MacLaurin, Clarke Velocities, and Powers

I've been reading various obscure sources recently as I investigate early interpretations of Newton and how they might have influenced Hume's understanding of space, time, body, souls, and scientific method. I came across this (to me) rather surprising passage from Colin MacLaurin's A Treatise of Fluxions in Two Books (1742). MacLaurin was a friend of Hume in Edinburgh, a key figure in the "Scottish Englightenment," and one of the premier mathematicians of his day. This passage comes from the book in which he attempts to explain recent developments in mathematics (especially in differential calculus) to an educated non-sophisticate.

It is indeed generally allowed, that if a body was to be left to itself from any term of the time of its motion, and was to be affected by no external influence after that term, it would proceed for ever with an uniform motion, describing always a certain space in a given time: and this seems to be a sufficient foundation for ascribing, in common language, the velocity to the body that moves, as a power. It is well known, that what is an effect in one respect, may be considered as a power or cause in another; and we know no cause in common philosophy, but what is itself to be considered as an effect: but this does not hinder us from judging of effects from such causes. However, if any dislike this expression, they may suppose any mover or cause of the motion they please, to which they may ascribe the power, considering the velocity as the action of this power, or as the adequate effect and measure of its exertion, while it is supposed to produce the motion at every term of the time. (Book 1, p. 54)

What interests me in this passage is his seeming indifference to whether something counts as a cause or an effect. On first reading, I thought MacLaurin was making the relatively uninteresting point that one thing (B) can be both an effect (of A) and a cause (of C). This is suggested when he says "what is an effect in one respect, may be considered as a power or cause in another." But his next sentence is more revealing. He offers that an unhappy reader may "suppose any mover or cause of the motion they please to which they may ascribe the power." This echoes his earlier line that the velocity of a moving body is "a power."

This is interesting because it is a break from a particularly influential reading of Newton that was popularized by Samuel Clarke. Clarke argued that since matter was incapable of self-motion, it had no power of its own. Therefore, to explain the introduction of motions into the world, we have to posit the existence of non-material souls. These souls can't be merely human, since motion shows up where humans aren't, so we have an argument for the existence of God (or at least other non-human souls; we get to God in needing to explain the existence of non-necessary matter). According to Clarke, Newton's world can't exist without God pushing matter around (or appointing lesser souls to do the pushing). (There is evidence to suggest that this was Newton's preferred hypothesis, when he felt like positing hypotheses.)

What MacLaurin is suggesting is quite radical in saying that one needn't invoke God to explain the power of motion. It's just as good from the perspective of Newtonian philosophy to refer to the powers of the bodies as it is to say that the bodies' motions are the effects of some other being. Whether velocity is the cause of the motion or the effect of the true cause of the motion makes no difference to studying velocity. This idea, which would not have been missed by Hume, not only does away with the need to posit "powers" in nature, it also does away with the need to invoke God to explain the Newtonian world. MacLaurin isn't suggesting here that one can do away with both of these hypotheses; he may have thought either one worked just fine, so he wouldn't push for either (in this place). It wouldn't be until Hume that we get a suggestion that one could do away with all these hypotheses and proceed in science and philosophy with mere regularity.

(Note: I'm not suggesting that it was Hume's reading of MacLaurin's Treatise of Fluxions that influenced his writings; Hume's Treatise, after all, was published first. But it is likely that MacLaurin was an important source of Hume's ideas about natural philosophy, if not while a student then at least in their friendship.)

how to explain the two Euclids

On occasion, I find myself talking about axioms and the notion of a paradigm-relative axiom.

The standard example I use is Euclidean geometry... which usually elicits judgmental stares from my students. (*sigh* Public school isn’t what it used to be, eh? But I digress...)

One of the axioms (forgive the loose description) is that if you draw two straight lines diverging from a single point, those lines will continue to diverge forever.

This is one of those propositions that seems so self-evident that the glow of its (alleged) necessity is like philosophical fairy dust... magical and dependable to boot!

But alas... the view is false.

Suppose that Euclid were born in a very different world, and by “world” I mean only a very small difference. Hold fixed all of the most fundamental laws of the cosmos. Change only one thing: the size of our earth. Imagine that our earth was only 500 feet in diameter. Euclid, had he been born on such a small planet, might have formed his conceptions of spatial representation on the basis of curved lines.

(To those careful readers out there who accuse me of an implicit (well, maybe explicit) empiricism... guilty as charged.)

If such had been the case, it would be very easy to see that two straight lines that diverge do not diverge forever. They actually converge again, say, at the south pole, which happens to be very close by. In this scenario, two straight lines can contain an area.

To the careful reader who thinks that I’m cheating a little bit in my characterization of straight lines...

Just conceive of a straight line as a line on which the shortest distance between two points on that line runs through that line. That’s true of traditional Euclidean lines as well as curved lines.

But... it’s also true of the lines on the alternate Earth that is only 500 feet in diameter. Let’s go back to the Euclid of this world... His geometry would consist of geometric figures such as squares and triangles whose angle sums would vary with the size of the figure.

Flat surfaces are one thing; curved surfaces are an entirely different matter. On the curved surface of a sphere, you can have a finite and endless line, and that requires a different kind of geometry.

Here’s the crazy cool reality: Our universe is composed of curved space(s).

Well, that’s a rather imprecise but free-of-jargon way of explaining this stuff.

two wacky ideas

Here are two quick thoughts about Spinoza before dinner.

The students find Spinoza both interesting and wacky.

I’ve posted a few other items in this blog about what I find interesting in Spinoza.

