Joe Niederberger posted Jun 16, 2013 11:15 PM: > R Hansen mentions: > >...abstract analytical (non visual) thinking. > > This just reminds me of something. A famous computer > scientist (not mathematician, unfortunately) once > confessed he could never find a relation "More > Abstract Than" on which to base a systems > architecture on, even though CS people had talked > about "levels of abstraction" for years prior. > (Parnas went on to write on of the classic papers in > the field after clearing away some built up B.S.) > i) Unless I misunderstand, the relationship "more abstract than" is simply a 'specification' of the relationship "more ... than" - and that is *fairly* well understood (*though probably more understanding is desirable, perhaps even required!)
I hasten to note that I had not earlier known either of David Parnas or of his classic paper (which I am now trying to read and understand). Offhand, it strikes me that Warfield's work in systems should effectively resolve many of the issues that arise. > > The above strikes me as somewhat the same. > Abstraction means to focus on some properties of > something while disregarding others. So Euclidean > lines and planes are already abstract. Much visual > thinking used in mathematics would be abstract by the > same token. Some thinking might be guided by > algebraic manipulation of expressions -- most people > would not call that visual thinking though the > expression on paper are visual objects. > > Ultimately, mathematics often strives towards the > "proof" as the gold standard. A proof, in the most > rigorous sense, is a formal object in a formal > system, which is abstract in one sense (if its tokens > and primitives are thought to *refer* to something > else -- the "abstraction" involved is distilling just > those properties of that something else one wants to > encode in the formal system.) On the other hand, if > the system is thought of as simply a game of symbols > and rules with no external referents, its a bit of a > misnomer to call it abstract, as it abstracts from > *nothing* in that case -- its just a game. The same > could be said for algebraic systems, if we really > view them as purely formal games. I know in > colloquial speech people would ordinarily refer to > any symbolic game of this type to be "abstract". But > from a different viewpoint, they are really > *concrete*, in that the symbols and rules are rigidly > fixed. > > Most real life proofs fall are simply reviewed > arguments that enough qualified people think could be > turned into a formal proof if push comes to shove. > Any kind of thinking, visual or otherwise, that leads > to a proof (formal or not) is mathematical, abstract > thinking, in the usual (colloquial) sense. > > Some people, I think, would regard expression > manipulation as "more abstract than" thinking about > lines and slopes and planes etc. I think I'll go with > Parnas on that issue though. > > Cheers, > Joe N > Yes, to much of your argument above - subject to the provisos that: a) I've not yet read David Parnas's paper and b) my note on Warfield's work above needs to be 'integrated' into whatever enhanced understanding I may obtain on reading the above-noted Parnas paper.