will mainly go over the four books directly connected to science, Against the Geometers (Πρὸς γεωμετρικούς), Against the Arithmeticians (Πρὸς ἀριθμητικούς), Against the Logicians (Πρὸς λογικούς) and Against the Physicists (Πρὸς φυσικούς).

  1. Against the Geometers
  2. Against the Arithmeticians

Against the Geometers

Against the Geometers is dedicated to counterarguments to the field of geometry, mostly specifically physical geometry, that is, the connection of abstract geometry to the physical world, mostly from the "main" tradition of geometry at the time, that of the Aristotelians as could be found in Aristotle's Physics. The basic text is that of Robert Gregg Bury's translation, who died in the United Kingdom in 1951, making it public domain as of 2021.

Since the Geometers, perceiving the multitude of difficulties which beset them, take refuged in a method which seems to be free from danger and sage, namely, to beg by "hypothesis" the principles of geometry, it will be well for us, too, to begin our attack against them with the argument against "hypothesis". For Timon, in his book Against the Physicists [A lost book], assumed that one ought to raise this question first of all, I mean, whether anything should be accepted from "hypothesis". Hence it is proper for us, in conformity with him, to do likewise in our treatise against these Mathematicians. And, for the sake of due order, one must premise that the word "hypothesis" is used in a number of different senses; but it will be enough now to mention three : in one sense it means the peripeteia (or "argument" or "plot") ["that part of a drama in which the plot is tied together and the whole concludes, the denouement"] of a drama, as we say that there is a tragic or comic "hypothesis", and certain "hypotheses" of Dicaearchus [A voluminous writer, disciple of Aristotle] of the stories of Euripides and Sophocles, meaning by "hypothesis" nothing else than the peripeteia of the drama. And "hypothesis" is used with another signification in rhetoric, as investigation of particulars, in which sense the sophists are wont to say often in their discourses, "One must posit the hypothesis". Moreover, in a third application we term the starting point of proofs "hypothesis", it being the postulating something for the purpose of proving something. Thus we say that Asclepiades made use of three "hypothesis" to demonstrate the initial condition which produces fever, the first, that there exist in us certain intelligible (or "non-perceptible") passages, differing from one another in size; the second, that particles of moisture and air are collected from all sides out of corpuscules perceived by reason and eternally in motion; the third, that certain unceasing effluvia are emitted from within us to the outside air, these being now more, now less, in number according to the condition prevailing at the moment.

Well then, "hypothesis" being now conceived in these three ways, we certainly do not now propose to inquire about the arrangement in dramas, nor about oratorical questionings, but about the "hypothesis" in the sense mentioned last, which was "the starting point of proof"; for this is the "hypothesis" which the Geometers adopt when they wish to prove anything geometrically. Consequently, we must state at once that since those who assume a thing by hypothesis are satisfied with mere assertion, without proof, for its confirmation, one will interrogate them, employing some such reasoning as this : Assuming a thing by hypothesis is either a strong and firm confirmation or unreliable and weak. But if it is strong, its contrary, when assumed by hypothesis, will also be reliable and firm, so that we shall be positing conflicting things simultaneously. But if the hypothesis is unreliable in the case of the man who assumes the contrary by hypothesis without proof, it will also be unreliable in the case of the other man, so that we shall posit neither of the things. Nothing, then, must be assumed by hypothesis. Moreover, the thing which is assumed is either true and such as we assume it to be, or false. But if it is true, let us not postulate it, fleeing for refuge to a thing which is highly suspicious - namely, hypothesis - but let us accept it straightaway, for no one assumes ex hypothesi things true and actual, such as "Now it is day", or "I am talking and breathing"; for the very obviousness of these facts does of itself make the statement firm and the assumption undisputed. So that if the thing is true, let us not postulate it as though it were not true. But if it is not true but is false, no help will emerge from the hypothesis; for though we assume it a myriad times, on rotten foundations, as the saying goes, will follow the conclusion of the inquiry which starts from non-existent principles. Moreover, if anyone shall maintain that the conclusions which follow from whatever assumptions are made are trustworthy, it is to be feared that he is destroying all inquiry. For example, each of us will assume that three is four, and, this being granted, will infer also that six is eight; for if two is four, six will be eight; but in fact, as the hypothesis grants, thee is four, therefore six is eight. Again, we shall postulate that what moves is at rest, and this being agreed, we shall infer that the flame is stationary; for if what moves is at rest, the flame is stationary; but what moves is at rest, therefore the flame is stationary. But just as the Geometers will say that these hypotheses are absurd (for the foundation must be firm in order that the inference which follows may be agreed), so too we shall refuse to accept any of their hypothetical assumptions without proof. Moreover, if the assumption, as assumed, is firm and trustworthy, let them not assume the things from which they will prove something, but the thing proved itself, that is, not the premisses of the proof but its conclusion; for the power for confirmation which their hypothesis possesses in the case of the things which reveal, the same power it will possess in the case of the things revealed by the proof. But if the conclusion of the proof without proof is untrustworthy, though it be assumed many times over, that which is assumed in order to demonstrate it will also be untrustworthy unless it be taught by means of proof. But in Heaven's name, they say, if what follows the hypotheses is found to be true, certainly the things assumed - that is, the things which it has followed - will be true. But this again is silly; for how do we know that that which follows certain things in a proof is in all cases true? For they will assert this as having learnt it either from the thing itself or from the premises which it followed. But they will not assert it from itself. For it is non-evident, and the non-evident is not of itself trustworthy; at any rate they essay to prove it, as though it were not of itself convincing. Nor yet from the premisses; for the whole controversy is about these, and while they are still unconfirmed the thing which is being proved by means of them cannot be firm. Further, even if the consequent is true, the antecedent is not inevitably true. For as the true naturally follows the true, and the false the false, so it is maintained that the true is a consequence of the false [ex falso quodlibet], for example, that "the earth exists", which is true, follows "the earth flies", which is false. Hence, if the consequent is true, the antecedent is not in all cases true, but when the consequent is true it is possible for the antecedent to be false.

