1. In the first lecture of his Commentary on Aristotle’s Physics* and in the course of an effort at distinguishing the science of physics from those of mathematics and metaphysics, Thomas Aquinas presents the well-known (at least within Aristotelian circles) distinction between the snub and the curved (Comment 2):

For the snub exists in sensible matter, and it is necessary that sensible matter fall in its definition, for the snub is a curved nose. And the same is true of all natural things, such as man and stone. But sensible matter does not fall in the definition of the curved, even though the curved cannot exist except in sensible matter. And this is true of all the mathematicals, such as numbers, magnitudes, and figures.

Again, he tells us shortly thereafter (Ibid., C: 3):

Now metaphysics deals with things of this latter sort [i.e., (Ibid., C: 2) “things which do not depend upon matter either according to their existence or according to their definitions”]. Whereas mathematics deals with those things which depend upon sensible matter for their existence but not for their definitions. And natural science, which is called physics, deals with those things which depend upon matter not only for their existence, but also for their definition.

Aquinas here has given expression to the traditional Aristotelian view that the objects of mathematics are natural or physical beings; this is the view which is appropriately known as physicalism in the philosophy of mathematics. And it is not only those who, like Thomists, can be classified as traditional Aristotelians, who find that understanding to be that of Aristotle. This we can read in Jonathan Lear’s Aristotle: the desire to understand** (pp. 231-232) that:

Mathematics, Aristotle argues, is directly about the changing objects of the natural world. There is no separate realm of numbers and geometrical objects. We shall see how he thought this was possible, but we ought to see immediately what an ingenious strategy it is: for it allows Aristotle simply to bypass the two problems which plagued Plato. There is no need for special mental access to the mathematical realm, for there is no special mathematical realm. There is just the realm of nature. Nor is there a special problem of how mathematics is applicable to the physical world: it is only if mathematics is about a removed realm that special problems of applicability arise. If mathematics is directly about the physical world, then of course it is applicable.

2. Now, to hold that mathematics, generically, “is directly about the changing objects of the natural world” or “deals with those things which depend upon sensible matter for their existence” is to hold, specifically, that both arithmetic, the science of the numerical, and geometry, the science of the, well, geometrical, are “directly about the changing objects of the natural world” or “[deal] with those things which depend upon sensible matter for their existence.” It is, however, necessary to distinguish the objects of arithmetic from those of geometry. That is, while the latter are, it seems to me, essentially related to physical objects in some complicated way which I am not yet fully ready to discuss, the former simply are not: the numerical need not restricted to the physical if beings are not exclusively physical.

This is hardly a new thesis. Thus we find Frege, in his The Foundations of Arithmetic*** (pp. 30-31), following Locke and Leibniz in going beyond John Stuart Mill’s physicalism:

Mill maintains that the truth that whatever is made up of parts is made up of parts of those parts holds good for natural phenomena of every sort, since all admit of being numbered.  But cannot still more than this be numbered?  Locke says: “Number applies itself to men, angels, actions, thoughts – everything that either doth exist or can be imagined.”  Leibniz rejects the view of the schoolmen that number is not applicable to immaterial things, and calls number a sort of immaterial figure which results from the union of things of any sort whatsoever, for example, of God, an angel, a man and a notion, which together are four.  For which reason he holds that number is of supreme universality and belongs to metaphysics.  In another passage he says: “Some things cannot be weighed, as having no force and power: some things cannot be measured, by reason of having no parts; but there is nothing which cannot be numbered.  Thus number is, as it were, a kind of metaphysical figure”.

Alfred North Whitehead’s An Introduction to Mathematics# (p. 2) puts it more simply: “The nature of things is perfectly indifferent, of all things it is true that two and two make four.” It makes no difference whatsoever whether the things in question are rocks or apples, and thus material, or angels or thoughts, and thus immaterial.

3. Now to Thomas: It is true that he very often expressed full agreement with Aristotle’s physicalism, as we saw above. There is, for but one more example, the case of his famous “deduction” of the Aristotelian categories in Book III (C. 322) of the Commentary on Aristotle’s Physics, in which he firmly bases quantity in matter.

Another mode [of predicating] is that in which that which is not of the essence of a thing, but which inheres in it, is predicated of a thing. This is found either on the part of the matter of the subject, and thus is the predicament of quantity (for quantity properly follows upon matter – thus Plato also held the great to be on the part of matter)….

