I’m sorry if I wasn’t clear. I endorse the “cynical view” that mathematics is fundamentally symbol manipulation. Fundamental in the sense that mathematics as we currently know it can be completely described by symbol manipulation without having to resort to notions such as “intuition.” Eliezer, apparently, disagrees with this philosophy on the grounds that it is robbing mathematics of something which he finds aesthetically pleasing.
jacinthebox
Karma: −6
You may have read—I’ve certainly read—some philosophy which endeavors to score points for counter-intuitive cynicism by saying that all mathematics is a mere game of tokens; that we start with a meaningless string of symbols like: …and we follow some symbol-manipulation rules like “If you have the string ‘A ∧ (A → B)’ you are allowed to go to the string ‘B’”, and so finally end up with the string: …and this activity of string-manipulation is all there is to what mathematicians call “theorem-proving”—all there is to the glorious human endeavor of mathematics.
The rejection of the definition of proofs as “mere” symbol manipulation is as insightful as the rejection of physics as the study of fundamental particles or of neurology as the study of neurons in the brain.
Your emotional appeal is about as moving (and nonsensical) as “It is not possible that all human intelligence, complexity, and emotion could be described by the mere atoms making up the neurons in our brains. What of the human heart and soul?”
The fact is that completely rigorous formal proofs are necessary to approach certain problems. It is not because certain mathematicians are “cynical” but because the approach of treating proofs as “mere” symbol manipulations bears fruit. But this does not invalidate the higher level picture that you are so attached to, just as the relativistic and quantum theories did not invalidate Newton’s.
I don’t consider the post to be an emotional appeal, just the phrase “and this activity of string-manipulation is all there is to what mathematicians call “theorem-proving”—all there is to the glorious human endeavor of mathematics.” I mean, there is essentially no content there. Just a rough outline of a philosophy, and then not even a legitimate attack on it, just a “if they are right, then mathematics doesn’t seem glorious to me any longer!”