What is classical mathematics?

Teaching math classically involves more than using the Socratic method. I promise that there's a reason why I spend so much time talking about Latin and Literature. See polymathclassical.com for more classical education resources.
Timestamps
00:00 Introduction & Definitions
02:40 Historical Overview
13:48 Classical Mathematics
17:55 Conclusions
I'm also on Odysee!
odysee.com/@pct:2

[@polymathclassical]

A slight correction: In the video, I list Cicero and Tully, as if they're two different people. As I recently discovered while researching something else, Tully was an anglicized form of Cicero's middle name (Marcus Tullius Cicero). Thus, references to Tully in the literature of the period are references to Cicero, which makes complete sense now that I go back and reread key passages from Ascham.

[@FuZiionx1]

0:06 0:06 inquiries

[@merrickdodge9760]

I love this! Looking back, it wasn’t until I got to grad school that I started learning math in a way similar to this classical approach you mentioned. I wonder how much more deeply I could understand things had I started learning this way sooner.

[@maycock86]

Interesting, informative, compelling, Great Video!

[@polymathclassical]

Thank you! Glad you enjoyed it!

[@danmurphy2577]

Great video! Correction, though: 47th proposition of Book 1 is the Pythagorean Theorem. The 48th is the converse of the Pythagorean Theorem.

[@polymathclassical]

Thank you! Yes, you're right. I misspoke. Thanks for the correction.

[@anotherelvis]

Skip to 13:48 to get the point in the video. He want to expose students to great articles in the history of math. (Similar to the way that we read the great authors when we study litterature)

[@annaclarafenyo8185]

It is very difficult to read the classics of mathematics without a foundation in modern textbook mathematics. Although it is important to read the classics, it must always be done after you learn the modern methods, because it cuts down the time to read the classics by a factor of 10, sometimes a factor of 100.

[@xDELFYonceagain]

Thank you for this video! God Bless.

[@polymathclassical]

You are so welcome!

[@annaclarafenyo8185]

Mathematics as a field underwent a revolution in the late 19th century, and the material from before around 1880 is completely trivial compared to what came later. While it is important to read the historical material as you suggest, it is very important to do so only after learning 20th century methods, because they are so much more powerful than the earlier methods. For example, Euclid's approach to the Pythagorean theorem is relatively dated, although it needs to be learned. It is important to learn the modern formulations of the theorem first, and then examine the classics. This is difficult to do. Also, your emphasis on the classical literature is not good, it is a type of traditionalism, which undercuts progress.

[@ace6285]

Disagree. Learning history accelerates insight into the current.

[@annaclarafenyo8185]

@@ace6285 In the case of mathematics, it's the opposite.

[@juliankleinactor]

I am a classically educating homeschool mama. My husband and I both LOVED math in school (we were both public-school educated), and he went on to get a degree in Physics and engineering, while I got a degree in Political Science and later worked in IT. I have struggled with how to educate my teenagers in math. I love your idea of foundational tools! My son just finished his 9th grade year. This summer we signed him up for geometry (with Thinkwell). My questions: 1. how can I know if we are giving him the "grammar," "vocabulary," and "translation exercises" that are foundational to geometry and will help him if he chooses engineering (likely)? 2. How can I give him the right foundation for physics and calculus through his geometry class? -- I'm really frustrated that I can't always see what needs to be really learned well and what can get glossed over in the different math curricula that I've tried! I feel like I do a better job with Algebra than the other branches. I need help! I'd love to hear your thoughts.

[@polymathclassical]

