**Gödel, Escher, Bach/**

__Chapter 3/ Figure and Ground: Primes vs. Composites.__**The "Tq" system**

As it is customary on each chapter, we have its respective dialogue and its respective exemplifying formal system, both convenient in the aid of our laymen understanding. The system introduced here is called the “Tq” system, were T will stand for “Times”, meaning multiplication. As we were told about each system, we must consider its rules, axioms, and theorems and work our ways with those. I am still working with the concepts, especially that called a “hyphen-string”, but for now my comprehension seems to suffice, I can only hope so.

The purpose of the “tq-system” seems to be that of identifying prime numbers, or rather separating composites and primes. The system itself is rather complex, it is curiously enough one of the Project Euler problems we tackled on the first term of the M.P.C at programming class. The rough basic idea behind it, I suppose, is generating a system, in this case “tq”, that is able to take each number and identify it as either prime or composite.

The proposed solution appears to be simplified by terms of divisibility. Ideally you would have the system which can take each number, starting with 2 (a prime), divide it and identify it, and keep on going on towards 3,4,5 etc. without backtracking.

The above statements were found in the book, and they seem pretty reliable, though I cannot claim complete cognizance on my part (As explained on the text).

**Figure and Ground**

Aside the whole “formality” of the chapter (its theoretical structure and process), it does always provide philosophical inquiries, specifically those pertaining to mind and body. That is an issue that I have mentioned on and off during the transgression of my readings, simply because I see it as so. One may argue whether such inquiries, as I mentioned, should be classified as philosophical or scientifically, and truly I believe those to be only certain approaches to the same. However, in this instance I will call it a philosophical matter, as it deals primarily with the “questions” and not the “answers”. The way in which the author directs us, indirectly perhaps, is with the appearances of both art and music, as it was distinguishes at the introduction, mainly using Escher and Bach as references; of course, Gödel will not be excluded, though one must consider the formal systems to be examples of his particular field, being such mathematics.

Particularly here, we are told of two new concepts; Form and Ground. To delve into these recurrent discussions, we begin with illustrations of Escher, and even some inclusions of the author’s works as well. The concepts help define certain differences, or rather certain spaces that are unseen within formal systems; namely axioms, theorems, and their respective negations (negation of axioms, negation of theorems). Though to our surprise we encounter some other phenomenon, which is the existence of ground, and in it unreachable truths and unreachable falsehoods. See the illustration below (Figure 1.) (excerpt from actual book) where Hofstadter (the author) gives us a vivid representation of what he means to tell.

Further more other terms come out such as cursive and recursive to help us understand more the establishments of form and ground. These terms were introduced before we are given the illustration below, using instead illustrations of Escher and another artist. The images can be spotted by the contrast, it usually resembles black and white, in a way in which they counteract showing different sides of a whole. Thre is a way however to join both, cursive and recursive for this matter, were both seemingly contrasting elements resemble the same. Now that one considers the value and use of the sketches and drawings, we must apply it to the usage of formal systems (which I believe is initially the main point).

Back to... primes?

Back to... primes?

So what is the whole purpose of this chapter, what message does it tries to convey, what is the meta-question? All are justly made questions, that should find an answer. For now, however, considering my levels of understanding, can only presume I know the answers. We dive deeper and deeper into the structure of formal systems, perhaps what we are truly intending to perceive is the work of Godel. To do so one must first take in the sequencing, order, and mannerisms, in other words one must learn the basic axioms, theorems, hybrid-strings, and rules that will evidently apply to the works of Godel, as it has been done with the works of Bach and Escher throughout the course of the reading of the book.

**Bach**

Bach’s intellectual importation to the use of formal systems, done in such a subtle manner, is beautifying and concrete at the same time. It seems he also made this contrast between melodies, making them work in conjunction though being in a sense opposites. I do not pretend to understand the works of Bach, at least not yet, nor do I pretend to be completely accurate on the previous linkage, but it is as far as I understood.

Contracrostipunctus

Contracrostipunctus

Yet another dialogue is presented to us, by now its seems evident that this will be the structure of the book (chapter, system, examples, dialogues). However I am unwary if the dialogues are used as a previous pondering tool for the upcoming chapter or is it a follow-up to the reading of the chapter, in other words, should the dialogue be read in conjunction to the coming chapter or the previously read one. I believe it to be the former, since in context it makes no sense, but no matter I will still speak of it in the latter structure, as it has been done thus far.

Now we find ourselves in the Tortuous home, with Achilles present as well. They have a conversation upon a friend of the Tortoise, the crab, and certain endeavors they have shared; specifically I suppose, a matter of a phonograph allegedly capable of reproducing all sounds, the crab states it is possible, the Tortoise differs and proves sit as well. I will not go into much detail upon the matter, but it is a significant dialogue nonetheless, and the longest one so far. In the discussion, practically the end of it, they speak of Bach's last fugue and how he left an encrypted message on it, acrostics they call them (in poetry at least), this is a very common usage of art. Now I wonder how this will relate to Godel, and mathematics in its whole, or even more so in formal systems.... I suppose I am left to find out on chapter III...