Appendix C. Popular Resumé Against the Platonic Ideas← | TOC |

## APPENDIX D

*The alleged Difficulty in the Connection of
Mathematics with the Doctrine of Ideas. (Cf. A, 9, 992b 12-17.)*

*Metaphysics*, *M*, 2, 1076b 11-39.

Yet, it is not even possible that there should be such
separate and independent^{188} entities.
For if, over and above the solids our senses perceive, there is to be a further
set of solids separate from and independent of the former, and logically prior
to them, manifestly there must also be separate and independent planes, over
and above the planes our senses perceive, and similarly in the case of points
and lines; it is all part of the same theory. But, so much being admitted, once
again, there must be yet further separate and independent planes, lines, and
points, over and above those contained in geometrical solid figure. For
isolated entities are logically prior to the same entities in combination; and
if bodies which are not perceptible to the senses are logically prior
to bodies which are so perceptible, it follows by the same argument that
independent, self-existing planes are logically prior to the planes of the
motionless^{189} solids. So
that these planes and lines are classes distinct from those postulated along
with the separate and independent solids. The latter are postulated *with*
the mathematical solid figures; the former are logically prior to these
figures.

Similarly once more, the planes just referred to will
contain lines, and by the same reasoning there must be yet other lines and
points prior to these, and besides the points of these "prior" lines
there must be yet other points, prior to them, but beyond which there is no
further prior class of points. Now, surely, this accumulation of entities
becomes an absurdity. For it follows that there is only one class of solids
besides those our senses perceive, but three such classes of planes (viz.,
those which are "beside" the sensible planes, those contained in the
mathematical solids, those which are "beside" these), four classes of
lines, five of points. Now which of all these are to be the objects of the
mathematical sciences? For it will surely not be said that it is the planes,
lines, and points which are in the motionless geometrical solid which are the
objects of these sciences, since it is always the logically prior classes which
are the objects of science.^{190}

The same argument is applicable also to the case of numbers.
For each class of points there will be a different corresponding
class of units,^{191} and so
again for each class of sensible objects, and again for each class of
conceptual objects.^{192} Thus the
numbers of classes of mathematical numbers will be infinite.

[Aristotle]^{193} *De
Lineis Insecabilibus*, 968a 9-14. (Text of Apelt in the Teubner series.) If
there is an "Idea of Line," and the Idea is the archetype of all the
objects which fall under the same concept, while the parts of an object are
logically prior to the whole which they constitute, the "Ideal Line"
must be indivisible. The same will be true of the "Ideal square" and
"triangle," and all the other figures, and universally of the
"Ideal plane" and "Ideal solid;" for otherwise it would
follow that there are things^{194} which are
logically prior to these entities.

*ib.* 969a 17-21. Those who construct indivisible lines
among the Ideas make an assumption--viz., in postulating Ideas of such
objects--which is perhaps of less extended scope than that now under
examination,^{195} and in
a sense they destroy the force of the very assumptions on which their
proof rests. For such arguments are, in fact, subversive of the Ideas.

### Taylor's Notes

(*Taylor's footnotes have been converted to endnotes*)

[188].κεχωρισμένας, "separated," i. e., existing as distinct objective entities, not merely as products of subjective mental abstraction without a real separate existence of their own. I have employed the double expression "separate and independent" to represent the one Greek word, which it is, however, very tempting to translate simply by "transcendent."

[189]. i. e., purely
geometrical, as distinguished from physical "bodies." The difference,
according to Aristotle, between the objects of Mathematics and those of Physics
is precisely that the former, though "inseparable from matter," are
not capable of motion, the latter are "inseparable from matter but not
incapable of motion." *Metaphysics, E*, 1026a 18.

[190]. The character of the
reasoning will become clearer if we consider the simplest of the cases
mentioned, that of the plane. Aristotle contends that on a realist theory, like
that of Plato, which regards the plane surfaces of Geometry not as mere logical
abstractions but as objective entities, there must be not only one but three
classes of such entities, over and above the perceptible surfaces of physical
bodies, viz.: (1) The single archetypal "Idea" of *the* plane,
i. e., the entity to which we refer in giving the definition of the plane as
such; (2) the entities which figure as constituents in the definition of the
geometrical solid, e.g., as determined or bounded by four planes; (3) the
infinitely numerous "mathematical planes" which appear in Geometry.
It is of these last that "physical" plane surfaces are immediately
"copies." Thus he arrives at the following series of entities as all
implied in the Platonic theory:

3 classes of plane, viz.:

(a) *The* plane as represented by its definition.

(b) The plane as a boundary of solids.

(c) "Mathematical" planes.

4 classes of line, viz.:

(a) *The* line as represented by its definition.

(b) The line as boundary, or rather as intersection, of planes.

(c) The line as intersection of planes which are boundaries of solids.

(d) "Mathematical" lines.

5 classes of point, viz.:

(a) *The* point as represented by its definition.

(b) The point as intersection of lines.

(c) The point as intersection of lines which are intersections of planes.

(d) The point as intersection of lines which are intersections of planes, which are boundaries of solids.

(e) "Mathematical" points.

I must leave the reader to decide whether the ingenuity of all this is not equalled by its perversity, merely observing that "by the same reasoning," there should be two and not, as Aristotle says, only one class of "solids" over and above "physical" solids.

[191]. Because a point is simply a "unit having position."

[192]. Because each object of sense or thought is a "unit," and also a "copy" of a simple "transcendent" unit.

[193]. The author of the
essay, though certainly not Aristotle, is almost equally certainly one of his
immediate disciples, possibly Theophrastus. See Apelt, *Beiträge zur
Geschichte der Griechischen Philosophie*, p. 269.

[194]. viz., the lines or planes into which the "Ideal plane," or "solid," if divisible, may, according to the Platonists under discussion, be divided.

[195]
The assumption under discussion is that there is a whole infinitely numerous
class of indivisible"mathematical" lines, or
"infinitesimal" lines, which are, in fact, the entities commonly
called *points*. "Aristotle's" objection, as Apelt (*Loc. cit.*
p. 274, note 2) explains, is that you cannot infer the indivisibility of
"mathematical" lines from the supposed indivisibility of the
"Ideal line;" on the contrary, the only valid ground for calling the
"Ideal line" indivisible would be your previous knowledge that
"mathematical" lines, as a class, are indivisibles. You have no
right, on Platonic principles, to assume an Idea except when you already know
of an existing class of corresponding individual things. There can be no idea
corresponding to any class which is inconceivable. Hence, if it can be shown
that all "mathematical" lines are divisible, there can be no reason
to postulate an "Idea" of the indivisible line.

Created for Peithô's Web fromAristotle on his predecessors; being the first book of his Metaphysics; tr. from the text edition of W. Christ, with introd. and notes by A. E. Taylor.Chicago, Open Court, 1907.Taylor's footnotes have been converted to endnotes. Greek unicode text entered with Peithô's Younicoder.

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