Prelection
Welcome to the Classical Liberal Arts Academy. This lecture is intended for students enrolled in the Academy’s Classical Arithmetic course. If you’re not enrolled and would like to join us for self-paced studies, visit classicalliberalarts.com. My name is William Michael. I’m the Academy headmaster, and I’ll be leading this study of Book One, Chapter One of Nicomachus’ Introduction to Arithmetic.
In this lesson, we study the opening chapter of Nicomachus’ Introduction to Arithmetic, which was written by Nicomachus of Gerasa, a Greek mathematician and philosopher who lived in the first or second century AD, so just after the Apostolic Age. And he wrote this foundational work on number theory in the Pythagorean tradition around 100 AD. This text is a systematic introduction. The title is the Introduction to the Mathematical Sciences, written to provide students with the theoretical foundations necessary for understanding numbers and their relationships as part of the classical quadrivium. And remember that arithmetic is the first art of the quadrivium. So this Introduction to Arithmetic is really an introduction to the quadrivium itself.
The opening chapter, which we’ll study in this lesson, serves as the foundation for the entire work, establishing why the study of mathematics is essential for the pursuit of wisdom and truth. Nicomachus’ work became one of the most influential mathematical texts in the medieval period, serving as a primary source for mathematical education in both—and this is important—in both Islamic and Christian schools. His approach to mathematics as a path to understanding eternal truth deeply influenced scholastic philosophy and the development of mathematical education in the Church until the modern period.
Opening Prayer
Lord our God, in your wisdom and love you surround us with the mysteries of the universe. Send your Spirit upon us and fill us with your wisdom and blessings. Grant that we may devote ourselves to our studies and draw ever closer to you, the source of all knowledge. We ask this through Christ our Lord. Amen.
Reading
Let’s read together this first chapter of Book One of Nicomachus’ Arithmetic. Remember the purpose of our first reading is just to become familiar with the content of the lesson. We’ll break it down and examine all of that content in a few minutes. Let’s read Chapter One.
The ancients who, under the leadership of Pythagoras, first made science systematic, defined philosophy as the love of wisdom. Indeed the name itself, Philosophia, means this, means the love of wisdom. And before Pythagoras, all who had knowledge were called wise indiscriminately: a carpenter, for example, a cobbler, a helmsman, and in a word, anyone who was versed in any art or handicraft. Pythagoras, however, restricting the title so as to apply to the knowledge and comprehension of reality and calling the knowledge of the truth in this the only wisdom, naturally designated the desire and pursuit of this knowledge philosophy, as being desire for wisdom. He is more worthy of credence than those who have given other definitions, since he makes clear the sense of the term and the thing defined.
This wisdom he defined as the knowledge or science of truth in real things, conceiving science to be a steadfast and firm apprehension of the underlying substance. And real things to be those which continue uniformly and the same in the universe and never depart even briefly from their existence. These real things would be things immaterial, by sharing in the substance of which everything else that exists under the same name and is so called is said to be this particular thing and exists. For bodily material things are, to be sure, forever involved in continuous flow and change in imitation of the nature and peculiar quality of that eternal matter and substance which has been from the beginning and which was all changeable and variable throughout.
The bodyless things, however, of which we conceive in connection with or together with matter, such as qualities, quantities, configurations, largeness, smallness, equality, relations, actualities, dispositions, places, times—all those things in a word whereby the qualities found in each body are comprehended—all these are of themselves immovable and unchangeable, but accidentally they share in and partake of the affections of the body to which they belong. Now it is with such things that wisdom is particularly concerned, but accidentally also with things that share in them, that is with bodies.
This is Chapter One of Book One of Nicomachus’ Arithmetic.
Division
We can divide this first chapter into six parts. The first part, where we have a definition of philosophy and wisdom, begins with the first line: “The ancients who under the leadership of Pythagoras.” The second part, which is Pythagoras’ restriction of the term “wise,” begins where we read “Pythagoras, however.” The third part addresses the nature of true wisdom and science, and this begins where we read “This wisdom he defined.” The fourth part tells of the distinction between material and immaterial things. This begins where we read “For bodily material things,” etc. The fifth part discusses the nature of mathematical objects, beginning with “The bodyless things, however.” And lastly, the sixth part, which discusses the proper object of wisdom, begins with “Now it is with such things.”
So we can divide the text into these six parts, and then we’ll work to understand each part and each sentence within each part, which is how we divide and study the text.
The purpose of this lesson, again, is to establish that true wisdom concerns itself primarily with unchanging, immaterial realities, and that mathematical objects belong to this realm of unchanging truth, making mathematics a proper path to wisdom. This is the Pythagorean method of following the study of mathematical things to wisdom.