Amy, That's a great question. To answer your first question, any decent geometry curriculum should teach the "grammar", "vocab", and "translation" to some degree. Otherwise, it would be impossible to progress. The tricky bit, however, is finding a way to integrate everything into a broader context, which is really what elevates math instruction (in my opinion) from mere training to true education. a) To answer your question more directly, I think proofs are super important to learn in geometry for 2 reasons: (1) they teach precise logical thinking & problem solving, and (2) they allow one to better appreciate the tremendous power of algebra. One has to know how to handle logical dependencies when writing a proof from scratch, and being able to clearly think through how one proposition relates to another is an important skill that translates well to many fields. b) Besides proofs, I think geometric constructions exercises are often overlooked but important. To perform constructions with facility, one must (again) internalize not only the various operations (like producing a perpendicular line), but also the many propositions that validate those operations. c) Proofs and Constructions are often not-emphasized in geometry curricula in order to make room for algebra. The reasoning, as far as I can tell, is that everyone wants to get students working problems on the Cartesian plane as quickly as possible. But that means that much of this gets taught twice, since Algebra 2 focuses on mapping equations (of various sorts) onto the Cartesian plane. d) To summarize, I wouldn't worry too much about your son being under-prepared for physics or calculus so long as he's using a decent curriculum, as most of that preparatory legwork will be done in Algebra 2. I would, however, try to find ways of emphasizing proofs and constructions, since I think those pay dividends far beyond the classroom. Physics and calculus rely heavily on a solid foundation in algebra (which is one of the reasons why algebra creeps into most geometry curricula). So, another thing to emphasize in geometry, besides what I've already mentioned, would be ratio. In one of his treatises, Francois Viete observes that proportions and equations are intimately related. A proportion, he says, may be resolved into an equation, and vice versa. We more or less take this for granted when we teach ratio and proportion, but we shouldn't underestimate the importance of helping students really grasp concepts of proportionality. Also, I'm sure it goes without saying--but I'll cover my bases here--I'm not suggesting that everything else may be glossed over--it really depends on the student. Some students grasp certain concepts quickly, while other struggle. But these are the things I'd try to pay attention to in geometry. I hope that helps! Sorry my reply is so long. Those were great questions! - Daniel polymathclassical.com

[@juliankleinactor]

@@polymathclassical This is very thorough and very helpful! To follow up: I see you are offering a summer class on Euclid's Elements. Would you say that you are focusing on Proofs and Constructions in this class? What about ratios?

[@polymathclassical]

@@juliankleinactor Yes! I offer a 6-week summer course in Euclid's Elements every year. We cover Book I of The Elements, which means that we spend every class meeting working through proofs and constructions. (Euclid doesn't tackle ratio until Book V.) My summer Euclid course would be a great way to bolster any year-long geometry course. (Plus, I think Euclid is a lot of fun!)

[@ryanmcmanus7273]

Personally, I found that the concept of vectors weren't brought early enough (only started to see it in college) and that if I was to have learned that and internalized it before I truly needed it that would've helped a ton. Additionally although constructions and proofs are good to learn I had a bad time with that class because the proofs that I did were not engaging enough and it seemed more like a memorization game. I wonder though how early can you effectively teach someone to think critically enough to create a proof.

[@andso7068]

Would you happen to have any recommendations for a returning student with a decent understanding of pre-algebra but has ambitions to understand math up to and beyond calculus and linear algebra but from a classical perspective?

[@polymathclassical]

@andso7068, great question. TRACK 1) If you wanted to trace the birth of algebra in the most thorough way, I'd begin by reading Diophantus alongside my guidebook _Diophantine Algebra_ (polymathclassical.com/curriculum-diophantine-algebra/). You could then proceed to read through Viete's _Preliminary Notes On Symbolic Logistic_ and the first 2 or 3 books of Zetetica (all contained in this book: www.amazon.com/Analytic-Art-Francois-Vi%C3%A8te/dp/0486453480). From there, things get tricky. Descartes should be one of the next stops on your trip, but you can't really appreciate what he's trying to do in his _Geometry_ without first reading bits of Apollonius' _Conics_ . That said, the first book of Descartes' _Geometry_ is relatively approachable, even without knowing anything about Apollonius. At this point, it becomes increasingly difficult to tackle the primary sources without first having mastered the mathematics in question. TRACK 2) If you wanted to do things an easier way, I'd suggest beginning with _Mathematics for the Nonmathematician_ by Morris Kline and _Journey through Genius_ by William Dunham. From there, I'd pick and choose a few interesting primary sources (based on what caught my attention in those two books), and read them alongside a standard math textbook (to fill in any gaps). For Calculus, it would be interesting to read Newton alongside a standard calculus text--or, for a more humanities-based approach to calculus, you might also look into Mitch Stoke's _Calculus for Everyone_ . Again, yours is a great question, and it's the same question I've been trying to answer for years, and no simple solution exists yet. But I hope what I've laid out above is helpful.

