I have been reading the famous Joseph Schoenfield's text "Mathematical Logic" and this may sound naïve but I can't make sense of his comments about finitary proofs. Can someone please explain to me what a "finitary" proof is?
To quote from Schoenfield's text:
"Proofs which deal with concrete objects in a constructive manner are said to be finitary. Another description, suggested by Kreisel, is this: a proof is finitary if we can visualize the proof. Of course neither description is very precise;"
What exactly is this supposed to mean? I can't think of a proof in mathematics that deals with concrete objects. All objects that are dealt with in mathematical proofs are abstract in nature. We're not talking about the notion of proof as used in a court room, where the objects in question are actual events and/or tangible evidence.
There is another question here which asks the same thing but the answers don't seem to help me. Instead, the answers suggest that if I can't think of a proof as an object in itself, then the above-quoted paragraph will not make sense. Again, I don't know how I'm supposed to look at a written proof as an "object in itself", and what exactly that is to accomplish.
If this suggests that my experience with proofs is lacking, that's a suggestion I absolutely reject. I hold a Master's of Science degree in pure mathematics and I've seen hundreds of proofs in lectures and textbooks. I know what a proof is, say, in analysis, but I couldn't tell you whether it's finitary according to Schoenfield's notion, or whether I'm looking at a written proof as an "object in itself" and what that is supposed to accomplish. Can someone shed light?