So now, by these arguments it has been sufficiently established that the Mathematicians do no good by assuming ex hypothesi the principles of proof and of each theorem, repeating the formula "Let it be granted". Passing on, let us show in the next place that the principles of their art are in fact false and incredible. Now many arguments can be used to prove this, as we said when commencing our exposition, but our doubts shall be cast on those principles the destruction of which will involve that of the rest. So, since their particular proofs cannot go forward when the principles are under suspicion, let us state suitable arguments against the principles.

To start with they tell us, as a primary and most fundamental fact, that "body" is that which has three dimensions - length, breadth, depth - and of these the first dimension, that of length, is from right to left, the third, that of depth, from before to behind. Thus there are six extensions of these three, two in each case, up and down of the first, right and left of the second, before and behind of the third. For they assert that the line is produced by the flow of the point, the surface by that of the line, and the solid body by that of the surface. So in describing these they say that "the point is a sign without parts or dimensions", or "the limit of a line", "the line is length without breadth", or "the limit of a surface", and "the surface is the limit of a body", or "breadth without depth". Taking these, then, in order, let us speak first about the point, next about the line, and after that about the surface and body; for if these are destroyed Geometry will not be an Art, as not possessing the conditions upon which success in its construction seems to depend.

Now the point, which they say is "a sign without dimensions", is conceived as either a body or incorporeal. And according to them it will not be a body; for things which have no dimension are not bodies. It remains, then, to say that it is incorporeal; but this again is incredible. For the incorporeal, as being impalpable, is conceived as generative of nothing, but the point is conceived as generative of the line; so the point is not a sign without dimensions. Moreover, if apparent things are "the vision of things non-evident", then, since in apparent things it is impossible to perceive a limit of anything or sign which is without dimension, it is plain that no such thing will be perceived in intelligible things either. But in fact, as I shall establish, it is impossible to perceive in things sensible anything without dimensions; so that it is also impossible in intelligibles. Now everything which is perceived in sensibles as the limit and sign of something is apprehended as being likewise the extremity of something, and also as being part of that whereof it is the extremity; if, then, we take it away, that from which it is taken will be diminished. And that which is part of a thing clearly helps to complete that thing, and that which helps to complete a thing will certainly increase its magnitude, and what serves to complete a magnitude necessarily possesses magnitude. Therefore every sign or extremity of anything in sensibles, as possessing magnitude, is not without dimensions. Hence, if we conceive the intelligible by transference from the sensible, we shall conceive it as being the sign and limit of the line, and also as helping to complete it, so that it too will certainly possess a dimension since it is productive of a dimension. Furthermore, they say that the straight line drawn from the center when it revolves describes a circle in the plane with its limit. Since then, the extremity of this straight line is a sign, and this by revolving measures out the circumference possesses a dimension; so the sign, too, which helps to complete it will possess a dimension. Moreover, it is held that the sphere touches the plane at one sign, and by rolling forward makes a line, the signs which make contact successively composing, as is evident, the whole line. Then, if the sign helps to complete the magnitude of the line, it too will possess magnitude. But it has been agreed that it does not help to complete the magnitude of the line; therefore it too will possess magnitude and will not be without dimensions.

But in answer to these objections Eratosthenes is accustomed to say that the sign neither occupies any space nor measures out the interval of the line, but by flowing makes the line. But this is inconceivable. For flowing is conceived as extension from a place to a place, as water extends. And if we shall imagine the sign to be something of that sort, it will follow that it is not like a thing without parts, but of the opposite sort, abounding in parts.

So much, then, concerning the point : in the next place let us see what ought to be said concerning the line; for this comes next in order after the point. Now even though it be granted that a point exists, the line will not exist. For if it is "a flux of the sign" and "length without breadth", it is either a single sign extended in length or a number of signs placed in a row without intervals; but it is neither a single sign extended in length, as we shall establish, nor a number of signs placed in a row, as we shall also show; therefore line does not exist. For if it is a single sign, this sign either occupies one place only or moves on from place to place, or is extended from a place to a place. But if it is contained in one place, it will not be a line but a point; for the line was conceived as the result of flux. And if it moves from place to place, either it moves - as I said before - by quitting one place and occupying another, or by keeping to one place and extending to another. But if it is by quitting one place and occupying another, again it will not be a line but a point; for as it was conceived as a point but not a line when it occupied the first place, so, by the same reasoning, it will be conceived as a point when it occupies the second place. And if it is keeping to one place and extending to another, it extends over place which is either divisible or indivisible. But if it extends over indivisible place, once again it will not be a line but a point or sign, for that which occupies indivisible place is indivisible, and that which is indivisible is a point and not a line. And if it extends over divisible places, then, since that which extends over the divisible has parts, since it is extended over all the place, and that which has parts wherewith it extends over the parts of the place is body, the sign will certainly be both divisible and corporeal; which is absurd. Consequently, the line is not one single sign. Nor yet is it a number of signs placed in a row. For these signs are conceived either as touching one another or as not touching. If as not touching one another, being intercepted they will be separated by certain spaces, and being separated by spaces they will no longer form one line. And if they are conceived as touching one another, they will either touch wholes as wholes or parts with parts. But if they shall touch parts with parts, they will no longer be without dimensions and without parts; for the sign which is conceived - shall we say ? - as midway between two signs will touch the sign in front with one part, and that behind with another, and the plane with a different part, and the other place with yet another, so that in very truth it is no longer without parts but with many part.

