The question which forms the title of this lecture might strike some as unnecessary and strange; but it impressed itself on me as a very important question just two days ago, after an intense conversation with my son Liam. His perspective is not always the same as mine, which is good, but he is always deeply perceptive, which is also good. So I became interested when recently I heard him say that he thought our community needs to think harder about the concept of form. Liam’s idea was that there is no such thing as form. That really made my ears perk up, as you can imagine. What he said surprised me; but after thinking about it, I realized that what he probably had in mind was not really the concept of form just as such, but the concept of form as it is sometimes conceived among students of traditional philosophy. So it is this that I want to address in this lecture. I want to discuss whether, and in what way, material things really do have a form. This is a challenging topic, and if there’s one thing I am sure of, it’s that I won’t do it justice. But if I can provoke a good discussion, perhaps I can believe I was successful.
Part I. The Opening Question, Concerning an Opinion Sometimes Heard Here at the College
So the question is this: do material things really possess this thing we call a form, a form which makes them to be what they are? But why ask this, and what does the question mean? Isn’t it quite obvious that material things have forms? Perhaps it is, but some preliminary distinctions could be in order. The word “form,” in English, means roughly the same thing as shape, though “form” is somewhat wider in its application. We ourselves, when we make artifacts, do so by forming them, or configuring them as we might say; this is how we make them to be what they are. And so in the Physics Aristotle proposes that, by analogy, we might say that nature as well “shapes” or forms things to be what they are. But it is important in the first place to recognize that this is to be understood by analogy, not by a purely univocal transference of words. The very being of natural things is “being” in a different sense than that in which our artifacts can be said to “be.” And so the form through which they exist is also different.
So is it really through a form that natural things come to exist? It seems hard to deny it. How else would one differentiate a rose from a tulip, except by its form? Or when you die, won’t worms eat your body, and make it into themselves? And so won’t your matter be transformed into the worm’s matter, by the imposition of a new form? And if this is true, why should anyone have a doubt about whether form is the real principle from which natural beings come to be what they are?
Well, we might begin to see a source of doubt if we consider an opinion which we encounter now and then, especially in the our examination of the history of atomic theory. The opinion in question is that substances must be unified by their form, in such as a way as to make them be entirely one thing. Frequently this opinion goes all the way to concluding that a true substance must be homogeneous. This is manifestly false in the case of animate substances, but many think it must at least be true of inanimate ones. Water, for instance, looks homogeneous to us, and so many think that how it looks must be how it is, especially if it is to be one thing. But this is the opinion which I found my son Liam to have serious doubts about. It cannot be, he said, that there are material forms of this sort; it flies in the face of too much scientific evidence, by which we now know that even inanimate material things are not only not homogeneous, but in fact extremely complex.
We do encounter the atomists, who argue to a certain complexity in material things. Most of our sophomore year’s natural science is taken up with them. But what do we do with them? The only atomists we read are the early ones, who are only just beginning to discover evidence for atoms. The evidence they possess for atoms is still weak enough so that we often think we can take refuge in general statements about the hypothetical nature of modern science. In other words, without much consideration, we are tempted to write modern science off, so that we can get back to this thing we call philosophy.
Some may find that description a little stark, but at any rate, right here at the start, I want to note parenthetically that such a dismissal would be far less likely if we did not often confuse experimental science with the most common philosophical account of contemporary science. That most common philosophical account is based largely on the very early and incomplete developments of science, along with an offshoot of Humean philosophy which came into vogue mainly through Ernst Mach. But if we look at contemporary science as it really is today, and take care to set aside accidental associations it has with various dubious philosophies, we find a completely wonderful and astonishing growth of understanding of the physical structure not only of material substances, but of the entire cosmos. And so while some of us discuss at the lunch table whether the hypothesis of atoms is viable, physicists and engineers around the world make nanotubes and other lovely little structures, even machines, out of actual atoms of various elements such as carbon.
So the first thing I want to propose is that we need to pay attention to what real science is doing. But the second thing I want to propose is that the opinion mentioned earlier, according to which forms must make things completely one, should not just be discarded without examination. It would be good for us to proceed carefully, and shun any tendency to disregard one extreme in favor of its opposite. The opinion mentioned above started from the thought that a true substance must be all one thing, and that therefore physical substances must be homogeneous. But if we are not to merely disregard this idea, what are we to do, on the other hand, with what has now become much more than a mere hypothesis, concerning the atomic structure of material things? Can we somehow hold both of these ideas as true at the same time?