So having divided the text, now we can move on to study in detail. And this is the real work of our lesson. We want to study each part as we’ve divided the text. We want to break down each part into each section, make sure we understand all the vocabulary, all of the ideas shared by Nicomachus, and take our time to really work through all of the content of the lesson for the sake of comprehension.
So let’s just begin, and we’ll work our way through it, taking whatever time is necessary. I’ll try to address every issue that I think, from my experience, might be a source of difficulty for students. So let’s work through the interpretation of our reading here.
Interpretation
Nicomachus begins by establishing the historical development of the concept of wisdom. What does it mean to be a wise man? That concept has changed through times, he says. “The ancients who under the leadership of Pythagoras”—and just chronologically, so you keep this in mind, Pythagoras is pre-Socratic, so before classical Greece. We’re going back to archaic or ancient Greece, before classical Greek philosophy—”the ancients who, under the leadership of Pythagoras, first made science systematic, defined philosophy as the love of wisdom.”
“Systematic” here, important term, means that knowledge is a certain kind of knowledge, a certain quality of knowledge. It’s knowledge that’s organized according to principles and methods, so not just familiar knowledge, but scientific or systematic knowledge, the kind of knowledge that results from careful study.
Philosophy, the word philosophy literally means the love or desire for wisdom, from the Greek word “philos,” which means loving, and “sophia,” which means wisdom. So philosophy literally means the love or desire for wisdom. And this first sentence establishes that before Pythagoras, knowledge was not organized in a systematic way. Knowledge was not organized into a coherent system of learning.
A good example of this that demonstrates this is to just think about the Sacred Scriptures. Think about the Old Testament. Think about the Law of Moses. Think about the histories, how things are written. Think about the Proverbs of Solomon or the Psalms of David. We see lots of knowledge, whether inspired by God or simply human knowledge outside of Sacred Scripture, but we don’t see systematic scientific treatises, which will develop later in Greek philosophy. Before Pythagoras, knowledge was not organized into a coherent system of learning. This is where true philosophy begins.
Nicomachus goes on to say, “Indeed the name itself”—that is, philosophy—”means this. And before Pythagoras, all who had knowledge, any kind of knowledge, were called wise indiscriminately: a carpenter could be called wise, a cobbler, a helmsman, anyone who was versed in any kind of art or handicraft might be referred to as wise.” The term “indiscriminately” means without making any distinctions or differences. Nicomachus shows that originally, anyone with practical skill was called wise, whether they were a woodworker or worked with leather or built ships or piloted ships or any other craft. Any kind of skill was considered wisdom. And this shows that wisdom was once understood as practical skill, skillfulness, rather than theoretical knowledge.
“Pythagoras, however, restricting the title so as to apply to the knowledge and comprehension of reality and calling the knowledge of the truth in this the only wisdom, designated the desire and pursuit of this knowledge”—knowledge of reality; we’ll talk about that in a minute—this knowledge alone was designated as philosophy, the desire for wisdom.
So here, “restricting” means limiting or narrowing the definition, not allowing the word to be used indiscriminately. Pythagoras made an important distinction: true wisdom is not practical skill, not any kind of knowledge, any kind of skill, but the understanding of reality itself, what things truly are. And a good way for Christians to think about this is: wisdom is the knowledge of things as God himself thinks of those things, true scientific knowledge, what things truly are. We have wisdom when we think about things the way that God thinks about them, the way that they truly are, not as they appear to us, but as they truly are.
The knowledge of the truth refers to understanding the fundamental nature of existence, and we’ll see that this is the theme of most of what we’ll study in classical philosophy: understanding the fundamental nature of existence.
Nicomachus goes on and says, “He”—that is, Pythagoras—”is more worthy of credence” or credibility, trust, “than those who have given other definitions, because he does two things. He makes sense of the word itself, philosophy, and also of the thing defined again.” Credence means belief or trust. His explanation is more credible because he makes sense of both the word or the name and the thing itself. His definition is more trustworthy because it clearly explains what the word means and what the reality is that the word describes or signifies.
So notice that as a principle here in philosophy, the best definition is one that explains both the word—like, where does the word itself come from? We can think of etymology—and what the reality is that the word signifies, what the essence is that’s named by that word.
“This wisdom,” Nicomachus writes, “he defined as the knowledge or science of the truth in real things.” So here we have a definition from Pythagoras of wisdom. Wisdom is the knowledge of the truth in real things. When he uses this word “science,” we’ve got to be very careful to distinguish this from the modern use of the word science. Pythagoras uses the word science to mean a steadfast and firm apprehension of the underlying substance of a thing, this true knowledge, this knowledge of the reality of things; that’s science. It also refers to a certain knowledge, a knowledge that is sure, not an opinion, not the consensus, but a demonstrable knowledge of a certain subject, not what we have in modern experimental science, as I said, which is largely theory and so on. This science is when we say certain knowledge, we mean knowledge that’s absolutely certain and sure that we know to be true. It can be demonstrated to be true.
