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Plato's Forms, Mathematics and Astronomy

Theokritos Kouremenos
Walter de Gruyter
Publication Date: 
Number of Pages: 
trends in Classics
[Reviewed by
Tom Schulte
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This work of philosophy is an exegesis of the works of Plato and related writings focusing on the evolution of notions around mathematical entities from music theory to the cosmos. A recent reading of the key works, especially The Repubic, Laws, Parmenides, Philebus, and Timaeus, will help one get more out of the book. A detailed Index of Passages allows using this volume to amplify the mathematical reasoning of nearly any work ascribed to Plato as well as several attributed to Aristotle, Xenocrates, Democritus, and more.

Fundamentally, this is an exploration of Mathematical Platonism suggesting that mathematical objects are abstract, lacking spatiotemporal and causal properties, and are eternal and unchanging. Pointedly pre-Kantian, I believe this is a rare work I have read in this area without the descriptive “a priori” in describing such knowledge independent of all experience. Still, the themes do not feel antique or lost in details of schools of thought from millennia ago no longer urgent to a lay reader with today a need to question. For, the point from Plato amplified here, is that the nature of mathematics is the gateway:

All branches of mathematics are chosen as the only appropriate propedeutics to philosophy because they in fact introduce forms.

(I will have to use that the next time a student asks, “When am I going to use this?”)

While I feel a broad reading in Plato is a recommended prerequisite, this book requires no significant philosophical or even mathematical background for the diligent reader seeking an introduction to Platonic Realism. As described here, in this branch of realism,

…mathematical objects exist objectively, independently of our thought and neither in space nor in time… As the objects mathematics is really about, forms are best viewed not as universals but as abstract particulars. Geometry does not study a single square: it assumes an indefinitely or infinitely large number of copies of each of its objects… In view of the imagery of the cave simile, these are in each case multiple 'shadows' cast by a unique form, which is approached by the mathematicians only via study of its ‘shadows’… Plato has Socrates locate the difference between mathematics and philosophy in the way each studies forms. Mathematics studies forms indirectly…. Ideally, philosophy has no need for crutches and approaches the forms in themselves… mathematics sees beings not in the state of wakefulness, as philosophy does, but as if in a dream.

So, what we have here is mathematics as the training ground for getting the mind ready to employ philosophy in understanding the nature of reality. Plato’s school, through its “mysterious identification of forms with numbers”, recommended mathematics as the essential underpinnings to using philosophy to truly “see” the forms directly. There is a tight coupling with astronomy here:

…Plato has Timaeus say that various astronomical phenomena have made it possible for us to develop the concepts of number and time and the inquiry into the nature of the cosmos, all of which have provided us with philosophy… [astronomy] sprang from marveling at the phenomena of the sky and wondering…

We truly only can see the shadows of these distinct, ancient objects. With mathematics we can know them.

This material, especially the first third of the book, feels at times overwhelmed by its own footnotes. It is not unusual for pages to have more content in footnotes than above. In the case of quotes from the original Greek and suggestions for further reading, the footnote feels appropriate. Many amplifications of points could have been included with the main content rather than being shunted to the small-point footnotes.

In this consideration of the Platonic worldview, there is only an onionskin of separation from considering a forms theory in the context of a contemporary philosophy. At times, the text returns instead to the Myth of Er and The Spindle of Necessity, where The Fates, the three daughters of the Goddess Necessity, keep the rims of the cosmos’ driving axle revolving. The hook, shaft, and whorl application can help convince us that probably Plato did truly recognize the retrograde motion of the Aristotelian planetary spheres. But can we envision a dedication to mathematics as making philosophical contemplation more rewarding and profitable? Have any other propedeutics proven worthier? The rest is left as an exercise for the reader.

Tom Schulte enjoys reading Plato periodically and his current favorite Plato quote is from The Republic: “Ruin comes when the trader, whose heart is lifted up by wealth, becomes ruler.”

See the table of contents in the publisher's webpage.