The report about his sudden fits of anger, his greed, and his debauchery, are probably derived from a very impure source: Athenaeus and Diogenes Laërtius can adduce as authority for them scarcely anything more than some abuse in certain letters of Dionysius the Younger, who was banished by Dion, with the cooperation of Speusippus. Having been selected by Plato as his successor as the leader (scholarch) of the Academy, he was at the head of the school for only eight years (347-339 BC.). He died, it appears, of a lingering paralytic illness, presumably a stroke. He was succeeded as the head of the school by Xenocrates.
With Plato, moreover, he distinguished between that which is the object of thought, and that which is the object of sensuous perception, between the cognition of the reason and sensuous perception. He tried, however, to show how perception can be taken up and transformed into knowledge, by the assumption of a perception, which, by participation in rational truth, raises itself to the rank of knowledge. By this he seems to have understood an immediate, (in the first instance aesthetic), mode of conception; since he appealed, in support of this view, to the consideration that artistic skill has its foundation not in sensuous activity, but in an unerring power of distinguishing between its objects, that is, in a rational perception of them.
Speusippus made still more kinds of substance, beginning with the One, and assuming principles for each kind of substance, one for numbers, another for spatial magnitudes, and then another for the soul; and by going on in this way he multiplies the kinds of substance.Nevertheless Speusippus also must have recognised something common in those different kinds of substances, inasmuch as, firstly, he set out from the absolute One, and regarded it as a formal principle which they had in common, and, secondly, he appears to have assumed that multitude and multiformity was a common primary element in their composition. But it is only the difficulties which led him to make this and similar deviations from the Platonist doctrine, of which we can get any clear idea, not the way in which he thought he had avoided those difficulties by distinguishing different kinds of principles. The criticism of Aristotle, directed apparently against Speusippus, shows how little satisfied he was with the modification of the original Platonist doctrine.
With this deviation from Plato's doctrine is connected another which takes a wider range. As the ultimate principle, Speusippus would not, with Plato, recognise the Good, but, with others, (who doubtless were also Platonists), going back to the older Theologi, maintained that the principles of the universe were to be set down as causes of the good and perfect, but were not the good and perfect itself, which must rather be regarded as the result of generated existence, or development, just as the seeds of plants and animals are not the fully formed plants or animals themselves.
Speusippus [supposes] that supreme beauty and goodness are not present in the beginning, because the beginnings both of plants and of animals are causes, but beauty and completeness are in the effects of these.
The ultimate principle he designated, like Plato, as the absolute One, but it was not to be regarded as an existing entity, since all entities can only be the result of development. When, however, with the Pythagoreans, he reckoned the One in the series of good things, he probably conceived it only in its opposition to the Many, and wished to indicate that it was from the One and not from the Many, that the good and perfect is to be derived. Nevertheless Speusippus seems to have attributed vital activity to the primordial Unity, as inseparably belonging to it, probably in order to explain how it could grow, by a process of self-development, into the good, spirit, etc.; for spirit also he distinguished from the one, as well as from the good; and the good from pleasure and pain. Less worthy of notice is the attempt by Speusippus to find a more suitable expression for the material principle, the indefinite duality of Plato; and his Pythagorizing mode of treating the doctrine of numbers which we can see in the extracts of his treatise on the Pythagorean numbers.