Concluding our intention, therefore, we affirm that it is most Peripatetic, and in accordance with the mind of Aristotle, that mathematical disciplines are in the first order of certainty – not by reason of most powerful demonstrations, which are not found within them, but soley by reason of the subject matter with which these disciplines deal. This was my primary intention to declare from beginning of this commentary.

Many other things could still be said in favor of this view, but I leave them to be handled by the more perceptive minds of those who read my work. It is enough for me that I am the first (to my knowledge) in this age to have perceived this truth and to have provided an occasion for others more learned than myself to pursue it further.

THE END

Alxenadri Piccolominei | Commentarium De Certitudine Mathematicarum Scientiarum

In this work, I propose to accomplish two things.

First, I intend to demonstrate, through both reasons and authorities, that mathematical demonstrations are not those potissimae which Aristotle constructs in the Posterior Analytics.

Second, so that we do not appear to undermine the opinion of Averroes regarding the second book of the Metaphysics cited above, I intend to assign the true cause why the mathematical disciplines can be said to be in the first degree of certainty.

Alxenadri Piccolominei | Commentarium De Certitudine Mathematicarum Scientiarum

There were those who argued that a demonstration propter quid only does not differ in species from an absolute demonstration, but only "by accident"—that is, relative to the observer. For example, if someone knows the Moon is eclipsed but is ignorant of the cause, a demonstration from the cause would be for them a demonstration propter quid only. However, if the same demonstration were presented to someone who did not even know the Moon was eclipsed, it would be for them a potissima demonstration, revealing both the fact and the cause. Since the same numerical demonstration could be called either potissima or propter quid depending on the person, they concluded the distinction is accidental rather than specific.

Alxenadri Piccolominei | Commentarium De Certitudine Mathematicarum Scientiarum

Regarding the two demonstrations, quia and propter quid, although their specific distinction is clear, Averroes notes in the Physics that when these two are joined, the resulting science is firmer than if derived from either alone. He asserts that Aristotle joined both processes in his discussion of the first eternal mover: first demonstrating the mover from motion (as an effect), and then, as if returning or regressing, concluding the nature of the motion from the mover.

This observation by Averroes gave rise to the debate over the "Regress." Can there truly be a scientific regression in demonstrating? Some denied this, reasoning that all scientific progress must move from the known to the unknown. They argued that once you have reached the cause via a demonstration quia, you already know it is the cause of that effect. Therefore, a second process—the demonstration propter quid—would be redundant and achieve nothing new. Since cause and effect are correlative, to know one as such is to know the other.

Others, devoted to Averroes, attempted to save his opinion by introducing a forced "negotiation" (negotiatio) or mental processing after the first step—a concept I have never found intelligible. It seems to lead only into obscure ambiguities.

Alxenadri Piccolominei | Commentarium De Certitudine Mathematicarum Scientiarum

When we say most powerful demonstrations consist of immediate premises—that is, indemonstrable ones, and of which one is not prior to the other—we ought not to understand this of both premises simultaneously, but of those premises for the most part which become the major in the first figure.

Alxenadri Piccolominei | Commentarium De Certitudine Mathematicarum Scientiarum

Proclus shows that there is a common science to which Geometry and Arithmetic are subalternated, possessing its own subject and principles. This is often overlooked by those less skilled in the discipline. This commonality is evident in theorems concerning proportional quantities found in the Elements, which apply universally to both branches.

Alxenadri Piccolominei | Commentarium De Certitudine Mathematicarum Scientiarum

I maintain, following Proclus, that while it may be impossible to find a demonstration based on a final cause in mathematics—because mathematical figures, insofar as they are such, do not perform actions from which a demonstration could be constructed (a point noted by Averroes in the second book of the Physics)—mathematics nevertheless possesses a significant good. While Philoponus and Alexander (cited by Simplicius) observe that mathematical proofs depend on formal definitions, the good found in mathematics is by no means to be despised.

First, mathematics subalternates Geometry and Arithmetic, and through them, Music, Astronomy, Perspective, Stereometry, and Mechanics; without these, nearly all the arts that contribute to human happiness would collapse. Therefore, mathematics brings the greatest utility. Furthermore, Proclus demonstrates that the mathematical sciences are not only useful but almost necessary for moral philosophy. They are also beneficial to divine philosophy, not only by training our intellect for the very abstraction that the Theologian requires, but for many other reasons detailed by the same Proclus. Their benefit to natural philosophy is self-evident and is explained at length by him. As for the utility they provide in leading armies, capturing and defending cities, the layout of camps, the science of navigation, trade, and architecture, I could argue extensively and in great detail; however, as this is peripheral to our current purpose, I believe I should pass over it for now. We may conclude, therefore, with the testimony of Simplicius, that a good—and indeed a most significant one—is found in mathematics. Even if they were not related to other
disciplines, one must admit they possess an intrinsic worth. Proclus argues this from the fact that once anyone has had even a brief taste of mathematics, they are captured by such delight that they set aside all other cares and no longer value other pleasures. This surely would not happen unless some profound good were enclosed within them.