The best way to deal with a problem like this is often to do our best to grasp the full extent of the apparently opposed thoughts. It is only by seeing into them as far as we can that we may be able to know what to make of them. And so, to begin, I want to propose that we should take seriously the thought that says that homogeneity, or better, being in all parts the same, seems to be a necessary characteristic of substances. As for the other side, I would like to suggest that it’s really not enough to affirm that atomism has gone beyond being a mere hypothesis. What we need to recognize, as completely as we can, is something much deeper and more significant: namely that modern science has revealed, not only through the atomic theory but through many other things, that the constitution of the material universe is profoundly structured. In other words, it isn’t just that there are atoms; it’s that for material substances to be what they are, to have the characteristics that they have and do the things that they do, they rely on a very deep, beautiful, complex, and intimate structure. I don’t want to lose time in proving this; if you want to prove it to yourself, it will suffice to do a few Google searches about the structure of DNA, say, or of the reverse sugar molecules that biologists invented to keep you from getting fat, or of nano-measuring machines made directly out of molecules, or carbon nanostructures, or drugs custom designed to handle the molecular structure of a virus, or any of 1,000 other possible examples.
Part II. Form and Quantity
What such examples reveal is that science has been making us more and more capable, at an astonishingly rapid pace, of really understanding how material processes work, in a way that might not have been dreamed of just one or two centuries ago. And one of the things that is striking about this – the thing I want especially to focus on in what follows – is that this ability relies very profoundly on measurement. The reason why we need to think about measurement is that this is the means by which we have learned to understand the structures of material things.
There are those, not a few, among our fellow scholars, who would have us believe that this focus on measurement is really science’s Achilles’ heel. Science, they say, never gets beyond the metric aspect of things, and therefore there is no danger of it intruding upon our philosophy. That, for example, is the fairly clear view of Fr. Weisheipl, or of Anthony Rizzi who recently founded the Institute for Advanced Physics. In this strain of thought, there are even some who claim that a study exclusively focused on the examination of the metrical aspect of things must perforce never extend beyond hypotheses; and so by that account science would not only fall far short of what we call philosophy, it would never even rise to the level of making safe assertions.
But these ideas are not merely false; they have long constituted, I believe, one of the greatest impediments to a serious philosophy of science. But yet they contain a resemblance to the truth, which is the key to most of what remains for me to say in this lecture. The resemblance results from the fact that quantitative considerations matter a great deal, much more than one might have thought at first.
The tradition of Scholasticism, following the intuitions of Aristotelian philosophy, saw quality as founded on quantity. The division between quantity and quality appears to follow upon the distinction between matter and form. Quantity, it appears, follows fairly immediately upon materiality, although it cannot be explained by matter alone. Quality, by contrast, seems to be the category of accident which follows more properly upon form.
Even when we did not have the tools of modern science at our disposal, it could still be affirmed with some reason that of these two categories of being, quantity is the more fundamental one, and quality is founded upon it. In the order of generation, matter and the more material aspects of things are first; the more formal comes later. And the dependence of quality on quantity seems to be borne out by examples which are fairly ready to hand. Insects are small, and elephants are large; it is unlikely, to say the least, that an insect, a mosquito, say, could have the dimensions of an elephant. There is an elegant little essay about this by the one-time communist and very fine writer, J.B.S. Haldane, called “On Being the Right Size.” Haldane describes many different ways in which physical beings depend on having a certain size in order to be as they are. A bee, for example, uses its wings in a quite different way from an albatross; so you will never get an albatross to fly the way a bee does. And so on.
But what the tools of modern science reveal is that this dependence of the qualitative on the quantitative extends beyond the wildest guesses of those who lived before these tools existed. It isn’t just that the mosquito must be small and the elephant large; it’s that inside the mosquito, first at the level of organs, then at the level of cells, then at the level of chemistry and molecular biology, then at the level of atoms, and then at the level of quarks, and so forth, there is a structure which is so complex and beautiful that it boggles the mind. And we should note well that by “structure,” here, we do not mean something inert; to understand what we really mean, we must first recognize the physical structure in question is invariably dynamic, that is, its parts always have characteristic functions and activities, even after we descend to the the level of the inanimate. There is usually one sort or another of movement involved in the each of the parts of any physical structure, a movement which is ordered to the perfection of the whole.
We should also note that, contrary to what some think, these structures have begun to be formed long before an animate being such as the mosquito ever comes into existence. It first begins to be formed, it appears, in the element factory known as a star; then it comes to be formed through chemical interactions, and finally through a long process of biological genesis. There is, in short, a whole history hidden inside that one little mosquito, the full dimensions of which extend out to the whole order of cosmology itself. And I would just note, in parentheses, that an examination of conscience may be in order for anybody who balks at this, on the ground that it seems to resemble evolution. What we should ask ourselves, I think, is whether we can be satisfied with anything less. We have no less than the authority of Aristotle and St. Thomas, if authority is what we need, to be convinced that the order of the cosmos as a whole is the most important thing, greater and more important than the order found in any of its parts.