[@Whatsit7777]

What resource(s) would you recommend for incorporating Diophantus, Descartes, Cardano, etc?

[@polymathclassical]

@Whatsit7777 That's a good question. Unfortunately, there aren't that many--which is why I'm working on developing my own resources (my Classical Math Two sequence aims to incorporate all three of those). In the meantime, I can recommend my own "Diophantine Algebra" text for Diophantus. I also recommend checking out Morris Kline's book "Mathematics for the Nonmathematician". Kline provides a general historic overview of mathematics, which I find really useful. I can also recommend William Dunham's book "Journey Through Genius". Dunham takes the reader through a selection of the greatest mathematical ideas and has a chapter on Cardano. Diophantine Algebra: polymathclassical.com/curriculum-diophantine-algebra/ Kline - Mathematics for the Nonmathematician: www.amazon.com/Mathematics-Nonmathematician-Morris-Kline/dp/0486248232 Dunham - Journey Through Genius: www.amazon.com/Journey-through-Genius-Theorems-Mathematics/dp/014014739X/ - Daniel

[@laalbujhakkar]

Please do a quick TLDR of what the video is going to be about. You may not realize this, but there are a billion videos on youtube and not a lot of time, so if you do get suggested by the algorithm, please don't expect people to stick around for a detailed treatment of things you are passionate about. I mean 14 minuts to get to what classical maths is, in a video that is titled "What is classical mathematics" ...

[@user-cl5sn1gg6i]

Great video, but one question; are those classical treatises still in print?

[@KristynSartin]

I learned a lot from your video. I’m curious on your thoughts about the negative consequences of using only European examples of math thinkers and doers. For example, with Algebra and our number system being rooted in Persia, are you really providing examples of creativity if you present the European re-presentation of Persian constructions? Also, I’m not very familiar with classical education. When you describe literature and Latin instruction, is it accurate to say that you de-emphasize students’ opportunities to write and create their own work? Is it not a detriment to not provide opportunities for students to be problem solvers themselves? Or to construct their own theorems or proofs? I know you said math should not be considered a mental exercise. However, when would students have the opportunity to become critical thinkers and problem solvers and creative with mathematics? Maybe I’m misunderstanding. I would appreciate some clarity here.

[@polymathclassical]

Those are great questions. Rather than respond here, I'm hoping to make a video soon to address them. I'll let you know once I do. Thanks for your comment!

[@polymathclassical]

@KristynSartin, I've just published a video addressing your questions. Thanks again for asking some great questions and giving me the chance to engage with them!

[@ericamcquown8990]

What age would you begin introducing the classic mathematic books?

[@polymathclassical]

Good question. I'd say 7-9th grade, depending on the student. Here's why: Given that many of the great mathematics texts were written to fellow mathematicians, I recommend that students have a solid foundation in PreAlgebra before being introduced to primary texts. In some cases, the math itself is really simple (like in Nicomachus's Introduction to Arithmetic), but is explained in relatively challenging language, so it helps for students to be capable readers as well. Thus, I reckon that most students will be ready between 7th and 9th grade.

[@antoniusnies-komponistpian2172]

I actually think that the emphasis on logic and proofs is very important to transport the beauty of mathematics. I like the attitude of mathematics that you actually need to prove something before you believe it and I think it's important to prove that to students. It does probably help many to tell stories about mathematicians who found those insights, but I personally don't need that. Rather than how other people got to insights, I care about how I can get to those insights. I also don't need to hear stories of Columbus to be curious about America, I need to learn about its impact on our lives and our impact on America. Learning is a lot about personal relevance, and mathematics have a lot more personal relevance than just beauty that is not transported sufficiently.

[@Oysters176]

I care more about ''visuals', Can you picture it and explain it simply? Can you develop an intuition of it that a child can understand and use it as a metaphor to help explain your ordinary world? Sure.. Study Euclid, but it's more important to develop intuitions about say Logarithms and Calculus can describe and predict your own development and experiences. How Trigonometry can explain cyclic behavior. Chaos Theory predict future recourse. Can Mathematics enhance one's reading of Literature and Movies?

