0. The purpose of the present post is to take a first step in spelling out the “neo-Aristotelian,” as its tagline characterizes it, point of view motivating this blog. First, the “Aristotelian” of the “neo-Aristotelian” remains apt, despite the “neo.” It is Aristotelian in the fundamental sense that at least some of its central bases are identical with at least some of the bases of Aristotelian theory which are generally recognized as characteristically and specifically Aristotelian and which have been adhered to by the great mainstream of traditional Aristotelianism. These foundations include the so-called “First Principles” of non-contradiction and excluded middle to which I gave a bit of attention in my first post to this blog, the “Introduction” of March 12, 2013. They also include the Aristotelian theory of act and potency about which I have not yet written, though I plan to do so in the future.
But the “neo” of the “neo-Aristotelian” is also significant, in that it points to the fact of some significant differences between this blog’s point of view and any point of view which could properly be identified as Aristotelian tout court. There are two primary such differences. One lies in the philosophy of mathematics, the other in the theory of universals. The present post, as its title announces, is devoted to my neo-Aristotelian critique of the Aristotelian philosophy of mathematics. I have begun drafting a post presenting my neo-Aristotelian critique of the Aristotelian theory of universals and I expect to have it ready to appear in the relatively near future. Though I will not in fact here explicitly point to any connection, I am publishing the present post on the philosophy of mathematics with my on-going reflections on the philosophy of Alain Badiou very much in mind; the same will be true of the future post on the theory of universals.
1. I will begin with the neo-Aristotelian critique of the Aristotelian understanding of one part of the philosophy of mathematics, that of the theory of the numerical. Below, in Section 5, I will give some thought to another part, that of the theory of that which has magnitude. Taking up the first, then, first, my neo-Aristotelian perspective is in brief one which rejects the explicit physicalism of the narrowly Aristotelian philosophy of the numerical. It does because, as Alfred North Whitehead’s An Introduction to Mathematics so simply puts it:
The nature of things is perfectly indifferent, of all things it is true that two and two make four. 
If, then, there are non-physical realities, it is not true of physical things alone that two and two make four. With this in mind, I too would pose the rhetorical question raised by Frege, as he follows Locke and Leibniz in criticizing, in his The Foundations of Arithmetic, Mill’s naturalism.
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.” 
Yet Aristotle can be subjected to the same criticism as Mill, fort he too holds that the quantitative is as such necessarily physical or material. The thesis basing this physicalism, that matter is the principle of number, shows up in the argument in the Metaphysics that there is only one heaven.
Evidently there is but one heaven. For if there are many heavens as there are many men, the moving principles, of which each heaven will have one, will be one in form but in number many. But all things that are many in number have matter. (For one and the same formula applies to many things, e.g. the formula of man; but Socrates is one. But the primary essence has not matter, for it is fulfillment. So the unmovable first mover is one both in formula and in number; therefore also that which is moved always and continuously is one alone; therefore, there is one heaven alone. 
2. Now it is true that Thomas Aquinas very often expressed full agreement with Aristotle’s physicalism in the theory of the quantitative and therefore specifically of the numerical. This is evident, to illustrate, in his famous “deduction” of the Aristotelian categories of his Commentary on Aristotle’s Physics, in which Aquinas firmly associates quantity with 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)…. 
3. But at other times Aquinas abandoned, if not with absolute, at least with very nearly absolute explicitness, Aristotle’s physicalism, recognizing the “metaphysical” number of Locke, Leibniz, and Frege. This is evident in his discussion, in the Summa Theologiae of the number of angels. Answering the objection that angels, being incorporeal, cannot exist in number, Aquinas tells us that “number” in relation to angels corresponds with a “‘transcendental’ sense of plurality”:
In the angelic world number does not mean discrete quantity resulting from the division of a continuum; it is the result of a distinction of form from form, answering to the ‘transcendental’ sense of plurality, as explained earlier. 
The earlier explanation of the distinction between transcendental number and the number restricted to material beings develops the distinction between the two modes of “division” more completely and then uses the distinction to explain how there can be the three decidedly not merely physical persons he believed to constitute the Trinity:
…[B]ear in mind that all plurality is the consequence of some division. Now there are two kinds of division. One, material division, which comes about by division of a continuum; from this number results, which is a kind of quantity. Hence number in this sense exists only in material things which have quantity. The other is formal division, which comes about by the opposition or diversity of forms; from this results that kind of plurality which is in none of the categories but is one of the transcendentals, in the sense that being itself is diversified by the ‘one’ and the ‘many.’ And only this kind of ‘many’ applies to spiritual realities.
Our view is that numeral terms, insofar as they enter statements about God [as three persons], are not based on ‘number’ taken as a special kind of quantity: then they would be used of God only metaphorically just as are the other properties of bodies, such as width, length and so on. They are taken from the ‘many’ which is transcendental. 