And if the signs as wholes should touch wholes, it is plain that signs will be contained in signs and will occupy the same place; an thus they will not be placed in a row, so as to form a line, but if they occupy the same place they will form one point. If, then, in order that the line may be conceived it is necessary that the sign, from which the notion of it is derived, should first be conceived, and it has been shown that the line is neither a sign nor composed of signs, then the line will be nothing.

Moreover, leaving aside the notion of the sign we can destroy the line directly and show its inconceivability. For tthe line, as one may learn from the Geometers themselves, is "length without breadth", but when we have examined the matter closely, we shall not find either amongst intelligibles or amongst sensibles anything that is capable of being perceived as length without breadth. Not amongst sensibles, since whatever sensible length we perceive we shall in every case perceive it as combined with a certain amount of breadth; nor amongst intelligibles, inasmuch as we can conceive one length as narrower than another, but when we keep the same length invariably and in thought cut slices from its breadth and keep doing this up to a point, we shall conceive the breadth as growing less and less, but when we reach the point of finally depriving the length of breadth we shall no longer be imagining even length, but even the notion of length will be destroyed. In general, also, everything conceived is conceived in two main ways, either by way of clear impression or by way of transference from things clear, and this way is three-fold, by similarity, or by composition, or by analogy.

Thus, by clear impression are conceived the white, the black, the sweet and the bitter, and by transference from things clear are concepts due to similarity, such as Socrates himself from a likeness of Socrates, and those due to composition, such as the hippocentaur from horse and man, for by mixing the limbs of horse and man we have imagined the hippocentaur which is neither man nor horse but a compound of both. And a thing is conceived by way of analogy also in two ways, sometimes by the way of increase, sometimes by decrease; for instance, from ordinary men :

Such mortals as now we see...

we conceive by way of increase the Cyclops who was

Less like a corn-eating man than a forest-clad peak of the mountains [From Homer's Odysseus ix 191]

and by way of decrease we conceive the pygmy whom we have not perceived through sense-impression. Now the modes of conception being so many, if length without breadth is conceived it must necessarily be conceived either by way of clear sense-impression or by way of transference from clear things; but it will not be conceived by way of clear sense-impression; for we have had no impression of any length without breadth. It remains, then, to say that it is conceived by way of transference from clear things; but this again is most impossible. For if it was conceived in this way, it was certainly conceived either through similarity or through composition or through analogy; but in none of these ways can it naturally be conceived, as we shall establish; therefore no length without breadth is conceived. For it is obviously impossible to conceive a length without breadth by way of similarity. For we have no length without breadth amongst things apparent by means of which we might conceive a similar length without breadth. For what is similar to anything is certainly similar to a thing known, and it is impossible to find a thing similar to what is not known. Since, then, we possess no clear impression of a length without breadth, we shall not be able to conceive anything similar to it. Nor yet is it possible for the Geometers to get the notion of it by way of composition; for let them tell us which of the things clearly known from sense-impression are we to compound with which so as to conceive length without breadth, as we did before, in the case of man and horse, when we imagined the hippocentaur. It remains, then, for them to take refuge in the third mode of conception, that of analogy, by way of increase or decrease; but this again is seen to be hopeless. For things conceived by analogy have something in common with the things wherefrom they are conceived, as for instance from the common size of men we conceived by way of increase the Cyclops and by way of decrease the pygmy, so that things conceived by analogy have something in common with the things wherefrom they are conceived. But we find nothing in common between the length that is without breadth and that conceived along with breadth, so that by setting out from the latter we might conceive length without breadth. But if we find nothing in common to them both we shall not be able to form the conception of length without breadth by analogy. Hence, if each of the concepts is conceived according to the modes described, and it has been shown that length without breadth is conceived according to none of them, then length without breadth is inconceivable.

Notwithstanding, even to arguments so clear as these the Geometers manfully endeavour to reply, as best they can, saying that length without breadth is conceived by way of "intension". Thus, when we have taken any given length along with a certain amount of breadth, they say that we diminish this breadth by "intension", intensifying ever more and more its narrowness, and so in the end we say that what is thus conceived by way of intension is length without breadth; for if the breadth is lessened little by little by being narrowed through intension, at some time it will come to be a length without breadth, the conception ending up in this. But surely, someone will say, we have proved that complete privation of breadth is also the abolition of length. Also, that which is conceived through the intension of something is not different from the thing preconceived but just that thing intensified. Since, then, we desire to conceive a thing by way of intension of its narrowness from that which has a certain amount of breadth, we shall not conceive length which is entirely without breadth (for that is different in kind), but we shall apprehend a narrow breadth, so that the conception ends up in the very least amount of breadth, but still breadth all the same, and after this the notion in the mind passes into something different in nature, which is neither length nor breadth. And if it is possible to apprehend length without breadth by privation of the breadth when we have conceived a certain length along with a certain amount of breadth, then it will be feasible in like manner when we have conceived flesh with the quality of vulnerability to conceive invulnerable and impassive flesh by privation of the quality of vulnerability; and it will be possible by privation of the solidity, to peceive a non-solid body. But this is perfectly impossible and contrary to the common notion of mankind. For that which we conceive as invulnerable is no longer flesh, for flesh includes the quality of vulnerability when conceived as flesh, and the non-solid body is no longer conceived as body, for body, qua body, is conceived as including the quality of solidity. Hence, the length conceived without breadth will not be length, for length, as length, is conceived as including the quality of having a certain amount of breadth.

