Imaginaries in Geometry
Pavel Florensky (Anna Maiorova and Karen Turnball, Translators; Andrea Oppo and Massimiliano Spano, Editors).
Milan IT: Mimesis International, 2021
Published in 1922, while mostly written in 1902 [Editor: When the author was 20!], Florensky’s Imaginaries in Geometry is not an easy text to read because its focus is mathematical in nature. There are philosophical and theological ideas addressed in it which will allow many readers to appreciate his thought. Nonetheless, the mathematical elements which serve as his proof of concept will not likely be understood by many of its readers. I found them to be almost incomprehensible. I say “almost,” because even though I was unable to comprehend his mathematical equations, I was able to use what mathematical knowledge I remember from my studies to follow along with the concepts being expressed. Most of his readers will be able to do so as well, allowing them to discern how the book connects to the greater philosophical picture which Florensky wanted to reveal through his work.
What is that picture? That mathematics and the sciences all serve to translate and interpret reality, each presenting its own worldview that accounts for some element of a greater whole. Just as those who translate poetry from one language to another will lose something in their translation, so something is lost in the translation of reality through their exploration in the sciences. Different models, different worldviews, just like different translations of a book, can and should complement each other. They each have something invaluable to tell us. What is important is that we do not merely embrace one model and try to reduce reality to what can be discerned by it. This is exactly what he felt was the error of the scientific positivism embraced by the Soviets:
We are also aware that, just as several translations of a poetic composition into another language or languages not only do not interfere with each other, but on the contrary actually complement one another, even though none of them can fully take the place of the original, in the same way, scientific representations of the same reality can and must be multiplied, and this does not obstruct the truth in any way. Knowing this, we have learned not to criticize any given interpretation for what it does not give us, but to be grateful each time we are able to find a way to use it [italics in the original]. (p. 20)
Different models will be superior to others in how they appropriate and engage reality. Some will be able to demonstrate more of it than others. Nonetheless, each of them has elements of reality which they cannot properly address. When we find ourselves being limited to one model, especially an inferior model, we must find a way to break with it so not to allow its limitations to negatively influence our engagement with reality:
However, we must point out the limitations of a given interpretation once we notice the exaggeration of this or that translation which tries to identify itself as the original text and take its place, i.e. which monopolizes some kind of essence and jealously excludes any other interpretation: in such a case, the only choice is to point the misguided interpretation towards its proper place and the true scope of its applicability [italics in the original]. (p. 20)
Florensky, throughout his life, fought against the limits imposed upon reality by scientific positivism. For example, he did this with his promotion of reverse perspective in his book, Iconostasis; likewise, we find him doing so with shorter works on the arts, such as the essays collected and published in Beyond Vision: Essays on the Perception of Art. He continued this assault on positivism with his mathematical works; he wanted to show in them how positivism failed to deal with all the data available in mathematics. In Imaginaries in Geometry, the issue he addressed was the way mathematics had failed to offer proper forms of spatial representation for imaginary numbers: “[. . . ] we must extend the domain of the two-dimensional images of geometry so that imaginary figures are also included in the system of spatial representation [italics in the original]” (p. 23).
Embracing non-Euclidean mathematics and advances in science, such as Einstein’s work with relativity, he formulated his own approach to geometry, which allowed imaginary numbers to receive spatial representation. In doing so, he created a new model of the universe, one which he thought turned out to be closer to Dante and medieval Ptolemaic cosmology than the worldview presented by positivism. Thanks to relativity, there was no reason why we could not center our map of the universe upon the Earth. Nonetheless, Florensky understood the distinction between what science had come to know and what Dante could have known in his time. What Dante possessed were intuitions which corresponded to discoveries in modern science. By showing the value of Dante’s cosmology, Florensky reaffirmed his notion that we can find a way to use and learn from all properly established worldviews.
