I’ve taught math classes before—trigonometry, statistics, calculus—and nowhere in those curricula do we find this sense of wonder.
And this is why math is universally despised. The way we teach it just sucks the soul out of it. Everyone knows about classical literature and classical music, but classical mathematics is sorely neglected. If we went back to studying Euclid, Apollonius of Perga, Aristotle, et al., I wager that we would see much more intellectual creativity. Primary sources beat textbooks every single time.
You make an important point about traditional mathematics contributing to intellectual creativity—after all, the quadrivium consisted of four "mathematical" subjects, but the "mathematical arts" of arithmetic, geometry, astronomy, and music are a far cry from algebra, (modern) geometry, trigonometry, and calculus. Modern math classes primarily teach students how to obtain correct answers by following established procedures. The traditional mathematical arts taught students how to think in beneficially abstract ways about the order, harmony, and proportion of the physical universe.
The problem is that 99% of people are only interested in *what* the result is, not *how* it was obtained. But once you understand the how, the possible applications are endless! It's an uphill battle though, since I don't think Euclid (much less any of the others) have been featured in mainstream education for over 100 years.
Check out their curriculum! It’s very heartening. Even better, visit some time if you can. It will restore your soul. If you have extra funds, they’re a great place to donate to.
Robert, I got this note just now from a priest friend in Toronto who asked me to pass it along.
~~
today on my substacks feed I got a triple reference to Kwasniewski, Keim, and you discussing number theory. This and symbolism are crucial to the recovery of premodern thought, as you and your cohort see. I'm writing to you because I am so computer illiterate that I don't know how to submit anything to whatever sort of entity forwarded the summary of your articles and comments to me.
Mr Keim's work is good, but in it he quotes something very puzzling about the number six, which seems to be a mistake--or possibly deliberate trolling on someone's part, viz., the claim that 6 represents imperfection. Far from it! 6 is (and is known as) the first of the "perfect numbers" in traditional number theory and in medieval lore. In witness to which I append a text about this standard teaching. If you could pass it along to Mr Keim it might be useful for him to consider:
“The number 6 was the first perfect number, and the number of creation. The adjective "perfect" was attached to numbers that are precisely equal to the sum of all the smaller numbers that divide into them, as 6=1+2+3. The next such number, incidentally, is 28=1+2+4+7+14, followed by 496=1+2+4+8+16+31+62+124+248; by the time we reach the ninth perfect number, it contains thirty-seven digits. Six is also the product of the first female number, 2, and the first masculine number, 3. The Hellenistic Jewish philosopher Philo Judaeus of Alexandria (ca. 20 B.C.-c.a. A.D. 40), whose work brought together Greek philosophy and Hebrew scriptures, suggested that God created the world in six days because six was a perfect number. The same idea was elaborated upon by St. Augustine (354-430) in The City of God: "Six is a number perfect in itself, and not because God created the world in six days; rather the contrary is true: God created the world in six days because this number is perfect, and it would remain perfect, even if the work of the six days did not exist." Some commentators of the Bible regarded 28 also as a basic number of the Supreme Architect, pointing to the 28 days of the lunar cycle. The fascination with perfect numbers penetrated even into Judaism, and their study was advocated in the twelfth century by Rabbi Yosef ben Yehudah Ankin in his book, Healing of the Souls.” ― Mario Livio, The Golden Ratio: The Story of Phi, the World's Most Astonishing Number
P.S. I can see that, considered from the perspective of grace presupposing and perfecting nature that 6 would be incomplete. But I think the traditional understanding of the integrity of creation in its own order (even fallen creation) would militate against identifying it with imperfection first and foremost. It is almost perverse to identify the first perfect number with imperfection!
Thanks for passing this along, Hilary. The main problem here is that medieval number symbolism was convoluted and not always consistent. The book that Hopper wrote on the topic is 272 pages! And he points out "the looseness with which these [number philosophy] principles were applied. It was never a matter of defining the meaning of a given number by checking it against these rules, but rather of selecting the rule which would provide the traditional or desired meaning."
The relationship between six and imperfection was one meaning that Dr. Haskell emphasized, and that's the one I included in the list (I also think the relationship six/seven–imperfect/perfect is interesting). Also, even numbers in general were considered less perfect than odd numbers, so six as an even number is like the less perfect counterpart of seven (and when "perfect" means "complete," then "less perfect" means "less complete," i.e., "incomplete/imperfect"; nine, despite being three times three, also had a status of "not quite complete" because it was one less than ten). Hopper specifically mentions six as a number that could be perfect according to one criterion (six days of Creation) and imperfect according to another (it was an even number). Six is also a circular number (like five), so six could in theory symbolize incorruptibility.
