المنهج العلمي الواقعي | علي آل شُبَّر

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

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

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

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

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

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

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

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

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