Addressing the Divine Comedy, Florensky noted how Dante and Virgil are turned upside down in their travels:
Both poets descend the cliffs of the funnel-like Inferno. The funnel ends at the narrowest circle of the Lord of Inferno. During their descent, both poets always remain vertical: keeping their heads towards the starting point of their journey, i.e. Italy, and their feet towards the center of the Earth. However, when they reach Lucifer’s lower back, both poets suddenly turn upside down, with their feet towards the surface of the Earth, whence they had entered the underworld, and their heads pointing in the opposite direction (Inferno, Canto XXXIV). (pp. 56-57)
Dante continued his journey, going through Purgatory and Heaven, advancing in a direct, straightforward fashion, until at last he found himself back home, with his feet properly situated on the ground. Florensky explained that this meant Dante’s journey is elliptical in fashion. If this were not the case, Dante would have found himself upside down when he returned home. This shows Dante held an insight into space which connects with relativity:
Dante’s space seems to be very similar to the elliptic space. This fact sheds unexpected light on the medieval view of the finite nature of the world. These general concepts of geometry have, however, recently found an unexpectedly concrete interpretation within the principle of relativity, and according to modern physics, space must be represented precisely as an elliptic space and recognized as being finite, just as time is finite and closed in upon itself. (pp. 58-59)
Through his discussion on Dante, Florensky was able to attack the exaggerated value placed upon the worldview presented by scientific positivism. This meant Florensky’s work was seen as a direct assault on the worldview held by the Soviet regime. Florensky criticized Copernicus and his cosmology, indicating that it must be seen as kind of metaphysics; in doing so, he repudiated those who believed scientific positivism was anti-metaphysical in nature (see p. 59). There is, to be sure, some weakness involved with his argument, for he ended up suggesting that the Ptolemaic universe, with a non-moving Earth, was superior to the Copernican worldview. It would have been better to say (contrary to Florensky’s assertion) that relativity allows both systems to complement each other instead of trying to have them oppose each other. Relativity allows us to follow Nicholas of Cusa and suggest that every point in the universe can be seen as a “center” of the universe instead of trying to find one fixed point which everything else encircles.
Florensky believed that our historical inability to spatially represent imaginary numbers lies behind many cosmological mistakes. Relativity provides us the way forward. With a better worldview, we will be able to find ways to understand even how bodies could go faster than light, as they will go from beyond the “real” universe into its “imaginary double.” They will break through “spaces” similar to the way bodies which go faster than sound break through “air.” Space collapses, but not the body, and the movement goes into the imaginary real realm of Dante’s Empyrean:
The imaginary realm is real, comprehensible, and in Dante’s language it is called the Empyrean. We can picture all space as double, made up of real and imaginary Gaussian coordinate surfaces that match the real ones, but the transition from the real surface to the imaginary one, however, is possible by fracturing the space and turning the body inside itself [Italics in the original]. (pp. 62-63)
Because Florensky’s scientific and technical skills could be put to use by the Soviet regime, he was given some level of freedom to speculate and discuss philosophical and theological notions which were otherwise frowned upon by the government. Despite this, what he wrote would be remembered, criticized, and eventually used as a part of the general charges made against him, even if, as Andrea Oppo’s introduction explains, many mathematicians, scientists, and literary figures such as Dmitrii Egorov, Aleksei Losev, and Mikhail Bulgakov read his works and embraced what Florensky suggested in them.
Some readers might find the translation a little off-putting because it does not engage inclusive language, but that is a minor (albeit real) issue. What is important is that the text as a whole is readable and engaging, even for those who do not normally read and study mathematical literature. Part of this, to be sure, lies with Florensky himself, and his own writing style, where, as a polymath, he liked to connect various aspects of his thoughts and beliefs together, showing how they intertwine. Some of it, nonetheless, lies with the translators and editors; they were able to render his complex ideas in English, a feat which is not easy for his less technically inspired texts, and so must have been much more difficult for something of this nature.
While this book is not for everyone, there is much in it to intrigue people coming from different disciplines of study. Those who study literature will find his interpretation of Dante interesting (even if they might think that he was reading too much into Dante and his cosmology in order to justify his defense of medieval cosmology). Those who study science, especially the philosophy of science, will find the work monumental as it demonstrates how there can be and should be more than one way to engage the sciences. Those who study theology should find the philosophical premises presented throughout the work insightful and helpful. Finally, those who study Florensky need to read this book, even if they do not grasp everything in it, for it presents a core of his own thinking, an epistemological hermeneutic which shapes many of his writings. Finally, the introduction and commentary by Andrea Oppo and Massimiliano Spano, respectively, do well to situate the text, not only in Florensky’s life and thought, but in the greater scientific and mathematical context which Florensky wanted to engage.
Imaginaries in Geometry can be purchased from Amazon.
Henry Karlson is an independent Byzantine Catholic scholar who holds an MA in Theology from Xavier University in Cincinnati. He blogs at A Little Bit of Nothing on the Patheos website. See his summary of his The Eschatological Judgment of Christ: The Hope of Universal Salvation and the Fear of Eternal Perdition in the Theology of Hans Urs von Balthasar. Follow him on Twitter @HenryKarlsonIII.