There is a difference between the mathematics of Descartes and later mathematical developments, which consists mainly in the fact that this later mathematics came to be understood as always having number as its object, even when the goal was to understand continuous quantity, or generally what pertains to geometry rather than to Euclidean arithmetic. While Descartes’ geometry suggests that we might try to think this way, it never quite does so itself; for while Descartes does extend the operations of arithmetic into geometry, he seems to do so only, arguably, through a kind of equivocation, by virtue of which we say that we can now multiply and divide not only discrete but also continuous quantities. But the later mathematics takes a more radical step: it claims that even the continuous can be described numerically, so that geometry can, at least to some extent, be subsumed under arithmetic.
If we are to study this newer kind of mathematics as something more than a technique, we shall have to wonder about this. How does this new, extended notion of number come about? Is it only justifiable as a practical expedient? Ptolemy, as we know, used numbers to measure angles and distances in space, without concern for an absolute precision which he knew to be neither attainable nor necessary. Is the “number line” used by contemporary mathematicians something like that — a practical expedient of sorts which really doesn’t aim at, or require, theoretical precision? Should we therefore see utility as the aim, finally, of this sort of mathematics? Or should we perhaps look for a middle position, by seeing modern mathematics as being neither pure theory, nor useful art, but contemplative art? Or should we look for something even deeper — such as perhaps a deeper connection between number and the continuous than we have so far been able to see?
The art of the calculus, which forms the larger part of what we will now study, was in some of its earlier forms presented more geometrico; but a numerical version of it soon seems to emerge almost spontaneously as the more perfect and elegant form, and so that has become the customary form of it. As we proceed, we shall therefore have many opportunities to reflect on the questions just posed; but nevertheless it would be well for us to begin with at least a brief consideration of them, and perhaps begin to suggest some tentative answers. One important reason for doing so is that we shall sooner or later discover that these questions are really at least in part metamathematical; that is, they belong at least partially to a level of reflection which bears on the very principles of mathematics, and which mathematics itself cannot, therefore, hope to fully resolve by itself. (But neither is that to say that reflections lying properly within the competence of mathematical investigation might not aid us, by disposing our minds to see something which extends beyond mathematics.)
Let us, for a moment, go back to Descartes. He finds a remarkable way to extend the operations of arithmetic into geometry. Yet his method is apt to leave us with some doubts. Our first doubts may arise from the fact that, although Descartes suggests a way to extend the arithmetic operations of multiplication and division into geometry by analogy, the extension seems to result in something only imperfectly accomplished. Descartes might really be said to have two definitions for multiplication and division, one for the continuous and another for the discrete; but the equivocation ends up hidden under algebraic symbols which look the same in either case, so that in fact we completely ignore whether we are applying one or the other of the two definitions at any given moment. For practical ends this may satisfy us, but in theoretical investigation it is crucial to be aware of equivocation when it arises.
And in any event, no matter what sort of similarities we see, it seems impossible finally to “multiply” a line by a line except by way of a metaphor. This will perhaps be clearer if we note, first, that in true multiplication the multiplier and the multiplied do not play the same role. There seems to be no difficulty with a line, or any sort of quantity at all, being the multiplied; but for anything but a number to be the multiplier seems problematic. Moreover, on reflection it appears to be this very fact which makes Descartes’ operation of multiplication produces ambiguous results. For if we multiply line a by line b (so that b is the multiplier, and a what is multiplied), we find that the result is not unique; it may vary from infinitely small to infinitely large, depending on an arbitrary choice of what Descartes calls the “unit.” But then we may also notice that the effect of this arbitrary choice is to make the multiplier be the bearer, as it were, of a different “number;” if, for instance, the multiplier is three times the unit, then what we really do is multiply by three; if four times the unit, then by four, and so on. Considering the matter this way, we can see that the ambiguity in the result derives, in a significant way, from nothing but the fact that the multiplier is not specified otherwise than accidentally when it is specified by a magnitude.
In conjunction with this, a further question arises. Can we associate a number with any magnitude whatsoever? What shall we say, especially, about magnitudes which are incommensurable with the unit? And even if we can can “associate” a number with a magnitude in every case, including the irrational magnitudes, what should we finally think about the arbitrariness of the unit? Still more: what should we understand by this word “associate”? Are we on the path of knowledge pure and simple, or is this some sort of art, as was already asked above?
