Introduction1

There is a growing body of literature detailing the

important role played by the so-called subalternate sciences

in the formation of the modern notion of mathematical

physics.2 Such investigations are crucial for fleshing out

the facts of continuity and discontinuity between ancient /

medieval thought and modern thought. Certainly,

postmodernity has done much to challenge the naïve

mathematical ontology of early modernity, these developments

are not quite so quick to come to common consciousness,

which even after the disruptions of early 20th century

physics is still able to breathe within a mathematicized

ontology—even if it is one that is far less imbued with the

naïve metaphysical claims of classical mechanics.

As a Thomist, I have used the expression “subalternate

sciences” with jejune bravado for some time. Having

1 Note to reader: This paper is something of a draft form, functioning as an outline for my presentation in person. This fact will be noted inplaces where citations are a bit thin.2 For just a mere sample, see the bibliography of recent dissertation byZvi Biener, “The Unity of Science in Early-Modern Philosophy: Subalternation, Metaphysics and the Geometrical Manner in Scholasticism,Galileo and Descartes,” (PhD Diss., University of Pittsburgh, 2008), 152-171.

1

assimilated nearly the entire corpus of Jacques Maritain, I

thought that these sciences—formally mathematical,

materially physical—were relatively easy things to explain.

I would trot out a few notes about astronomy and optics,

note that they were not merely mathematical, and—voila—you

had a new formal object. Little else seemed necessary, and

I was able to spew arrogant verbiage concerning the

relationship of sciences.

Along these lines, allow me to recall a shattering (and

justly due) critique of my verbalism made by Msgr.

Sokolowski during my first year of studies. The critique

ran—in my own light-hearted paraphrase: “Subalternation: You

keep using that word. I do not think it means what you think

it means.” After licking my wounds, I realized that I

really had never brought to my own intuitive awareness what

all is involved in the problem of subalternation. I had a

superficial grasp of the sources in my own tradition but had

never thought of it as a true problem or question.

Today, I want to present this as a problem—not as an

historical phenomenon, nor as a complete theoretical

2

package. I want to travel side-by-side with you down this

road for a while. I will be using my own vocabulary—the

hoary language of scholasticism. However, I promise that I

will make distinctions as we go. In the spirit of Msgr.

Sokolowski’s own dicta: “I don’t plan to stare long and hard

to find a disembodied essence; instead, I hope to make a few

distinctions.” I hope to do that today along with you.

Though clearly following a preset path, my aim is to bring

to our intuitive awareness the problem posed by the

subalternate sciences. As we scholastics say, I will answer

the question an sit—are there subalternate sciences? In so

doing, I hope to give an initial grasp of the all-important

question quid sit—what is the nature of such sciences?

I will take as our banner words that recently struck me

—independently of our conference—while reading Msgr.

Sokolowski’s Presence and Absence:

To name constructs in science is even more complicated in its manipulation of absence. Such scientific referents, though not perceivable in themselves, are to be related to observable objects and events; they enter indirectly into ourcontext of experience and speech. . . . Constructs are usually acknowledged to be unlike

3

those things with which we are familiar, and yet they have effects we can register. There is a curious mixture of the real and the constructed, an unusual combination of both presences and absences, in the establishment of such objects of reference.3

Though I will be progressing as a Thomist, I hope to be

speaking with a fledgling philosopher’s voice—one that does

homage to a truth that we can share together.

Subalternation—Basic Remarks

To begin our reflections, we must choose an appropriate

point of departure. For our talk, I am going to take the

example of classical mechanics. I do this for historical

reasons but likewise so as not to ape at being something

more than I am. Acknowledging the limits of my grasp of

20th and 21st century physics, I would like to start with an

area that has resonance with all of us who have had some

high school physics. The world in which we begin is one

that is “of the middle dimensions”—one that is quite

amenable to representation in imagination. Additionally,

this choice of classical mechanics is historically

3 Robert Sokolowski, Presence and Absence: A Philosophical Investigation of Language and Being (Bloomington, IN: Indiana University Press, 1978), 39-40.

