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Month: March 2026

Even & Odd.

Published on March 26, 2026

Nicomachus begins by teaching that number is of 3 kinds. He says it is a: “limited multitude or a combination of units or a flow of quantity made up of units”. E.g. the number 3, 4 plus 4, 5 in motion. He then provides an ordinary definition of even being: “that which can be divided into two equal parts without a unit intervening in the middle” E.g. 4 is divided by 2 two times. Concerning the odd number species he says: “that which cannot be divided into two equal parts because of the aforesaid intervention of a unit” E.g. 3 is divided by 2 one time with 1 remaining, i.e. two unequal parts.

He goes deeper with both species according to Pythagorean doctrine. Concerning an even number, we learn that it admits of division into the greatest and smallest parts at the same operation. I.e. ‘a half’ is the greatest magnitude, and the ‘number 2’ is the least multitude possible post-division. Indeed, Nicomachus says that this is: “in accordance with the natural contrariety of these two genera”. On the other hand, odd is that which doesn’t allow this to be done to it. An odd number is divided into two unequal parts. E.g. 3 is divided by 2 two times with 1 remaining.

At this point he introduces a more ancient definition of both respective species. The ancients considered even to be that which can be divided into two equal and two unequal parts. E.g. 4 is divided by 2 two times, but 8 is divided by 6 is one time with 2 remaining. However, concerning the number 2, he says: “… except that the dyad, which is its elementary form, admits but one division, that into equal parts’. Interestingly, he teaches that in any division with even numbers, brings to light only one species of number, however it may be divided, independent of the other. I.e. even divided by even equates to even in parts. E.g. 22 is divided by 6 three times with 4 remaining, both the 22 and the 6 are even numbers.

Juxtaposed to this, concerning the ancient definition of odd, he says: “a number which in any division whatsoever, which necessarily is a division into unequal parts, shows both the two species of number together”. He goes on to explain that there is never an instance where there is not some intermixture of even and odd together. E.g. 17 is divided by 6 two times with 5 remaining, the two instances of 6 being the even numbers, with the 5 being the odd number, to complete the sum of 17. 

He concludes with an interesting image of the positioning of even and odd with respect to each other. He says: “the odd is that which differs by a unit from the even in either direction, that is, toward the greater or the less, and the even is that which differs by a unit in either direction from the odd, that is, is greater by a unit or less by a unit”.

EAR

Nicomachus, Introduction to Arithmetic. Book 1, Chapter 7.

Indefinite contingency.

Published on March 26, 2026

This was again, a very difficult lesson to sift through. As I took notes, I was tracking and laying out what Aristotle was saying, but I was not comprehending the substance of the message at all. The key that seemed to unlock the fog over the text was coming to understand the distinction between definite and indefinite contingencies. I meditated and prayed on these two terms as I drove home from work yesterday. My mind focused on indefinite and trying to understand why Aristotle would using walking and an earthquake to demonstrate his point. I began to ask myself the questions like: “Is walking indefinite? How can it be indefinite if what walks, must walk, before walking to be known and apparent? Would that not be a definite act? Indefinite seems to imply perpetuity in action. Yet the earthquake does not happen without a prior cause and not indefinitely? Is it the earthquake itself that which is contingent here or something else?” Then I locked in on the term ‘for’ which was used in both examples. It was walking to the animal, and the time and place to the earthquake which was indefinitely contingent. So, then what is indefinite must be related to the predicate, if that were true, then that meant what is definite is concerning the subject. I took this interpretation to the tutor and received affirmation that this aligns with what Aristotle is teaching.

EAR

Aristotle, Prior Analytics. Book 1, Chapter 13.

Root.

Published on March 7, 2026

Nicomachus seems to divide his exposition on Arithmetic into three parts: viz. its natural superiority, ontological priority, and the posteriority of foregoing Mathematical sciences that are subject to it.