When he refers to the “steadfast and firm apprehension,” he means a secure and unchanging grasp or understanding, so not a flimsy or unstable familiarity with the thing, but a secure and unchanging, steadfast and firm apprehension. And then he refers to “underlying substance” in that sentence. Underlying substance—and what he’s talking about here is what makes a thing what it truly is, that is its essential nature, its essence. Understanding about any subject what it truly is, that is, what is the essence of that thing? This is the quality of knowledge that alone should be considered wisdom, according to Pythagoras, and really according to classical philosophy.
He goes on to talk about real things. And by real things, what he means is sort of the opposite of what we normally mean by that phrase in modern society. He means those which continue uniformly and the same in the universe and never depart even briefly from their existence. These real things, he says, would not be material things, but immaterial things. This is where we have a very significant difference between classical philosophy and modern thinking. In the modern world, when we say something is real, we mean that it’s sensible; we can perceive it with the senses. People mock religion and say, “Oh, you with all of your, you know, your fake knowledge and all of this religion; I want to see something real. I want to feel it, taste it, smell it, hear it,” and so on. That’s what a real thing is. Pythagoras said exactly the opposite. He defined real things as those which are immaterial, because they are always the same. Material things, he argues, as we’ll see, are always changing. Therefore they themselves are not reality. They’re not the real things.
When he says “uniformly,” he means consistently without variation, that which does not change. And so what we see here is Nicomachus is establishing that truly real things are those that never change. They are eternal and immaterial. They are not made of physical matter. And this reflects this Platonic understanding. Remember, Pythagoras lived before Plato, but this book is being written by Nicomachus, who lived after Plato and Aristotle. So we’ve got this Platonic philosophy that we can’t allow to be mixed with Pythagorean philosophy. There are two different time periods, but this all comes together into a single system of philosophical understanding, which is being taught here by Nicomachus.
So we see here a Platonic understanding that material things are constantly changing, and therefore they are less real than unchanging, eternal forms. And we’ll continue to talk about this Platonic understanding as we work through classical philosophy.
Nicomachus goes on to say, “For bodily material things are, to be sure, forever involved in continuous flow and change.” Bodily material things are forever involved in continuous flow and change. “This is in imitation of the nature and peculiar quality of that eternal matter.” Notice that phrase, “eternal matter”—that’s a controversial issue in ancient philosophy—”eternal matter and substance which has been from the beginning.” Physical things, Nicomachus says, are always changing. They grow; they’re even born. They decay; they move. They’re transformed. Physical things imitate eternal realities, but they are themselves not eternal. Physical things are not eternal. Physical things are not, quote unquote, real.
This reflects for us this ancient understanding that the material world is a reflection of some higher, unchanging reality. So we can see where Platonism comes from. And as we get into Aristotelian and Scholastic philosophy, we’ll want to be careful to make distinctions here, because some of this is later disproven by Aristotle. But it’s important to see this general concept that the physical world, contrary to our initial thoughts or initial appearances, the physical world is not the reality. The reality is that which never changes. The knowledge of that which never changes is true scientific knowledge, according to Pythagoras.
So you can see the distinction between the thinking of the common people who live according to the senses, who say things like “seeing is believing,” the way of thinking according to common sense, the thinking of the common people, and the thinking of the wise man. The wise man looks beyond the appearances of material things and contemplates that which is immaterial, which is unchanging. And we’re going to talk about what exactly these unchanging realities are as we get into the study of arithmetic.
Here where he says, “The bodiless things of which we conceive in connection with or together with matter.” So notice that the bodiless things, the things that are reality, they combine with matter. They participate in matter. So they have an appearance in the material world. But our desire in philosophy is to abstract them—the immaterial things, the unchanging things—to abstract them from the material things, and study them in and of themselves.
And he lists these bodiless things. What are these bodiless things? He says the bodiless things of which we conceive in connection with or together with matter, such as—and here he lists some: qualities, quantities, configurations, largeness, smallness, equality, relations, actualities, dispositions, places, times—all of these are themselves immovable and unchangeable. These are the kinds of things that he’s telling us Pythagoras recognized as the real things in the world.
Bodyless things are immaterial realities that we can think about. We have the ability to contemplate them, even though they don’t exist in a way that our senses can perceive them. The faculty of reason allows us to contemplate them: immaterial, bodiless things, invisible things, imperceptible things. We’re able to contemplate them, and we see here the foundation of contemplative life, a life whereby we think about and reason about immaterial realities, things that are immovable and unchanging, which we could say are eternal mathematical concepts.
