Tag: Number

Aristotle on the Platonic Good

“Philosophy begins with wonder not that there are things rather than not, but that they are as they are” (Gwenaëlle Aubry, Dieu sans la puissance, 2nd ed., ch. 1, p. 33, my translation throughout).

Here Aubry refers to Aristotle’s famous statement in book capital Alpha of the Metaphysics that philosophy begins in wonder. In the 20th century, Heidegger emphasized the question why there is something rather than nothing. Aubry is pointing out that that is not Aristotle’s question at all. As detailed in many posts here, it is the more particular what-it-is of things that Aristotle is mainly concerned to explain.

“Two texts, in Lambda and Nu, echo the critique of capital Alpha, and each time the insistence of Aristotle is the same, in underlining that he is, of all the philosophers, the first and the only to have posed the good as a principle” (ibid).

Of Aristotle’s four causes, the material cause and the source of motion were generally recognized by the pre-Socratics. The Pythagoreans and Platonists added something approximating to Aristotle’s formal cause. But Aristotle insists that even the Platonists made no real use of a concept of that-for-the-sake-of-which, and confusingly treated the good as a formal cause or as a source of motion instead.

In book Nu of the Metaphysics, according to Aubry he says that Plato’s successor Speusippus and the Pythagoreans “agree with the mythologists in seeing the good not as a principle but as an effect of order…. Beyond Speusippus, the allusion is to Plato…; the error of Plato is not in having posed the good as a principle in making it an attribute of the One, but in having made the One itself the principle” (p. 39).

The relevant passage in book Nu says, “Things that come down to us from those who wrote about the gods seem to agree with some people of the present time who say that the good and the beautiful are not sources but make their appearance within the nature of things when it has advanced. (They do this out of caution about a true difficulty which follows for those who say, as some do, that the one is a source. The difficulty is not on account of reckoning what is good to the source as something present in it, but on account of making the one a source — and a source in the sense of an element — and making number out of the one)” (Metaphysics, Sachs tr., pp. 291-292).

Aubry’s point rings true. Elsewhere Aristotle goes on at length about Speusippus’ and the Pythagoreans’ insistence on the central role of number, which gestures in the direction of a formal cause but is actually treated by them more like a kind of material cause. And when Speusippus and the Pythagoreans talk about the One, they seem to literally mean the number one. As Aristotle points out repeatedly, it is nonsense to make the number one the source of all things, and this also doesn’t explain anything.

I myself for many years simply accepted at face value an identity of the Platonic Good with the One as the source of all things, while downplaying the One’s connection with the mere number one, and emphasizing a sort of negative theology in the style of Plotinus, which eliminates all positive attributes of the One. This really just comes down to saying there is a source of all things, while leaving unclear the way in which it is a source. It also assumes the Platonic thesis — rejected by Aristotle — that there is a single form of the Good.

For Aristotle, the good is said in as many ways as being is. What is essential is not this good or that good, but the relevance of value and valuation to all judgment whatsoever and all doing whatsoever. That relevance appears concretely as that-for-the-sake-of-which, or “final” causality. This was Aristotle’s huge innovation.

Aubry reviews Aristotle’s critique of the Platonic Good in both the Nicomachean and the Eudemian Ethics, which focuses on its status as a universal or an Idea.

“Aristotle underlines that there is not a unique science of the Good, but multiple sciences having for their object different goods” (p. 44).

“To the Good as a genus — undiscoverable — and as Idea — useless and void — Aristotle opposes a good that is prakton [practical, in the ethical sense], a good realizable by the human, determined as the first term of a hierarchy of goods and ends, … which the Nicomachean Ethics calls ‘politics’…. If the critique of the Platonic Good leaves open this path, which the Ethics explore, it does not close off that which consists in posing a Good as principle, the relation of which to particular goods remains to be determined, and of the sort that it is neither that of a genus nor of a species, nor that of an Idea to its participants” (p. 45).

Once again, it will be only the causality of that-for-the-sake-of-which and the related details of act and potentiality that truly explain this relation.

Next in this series: Being and Becoming

Figurative Synthesis

I wanted to extract a few more key points from Beatrice Longuenesse’s landmark study Kant and the Capacity to Judge. She strongly emphasizes that judgment for Kant refers to a complex activity, not a simple reaching of conclusions. She especially stresses the role of a capacity to judge that precedes any particular judgment and is grounded in a synthesis of imagination. (See Capacity to Judge; Imagination: Aristotle, Kant; Kantian Synthesis.)