But although the inconceivability of the thing has been established in a variety of ways, and the Geometers are in a state of no little confusion, yet Aristotle affirms that the length without breath they talk of is not inconceivable but can come into our minds without any difficulty. He bases his argument on an obvious and clear example. Thus we perceive the length of a wall, he says, without thinking simultaneously of its breadth, and therefore it will be possible also to conceive of the "length without any breadth" talked of by the Geometers, seeing that "things evident are the vision of things non-evident"; but he is in error, or perhaps humbugging us. For whenever we conceive the length of the wall without breadth, we do not conceive it as wholly without breadth but without the breadth which belongs to the wall. And thus it is possible for us to combine the length of the wall with a certain amount, however small, of breadth to form a conception of it; so that in this case the length is perceived not without any breadth at all, as the Mathematician claim, but without this particular breadth. But Aristotle's problem was to prove not that the length talked of by the Geometers is devoid of a certain breadth, but that it is wholly deprived of breadth; and this he has not proved.

So much then, concerning these matters; and seeing that the Geometers declare that the line, which is "length without breadth", is also "the limit of the plane", come and let us raise doubts in a more general way concering both lines and planes; for thus the statement about body will become easy to refute. If, then, the line, being length without breadth, is the limit of the plane, it is evident that when a plane is set beside a plane either the two lines will be parallel or both will become one. And if the two lines become one, since the line is the limit of the plane, and the plane the limit of the body, as the two lines become one the two planes also will simultaneously become one plane, and when the two planes have become one, the juxtaposition will not be juxtaposition but unification. But this is impossible. For while juxtaposition can become unification in some cases, as in that of water and things like it, in some cases it cannot; for when stone is set beside stone and iron beside iron and adamant beside adamant they are not unified in respect of their lines. Consequently, the two lines will not become one line. Moreover, if there is unification of the two lines which have become one and natural junction of the bodies, the separation ought to take place when they are pulled asunder, not at the same limits but now at one part and now at another, so that as a result they perish. This, however, is not found to occur, but the limits of the bodies both before the juxtaposition and after the separation are just the same as they originally appeared to be during the juxtaposition. So the two lines do not become one. If, however, the two lines do become one, the bodies set beside each other will have to be less by one extremity; for the two have become one, and this must have one limit and extremity. But the bodies set beside each other do not become less by one extremity, so that the two lines will not become one line. And if the two lines are parallel in the juxtaposition of two bodies, that which results from the two lines will be greater than the one line. But if that which results from the two lines is greater than the one line, one of the two will have breadth, which along with the other will make the dimension greater, and thus the line is not length without breadth. Of two things one, then, we must either do away with the evidence of the senses, or, if this remains unshaken, we must disallow the notion of the Geometers which leads them to suppose that the line is "length without breadth".

This, then, is what we have primarily had to say against the Geometers' principles; so now let us pass on and show that on their ow assumptions it is not possible for their investigation to go forward. Thus, they are fain to believe, as we said above, that the straight line by revolving describes circles with all its parts; but the view that the line is length without breadth is in conflict with this most convincing theorem. Let us probe the matter in this way. If, as they say, every part of the line has a sign, and the sign as it revolves describes a circle, then, whenever the straight line by revolving and describing circles with all its parts measures off the distance from the center to the outermost circumference of the plane, it will be necessary, according to them, that the circles described should be either continuous with one another or separate from one another. But if they are separate from one another it will follow that there is a certain part of the plane which is not encircled, and a part of the straight line which moves over this interval but does not describe a circle. But this is absurd. for either the straight line has no sign in this part, or having one does not describe a circle; but each of these alternatives is contrary to geometrical doctrine; for they assert that every part of the line has a sign, and also that every sign when revolving describes a circle. And if they suppose that the circles are continuous with one another, they are continuous either in such a way as to occupy the same place or so as to be ranged in order one beside another with no sign falling between; for every sign which is conceived as falling between must of itself describe a circle. But if they all occupy the same place, there will be one circle, and therefore the circle which is greater and outermost and inclusive of them all will be equal to the smallest circle which is at the center; for if the outermost circle, that which is on the very circumference, occupies a greater distance, and the innermost circle at the center occupies a little distance, and all the circles occupy the same place, then that which occupies the greater distance will be equal to that which occupies the least distance, which is absurd. So, then, the circles are not continuous in such a way as to occupy the same place. And if they are parallel so that no indivisible sign falls between, they will fill up the breadth from the center to the circumference. But if they fill it up, they occupy some breadth. Yet thesecircles are lines. Lines, therefore, possess a certain breadth and are not "without breadth".

Starting with the same theory we shall construct a confutation similar to that already stated. Since they assert that the straight line which describes a circle describes the circle of itself, we shall reply with the objection, if the straight line which describes a circle is by nature such as to describe the circle of itself, the line is not length without breadth; but in fact, as they assert, the straight line which describes a circle does describe the circle of itself; therefore the line is not length without breadth, this being the consequence of their theory, as we shall show. For when the straight line drawn from the center revolves and of itself describes a circle, the straight line then either moves over all the parts of the breadth within the circumference, or not over all but over some. And if it moves over some, it does not describe a circle, as it moves over some parts but not over others. And if it moves over all, it will measure out all the breadth of the circumference, and measuring out breadth it will possess breadth; for that which is capable of measuring out breadth must possess breadth wherewith it measures. Therefore, if the straight line in describing a circle measures out all the breadth and possesses breadth, the line is not "length without breadth".

The same thing will be shown more clearly when the Geometers state that when the downward side of the square is drawn it measures out the plane bounded by the parallel lines. For if it is length without breadth, the downward side of the square when drawn will not of itself measure out the plane surface of the square bounded by the parallel lines; for that which is capable of measuring out a breadth must possess breadth. And if it measures out, it certainly possesses breadth. So that, once again, either this theorem of the Geometers is false, or the concept "length without breadth" is nothing.