Often the decisive thing needed, if one would begin to see what is true, is to see what questions need answering; for an answer without a well perceived question is apt, as the saying goes, “to fall on deaf ears.” It may help us to see what questions to ask if we go back much further than Descartes, to a moment when more fundamental things still vied for greater attention in the minds of philosophers. We have in mind the moment when the Pythagoreans discovered the irrational magnitudes. Legend has it that this discovery was a great scandal to the Pythagoreans — so great that, at least by one version of the legend, the philosopher who divulged the fact was subsequently drowned at sea.
This concern for the rationality of things is apt to strike us nowadays perhaps as quaint. It is even difficult, looking at the matter from the perspective of contemporary mathematics, to hear in the term “irrational” much more than a technical term or perhaps a metaphor. But the Pythagoreans clearly intended nothing like a mere technical term; in its original use, “irrational” as applied to magnitudes referred to a real, or at least perceived, defect in the intelligibility of quantity; and that, in turn, seemed to betoken a peculiar lack of connaturality between the real world and our own minds.
We may understand this still better by reflecting for a moment on what we understand quantity to be, taken generally. As some philosophers have noted, quantity seems to have both a material and a formal aspect: taken materially, it seems to be what is responsible for making material things exist “part outside of part;” but taken formally, it is what makes things have a determinate measure. Measure, here, refers to the ability of a whole to be measured, or made determinately knowable through its parts. This seems to happen perfectly in the case of number; for there the unit always measures perfectly. By “perfect,” we mean not only that the measure never leaves a remainder, but also something which is less easy to grasp but is yet in a way more fundamental: namely that the unit’s ability to measure perfectly stems from the fact that the unit itself is not strictly a quantity, except by extension of meaning. For if the unit were a quantity, then evidently it itself would be in need of measure before it could serve as a measure. With magnitude, by contrast to number, there seems to be a possibility of measurement at least in some cases; but even in those cases, we are never able to reduce the measure of a magnitude to something absolute and final, as we are with number, since the parts of magnitudes are never themselves anything but magnitudes.
But the Pythagoreans evidently assumed that magnitudes would always have a perfect measure, at least in a relative sense; that seems to be the import of their initial assumption that all magnitudes would be commensurable. On reflection, we can see that this is perhaps not an unreasonable surmise, at least at first, especially if we assume that the determinateness of quantity — in other words its more formal aspect, as described above — lies precisely in its measure. And we may note further that commensurability and rationality are two words which are used to name the same reality; as if to suggest that what lacks a measure, or at least a “co-measure,” must by that very fact be “irrational.” In the discovery of the irrational, therefore, the Pythagoreans thought that they had uncovered a real sort of deficit in the rationality of the world.
But the fact is, of course, that this irrationality or incommensurability is something quite real; it seems we can have no doubt about its existence. What, if anything, should we really infer from this? Is it finally right to infer that magnitude really lacks something of the rational?
A first thing to note is that this deficit of rationality is perhaps not so unfamiliar as it might seem at first. If there is a deficit of rationality in magnitude as opposed to number, that would seem to be a result of the fact that magnitude is a form of quantity which is more immersed, so to speak, in matter. Number, by contrast, seems to be a more abstract kind of being. One reason for saying this can be found in what we noted a moment ago about the unit: with numbers, we are simply counting ones, or in other words the undivided, conceived in abstraction from whatever it is that might be undivided. But with magnitude, we cannot abstract in this way, and so the materiality of magnitude seems to be what brings about its lesser intelligibility. But then it seems that this happens not just with magnitude, but with anything in which matter is a principle. This, indeed, came to be understood more and more among philosophers (perhaps partly as a result of the Pythagorean discovery).
We may recall, for example, the statement of Timaeus, who suggests that it might really be impossible to give strictly theoretical account of the material world itself; hence, rather than trying to offer a theory of the genesis of the world, he tells a “likely story.” Later philosophers — Aristotle in particular, and Aquinas following him, developed the conception of matter significantly, but nevertheless continued to see in it a principle of defective intelligibility. Abstraction from matter therefore came to be seen as an essential principle of scientific conception, whenever the object to be known has a material existence.