Plato, rightfully recognizing the dignity and utility of the mathematical faculty, wished it to be guarded by the inscription at the Academy: "Let no one ignorant of geometry enter." Simplicius also notes, following Plato's thought, that Geometry makes all other sciences more perfect and brings them great light.

Alxenadri Piccolominei | Commentarium De Certitudine Mathematicarum Scientiarum

Every mathematical faculty is arranged in this order: first, principles are assumed namely, axioms, definitions, and postulates. Then, if any Problem is required for the construction of Theorems, it is introduced. According to Proclus, a Problem differs from a Theorem in this way: a Problem is that in which something that did not previously exist is proposed to be found and constructed. A Theorem, however, is that in which something within an already established figure is demonstrated to be true or not true. It follows that Problems are proposed for the sake of constructing Theorems.Indeed, in Theorems, superior Problems are never cited except in the construction phase of the Theorem, just as occurs with postulates.

Alxenadri Piccolominei | Commentarium De Certitudine Mathematicarum Scientiarum

First, it must be maintained that among the four genera of causes, the mathematician cannot demonstrate via efficient or final causes. Regarding the efficient cause, no one doubts this, since the mathematician does not consider motion except metaphorically—and as Aristotle and Averroes observe in the Posterior Analytics, De Caelo, and De Anima, we must not demonstrate through metaphors.

Regarding the final cause, some have labored greatly to show that "the good" (and thus an "end") is found in mathematics, since the end is interchangeable with the good according to the first book of the Ethics. But all their labor is in vain. They are deceived by the belief that for a "good" to result from a science is the same as demonstrating through a final cause. These are very different things.

...

It does not follow that they demonstrate through an "end," especially since Aristotle explicitly states in the third book of the Metaphysics, Chapter 3, that there is no "end" in mathematics—a point confirmed by Alexander and Averroes. They cannot say otherwise, for mathematics deals with quantity, which is not among the active powers. Even if we were given the faculty of imagination, we could not discover for the sake of what or for what end alternate angles in parallel lines are equal. We conclude, therefore, that mathematical demonstrations cannot be given through efficient or final causes.

Alxenadri Piccolominei | Commentarium De Certitudine Mathematicarum Scientiarum

[Major Premise]

Every most powerful demonstration uses either the definition of the passion or the definition of the subject as its middle term.

[Minor Premise]

Mathematical demonstrations do not have such a middle term.

The major premise is accepted by everyone: though some believe the subject's definition is the middle and others the passion's, all concede it must be one of the two.

The minor is proven by induction through all the theorems of Euclid, Theodosius, Archimedes, and others.

Alxenadri Piccolominei | Commentarium De Certitudine Mathematicarum Scientiarum

For example, if we examine the much-cited Theorem 32 of Euclid’s first book, we find that the exterior angle, which is used as the middle term to declare the passion—that the triangle has three angles equal to two right angles—is the definition of neither the triangle (the subject) nor the passion itself. Neither the triangle nor "having three angles equal to two right angles" requires an exterior angle in its definition. The triangle exists, and has those angles, even if the exterior angle is not present. This will be clear in almost all other theorems and problems of Euclid; thus the minor is proven, and consequently our conclusion.

Alxenadri Piccolominei | Commentarium De Certitudine Mathematicarum Scientiarum

[Major Premise]

Furthermore, every most powerful demonstration has a middle term that is the immediate cause of the effect (the passion).

[Minor Premise]

But no mathematical demonstration is found to be such.

The major is evident from Aristotle in the second book of the Posterior Analytics, who says that while an effect—such as the shedding of leaves in trees—may have several causes—being broad-leaved and the freezing of moisture only one is the proper, immediate, and convertible cause, namely the freezing of moisture.

The minor is proven because mathematical passions cannot flow from an extrinsic cause, as we declared slightly above.

And as for the form—since we have already excluded matter how will they depend on the form, if in quantity there is no action or principle of action?

Thus, no one can say how, in the nature and form of a triangle, there exists the fact that the exterior angle is greater than either opposite interior angle—a passion proven by Euclid in Book I, Proposition 17.

[Proposition 16 in Heiberg]

Alxenadri Piccolominei | Commentarium De Certitudine Mathematicarum Scientiarum

[Major Premise]

Thirdly, we may reason as follows: for a passion in a subject, there ought to be only one immediate and true middle term from which a most powerful demonstration is constructed.

[Minor Premise]

But mathematical passions do not have such unique, immediate middle terms.

The major is clear because the middle is the cause; therefore, there is only one true middle because there is only one proper cause for each thing according to Aristotle in the second book of On Generation and the second book of the Physics.

The minor is proven because mathematical passions are discovered in their subjects without any fixed order of priority. The true order of passions arises from their flowing from the subject and its form. When such a flow occurs, an order of natural priority is established, because one thing, insofar as it is one, can only immediately produce one thing. Mathematical passions, however, cannot have such an order, process, or flow from the subject, because quantity is not among the active principles according to Averroes in the fourth book of the Physics, commentary 84.