[@HickoryDickory86]

Or, just teach the actual classical mathematical arts of the Quadrivium from the actual classical texts that were standard in Late Antiquity and the Middle Ages (all of which have readily available English translations). Arithmetic • Nicomachus of Gerasa's _Introduction to Arithmetic_ **or** • St. Boethius' _De arithmetica_ (his Latin translation and edition of Nicomachus' earlier Greek text) Geometry • Euclid's _Elements_ Music • Nicomachus of Gerasa's _Manual of Harmonics_ **or** • St. Boethius' _De institutione musica_ (again, his Latin translation and edition of Nicomachus' earlier Greek text) And Astronomy • Claudius Ptolemy's _Almagest_

[@HickoryDickory86]

That said, I appreciate the other thoughts presented in the video. Like exposing them to various examples of different discoveries of mathematics and the beauty of it.

[@polymathclassical]

That's a great idea! The problem I'm working on at present is how to present high school mathematics _as students need it today_ using classical texts as a framework. It's not so hard to do with Geometry, since Euclid's _Elements_ is a single text and pretty thorough. It's much more difficult to do with Algebra, since Algebra was developed in bits and pieces across various treatises over many centuries. Ideally, students should take Calculus at some point, so the curriculum has to be arranged so as to make that possible. Essentially, this makes places us in the difficult position of having to balance the teaching of primary texts with decontextualized math instruction in order to make 1) the primary texts intelligible and 2) give students the math they need for college, etc. I wish we lived in a world where I could do exactly what you outline. That would be spectacular. I'm just not sure how to make that work for high school, given the expectations and requirements of high school math. Your outline would be great for a supplementary course of study, or as part of an undergraduate program _a la_ St. John's College, etc. Thanks for your comment!

[@HickoryDickory86]

@@polymathclassical William Michael, of the Classical Liberal Arts Academy, is actually transitioning his school's curriculum for Algebra to be based on and work through Leonhard Euler's _Elements of Algebra_ (1765). I have a copy of it myself (the Cambridge Library Collection edition), but just looking at it makes my brain go fuzzy, so best wishes to him in his endeavors. 😂 He might be on to something, though. Any way to make such a fundamental text for modern mathematics more accessible is appreciated, and having two great teachers both coming up with their own ways to do so benefits everyone. So there's a thought. I'm at a loss for calculus, though, as I never took it formally and it still frightens me (as all math does on a primal level, if I'm being honest; I'm much more a history and theology person).

[@polymathclassical]

@@HickoryDickory86 I've also been considering using _Euler's Elements of Algebra_ as a core text, so that's good to know that I'm not the only one with that idea. I might consider that further, then. I haven't thought ahead to Calculus much, but I've heard great things about Mitch Stokes' book _Calculus for Everyone_ . I actually own a copy, but haven't had a chance to read through it yet. Thanks for your valuable input!

[@mrpocock]

Seems like you are describing duolingo :)

[@palfers1]

Almost (but not quite) an American Monty Python!

[@meofamily4]

657 subscribers. That is as it should be.

[@Ozymandi_as]

Ooh, that's a burn. Total KO. Rapier-like. Gotcha. 645 more than you I notice. Although it's not really a fair comparison, since you haven't made any videos and tried to find an audience for them.

[@meofamily4]

You mistake, @@Ozymandi_as , the second comment I made, as the only one. The statement to which you reply followed the discussion of specific shortcomings which I consider to be egregious. "It is just false to maintain that, since we speak English, what happened in 16th-century England has had the decisive effect on U.S. education. It is false because our educational system is primarily modeled on 19th-century German education. From Kinder-garten through the research university, our educational system of the period following the Civil War (the one in the United states, not the one in England) was shaped by educational authorities who consciously modeled, with variations, what the Germans were doing. " Next, the entire claim that classical education was reformed by focussing on Latin in 16th-century England is ridiculous. ALL European education was based on Latin, going back to the Roman Empire. " The Bible was in Latin. Church schools were the only schools. " Man, this whole video stinks. " When the Italian Renaissance revivified Platonism, intellectuals across Europe engaged more seriously in mathematics, since Plato taught that the divine was approached by means of mathematics -- a doctrine articulated eloquently by Galileo. Nothing to do with English educational reform. " Finally, the most ridiculous aspect of this hopeless, clueless video is the failure to understand that mathematics is involved in SOLVING PROBLEMS. If you can solve a set of challenging problems well, you're a trained mathematician. Sigh."

