(Link to notes on previous chapter.) (Link to book on Goodreads.) (Link to notes on next chapter.)

Before we get into the chapter proper, thought I would say something more general – one of the themes I get from the book (and especially from this chapter) goes roughly as follows:

1. Once upon a time, the educational curriculum was (eventually) divided into these subjects.

2. Over time, knowledge grew. At first, the new knowledge was naturally seen as part of one of the existing subjects (makes perfect sense).

3. Eventually, the new knowledge became so extensive that it was divided into new and independent-seeming fields of study (makes perfect sense).

4. Let us go recover the original synthesis.

There are certainly many benefits to this approach – indeed, the intentionality it lends to a curriculum is itself refreshing. I think if you dropped in most schools in the US today and asked why the school teaches the subjects it teaches, you would actually hear a lot of answers that deflect the “blame” for the curriculum to someone else – “the state makes us teach these subjects” or “it’s what the colleges want” or “it’s what the employers want”. Certainly the liberal-arts wing of LCC doesn’t feel like it controls its own curriculum at all, we can’t develop a new course if we can’t show how it will transfer to the four-year schools. A classical school could actually give a considered answer to the question.

I suppose what has repeatedly bothered me is just that… perhaps speaking as a scientist, I like precisely defined terms, and I like tight categories, and in their quest for unity the authors quite *encourage* category-leakage. That’s not all bad either of course but… eh. “Astronomy is about mathematically modeling a complex system” (true), “and macroeconomics (or whatever) is about mathematically modeling a complex system” (true), “so you see, macroeconomics is astronomy!”. OK, they don’t quite say that… but they say things that approach that. I would be more comfortable with, “this field of study X over here, which really didn’t exist in anything like its modern form when the Quadrivium was developed, is most similar to astronomy in that both are focused on the mathematical modeling of a complex system. Therefore modern classical schools that wish to teach X should feel free to do so, placing it in the curriculum in a position analogous to where astronomy was placed historically, recognizing its practice will train students in similar ways.” That would make me feel better – but of course, they are trying to recover a synthesis as well. I suppose I am wary because I know well that imprecisely defined terms lead to imprecise thinking, and I am forever trying to drive students to think more carefully. Words mean something, not everything, by golly!

**But anyway, onto this chapter proper.**

This much longer chapter is on the Quadrivium, the four mathematical liberal arts – arithmetic, geometry, astronomy, and music. While we tend to think of **arithmetic** as “operations with numbers”, and **geometry** as “operations involving shapes”, or something like that, the authors reference the opinion of Nichomachus:

Arithmetic, he says, deals with the relationships among discrete quantities, while geometry investigates the nature of extended continuous quantities.

I appreciated this definition because it means that a subject like Calculus would fit very nicely within the “geometry” section of a classical curriculum – which any physicist would tell you makes a whole lot of sense. I told one of our math professors once “I don’t know math, I only know shapes”, which rather charmed him – because certainly to a physicist, that is what Calculus *is*, a way to divide big shapes that are too complex to handle by themselves into smaller shapes and then add them up again. Sometimes students get lost in the formalism of Calculus (or want to remain in the comfort of the formalism) and forget what they’re actually doing – physicists are more likely to forget the formalism (oh yeah) but, fortunately, still remember what we’re actually doing, so that we can “rediscover” the formalism when we need it. It is a typically arrogant comment from us, but we do sometimes say “students don’t really understand Calculus until they take Physics”.

Specifically, they recommend the continued use of Euclid’s original *Elements*, which is especially notable for something that will be familiar to many geometry students – proofs.

Each [proposition] is carefully proven because the emphasis is not on the geometrical facts themselves, but on why the geometrical facts are true.

Therefore,

The emphasis on proofs in geometry forms wisdom in students to such a degree that it has for thousands of years captured the imagination of a multitude of philosophers from Plato to Descartes.

Any mathematics professor you talk to will tell you (though he may find it difficult to articulate just how or why), learning mathematics helps you learn to think with care and rigor.

**But then, on to astronomy…**

We know what astronomy is – astronomy is the study of the motions of the stars and planets. But more generally and more importantly for the classical curriculum, they say:

Astronomy was the best example of a mathematical system devised to contain a vast amount of data.

