Quote:
Originally Posted by well named
uke_master: I absolutely agree that it's the underlying concepts that are important. I don't really have an argument in favor of one word for another. I think people whose math understanding exceeds a certain threshold probably understand either way. I chose to say symbolic logic in case people who don't understand math very well (I should probably include me here but a man's reach should exceed his grasp... :P) might get the wrong image from "math". Anyway, that's my convoluted reasoning for the word choice.
I suppose I also had in mind that part of the wonder of it, to me, is that the abstraction into "mere" symbols operated on by computable (so to speak) rules of inference is even possible. And I also wanted to highlight that part by way of emphasizing that this is one reason why "the fact that the number 9 appears everywhere, or a certain numeric ratio" interests me less than the fact that the process of reasoning works so well to represent reality. As you say it's not the specific symbols so much as the "concepts", and not even the "concepts" in and of themselves but the sorts of relationships that they have to one another
You should be careful here, as actually you are advancing a controversial theory about the foundation of mathematics known as
logicism. This was a common view among leading mathematicians and logicians in the late nineteenth- and early twentieth-century but, especially as a result of Godel's second incompleteness theorem, was largely rejected by mid-century (although has been making a comeback lately).
Before Frege and Dedekind, people typically thought of mathematics as an investigation into the fundamental structure of the universe. For instance, Kant used Euclidean geometry as a paradigmatic example of synthetic a priori knowledge (synthetic, so a claim about the world rather than just a concept). Then along came Frege, who invented mathematical logic and argued that the basic principles of mathematics are all reducible to logical principles (Russell and Whitehead's
Principia Mathematica is the most complete attempt to prove this rigorously). Then, when Godel proved that it was impossible to develop a logical system that is robust enough to justify arithmetic without including claims that are themselves not provably consistent, most people took this to mean that the logicist theory of mathematics, at least in the versions put forward by Frege and Russell, was a failure. Godel himself ended up arguing for a version of mathematical Platonism and
other attempts to understand the foundation of math became more popular.