But for now I will set that aside and come to what I should like to propose as the heart of my thesis. I mentioned a moment ago that Scholastic thought has always acknowledged a dependence of the qualitative on the quantitative. There are many things, nevertheless, which we may recognize without really grasping their full implications. This brings me to what my son Liam wanted to say about form. He proposed, seemingly rather starkly, that there is no such thing as form in material things. But I believe what he meant is that there is cannot be a form in the manner frequently assumed; and I think he is absolutely right. What do I mean by “the manner frequently assumed”? What I mean is that we can cheerfully assert that quality, and therefore also substance, depends on quantity, but yet not see what this really means. What it means – what science proves over and over again – is not just that quality and substance depend on form externally as it were, but that they depend on it much more internally, which is to say structurally. In other words, in material things, form turns out not only to be compatible with an internal structure and heterogeneity, but to depend on it profoundly. I want to say in effect that in material things, to a surprisingly large extent, form IS structure. And so a conception of form which unifies things to the exclusion of a structure is a false conception.
You will perhaps recognize that this solves some problems, but raises others. The biggest problem that it solves is that very Scholastic principle that I have been referring to, which is that quality and substance, the more formal principles, depend on quantity. Now we can start to affirm that we know a little better what that really means. What it means is not just that things have to “be the right size,” but rather that quality and substance depend on quantity internally, because it is quantity that makes structure possible; and structure is, if you will, the intermediary between matter and whatever more abstract kind of form we may have yet to consider. And what I want to insist on again is that this structure is not a negligible thing; in fact it is so important that scientists spend a very large portion of their time examining it. Without it we could know, did know, only the first rudiments of how material things are made. And so this is why the metric part of scientific investigation acquires such a prominent aspect; it isn’t because that is all that the scientists are interested in or that they arbitrarily restrict themselves to it; on the contrary, it is because that is the very condition upon which an understanding of material forms hinges. In various places, Aristotle notes that there is a real difference between a mere dialectical or logical investigation of physical reality, and a truly physical one. The latter, as Aristotle understands it, depends on a sufficient accounting of the material aspects of things so that we can begin to see how forms are truly materialized. Now we can see perhaps a little better how this materialization of forms really happens. It happens especially through the understanding of quantitative structure.
There is a negative way to put what I have been saying, which will help to further highlight its importance, and perhaps also reveal some of the difficulties I want to discuss. The negative way is this: if we fail to see structure as mediating between matter and form, the result must finally be that we won’t truly see much at all about how matter and form are united. But if, in addition, we fail to recognize our own ignorance in this regard, we are liable to imagine that there isn’t really much, after all, that is needed for a form to exist in matter. We will suppose them to be together just as we think them together. But the result of this will be, in effect, that we won’t grasp what material forms are really like. We will be prone to imagine that material forms can be quite perfectly undifferentiated, like an angel, which has no material parts at all. We will then perhaps imagine that dogs, for example, have some essence which for some reason no one has been able to grasp, even after several millennia; we will imagine that if we could grasp it, it would lurk before our mind’s eye like a sort of ghost wearing its genus and specific difference, perfectly welded together, and having only one almost invisible joint.
But from this, perhaps, one sees the primary difficulty which also arises in what I am claiming. It is none other than the difficulty I proposed at the beginning of the discussion. How can we, at the same time, affirm the unity of substances, even while we acknowledge their very complex structure and the multiplicity of parts which that structure entails?
Part III. Indeterminacy in physics: DeKoninck on the Principle of Indeterminism
The solution to this problem demands that we be open to possibilities; not just any possibilities, but the very possibilities suggested by our difficulty itself. That is always how one solves difficulties; one lets them be, and then asks what it is that one wasn’t seeing, which is what really gives rise to the difficulty in the first place. If it is true, as I have been insisting, that the structures of material things are deep and complex, then perhaps we should be willing to let go of the requirement that substances be unified in the way we were imagining. And if we are to let go of this, perhaps we shall perhaps have to let go of the premise that led to it, which is that material substances really are substances in the way that we were imagining. What will enable us to let go of these things quite happily is a proper view of the whole cosmological order, or at least openness to such a view. In other words, we must pose a question to ourselves, namely: what sort of cosmological order would it be, in which there is not only no detriment, but perhaps indeed an advantage, to having the various substances around us be not so determinately unified as we have been inclined to imagine?
For an answer, perhaps I can illustrate what I have in mind by an analogy. In the animal kingdom, we find all sorts of varieties, and all sorts of perfections. Some animals can run extraordinarily fast. Some can hardly run at all, but they can fly. Some have acute senses of one sort, and some of another. Some are good at communicating at long distances, but can’t communicate very much; others can imitate all sorts of sounds, but not at very great distances. Some have extraordinary means of self-defense, or of hunting, and so on.