4

appropriate, for the scientiae mediae exercised their most

forceful claims precisely in this form in early modernity.

Finally, such a starting point appropriately begins with

what is more knowable to us.

Consider something as simple as the equation for

determining the distance traveled by an object over a time

interval under the influence of an accelerative force. It

is nothing profound to note that this provides a

mathematical representation of visible and tangible

phenomena. Already in the background—and we must be aware

of this “static”—there is a massive issue playing upon our

conceptual equipment, providing two different roads: (1) an

analysis based upon primary and secondary qualities (or some

related version thereof); (2) an analysis based upon proper

and common sensibles. Indeed, it could well be argued that

this is a primary theme in the clash of Aristotelian natural

philosophy and early modern natural philosophy.4 These

4 Likewise, Deely has perspicuous remarks regarding the modern-nominalist understanding (or misunderstanding) of the category of relation as being indifferent to real or intentional existence. Still, regarding sensation, see John Deely, Descartes & Poinsot: The Crossroad of Signs and Ideas (Scranton, PA: University of Scranton, 2008), 67-102.

5

matters will become indirectly important—but that path would

require quite broader considerations.

Instead, we will consider two scientists: (1) the

mathematician and (2) the young physics student, beginning

with the former. We provide her with the equation, giving

her values for a and t. Although most of the reasoning

functions through various acquired / developed mathematical

methods, we should note immediately that the middle terms of

such reasoning are all purely formal. Every operation—even

though concatenated in form—progresses according to the

formal notions of the numbers involved. One-half is multiplied

by a, providing a formal equivalence; t is squared; and so

forth. In all cases, the reasoning is from one numeric form

to another.5 We then provide her with an equation for

computing a based upon to variables: t0 and t1. The reasoning

will proceed much akin like before. For now, I will merely

emphasize that all of the middle terms pertain to formal

causality. There is no question of efficiency or finality

that is involved in the reasoning concerning the equations.

5 Hence, the old Aristotelian dictum that reasoning in mathematics is always according formal causality alone.

6

Now, consider the physics student who has two

synchronized clocks, both of which are attached to light

detectors checking the time of descent of a generally

smooth, relatively aerodynamic tennis ball from the roof of

her high school. She is armed with the equation, but her

teacher has presented it as an hypothesis to be tested.

Presume that the first five attempts proceed exactly as

expected by the equation (within a reasonable range of

error). On a very simplistic reading, the student could

well think that she is doing nothing more than registering

the mathematical points applicable to calculating t and

noticing that they are exactly the same as expected based

upon the value provided for g. She blames her equipment for

the minor errors.

However, a closer inspection should begin to bring to

light “wrinkles in the mathematics.” First, there is the

fact of measurement—this is quotidian but let us not forget it.

There is some transition that occurs between the “proper

sensibles” and the mathematical quantities involved—though

the mechanics of this transition is effected by the sensors.

7

The first obvious transition is the numbering of motion and

explicit referencing of that numbering against a

quantitative standard. The data of local motion is

“bleached” into quantity—but this “bleached value” is not

merely a numeric concept. It is a number that represents

some non-mathematical set of events for measuring the time.

Thus, our first observation: We are not dealing with pure

quantities but instead are dealing with a description that

has some conceptual content that is external to mathematics:

i.e. quantity as related to the measurement of a body’s

motion. What we appear to have is some sort of

“mathematical reading of the sensible.” 6 The concepts that

we are already explicating—in virtue of their reference

outside of mathematics—are not strictly equivalent to those by

the mathematician in pure mathematical exercise. They

retain a predominately mathematical formality but there are

elements involved that are clearly non-mathematical: notions

of causality, time, motion, and so forth. These are

mathematical models—but models clearly referred to things

6 See Jacques Maritain, The Philosophy of Nature, trans. Imelda C. Byrne (NewYork: Philosophical Library, 1951), 27.

8

that are non-mathematical. Perhaps this is an anodyne

observation, but please just note it with me.

