Thanks for the very apt segue from Jon Barwise —
Modern mathematics might be described as the science of abstract objects, be they real numbers, functions, surfaces, algebraic structures or whatever. Mathematical logic adds a new dimension to this science by paying attention to the language used in mathematics, to the ways abstract objects are defined, and to the laws of logic which govern us as we reason about these objects. The logician undertakes this study with the hope of understanding the phenomena of mathematical experience and eventually contributing to mathematics, both in terms of important results that arise out of the subject itself (Gödel’s Second Incompleteness Theorem is the most famous example) and in terms of applications to other branches of mathematics. (Barwise p. 6)
When it comes to mathematics as the science of abstract objects I have my personal favorite classes among its abstract gardens and zoos. One order of particular interest in the great chain of abstract being descends from the family of mathematical relations to the genus of triadic relations to the species of triadic sign relations.
By a curious turn, but no real surprise when we stop to think about it, sign relations, with their object, sign, and interpretant sign domains, come into being whenever we reflect on the systems of signs we use to describe any universe of objects, abstract or otherwise, and thus they are just the tickets we need to enter that “new dimension” of mathematical logic.
- Barwise, J. (1977), “An Introduction to First-Order Logic”, pp. 5–46 in Barwise, J. (1977, ed.), Handbook of Mathematical Logic, Elsevier (North Holland), Amsterdam.
- Eisele, C. (1982), “Mathematical Methodology in the Thought of Charles S. Peirce”, Historia Mathematica 9, pp. 333–341. Online. PDF.