Gödel, Escher, Bach: An Eternal Golden Braid by Douglas R. Hofstadter Page A

in inventing the pq-system was to imitate additions: I wanted every theorem] to express a true addition under interpretation; conversely, I wanted every true addition of precisely two positive integers to be translatable into a string, which would be a theorem. That goal was achieved. Notice, then fore, that all false additions, such as "2 plus 3 equals 6", are mapped into strings which are well-formed, but which are not theorems.

    Formal Systems and Reality

    This is our first example of 'a case where a formal system is based upon portion of reality, and seems to mimic it perfectly, in that its theorems a] isomorphic to truths about that part of reality. However, reality and tt formal system are independent. Nobody need be aware that there is a isomorphism between the two. Each side stands by itself-one plus or equals two, whether or not we know that -p-q-- is a theorem; and -p-q-- is still a theorem whether or not we connect it with addition.
    You might wonder whether making this formal system, or any form system, sheds new light on truths in the domain of its interpretation. Hat we learned any new additions by producing pq-theorems? Certainly not but we have learned something about the nature of addition as process-namely, that it is easily mimicked by a typographical rule governing meaningless symbols. This still should not be a big surprise sing addition is such a simple concept. It is a commonplace that addition can I captured in the spinning gears of a device like a cash register.
    But it is clear that we have hardly scratched the surface, as far formal systems go; it is natural to wonder about what portion of reality co be imitated in its behavior by a set of meaningless symbols governed I formal rules. Can all of reality be turned into a formal system? In a very broad sense, the answer might appear to be yes. One could suggest, for instance, that reality is itself nothing but one very complicated formal system. Its symbols do not move around on paper, but rather in a three-dimensional vacuum (space); they are the elementary particles of which everything is composed.
    (Tacit assumption: that there is an end to the descending chain of matter, so that the expression "elementary particles" makes sense.) The "typographical rules" are the laws of physics, which tell how, given the positions and velocities of all particles at a given instant, to modify them, resulting in a new set of positions and velocities belonging to the
    "next" instant. So the theorems of this grand formal system are the possible configurations of particles at different times in the history of the universe. The sole axiom is (or perhaps, was) the original configuration of all the particles at the "beginning of time". This is so grandiose a conception, however, that it has only the most theoretical interest; and besides, quantum mechanics (and other parts of physics) casts at least some doubt on even the theoretical worth of this idea. Basically, we are asking if the universe operates deterministically, which is an open question.

    Mathematics and Symbol Manipulation

    Instead of dealing with such a big picture, let's limit ourselves to mathematics as our "real world". Here, a serious question arises: How can we be sure, if we've tried to model a formal system on some part of mathematics, that we've done the job accurately-especially if we're not one hundred per cent familiar with that portion of mathematics already? Suppose the goal of the formal system is to bring us new knowledge in that discipline. How will we know that the interpretation of every theorem is true, unless we've proven that the isomorphism is perfect? And how will we prove that the isomorphism is perfect, if we don't already know all about the truths in the discipline to begin with?
    Suppose that in an excavation somewhere, we actually did discover some mysterious formal system. We would try out various interpretations and perhaps eventually hit upon one which seemed to make every