Also, they say that the cylinder touches the plane along a straight line and when rolling forward, by the placing of straight lines in turn, one after another, measures out the plane. But if the cylinder touches the plane along a straight line and when rolling measures out the plane by placing its straight lines in turn, one after another, the plane certainly is composed of straight lines and the surface of the cylinder, too, is made up of straight lines. Hence, since the plane possesses breadth, and the surface of the cylinder likewise is not without breadth, and what is productive of breadth must itself possess breadth, it is plain that the straight lines too, as they serve to fill up the breadth, necessarily possess breadth, so that no "length without breadth" exists, and consequently no line.

And even if we should grant that the line is "length without breadth", the consequences of this will be even more hopeless than those stated. For as the sign when it has flowed makes the line, so also the line when it has flowed makes, according to them, the plane, which is, they say, "the limit of the body", possessing two dimensions, length and breadth. If, then, the plane is the limit of the body, the body certainly is limited; and if so, when two bodies are set beside each other, then either the limits will touch the limits or the things limited the things limited, or the things limited will touch the things limited and also the limits the limits, as though, in the case of a jar, we were to conceive the external earthenware as the limit, and the wine within it as the thing limited. When, then, two jars are set beside each other, either the ware will touch the ware or the wine the wine, or the ware will touch the ware and also the wine the wine. But if the limits touch the limits the things limited (that is, the bodies) will not touch each other, which is absurd. And if the things limited (that is, the bodies) shall touch each other, and the limits shall not touch each other, the bodies will be outside their own limits. And if both the limits touch the limits and the things limited the things limited, we shall be multiplying the difficulties; for where the limits touch each other, the things limited will not touch each other, and where the things limited touch, the bodies will be outside their own limits, since the surface is the limit and the body the thing limited. Also, the limits are either bodies or incorporeal. But if they are bodies, the Geometers will find that it is false that the surface is without depth. For if it is corporeal, it will of necessity have depth; for every body must have depth. Then, too, it will not touch anything but will all be infinite in magnitude. For if it is body, since every body has a limit, that limit too, being a body, will have a limit, and likewise this last one, and so on ad infinitum. And if the limit is incorporeal, since the incorporeal cannot touch or be touched by anything, the limits will not touch each other, and as they do not touch neither will the things limited touch each other. So, even if we grant that the line is "length without breadth", the account given of the plane surface is dubious. And these things being dubious, along with them doubt is cast - even if we do not affirm it - on the solid body, seeing it is composed of these.

Let us consider the matter in this way : If body is, as the Geometers assert, that which has the three dimensions, length, breadth and depth, either the body is separable from these, so that the body is one thing and the length, breadth and depth of the body is something different, or else the aggregation of these is the body. But that the body should be separated from these is not credible; for where neither length nor breadth exists, there it is impossible to conceive body; and if the aggregation of these is conceived as body, and there is nothing else besides these, then, since each of these is incorporeal, the united assemblage of these incorporeals will necessarily be incorporeal. For just as the combination of the points and the conjunction of the lines, which are by nature incorporeal, do not make a solid and resistant body, so too the union of breadth and length, and depth as well, being incorporeal, will not make a solid and resistant body. But if the body is neither separate from these nor identical with these, the body is - so far as the Geometers' account goes - inconceivable. Furthermore, if the conjunction of length and breadth and depth makes body, either each of these is conceived as containing in itself corporeality and what we may call "the corporeal reasons" before the conjunction, or else body is constructed after these have come together. But if each of these is conceived as containing corporeality before the conjunction, each of these will be body, and body will not come into being after the conjunction. Moreover, since body is not length alone, nor breadth by itself, nor exclusively depth, but the three together, length and breadth and depth, and each of these includes corporeality, each of them will possess the three, and the length will be not length only but also breadth and depth, and the breadth will be not breadth only but also length and depth, and similarly the depth will also be length and breadth. But this is most completely illogical. And if the composition of body is conceived as taking place after these have come together, then either the original nature of those things which have come together remains, that of length as length, of breadth as breadth, of depth as depth, or it is changed to corporeality. But if their original nature remains, since they are corporeal they will not form a different body, but even after their conjunction they will remain incorporeal, being incorporeal by nature. And if after coming together they change to corporeality, then, since that which admits of change is ispo facto corporeal, each of these will be body even before their coming together, and thus too incorporeal will be body. Also, just as the body when it has changed exchanges one property for another, but none the less remains body, for example, white to become black, and sweet to become bitter, and wine to become vinegar, and lead to become white lead, and bronze to become rust, exchange one property for another yet do not cease to be bodies, but the black, when from white it has become black, and the bitter, when from being sweet it has become bitter, and the vinegar, when from being wine it has become vinegar, all remain bodies, so these dimensions also, if they change, will change from one sort of incorporeal to another, but none the less will remain incorporeal; for they will not go out of their own proper nature. If, then, it is not possible to conceive the body either before the coming together of these dimensions or after their coming together, and besides these no other alternative can be conceived, body is nothing. And further, if neither length is anything, nor breadth, nor depth, that which is conceived as participating in these will not be body; but length is not anything, nor breadth, nor depth, as we have already pointed out; therefore that which is conceived as participating in these will not be body.