At issue here is the very capacity for discursive thought, as well as “the manner in which things are given to us” (p. 225, emphasis in original), which for Kant involves what he called intuition. (See also Beauty and Discursivity).

Through careful textual analysis, Longuenesse argues that Kant’s claim to derive logical categories from forms of judgment makes far more sense than most previous commentators had recognized. For Kant, she argues, the “forms of judgment” are not just logical abstractions but essential cognitive acts that reflect “universal rules of discursive thought” (p. 5).

She recalls Kant’s insistence that the early modern tradition was wrong to take categorical judgments (simple predications like “A is B“) as the model for judgments in general. For Kant, hypothetical and disjunctive judgments (“if A then B” and “not both A and B“, respectively) are more primitive. These correspond to the judgments of material consequence and material incompatibility that Brandom argues form the basis of real-world reasoning.

Another distinctive Kantian thesis is that space and time are neither objective realities nor discursive concepts that we apply. Rather, they are intuitions and necessary forms of all sensibility. Kantian intuitions are produced by the synthesis of imagination according to definite rules.

“[I]ntuition is a species of cognition (Erkenntnis), that is, a conscious representation related to an object. As such it is distinguished from mere sensation, which is a mere state of the subject, by itself unrelated to any object…. One might say that, in intuition, the object is represented even if it is not recognized (under a concept).” (pp. 219-220, emphasis in original).

Before we apply any concepts or judgments, “Representational receptivity, the capacity to process affections into sensations (conscious representations), must also be able to present these sensations in an intuition of space and an intuition of time. This occurs when the affection from outside is the occasion for the affection from inside — the figurative synthesis. The form of the receptive capacity is thus a merely potential form, a form that is actualized only by the figurative synthesis” (p. 221, emphasis in original).

“[A]ccording to Locke, in this receptivity to its own acts the mind mirrors itself, just as in sensation it mirrors outer objects…. Kant shares with Locke the conception of inner sense as receptivity, but he no longer considers the mind as a mirror, either in relation to itself or in relation to objects…. Just as the thing in itself that affects me from outside is forever unknowable to me, I who affect myself from within by my own representative act am forever unknowable to me” (p. 239, emphasis added).

The point that the mind is not a mirror — either of itself or of the world — is extremely important. The mirror analogy Kant is rejecting is a product of early modern representationalism. We can still have well-founded beliefs about things of which we have no knowledge in a strict sense.

“Kant’s explanation is roughly this: our receptivity is constituted in such a way that objects are intuited as outer objects only in the form of space. But the form of space is itself intuited only insofar as an act, by which the ‘manifold of a given cognition is brought to the objective unity of apperception’, affects inner sense. Thanks to this act the manifold becomes consciously perceived, and this occurs only in the form of time” (p. 240, emphasis in original).

She develops Kant’s idea that mathematics is grounded in this kind of intuition, ultimately derived from the conditions governing imaginative synthesis. In particular, for Kant our apprehensions of unities and any kind of identification of units are consequences of imaginative synthesis.

“Extension and figure belong to the ‘pure intuition’ of space, which is ‘that in which the manifold of appearances can be ordered’, that is, that by limitation of which the extension and figure of a given object are delineated. Therefore, space and time provide the form of appearances only insofar as they are themselves an intuition: a pure intuition, that is, an intuition preceding and conditioning all empirical intuition; and an undivided intuition, that is, an intuition that is presupposed by other intuitions rather than resulting from their combinations” (p. 219, emphasis in original).

“According to Locke, the idea of unity naturally accompanies every object of our senses, and the idea of number arises from repeating the idea of unity and associating a sign with each collection thus generated by addition of units…. But for Kant, the idea (the concept) of a unit is not given with each sensory object. It presupposes an act of constituting a homogeneous multiplicity…. Thus the idea of number is not the idea of a collection of given units to which we associate a sign, but the reflected representation of a rule for synthesis, that is, for the act of constituting a homogeneous multiplicity. When such an act is presented a priori in intuition, a concept of number is constructed.” (p. 260, emphasis in original).

“Mathematics has no principles in the absolute sense required by reason. Axioms are not universal propositions cognized by means of pure concepts. They may be universally and apodeictically true, but their truth is based on the pure intuition of space, not derived from pure concepts according to the principle of contradiction” (p. 287).

Incidentally, Longuenesse thinks it does not follow from Kant’s account that space is necessarily Euclidean, as many commentators have believed and Kant himself suggested.