## Equality Is Not Identity

I want to record here an account by Collingwood of Aristotle’s theory of knowledge. The passages quoted below are relevant, both to something I have learned from reading Euclid with students, and to the considerations of consciousness that led to my recent article “Body and Mind.”

From the mathematical side, the basic point is simple: for Euclid, equality is not identity or sameness. These matters (and a lot more) are treated at greater length in a paper that I have been working on. An equilateral triangle is a triangle whose sides are equal to one another; but obviously those three sides are not identical with one another or the same as one another. Equal bounded straight lines (our “line segments”) might be said to have the same length; but a straight line is different from its length. (The straight line itself has position as well as length.)

What we call ratio is λόγος in Greek, and Euclid never describes two ratios as being equal to one another: what he defines in Book V of the Elements is when two ratios are the same as one another. According to Collingwood, for Aristotle, something like the frequency of a ringing bell is also a λόγος. (I should think that, strictly speaking, the λόγος in this case would be the ratio of one frequency to another.) When we hear the ringing of a bell, what this means is that some part of us resonates with the same λόγος (and not another one that is similar or equal). The words quoted below spell this out.

The Idea of Nature consists of some of Collingwood’s lectures, edited after his death in 1943. I believe the book has been kept in print since its original publication in 1945. It has three parts, on the Greek, Renaissance, and Modern views of nature, respectively. The first part has three chapters, on the Ionians, the “Pythagoreans” (including Plato), and Aristotle, respectively. The sections of the chapter on Aristotle are:

1. Meaning of φύσις
2. Nature as self-moving
3. Aristotle’s theory of knowledge
4. Aristotle’s theology
5. Plurality of unmoved movers
6. Matter

The second of these sections ends as follows:

The seed only grows at all because it is working at becoming a plant; hence the form of a plant is the cause not only of its growing in that way but of its growing at all, and is therefore the efficient as well as the final cause of its growth. The seed grows only because it wants to become a plant. It desires to embody in itself, in material shape, the form of a plant which otherwise has a merely ideal or immaterial existence. We can use these words ‘want’ or ‘desire’ because although the plant has no intellect or mind and cannot conceive the form in question it has a soul or ψυχή and therefore has wants or desires, although it does not know what it wants. The form is the object of these desires: in Aristotle’s own words, it is not itself in motion (for it is not a material thing and therefore of course cannot be in motion) but it causes motion in other things by being an object of desire: κινεῖ ὡς ἐρώμενον (1072b3).

Now the desire of the material thing is a desire to embody this form in its own matter, to conform itself to it and to imitate it, as well as possible, in that matter. The form, in order to excite such desire, must already be in its own right something worth imitating, something having an activity of its own which is inherently valuable. What kind of activity can we ascribe to the immaterial being which is in this sense the unmoved first mover of the natural world?

The next section proceeds to answer this question. In its entirety, the section is as follows.

In order to answer this question we must turn to Aristotle’s theory of knowledge. Long before his time the Greeks had discovered that sound is a rhythmical vibration set up by a sonorous body and transmitted by the air to the mechanism of hearing. The essence of this mechanism is that it is a part of the organism which picks up the vibrations from the air and vibrates itself in the same rhythm. Any sound having a rhythm which our ears cannot reproduce in themselves is inaudible to us. To reproduce in myself a rhythmical vibration of this kind, and to hear a sound, are the same thing; because, for the Greeks, the soul is nothing but the vital activities of the body, and therefore the gulf which exists in modern thought between the bodily vibrations of the aural mechanism and the mental sensation of sound was for them non-existent. Now, the bronze of the bell, and the gases of the air, do not enter into my organism; but the rhythm of their vibrations does enter into it; and it is precisely this entrance of the rhythm into my head which is my hearing of the sound. But a rhythm is a Pythagorean or Platonic form; it is an immaterial thing, a type of structure, or in Aristotle’s language a λόγος. To hear a ringing bell, then, is to receive into one’s own organism the λόγος of the ringing bell without its ὕλη; and this, generalized, gives us Aristotle’s definition of sensation. The ringing of the bell, its rhythmical vibration, reproduces itself in my head; and that is hearing. Similarly with sight and the other senses. In every case there is a perceived object, which is a certain kind of matter possessing whether permanently or temporarily a certain form: to perceive that object is to reproduce the form in ourselves while the matter remains outside ourselves. Hence Aristotle’s definition of sense as the reception of sensible form without its matter.

This is not a representational or copy-theory of perception. It would be false to say that on Aristotle’s view what we hear is the ringing in our head, which resembles the ringing of the bell in pitch and tone. For the note of the bell is nothing but a λόγος or rhythm: it simply is the rhythm of 480 vibrations a second or whatever it may be. Consequently the note ringing in our head is not another note like that of the bell, it is the very same note; precisely as the equation $(x+y)^2 = x^2 + 2xy + y^2$ is the very same equation when $x = 2$ and $y = 3$ that it is when $x = 3$ and $y = 4$. The note is not matter, it is form; true, a form which, to exist, must exist in some matter; but it is the same form in whatever matter it exists.

Now sensation is a kind of cognition; not a perfect kind, because in hearing the bell we only hear its note, and do not hear its shape or colour or chemical composition. But to this extent it is a fair example of cognition, that what we do hear is a form and that the way in which we hear it is by receiving that form into our organ of hearing. Suppose now there were a kind of knowledge whose object was a form not embodied in any matter: for example, the form of the good, assuming that there is such a thing. If we apprehend that form by thought, we can only do so by receiving it into our mind, experiencing it as a way in which our mind is organized for the time being, just as we hear a note by experiencing it as a way in which our ear is organized for the time being. In the case of the bell, the bronze remains outside us; but in the case of the good, where there is no matter, only form, nothing remains outside us; the entire object reproduces itself (not a copy of itself, but its very self) in our intellect. Hence, as Aristotle puts it, in the case of objects where there is no matter, the knower and the known are identical.

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