In any case, thanks to your priest friend for taking the time to write and send this. I'll add a note to the article directing readers to these comments if they want to dive deeper.
I'm wondering in which book you found a "penitential psalms section," as the penitential psalms aren't sequential but are scattered throughout the Psalter.
A section for the seven penitential Psalms was a standard component of Books of Hours. Most Books of Hours included them. These books were not psalters—i.e., their purpose was not to provide all of the Psalms or to provide Psalms in numerical order. Psalms were organized according to devotional purposes.
Well, you continue to blow my mind! I knew numbers were symbolic in past times, but I've never had it explained like this. Lord, if we could only think this deeply, what a world of wonder is opened to us. We have lost so much of what is truly important. I feel like my brain is slowly being rewired. At least I hope so!
Yes, exactly, "a world of wonder"—that's what we've lost in so many ways, and what we need to recover! And as far as rewiring our brains (in beneficial ways of course!), that is one of our fundamental objectives here, just as I say in the tagline for this newsletter: "Medieval spirituality for the postmodern world—sharing authentic texts, serious scholarship, and sincere commentary *to help us rediscover reality*."
St Benedict and the other early monks (including the celtic monks) didn't just recite the psalms, they learned them by heart. Numbers were not important - as you rightly say, the cantor intoning the first line would give a clue and for the few with identical beginnings such as psalm 6 (Domine, ne in furore tua)/37 their position in the liturgy would give the clue as to which one it was. St Benedict did use the numbers when he prescribed their order in the Rule, though, so they clearly were used in some way.
Yes, though it's difficult to make firm conclusions from the existing manuscripts, it seems to me that Psalm numbers were more likely to be used in "scholarly" or monastic manuscripts (though they were still rare), and they appeared less frequently (almost never) in books intended for the laity. If you look through the article by Peter Kwasniewski that I mentioned, you'll see examples of Psalm numbers used in a scholarly codex.
Thanks for all of this. In my experience Psalm numbering is at least somewhat rare in all classes of surviving medieval manuscripts, but when I say "prayer books," I mean the medieval equivalent of the books that we nowadays call "prayer books" (i.e., books containing a collection of prayers intended primarily for the laity). In other words, I'm referring to medieval Books of Hours, which were extremely popular in the later Middle Ages and widely used by the literate classes. Books of Hours did not, as a rule, include Psalm numbers. Presumably there are exceptions to this rule, but I have examined many such manuscripts, and I do not recall encountering a single example of numbered Psalms.
Of course it is difficult to know what exactly medieval people did with their prayer books, but I would have to disagree with this statement: "the vast majority of manuscript Book of Hours, certainly all the illuminated ones, were purely for display/representative purposes." For example, Dr. Rachel Fulton Brown is a leading scholar in this field, and she describes the "hours of the Virgin" (which was a core element of Books of Hours) as "that cycle of psalms, chants, lessons, and prayers said daily throughout the High Middle Ages and later by every man, woman, or child who could read and … even by many who knew the texts only by heart." She also affirms that "the Psalms were at the heart of medieval Christian life and thought. Monks chanted them daily in the Divine Office, lay people recited them in the offices of the Virgin Mary and of the Dead, children learned them as the basis of their ABCs." Many Books of Hours have survived, not just sumptuously illuminated ones, and I see no reason to assume that most of these were merely for display.
I also don't see why the many abbreviations would cause a problem—any poetic prayer said (or sung) frequently will become partially memorized, so the abbreviated text functions as a supplement to the memorized words.
"Either way, Books of Hours usually presented the psalms arranged according to st Benedict's division, not in the 1-150 order": I'm not sure now if we're talking about the same class of books. Books of Hours did not use the Benedictine division of Psalms into canonical hours (Prime, Terce, etc.). Typical sections included "Hours of the Virgin," which was divided into canonical hours but these were "little hours," with different content; "Hours of the Cross" or "Hours of the Holy Spirit," which also had hours but without Psalms; the "Seven Penitential Psalms"; and sometimes the "Fifteen Gradual Psalms."
"I do appreciate the fact that there still a lot to understand and verify in all this – a truly fascinating field of investigation": I completely agree! And I appreciate your contributions and expertise.
I’ve taught math classes before—trigonometry, statistics, calculus—and nowhere in those curricula do we find this sense of wonder.
And this is why math is universally despised. The way we teach it just sucks the soul out of it. Everyone knows about classical literature and classical music, but classical mathematics is sorely neglected. If we went back to studying Euclid, Apollonius of Perga, Aristotle, et al., I wager that we would see much more intellectual creativity. Primary sources beat textbooks every single time.
You make an important point about traditional mathematics contributing to intellectual creativity—after all, the quadrivium consisted of four "mathematical" subjects, but the "mathematical arts" of arithmetic, geometry, astronomy, and music are a far cry from algebra, (modern) geometry, trigonometry, and calculus. Modern math classes primarily teach students how to obtain correct answers by following established procedures. The traditional mathematical arts taught students how to think in beneficially abstract ways about the order, harmony, and proportion of the physical universe.
The problem is that 99% of people are only interested in *what* the result is, not *how* it was obtained. But once you understand the how, the possible applications are endless! It's an uphill battle though, since I don't think Euclid (much less any of the others) have been featured in mainstream education for over 100 years.
All freshmen at Thomas Aquinas College in California study from Euclid’s Elements. They have an amazing curriculum using primary source materials.
I stand corrected. That's wonderful to hear.
Check out their curriculum! It’s very heartening. Even better, visit some time if you can. It will restore your soul. If you have extra funds, they’re a great place to donate to.
Robert, I got this note just now from a priest friend in Toronto who asked me to pass it along.
~~
today on my substacks feed I got a triple reference to Kwasniewski, Keim, and you discussing number theory. This and symbolism are crucial to the recovery of premodern thought, as you and your cohort see. I'm writing to you because I am so computer illiterate that I don't know how to submit anything to whatever sort of entity forwarded the summary of your articles and comments to me.
Mr Keim's work is good, but in it he quotes something very puzzling about the number six, which seems to be a mistake--or possibly deliberate trolling on someone's part, viz., the claim that 6 represents imperfection. Far from it! 6 is (and is known as) the first of the "perfect numbers" in traditional number theory and in medieval lore. In witness to which I append a text about this standard teaching. If you could pass it along to Mr Keim it might be useful for him to consider:
“The number 6 was the first perfect number, and the number of creation. The adjective "perfect" was attached to numbers that are precisely equal to the sum of all the smaller numbers that divide into them, as 6=1+2+3. The next such number, incidentally, is 28=1+2+4+7+14, followed by 496=1+2+4+8+16+31+62+124+248; by the time we reach the ninth perfect number, it contains thirty-seven digits. Six is also the product of the first female number, 2, and the first masculine number, 3. The Hellenistic Jewish philosopher Philo Judaeus of Alexandria (ca. 20 B.C.-c.a. A.D. 40), whose work brought together Greek philosophy and Hebrew scriptures, suggested that God created the world in six days because six was a perfect number. The same idea was elaborated upon by St. Augustine (354-430) in The City of God: "Six is a number perfect in itself, and not because God created the world in six days; rather the contrary is true: God created the world in six days because this number is perfect, and it would remain perfect, even if the work of the six days did not exist." Some commentators of the Bible regarded 28 also as a basic number of the Supreme Architect, pointing to the 28 days of the lunar cycle. The fascination with perfect numbers penetrated even into Judaism, and their study was advocated in the twelfth century by Rabbi Yosef ben Yehudah Ankin in his book, Healing of the Souls.” ― Mario Livio, The Golden Ratio: The Story of Phi, the World's Most Astonishing Number
P.S. I can see that, considered from the perspective of grace presupposing and perfecting nature that 6 would be incomplete. But I think the traditional understanding of the integrity of creation in its own order (even fallen creation) would militate against identifying it with imperfection first and foremost. It is almost perverse to identify the first perfect number with imperfection!
Thanks for passing this along, Hilary. The main problem here is that medieval number symbolism was convoluted and not always consistent. The book that Hopper wrote on the topic is 272 pages! And he points out "the looseness with which these [number philosophy] principles were applied. It was never a matter of defining the meaning of a given number by checking it against these rules, but rather of selecting the rule which would provide the traditional or desired meaning."
The relationship between six and imperfection was one meaning that Dr. Haskell emphasized, and that's the one I included in the list (I also think the relationship six/seven–imperfect/perfect is interesting). Also, even numbers in general were considered less perfect than odd numbers, so six as an even number is like the less perfect counterpart of seven (and when "perfect" means "complete," then "less perfect" means "less complete," i.e., "incomplete/imperfect"; nine, despite being three times three, also had a status of "not quite complete" because it was one less than ten). Hopper specifically mentions six as a number that could be perfect according to one criterion (six days of Creation) and imperfect according to another (it was an even number). Six is also a circular number (like five), so six could in theory symbolize incorruptibility.
