QUADRIVIUM, OR THE TALE OF FOUR SISTERS

The Pythagorean scientist-philosopher Archytas of Tarentum (428-347 BC) was of course a post-Socratic, but for the reasons explained earlier he now gets this pre-Socratic entry, yes, as another notable follower of the great pre-Socratic Pythagoras.

He was a philosopher, a mathematician, an astronomer, a musicologist, a statesman, a strategist, and even a commander-in-chief. He lived during Plato’s lifetime, and he was instrumental in the latter’s life and wellbeing. He is known to have been the last prominent figure in the original Pythagorean tradition (as opposed to the later Pythagorean tradition known as neo-Pythagoreanism), and the dominant political figure in his native Tarentum, having been elected strategos seven consecutive times. It was Archytas who sent a ship to rescue Plato from the tyrant of Syracuse Dionysius II in 361 BC, but the personal and philosophical connection of the two philosophers is said to have been somewhat complex, and there is a record of numerous instances of considerable disagreement between the two men.

A great number of works were forged in Archytas’ name after his death, but, apparently, the four fragments attributed to him by all classical scholars, are indeed his genuine surviving works.

There is overwhelming evidence that he was a very great scientist. He is credited with being the first to find an ingenious solution to the classic problem of antiquity of doubling the cube. As a mathematician, he was the first to group together four sciences, logistic (arithmetic), geometry, music, and astronomy. (These four would later become known as the quadrivium, courtesy of Boethius, Archytas’ junior by a thousand years.) Among the multitude of fragments spuriously, and sometimes blatantly falsely, attributed to Archytas, there are yet some which are likely to be his own, and one of these talks about the four “sisters-sciences” of the Boethian quadrivium:

“Mathematicians seem to me to have excellent discernment, and it is not at all strange that they should think correctly about the particulars that are; for inasmuch as they can discern excellently about the physics of the whole, they are also likely to have excellent discernment of the particulars. Indeed, they have transmitted to us a keen discernment about the velocities of the stars, about their risings and settings, about geometry and arithmetic, astronomy, and not least of all, about music. Methinks that all these sciences are sisters, for they do concern themselves with the two original and related forms of being [namely, number and magnitude].”

In music and harmony he was the most outstanding Pythagorean theorist, having developed a mathematical rationalization for the musical scales, intervals and chords. He was probably the first to describe three kinds of proportions employed in music: arithmetical, geometrical, and harmonic.

In mechanics, according to Diogenes Laertius, he was “the first to give order to the science of mechanics, to put it on a mathematical foundation, the first to reduce mechanical movements to geometrical drawing.” He is also known to be a groundbreaker in the science of optics.

His chief scientific principle in all sciences was that everything in the universe could be described in terms of ratios and proportions, and for this reason, logistic (arithmetic) was for him the queen of all sciences.

Precious little is known about his cosmology, but to him belongs the classic argument for the infinity of the universe:

“If I were at the outside, say at the heaven of the fixed stars, could I stretch my hand or my stick outward or not? To suppose that I could not would be absurd, but if I could stretch it out, that which is outside must be either body or space (it makes no difference which it is, as we shall see). We may, then, in the same way get to the outside of that again and so on, asking on arrival at each new limit the same question; and if there is always a new place to which the stick may be held out, this clearly involves extension without limit. If, now, what so extends is body, the proposition is proved; but even if it is space, then, since space is that in which body is or can be, and in the case of eternal things we must treat that which potentially is as actually being, it follows equally that there must be body and space extending without limit.”

So far we have been dealing mainly with Archytas the scientist, but his science ever so smoothly transforms itself into what we normally understand as philosophy. Here is Archytas the political scientist and ethicist:

“When mathematical reasoning has been found, it checks political faction and increases concord, for there is no unfair advantage in its presence, and equality reigns. With the mathematical reasoning we smooth out differences in our dealings with each other. Through it the poor take from the powerful and the rich give to the needy, both trusting in it to obtain an equal share.”

There is another fragment where Archytas offers his explanation of how knowledge is obtained:

“To become knowledgeable about things that one does not know, one must either learn from others or find out for oneself. Now, learning is derived from someone else and is foreign, whereas finding out is of and by oneself. Finding out without seeking is difficult and rare, but with seeking it becomes manageable and easy, though someone who does not know how to seek cannot find.”

With this discussion of Archytas we are now closing our Pythagorean series, and moving on to other things.