In conclusion, then, the Pythagorean discovery forces us to reckon with a certain materiality associated with continuous quantity itself. This materiality results in a certain indeterminacy. We mean this not as if to suggest that magnitudes could have a determinate measure which varies from moment to moment; for evidently magnitude as such does not vary. Rather, by “indeterminacy” we mean here something more radical: namely that the presence of matter may prevent there being, even momentarily, such a determinate and intelligible form as we find in numbers. We may in fact observe, following the analysis of Aristotle in the Physics, that since the divisions of magnitude only exist potentially, it has no real divisions in it of such a sort as a number has; and if it should turn out, then, that the divisions which it has only potentially — that is to say, which we might make in it — are not fully capable of bringing about a perfect measure in it, perhaps we should not finally be too surprised.
This indeterminacy which results from matter (or at least is associated with the presence of matter) seems, in the case of magnitude, to have an interesting manifestation in Book X of Euclid. There Euclid tries to classify and name the kinds of irrationals. This endeavor soon looks, however, as if it will never end. As we descend further and further into the abyss of material indeterminacy, perhaps we should also not be surprised if we find that there is no end, since indeterminacy seems antithetic to the very notion of a perfect formal schema.
But this is getting a little ahead of ourselves. For the very ratios of which Euclid speaks in Bk. 10 constitute, in effect, a proof that the indeterminacy of magnitude in comparison with number is real, but yet also not the end of the story. To see this well, we must have a very clear grasp of the relation between three things, namely ratio, measure, and proportion.
First let us consider the connection between ratio and measure. We should recall to begin with that ratio is a kind of relation between magnitudes. But there are two kinds of relations in general, namely compound relations and simple ones; or, to put it another way, sometimes things are related directly, whereas sometimes they are related indirectly. For example, the relation of sibling to sibling is, in its essential account, a compound relation, for it reduces to a combination of two relations between child and parent. We might say that “child to parent + parent to second child = sibling to sibling.” It must be understood that this is not merely true of siblings, but the very thing which defines this relation. The relation of child to parent is thus prior and more fundamental in account than that of sibling to sibling.
In like manner, we can now observe that the relation of a number to a number is also a compound relation, like the relation of siblings, and is in every case defined in terms of a more fundamental relation, just as with siblings. The more fundamental relation is that of measure and measured.
Ratios of numbers are founded on the relation of measure and measured in two ways. Sometimes a number stands directly to a number as measure to measured, or in the same relation as the unit to (another) number. But it happens more often that one number does not simply meaure another, but both have a relation to each other through their common measure. These two possibilities are, evidently, what Euclid is referring to when he says that a number has to a number a ratio by being either “part or parts” of the other.
Thus the relations of numbers to numbers are always through their more fundamental relation to the unit which measures them, just as the relation of siblings is through their more fundamental relations to the parents. So the relation of a to b is derived from the relation of a to the unit, combined with the relation of the unit to b. In fact, this can be seen to be true even when one number measures another; for even there, the possibility of one number being in a relation to the other evidently derives from the fact that both numbers are measured by the unit.
Seeing that the relation of measure to measured is the more fundamental relation which gives rise to ratio, we can infer that this will be true of magnitude as well, at least to the extent that magnitudes are capable of having any measures and ratios at all. And so the existence of ratios in magnitudes may become, in effect, a proof that they have a measure.
But is this even true when we say that magnitudes are “incommensurable” — or should we acknowledge, rather, that “incommensurable” means nothing if not that magnitudes sometimes lack both an absolute and a common measure? In conjunction with this question, we should perhaps wonder about whether arguing from ratio to measure would not be backwards, if indeed ratio is defined by measure rather than vice versa.
It is here, however, that we must have a clear grasp of the connection between ratio and proportion. At first this seems to be a fairly simple matter: we say that two pairs of magnitudes are “in proportion” when they have the same ratio. That is to say that proportion is defined in terms of ratio.
Euclid does, at least at a superficial glance, seem to define ratio before proportion. He says in Definition 4 of Book V that magnitudes “are said to have a ratio which are capable, when multiplied, of exceeding one another.” And then after this, he defines proportion. But several things are noteworthy, not to say even peculiar. First, Euclid doesn’t exactly define ratio itself in Definition 4; he seems to only be speaking about existence, rather than what a ratio would be. And if the definition were to be taken as saying what a ratio is, it would seem to be much too vague. And lastly, Definition 5 does not, as one might expect, say anything directly about how a proportion will be a sameness of ratio; it does not seem, in other words, to define proportion by ratio.