The truth of this is seen in the fact that mathematicians demonstrate the same passions about the same subjects using various assumed middle terms. Theon and Euclid show that a triangle has three angles in one way; Campanus in another; and Proclus in yet another.

...

Themistius also in the first book of the Physics, commentary 89, explicitly affirms that the same conclusion in mathematics can be demonstrated through several different premises. Simplicius promises the same in the second book of the Physics. If someone should say that although several demonstrations of the same passion are possible, only one will be the most powerful through the immediate middle, we can respond that this is false. Proclus openly asserts that diverse, equally perfect demonstrations can be made for mathematical passions. Furthermore, we have Plato (as cited by Philoponus in the first book of the Posterior Analytics), who always proposed some Problem to be demonstrated to his students. No matter how diversely they demonstrated it, provided they did demonstrate it, he praised them equally—as he did with the discovery of the two mean proportionals for the duplication of the cube. Plato knew, therefore, that the nature of the mathematical faculties was such that their passions could be demonstrated in various ways.

Alxenadri Piccolominei | Commentarium De Certitudine Mathematicarum Scientiarum

Since the entire weight of the previous argument regarding the minor premise depends on the fact that quantity is not an active power, this point must be examined more closely.

I maintain that since all action occurs by virtue of a form introduced into primary matter—for primary matter, as such, performs no action—it follows that for a subject to act, it must be endowed with a substantial form.

...

Since this is how the matter stands, and since quantity is the most imperfect of all accidents—for it alone among accidents follows matter eternally without the mediation of a form, according to Averroes in De substantia orbis, the first book of the Physics, commentary 63, and the Epitome of Metaphysics—it follows that no principle of action can be attributed to quantity.

...

This happens because quantity is coeternal with matter, yet it is indeterminate and naturally prior to every substantial form. We do not accept that chimerical "form of corporeity" which some place in the predicament of substance, nor that "metaphysical body" which Albertus Magnus posited in imitation of Avicenna. No Greek author posited such a thing, nor did Averroes.

Therefore, among legitimate Peripatetics, the form of corporeity is simply indeterminate quantity within primary matter. Matter considered in this way, prior to every substantial form, is what Ammonius calls the underlying subject. Thus, quantity cannot be a principle of action.

Alxenadri Piccolominei | Commentarium De Certitudine Mathematicarum Scientiarum

I could bring forward many more reasons to conclude that mathematical demonstrations are not most powerful, but these should suffice, especially given the remarkable authorities we have. First, we have Proclus, that illustrious man in mathematics, who says in the first book of the Elements, page 21, that several Elements—that is, various Euclidean propositions—can be elements to one another. This would surely not be possible if they were demonstrated through true causes, for nothing can be its own cause, and a single thing has only one definition according to Aristotle in the sixth book of the Topics, Chapter 3.

Alxenadri Piccolominei | Commentarium De Certitudine Mathematicarum Scientiarum

We concede that the mathematical disciplines hold the first rank of certainty, but we deny that the cause for this rank was correctly identified by the Latin commentators. What, then, is the true cause of this certainty? It is precisely that which Aristotle posits in the sixth book of the Ethics and the seventh book of the Metaphysics, and which the Greek authors have confirmed. When Aristotle asks in the Ethics why children can become mathematicians but cannot become prudent, wise, or natural philosophers, he immediately identifies the cause: mathematics is based on abstraction, whereas the principles of other faculties are assumed through experience. Children lack experience, but they are perfectly suited for abstraction. These are Aristotle’s very pregnant words. Natural principles and natural things—as well as metaphysical ones—are known from their effects through long experience perceived by the senses; this requires a long time, immense labor, and constant observation. It is no wonder children are denied entry into these fields, as they cannot be experienced due to their age. Mathematical things, however, because they exist by abstraction, offer themselves fully and intimately to our senses. They reveal themselves entirely: not only their passions but even their subjects and forms are perfectly manifest to our sense, because they are all quantities.

...

It is clear, then, from Aristotle's words, that the cause of mathematical certainty is the subject matter. Simplicius feels the same in the first book of De Anima, saying the cause of mathematical certainty is that they deal with quantity; for quantities are sensible things and have sensible causes and are therefore known to us. Aristotle confirms this in the seventh book of the Metaphysics, saying natural things cannot be abstracted like mathematical ones because they have determinate matter, actuated and limited by a specific form—a limitation we cannot know without long use and observation. Mathematical things, being abstractable, do not claim such limited matter for themselves.

A circle does not require gold, wood, or any other determinate material. Since the ease of abstraction arises from the greater or lesser determination and limitation of matter, it follows that those things that are not determined by any matter in act, but are coeternal with naked matter, are the most abstractable. Thus, they are easy to know, certain, and manifest.

Quantity, because it is a "common sensible" and is not tied to any limited matter, holds no secrets; it explains and manifests itself to us entirely.

Alxenadri Piccolominei | Commentarium De Certitudine Mathematicarum Scientiarum