On the sixth attempt, several imperfections on the

surface of the ball happen to cause enough drag to cause

noticeable deviation in the outcome. Perplexed by this

occurrence, the student considers the equation and is faced

with a crucial question: “What is the arbiter of truth of

the matter—the equation or the gathered data?” Even if she

decides that the error is to be blamed on the experiment and

experimenter, still, she must decide what plays the role of

final arbiter in judgment: the purely mathematical reasoning

or the quantitative measuring of a non-mathematical reality.

This scenario is fabricated and simplified, but

consider it in contrast with a simple

“applied” mathematical question like: “A cat is five pounds.

It doubles in weight. What is its new weight?” Although

measurement is presupposed here, we do not say, “Ten is

correct because that is what will be measured.” We merely

consider the quantified substance involved—only in terms of

quantified substance. The mind “launches” into the

9

quantitative and can remain there to determine the truth of

the question.

For our young physicist, a two-fold motion occurs in

her reasoning. From the qualitative order, quantitative

measurements are taken. Using this data, a mathematical

model is created or utilized. However, to determine the

truth of the model, the mind cannot remain purely within the

mathematical realm. The final arbiter of truth is not

quantified substance—i.e. purely mathematical concepts.

Instead, the arbiter is: The motion in reality, considered

as being quantifiable.

The experiment can be expressed, reasoned through, and

even altered in a wholly mathematical manner—the reasoning

formally is mathematical for a great deal of the construction and

verification. However, the truth value of the mathematical

symbolization—e.g. “What is the time for this projectile to

fall?”—relies not merely upon the canons of mathematical

reasoning. It likewise is judged by reference to causality

(efficient and final) not captured in the pure formality of

mathematics. Thus, our second observation: The terminus of

10

reasoning does not seem here to be the same as that found in

the mathematician’s reasoning. However, what is the nature

of the difference—to the degree that it does in fact differ?

Abstraction

We are faced with an aporia that is important to

understanding Maritain’s work on this topic. We have

considered two types of inquiry—the mathematician and the

classical mechanist. The most important question requiring

an answer is: “Are these two modes of knowing reducible to

each other?” This question seems simplistic, but we must

beware. Greater minds than ours have been perplexed about

these matters. Likewise, as Fr. Lawrence Dewan reminds us—

the Presocratics are always among (and in) us.7 Let us also

be sure that the atomists—both ancient and modern—are among

us as well.

7 See Lawrence Dewan, St. Thomas and Form as Something Divine in Things (Milwaukee:Marquette University, 2007), 29: “It is my contention that there is sucha thing as ‘perennial presocratism.’ The presocratics, like the poor, wehave always with us. And they are to be found among those who are very successful scientists (as was the case with the ancient Greek Presocratics; thus Thales was famous for his prediction of the solar eclipse of 585 B.C.). The ancient scientists were also metaphysicians, making judgments about the being of things…”

11

At this point, I must make a stand, particularly to

make sense of the tradition out of which I write. As is

likely well known by many in the room, the traditional

Thomistic explication of speculative knowledge traces out

three fundamental domains: (1) the natural, (2) the

mathematical, (3) and the metaphysical. There is a complex

history to this division, as well as its relationship to

Aristotle’s own dicta—indeed, the history is very complex

and beyond the scope of our talk.