But there is one animal that seems to have very few innate perfections, namely man. When my oldest son was first born, I watched him lie on the table at the hospital with his arms and hands moving every which way, with apparently no rhyme or reason. He had no observable control over his own body movements. Jokingly, I told my wife that he seemed dumber than a dog. She didn’t appreciate that. In reality, of course, there is a very good reason why this one animal species, man, is so entirely lacking in material determinations. It’s that we possess reason. If we had the material determinations that animals have, they would stand in the way of reason. Reason requires, on a certain level, a kind of indeterminacy in the bodily dispositions, precisely so that they can be open to the perfectibility of reason, which is an infinitely greater perfection than that of animals. If we were born with an instinct to bark, for example, it would be exceedingly difficult for us to learn to speak English.
And so what I will propose is that the same is true not only of the human body, but of the whole cosmos. What do I mean by “the same”? I mean two things: first, that the cosmos, like the human body, is built up, to an exceedingly intricate degree, of material complexities on all levels; and second, that this being built up out of parts made of parts entails, just as we feared it did, a certain defect of unity and determinateness in what we are wont to refer to as the “essences” of things. But, just as with the body, this turns out to be something that we ought not to have feared; for it turns out to be the very condition required for the ultimate unity of things.
I have not wished, in this lecture, to appeal much to the authority of anyone, because it seems better for us to think on our own feet, and remember what it is like to understand things ourselves, rather than say them because someone else has said them. Nevertheless, at this point, it would feel wrong to me if I were to pass over the fact that I am hardly the first one to have thought about these things, and surely not the best either. Charles De Koninck wrote a great deal about them, and all those who want to think about them well should read what he says. I will here just note a couple of things that he says. Historically, he notes that late scholasticism came to imagine that the order of the cosmos should be understood without reference to anything beyond itself; and this, De Koninck says, led to what he called “bad angelology.” Lacking any sense of the subordination of the material order to the angelic, philosophers began ascribing to the material order itself what really only belongs to the angelic order. In particular, there came to be a tendency to search for the “essences” of things in a manner which would really only be appropriate to the essences of angels. De Koninck goes so far as to say that material things are not definable in the way that immaterial things would be, except on a very universal level. We can see that the order of the sensate, for example, is essentially different from the order of the plant, and that again from the order of the inanimate; but within these orders, what we largely find is an infinite variety of instrumental developments, which variety reveals itself, on investigation, to be quite entirely a function of the indeterminate necessities imposed by the material environment. De Koninck doesn’t see this as only a conclusion to be drawn from science; he sees it, at least in retrospect, as a conclusion which a philosopher can recognize as simply an implication of materiality itself. For the material, he says, is like the continuous, divisible in infinitely many ways.
Part IV. Cosmology as the solution: man is the head, not the heavenly bodies. The heavenly bodies are element factories. And the whole is historical.
You may still be aware of the fact that I haven’t yet fully answered the opening question. How are we to reconcile the demand that the substantially one be the same in all of its parts, with the manifest complexity which the structure of material beings evidently cannot do without? Or, to put it another way, how in the world can the forms of material things be unified after the manner of the angelic forms, and simultaneously built up out of profoundly material and complex structures? It bears noting, before I propose my answer, that many Catholic philosophers have suggested what amounts to the same view, fundamentally, as that of the radically reductionist scientism which tries to reduce everything to raw material complexity. Both of these apparently opposed camps respond to the question in what amounts to the same way, by simply conceding that the difficulty is unanswerable; you have to choose between bad angelogy on one side, and radical reductionism on the other. I was just the other day reading a book by the Nobel Prize physicist Frank Wilczek, who chooses the route of radical reductionism. He goes so far as to mention that he was once a Catholic, until he saw the light. But it does seem to me to be a bad strategy, if you happen to lie in the opposing camp, to just agree that these are the only choices.
So what other choice is there? I think that De Koninck’s mention of angels is a key. In the material universe, it turns out that man is the being most like the angels. For it is man alone who has a soul that reaches towards the realm of the immaterial, while nevertheless maintaining its essential connection with the material. It is in man, therefore, that we find the closest approach to the unity we have been seeking. The soul has no intrinsic differentiation, but a sort of extrinsic one through the body. The human person reaches towards the sort of unity our Sophomores can never find when they are asked to examine how a substance such as water can be one thing.
But then of course a natural question is what to make of these lower beings, such as water. The answer, I propose, depends again on how we more or less consciously envision the unity of the entire material cosmos. This could well lead into an entirely new lecture, but I will just briefly propose two radically different pictures of what the material cosmos is really like. In both pictures I will take it as granted that the end of the cosmos is something in the order of the contemplative; that is, the cosmos exists in order to be beautiful and orderly, just as its name suggests. But there are many kinds of beauty and order. In particular, there is the beauty of the seen, versus the beauty of the heard or acted out. In the fine arts, painting and sculpture stand in a very different sort of category from music, drama, and poetry. The former are motionless in their manner of depiction, even when they happen to depict things that move; whereas the latter highlight movement, development, and unfolding as the very form of their depiction.