Thus, as regards the principles of geometry, the result is that they are unfounded; and as these are abolished no other geometrical theorem can subsist. For the theorem, of whatever sort it be, must be proved by a diagram, but we have shown that the generic line is nothing, and from this it follows that none of the specific lines exist, whether one assumes a straight one, or a curved one, or one of some other form. Hence, it might, no doubt, have sufficed to finish at this point our confutation of the Geometers; however, we shall contend against them further and try to show that, even if we disregard the principles of geometry, the Geometers are unable to construct or prove a theorem. Before this, however, no little can be said against their underlying principles, as, for instance, when they declare that "a straight line is that which is equally placed with its parts". For, to pass over all other objections, this one is obvious, that the generic line being non-existent, the straight line will not exist; for just as "man" does not exist if "animal" is non-existent, and "Socrates" does not exist if "man" is non-existent, so if the generic line is destroyed the plane straight line is destroyed along with it. Moreover, the term "equal" is used in two senses, in one sense as "equal in magnitude" and neither exceeding nor being exceeded by that to which it is said to be equal (as we say that the staff of a cubit's length is equal to a cubits length), in another sense of "that which has its parts placed equally", that is to say, "the even"; thus, for instance, we call a pavement "equal" instead of "even" (or "level). The term "equal", then, being applied in two ways, when the Geometers in describing the straight line say that "a straight line is that which lies equally with its parts", they are taking the term "equal" either in the first signification or in the second. But if it is in the first, they are perfectly senseless; for there is no sense in saying that the straight line is of equal magnitude with its parts, neither exceeding these nor being exceeded by these. And if it is in the second sense, they will be proving the matter in question by means of itself, seeing that they establish the fact that it is straight from the fact that it has its parts lying evenly and in a straight line, whereas it is not possible to learn hat a thing lies in a straight line without having sensed the straight line. But they are far more absurd when they give the following definition, "A straight line is that which revolves equally with its limits, or this "which in revolving round its limits touches the plane with all its parts". For, firstly, these descriptions are subject to the doubts already expressed by us; and secondly, as the Epicureans affirm, the straight line of the void is, indeed, straight, but does not revolve because the void itself does not admit of motion either as a whole or in part. And the last description falls also into the vice of circular reasoning, which is most unsound. For they both explain the plane by means of the straight line and the straight line by means of the plane; for they say that the straight line is that which touches the plane with all its parts, and the plane is that which, when the straight line is drawn over it, it touches it with all its parts, so that in order to get to know the straight line we must first get to know the plane, and in order to do this, we must necessarily know beforehand the straight line; which is absurd. And, in sum, he who explains the straight line by means of the plane is doing nothing else than establishing the stragiht line by means of the straight line, since, according to them, the plane is many straight lines.

The argument about the angle will be of much the same kind as that about the straight line. For again, when in describing the angle they say that the angle is "the minimum under the inclination of two lines which do not lie parallel", they mean by "minimum" either the indivisible body or what they call the sign or point. But they will not mean the indivisible body, since this cannot be divided into two parts, whereas, according to them, the angle is divided to infinity. And besides, in the case of the angle, one, they say, is greater, another lesser; but nothing is smaller than the minimal body, for if so it, and not the body, would be the minimum. It remains then to say that it is what they call the sign; but this itself is also dubious. For if the sign is in every way wholly without dimensions, the angle will not be divided. Moreover, no angle will be greater or lesser; for in things which have no dimension there will be no difference in respect of magnitude. Besides, if the sign falls between the straight lines, it divides the straight lines, and as dividing it will not be without dimensions. But, in sooth, some of them are wont to say that the angle is "the first interval under the inclination". Against whom

By nature simple is the tale which truth doth tell.

For this interval is either without parts or with parts. But if it is without parts, they will find themselves beset in consequence with the difficulties already stated; and if it has parts, none of them will be "first"; for another will be found to be prior to that assumed to be "first" because of the division of existents ad infinitum which is approved by them. I forbear to argue that such a notion of the angles is in conflict with another piece of their technology. For in their classification they say that one class of angle is "right", another "obtuse", another "acute"; and that, of the obtuse angles, some are more obtuse than others, and so likewise with the acute angles. But if we affirm that the angle is "the least interval under the inclination", such differences in angles will not be preserved, in so far as they both exceed one another and are exceeded by one another. Or, if they are preserved, the angle is destroyed, not possessing a fixed standard by which it can be distinguished.

Such, then, are the arguments we must use against them with respect to the straight line and the angle; and in defining the circle they say "The circle is a plane figure enclosed by one line, and the straight lines from the center which fall on this are equal to one another", talking idly; for when the sign and the line and the straight line, and the plane, too, and the angle are destroyed, the circle cannot be conceived.

But in order that we may not seem to be sophistical people and to expend all the reasoning in our refutation on the principles of geometry alone, come and let us pass on, as we previously promised, and investigate the theorems which come after their principles. When, then, they say that they will "bisect the given straight line", they mean that they are bisecting either that given on the board or that which is conceived by transference from it. But they will not mean that they are bisecting that given on the board; for this appears to possess sensible length and breadth, whereas, according to them, the straight line is "length without breadth", so that the line on the board, not being a line according to them, will not be bisected like a line. Nor, indeed, will the line which is conceived by transference from that on the board. For let us assume, for the sake of argument, that it is composed of nine points, four being numbered from the one extremity and four from the other and one point occupying the middle place between the two sets of four. Then, if the whole line is bisected, the secant will strike either between this fifth point and one of the sets of four or on the fifth point itself so as to divide it in two. That the secant should strike between the fifth point and one of the sets of four is, however, illogical; for the sections will be unequal, one being made up of four points and the other of five. But the dividing the point itself into two is much more illogical than the former alternative; for they will no longer be leaving the sign without dimensions, as it is divided into two by the secant. And the argument is the same when they say they are cutting the circle into equal parts. For if the circle is cut into equal parts, then, since it has the center (which itself is a point), in the very middle, the center will certainly be annexed either to this section or to that, or else it will itself be cut in two. But the fact of its being annexed to this section or that makes the bisection unequal; and that it should itself be bisected is in conflict with the fact that the sign is without dimensions and without parts. Also, the secant which cuts the line is either a body or incorporeal. But it cannot be a body; for, if so, it will not cut a thing without parts and incorporeal and on which it cannot strike; nor yet can it be incorporeal. For this, again, if it is a point, will not cut owing to its being without parts and striking on what is without parts; and if it is a line, again it does not cut since it must cut with its limits, and its limits is without parts. Besides, the limit which cuts bisects the line either by falling between the two points, or by striking on the middle of the sign. But that it should strike on the middle of the sign is a thing impossible. For, as we have said before, that on which it strikes will have to possess parts and be no longer without dimensions. And that it should strike between the two points is much more irrational. For, firstly, no limit can fall in the middle of what is continuous; and secondly, even if we allow that such a thing is possible, it must move apart the things between which it posts itself, if they are continuous; but these are immovable. So then, the account given of the secant is dubious. Moreover, even if we grant them that substractions are made in the case of these sensible lines, even so they will be unable to make progress. For the substraction will either be from the whole line or from a part, and the part substracted will be either an equal part from an equal, or an unequal from an unequal; but none of these is feasible, as we have established in our treatise Against the Grammarians and in that Against the Physicists; therefore it is not possible for the geometers to substract or cut off anything from the line.