Finally, in his Commentary on the Metaphysics, Aquinas expresses his clear recognition of the fact that simple division (or 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. 
Conjoining what Aquinas has said in the Summa with what he says in the Commentary on the Metaphysics, it follows that the division, or distinction, of one being from another is both necessary and sufficient for multitude and that “transcendental” number is therefore to be attributed to any multitude whatsoever; for any two “particular” physical or material beings, it is a “fact that this being is not that being,” just as for any two non-physical or immaterial beings, it is a “fact that this being is not that being.”
It further follows, though Aquinas himself perhaps did not draw this conclusion with absolute explicitness, that the materiality or immateriality of the beings constituting a multitude has absolutely no bearing on the number of the beings as such; with Whitehead, again, “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 beings in question are apples, and thus material, or angels or thoughts, and thus immaterial. Consequently, though physical existents are indeed numerous, a kind of number that is restricted to physical multitudes is at best a needless duplication of “transcendental” or “metaphysical” number and can simply be dropped.
4. In the first post of this blog, the “Introduction” of March 12, 2013, mentioned in this post’s first paragraph, I let the following definitions, of “2” and “3,” serve as indicators of the difference between the philosophical perspective motivating After Aristotle and that or those of Aristotle and the Aristotelian tradition:
(x)(y)(2xy ↔ ~Ixy)
(x)(y)(z)(3xyz ↔ (~Ixy & ~Iyz & ~I xz))
For any existent x and any existent y, x and y are two existents if and only if x and y are not identical.
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, y and z are not identical, and x and z are not identical.
Let me now state that I find it hard to believe that the Aquinas who asserted that “two particular beings are said to be divided by reason of the fact that this being is not that being” would have hesitated in his acceptance of the definition of “2” just given. I find it equally hard to believe that he would have hesitated in front of the definition of “3.”
Let me also note that the theses expressed in the two definitions just given are true whether or not there are non-physical existents.
5. As we turn from the theory of the numerical to the theory of that which has magnitude, I have to confess that anything like a full exposition of the theory which I have in mind will have to await a significant deepening of my understanding. I can, however, provide something of the beginning of such an exposition, in the form of a bringing to attention of some texts of Aquinas showing that just as he at times went beyond Aristotle’s physicalist conception of the numerical, so too he at times went beyond Aristotle’s physicalist conception of magnitude.
Let us, first, however, recall that Aristotle made both his physicalist conception of magnitude and a significant theological consequence absolutely explicit in a famous passage of the Metaphysics denying magnitude and infinite magnitude of the divine:
It is clear then from what has been said that there is a substance which is eternal and immovable and separate for sensible things. It has also been shown that this substance cannot have any magnitude but is without parts and indivisible. For it causes motion for an infinite time, but no finite thing has infinite potency. Since every magnitude is either infinite or finite, this substance cannot have a finite magnitude because of what we said, and it cannot be infinite in view of the fact that there exists no infinite magnitude at all. 
Now in his commentary on the passage, Aquinas faithfully followed Aristotle in his exposition. It is important, however, to note that his introduction very carefully says:
He [Aristotle] says that it has been proved in book VIII of the Physics [266a26-b6] that this kind of substance can have no magnitude but is without parts and indivisible. 
He, i.e., Aquinas, does not say:
It has been proved in book VIII of the Physics [266a26-b6] that this kind of substance can have no magnitude but is without parts and indivisible.
In other words, he reproduces both Aristotle’s denial of the infinite magnitude of the divine and the argument on behalf of the denial without either rejecting or accepting them as his own.
While, however, divine infinity is clearly out of place in Aristotle’s own theology, it is just as clearly in place in that of Aquinas. In Aquinas’s version of Aristotelian theory, the divine being must be “limitless [infinitus]” . It is, of course, true that Aquinas refused to consider bodily extension or magnitude as literally attributable to God. Thus in the Summa Theologiae’s discussion of divine simplicity, he tells us that “the Scriptures make use of bodily metaphors to convey truth about God and about spiritual things.” He continues:
In ascribing, therefore, three dimensions to God, they are using bodily extension to symbolize the extent of God’s power: depth, for example, symbolizes his power to know what is hidden; height, the loftiness of his power above all other things; length, the lasting quality of his existence; breadth, the universality of his love. 
But that a non-physical being cannot possibly have bodily extension, or physical quantity, does not, for Aquinas, entail that such a being cannot have magnitude. He does in fact explicitly attribute quantity to God, literally and without qualification, telling us not merely that God is an infinite being, and not merely that God is thus a quantitative being, but that the divine essence and the divine quantity are identical. He puts it as follows, speaking of the quantity which he thought that the three persons of the Trinity in which he believed had in common: “Quantitas … in divinis non est aliud quam earum essentia,” i.e., “… quantity in the divine persons is nothing other than their essence.” 