[@tomellman2418]

Say something please! You’ve gone on and on and haven’t said anything.

[@polymathclassical]

If the theoretical backdrop isn't helpful to you, feel free to skip to the conclusion (starts around 18:00).

[@bashisobsolete.pythonismyn6321]

Ascham pronounced Ask not Ash. (yeah, i'm that guy, lol). great explanation of a topic dear to my heart.

[@polymathclassical]

Thanks for the comment. That's the one downside to getting one's information primarily out of books--there's no one around to tell you how names are pronounced!

[@rv706]

England?? What about Italy? I get it's about education and not 'research', but none of the English people you mentioned at 3:36 was a great mathematician.

[@polymathclassical]

Precisely. I'm not discussing mathematicians here, but rather the theorists who gave us classical education. Later in the video, I situate mathematics within the classical framework. Thanks for your comment!

[@kylecow1930]

im not a fan of this approach, learn maths rigorously yeah, but historically? we're better at maths now than we were like noone should learn Galois theory as Galois wrote it. The modern treatments are far superior in generality and application to actual maths

[@jamessheffield4173]

Renaissance humanism was a worldview centered on the nature and importance of humanity, that emerged from the study of Classical antiquity. Bing search

[@the_nuwarrior]

Mathematics does not work like that, it is impossible to learn calculus by reading the principia

[@chrisikaris5891]

I am not so sure this approach would work very well in public schools, only in private schools with highly selective admissions criteria and highly trained and educated teachers. So 99 percent of American children will not and cannot receive it. They are simply not ready or able to begin with. Not to mention the consensus against "Eurocentric" learning based on the work of "Dead White European Males" because of the radically different demographics of the country. Nowadays the overwhelming trend is toward "multiculturalism", "diversity," "equity", and "decolonization" of the curriculum. Even many expensive private schools are going in this direction. And to top it off, the classroom of today is dominated by kids who have their minds (and eyes) focussed on what is on their cellphones, or literally have buds in their ears, or head phones over their ears, listening to their favorite music. I suppose the speaker has never witnessed or heard of this happening in schools, but it does. And, btw, there are still beatings taking place in schools. It is just the teachers and staff members who are receving the beatings. All of this is to say that the speaker has the best of intentions, quite noble actually, but he does not seem grounded in, well, what is happening on the ground. But more power to him, resisting the forces of social and cultural dissolution and chaos. I wish him well.

[@diskgrinder]

No maths, move on. This is a waste of time

[@eduardopereira4761]

Dude, you sound like a psychologist!(it's not a compliment!)

[@SkeletalBasis]

Utter poppycock. Does this guy know any math? Has he ever taught anything? Aside from badgering people who do and who have? One great thing about "math" is that it becomes progressively clearer what it's actually about. That's why it makes literally no sense to go back looking for some lost essence.

[@meofamily4]

It is just false to maintain that, since we speak English, what happened in 16th-century England has had the decisive effect on U.S. education. It is false because our educational system is primarily modeled on 19th-century German education. From Kinder-garten through the research university, our educational system of the period following the Civil War (the one in the United states, not the one in England) was shaped by educational authorities who consciously modeled, with variations, what the Germans were doing. Next, the entire claim that classical education was reformed by focussing on Latin in 16th-century England is ridiculous. ALL European education was based on Latin, going back to the Roman Empire. The Bible was in Latin. Church schools were the only schools. Man, this whole video stinks. When the Italian Renaissance revivified Platonism, intellectuals across Europe engaged more seriously in mathematics, since Plato taught that the divine was approached by means of mathematics -- a doctrine articulated eloquently by Galileo. Nothing to do with English educational reform. Finally, the most ridiculous aspect of this hopeless, clueless video is the failure to understand that mathematics is involved in SOLVING PROBLEMS. If you can solve a set of challenging problems well, you're a trained mathematician. Sigh.

[@CliffSedge-nu5fv]

Sounds too much like an ignorant child pretending to know something. Disliking and moving on.