There really are two things to be considered in this

approach to the matter. First, there is the relationship

between universal and particular intentions. The classic

Thomist discussion of “abstraction” has always included this

element—not without the risk of nominalist errors. The idea

of the “first degree of abstraction” is said to capture what

it means to consider an actuality in separation from its

individuating designated matter. That which makes a given

set of individuals to be a human being (form) is not the

same as that which makes them to be individuals (designated

matter). There is an actuality that can be considered “in

12

abstraction” from individuating characteristics but not at

all from the notions of materiality in general.8

The classic example is the consideration of “bone” from

“this bone”9—one duly noted as being deficient. When we

talk about “what something is,” we invariably discuss what

it does—or what is its specific act character. Specific

determination is a matter of finding the particular nexus of

acts that help to provide “proper accidents”—as we

scholastics are wont to say. The search for such complexes

of actualities is quite difficult both to find and to

8 Very well thought-out (though characteristically semiotic) treatment of abstraction can be found in John Deely, Intentionality and Semiotics: A Story of Mutual Fecundation (Scranton, PA: University of Scranton, 2007), 81-114. However, I would likewise recommend to the reader the very accessible essay Yves Simon, “Nature and the Process of Mathematical Abstraction” in Philosopher at Work, ed. Anthony O. Simon (Lanham, MD: Rowman & Littlefield Publishers, 1999), 113-133. Though the essay as a whole is applicable to our topic in a number of ways, I turn the reader’s attention here to the very brief treatment of Cajetan’s distinction between formal and total abstraction. The theme comes up on a number ofoccasions through Maritain’s corpus and is important for understanding abstraction within the “longer narrative” of Thomistic logic. Although the issue is raised in a very different vocabulary and a very different context, it is likely helpful for the non-Thomist to read the 2nd logical investigation in Edmund Husserl, Logical Investigations, ed. Dermot Moran, trans. J. N. Findlay (Vol. 1. London: Routledge, 2008), 237-313. For those of an analytical extraction, I can think of no more valiant attempt at this issue than the recent work done by Michael Thompson in Life and Action. See Michael Thompson, Life and Action: Elementary Structures of Practice and Practical Thought (Cambridge, MA: Harvard University Press, 2008),1-82.9 E.g. See Aquinas, De Trinitate, q.5 a.2,3.

13

describe. There is an interplay between discussion of

“bobcats” and “these bobcats,” but the types of intentions

proper to each are different. Indeed, even here, there is

interplay between the universal and the selected

particulars.10 However, we must set aside such weighty

matters for today.

The very, very basic question at the level of “nature”

is: What are the principles for the type of motion or motions

in question? All four classical causes are considered: The

matter in which change occurs; the nature of “what” it is

becoming; the agent that is making it to become such; and

that “what” considered as the explanation for the motion

(i.e. qua final cause).

10 There really is a massive question here about the very mode of defining that is proper to considering things essentially as opposed to considering them in terms of their sensible qualities. In a topic related to the overall question of subalternate sciences but separate invery important ways, Maritain considered this under the rubric of “empiriological” concepts. Like so much in Maritain, it is easier to present his thought through the prism of his much more pedagogically able student Yves Simon. I wholly encourage the reading of Maritain, but to do so properly, you have to “sign up” for the wild ride of his narrative as well as the grand, flowing undercurrents of traditional Thomism without which his very vocabulary becomes easily misinterpreted.(I speak from personal experience.) See Yves R. Simon, “Maritain’s Philosophy of the Sciences,” The Thomist 5, no. 1 (1943), 85–102.

14

Mathematics, does not consider the fourfold structure

of causes. There is never a case where one mathematical

object “becomes” another. One triangle is different from

another, and while each might share properties, there is no

becoming in realm of mathematical abstraction. The “logic”—

I mean the term broadly—of mathematical relations is beyond

my pay scale, but given the appropriate relational schemes,

one species is deduced from another by means of formal

causality alone. We do not explain motion in mathematics;

we solve equations—the relations between specific

intelligibilities.

Three Modes of Subalternation

Now, what are we to make of the mathematician, the

“natural philosopher,” and our young classical mechanist?

The “natural philosopher”—taken in the simplest sense—

considers the general notions of becoming. The basic

question asked in the Physics is “what is mobile being?” All

four causes can be considered: “(1) In what does this motion

occur? To what end—(2) as end, (3) as form? (4) What is

15

causing it?” The mathematician is concerned with ideal (or

semi-ideal11) relations that hold between quantified

substances. The truth of such judgments is determined by

relations of formal causality, “Can these species of

quantity be related in that sort of manner?” The young

classical mechanist seems to be somewhere in between—neither

purely mathematical, nor purely physical. Both canons seem

to be involved. The science clearly depends upon

mathematics, but it neither defines its terms nor offers

complete explanations in exactly the same manner as does

mathematical reasoning.

It is here (along with questions of theology,

actually), that the notion of subalternation arises in the

Cursus Philosophicus of John of St. Thomas. The locus classicus is

q.26, a.2 of his treatment of material (posterioristic)

11 This is a fascinating question merely for the Thomist, let alone in the general context of the philosophy of mathematics—which is above my pay grade at this time. There is an elaborate relationship between thought and reality in mathematics, and one must be very careful not to simplify matters. Regarding the Thomistic position, an overview can be found in the article by Simon cited above in note 8. Likewise, see Armand Maurer, “A Neglected Thomistic Text on the Foundation of Mathematics,” Mediaeval Studies 21, no. 1 (1956): 185–192. Armand Maurer, “Thomists and Thomas Aquinas on the Foundation of Mathematics,” The Review of Metaphysics 47, no. 1 (1993), 43–61.

16

logic.12 In the text, he clearly is taking part in a

longer-reaching conversation of Iberian scholasticism;

however, the relationship between physics, mathematics, and

metaphysics had been greatly discussed and debated from the

time that the Posterior Analytics and the Physics had arrived in

the West.13 In this selection, he takes over a threefold

division of subalternation. It might be summarized as

follows:

1. Subalternation according to end: For this, the examplesremain in the practical order, such as the virtue of temperance being governed by religion (as a species of justice) or military science by politics. I cannot comment at length on this here, but this type of relationship is analogous to the relationship between (e.g.) natural philosophy and metaphysics or even, metaphysics and supernatural theology.

2. Subalternation on account of principles: Here, it is a question of a science that cannot come to self-evident principles by which it can constitute its demonstrations. There is a distinction here between the case of simpliciter and secundum quid subalternation, but we cannot be detained with his full treatment. Thepoint is that such subalternation, when taken in its

12 The text is available in English as well. See John of St. Thomas, TheMaterial Logic of John of St. Thomas, trans. Yves R. Simon, John J. Glanville, and G. Donald Hollenhorst (Chicago: University of Chicago, 1965). For an in-depth treatment of this question in a fuller manner, see Bernard Mullahy, “Subalternation and Mathematical Physics,” Laval théologique et philosophique 2, no. 2 (1946): 89–107. 13 For an accessible overview of the earlier period of this history, seeJames A. Weisheipl, “Classification of the Sciences in Medieval Though,”Mediaeval Studies 27, no. 1 (1965): 54–90.

17

strongest (simpliciter) sense, implies that a science cannot provide the totality of its principles of demonstration. For our purposes: Is classical mechanics merely a case of natural philosophy borrowingfrom mathematics to explain its own proper domain of “general concepts of change”? The Aristotelian philosopher answers that this is certainly not needed simpliciter by natural philosophy. Nature, matter, form, privation: These are dialectically proven self-evident principles that can be used to explain change qua mobile (not qua mobile as measurable).

3. Subalternation on account of the subject: This is the mysterious third domain, the one that John of St. Thomas states is subalternation in the most proper sense. Here, the superior science is not considered initself—it is not a matter of mathematics in a pure sense. Instead, the subject of the science is a composite of two formally different aspects of considering reality. His examples are typical of the period: optics is the science that adds “visual” to themathematical concept of “line,” and harmony adds “sonorous” to the mathematical concept of “number.” The accidental character must be the principle of scientific truths, thus disqualifying—in a homely image—the “hot or cold” line. Note that he here passes overany discussion of the criteria for determining such proper differentia but presents the cases of optics andmusic as non-problematic examples.

Within traditional Thomism, what generally occurred was

an acknowledgment of such sciences, but they always retained

something of an “undifferentiated” status. The staunch

defense of the “degrees of abstraction” is somewhat

stretched by such intermediary formal objects—really neither

18

mathematical nor physical in their “abstraction.” In II

Physics, Aristotle had referred to these as the more physical

of the mathematical sciences. Somewhat skewing the sense of

the text, Thomas referred to them as more physical than they

were mathematical.14 From the viewpoint of Aristotle’s

dictum, we must say that a “mathematical science that is

physical” is ultimately “mathematical” in some very

different way than purely mathematical science. The same

goes in the inverse for the Thomists as well: a “physical

science with mathematical characteristics” is not exactly

the same as natural philosophy purely considered.

The natural philosopher speaks of “motion”—and does so

completely within his conceptual sphere. It is a very

simple sphere indeed, but one that does not intrinsically

require a mathematical vocabulary to explain motion.

However, as soon as we begin to consider “mobility qua

14 Aquinas, In II Phys., n.164: “Huiusmodi autem scientiae, licet sint mediae inter scientiam naturalem et mathematicam, tamen dicuntur hic a philosopho esse magis naturales quam mathematicae, quia unumquodque denominatur et speciem habet a termino: unde, quia harum scientiarum consideratio terminatur ad materiam naturalem, licet per principia mathematica procedant, magis sunt naturales quam mathematicae” (emphasisadded).

19

geometrically measurable,” the concepts involved cross

between pure geometry and natural philosophy.

Such a science is neither natural philosophy, nor

mathematics—but there is an order and a relationship

involved. Clearly, quantity plays its role in the

determining of qualitative change in the material order—and

no Thomist should forget that. The mathematical reasoning

of (e.g.) optics or classical mechanics is not merely a

“saving of appearances.” However, the explanations are

neither of motion nor of geometry; they are explanations of

motion as intelligible through measurability. (Or, to view the matter

from the point of view of the mathematical order: Quantity

as revelatory of the material conditions corresponding to

the qualitative changes formally operative in a given

material being.)

The “sub-” prefix of “subalternation” pertains to the

necessarily and formally mathematical character of the reasoning

of these sciences. They would have no proper reasoning—they

would not exist at all—if they did not have mathematical

principles. However, they do not look to “apply” mathematics

20

to reality in a purely extrinsic manner, either. Their aim

is to explain certain physical / mobile intelligibilities—

e.g. the nature of light, of force applied to bodies, of the

distinction of colors, etc—through mathematical models. I

emphasize “through,” for it implies that they must pass

through such models to end up somewhere—namely, in a

terminal sense judgment that dictates the truth of the

model. It is very easy to forget that this is necessary,

for we often apply mathematics to reality directly:

counting, numbering, etc. However, these instances all

reflect considerations of quantity as such. Thus, the

truth-value is mathematical—i.e. in terms of quantified

substance as such.

The case of the mixed sciences is different. The truth-

value is dictated by something extra-mathematical. The

observation of effects confirming theories concerning the

Higgs particle are a striking example of this kind of

reasoning—extraordinarily mathematical but never accepted as

true until observed. It provides a mathematical exegesis of

mutable reality, but as an exegesis, it is only true as

21

conforming to that reality. No amount of purely mathematical

certainty can create an ounce of demonstrative certainty in

this science.

The temptation, of course, will be to view this

mathematical reasoning as being a true cause of the

realities in question. The fact that the mind moves more

geometrico in much of its reasoning might lead one to believe

that it is only a branch of geometry. This was particularly

the temptation of the (e.g.) Newtonian world-view of

physical reality, likely brought to existence, in part, by

certain views on the subalternate sciences that did not

acknowledge the implications of their mediate status. The

historians of thought can provide you with much in that

regard. For my part, I hope that I have provided an

entryway into some of the terminology of my particular

tradition and now open the floor for questions that can

perhaps address more specific issues.

22