Not only that but the obviousness of "and so on" is often elided. "And so on" assumes unbounded time and/or resources and also assumes you know intimately the (undoubtedly recursive) algorithm and can codify it. These proofs often blatantly ignore that _in practice_ someone or something must perform this algorithm. (Er, not sure if I used elided correctly back there.) Anyhow, if you have an innate distaste for these types of proof then rest assured that you are not alone.
Also check out intuitionism by keerazy Dutch mathematician L.E.J Brouwer which asserts that we must essentially modify some preconceived logical tenets/laws in the face of infinity.
Apologies if you were aware of all this already. I have found it very helpful to respect my suspicions. Regardless of what people may tell you, this stuff is neither simple nor straight-forward once you start thinking deeply enough about the minutiae.
Not only that but the obviousness of "and so on" is often elided. "And so on" assumes unbounded time and/or resources and also assumes you know intimately the (undoubtedly recursive) algorithm and can codify it. These proofs often blatantly ignore that _in practice_ someone or something must perform this algorithm. (Er, not sure if I used elided correctly back there.) Anyhow, if you have an innate distaste for these types of proof then rest assured that you are not alone.
You may check out the esoteric philosopher René Guénon, R. (1946) The Metaphysical Principles of the Infinitesimal Calculus http://books.google.com/books?id=9KyLPwielTEC
Also check out intuitionism by keerazy Dutch mathematician L.E.J Brouwer which asserts that we must essentially modify some preconceived logical tenets/laws in the face of infinity.
See also intuitionistic logic, intuitionistic type theory and constructivism (mathematics). http://en.wikipedia.org/wiki/Constructivism_(mathematics)
Apologies if you were aware of all this already. I have found it very helpful to respect my suspicions. Regardless of what people may tell you, this stuff is neither simple nor straight-forward once you start thinking deeply enough about the minutiae.