The two cosmos’s that we might envision are just like these two kinds of arts. The first kind of cosmos does, indeed, have motion within it, but its end is nevertheless to stay the same always, and to make the impression of its sameness on the one who contemplates it. The other cosmos, by contrast, also has motion and time within it, but they are motion and time of an entirely different kind, with an entirely different meaning. The second cosmos has a history, a history in fact as its very form, whereas the first one has no history. Within the first, hour succeeds hour and day succeeds day, but to no end other than the sameness that both contain. Its motto might be the old French adage, “Plus ca change, plus c’est la meme chose;” or, “The more it changes, the more it stays the same.”
In the first cosmos, if there are species or kinds of things, their contemplative significance must be like the contemplative significance of pictures in a museum; those pictures do not employ mobility, because they have no developmental story to tell; rather, the presumed end is to impress us with the individual beauty of each separate picture. A cosmos of this sort could indeed reflect the beauty of God to some extent, though not as perfectly as we might hope.
But the most salient fact is that this is not the sort of cosmos we really inhabit. The one we really inhabit is one where the formal order of the whole is not iconographic, but consists in development. We are learning, day by day, more and more about how the cosmos has developed over a very long history. The universe of the pagans, including Aristotle, had as yet revealed no very convincing signs of this development, and that fact, combined with the difficulty of envisioning an unchanging divinity engaging with a radically historical world led to the conclusion that the real cosmos was the first one I just described, rather than the second. In that cosmos, the one I am calling iconographic, it would make no sense for the material species of things to lack complete substantial determinateness, because they could not be seen as ordered, in their essence, to the development of anything beyond themselves. In the real cosmos about which we are now learning, by contrast, material things, in every order below the human, has essentially the nature of a part. As one rises to higher and higher levels of perfection, the ordering of part to whole becomes more perfect, and the material indeterminacy therefore less and less pronounced. But it is not until the human soul is given to man that the structural complexity of which I have been speaking reaches towards its full completion, meaning, and determinacy.
Part V. Indeterminacy in Music and Mathematics
It may be difficult, to be sure, to change our vision of things quickly, and in so radical a fashion as I am proposing. But to make it a little easier, I will devote the last part of my lecture to the consideration of some signs and effects of that changed vision. For us the greatest effect is, I think, a possibility for coherence in our own vision of intellectual life which we could not even approach otherwise. I mean this to be understood quite concretely; for example, if you ask, why should we study music, mathematics, science, philosophy, theology, and both intellectual and moral history at Thomas Aquinas College in just the way that we do, I think that the answer must be that we can’t know why, and won’t know why, for as long as we do not set ourselves the task of gaining a real grasp of the coherence of the objects of these disciplines, both within themselves, and among each other. To discuss all of that at length would exceed our time and ability. But I think I can offer you at least an example or two of what I am talking about. Let me begin with music. I could equally begin with any of the arts, because what I mean to talk about is really art as such. The fact that man is at the head and not the foot of the material cosmos, and the correlative fact that the lower levels of the material cosmos lack determination, is reflected in the arts; because it partly through the arts that we bring the cosmos to its perfection. Since the art of music is the one which we attend to most in our curriculum, I will say a few words about it.
I only want to note a couple of things. The first is that music has a history. And its history has a direction, which I think we can say parallels the direction of thought itself. It is fairly early on in the history of music that it is seen as exemplifying, perhaps indeed being an ideal exemplification, of the Platonic vision of the intelligible forms. Timaeus even makes the fundamental musical intervals be the embedded form of the cosmos itself.
But two things happen subsequently. The first is that the supposedly eternal forms of music undergo a rather surprising development, which continues all the way from ancient times until now. If one compares the music of the ancient Greeks with that of the highest forms of music that we now know, one may easily be astonished at the difference. One is inclined to say that the forms of music that lie at opposite ends of this development don’t even try to achieve the same thing. The ancient music confines itself to far more rudimentary forms. About them one can acknowledge that they have always existed, for at least as long as we have, and have never gone away. Their permanence seems hardly a matter for doubt. If one wants something to reflect an allegedly permanent cosmos, there it is. But on the other hand, no one can deny that those permanent forms are also very rudimentary. They cannot even begin to narrate a story in the way that a concerto by Bach does, for example.