Here’s something I find wacky... Spinoza is a necessitarian, which is to say, he believes that all events that occur are not only necessitated by the past but also necessary.

Take any proposition P about an event where some attribute is exemplified by a particular. What explains P? If you’re a red-blooded rationalist, then you’re going to cite another proposition Q that explains P. “Explains” means.... what? Maybe it means that given the history of the world, including the physical laws up to that point, P is necessitated by Q, plus the operative laws at that slice of the universe. “Necessitated” means... what? If we limit the description to the previous sentence, then it just means that P was inevitable given the history of the world. So... “necessitated” in this context means inevitable.

But... to say that P is necessitated is not the same as saying P is necessary.

Spinoza is really clear that he believes P, along with any proposition describing an event, is necessary.

Here’s one way to get to the necessity of P. Take the entire series of explanatory propositions of any given event: R, S, T1...Tn, where each in the series explains the prior one. Then, make a massive conjunction of them all; call it C.

Ask yourself, what explains C? Or alternately, why did C obtain?

Nothing from inside the series can do so. For Spinoza, nothing contingent outside of the series can explain C, since there aren’t any contingencies.

The question cannot be answered. This, however, is a fate worse than death to a red-blooded rationalist like Spinoza. The only other alternative is that somehow the very nature of the series is such that it is self-necessitating.

So... on to the first wacky thing: This is wacky in exactly the same way that it’s wacky to say that it’s God’s nature to be a necessary being. What’s wacky is to think that this delivers some item of meaningful knowledge (my empiricist tendencies shine through, eh?). Musing on this in this way, however, does help me see another angle on why Spinoza would (refer to previous post) use a locution like: “God, or in other words, Nature.”

On to the second wacky thing: Even really dull students of philosophy get what I’m about to say, and so I’m almost ashamed to put it to paper, since it’s rather like pronouncing that “water is wet.” But here goes... Believing that this is the only possible world is tantamount to claiming that nothing (that is in fact) false is possible (or possibly true). Wow, that’s nuts.

Regardless, I still love Spinoza and wish I had one-hundredth of his intellect.

Spinoza the mystic

Recently in class (and hopefully again this week), we talked about whether Spinoza should be considered a strange theist (e.g., a pantheist or panenthiest) or an atheist. Either is consistent with his monistic metaphysics.

Hard to say...

He’s got this great way of referring to his view of reality: “God, or in other words, Nature.”

Recall that Descartes was a substance-type dualist and a substance-token pluralist. Spinoza is a substance-type monist and a substance-token monist! There is only one kind of thing, and there is, in fact, only one thing.

This one thing is “God, or in other words, Nature.”

So... what is this God like? The Appendix to Ethics part I is a wonderful tirade against traditional notions of a personal God.

He gives an argument against taking God to be a personal agent. If God is perfect, then it follows that he lacks nothing. If something intervenes owing to intentional purposes, the reason that such intervention occurs is due to a lack in the something that intervenes. The idea is that one could only hold a view of an interventionist God if one gives up the view that such a God is perfect. So, Spinoza preserves perfection at the price of God having one of the marks of personhood — viz., intentional, purposive action. The real target obviously is any traditional notion of God that is drawn from religions.

What doesn’t appear very defined at the end of this argument is the notion of perfection that he attributes to God. And here is where it begins to become clearer why he says of reality that it is “God, or in other words, Nature.”

If God were to exist as religions proclaim, it would make impossible any type of explanatory rationalism, whereby all events that occur in the spacetime world could be explained by reference to a (hopefully) small class of deterministic laws. I think that the real awe-inspiring prospect of a completed explanatory rationalism is what Spinoza has in mind when he attributes perfection to God (or Nature). For us today, I imagine it’s the same sensation of awe and hints of viewing perfection when we think about the prospects of a completed physics.

It’s not really a coherent merging of the concepts of God and Nature, since the conceptual hangover of traditional religious notions of God (e.g., the inspiration of awe) begin to obscure the impersonal but also awe-inspiring, even terrifying, aspects of the Natural world. Something that seems correct gets transposed, but can you really transpose the thing you want without also dragging along some residue of the other thing you don’t want?

Still, there’s a tense resonance that Spinoza was onto, and I probably would be where Spinoza is if I did not hold onto some of my more traditional religiously-inspired beliefs about God being personal.

I asked the students why Spinoza’s treatise is called Ethics. Here again is another tense and incoherent resonance between two views that seem right in their own ways. If something happens according to the will of (a traditional religious) God, then it follows that, in some respect, the thing that happens is “good” (though maybe not “good” for the collection of things implicated in that one event). In the same way, if something occurs by the will of Nature (that is, according to laws of determinism... Spinoza would go further and argue fatalism), and if one conjoins this alleged fact with the view of Nature as being perfect, then it follows, in some sense, that this occurrence is “good” or “fitting” or “perfect.”

The lesson drawn for agents is the same in both cases: contentment is the correct response.

In the first case, it’s pious contentment as a way of surrendering to the will of God. In the second case, it’s rational contentment as a way of surrendering to the perfect, law-like, and inevitable workings of Nature. Thus it makes sense why Spinoza’s musings, though totally saturated with metaphysics, are actually for him an investigation into ethics. It amounts to Stoicism for the seventeenth-century.

In the end, his use of “God” and “nature” may be an incoherent juxtaposition of vocabularies that are too different in their semantic histories. It is nevertheless a gem of linguistic play that makes progress in the way a pinball makes progress through a series of bumpers. It’s almost never a straight line and it’s not clear that the ball is going anywhere but you’ll play every time.