On Difference in
The curve is a translatable point of a form which possesses more importance than the snub, in that the snub is a form which can not be translated to any part of a form (Aristotle, Physics, II. 2, 194a2-6). Explaining such a curve proves to be Aristotle’s aristeia with which he criticizes Plato. Plato is being criticized through Aristotle’s “reverse of geometry”, which, since geometry “considers natural lines, but not as natural,” affirms the forms we see, however optical, are merely illusory perceptions of the mathematical (Physics, II. 2, 194a9-12). Plato claims that the mathematical is the highest form, “never permitting anyone to propose for discussion numbers attached to visible or tangible bodies” (Plato, Republic, VII. 525d).
Yet, if the forms of geometry and mathematics “come nearest to the study of nature, like optics, harmonics, and astronomy”, they must be perceptions—that which is visible—of perfection by which we can measure nature (Physics, II. 2, 194a8-9). And Plato must be incorrect in the afore-quoted statements because he is postulating that calculation is the highest of all transcendental arts and can only be justifiably applied to that which is not visible or tangible. But Aristotle, in his anecdote of curve and snub, criticizes Plato almost directly and brings attention to the division Plato makes between the visible and the mathematical:
Nor does he consider things which supervene as supervening on such bodies. That is why he separates them; for they are separable in thought from change, and it makes no difference; no error results. Those who talk about ideas do not notice that they too are doing this: they separate physical things though they are less separable than the objects of mathematics. That becomes clear if you try to define the objects and the things which supervene (Physics, II. 2, 193, b32-194a3).
The way in which one can see the supervening of things with which Aristotle criticizes Plato is the analogy of the circle. A circle can be thought of as a form which contains an infinite potential of curves, itself being the culmination of geometry without any straight lines. Given the nature of the circle, a curve of any extent can be fitted to the circle in any position and blend in to its borders seamlessly as a curve upon a curve. However, snub is that which is unable to conform to the circle and thereby not be universal. The curve supervenes the line of the circle but the snub can not do so. This criticizes Plato in a way that asserts “there are two sorts of thing called nature, form and matter” as opposed to accepting the one thing Plato sees—mathematics (Physics, II. 2, 194b12-13). Why does Plato separate the mathematical from the visible? For in thought, Aristotle said above, that which is thought is separated from that which is change.
“Inquiring what snubness is,” Aristotle veers away from Plato’s emphasis on rationalizing his own theories by an ethereal underlying force—mathematics, that which is in the mind—and says “we should consider things neither without their matter nor in accordance with their matter” (Physics, II. 2, 194b13-15). It is the discrimination between the matter of forms which preoccupies Plato and “those who fix on some such element . . . represent it . . . as the entire reality . . . and say that other things are merely . . . dispositions”; Plato, Aristotle can be seen to describe, is likely to abandon reality in order to leave reason to ambiguity (Physics, II. 1, 193a23-26). Such is the way in which the criticism is made. Aristotle feels that the discrimination Plato makes between the vulgarity of the tangible and the beauty of the imaginative, internalized logic of the contemplation of matter and form is a petty attempt “to show what is plain by what is obscure” (Physics, II. 1, 193a5-7). Aristotle criticizes Plato by saying that the snub—which represents the kind of distinction Plato stands for—“is a sign of inability to discriminate between what is self-evident and what is not” (Physics, II. 1, 193a5-7).
Referring back to the example of the circle and the application to it of curve and snub, one must agree that since the curve is found in every potential locality of the line of the circle, such a potentiality is self-evident. However, since snub, as a nose, can not be applied to the circle, then that which is snub—Plato in his consideration of matter—is not self-evident. Everything which is snub, “they none of them have in themselves the source of their making”, while curve being the making of the circle does, “but in some cases . . . the source is in something else and external,” just as snub must always remain external to the circle (Physics, II. 1, 192b29-31). The difference here is not merely between the shape of curve and snub, but too, in the debate of the relation between matter and form.
Because “it belongs to the same study to know the end or what something is for” and curve is the symbol of potentiality, manipulative ability, its end can be quite changed though ultimately, produced (Physics, II. 2, 194b27). Snub can not be changed, “for whenever there is a definite end to a continuous change, that last thing is also what it is for” and snub never becomes what it is for (Physics, II. 2, 194b29-31). The way in which Aristotle is criticizing Plato and doing so through his demonstration of curve and snub, is that curve has the potentiality of motion, or transition, while snub intercepts and disengages; this attacks what Plato sees as a necessary ignorance of an explanation of the evolutionary ability of geometry and the nature of form; “accounts are for the sake of knowing what always is, not what comes into being and passes away . . . for geometry is knowledge of what always is” (Republic, VII. 527b).
Aristotle. “Physics.” Foundation Year Programme Handbook. Halifax: University of King’s College, 2006-07.
Plato, Republic. Trans. G. M. A. Grube. Indianapolis: Hackett, 1992.