Thomistic Philosophy (vols. 1-4)

Description

When still in print, Henri Grenier's manual of Thomism was widely used in seminaries and universities. Those discovering this work for the first time have described it as "the answer key" to St. Thomas Aquinas. Because of the importance of Thomistic philosophy, Grenier's multi-volume manual is being brought back into print. This is a re-print edition of the first edition set, so there are three volumes total: Volume I: Logic and Philosophy of Nature Volume II: Metaphysics Volume III: Moral Philosophy **

PDF p. 34 describes the distinction between instrumental and formal signs even more clearly than Poinsot in his Summulæ:

27. Division of the sign. — The sign is divided both as regards its relation to the cognitive power and as regards its relation to the thing signified.

1° As regards its relation to the cognitive power, it is divided into instrumental sign and formal sign.

An instrumental sign is a sign which, from previous knowledge of itself, represents something other than itself ; v.g., a picture of Jupiter is an instrumental sign, because the picture does not represent Jupiter to the cognitive faculty before it is itself known. Similarly smoke does not represent fire to the cognitive faculty, unless the smoke itself is first known. A picture of Jupiter is a sign which is an image; smoke is a sign which is not an image.

A formal sign is a sign which, without previous knowledge of itself, represents something other than itself. A concept is a formal sign, for, when we conceive a thing, we know the thing before we know its concept. A formal sign is an image of a thing; but not every image of a thing is a formal sign.

Concepts and phantasms are formal signs. All other signs are instrumental signs.

PDF p. 138 on subalernation seems to disagree with Duhem (last sentence):

e) Experimental knowledge, which is subalternate to a superior science in as much as it attains sensible things by its own motion, has its own proper principles, and is inductive ; but such knowledge does not become explanatory , except in so far as it borrows principles from its subalternating science.
This is the reason why nowadays experimental “science” which is not physico-mathematical is called merely descriptive , and not explanatory.

Mathematical objects, as objects (e.g. numbers themselves), may not be signs (unless one reduces them, quite erroneously, to their numeral denotations) in the instrumental and/or formal sense. As elementary as it may sound to point out, numbers and their designating conventional representations are not, in fact, one and the same—just like atoms and their conventional chemical denotations are not one one and the same. They may nevertheless be signs, in a supraformal, “Platonist” sense, which does not seem to be included in the particular scope of your question (bearing on more restricted epistemic and semiotic matters). As for mathematical signs per se, again distinct from mathematical objects, they can either be “instrumental” and/or “formal,” as is accurately highlighted, especially in that quote from Henri Grenier’s 1950 manual of Thomism.

The distinction is key, and its confusion lies at the core of the early 20th century crisis associated with Hilbert’s program, which initiated the linguistic turn of logic (and of mathematics conceived as resulting from a finite number of logical processes), opposing formalism (the conveyor of instrumental accuracy) and semantic content (the meta-conventional power of mathematical objects and operations to mean something in relation to the extra-linguistic domain of things). His view was that to prove its consistency, mathematics itself had to be treated as a semantically free subject consisting only in axioms, rules of inference, and formal proofs. Thus, the focus needed to be switched from the objects of mathematics… to its symbols or signs (= its linguistic, instrumental elements). The language itself with its symbolism accordingly became the quid , in place of the actual referents signified or variously represented by way of linguistic symbols.

Mathematical laws are yet another, more subtle aspects of your question, because they are expressed using mathematical objects and signs. But they are not the same as the two latter (laws, objects, and signs are to be properly distinguished, which is not as straightforward a thing to do as it may appear to people less or little acquainted with problems pertaining to the philosophy of mathematics). Notice though that mathematical laws need not be meaningful. For they essentially say nothing beyond the strict order that is observed in a system resting and deriving its lemmas upon their indisputable postulation. Hence, they escape the instrumental vs. formal question. One may just say that they exist. They certainly manifest themselves both in the extra and intra mentis order, but stand irreducible to either. The following famous remark by Einstein actually gets at the heart of this curious and elusive quiddity of mathematics, beyond its objects and (instrumental and/or formal) signs:

“How can it be that mathematics, being after all a product of human thought independent of experience, is so admirably adapted to the objects of reality?”

Some mathematicians and philosophers assume and argue that mathematics is only a tool, “a product of human thought” expressed by way of a symbolic language fashioned by men through some materialistic “evolution” devoid of inspiration. From that perspective, mathematical symbols are only enough, and the formal rules of articulation between them is all you need to give them some instrumental power, as we see in whatever algorithm, with no other meaning and connection to reality than its own linguistic functionality.

What I was trying to convey in the previous note I sent yesterday, reading Duhem’s remarks itself and a little between its lines, was that mathematics is not merely an emanation of the physical world via the process of deduction and abstraction from empirical experience. To argue otherwise is to typically fall into the erroneous reduction of a false realism, forgetting that the structure of the knowing intellect itself is comprised in and belongs to the real world (as I have tried to illustrate in a recent note). Nor is it a pure emanation of thought (the error especially characterized by idealism and logicism).

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