Great stuff overall! I don't mind Farley's take (as you portray it), and find it a lot better than most as far as broad strokes to large audiences go, although his take could be more nuanced. As far as your take goes:
Not at all convinced we should take "ἀριθμῷ δέ τε πάντ᾽ ἐπέοικεν" in any other way except, "but all resembles number" and just explain what ἀριθμός means. I'm also concerned by construing it in an identity sense like, "all is number," because of "ἐπέοικεν". And I would wonder if there is a fuller quote/sentence with context (though Reddit reports one, it is faulty and incorrectly cited), due to the "δέ τε" which would make me thinks it would have originally been a "but and" or "and also" or "this ____ but in addition, this ____" sort of clause. But that is trifling.
I disagree with your comment on time regarding music. To say time doesn't come into the picture, but motion does, and time can't because it is not taken substantially by the ancients and medievals in question here seems a bit faulty. It can be spoken of even if it does not exist as a thing. And especially if we grant that time is the "number of motion with respect to before or after" (Aristotle, Physics IV, as you will have already recognized), it seems to be some major aspect of rhythm and perceived harmonies (where harmony is primarily linear) which does not come into play with Arith. and Geo. thus far in the quad, so is a a distinguisher for music, by which harmonia is best seen. NOW, if we mean harmonia strictly, this is less so, and a study of "multitudes insofar as they are in relation" is a better way to construe it.
Also, I have no problem with the medievals as far as the liberal arts go, especially since that's when we really see things formalized, but I wouldn't mind consulting Presocratics/Plato/Nicomachus/Iamblichus/Plotinus/Aristides Quint. on the quadrivium a bit more. They sometimes conceive of things with differing metaphysics or methods that can be nice to have, especially when thinking about the symbolic side of things, and are majors players in the development of ancient quadrivial stuff. And help when thinking of limit/unlimited. Then again I'm not really a pythagorean nor a neoplatonist in a lot of senses anyway.
Indeed, one of the things I regret about this article is putting my comments about time so sharply (namely saying "time doesn't come into it at all"). As i have thought more since writing this, I've clarified my position. If we take time as a number (i.e., a measure--for that is what quantity is) it seems somewhat strange to say "number in time" as this would be to say that music is "number in number of motion."
Still, from a historical position, which is where I was mainly coming from in this article (in order to try and arrive at the truth possessed by our fathers), the division of space, time, and space and time is at the very oldest Newtonian. Our fathers did not think of "space and time" and even if taken in a more ancient sense (place and motion and/or succession of being), the words do not convey that to the modern student. Thus, though the modern division can be explained in a true sense, it obscures the very nature of classical mathematics and physics. It's late and I'm not sitting in front of all my sources right now, but either in Boethius, Hugh, or Thomas, one explicitly denies that the science of music considers the mobile mathematically (even though the art does). Cosmology is this science, So even if we bring in motion to music (as say, the euclidean sectio canonis and Augustine do), we must be careful not to make music a science of the continuous (it becomes indistinct from Cosmology if we do so) and likewise careful to keep it in the study of the discrete.
You may be right about pythagoras—your greek is far better than mine, but Klein was a formidable scholar. There is little context. I have gone to the source material, but it's an isolated quote in the Life of Pythagoras. It basically says "they taught this" and moves on. If we do take it as "All resembles number," I think that actually strengthens Klein's interpretation (his "better translated as" may very well mean "in order to convey the sense") as all things are not themselves the multitudes, but are rather measurable by multitude because they resemble i.e., bear some likeness to number.
Really appreciate this interaction! I'm, like many, trying to find an education. Much of what you write is a helpful drilling down to the particulars. Also, as I've read through the relevant works, it seems that the conversation and debate is where a significant portion of the wisdom is found. So, here we go!
I agree that I'm not careful with definitions. I'm writing with current vocabulary. Your pull into more precise definitions is appreciated. I do not believe that I severely misunderstand the ancients and medievals, having spent my fair share with them, but I did not write this essay with the precision of the language that they would have. Point taken. Writing post-Heidegger about medievals is a balancing act that I have not mastered. Many readers do not have the metaphysical vocabulary necessary, but that doesn't excuse my sloppy definitions.
Some I will stand behind. I believe words like information have changed meaning in modern English such that they no longer denote what the medievals meant. But over all, I ought to shore up.
And many of your distinctions are exactly what I am working through as I work through the traditions.
At the heart of what I am after is a restoration of the Quadrivium as something other than what we think of as Math and Science curriculum. In the same way the trivium is not grammar and literature curriculum. It is the recovery of the art of knowing. This whole project, for me, began as I worked through Aristotle's divisions of knowledge. The realization that we lack care in our distinctions opened an attic of dust covered books that I have been working my way through in order to understand the quadrivium. Part of writing about it is to sort my own thoughts. As Bacon quipped, writing makes a man exact. But the other part is to find those already having the discussion. So, I'm halfway there (and living on a prayer)
To the meat: Though I am not surprised to find a definition I am using to not be ancient, I am surprised that it is not because it is extremely prevalent. I stepped into a conversation in the present and am, in many ways, reading my way backwards. So I appreciate what you are saying, but I also am stepping into the contemporary conversation and, as is my habit, trying to pull together what I am learning as I go. I also fully admit that much of my reading has focused on alchemical writings as much as philosophical.