Firstly, he describes Arithmetic as being the origin, root, and mother science of all and any mathematical conception that has or could ever exist. Indeed, he says: “… it existed before all the others in the mind of the creating God like some universal and exemplary plan”. He goes on to assert that – because of this – it served as the archetypal design for the ordering of material creations to attaining their proper ends. 

Next, he transitions into examining the ontological priority of Arithmetic. We learn that it abolishes other sciences but is itself not abolished by them. He demonstrates, with appeals to natural philosophy: “For example, ‘animal’ is naturally antecedent to ‘man’, for abolish ‘animal’ and ‘man is abolished; but if ‘man’ be abolished, it no longer follows that ‘animal’ is abolished at the same time…. Hence arithmetic abolishes geometry along with itself, but is not abolished by it”.

Finally, he concludes that the ontological posteriority of the rest of the Quadrivium implies the older with themselves but are not implied by the older. Nicomachus affirms this by saying: “Conversely, that is called younger and posterior which implies the other thing with itself, but is not implied by it”. He goes on to give different examples: how musician implies man, but man is not implied by it, how horse implies animal, but animal is not implied by it, etc. He ties all this together conclusively by teaching: “For how can ‘triple’ exist, or be spoken of, unless the number 3 exists beforehand, or ‘eightfold’ without 8? But on the contrary 3, 4, and the rest might be without the figures existing to which they give names”.

EAR

Nicomachus, Introduction to Arithmetic. Book 1, Chapter 4.

FOV.

Published on March 6, 2026

I had the prior understanding from my brief study with Euler that magnitude was any kind of quantity. However, Nicomachus makes it extremely clear that magnitude is what can be measured, and multitude is what can be counted. Learning that distinction, has done something to my mind that I am struggling to explain. When I see a tree, I am contemplating its wholeness, its completeness, and its size. Yet, when my field of view takes in the forest, my mind switches from this lens to one concerned with multitude. Then I find myself dumbfounded by the number of trees that are actually in this forest, and if the quantity could ever be known by a single man. This is fascinating, and I’m really curious to know if both these will be covered in this treatise or one of them.

EAR

Nicomachus, Introduction to Arithmetic. Book 1, Chapter 2.

Sub-division.

Published on March 6, 2026

I did not know that the Quadrivium is sub-divided into two groups for quantity and size. I had never made that connection before. At first glance, with the modern callouses over the eye of my soul, it seemed to me, prior to this lesson, that these sciences were all exclusive from one another. As I think about it now, I think this is a result of looking at these only through the temporal things which subsist in these sciences, just as the interlocutor does from the Republic. If we were to assume materiality as our first principle, then what does transactional flow, configurative partitioning, beautiful entertainment, and shipping cadences have in common essential? Nothing, I would say at first, because all of these bodily things are very different from one another; therefore, these must not be related essentially, but superficially. On the contrary, this must be a flawed misconception, and just coming to know that they are related immaterially by means of quantity and size is mind-blowing to be frank.

EAR

Nicomachus, Introduction to Arithmetic. Book 1, Chapter 3.

Minor.

Published on March 6, 2026

There is a distinction that has remained with me from the past three lessons. It is that the minor premise alone being necessary, with the exception of Darapti, seems to never yield a necessary conclusive proposition. The tutor confirms my interpretation of why this is the case. I reason that it is because the lesser extreme is posterior to the greater extreme in a priori, therefore what is necessary of a lesser in any figure, necessarily has no bearing on its syllogistic relationship with a greater. As I think about this, it seems to me that if we forget the hierarchical flow of terms in the different figures of syllogisms, then we are certainly tempted into thinking of everything being autonomous to itself and having prior causes to it. I don’t know why, but I can’t help but wonder if this line of thinking is a rotten fruit of the post-modern intellectual movement. If so, then it would explain why everything for a rational mind seems arbitrary, disconnected, and chaotic. There is essentially no flow, no connection, no relationship, and frankly an abyss of pointlessness without a mind that is perceptive in Prior Analytics. This seems to be a recipe for a possible existential crisis.

EAR

Aristotle, Prior Analytics. Book 1, Chapter 11.