**Gödel, Escher, Bach/**

__Chapter 2/ Meaning and Form in Mathematics.__**The "Pq" system.**

A new system is introduced, a new formal system that is, and we are given the “privy” to work it out ourselves. This system goes by the name of the “Pq System”, it is basically supposed to be simpler than the formerly presented “MU” system. However, here, the axioms may vary, and the theorems as well. Nothing really made sense to me at first, until afterwards on the chapter when the author, Hofstader, goes into the worth of meaning and truths. The pq system in itself allegedly functions in regards to additions (p = “+”, - = number, and q = “=”), so if you were to have –p---q----, that is a theorem because 1 + 3 = 4.

Other than the shallow concepts of this pw system, we are pulled by this intriguing idea of what is reality, and whether or not the universe as we know it, or rather life itself, can be defined by a formal concept such as the pq and MU. Personally, I highly doubt so, and am an advocate for its contrary, thinking mathematics was a human invention rather than a discovery,nevertheless we are here to discuss Hofstaders view and input on the matter. A great line segment of this chapter says the following: “Basically we are asking if the universe operates deterministically, which is an open question.” I would argue if the universe “operates” at all, but here we are not meant to ponder on that term specifically, instead we are given deterministically as a word of interest, referring to formal systems. I wonder if it is implied within that question, that human beings are made solely out of matter.

**Isomorphism**

An isomorphism is another term that is used in this chapter, it is meant to define thos patterns we identify in these kinds of systems, it is our forms of interpretation, but one must be careful in identifying isomorphism and actual truths, here lies the question. Hofstader gives us now two significant ideas, that of good interpretation of meaning and that of wrong insignificant and unrelated interpretations. One must be careful in distinguishing which is which.

From there on we end up in discussing Euclids logic, in regards to “proofs” such as that which backs up the argument that prime numbers are infinite. First off this will require numbers to be infinite (just a note). The thing happens as well when multiplying a number such as 3948239048 X 123123455, it is supposed to be possible given the proper multiplication formula, but who will generate the squares necessary to confirm such a statement. The same happens with prime numbers: who will keep identifying primes over and over until infinity?

Thus, we end up in another sort of morbid loop, upon which Bach’s canons and Escher’s sketching can once again relate, mathematics is definitely an intellectually stimulating discipline.

**Sonata for Unaccompanied Achilles**

The chapter ends with another dialogue between Achilles and the Tortoise, but here we only hear Achilles voice as he is speaking on the telephone and we hear only his end. There are two riddles presented on this dialogue, whose answers I failed to unravel. The first is identifying a word with the letters ACAD on it, the second is identifying a word (other than HE) that begins with HE, and ends with HE. In regards to the latter, it was my understanding that the Tortoise replied “HE” as the answer, but Achilles negated it implying it was too obvious ant that another answer must be given, the Tortoise figures it out, but since we only hear Achilles end, we do not hear the actual answer.