I am approaching the quadrivium as a humanist more than a mathematician (to my shame), but, I am looking to understand, so your distinction between discrete quantities and number in itself is interesting. The way I would see that distinction is between the abstract and the particular. The study of arithmetic would be the study of both as well as the study of the relation of the abstract to the particular. Correct? My definition is partial, not incorrect, if I am reading you correctly.
On to Geometry. Yes, space is a foreign concept to the ancients. but it is not to us. So I believe my definition and your definition are a difference of vocabulary. Not to say they are not substantive, and that my vocabulary is not the one in need of refining, but the substance of geometry as the art of enumerating magnitudes (what many would now call space) is at least a starting point. Correct?
Now, music, this is where we may have a disagreement. I'm not sure. I do believe that time is one important aspect of what is being enumerated in music. The divisions of time by rhythm are central to writings on music, as well as the discussions of the music of the spheres. Now, my philosophical definition of time may, perhaps, be what you see as the problem. I'm still unsure about the metaphysical nature of time. I'm stuck between competing voices at the moment and that may be coming through, but the vibrations that produce sound as well as the divisions of rhythm can be neither produced or experienced without time.
On astronomy, I don't disagree with Boethius, but believe astronomy as a quadrivial science, came to include much more than that. Bede's work on the keeping of time has influenced my understanding of astronomy as a broader discipline.
And I simply disagree with Thomas (with sufficient shamefacedness, since I understand I am disagreeing with my superior) on the use of the imagination. Logocentric reason does not hold the central place in reasoning anymore and, as Christ is both the Divine Logos and the Divine Image at the same time, I do not believe we need set asunder the imagination and reason. We are, as a rule, more prone to make the opposite error in our day, so we could use the ballast of The Angelic Doctor, but I stand by my statements about the imagination.
Sir, I appreciate your response immensely! This is an incredibly valuable conversation. Thank you!
Thank you for this charitable and well thought response. I'll just go paragraph by paragraph.
Yes, really defining the particulars and understanding what the medieval philosophers recognized the quadrivium to be is precisely what I'm after--and I haven't fully grasped them yet either. On the severely comment, perhaps I spoke too strongly though what I was getting at is any error which fails to actually grasp the position. We are all learners here; you are right when you say that conversation around the primary texts is what will move us forward.
Indeed, there is room for broader brush strokes, especially when communicating to a more general audience. I don't actually fault you for that--my comments on e.g., information and the like were intended to use your broader strokes as a heuristic to point to the more precise understandings of the ancients. When I said that I laud your distinction between information and formation--I really meant that, and really, we were saying more or less the same thing.
"To the heart" Yes and amen--the quadrivium is not about what we think of as the maths and sciences today, but it is what they thought of as mathematics (and in the case of music* and more especially astronomy, natural philosophy mathematically approached, which actually makes them very much like modern physics). As a die-hard thomist (as die-hard as one can be without being RC), I hold (and would say I know) that the categories rightly divide the world according to how it may be known. And so when I make distinctions contra modern understandings I have two primary reasons for doing so. First, the modern understanding of x may be true, but it is not the same as what the ancients pursued. Second, the modern understanding of x may in fact be in error concerning reality, and I believe the ancient's position to be the true one. More on this later.
"To the meat," I too was surprised that the space and time def is so prevalent yet I have been unable to find it in the tradition. I've done some digital searches through google books and there seems to be a kind of definition creep. But I think it really comes from the fact that the classical education movement wasn't started by hardcore scholars, other than being inspired by Sayers' book—who simply says it need not concern the reader and didn't bother to even define it. And even that book is rather imaginative and playful, which is fine if it's not taken as a precise and sufficient explanation of the tradition. My mentor and friend Greg Wilbur has also been very interested in the alchemical side of things, and I will admit—there may be presentations of the quadrivium in those writings that run along different lines. I have submitted myself to Boethius, because he is taken as THE authority on the matter by the medievals, and he coined the term.
So my critique of the common definition of Arithmetic is two-fold. First, is terminological because of the use of number today. Number today roughly means mathematical entity as a genus, so that includes 1 and .00003. That these are real categorizations of quantitative being is granted, but it would not fall under the understanding of discrete multitude, which is not infinitely divisible, and therefore I use multitude to help in translation. Etymologically, number is a perfectly correct translation and from here on in this comment I will use them interchangeably for the fun of it lol. Second, I made a distinction that was perhaps not very clear. The quadrivial sciences sit in one way on the same abstractional level as each other (even though there is a real ontological hierarchy to the various mathematical objects): they all consider that category of being, which can be understood apart from matter but cannot exist apart from it. So, when Boethius says Arithmetic considers numbers (or multitudes) sunt per se to distinguish it from music which considers numbers sunt ad aliquid he is not distinguishing them by abstraction, but rather in what respect they are considered. Arithmetic considers 3 as 3, Music considers 3 as the sesquialter of 2.