But it is equally true that Thomas also on occasion, critical occasion, expresses a very different thesis. First, because he, unlike Aristotle, believed in a triune God and three decidedly not merely physical Persons of the Trinity, Thomas had to attribute plurality and number of God. In the Summa theologiae## he offers an explanation (Ia, Q. 30, art. 3) of how that can be (in my translation):

Now we say that numerical terms, insofar as they are predicated of the divine, are not derived from the number which is a species of quantity: since they would thus not be said of God unless metaphorically, just as would be the other properties of bodies, such as width, length and the like. They are derived from the multitude which is transcendental.

He tells us (Ibid.) more fully that:

…It is to be considered that all plurality is the consequence of some division. Now division is twofold. One, material, is the division of a continuum; from this results the number which is a species of quantity. Hence number in this sense does not exist unless in material things which have quantity. The other is formal division, which is brought about by opposite or diverse forms; from this results that multitude which is not in any of the categories but is one of the transcendentals, being is divided by the one and the many. And this kind of multitude alone happens to be in immaterial things.

God is not, for Thomas, the only immaterial reality of which transcendental plurality and number can be predicated. In the Summa Theologiae’s discussion (Ia, Q. 50, 3 ad 1) of the multitudes of angels Thomas answers the objection that angels, being incorporeal, cannot exist in number by saying that the number or plurality appropriate to angels is, again, transcendental plurality and number:

…It is to be said that in angels number is not that of discrete quantity, caused by the division of a continuum; but is caused by a distinction of forms, insofar as multitude is one of the transcendentals….

Finally, Thomas sees that immaterial things cannot be the only realities of which transcendental multitude and number can be attributed. He tells us in his Commentary on the Metaphysics### (Bk. X, L. 4, c. 1997), that the division caused by contradiction, or simple distinction, as I would prefer to put it, such that one reality is not identical to another, is the foundation of transcendental multitude.

Now the division which is implied in the notion of that kind of unity which is interchangeable with being [and thus in that of the corresponding multiplicity] is not the division of continuous quantity, which is understood prior to that kind of unity which is the basis of number, but is the division which is caused by contradiction, inasmuch as two particular beings are said to be divided by reason of the fact that this being is not that being.

4. a. The first of three closing thoughts: Aristotle himself, insofar as he was committed to the existence of a plurality of non-physical beings, should have recognized the metaphysical or transcendental number Thomas recognized.

4. b. The second of the three closing thoughts: the metaphysical or transcendental numbers which Thomas has introduced into the Aristotelian perspective are readily and quite tidily defined in the contemporary idiom; thus “two”:

For any existent x and any existent y, x and y are two existents if and only if x and y are not identical.

Thus “three”:

For any existent x, any existent y, and any existent z, x, y, and z are three existents if and only if x and y are not identical, x and z are not identical, and y and z are not identical.

And so forth, though the formulations do rapidly become quite lengthy.

4. c. The third of the three closing thoughts: I noted above that the objects of geometry are essentially related to physical objects in some complicated way which I am not yet fully ready to discuss. But there is more, for magnitudes are not thus essentially related to physical objects, if indeed there are non-physical objects, such as Thomas’s God and angels or the minds of many a contemporary thinker, and such non-physical objects to have magnitude. The non-physical, or transcendental, magnitude of non-physical objects is another topic which I am not yet fully ready to discuss but which very much bears being mentioned.

Until next time.

* Thomas Aquinas, Commentary on Aristotle’s Physics, translated by Richard J. Blackwell, Richard J. Spath, and W. Edmund Thirlkel, introduced by Vernon J. Bourke, and with a foreword by Ralph McInerny. (Notre Dame, Indiana: Dumb Ox Press, 1999), originally published by Yale University Press, 1963.

** Jonathan Lear, Aristotle: the desire to understand (Cambridge and New York: Cambridge University Press, 1988).

*** Gottlob Frege, The Foundations of Arithmetic. A Logico-Mathematical Enquiry into the Concept of Number. Translated by J. L. Austin. (2nd revised edition; Evanston, Illinois: Northwestern University Press, 1978).

# Alfred North Whitehead, An Introduction to Mathematics (New York: The Macmillan Free Press, 1967).

## I have used the Latin text of Thomas Aquinas, Summa theologiae. Thomas Gilby, O.P., general editor (60 vols; Cambridge, England: Blackfriars, 1964-1976), Vol. VI, 1a. 27-32, The Trinity, edited, translated, introduced, and with notes by Ceslaus Velecky, O.P., and Vol. IX, 1a. 50-64, Angels, edited, translated, introduced, and with notes by Kenelm Foster, O.P.

### Thomas Aquinas, Commentary on Aristotle’s Metaphysics, translated by John P. Rowan and with a preface by Ralph McInerny. (Notre Dame, Indiana: Dumb Ox Press, 1995), originally published by Henry Regnery company, 1961.