But alongside this apparently evolving perfection, we observe as well that the modern music departs more and more from the postulate of an abstractly perfect and unchanging musical form. It seems to give it up as unattainable, but then later as something which only the immature soul was seeking to attain in the first place. In his wonderfully playful depiction of music’s history, Peter Kalkavage, whom our juniors read, helps us to think about this by putting ourselves in the mind of the ancient Pythagoreans, when they discovered the absolute impossibility of making a perfect musical scale. Students often find it strange when Kalkavage suggests that one might attribute this to “tragic necessity.” But this deserves to be taken quite seriously. Not, of course, that music really does turn out to be tragic in its essential meaning; but the reason why it doesn’t, the reason why that conclusion isn’t warranted, is the very thing which the ancients did not and could not see about the relation of man to the cosmos: the fact, that is, that man stands at the summit of the cosmos, not at its lowest point. Art, therefore, including music, has a different significance than it would otherwise have. Art does not exist in order that we might conform ourselves materially to permanent art forms residing somewhere in the heavenly spheres. If art were of that sort, it would finally have to be subsumed under nature itself. The truth is the reverse; the natural world exists for man. Natural beings, and the arts with which we perfect them, exist in order that reason’s creativity might shape the material world into the most perfect reflection of what is, in its ultimate and highest principles, immaterial. When we make music, we therefore take advantage of objectively perfect mathematical ratios, as well as the physically real phenomenon of overtones. But beyond that, music is very much an art, which is to say very much our own creation. Its object is indeed to imitate, but what it imitates most formally is things in our own soul, not things in the natural world. And so there can also, to be sure, be aberrations in the historical development of music and the arts, in which it is pretended that reason can be an absolute arbiter of what is true and real, rather than only a relative one. But one cannot correct that aberration by insisting on the opposite extreme.
V. b. Mathematics
What I have intended to be describing in this lecture could be approached in a different way, through a more focused examination of the idea of history, and especially intellectual history. My purpose has really been to disagree, quite entirely, with the view that history has little to do with intellectual life. I am arguing that the indeterminacy of material being, of the forms of material things, only makes sense in light of a history which culminates in man. This intellectual history might be described as a history of the gradual entrance of mind into matter; or, alternately, of the gradual perfection of matter by mind. One can see quite easily, perhaps, that the history of early philosophy, especially Plato and Aristotle, made some of the first decisive steps down that road. For it was Plato who recognized the importance of the intelligible order; but he could not see well how to integrate that order with the order of the material. Aristotle stood on Plato’s shoulders, as the saying goes, and recognized that privation and material perfectibility are not the same thing; he thereby was able to see a reconciliation between the intelligible order and the material order, and thus he inaugurated, in a decisive way, the study of the physical world as a real integral part of our contemplative endeavor. But that is not to say that Aristotle saw all the way, or could see all the way, or even close to all the way, into what it would mean for intelligible forms to be put back into matter. For it is relatively easy to merely assert that there are intelligible forms which are enmattered, without really seeing that this entails some serious qualification concerning the nature of those intelligible forms. We may smile when, in the politics, Aristotle counts up the arts of acquiring wealth, and arrives at the nice clean number three. That’s very Platonic of you Aristotle, but wait until you find out about cell phones, genetic modification, and tractors with GPS guidance systems!
Yet though we smile, we ourselves often continue, even now, and despite a great deal of evidence to the contrary, to just assume that the criteria for science laid out in the Posterior Analytics — criteria taken not primarily from the real science already possessed of things, but rather from a grasp of the character of the human soul and the metaphysics of causality — should simply be stamped onto material things, so that we should have no need of reexamining the way in which material things are knowable.
There is one place above all where the temptation to do this exists, and that is mathematics. Mathematics often appears as the great bulwark against any doubt that the material cosmos has purely intelligible forms, capable of a strict scientific analysis. And if we have perfectly determinate and scientifically describable essences there, we easily are led to believe that we shall find them elsewhere as well. So I want to take a few final moments in my lecture to say why I think that we are misled by our conception of mathematics, and geometry in particular.
A first thing to note about it is that we should rather certainly expect a different account of mathematics from Aristotle than we did from Plato; yet we often don’t. What I mean is this: Plato’s view of the world, where everything material is seen as an imperfect copy of eternal forms, naturally conduced to thinking of mathematics as higher, in the order of the intelligibles, than physics. Aristotle, however, recognized that mathematics does not become more abstract than physics because of its object being some super-physical form, but rather because of a human act of abstraction. In other words, Aristotle says that what mathematics studies really is quantity, which is nothing but an aspect of the physical and not above the physical. He saves the apparent abstractness of mathematics by attributing this not simply, in the manner of Plato, to some objective order of intelligible being, but rather to the power of the human mind to bring objects nearer, somehow, to its own level of intelligibility.