Many today would consider magnitudes as taking up space, and view geometry as considering space quantified. However, as an Aristotelian (and the Platonists agree on this as well), I hold that absolute space (which is Newtonian in origin as far as I know) is nonsense, rather, there are magnitudes (bodies) which have extension. Place (often poorly translated in sources as space) is defined by the limited extension of bodies. That is, place is a measure of bodily extension. So there is a real difference, even though it seems to translate well.
Now, music, this is where we may have a disagreement. I'm not sure. I do believe that time is one important aspect of what is being enumerated in music. The divisions of time by rhythm are central to writings on music, as well as the discussions of the music of the spheres. Now, my philosophical definition of time may, perhaps, be what you see as the problem. I'm still unsure about the metaphysical nature of time. I'm stuck between competing voices at the moment and that may be coming through, but the vibrations that produce sound as well as the divisions of rhythm can be neither produced or experienced without time.
Time, like place, is a measurement. That is to say, “time is a number of motion.” The now (or the moment) is to motion as a point is to magnitude: It has no part, but it is also what we measure through, toward, and from. Thus, to say “number in time” is to say “Number in number.” Now as regards music. Note that all of these sciences understand something which is a principle of reality. A sheep is a discrete unit which is indivisible (its constituent parts are not a sheep but something else entirely) and wood is a continuous body which is infinitely divisible (its constituent parts are still wood). Now, sheep do not fall under the definition of unit or multitude even though each one is a unit and a flock is a multitude. Neither does matter fall under the definition of a line or point. So also music, even though it considers various really existing things, which are multitudes in relation (such as pitches in a scale, the human body to the soul, or the heavens in their courses), is not defined by its materially existing objects (sounds which are subject to motion, parts of the human individual, the planetary objects). It is rather defined by that category of being by which they are considered and are existing. This is in part a new discovery for me (as in, since writing this article), so I have yet to run this by some of my mentors and colleagues. Boethius’ three kinds of music are these really existing musics which are multitudes in relation, ruled by the science of music. So yes—these three musics cannot be nor be perceived without motion, but the science of music itself doesn’t include motion in its definition.
Astronomy really stands out because it considers magnitudes as mobile, which means that its unique objects do in fact include matter in definition. In fact, time does come into it, in that it is concerned with measuring motion. In prepping this answer I was rereading Hugh, and he says this “Astronomy is the discipline which examines the spaces [here is example of bad translation], movements, and circuits of the heavenly bodies at determined intervals.” Book 2 Chapter 15. Note that the focus is still the proportion/measure for him, not the time in itself. Thomas says that Astronomy (actually along with music--whether multitudes in relation should be considered a separate science from Arithmetic is a long standing debate before Boethius coined the term quadrivium, and I think Thomas is just following Aristotle here instead of following Boethius' definition) is an intermediate science which considers objects that would normally fall under (aristotelian) physics, but he says it is distinct from it because it demonstrates its conclusions mathematically.
Ok, It's very late. I must sleep. I want to respond to your comments about imagination at some point more fully, but I should say that I was, just like with "information," using your contemporary use of the term imagination (at least what I perceived to be your use of it) as a heuristic for bringing in Thomistic distinctions. The imagination is something very particular for Thomas. It is pretty much just stored sense knowledge, and he would agree with Hugh who writes “Mathematics never operates without the imagination, and therefore never possesses its object in a simple or non-composite manner. [...] it was necessary that [logic and mathematics] base their considerations not upon the physical actualities of things, of which we have deceptive experience, but upon reason alone, in which unshakeable truth stands fast, and that then, with reason itself to lead them, they descend into the physical order.”
So would you put the wisdom of the quodrivium as achieving its summit in a more 'metaphysically-tuned' approach to metaphysics proper (and from there higher theology)? I.e., would you say one of the greatest glories of the quadrivium is metaphysics?
I ask because I always struggle to see the location of metaphysics in the quadrivium. Because I worry that, "although the Quadrivium does help us behold, measure, et cetera, and as Boethius says, is the path through which those seeking wisdom (again, knowledge of first causes, id est, metaphysics) must first proceed"—metaphysics is also that which "provides the principles from which the sciences of arithmetic... are pursued." So what comes first? Quadrivium or metaphysics? Substance is necessary for quantity and quality, but in the order of knowledge, do we not move from quality and quantity to substance (ref. Thomas somewhere, I think)? So I have a question, in terms of a true classical education, and in relation to the life of the developing student, do you think that metaphysics stands as the backdrop to the quodrivium and functions as a certain hidden but constantly-being-revealed-wisdom standing behind and supporting the quodrivium? Or should the tenets of metaphysics serve as the principle of the quadrivium as the student moves through the four branches? Math or Metaphysics first Noah?
I ask as a Classical educator who is a curriculum developer and desires to include a far more classical approach, something even perhaps radical. Our education council is making our ninth graders read selections of Aristotle's metaphysics and do a debate on 'Aristotle vs. Plato on the forms', so I do not want to sacrifice an opportunity to provide them a truly classical mathematics as well.
