Tuesday, July 24, 2007

Space and Concepts of Space

In his Physics, Aristotle, famously, denies that there can be such a thing as an actually infinite magnitude. That seems plausible enough, but he goes on to deny that anything can be actually infinite, which is a slightly more problematic claim. The most notorious example of the infinite, the continuum, was only potentially infinite for Aristotle. He held that a potentiality is to be understood in terms of the activity that would be required in order for the thing to become the case, so to say that there are infinite number of points on a line segment AB is only to say that the line AB may be arbitrarily bisected at any point. Does the mathematician need the concept of the infinite to do the work of mathematics? Not according to Aristotle:
This reasoning does not deprive the mathematicians of their study, either, in refuting the existence in actual operation of an intransversable infinite in extent. Even as it is they do not need the infinite, for they make no use of it; they need only that there should be a finite line of any size they wish (Physics III.7).
This last claim is clearly false in actual practice, because Euclid, writing not long after Aristotle, made no secret of the fact that he thought that it was necessary to postulate infinitely long lines in certain proofs, and it is not clear how the postulates he had in mind could be proved without making an appeal to a possibly infinite line.

Aristotle's mistake is interesting because it raises an extremely important question in the history of mathematics: how did Aristotle conceive of the relation between space and the science of geometry? Kant, notoriously, held geometry to be the science of space, only to be proven wrong with the advent of non-Euclidean geometries in the 19th century.

Because he thought the cosmos itself to be finite (because there are no infinitely large magnitudes), Aristotle could not countenance the idea that a line might extend beyond the boundary of the cosmos, but why couldn't one imagine a line that theoretically extended beyond the cosmos just for the purposes of constructing a proof of some postulate? Perhaps because such an assumption would raise the question of what sort of "space" there might be "outside" the cosmos, a possibility rather like division by zero as far as Aristotle was concerned, or would have been concerned if he had had a concept of zero.

11 comments:

Apollodorus said...

Alright, fine. I know it's "notorious," but I don't understand this whole refutation of Kant by means of non-Euclidean geometry thing. Maybe it's because I had an art teacher for geometry in my fundamentalist high school and my Intro to Modern class at OU spent so much time on Hume that we 'did' Kant in a week. But I don't get it. Would you mind explaining it briefly, or telling me where I can find a good explanation that doesn't involve reading all of the Critique of Pure Reason and learning the history of mathematics?

Scott Carson said...

Briefly the idea is that, for Kant, space just is Euclidean in its ontology, that is, it is uniform throughout, hence the Euclidean "description" of the properties of space is an instance of synthetic a priori truth.

What happened, though, is that the invention of non-Euclidean geometries, followed by the discovery of the curvature of space near massive objects, showed that space is not, in actual fact, uniform throughout in the way stipulated by Euclid. Hence Euclidean geometry is either incomplete or false, a model that models only certain kinds of space. But Kant thought it modeled space as such.

This was a crucial development, because it shows that even a science like mathematics can be falsified by experimental data, something that Kant did not think possible.

CrimsonCatholic said...

I've got this notion (for which I've been roundly abused by at least one Thomist) that all of Aristotle's errors could be boiled down to his misconception of infinity, and that St. Thomas actually said so.

My theory is this: Aristotle was so focused on motion as a model of act that he never quite got past the limitations of the analogy. Clearly, no material thing could be infinite, because it would only be real by its delimitation (i.e., form). But Aristotle could imagine successive operations, the sort of privative infinity when you can always add or divide or move one more. He closed up the circles with the celestial bodies (so that they could go on forever), had the Prime Mover as the source of the (circular) celestial motion, and then the material bodies moved according to their particular finitude. Then, assume all that is is what exists, and boom! Everything's explained.

I think that if Aristotle had seen the possibility of a qualitative infinity, i.e., a degree of perfection, then he might have seen the crack in the foundation of this edifice. He recognized that there was a difference between the essence of a thing and its existence, because he recognized that things really did have the capacity to become mobile, but he didn't get to a true explanation of the existence of the things themselves. Of course, his hostility to forms might have made him a bit reactionary; St. Thomas had the benefit of Pseudo-Dionysius to moderate this influence. But it seems that Aristotle couldn't get his head around the idea that perfections could increase in degree like actions, which prevented him from leveraging the concept of act and potency stripped from its association from motion as an explanation for existence itself, a la St. Thomas.

But like I said, there are some Thomists who are adamant that St. Thomas's philosophy just doesn't work absent revelation, and that his metaphysical method was alien to Aristotle's on that account. Honestly, I lack the expertise to make a judgment, but something about that argument just seems fundamentally contrary to the spirit of St. Thomas's commentaries on Aristotle. It just doesn't seem right that he is taking the Hebrew YHWH as a basis for critiquing Aristotle's entire project.

The reason I mention this is that it strikes me as an even stronger critique of Kant from the perspective of the philosophia perennis. If Aristotle was more or less on the right track, having more or less tripped at the end from failure of imagination, then the Kantian separation between things as they are and a priori truth looks even more goofy. You can point to St. Thomas as having empirically corrected Aristotle according to Aristotle's own method, well before the conceivability of non-Euclidean geometry put paid to Euclidean geometry as a Kantian a priori truth. It destroys the all-too-common idea, that we've learned so much more about metaphysics since the Enlightenment.

That's probably far too much for any comment box, but I figured if anyone would know this subject, it would be you!

Anonymous said...

"This last claim is clearly false in actual practice, because Euclid, writing not long after Aristotle, made no secret of the fact that he thought that it was necessary to postulate infinitely long lines in certain proofs, and it is not clear how the postulates he had in mind could be proved without making an appeal to a possibly infinite line."

I don't believe that the above is correct. Euclid never postulates infinitely long lines but only lines of indefinite length, i.e., lines whose length is not specified but which are sufficiently long for his purposes. I read through the Elements some time ago so it is quite possible that my memory fails me. Nevertheless, I am not aware of any proposition in the Elements that requires the notion of an actually infinite line.

"What happened, though, is that the invention of non-Euclidean geometries, followed by the discovery of the curvature of space near massive objects, showed that space is not, in actual fact, uniform throughout in the way stipulated by Euclid. "

The above statement seems to me to be based on a confusion between the science of nature and the science of mathematics. St. Thomas, in his "Divisions and Methods of the Sciences" clearly distinguishes the two. The science of mathematics abstracts from the material and the sensible and considers only the quantitative properties of things (magnitude, number). The science of nature, on the other hand, must consider the material and sensible properties of things. Modern physics, though it is a mixed science which looks at nature through a mathematical lens as it were, nevertheless is firmly within the realm of the physical and must always consider the material and sensible properties of things. This is why physics involves measurement and error while mathematics does not.
"Space", I would argue, is a purely mathematical concept. As such, it is an abstraction from the material and sensible. It follows from this that no supposed "discovery" of modern physics can really touch our concept of space. Indeed, the very notion of a curvature of space implies some Euclidean backdrop from which the curved space deviates. The point here is that even if it were true that the space around massive bodies is curved, one could still abstract a Euclidean space from that curved space. And mathematics is all about such an abstraction.

Ed

Scott Carson said...

Ed

Thanks for the comment. My point, however, still stands: in order for Euclid's proofs to work, lines would have to extend beyond the physical kosmos (as Euclid himself says), and yet Aristotle explicitly says that geometers don't need to posit such lines. For him the point is an important methodological one, since he wants to deny that there can be meaningful references to points outside the kosmos, whether those points are actual physical points or merely potential points of reference for the purpose of mathematical proof.

I'm not sure I understand your discussion of space as a purely mathematical concept. I'll grant you that it has certain applications that are conceptual in nature, but the suggestion that space is in itself nothing more than a concept is rather mysterious to me.

To the extent that I think I understand what you're getting at, I think it's safe to say that Kant did not share your view. But it may not matter much: if you are correct in your characterization of Euclidean geometry, then Kant's hypothesis is false; if I am correct in my characterization of it, then Kant's hypothesis is false.

St. Thomas's division of the sciences in the S.T. and S.C.G. is interesting and valuable, but of course it is, after all, merely one view among many possible views about the nature and structure of the sciences. In particular, there is no need to infer, from a difference in perceptible vs. intelligible objects of knowledge, any incompatibility in the modeling of ontological correlates.

Scott Carson said...

Jonathan

Thanks for your comment--you seem to know more about this than I but what you say makes a great deal of sense to me. I'm assuming, when you say that Aristotle's potency/act distinction is grounded in his conception of motion, that you mean by "motion" something like kinĂªsis, or at the very least something broader than phora, and that seems right to me.

The principal difference might lie, as you seem to suggest, in the Aristotelian critique vs. the Neoplatonic acceptance of a certain role for form; it would be interesting to hear some more details about that, because in some ways Thomas' essentialism seems more like Aristotle's than the Neoplatonists', and I confess that I'm mystified as to how Thomas's metaphysics could really be as dependent on revelation as the critics you mention claim. The human body/soul unity, for example, seems to me to be something that could be made sense of independently of revelation, it would just depend on what sort of ontological correlates you think the terms refer to. Revelation gives a particular meaning to the metaphysics, but surely the metaphysics could stand without the revelation (though not without some kind of interpretation..

CrimsonCatholic said...