Aquinas answers the obvious, to one steeped in Aristotelian thought, objection to that thesis, an objection based in the claim that “in the divine persons there is no continuous quantity, whether intrinsic, i.e. magnitude, or extrinsic, i.e. place and time,” by distinguishing between two sorts of magnitude, the one a dimensional or extensive quantity belonging only to physical beings and the other a virtual or intensive quantity which can belong to non-physical beings:
Quantity is of two sorts. The first is a quantity of mass, dimensional quantity; possible only among bodily beings, it does not apply to the divine persons. But there is a second sort, a virtual quantity, the measure of which is superiority in having some nature or form. This is the sort of quantity intended in describing something as more or less hot from its possessing heat in greater or lesser intensity. The first measure of this sort of quantity is its root, i.e. the superior possession of a form or nature; according as it is referred to as spiritual greatness, even as heat is termed great because of its high degree of intensity. That usage explains Augustine’s stating that with things that are great but not in the sense of bulk, the greater is the better; for the better means the more excellent. 
6. One primary difference then, between the properly or narrowly Aristotelian theory of that which has magnitude and the neo-Aristotelian theory to which I adhere is that the latter is not declaredly physicalist. With one major caveat, then, I can affirm that my neo-Aristotelian theory of that which has magnitude is in the end the same as that of Aquinas, at least on that score. The caveat is this: the truth of the two theses, that not only the physical or material is subject to number and that not only the physical or material possesses magnitude, depends upon the truth of the additional thesis that there is the non-physical or non-material. In the yet short life of this blog, I have not even taken up the question of the existence of the non-physical, much less demonstrated, say, that there is a divine being.
 Alfred North Whitehead, Science and the Modern World (New York: The Macmillan Free Press, 1967), p. 29.
 Gottlob Frege, The Foundations of Arithmetic. Translated by J. L. Austin (2nd Revised Edition; Evanston, Illinois: Northwestern University Press, 1978), pp. 30 – 31.
 Aristotle, Metaphysics, in Jonathan Barnes, editor. The Complete Works of Aristotle. The Revised Oxford Translation (2 vols.; Princeton, New Jersey, and Chichester, West Sussex, United Kingdom: Princeton University Press, 1984), Vol. Two, pp. 1697-8, Bk XII, 1074a31-38).
 Thomas Aquinas, Commentary on Aristotle’s Physics. Translated by Richard. J. Blackwell, Richard J. Spath, and W. Edmund Thirlkel, introduction by Vernon J. Bourke, and foreword by Ralph McInerny (Notre Dame, Indiana; Dumb Ox Books, 1999), Book III, C. 322. This was originally published as Thomas Aquinas, Commentary on Aristotle’s Physics. Translated by Richard J. Blackwell, Richard J. Spath, and W. Edmund Thirkel (New Haven, Yale University Press, 1963).
 Thomas Aquinas, Summa Theologiae. Thomas Gilby, O. P., General Editor. (60 vols.; Cambridge, England: Blackfriars, 1964 – 1976), Vol. IX, Angels, Edited, translated, and introduced by Kenelm Foster, O. P., Ia, Q. L, 3 ad 1.
 Thomas Aquinas, Summa Theologiae. Thomas Gilby, O. P., General Editor. (60 vols.; Cambridge, England: Blackfriars, 1964 – 1976), Vol. VI, The Trinity, Edited, translated, and introduced by Ceslaus Velecky, O. P., Ia, Q. XXX, 3)
 Thomas Aquinas, Commentary on the Metaphysics. Translated and introduced by John P. Rowan, with a preface by Ralph McInerny. (Notre Dame, Indiana; Dumb Ox Books, 1995 ), Book XII, C. 2548. This was originally published as Thomas Aquinas, Commentary on the Metaphysics of Aristotle. Translated by John P. Rowan. (2 vols. Chicago, Henry Regnery Company, 1961).
 Aristotle, Metaphysics, Bk. XII, Ch. 7, 1075a5-11.
 Aquinas, Commentary on the Metaphysics, Book XII, C. 2548.
 Thomas Aquinas, Summa Theologiae. Thomas Gilby, O. P., General Editor. (60 vols.; Cambridge, England: Blackfriars, 1964 – 1976), Vol. II, Existence and Nature of God, Edited, translated, and introduced by Timothy McDermott, O. P., Ia, Q. VII, 1)
 Ibid., Ia, Q. III, 1 ad 1.
 Thomas Aquinas, Summa Theologiae. Thomas Gilby, O. P., General Editor. (60 vols.; Cambridge, England: Blackfriars, 1964 – 1976), Vol. VII, Father, Son, and Holy Ghost, Edited, translated, and introduced by Thomas C. O’Brien, Ia, Q. XLII, 1.
 Ibid., ad 1.