I really like what you are saying and agree wholeheartedly.
I just wonder that if metaphysics is kept back from them—even early on—or at least, if the certain math teacher is not a raging realist, could not the Cartesian dream our world suffers from (simply and implicitly by being a 'modern' world) sweep up the souls of these young math students and wisper to them they are doing 'a priori' judgements ripped clean from the 'thing themselves,' convincing them that they are 'intellects functioning in a void... cut away from senstation' (Wilhelmsen, Mans knowledge of reality). I hate that. The human intellect (and I emphasize the 'human' in that, but comments won't let me emphasize) knows the universal of 'quantity' or 'magnitude' because he or she has sensed their respective 'secondary substance' in the matter! Thus, one must start by convincing a student of the intellect-to-thing-through-sense power of the human soul to do proper metaphysically-true-quodrivium-activities. The sad thing is, we have many students who could easily be tempted by a modern sentiment on this issues. These types of principles do affect teaching in the classroom. Therefore—in this line of reasoning—you have to do metaphysics to do math. But what does it mean to 'do metaphysics'! Assume the first principles, hoping the student will find them OR point them out explicitly? Probably the latter!
By way of authority, first. The order of learning given by Aristotle and largely followed by the tradition, as I believe you know is: Logic, then Mathematics, then Physics (which requires experience), then Ethics, and finally when the student is old and his passions are in control, Metaphysics. This is really a rather shocking order, for the order of learning does not follow the order of knowing, that is, physics is by nature better known for its objects fall under the senses—this despite Aristotle's famous philosophic method which starts with those which are better known to us by nature (i.e. sensibles) and proceeds to those which are clearer in themselves but less known to us by nature (principles of being, first causes, etc.)
It is of the utmost importance to realize that this order is not for little boys and girls, nor is it for the common man. Still, it is not the educational order for a grown man either, but for something like the adolescent through adulthood. Logic (which the trivium concerns) and Mathematics (which the quadrivium concerns) are those first serious studies which prepare the serious student for the rest of the philosophic journey. But note that though the disciplines are scrambled, the individual studies of them still follow Aristotle's method of clearer to us proceeding to clearer in itself.
Porphyry, for example, defers in the Isagoge from discussing subsistence: "For instance, I shall omit to speak about genera and species, as to whether they subsist (in the nature of things) or in mere conceptions only;" Or again, before the technical study begins, every young child who is taught to count is starting with this sensible and that sensible. Or again, he draws a line from a given point, or a circle in the dirt when playing jacks or to mark a finish line for a race. Classical education should be starting the student in the world of the senses and guiding him by the hand into the world of the forms. In this way, the stereotypical homeschool nature walk gets something right.
Now more directly to your questions. You quote me quoting Boethius that the quadrivium (along with trivium) leads us to the principles of metaphysics and metaphysics provides the principles for the pursuit of the lower sciences. To clarify, the principles it prepares us for are principles of being, because in considering mathematical entities, acts of the mind, etc. we are considering the varying levels of being. So just as considering sensibles is necessary to abstract forms, so considering forms (and other causes) is necessary to come to knowledge of metaphysical principles e.g., act and potency
This is indeed perplexing at first, but it is good to keep in mind 2 things. 1st, that the higher sciences are always said to provide the principles of lower sciences. 2nd, that the investigation of a science does not start from its principles, but from what is more knowable by nature. I do not think I was super clear, and I was probably confusing here in the article on this issue. In fact, there is much “woodshedding” (to borrow a term from music practice culture) that I still have to do to become laser clear on this issue, as you will see in the following paragraph (where I’m still working out some of my thought on this).
Boethius says (translating Nicomachus), for example, that “wisdom gives name to a science by virtue of essentia whether in reality or only in name” (in nicomachus ουσιν whether in reality or equivocally named, so I take this to be quoting Categories on first and second substance). To speak roughly, as we proceed to those less knowable to us by nature but more knowable in themselves, we then look back down and can give name (i.e. λογος, ratio, definition) to what we proceeded through. There’s a way in which just as in proceeding from sensibles to their principles we understand the sensibles better by way of their definitions, so in proceeding from principles to first (or higher) principles we come to know the lower principles. There’s a starting in sense that in reaching the heights descends back into sense.
Ok it’s getting late, and I’m reaching the limit of what I’m able to say from working knowledge and memory—I know this doesn’t answer everything you asked. In conclusion, mathematics needs to be taught in such a way to allow the student to ascend in knowledge through the chain of being, from lower forms to higher forms. Recovering multitude vs magnitude, continuous vs discrete, euclid’s definitions, etc., is part of the answer—these definitions don’t implant knowledge in the student’s head, but they do present judgments that may not have been considered by the student before (likely have not). Teachers must be raging realists. Teaching, is as Ryan likes to say, all about providing the right phantasm. You are guiding the student in a consideration of reality.
Great stuff overall! I don't mind Farley's take (as you portray it), and find it a lot better than most as far as broad strokes to large audiences go, although his take could be more nuanced. As far as your take goes:
Not at all convinced we should take "ἀριθμῷ δέ τε πάντ᾽ ἐπέοικεν" in any other way except, "but all resembles number" and just explain what ἀριθμός means. I'm also concerned by construing it in an identity sense like, "all is number," because of "ἐπέοικεν". And I would wonder if there is a fuller quote/sentence with context (though Reddit reports one, it is faulty and incorrectly cited), due to the "δέ τε" which would make me thinks it would have originally been a "but and" or "and also" or "this ____ but in addition, this ____" sort of clause. But that is trifling.
I disagree with your comment on time regarding music. To say time doesn't come into the picture, but motion does, and time can't because it is not taken substantially by the ancients and medievals in question here seems a bit faulty. It can be spoken of even if it does not exist as a thing. And especially if we grant that time is the "number of motion with respect to before or after" (Aristotle, Physics IV, as you will have already recognized), it seems to be some major aspect of rhythm and perceived harmonies (where harmony is primarily linear) which does not come into play with Arith. and Geo. thus far in the quad, so is a a distinguisher for music, by which harmonia is best seen. NOW, if we mean harmonia strictly, this is less so, and a study of "multitudes insofar as they are in relation" is a better way to construe it.
Also, I have no problem with the medievals as far as the liberal arts go, especially since that's when we really see things formalized, but I wouldn't mind consulting Presocratics/Plato/Nicomachus/Iamblichus/Plotinus/Aristides Quint. on the quadrivium a bit more. They sometimes conceive of things with differing metaphysics or methods that can be nice to have, especially when thinking about the symbolic side of things, and are majors players in the development of ancient quadrivial stuff. And help when thinking of limit/unlimited. Then again I'm not really a pythagorean nor a neoplatonist in a lot of senses anyway.
I'm glad we agree one isn't a number :)
Thanks for a great take on the quad!
Indeed, one of the things I regret about this article is putting my comments about time so sharply (namely saying "time doesn't come into it at all"). As i have thought more since writing this, I've clarified my position. If we take time as a number (i.e., a measure--for that is what quantity is) it seems somewhat strange to say "number in time" as this would be to say that music is "number in number of motion."
Still, from a historical position, which is where I was mainly coming from in this article (in order to try and arrive at the truth possessed by our fathers), the division of space, time, and space and time is at the very oldest Newtonian. Our fathers did not think of "space and time" and even if taken in a more ancient sense (place and motion and/or succession of being), the words do not convey that to the modern student. Thus, though the modern division can be explained in a true sense, it obscures the very nature of classical mathematics and physics. It's late and I'm not sitting in front of all my sources right now, but either in Boethius, Hugh, or Thomas, one explicitly denies that the science of music considers the mobile mathematically (even though the art does). Cosmology is this science, So even if we bring in motion to music (as say, the euclidean sectio canonis and Augustine do), we must be careful not to make music a science of the continuous (it becomes indistinct from Cosmology if we do so) and likewise careful to keep it in the study of the discrete.
You may be right about pythagoras—your greek is far better than mine, but Klein was a formidable scholar. There is little context. I have gone to the source material, but it's an isolated quote in the Life of Pythagoras. It basically says "they taught this" and moves on. If we do take it as "All resembles number," I think that actually strengthens Klein's interpretation (his "better translated as" may very well mean "in order to convey the sense") as all things are not themselves the multitudes, but are rather measurable by multitude because they resemble i.e., bear some likeness to number.
Nice. Yes.
Really appreciate this interaction! I'm, like many, trying to find an education. Much of what you write is a helpful drilling down to the particulars. Also, as I've read through the relevant works, it seems that the conversation and debate is where a significant portion of the wisdom is found. So, here we go!
I agree that I'm not careful with definitions. I'm writing with current vocabulary. Your pull into more precise definitions is appreciated. I do not believe that I severely misunderstand the ancients and medievals, having spent my fair share with them, but I did not write this essay with the precision of the language that they would have. Point taken. Writing post-Heidegger about medievals is a balancing act that I have not mastered. Many readers do not have the metaphysical vocabulary necessary, but that doesn't excuse my sloppy definitions.
Some I will stand behind. I believe words like information have changed meaning in modern English such that they no longer denote what the medievals meant. But over all, I ought to shore up.
And many of your distinctions are exactly what I am working through as I work through the traditions.
At the heart of what I am after is a restoration of the Quadrivium as something other than what we think of as Math and Science curriculum. In the same way the trivium is not grammar and literature curriculum. It is the recovery of the art of knowing. This whole project, for me, began as I worked through Aristotle's divisions of knowledge. The realization that we lack care in our distinctions opened an attic of dust covered books that I have been working my way through in order to understand the quadrivium. Part of writing about it is to sort my own thoughts. As Bacon quipped, writing makes a man exact. But the other part is to find those already having the discussion. So, I'm halfway there (and living on a prayer)
To the meat: Though I am not surprised to find a definition I am using to not be ancient, I am surprised that it is not because it is extremely prevalent. I stepped into a conversation in the present and am, in many ways, reading my way backwards. So I appreciate what you are saying, but I also am stepping into the contemporary conversation and, as is my habit, trying to pull together what I am learning as I go. I also fully admit that much of my reading has focused on alchemical writings as much as philosophical.
I am approaching the quadrivium as a humanist more than a mathematician (to my shame), but, I am looking to understand, so your distinction between discrete quantities and number in itself is interesting. The way I would see that distinction is between the abstract and the particular. The study of arithmetic would be the study of both as well as the study of the relation of the abstract to the particular. Correct? My definition is partial, not incorrect, if I am reading you correctly.
On to Geometry. Yes, space is a foreign concept to the ancients. but it is not to us. So I believe my definition and your definition are a difference of vocabulary. Not to say they are not substantive, and that my vocabulary is not the one in need of refining, but the substance of geometry as the art of enumerating magnitudes (what many would now call space) is at least a starting point. Correct?
Now, music, this is where we may have a disagreement. I'm not sure. I do believe that time is one important aspect of what is being enumerated in music. The divisions of time by rhythm are central to writings on music, as well as the discussions of the music of the spheres. Now, my philosophical definition of time may, perhaps, be what you see as the problem. I'm still unsure about the metaphysical nature of time. I'm stuck between competing voices at the moment and that may be coming through, but the vibrations that produce sound as well as the divisions of rhythm can be neither produced or experienced without time.
On astronomy, I don't disagree with Boethius, but believe astronomy as a quadrivial science, came to include much more than that. Bede's work on the keeping of time has influenced my understanding of astronomy as a broader discipline.
And I simply disagree with Thomas (with sufficient shamefacedness, since I understand I am disagreeing with my superior) on the use of the imagination. Logocentric reason does not hold the central place in reasoning anymore and, as Christ is both the Divine Logos and the Divine Image at the same time, I do not believe we need set asunder the imagination and reason. We are, as a rule, more prone to make the opposite error in our day, so we could use the ballast of The Angelic Doctor, but I stand by my statements about the imagination.
Sir, I appreciate your response immensely! This is an incredibly valuable conversation. Thank you!
Thank you for this charitable and well thought response. I'll just go paragraph by paragraph.
Yes, really defining the particulars and understanding what the medieval philosophers recognized the quadrivium to be is precisely what I'm after--and I haven't fully grasped them yet either. On the severely comment, perhaps I spoke too strongly though what I was getting at is any error which fails to actually grasp the position. We are all learners here; you are right when you say that conversation around the primary texts is what will move us forward.
Indeed, there is room for broader brush strokes, especially when communicating to a more general audience. I don't actually fault you for that--my comments on e.g., information and the like were intended to use your broader strokes as a heuristic to point to the more precise understandings of the ancients. When I said that I laud your distinction between information and formation--I really meant that, and really, we were saying more or less the same thing.
"To the heart" Yes and amen--the quadrivium is not about what we think of as the maths and sciences today, but it is what they thought of as mathematics (and in the case of music* and more especially astronomy, natural philosophy mathematically approached, which actually makes them very much like modern physics). As a die-hard thomist (as die-hard as one can be without being RC), I hold (and would say I know) that the categories rightly divide the world according to how it may be known. And so when I make distinctions contra modern understandings I have two primary reasons for doing so. First, the modern understanding of x may be true, but it is not the same as what the ancients pursued. Second, the modern understanding of x may in fact be in error concerning reality, and I believe the ancient's position to be the true one. More on this later.
"To the meat," I too was surprised that the space and time def is so prevalent yet I have been unable to find it in the tradition. I've done some digital searches through google books and there seems to be a kind of definition creep. But I think it really comes from the fact that the classical education movement wasn't started by hardcore scholars, other than being inspired by Sayers' book—who simply says it need not concern the reader and didn't bother to even define it. And even that book is rather imaginative and playful, which is fine if it's not taken as a precise and sufficient explanation of the tradition. My mentor and friend Greg Wilbur has also been very interested in the alchemical side of things, and I will admit—there may be presentations of the quadrivium in those writings that run along different lines. I have submitted myself to Boethius, because he is taken as THE authority on the matter by the medievals, and he coined the term.
So my critique of the common definition of Arithmetic is two-fold. First, is terminological because of the use of number today. Number today roughly means mathematical entity as a genus, so that includes 1 and .00003. That these are real categorizations of quantitative being is granted, but it would not fall under the understanding of discrete multitude, which is not infinitely divisible, and therefore I use multitude to help in translation. Etymologically, number is a perfectly correct translation and from here on in this comment I will use them interchangeably for the fun of it lol. Second, I made a distinction that was perhaps not very clear. The quadrivial sciences sit in one way on the same abstractional level as each other (even though there is a real ontological hierarchy to the various mathematical objects): they all consider that category of being, which can be understood apart from matter but cannot exist apart from it. So, when Boethius says Arithmetic considers numbers (or multitudes) sunt per se to distinguish it from music which considers numbers sunt ad aliquid he is not distinguishing them by abstraction, but rather in what respect they are considered. Arithmetic considers 3 as 3, Music considers 3 as the sesquialter of 2.