But a proportion, as we said, is a kind of sameness. And in our knowledge of reality, sameness plays a very important role, for it is often through the sameness of things that we discover the very existence of forms or species which are said to be “the same.” We infer that there is such a thing as a buffalo (and that it is not merely, perhaps, an aberration from the form of a cow) when we see many which are the same; or we might infer that there is such a thing as a musical art when we find that we can make many melodies which share a common matter and a common way of being formed. Evidently, though, this is what we should characterize as argument from “the better known to us,” as opposed to argument from principle; for argument from principle would proceed in the opposite direction, from apprehension or definition of the form or species itself to apprehension or definition of a likeness.
Immediately, then, we may see a cause to wonder if the order of discovery in the geometric account of ratios involves something like this as well. A summary of what we have seen so far makes it apparent that this must in fact be true. For, as we saw, it is the relation of measure to measured which is really the foundation of the relation of “ratio.” But in magnitudes, we discover first that there is never an ultimate measure; and second, that there is often not even a relative measure. From this, the question arises as to whether the relatively material character of magnitude, as opposed to number, might entail that there simply cannot be a measurable determinacy such as one finds in number; magnitudes will then turn out to be in some way indeterminate in comparison with number. But then we discover, marvelously, that pairs of magnitudes can be compared indirectly through their “equimultiples.” This is, in effect, a “via negativa” for comparing pairs of magnitudes to each other through their comparison to numerical ratios. For example, if a, b, c, and d are magnitudes of such a sort that the triples of a and c “like exceed” the quadruples of b and d, we could say that the relation of a to b and c to d are both such as to be “greater” than the relation of 4 to 3. And if we find a similar likeness no matter what numerical relation we choose — and if we realize that numerical relations can be made to have as much precision as we should like to have (though not infinite precision) — then we find ourselves inferring that there is a relation of the magnitudes themselves which merits the name of “proportion.”
But this is to say, in effect, that there are two ratios in the magnitudes, and they are the same! We have discovered a proportion; but we have not discovered it according to the order of being itself, but rather according to an order of intelligibility “quoad nos.” To discover it according to the order of being itself, it appears that we should have to have an intelligence which exceeds the human; for we should have to see, with our minds, the ratios themselves of a to b, and of c to d, so as then to see that they are the same.
But now, indeed, we may complete a final step, by realizing that this existence of ratios which we cannot exactly see is also evidence of the existence of a measure which we cannot exactly see; for it remains true that the relation called “ratio” must be founded on the still more fundamental relation of measure and measured. The determination of a magnitude which we are calling its measure will, evidently, have to be seen as being a “measure” by a kind of analogy (rather than by pure univocal identification) with the measure of a number. But we should remember that it is often through analogous predication, and not otherwise, that the most important insights are reached.
And from this, we may finally see, as from a higher vantage point, what it was that Descartes was really trying to do — beginning to do, in fact, but imperfectly. The possibility of “multiplying lines,” as Descartes does, really derives from the fact that lines have not only ratios, but that they have, more fundamentally, measures (at least in a relative sense). And as mathematicians gained experience and familiarity with the Cartesian method of multiplying and dividing lines, it seems to have become apparent, as if unconsciously, that this measure of a line was what really was at work in the Cartesian operations. The “number” of a line is then nothing but its measure. And so here, we may see a sort of reversal of the expected order of definition: for as ratio derives from measure, so ratio ends up, in this case, being turned back into its principle, since we are unable to apprehend the principle in any other way. By this we mean to say that the ratio of a magnitude to the unit now stands in for the “number” of that magnitude. Thus, for instance, √2 refers to the ratio which the hypotenuse has to the side of the square, but now conceived of as not merely a ratio, but as the measure of the hypotenuse from which that ratio derives.
From this we can perhaps begin to see that what is being aimed at is not merely a practical expedient; but it does seem to involve some art, in order to reach beyond what we are able to see by pure theorizing. Many important questions bearing on the character and meaning of mathematics may arise from this, as we study it further. For example, we have seen in both Viete and Descartes that algebraic symbols are now being used to facilitate mathematical calculation. Could we now perhaps see that they are not merely for practical calculation, but also for (among other things) artfully representing these “numbers” which we cannot see directly? In the literary arts, artfully contrived symbolic representations (taken in a wider sense) are commonly understood to enable us to see things which we might not otherwise be able to see easily or at all. Is mathematics then becoming a little more like the literary arts, which are a more familiar kind of contemplative art — and is this being facilitated through the symbolic representations of algebra? If so, how far will it be necessary or advisable to extend this sort of art? Is the connection between art and science then perhaps more intimate than we might have surmised at first? These and other important questions will arise as we continue our study.