Against the Arithmeticians

Against the Arithmeticians is dedicated to counterarguments to arithmetics, by which is mostly meant the doctrines of the Pythagoreans about the nature of numbers.

Since one kind of quantity, which is called "magnitude", and which is the chief concern of geometry, belongs to continuous bodies, and another kind, which is number, the subject of arithmetic, belongs to discontinuous, let us pass on from the principles and the theorems of geometry and examine also those which deal with number; for if this is destroyed, the art which is constructed to handle it will not exist.

Now, speaking generally, the mathematical Pythagoreans ascribe great power to numbers, as though the nature of all things was governed in conformity with them. Hence, they constantly kept repeating :

All things, too, are like unto number.

And they swear not only by number but also by Pythagoras, the man who showed it to them, as though he were a god because of the power of arithmetic, saying :

Nay, by the man I swear who bequeathed to our soul the Tetraktys, Fount containing the roots of Nature ever-enduring.

And "tetraktys" was the name given by them to the number ten, it being composed of the first four numbers. For one and two and three and four make up ten; and this is the most perfect number, since, when we have reached it, we revert again to the one and make our numerations afresh. And they have called it the "fount containing the root of Nature ever-enduring" because, according to them, the reason of the structure of all things resides in it, as for instance that of the body and soul; for it will suffice to mention these by way of example. Now the monad (or one) is an underlying principle which produces the structure of all the other numbers, and the dyasd (or two) is productive of length. For as in the case of the geometrical principles we explained first what the point is, and next, after it, the line which is length without breadth, similarly, in the case before us, the monad corresponds to the point and the dyad to the line and length; for thought in conceiving this moves from some place to some place, and this is length. And the triad (or three) is set over breadth and the plane; for the mind has moved from here to there and on again to some other place, and when the distance in breadth is added to the distance in length the plane is conceived. But if, in addition to the triad one imagines a fourth monad, that is, a fourth sign, the pyramid is formed, a solid body and figure; for it possesses length and breadth and depth; so that the formula of the body is comprised in the number four. And so also is that of the soul; for they declare that as the whole Universe is governed according to harmony, so too the living creature is ensouled. And the perfect harmony is held to consist in three symphonies - that of the "By-Fours" and that of the "By-Fives" and that of the "By-Alls". Now the "By-Fours" symphony consists of the "epitrite" (4:3 ratio), and that of the "By-Fives" in the ratio 3:2, and the "By-Alls" in the ratio 2:. The number called "epitrite" is that composed of a certain number taken as a whole plus its third part - which is the ratio of eight to six; for the eight includes the six plus the third part of it, that is the dyad. And a number is said to be in the ratio 3:2 when the number includes a number plus its half, the relation of nine to six; for the nine is composed of the six plus its half, that is, three. And that called "double" is that which is equal to two equal numbers, the relation of four to two; for it includes the same number twice. Such, then, being the facts, and there being, according to the original assumption, four numbers - one, two, three and four - in which is included, as we said, the form of the soul according to the harmonical formula, the four is double the two and the two double the monad, and therein consists the "By-Alls" symphony; and the three is to the two in the 3:2 ratio (for it includes the two itself plus its half, and this it supplies the "By-Fives" symphony); and the four is to the three in the "epitrite" or 4:3 ratio, on which is based the "By-Fours" symphony. So that naturally the number four is called by the Pythagoreans the "fount containing the root of nature ever-enduring".

From what has been said by way of brief illustration it is clear that they ascribed much power to numbers; for the account they give of numbers is voluminous, but forbearing for the present to dwell on it, let us take up the confutation, beginning our argument with the monad, which is the principle of all number and with the destruction of which number ceases to exist.