I'm assuming, when you say that Aristotle's potency/act distinction is grounded in his conception of motion, that you mean by "motion" something like kinĂªsis, or at the very least something broader than phora, and that seems right to me.

Exactly! Glad to hear I'm not completely off the beam.

The principal difference might lie, as you seem to suggest, in the Aristotelian critique vs. the Neoplatonic acceptance of a certain role for form; it would be interesting to hear some more details about that, because in some ways Thomas' essentialism seems more like Aristotle's than the Neoplatonists', and I confess that I'm mystified as to how Thomas's metaphysics could really be as dependent on revelation as the critics you mention claim.

Funny you should mention that, because that's exactly why I asked you the question I did. If you're curious, there's a book by Fran O'Rourke titled Pseudo-Dionysius and the Metaphysics of Aquinas that delved into this issue in a way that I haven't seen anywhere else. I think it makes a very good case for Thomas giving a decidedly Aristotelian gloss to the Pseudo-Dionysian Neoplationism. I believe you've read Bradshaw's Aristotle East and West, and he points out some respects in which the East was conscious of Aristotle (particularly dynamis and energeia). But the East basically takes those Aristotelian concepts and reads them in a Platonic metaphysics, while Thomas does the reverse.

A couple of other works that I've found helpful on how the Latins gloss Eastern works include Nancy J. Hudson, Becoming God: The Doctrine of Theosis in Nicholas of Cusa, and James J. McEvoy (ed.), Mystical Theology: The Glosses by Thomas Gallus and the Commentary of Robert Grosseteste De Mystica Theologia. The latter is particularly interesting, because it basically documents the second reception of Pseudo-Dionysius in the West. The first reception through John Scotus Eriugena was more or less a non-factor, but the second reception fell squarely within the dominant Western monastic tradition (influenced heavily by the Augustinianism of St. Bernard of Clairvaux and the Victorines Hugh and Richard). That provides an excellent rationale for why St. Thomas's ideas with respect to Ps.-D. weren't just pulled out of the air, but the explanation would be even better if the exact innovations and corrections that St. Thomas made could be identified.

That's why I wanted to get a "gut check" on the plausibilty of St. Thomas's correction of Aristotle being in the concept of infinity, because his innovation in that regard can be documented (and it fact, it has been documented by Leo Sweeney). And I'm pleased to see that the notion at least passes the "straight face" test. I don't pretend to have proved anything, but my theory sure seems plausible at this point.

Anonymous said...

Scott,
Thanks for your reply to my post. I’m not sure I can contribute much to this discussion. I’m not a philosopher by training and I may be missing the point of what you were getting at. Nevertheless, I should clarify a point which was very misleading in my last post. I stated that space is a purely mathematical "concept." What I should have said is that it is a purely mathematical "object." Mathematics deals with magnitude and number, objects which really exist in the world but not in the manner in which mathematics considers them, i.e., free from material conditions. Space, I would contend, is such a mathematical object. It is pure extension free from the conditions of matter. I hope that clarifies what I meant.

God bless,

Ed

CrimsonCatholic said...

Mathematics deals with magnitude and number, objects which really exist in the world but not in the manner in which mathematics considers them, i.e., free from material conditions.

I would think that non-Euclidean space also deals with quantity and magnitude as abstracted from matter, perhaps even more clearly so than Euclidean space, which is tied up with a Newtonian account of motion. Even for St. Thomas, magnitude (dimension, geometry) is closer to matter than quantity (number, arithmetic), so it wouldn't even be surprising to me that there are more quantitative accounts of space opened up by observing more things than triangles inscribed in circles and whatnot. And that's effectively what happened with relativity; the actual shape traced by Mercury's orbit was inexplicable by the concepts drawn from objects in Euclidean motion. You seem to have the situation backward, as if we believe that space in non-Euclidean because we are too much tied up in matter. If anything, it appears to have been the other way around; scientists were previously unable to purely abstract magnitude qua magnitude from what they observed without abstracting all the way to number and then intentionally generating quantitative models of a variety of spaces. Consequently, when the relevance of Lorentz transforms for space was perceived, it was really just a recognition of what we already knew to be possible from our reasoning based on pure quantity.

Effectively, that's the same thing that I was arguing with respect to Aristotle. He should have known better, but because his thinking was insufficiently abstracted from the material context in which he encountered it, he was unable to perceive the full implications of his knowledge. In Thomist terms, Aristotle's "phantasm" was thinner than it ought to have been; he didn't understand what he knew as well as he ought to have. That's the way that our experience can sometimes be a limiting factor. If not confronted a clear manifestation that we are wrong, sometimes our knowledge is not as deep as it ought to be. In the case of Euclidean space, for example, we know the limiting case (low-velocity, low-mass) accurately, because it is most familiar to us, but we err when more careful abstraction is required.