In any case, thanks to your priest friend for taking the time to write and send this. I'll add a note to the article directing readers to these comments if they want to dive deeper.
I'm wondering in which book you found a "penitential psalms section," as the penitential psalms aren't sequential but are scattered throughout the Psalter.
A section for the seven penitential Psalms was a standard component of Books of Hours. Most Books of Hours included them. These books were not psalters—i.e., their purpose was not to provide all of the Psalms or to provide Psalms in numerical order. Psalms were organized according to devotional purposes.
This was a fun one! And the numbers and their significance was good to know! Thanks!
Well, you continue to blow my mind! I knew numbers were symbolic in past times, but I've never had it explained like this. Lord, if we could only think this deeply, what a world of wonder is opened to us. We have lost so much of what is truly important. I feel like my brain is slowly being rewired. At least I hope so!
Yes, exactly, "a world of wonder"—that's what we've lost in so many ways, and what we need to recover! And as far as rewiring our brains (in beneficial ways of course!), that is one of our fundamental objectives here, just as I say in the tagline for this newsletter: "Medieval spirituality for the postmodern world—sharing authentic texts, serious scholarship, and sincere commentary *to help us rediscover reality*."
St Benedict and the other early monks (including the celtic monks) didn't just recite the psalms, they learned them by heart. Numbers were not important - as you rightly say, the cantor intoning the first line would give a clue and for the few with identical beginnings such as psalm 6 (Domine, ne in furore tua)/37 their position in the liturgy would give the clue as to which one it was. St Benedict did use the numbers when he prescribed their order in the Rule, though, so they clearly were used in some way.
Yes, though it's difficult to make firm conclusions from the existing manuscripts, it seems to me that Psalm numbers were more likely to be used in "scholarly" or monastic manuscripts (though they were still rare), and they appeared less frequently (almost never) in books intended for the laity. If you look through the article by Peter Kwasniewski that I mentioned, you'll see examples of Psalm numbers used in a scholarly codex.
Thanks for all of this. In my experience Psalm numbering is at least somewhat rare in all classes of surviving medieval manuscripts, but when I say "prayer books," I mean the medieval equivalent of the books that we nowadays call "prayer books" (i.e., books containing a collection of prayers intended primarily for the laity). In other words, I'm referring to medieval Books of Hours, which were extremely popular in the later Middle Ages and widely used by the literate classes. Books of Hours did not, as a rule, include Psalm numbers. Presumably there are exceptions to this rule, but I have examined many such manuscripts, and I do not recall encountering a single example of numbered Psalms.
Of course it is difficult to know what exactly medieval people did with their prayer books, but I would have to disagree with this statement: "the vast majority of manuscript Book of Hours, certainly all the illuminated ones, were purely for display/representative purposes." For example, Dr. Rachel Fulton Brown is a leading scholar in this field, and she describes the "hours of the Virgin" (which was a core element of Books of Hours) as "that cycle of psalms, chants, lessons, and prayers said daily throughout the High Middle Ages and later by every man, woman, or child who could read and … even by many who knew the texts only by heart." She also affirms that "the Psalms were at the heart of medieval Christian life and thought. Monks chanted them daily in the Divine Office, lay people recited them in the offices of the Virgin Mary and of the Dead, children learned them as the basis of their ABCs." Many Books of Hours have survived, not just sumptuously illuminated ones, and I see no reason to assume that most of these were merely for display.
I also don't see why the many abbreviations would cause a problem—any poetic prayer said (or sung) frequently will become partially memorized, so the abbreviated text functions as a supplement to the memorized words.
"Either way, Books of Hours usually presented the psalms arranged according to st Benedict's division, not in the 1-150 order": I'm not sure now if we're talking about the same class of books. Books of Hours did not use the Benedictine division of Psalms into canonical hours (Prime, Terce, etc.). Typical sections included "Hours of the Virgin," which was divided into canonical hours but these were "little hours," with different content; "Hours of the Cross" or "Hours of the Holy Spirit," which also had hours but without Psalms; the "Seven Penitential Psalms"; and sometimes the "Fifteen Gradual Psalms."
"I do appreciate the fact that there still a lot to understand and verify in all this – a truly fascinating field of investigation": I completely agree! And I appreciate your contributions and expertise.