Meaning and Form in Mathematics
A new formal system (the pq- system) is presented, even simpler than the MIU-system of Chapter I. Apparently meaningless at first, its symbols are suddenly revealed to possess meaning by virtue of the form of the theorems they appear in. This revelation is the first important insight into meaning: its deep connection to isomorphism. Various issues related to meaning are then discussed, such as truth, proof, symbol manipulation, and the elusive concept, “form”. (p. viii)
While you read
- How many axioms are there in the pq-system?
- Suppose that
--p--q--is a theorem. What theorem can you immediately deduce from it?
- 2 + 0 = 2 is a true statement of addition, but
--pq--is not a theorem in the pq-system. Why not?
p--q--a theorem of the system? Should it be? What's the problem with the axiom schema that leaves room to be uncertain about this?
- Make a system isomorphic to the pq-system that has only one axiom.
- Consider this system:
- Rule: If
AsBis a theorem, then
-AsBAA-is a theorem.
- What is this isomorphic to? How did you figure that out?
An isomorphism in the discovery of calculus
Readers of GEB have also recommended reading "The Information" by James Gleick. Reddit user colo90 pointed out a passage of that book that would fit in well in this chapter of GEB, discussing how Newton and Leibniz both discovered calculus but gave it different notations:
- Fundamentally, there was only one calculus. Newton and Leibniz knew how similar their work was—enough that each accused the other of plagiarism. But they had devised incompatible systems of notation—different languages—and in practice these surface differences mattered more than the underlying sameness. Symbols and operators were what a mathematician had to work with, after all. Babbage, unlike most students, made himself fluent in both—“the dots of Newton, the d’s of Leibnitz”—and felt he had seen the light. “It is always difficult to think and reason in a new language.”
- —James Gleick, The Information: A History, a Theory, a Flood
(This section is for adding your thoughts about the chapter. Sign what you write with your user name. Others may edit this section for length later. More free-form, unedited discussion can take place in the comment section below.)
- Hofstadter recommends that you read The Decipherment of Linear B, by John Chadwick, and I second the recommendation. It describes one of the most remarkable recent accomplishments in linguistics in a way that lets you feel the "a-ha" moment as well. --Rspeer