Now Plato, in formulating in rather Pythagorean fashion the concept of the one, declares that "One is that without which nothing is termed one", or "by participation in which each thing is termed one or many". For the plant, let us say, or the animal, or the stone is called one, yet is not one according to its own proper description, but is conceived as one by participation in the One, none of them being actually the One. for neither plant nor animal nor stone nor any other numerable object is the essential One. For if a plant or an animal is the One, what is not a plant or an animal will certainly not be termed one; but a plant is termed one, as is an animal and countless other things; therefore none of the numerables is the one. But that by participation in which each thing is by itself each one thing, and a plurality by aggregation, is the One and Many of the individual things. But this Plurality, again, is none of the many things, such as plants, animals, stones; for it is by participation in it that these things are termed "many", but the Plurality itself is not one of them. Such, then, is the Idea of the One as conceived by Plato; so let us subjoin our argument. Either the Idea of the One is different from the particular numerables, or it is conceived along with those things which participate in it. But it does not subsist by itself, since no One other than the particular numerables is conceived as subsisting. It remains, then, to say that it is conceived as included in those things which partake of it, which, again, is dubious. For if the numerable log is one by participation in the Monad, what is not a log will not be termed one; but, as has been shown above, it is so termed; therefore the Monad, by participation in which each of the particular numerables is called a monad, does not exist. Further, that in which many participate is Many and not One, and the numerables are both many and infinite; each of the numerables, therefore, is not one by participation in the Minad. So, just as the generic Man - whom some conceive as "a mortal animal" - is not Socrates or Plato (for, if so, nobody else will be termed man), and does not subsist of himself nor together with Plato and Socrates (for then he would have been observed as a man), so likewise the One, not being conceived either as subsisting by itself or along with the particular numerables, is ipso facto inconceivable. And the same must be said of the Two and the Three, and in general - not to make a long story of it - of every number. One may also propound the following argument : The Idea of the One, by participation in which a thing is termed one, either is one Idea, or there are several Ideas of the One. But if it is one, many do not participate in it; for (to explain the point clearly) if A possesses the whole of the Idea of the One, B, which does not participate in it, will no longer be one. Nor yet is it multipartite, so that the things participating in it might be many; for, firstly, each thing will be participating not in the idea of the One but in a part of it; and secondly, the Monad, according to them, is conceived as indivisible and without parts. And if there are several Ideas of the One, each of the numerables ranked as unities (whether it be a one or a two, both taken singly) participates in a certain common Idea, or it does not participate. But if it does not participate , all things, apart from participating in an idea, will have to be ranked as unities, a conclusion which they reject. And if they participate, the original difficulty will recur; for how will the twos participate in one Idea?

So much, then, concerning the monad, and if it is destroyed all number is destroyed; all the same, let us subjoin an attack on the dyad. For it is formed in a doubtful way by the conjunction of the monads, even as Plato formerly expressed doubts about it in his book On the Soul. For when a monad is set beside another monad, either something is added by the juxtaposition or something is subtracted, or nothing is either added or subtracted. But if nothing is either added or subtracted, the dyad will not exist through the juxtaposition of the one monad with the other. And if something is substracted through the juxtaposition, there will be a diminution of the one and one, and a dyad will not be formed. And if something is added, the two will become not two but four; for the additional dyad plus the monad and the second monad make up the number four. Therefore nothing will be a dyad. And the same difficulty will exist in the case of every number, so that owing to this number is nothing.

Since, however, number is conceived as a result of the addition or subtraction of the monad, it is plain that if we shall establish that each of these two processes is impossible, the reality of number, too, will be abolished. Let us, for instance, deal first with subtraction, using the method of demonstration by examples. The monad, then, which is being subtracted from the decad assumed is subtracted either from the whole decad or from the nine left over; but it is not subtracted from the whole, as we shall establish, nor from the nine, as we shall demonstrate; nothing, therefore, is subtracted from the decad assumed. For if the monad is subtracted from this as a whole, either the decad is other than the particular monads or the aggregate of these is termed a decad. But the decad is not other than the particular monads; for if these are destroyed the decad does not exist, and similarly if the decad is destroyed the monads no longer exist. And if the decad is the same as the monads, that is to say, if the particular monads are the decad, it is plain that if the subtraction of the monad is from the decad, it will be subtracted from each monad (for the particular monads are the decad), and thus it will no longer be a subtraction of the monad but of the decad. Consequently, the monad is not subtracted from the whole decad. Nor, indeed, is it subtracted from the nine left over; for how will the assumed nine be still preserved after the subtraction? But if the monad is not subtracted either from the decad as a whole or from the nine left over, no number subsists through subtraction. Besides, if the monad is subtracted from the nine, it is subtracted either from the whole or from its last monad. But if the monad is subtracted from the whole nine, there will be a subtraction of the nine; for that which is subtracted from each monad makes up the number of the nine, as the particular monads are nine. And if the subtraction is from the last monad, then, firstly, the last monad, which is indivisible, will be shown to be divisible, which is absurd; and secondly, if the monad is subtracted from the last monad, the nine will no longer be able to remain complete. Further, if the subtraction of the monad is from the decad, it is from the decad either as existent or as non-existent; but it will not be from the existent (for so long as the decad remains a decad nothing can be subtracted from it as a decad, for if so it will no longer be a decad), nor from the non-existent decad; for from what is non-existent nothing can be subtracted. And of course it is impossible to conceive anything other than existence and non-existence; therefore nothing is subtracted from the decad.

Now by these arguments it has been shown that it is not feasible to conceive any number by subtraction; and that it is not feasible by addition either is easy to show by continuing to raise difficulties of a like kind. For, again, if the monad is added to the decad, one must say that the addition is made either to the whole decad or to the last part of the decad. But if the monad is added to the whole decad, then, since the whole decad is conceived along with all the particular monads, the addition which is being made of the monad will have to be an addition to all the particular monads of the decad, which is absurd; for it will follow that by the addition of the monad the decad becomes twenty, which is a thing impossible. We must say, then, that the monad is not added to the whole decad. Nor yet to the last part of the decad, since the decad will not be increased owing to the fact that the increase of the one part is not ipso facto an increase of the whole decad. Generally, too, and finally, the monad is added to the decad either remaining as it is or not remaining. But it will never be added to it while it remains, since in that case it will no longer remain a decad; nor yet while it does not remain, for it is absolutely impossible for an addition to be made to it if it does not remain.

But if number is conceived as subsisting through addition, as I said, and subtraction, and we have shown that neither of these exists, one must declare that number is nothing. Hence, now that we have stated at length all these skeptical arguments against the Geometers and Arithmeticians, let us start afresh and deliver our attack on the Mathematici (or "Astrologers").