Many today would consider magnitudes as taking up space, and view geometry as considering space quantified. However, as an Aristotelian (and the Platonists agree on this as well), I hold that absolute space (which is Newtonian in origin as far as I know) is nonsense, rather, there are magnitudes (bodies) which have extension. Place (often poorly translated in sources as space) is defined by the limited extension of bodies. That is, place is a measure of bodily extension. So there is a real difference, even though it seems to translate well.
Now, music, this is where we may have a disagreement. I'm not sure. I do believe that time is one important aspect of what is being enumerated in music. The divisions of time by rhythm are central to writings on music, as well as the discussions of the music of the spheres. Now, my philosophical definition of time may, perhaps, be what you see as the problem. I'm still unsure about the metaphysical nature of time. I'm stuck between competing voices at the moment and that may be coming through, but the vibrations that produce sound as well as the divisions of rhythm can be neither produced or experienced without time.
Time, like place, is a measurement. That is to say, “time is a number of motion.” The now (or the moment) is to motion as a point is to magnitude: It has no part, but it is also what we measure through, toward, and from. Thus, to say “number in time” is to say “Number in number.” Now as regards music. Note that all of these sciences understand something which is a principle of reality. A sheep is a discrete unit which is indivisible (its constituent parts are not a sheep but something else entirely) and wood is a continuous body which is infinitely divisible (its constituent parts are still wood). Now, sheep do not fall under the definition of unit or multitude even though each one is a unit and a flock is a multitude. Neither does matter fall under the definition of a line or point. So also music, even though it considers various really existing things, which are multitudes in relation (such as pitches in a scale, the human body to the soul, or the heavens in their courses), is not defined by its materially existing objects (sounds which are subject to motion, parts of the human individual, the planetary objects). It is rather defined by that category of being by which they are considered and are existing. This is in part a new discovery for me (as in, since writing this article), so I have yet to run this by some of my mentors and colleagues. Boethius’ three kinds of music are these really existing musics which are multitudes in relation, ruled by the science of music. So yes—these three musics cannot be nor be perceived without motion, but the science of music itself doesn’t include motion in its definition.
Astronomy really stands out because it considers magnitudes as mobile, which means that its unique objects do in fact include matter in definition. In fact, time does come into it, in that it is concerned with measuring motion. In prepping this answer I was rereading Hugh, and he says this “Astronomy is the discipline which examines the spaces [here is example of bad translation], movements, and circuits of the heavenly bodies at determined intervals.” Book 2 Chapter 15. Note that the focus is still the proportion/measure for him, not the time in itself. Thomas says that Astronomy (actually along with music--whether multitudes in relation should be considered a separate science from Arithmetic is a long standing debate before Boethius coined the term quadrivium, and I think Thomas is just following Aristotle here instead of following Boethius' definition) is an intermediate science which considers objects that would normally fall under (aristotelian) physics, but he says it is distinct from it because it demonstrates its conclusions mathematically.
Ok, It's very late. I must sleep. I want to respond to your comments about imagination at some point more fully, but I should say that I was, just like with "information," using your contemporary use of the term imagination (at least what I perceived to be your use of it) as a heuristic for bringing in Thomistic distinctions. The imagination is something very particular for Thomas. It is pretty much just stored sense knowledge, and he would agree with Hugh who writes “Mathematics never operates without the imagination, and therefore never possesses its object in a simple or non-composite manner. [...] it was necessary that [logic and mathematics] base their considerations not upon the physical actualities of things, of which we have deceptive experience, but upon reason alone, in which unshakeable truth stands fast, and that then, with reason itself to lead them, they descend into the physical order.”
So would you put the wisdom of the quodrivium as achieving its summit in a more 'metaphysically-tuned' approach to metaphysics proper (and from there higher theology)? I.e., would you say one of the greatest glories of the quadrivium is metaphysics?
I ask because I always struggle to see the location of metaphysics in the quadrivium. Because I worry that, "although the Quadrivium does help us behold, measure, et cetera, and as Boethius says, is the path through which those seeking wisdom (again, knowledge of first causes, id est, metaphysics) must first proceed"—metaphysics is also that which "provides the principles from which the sciences of arithmetic... are pursued." So what comes first? Quadrivium or metaphysics? Substance is necessary for quantity and quality, but in the order of knowledge, do we not move from quality and quantity to substance (ref. Thomas somewhere, I think)? So I have a question, in terms of a true classical education, and in relation to the life of the developing student, do you think that metaphysics stands as the backdrop to the quodrivium and functions as a certain hidden but constantly-being-revealed-wisdom standing behind and supporting the quodrivium? Or should the tenets of metaphysics serve as the principle of the quadrivium as the student moves through the four branches? Math or Metaphysics first Noah?
I ask as a Classical educator who is a curriculum developer and desires to include a far more classical approach, something even perhaps radical. Our education council is making our ninth graders read selections of Aristotle's metaphysics and do a debate on 'Aristotle vs. Plato on the forms', so I do not want to sacrifice an opportunity to provide them a truly classical mathematics as well.
