Lecture 4 — Other natural arguments against Plato's opinion. Pythagorean opinion refuted

422 Further, to assume, on the one hand, that the planes can only be put in linear contact would be ridiculous. For just as there are two ways of putting lines together, namely, end to and side by side, so there must be two ways of putting planes together. Lines can be put together so that contact is linear by laying one along the other, though not by putting them end to end. But if, similarly, in putting the lanes together, superficial contact is allowed as an alternative to linear, that method will give them bodies which are not any element nor composed of elements.

423 Again, if it is the number of planes in a body that makes one heavier than another, as the Timaeus explains, clearly the line and the point will have weight. For the three cases are, as we said before, analogous. But if the reason of differences of weight is not this, but rather the heaviness of earth and the lightness of fire, then some of the planes will be light and others heavy (which involves a similar distinction in the lines and the points); the earthplane, I mean, will be heavier than the fire-plane.

424 In general, the result is either that there is no magnitude at all, or that all magnitude could be done away with. For a point is to a line as a line is to a plane and as a plane is to a body. Now the various forms in passing into one another will each be resolved into its ultimate constituents. It might happen therefore that nothing existed except points, and that there was no body at all.

425 A further consideration is that if time is similarly constituted, there would be, or might be, a time at which it was done away with. For the indivisible now is like a point in a line.

426 The same consequences follow from composing the heaven of numbers, as some of the Pythagoreans do who make all nature out of numbers. For natural bodies are manifestly endowed with weight and lightness, but an assemblage of units can neither be composed to form a body nor possess weight.

427 Therefore, it is clear from what has been said that generation neither applies to everything nor to nothing.

568. Having given a first argument against Plato's opinion that bodies are generated from surfaces, Aristotle here presents a second argument. To understand this argument it should be known that Plato, because he did not distinguish between the "one" which is the principle of number, and the "one" which is convertible with being, which signifies the substance of the thing, posited as a consequence that the "one" which is the principle of number is the substance of a thing, and consequently, that all things are numbers. Accordingly, he posited the dimensions of continuous quantity to be certain numbers "having position." According to him, therefore, a point is unity having position, and so for all the others. And because he attributed duality to matter and unity to form, he estimated that the forms of all bodies should be taken according to the notion of the figures according to which the bodies are terminated. Now the ultimate terminations of dimensions are points, which are "positioned unities," as has been said. Therefore he attributed the various bodily figures to the various bodies: such as that of the pyramid to fire, that of eight bases [faces, in this case, the octohedron] to air, a figure of 20 bases [the icosahedron] to water, the cube to earth, a 12-based figure [the dodecahedron] to the aether, i.e., to the heaven. Now it is clear that bodily figures are constituted of surfaces, insofar as they are conjoined according to contact along lines—thus constituting corporeal angles. And therefore, assigning the formal composition to bodies, Plato said that bodies are composed of surfaces joined along a line.

569. Against this, therefore, Aristotle objects (299b23), and says that an impossibility follows, if it is assumed that surfaces are composed or joined to form a body by mere linear contact. This he manifests with the example of a line. For a line can be joined to another line in two ways: One way is according to length, and which is according to contact by point, insofar, namely, as the length of one line is joined to the length of another at a point, whether it makes an angle with it or not. The other way is according to width, which consists in adding one whole line to another whole line in the direction of width. Similarly, surface must be joined to surface in two ways: namely, according to depth—for example, if one whole surface should be placed under another; and according to linear contact, whether they form a corporeal angle or not. To explain further what he says, he adds that line can be joined to line by being placed under it, and not only as being added to it according to linear contact [at a point].

570. Therefore, because there are two ways in which surfaces can be joined and because it is according to one way, namely, according to linear contact, that joined surfaces form all the elements, it will follow that, if they should be joined in the direction of width, i.e., by placing one surface under another, the resultant of surfaces so composed will be a body that is neither an element nor composed from elements. That it is not an element is plain, since all the elements are constituted according to the other mode of conjunction of surfaces. That it is not made from elements is plain, since this combination of surfaces, resulting from superposition, seems to cause a body's depth, which is its substance, while the other combination of surfaces establishes the body with respect to figure, which is a form added to bodily substance. Hence, the combination according to sub-position will be prior, and what results from this composition seems to be compared to what results from the other kind of composition, as matter is compared to form. But according to the opinion of Plato, a body's nature is to be composed of surfaces [in linear contact]. It follows, therefore, that what precedes all the elements, as though it were their matter receiving all their figures, or forms, is a body. And Plato regarded this as something unacceptable—for he did not state the first matter to he a body, as certain of the early natural philosophers did.

571. Then, at again, if it is the number (299b31) he presents a third argument, as follows: since bodies are constituted out of surfaces, some of which bodies are heavier than others, this can come about in two ways. One way is that a body would be made heavier from the fact that it is composed of more surfaces, as is said in the Timaeus. And from this it will follow that surfaces are heavy—because an excess in heaviness cannot be produced except by something that is itself heavy, as was said above. And from this it will further follow that lines and points have heaviness — for these are proportionately related, that is, as surfaces are to bodies, so lines are to surfaces and points to lines. But that points should have heaviness has been disproved above. The other way is that heavy bodies do not differ from light in this way, i.e., by the multiplication of surfaces, but on account of earth's being composed of heavy [surfaces] and fire of light. And thus it will follow that some surfaces are light and some heavy, and the same for lines and points—because a surface of earth will be heavier than one of fire. And so the same impossibility will arise as before.

572. Then, at in general, the result (300a7), he presents a fourth argument, saying that according to the position of Plato it follows that no magnitude exists, or that every magnitude can be taken away i.e., cease to exist. For point is to line as line is to surface and surface to body. Consequently, if a body is composed of surfaces, it will be able to be reduced to surface, and, by the same token, all magnitudes will be reduced to the first things, i.e., to points. Thus it would follow that there would be no body but points only. Nor is it similar if one should wish to argue that it can occur that there be no mixed [composite] bodies, since they can be resolved into the elements out of which they are composed—since such bodies are subject to the heavenly bodies, which produce a mixture in them. But points are not subject to any higher principles which would induce in them the necessity of composition.

573. Then, at a further consideration (300a12), he presents a fifth argument, saying that if time is such as to be composed of instants, in the same way as a body is of surfaces, or a line of points (all of which involve the same notion, as is proved in Physics VI), it follows that time too can be entirely destroyed by being reduced to its indivisibles—for the now is the indivisible of time, just as the point is the indivisible of line.

574. Then, at the same consequences follow (300a15) he assimilates the foregoing position to that of the Pythagoreans. And he says that the same irreconcilable factors follow for those who claim that the heaven is constituted from numbers. For some Pythagoreans posited that all nature is constituted from numbers for the aforesaid reason, and Plato followed them. But now the Philosopher disproves this: For natural bodies have heaviness and lightness. But units joined one to another cannot form a body that is continuous; rather they form something discrete. Nor do they possess heaviness, because they abstract from position, and, consequently, from place.

Finally, in summary he concludes (300a19) that there is neither generation of all things nor of none. That it is not of none is evident to sense. But that it is not of all things is plain from the fact that it is impossible for there to be generation of every body—which would indeed be the case if a body were generated from surfaces.