This term “mathematical” is important in ancient philosophy. Mathematical concepts like quantity, equality, geometric shapes—again, not visible, but contemplated according to their definitions. These are some examples of these immaterial realities. Though we see them, we see them in physical objects. The concepts themselves exist outside of those physical objects and themselves never change. For example, the idea of the number three: we may see a multitude of three things, $3 for example, but the concept of three is a concept that doesn’t change. It’s an immaterial concept. It’s an unmovable, unchanging concept. The same is true of a triangle. Once we establish the definition of a triangle, we may see the triangle imitated in physical things. But the concept of triangle itself is separate from—or abstract, separate from, drawn away from—the material thing, and exists itself and can be contemplated by the mind. It’s always the same.
And so you can see how this is what we study in arithmetic, where we study the science of number. In geometry, where we get into the science of these figures.
Nicomachus goes on to say, “It is with such things”—these immaterial realities, these bodiless things—”it is with such things that wisdom is particularly concerned,” true wisdom, “but only accidentally with the things that share in them, that is with bodies.”
So do we have an interest in bodily things in philosophy? Yes, but in a secondary way, in an accidental way. Our primary concern is with the bodiless things, the immaterial realities, which are unchanging. So we’re not rejecting bodiless things completely. We’re not saying the knowledge of bodies is completely useless, but we’re saying it’s secondary to the unchanging things, to reality. True wisdom studies these unchanging mathematical realities primarily. That’s our primary interest in the pursuit of wisdom, and we study physical things only secondarily, insofar as they participate in or imitate or reflect these eternal truths.
So we see as we gain wisdom, we learn about these invisible realities, these bodiless things, these mathematical ideas, and we then observe them. We recognize them in the material world, and that allows us to see order and significance in the material world. That’s the goal of mathematical studies. True wisdom, again, sees these mathematical realities primarily, and physical things secondarily.
This establishes for us, as we get into the quadrivium here, this establishes mathematics, which is the whole Pythagorean method. It establishes the study of mathematics as a proper path to wisdom, because mathematics, in this classical sense, deals with unchanging, eternal realities, and the knowledge of such things is wisdom, and the love of wisdom is philosophy. So we’re getting here into a true understanding of philosophy. Philosophy is the love of wisdom. Wisdom is the science of reality, the science of the truth of things in reality.
So we’ve gone there through all of the text of this first chapter in Nicomachus’ Introduction to Arithmetic.
What I’d like to do at the end of these expositions is provide a list of simple assertions. And I recommend you take some time to jot these down, simple assertions that would be good memory points to commit to memory that sort of summarize the key concepts of the lesson here. And I’m going to give you, I don’t know, between maybe five and 10 for each lesson that we study.
Memory Work
So having studied all of that, let’s establish some key points to commit to memory, and you can just listen to these, but you might want to hit pause and copy these for the sake of memorization.
- Philosophy means the love of wisdom.
- Pythagoras first made science systematic.
- True wisdom is the knowledge of unchanging reality.
- Science is steadfast and firm knowledge of underlying substance.
- Real things are immaterial and never change.
- Material things are always in flow and change.
- Mathematical objects are immaterial and unchanging.
- Wisdom is primarily concerned with unchanging things.
- Mathematics is a path to wisdom because it studies eternal truths.
Conclusion
So in this class, in this lecture, we’ve studied Book One, Chapter One of Nicomachus’ Introduction to Arithmetic. We’ve discussed the definition of philosophy as the love of wisdom. We have talked about Pythagoras’ restriction of wisdom to knowledge of unchanging reality, the distinction between material and immaterial things, the nature of mathematical objects as unchanging realities, and the connection between mathematics and the pursuit of wisdom, which is true philosophy.
So having gone through the lesson, now it’s time for you to reread and study the lesson for yourself, develop your understanding of the lesson, and demonstrate that understanding through the graded assessments that are provided in the course on the Academy Study Center. So it’s time for you to develop your own comprehension. Comprehension means taking hold of the knowledge. It’s time for you to work now. Go back through this lesson, reread it, study it carefully, use this exposition to help you, and then use the assessments as a way of developing your mastery and demonstrating that mastery by submitting to that assessment.
If you have any questions or need any help, visit the Academy Help Center at classicalliberalarts.com/support. You can post any questions you have there. There’s a ticket form. You can submit questions for responses. There are articles where many questions are already answered, but classicalliberalarts.com/support is the Academy Help Center.
And once again, my name is William Michael. I’m the headmaster of the Classical Liberal Arts Academy, and I hope that this lesson has been helpful.
God bless your studies.
Mr. William C. Michael, O.P.
Headmaster
Classical Liberal Arts Academy