I really like what you are saying and agree wholeheartedly.
I just wonder that if metaphysics is kept back from them—even early on—or at least, if the certain math teacher is not a raging realist, could not the Cartesian dream our world suffers from (simply and implicitly by being a 'modern' world) sweep up the souls of these young math students and wisper to them they are doing 'a priori' judgements ripped clean from the 'thing themselves,' convincing them that they are 'intellects functioning in a void... cut away from senstation' (Wilhelmsen, Mans knowledge of reality). I hate that. The human intellect (and I emphasize the 'human' in that, but comments won't let me emphasize) knows the universal of 'quantity' or 'magnitude' because he or she has sensed their respective 'secondary substance' in the matter! Thus, one must start by convincing a student of the intellect-to-thing-through-sense power of the human soul to do proper metaphysically-true-quodrivium-activities. The sad thing is, we have many students who could easily be tempted by a modern sentiment on this issues. These types of principles do affect teaching in the classroom. Therefore—in this line of reasoning—you have to do metaphysics to do math. But what does it mean to 'do metaphysics'! Assume the first principles, hoping the student will find them OR point them out explicitly? Probably the latter!
Thoughts?
Ok this is totally insufficient but here goes:
By way of authority, first. The order of learning given by Aristotle and largely followed by the tradition, as I believe you know is: Logic, then Mathematics, then Physics (which requires experience), then Ethics, and finally when the student is old and his passions are in control, Metaphysics. This is really a rather shocking order, for the order of learning does not follow the order of knowing, that is, physics is by nature better known for its objects fall under the senses—this despite Aristotle's famous philosophic method which starts with those which are better known to us by nature (i.e. sensibles) and proceeds to those which are clearer in themselves but less known to us by nature (principles of being, first causes, etc.)
It is of the utmost importance to realize that this order is not for little boys and girls, nor is it for the common man. Still, it is not the educational order for a grown man either, but for something like the adolescent through adulthood. Logic (which the trivium concerns) and Mathematics (which the quadrivium concerns) are those first serious studies which prepare the serious student for the rest of the philosophic journey. But note that though the disciplines are scrambled, the individual studies of them still follow Aristotle's method of clearer to us proceeding to clearer in itself.
Porphyry, for example, defers in the Isagoge from discussing subsistence: "For instance, I shall omit to speak about genera and species, as to whether they subsist (in the nature of things) or in mere conceptions only;" Or again, before the technical study begins, every young child who is taught to count is starting with this sensible and that sensible. Or again, he draws a line from a given point, or a circle in the dirt when playing jacks or to mark a finish line for a race. Classical education should be starting the student in the world of the senses and guiding him by the hand into the world of the forms. In this way, the stereotypical homeschool nature walk gets something right.
Now more directly to your questions. You quote me quoting Boethius that the quadrivium (along with trivium) leads us to the principles of metaphysics and metaphysics provides the principles for the pursuit of the lower sciences. To clarify, the principles it prepares us for are principles of being, because in considering mathematical entities, acts of the mind, etc. we are considering the varying levels of being. So just as considering sensibles is necessary to abstract forms, so considering forms (and other causes) is necessary to come to knowledge of metaphysical principles e.g., act and potency
This is indeed perplexing at first, but it is good to keep in mind 2 things. 1st, that the higher sciences are always said to provide the principles of lower sciences. 2nd, that the investigation of a science does not start from its principles, but from what is more knowable by nature. I do not think I was super clear, and I was probably confusing here in the article on this issue. In fact, there is much “woodshedding” (to borrow a term from music practice culture) that I still have to do to become laser clear on this issue, as you will see in the following paragraph (where I’m still working out some of my thought on this).
Boethius says (translating Nicomachus), for example, that “wisdom gives name to a science by virtue of essentia whether in reality or only in name” (in nicomachus ουσιν whether in reality or equivocally named, so I take this to be quoting Categories on first and second substance). To speak roughly, as we proceed to those less knowable to us by nature but more knowable in themselves, we then look back down and can give name (i.e. λογος, ratio, definition) to what we proceeded through. There’s a way in which just as in proceeding from sensibles to their principles we understand the sensibles better by way of their definitions, so in proceeding from principles to first (or higher) principles we come to know the lower principles. There’s a starting in sense that in reaching the heights descends back into sense.
Ok it’s getting late, and I’m reaching the limit of what I’m able to say from working knowledge and memory—I know this doesn’t answer everything you asked. In conclusion, mathematics needs to be taught in such a way to allow the student to ascend in knowledge through the chain of being, from lower forms to higher forms. Recovering multitude vs magnitude, continuous vs discrete, euclid’s definitions, etc., is part of the answer—these definitions don’t implant knowledge in the student’s head, but they do present judgments that may not have been considered by the student before (likely have not). Teachers must be raging realists. Teaching, is as Ryan likes to say, all about providing the right phantasm. You are guiding the student in a consideration of reality.