Dimostrationi Harmoniche

CLAV. We shall say, then, that every time I say that the Premises of a Demonstration must be better known than the conclusion, and precede it, we must always understand this according to the proceeding of nature. Aristotle argued that while some things are "better known to us" through our senses, scientific premises must be "better known by nature"—that is, fundamental truths. GIOS. That is exactly how it is. ADRI. Therefore, in such a Syllogism Syllogisma logical argument where a conclusion is drawn from two premises, we follow the order of nature and its own progression. GIOS. Without any doubt. But this does not happen many times. FRAN. For what reason? GIOS. Because of our imperfection; from which it arises that, similarly, philosophers rarely put these syllogisms into practice. CLAV. This we understand well; but it remains for you to go on declaring what is meant when the "middle term" taken in the demonstration is so closely joined to the "major term" In logic, the middle term links the subject and predicate of the conclusion that nothing else can be found in between; or rather, it is the middle between them by its rank in the category. You can understand this occurring (to give an example) between the Definition and the Defined. For there is no "middle" between "Man," which is the defined, and "Discursive Animal" original: "Animale discorsivo"; meaning a rational being capable of speech and reasoning, which is the definition, by which one could show that Man is such a thing. One cannot demonstrate the definition of things. CLAV. The example you have brought has illustrated this reasoning in such a way that, until now, what you have said is understood very well. Therefore, continue with the rest. GIOS. Beyond this, the Premises must be primary; that is, they must be such that there is not found in any science (so to speak) a higher or better-known proposition than them. It is necessary that they be taken as known, without any proof. ADRI. Therefore, for such a reason, they shall be, or shall be called, Indemonstrable. GIOS. That is very right, Monsieur; for since that which is demonstrated must be born from preceding and better-known Premises, if all the Premises always had to be demonstrated—and also the premises of the premises—our proceeding would have to climb upward forever toward the "better known" and "more preceding" in an infinite loop. Whereupon, never being able to arrive at those Premises which were known for themselves, nor Aristotle, Physics, Book 3, Chapter 6 being able to traverse the infinite, it would be necessary to stop at some of them. These, by depending on higher premises, would not be known by us for themselves; and consequently, the conclusions that arose from such premises could not be made manifest. From this would follow what many have imagined: that no proposition could ever be demonstrated. FRAN. This discourse has been very useful to me, because it has brought back to my memory many things which already (for not attending to these studies) had left my mind. And I remember that I heard it said many times: that in any Science, before one comes to discourse upon anything within it, certain manifest propositions are supposed, which must not be denied by anyone who wishes to practice in that Science. GIOS. You remember very well, by my faith! And I rejoice that I will not have the trouble of repeating such things, nor many others that coincide with Demonstration for our sake, in which I see you and the others are already instructed. However, following where I left off, I will say that some call such Principles "Positions"; and these Positions, or Principles, are of several kinds. For some are called Common Principles, and some are called Proper Principles. The Proper ones are those that serve a particular Science. You must not think that the Proper Principles of one science are the same principles of another, for you would be in error. But you must know (as I have also said above) that from the different kinds of knowable things, different Sciences are born. Thus, just as Continuous Quantity such as a line or a surface in geometry is different in kind from Discrete Quantity such as individual numbers in arithmetic, so Arithmetic is different from Geometry. And just as Magnitude is different from Number, so the principles of Geometry (with which its conclusions are demonstrated) are different from those of Arithmetic. Hence, the Proper Principles of Geometry are (to give you an example) these: A line can be drawn from one point to another; the continuous is infinitely divisible; and other similar ones. But those of Arithmetic are: Number is an ordered multitude of Units; the parts of a number do not join at a common boundary; Numbers proceed beyond Unity to infinity; and the others. And those of Music are: An Interval is the relationship of the spaces between a low and a high sound; and other similar ones, as you will soon see. And these are called Proper Principles. But the Common ones are so well known that they can be supposed not only in this or that Science, but in all the other Scien-