Jeffrey C. Kalb, Jr.
Thomism, Mathematics and Science


By detailing some of my interests and writings, I hope to hear from those whose interests bear some resemblance to my own. I wish to thank Dr. Arraj for his kindness in giving individuals such as myself a forum for our ideas.

My Road to Thomism

I was born July 7, 1966. I studied materials science and electrical engineering at Rensselaer Polytechnic Institute in Troy, NY. I graduated with a B.S. in materials science in 1988 and then received a M.S. in materials science and a M.S. in electrical engineering in 1989 while studying under a graduate fellowship from the National Science Foundation. Somewhat disappointed with the state of modern physics, increasingly hostile to its foundations, and lacking the patience of a good experimentalist, I elected to terminate my fellowship and take employment as an engineer in the semiconductor industry. In 2001 I enrolled at the University of Arizona, where I am pursuing an M.A. in classics with a philology concentration. As regards my work in philosophy I have no properly academic credentials, so my writing will have to speak for itself.

I began to study natural philosophy in response to an insight that both quantum and statistical mechanics are ultimately unverifiable in the empirical sense. For the last twelve years I have expanded my study, branching into many other areas, primarily within the triangle defined by metaphysics, mathematics, and natural philosophy. My mathematical considerations led me to conclude to the existence of form, which is implicitly denied by Cartesian mathematical analysis and its progeny. As I was at the same time rediscovering and deepening my Catholic faith, I was led to St. Thomas as the Church’s doctrinal norm. I was fortunate in having picked up a copy of Etienne Gilson’s "The Christian Philosophy of Saint Thomas Aquinas." It was difficult for me at first. The terms were opaque after the first reading: substance, accident, essence, quiddity, prime matter, substantial form, act-of-existence, etc. But I gradually absorbed this previously alien perspective, only to find that my own speculations shed some light on contemporary thomistic philosophy.

I consider myself to be part of a wider movement to restore the cultural and liturgical patrimony of the Catholic Church, especially the Tridentine Latin Rite. This extends to political reform to curb personal license in view of the true liberty described by Leo XIII, and includes a ressourcement of traditional mores, art, and architecture. This is primarily a movement of young intellectuals today, but it will make itself felt as these enter into positions of influence and authority.

My Interests

My thinking is influenced by Etienne Gilson, Jacob Klein, David Rapport Lachterman, St. Bonaventure, Jacques Maritain, St. Augustine, Plato, Blessed John Duns Scotus, Proclus Diadochus, Antonio Rosmini-Serbati, and Gregory Palamas. However, I try to judge all in the light of St. Thomas.

My interests include: The Trinity, the Incarnation and Redemption, the theory of universals, the analogy of being, points of debate between Orthodox and Roman Catholic theology, the liberal arts, the theory of color, the ontological and epistemological foundations of geometry, music, and arithmetic, and Marian theology.

Some Characteristic Views of Mine

The importance of mathematical study: Traditionally, the liberal arts were the portals to philosophy, and in the Theaetetus Plato goes so far as to call geometry a form of philosophy. It was the corruption of the mathematical arts that infected physics, and this in turn corrupted modern philosophy. St. Thomas spoke very little about mathematics, and it is an area that is ripe for development. One of my projects is to introduce a new Quadrivium: Arithmetic, Planimetry, Music, and the study of the Form of Corporeity, a Scotistic borrowing reconciled to Thomism. This would replace the mathesis universalis of Descartes, which poses as metaphysics, but is in truth a confused theory of music. There is much to be learned from modern mathematics. But there is also much that must be altered or rejected.

The order of mathematical intentionality recapitulates the order of corporeality. – The totality of mathematical forms is of itself sufficient to describe body as body. It is insufficient to study body as living body or body as sensible body, but in its own domain it is perfectly competent. It is a terrible mistake to effect an artificial separation between mathematical physics and natural philosophy. Mathematical physics is a part of natural philosophy; it is simply not the whole of it. If, as St. Thomas teaches, mathematicals have both an intelligible matter and form, then there is no reason to exclude mathematical physics from natural philosophy on the basis of its dealing with "mere quantity." (It might be claimed that only a distinction has been made, but in practice it has been a separation. Natural philosophy was redefined as the branch of metaphysics that defines motion generically, and the metaphysician was thereafter free to philosophize without an eye toward mathematical physics.) By distinguishing the various intentional forms proper to each branch of mathematics and ordering them into a whole, it is possible to constitute the intelligible analogue of body. Hence my claim at the head of this paragraph.

Occasional Causality – I take this to be the most general structure of efficient causality. The form is: "The coincidence of A and B is the occasion for C to exercise its efficiency in act D." Do not confuse this with Occasionalism, which treats the coincidence of two causal chains (i.e. chance) as though the act in one chain were in some way the cause of the coincident act in the other chain. What is being said here is that a coincidence of acts is the sine qua non of the exercise of an agent’s efficiency. Through this structure of causation, I believe it possible to demonstrate the necessity of a first cause in any chain of efficient causes. The traditional statement is that as every effect requires a cause, so there must be a first cause for the subsequent ones to be actuated and the final effect produced. But this is merely to state the conclusion, not to demonstrate it. A very special logic is required to demonstrate that there must be a first cause. Not every causal scheme admits such a demonstration. The simple "A is the efficient cause of B" does not. In order to demonstrate this, it must be shown that every infinite chain of causes implies the existence of a cause prior to them all, taken in toto. This can be done, but it requires both an extrinsic formal cause (efficient cause) and an extrinsic material cause (occasion). I call the first formal because, like the intrinsic formal cause, it is the principle of determination. And I call the second material because, like the intrinsic material cause, although necessary, it does not supply a determination. This scheme answers in a broad sense to the mediaeval distinction between essential causes and accidental causes.