If we see this, we might be inclined go still further, and suspect that, according to its object as opposed to its subjective conditions, mathematics is not only not one of the highest studies, but rather among the very lowest; for quantity is very near to matter itself. And then, if my claim is true that the most material things are among the least scientifically determinable things, we should expect mathematics to be a rather paltry study indeed. How does it come about, then, that it seems to us so perfect, and even worthy of being seen as a paradigm of what science is supposed to be?
If one looks more closely at mathematics, one can not only see that it doesn’t have a perfect scientific order, but also that when it does appear to have it, it turns out that there is more of the human in it than one thought. And this is not because we unwittingly or unnaturally insert ourselves where we should have refrained from intruding; rather, it is because the human act of counting and measuring is very near to the proper end of quantified matter itself. I might easily spend a whole lecture describing this alone, but it the few moments remaining I will give you just a couple of small illustrations of what I mean.
In Apollonius I.11, Apollonius reasons to what purports to be a definition of the conic section called the parabola. Interestingly, this definition is only articulable through the famous artifact known as the upright side or the latus rectum. It is quite apparent that the upright side, through which Apollonius defines the parabola, is not a naturally given mathematical principle, but an artifact. It is invented, quite out of whole cloth, for the express purpose of defining the parabola. What is more, one discovers a few pages later another proposition, (I.20), which Apollonius proves from I.11, in what purports to be good scientific order, wherein one starts with the essential, and uses it to prove the necessary but non-essential. But it is most interesting to observe that how Apollonius ever discovered I.11 in the first place remains completely hidden to us; hidden, that is, until we realize that 1.20 could easily be proven on its own, with absolutely no need of 1.11, and that 1.11 can then very easily be deduced from I.20. There can be little doubt but that this is in fact how Apollonius himself discovered these propositions. And there can be little doubt either, but that this sort of thing is the reason why later geometers, such as Viete and Descartes, came to accuse the earlier geometers of hiding their tracks.
But the meaning of all this might remain hidden to us as long as we fail to wonder whether the “true scientific order” of mathematics is absolutely real or not. It is customary to assume that Apollonius’s procedure must be the truly scientific order, and that of the later analytic mathematicians be the order in the opposite direction, referred to as “analytic.” But that clearly isn’t the whole truth, because, as I said, the order Apollonius follows starts with an artifact, instead of what is simply given, as its principle. And the effect of this is that we cannot directly see how such a proposition turns out to be true directly from what is given. All of this, however, begins to make sense if it occurs to us that even here, as with music and the other arts, something of human inventiveness is helpful to complete what is otherwise somewhat indeterminate. When we begin to recognize this sort of indeterminacy not only here but in many places in mathematics, the historical development of mathematics itself begins to make more sense.
But it is natural to wonder what, in geometry, could give rise to such indeterminacy as I am claiming it has. Is there really anything in its principles to warrant such a claim? I won’t answer this completely, but here perhaps is the beginning of an answer.
Everything we study in geometry follows upon the nature of continuous magnitude, since that is the subject. St. Thomas observes that we can consider magnitude in its material aspect, as giving rise to divisibility, or in its formal aspect, as giving rise to determinate measure. In its material aspect, it is provable that magnitude is infinitely divisible, and from this one can infer that its divisibility involves the distinction between the potential and the actual. The potential divisibility of magnitude must, however, be correlative to some act, and therefore an understanding of what mathematics is must, in the end, address above all the question as to where that actuality originates. Historically, with the discovery of the calculus especially, that question came very much to the fore among mathematicians themselves. Their answer was very interesting. But before I get to that, I want to note a couple of very striking things about this question which one can observe even in mathematics as it was practiced among the Greeks.
The first very striking thing is that the principle wherein this actuality originates was, among the Greeks, thought to involve the circle. The very first three propositions in Euclid show this. They reveal a very fundamental groundwork for the whole edifice of Euclidean geometry, by showing us how we are to think about the principle by which magnitudes are determinable to an actual size. But this means that what mathematicians now refer to as the metric of quantity, even in a single dimension, is not intelligible without reference to higher dimensions, because the circle itself is a two-dimensional being. We might have thought that the characteristics of beings of higher dimension in geometry are simply derived from the characteristics of the first dimension, which latter is to be understood apart, all by itself. But if we said this, we would be wrong, at least if we gauge ourselves by what Euclid does.
This is startling all by itself, and it has startling consequences. One can demonstrate from it, for example, that it is not simply given that a line has to a line the ratio that a square has to a circle. So if I take line A to represent the area of a circle, there may be no line which can represent the area of a square. Thus the potentialities which are actualizable in quantity turn out to be intimately linked to dimensionality, and inherently dependent on how many dimensions we are given. The vision of magnitude as only accidentally dimensional would therefore turn out to be entirely a mistake.
But what is then even more startling is that we really have no idea — no one has ever understood — what gives rise to dimensionality itself. One thing we can be fairly sure of, though, is that multidimensionality doesn’t come from mathematics; mathematics adopts it, and the only place it can adopt it from is physics. It seems uncommon among those who follow Aristotle to suppose that mathematics is subordinate to physics, but this is what we shall have to conclude on these suppositions, and in fact there are other reasons to think this which I won’t go into right now. The conclusion, though, is this: mathematics on this account is certainly not above physics in the order of intelligibles; rather, it is just the reverse. And so the illusion that mathematics escapes the indeterminacy of physical being, and rises higher in the order of disciplines, seems to fall to the ground.
But it is both easy and common to wonder if we really have to say all this. Why can’t we just postulate that all possible quantities exist in all dimensions, because we say so? One would, of course, have to ask whether this isn’t letting a tautology pull itself up by its own bootstraps, but I won’t dwell on that. This postulate is, in effect, what Dedekind proposes, in his very penetrating examination of what continuity is. He sees that the efforts of earlier theorists to remove mathematics from any subordination to physics are a failure, and so he concludes that the only possibility left is to define the continuous as a logical function, which is to say that we define it by the human power of reason rather than by anything objectively found in the world itself, taken apart from our own reason. The fact that this human power of reason seems too indefinite to give us a real definition does not deter Dedekind; he knows well enough that the alternative will be to return to the physical account, and there he will not go. He also knows, significantly, that all the traditional mathematical relationships are preserved in his account, and in fact rendered more certain; not indeed in the way that they were understood before, but arguably in a new and better way.
Nevertheless, it could be objected that Dedekind’s account removes mathematical objects entirely from the sphere of the knowable, since we end up knowing nothing but ourselves in our own act of knowing, which finally would make our knowing turn back on itself in an unintelligible manner. But I think that we have once again here a case of opposite extremes, between which we need to see the middle. Dedekind places the meaning of the possibilities for actualizing the continuous completely in the act of human reason. Euclid implicitly places it in a physical principle which mathematics adopts. The truth is likely, it seems, to be neither of these things without qualification. The continuity of quantity achieves its ultimate reason for existence in the making of physical things which culminates in human reason itself, where counting achieves its perfect meaning. In short, geometry undergoes a kind of arithmetization, by which the indeterminacies found in physical quality are filled up not by something discovered, but rather something made. The intelligibility which results is analogous, in a way, to what the fine arts such as music do in their own sphere.
I am sorry to be compressing a great deal into rather few words. But another way to summarize this is to note that theorists such as Dedekind finally resorted to a certain variety of nominalism as the account of what their discipline is. At the opposite extreme from them, there remain those who are convinced that words must have meanings taken entirely from what is given outside of ourselves. Mathematics, of course, is by no means the only discipline in which this polarity of interpretations has come about; in fact it has come about in just about every discipline. But I maintain that neither extreme is correct. The truth is that nominalism cannot be avoided as long as the only alternative allowed is semi-Platonic forms as the objects to which our words refer.
V I. Conclusion
The ancient and medieval view of the world still tended towards the idea that the account of the world would be atemporal. The medieval Christian philosophers, including St. Thomas, did hold from faith that the world had a beginning; nevertheless, one can say that to have a beginning and to have a beginning essentially are two different things. In the middle ages, there was as yet no strong evidence from natural reason that the world was essentially historical; and consequently, the Aristotelian worldview of history as outside the range of the theoretical remained fairly intact.
Today, by contrast, it cannot appear thus to anybody who looks closely at what we ahve learned. In the conclusion to the remarkable book, Introduction to Christianity, Cardinal Ratzinger offers this as a description of the cosmos we have increasingly come to understand ourselves to inhabit:
If, he says, the cosmos is history, and if matter represents a moment in the history of spirit, then there is no such thing as an eternal, neutral combination of matter and spirit; rather, there is a final “complexity” in which the world finds its omega and unity. In that case there is a final connection between matter and spirit in which the destiny of man and of the world is consummated, even if it is impossible for us today to define the nature of this connection. In that case there is a “Last Day,” on which the destiny of the individual man becomes full because the destiny of mankind is fulfilled. The goal of the Christian is not private bliss but the whole. (p. 358)
If there was a time when it was plausible to suppose that both the human and the natural world was static in its essential features, that it was created to remain always more or less the same, it is now apparent that that vision cannot encompass the whole of our experience — not just on the level of experimental science (in the infamous debates over evolution, for example), but equally, if not more importantly, in our understanding of human culture, and finally in our understanding of the cosmic bridge from what is material to what is metaphysical. This may strike us as a fairly new development, as indeed it is.
Yet there is one very significant strain of life and thought which tends very strongly in the same direction as much of modern thought, but greatly predates it in its origin. That strain is Christianity.