Anonymous said...

"I would think that non-Euclidean space also deals with quantity and magnitude as abstracted from matter, perhaps even more clearly so than Euclidean space, which is tied up with a Newtonian account of motion."

Jonathan,

I do not deny that non-Euclidean geometries deal with quantity and magnitude as abstracted from matter. That is, after all, what mathematics is all about. What seems to me to be incorrect, however, is the assertion that discoveries in physics have shown Euclidean geometry to be false. Historically, non-Euclidean geometries did not arise as a consequence of discoveries in physics, but rather, as a consequence of various revisions of Euclid’s 5th postulate. If the 5th postulate is true then Euclidean geometry follows; if false, it does not follow. Either way, I don’t see how physics, which studies the world of moving bodies, can have anything to say about it. As for your assertion that Euclidean space is tied up with a Newtonian account of motion, again, it seems to me that this rests on a conflation of physics with mathematics. Mathematics, in itself, does not deal with motion. It abstracts from all material conditions except magnitude and number.

“Even for St. Thomas, magnitude (dimension, geometry) is closer to matter than quantity (number, arithmetic), so it wouldn't even be surprising to me that there are more quantitative accounts of space opened up by observing more things than triangles inscribed in circles and whatnot. And that's effectively what happened with relativity; the actual shape traced by Mercury's orbit was inexplicable by the concepts drawn from objects in Euclidean motion.”

I’m not certain what you mean by “Euclidean motion.” As I’ve stated, mathematics does not deal with motion which is the proper subject matter of physics. I gather, however, that you’re drawing a distinction between the Newtonian or classical account of gravitation and that of General Relativity. I grant that General Relativity is the best explanation of the appearances that we have today (though it is not quite accurate to say that the precession of Mercury’s orbit is inexplicable in classical terms --- it was explained by a fellow named Paul Gerber in 1898 using classical physics coupled with the assumption that gravity propagates at a finite speed). Nevertheless, I don’t believe that the success of the relativistic theory of gravitation tells us anything about the validity or invalidity of Euclidean geometry. First, there is no reason to assume that Einstein’s application of Riemannian geometry to his construct of spacetime is anything more than a convenient mathematical tool for obtaining correct results. Second, in Einstein’s own popular work on relativity, he uses the analogy of the surface of a sphere to explain what he means by a curved spacetime. But the surface of a sphere is not non-Euclidean in the sense of being beyond the reach of or contrary to Euclidean geometry. One can study all the pertinent properties of the surface of a sphere from a purely Euclidean point of view (in fact, one must generally use a differential form of the Pythagorean theorem to determine the distances between points on a sphere --- a theorem proven only on the basis of Euclidean assumptions). Moreover, the curvature of the sphere can only be understood as a deviation from straightness or a curving into a third dimension. This implies the existence of a tangent space which is “flat” or Euclidean. The same holds for spacetime which, in the general theory of relativity, is always locally flat. But if this is so, then it follows that one can always abstract a Euclidean substratum from a so-called non-Euclidean space.

“You seem to have the situation backward, as if we believe that space in non-Euclidean because we are too much tied up in matter. If anything, it appears to have been the other way around; scientists were previously unable to purely abstract magnitude qua magnitude from what they observed without abstracting all the way to number and then intentionally generating quantitative models of a variety of spaces. Consequently, when the relevance of Lorentz transforms for space was perceived, it was really just a recognition of what we already knew to be possible from our reasoning based on pure quantity.”

I don’t really understand what you’re saying here. Could you clarify?

Ed De Vita

CrimsonCatholic said...

What seems to me to be incorrect, however, is the assertion that discoveries in physics have shown Euclidean geometry to be false.

I don't think anyone said that Euclidean geometry was false. What Dr. Carson said was that Kant's description of space (and Euclid's description of space, if we assume that Euclid had a similar notion of physical motion) was false as applied to the mixed scientific judgment. Clearly, space is real, not an abstraction, and since it is not a mathematical object, it can be the object of a mixed science. My point was only that if one fails to properly abstract to the mathematical level, one will likely stumble in the judgments of a mixed science that makes judgments about mathematical concepts applied to real things. That seems to have been Kant's problem; he couldn't even conceive non-Euclidean space because he was too caught up with his own experience (being that material objects don't naturally curve as they extend).

Either way, I don’t see how physics, which studies the world of moving bodies, can have anything to say about it.

Physics technically studies mobile being, which is anything that changes in any way (technically, subject to generation, motion, or corruption). Space is mobile being, because it hasn't always existed (it was created). Hence, space can be the object of physical study just like any other object. Consequently, a particular judgment about space according to a mixed science (like physics) can err, and indeed, Kant's did.