The Axiom of Choice - Part 1

One of the most surprising and astonishing axioms in mathematics, simply by the means of its assertion is the axiom of choice. Its assertion is extremely elementary, and yet this is precise fact, this nature of it portrays deep profound properties of thought, mathematical thought in particular. It has many formulations. One of them basically states that given a collection, or a set of non-empty sets, it is possible to choose one element from every one of those sets, in this way forming a set which has one element in common with every set in this set of sets. The sets in the set of sets, can be sets of anything, other sets or numbers (which are sets as well), etc. At this point the statement of the axiom seems empty, its truth being soo implicit. After all, if you have a collection of sets, which have say at least one number in them, it seems obvious that you can choose an element from the first set, then the second, and so on. Every set in this collection of sets is non-empty by assumption and so has at least one element, which we can choose. So what wrong? One inconsistency in the former logic was the use of the word 'first'. This word, albeit invisibly, implies a great deal. Namely, it implies that one has determined an ordering on the set of things from which one chooses a first element. For otherwise, 'first' has no meaning. What does 'first' mean? That there is nothing before that first element? Then what does 'before' mean? To define these terms would be to define an ordering on the set, or essentially a set of rules which define a first element, and the next, etc. So that can be very simply. Lets say for instance we are talking about a set of apples. I can throw these apples on the ground, and then I can specifically define the first element to be the one closest to me in distance, the second element is closes to me in distance if the first were removed and so on. Then in this way I have an ordering and so I can take the first set and choose an element, which must be possible since its non-empty. Here I do not specify the ordinal position of the element I am choosing, simply that I am choosing an element and so the discussion of ordering does not apply here. What is wrong here? The next inconsistency arises when we speak about sets which have an infinite number of elements. More specifically, sets, the elements of which cannot be counted or basically aligned with any number. More in part two...

Walt Whitman

...read these leaves in the open air every season of every year of your life...and your very flesh shall be a great poem and have the richest fluency not only in its words but in the silent lines of its lips and face and between the lashes of your eyes and in every motion and joint of your body...

Beauty Part 1 - The epistemology of beauty

This is a first in a series of posts by which I will attempt to define and convince myself of the truth of the definition of what beauty is. I wish to do this because beauty is what I value most in life, be it visual, musical, mathematical, etc. Maybe this is due to my being very passionate about certain things. This leads me to relate beauty to passion, the former being a manifestation of the latter. Once again I notice the reccuring theme of one type of information - beauty being the representation of another - passion. Often times, beauty seems to represent many different things, the concept itself being so abstract, yet I still feel it can be defined and its concept can be put onto solid ground. Offcourse I can always say that beauty is that which one is attracted to, but that to me does not truly encompass its essence. Moreover, I believe beauty can be defined in fairly objective way, so that what someone can be attracted to may not be beauty. I keep some conviction in the phrase 'beauty is in the eye of the beholder' but I also think many of these beholders do not know what beauty is, or hold plagued views of what it is. They are plagued by the same forces that create those qualities of mankind that all regard with contempt, those forces which give birth to minds who do not create, but instead destroy. Beauty is universal by the same means that the laws of mathematics are, and to that extent I believe it possible to convice those who have defective views of the ones that are true - just as someone can prove to another a mathematical theorem, by stepping through the elementary laws and concepts. To this end, I behold the creed that all human beings share a certain degree of understanding, through atomic rules of logic which are inherent from the physical laws that govern our biological existence. One such universal rule might be the common understanding that a set of 3 things combined with a set of 4 things, forms a set of 7 things, countable not neccesarily by a number, which is offcourse a far more developed concept, but by means of matching the elements of this set with something else, and getting a consistent result. So, given this collection of elementary principles, I am further assured that beauty can be defined.

The Secret Garden

There was every joy on earth in the secret garden that morning, and among those joys was a delight more delightful than all because it was more wonderful.

Mathematics

From a philosophers, or more precicely from the most general point of view (in the desire to arrive at something all-encompassing), mathematics is the search for definable structures that emerge from the interactions of a set of axioms and rules of derivation. The important term to note is interaction, and 'action' in particular, because it specifies that a certain action is part of what mathematics is. In other words, mathematics is not static, for otherwise the mere existence of a set of rules of derivation and axioms would be mathematics, which is obviously not the case. The practical observation is that mathematicians enact the process, and study the interactions, which they in turn generate. The backward-forward correlation pattern emenates once again. A further observation is that the type of action can be defined in rather absolute and concise way. This action is the transformation of one set of information into another. Thes existence of these transformation, and particularly their existence as the clockwork of mathematics shines throughout every theorem, or theory at the most elementary and abstract level, converging into nothing and diverging into infinity.

Music

Music is the strife to discover what music is. The manifestation of beauty through an essence woven into the fabric of time, constantly arriving, constantly receding and leaving an ethereal after-glow which echoes across the neural web to finally become conceived, grasped and defined. One enjoys music by sensing and almost predicting its motions, creating tension and release, which is a recurrent pattern, emerging from the very structure of our universe.