Therefore, as I alluded above, other subjects that also fit this description could perhaps be taught by classical schools in a similar place in the curriculum.

They then go into a pretty interesting discussion about the historical tension between **nominalism and realism **in science. Roughly:

1. Early on in the natural sciences, scientists tended to be nominalists, which means they wanted to develop systems that encompassed all the data and had predictive power, but didn’t necessarily correspond to anything real. An example might be ancient Greek astronomy with its epicycles upon epicycles – the system had predictive power for where the planets would appear in the sky, but the ancient astronomers didn’t necessarily believe the epicycles corresponded to anything real. It was a mathematical trick, if you will.

2. Over time, scientists became realists, who believed that our models actually correspond to reality. The big example they give is Galileo who got in so much trouble, not because he employed a model in which the Sun was at the center of the solar system (as others had before him), but because he believed that model was also *reality*.

3. Today, there is a concerning trend by some back toward nominalism:

Thinkers such as Werner Heisenberg and Niels Bohr, otherwise excellent scientists, claim that science is not interested in truth but only in predicting events.

Well… I’m going to avoid turning this into a giant post about the nature of science (perhaps another time!). But my mind actually immediately ran to quantum mechanics, the science of the very small (think electrons, for example). The “problem” with quantum mechanics is that we are now observing particles that:

1. Behave in ways difficult for our intuition to understand. Is it a particle? Is it a wave? Yes. It has behaviors that, in our macroscopic world, we associate with one or the other, but not both. It does both.

2. We can only sense the particle with complex instruments, never observe it directly in anything approaching the way I might watch you toss a ball to a friend.

In that context, physicists will ask themselves questions like, “do our strange mathematical models merely predict how the particle will behave, or do they actually correspond to reality?”. And some physicists (the true nominalists you might say), will respond “who cares? If the model makes correct predictions, we’re done”. More commonly though I think you might hear someone object to the question itself – we are now in a realm of the universe so far beyond our normal experience that the mathematical models are the only way we can really understand what is happening anyway – so if you understand the mathematics, you understand what is really happening and should stop worrying about it. But the nominalist/realist distinction almost seems to be breaking down on these tiny spatial scales because they are just so far removed from normal human experience… it’s almost impossible (or actually impossible) for us to know if our models are “real” or not. Eh.

**Finally, to music…**

We all know what music is! But again they are going to define it more broadly – really as the study of mathematically proportional relationships (and if you know the physics of how musical instruments make sounds – absolutely it is all about mathematical relationships). Specifically they note the historical distinction between…

*musica instrumentalis*: the heard harmonies of choirs, musical instruments, and the like*musica mundana*: mathematical proportionalities woven into the world*musica humana*: mathematical proportionalities that exist in human society

There is a lot of science mentioned in this section which I will only comment upon a bit for now. They quote Paul Dirac:

It is more important to have beauty in one’s equations than to have them fit experiment.

I am sure most scientists would disagree with that statement, taken literally – but would recognize all the same that the right answers in science, the ones that actually correspond to reality, very often have a mathematical beauty to them. Most scientists could also probably not tell you why that should be, only that it is so. Students even get a taste of this in the work they do in introductory college classes (and probably earlier, but that’s what I teach).

Finally, they do express some more concerns about modern science:

Considering the current milieu, one may wonder whether the skepticism, instrumentalism, and ultimately nominalism pervasive in modern science has already endangered its own foundations. Christian classical math and science departments must resist these tendencies.

Personally, I do think skepticism is a virtue when doing scientific work, a recognition of the limitations and failures of other human beings also doing science (and just the difficulty of the work itself). But I understand their concern here – they are rebelling against a science that, “officially”, never reaches any certain conclusions, is always tentative, and is therefore never a sure guide to any truth. I do think there are many practical benefits to seeing science that way (another post, perhaps!), but they really want to raise science above that and say “yes, it is a reliable guide to the true and beautiful”. And it is… I think the best thing that can be quickly said is just that the tension of both perspectives should be kept in mind when teaching science – the benefits of skepticism and an awareness of our uncertainty, without minimizing science into a “we can never really know anything – or don’t even care if we *know* as long as we can make predictions” sort of mode.

Next time… philosophy.

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