Apprehension versus Intuition – It is sometimes said that we have an intuition of being (Maritain) or an innate intuition of the idea of being (Rosmini). I do not subscribe to these views. The agent intellect (intellectus agens) possesses an innate formality for apprehension prior to judgment, and this is the formality of being. It must be presupposed for the very possibility of a judgment reaching out to the act-of-existence (esse). The attributes that Rosmini attaches to the "idea of being" really apply to this formality of being. Apprehension, or laying hold of a thing, is different than intuition. Intuition often follows upon and perfects apprehension, as in the case of corporeal vision, but this does not apply to being according to our natural mode of knowing. As Saint Thomas says, knowledge begins in apprehension and concludes in judgment.

Transcendentals – There are many subsequent formalities that determine the intellectual apprehension of an object. Some of these, such as those of geometry, determine the apprehension to a particular form. Those that transcend the categories are called transcendentals. These include Unity, Goodness, Truth, and Thing (Res). Every judgment presupposes the innate formality of being. So we say: "It is one. It is good. It is true. It is a thing." Whereas the absolute judgment of existence says merely: "It is." One can see that the transcendental formalities add something to the innate formality of being, and that this is reflected in our language. There is a logic, in the wide sense of the word, corresponding to each transcendental formality, which serves to distinguish the several adjunct sciences.

Esse et ordo convertuntur. The transcendentals can be defined as notions that transcend the Aristotelian categories. The study of order holds a pivotal position in any doctrine of the transcendentals, particularly in the Trinity, where there is no distinction in the order of act and potency, but a real distinction of divine Persons founded on the order of origin. One may legitimately ask: How are esse, essentia, forma, suppositum, and potentia activa, convertible with ordo? Further, how does their convertibility with order determine their mutual relations and therefore a coherent metaphysics? I have developed a theory of order, a taxology of being, that in turn yields a taxonomy of being in remarkable conformity with the metaphysics of St. Thomas. I have found this approach more fruitful than the application of Aristotle’s Organon. By elaborating the Trinity in this manner, created being can in turn be elucidated and the causal structure of creation uncovered. The real relation of creature to God is found to be threefold, not as three relations to the three divine Persons, but as three different ways of relating to the unique divine Essence: a vestige of the Trinity in the creative act. More recently, I have found that this New Organon shows clearly the real foundation of ens commune – without overturning the analogy of being – and answers in detail Heidegger’s critique of onto-theo-logy from within a thomistic framework.

Completed Writings

I retain copyright over this material, but I extend complete license to reproduce and circulate it.

A Short Treatise on the Mathematical Principles of Harmony (22 pages) – This is a preliminary work addressed primarily to mathematicians and physicists. The central premise is that music has been ill defined. It is the science of the measurable. It thus stands mid-way between arithmetic (the science of the countable) and geometry (the science of the extended). I divide music into two parts, meter and harmony. Meter treats measurable quantity by means of number. Harmony treats measurable quantity through species of numbers (e.g. the odd). Therefore, I call the theory of harmony eidetic. The first part of the work investigates the foundations of harmony, deriving the unison, octave, perfect fifth and fourth, and major and minor thirds and sixths from infinite ensembles of natural numbers. An ordering principle holds such that the inclusion of one species within another yields harmonious ratios. It explains such interesting qualitative features as the "sameness" of tones separated by an octave, and the distinction between major and minor intervals. And it predicts new, as yet unexplored, intervals. The second part of the work traces the connections between music and fractal geometry, quantum mechanics, statistical mechanics, and the psychophysical function. A complete understanding requires knowledge of infinite series, least common multiples, and some basic limit theory of the sort used in differential calculus.

Rhetoric and the Language of Nature (6 pages) – This short essay investigates the consequences of the ubiquitous modern expression, "The book of nature is written in the language of mathematics." If mathematics does not concern form, but rather language, if it belongs not to the Quadrivium, but to the Trivium, what are the implications for subsequent philosophy? The essay traces in outline those consequences.

Writings in Progress

Mathesis, Self-Knowledge and Apodictic Certainty – This work will examine the views of both ancient and modern philosophers about the origin and mode of universals, using planimetry as a test case. It will propose a new kind of abstraction that can account for the necessity intrinsic to mathematical reasoning, overcoming the critique of induction. And it will show how mathematics, studied properly, leads to self-knowledge. It will also defend the epistemic credentials of Euclid’s fifth postulate in view of this new manner of abstraction. Finally, it makes a preliminary enquiry into the role of sentiment as a mediator between will and appetite.

An Enquiry into Metaphysics – The work will begin by investigating the analogy of being in the light of being’s convertibility with order. It will examine the real foundation for analogy and show in what limited sense analogy admits a kind of univocity. It will examine the role of the supposit (hypostasis) in creation metaphysics, and the nature of communicativity and receptivity. Using angels and material being as test cases, it will use the thomistic metaphysics of participation to elucidate and confirm the "uncreated energy" of God as posited by the Greek Fathers and Orthodox theologians, taking Gregory Palamas as the foremost proponent of this doctrine. It will examine the manner in which prime matter shares in existence through substantial form. Finally, it will investigate divine grace and the beatific vision in relation to these previous topics.


For copies of manuscripts or to express comments, I can be reached by email at: