In the Abstract to Psychoanalysis and Socratic Education*** by Trevor Pateman (article available as a .rtf document), it is written that:
"A range of concepts are introduced to argue similarities between Socratic Education and Freudian psychoanalysis. The concepts are these: the talking cure; floating attention; knowledge and acknowledgment; judgment and intuition; (prior) theoretical understanding; attending for truth; acting in role; play; negative dialectics; the training of the self ... "
What interests us particularly is Bateman's discussion of judgment and intuition, the former - in his definition - involving what we know or think to to know in an appeal to shared knowledge and the latter - in his definition - involving the subjective expression of how things look or feel to us as individuals. Bateman writes:
"The exercise of judgment involves appeal to what I know or think I know at some articulate level of consciousness. Typically, judgment appeals to shared knowledge: what everyone knows or thinks. So rationalization and self-deception find ready-made support in all kinds of conventional wisdom...."
In contrast, intuition is the expression of a personally experienced connection, drawing on a reservoir of inarticulate consciousness. Intuition is the expression of how things look or feel to me.... [I]ntuition will get us to a (correct) result well before we have the means to judge its correctness ... [M]athematicians have the concept of a hand proof. In a hand proof, there is no (real) proof, just a lot of handwaving. But it gestures to an intuition that if we set out in the general direction indicated by the hand proof, we will get to the proof we want to reach. Intuition is then like a compass. [emphasis added]
But intuition does not always work like this; sometimes it leads us astray. Shown the Muller-Lyer lines, I may intuit that one is longer than the other, but I am actually wrong; judgment is against me. But it still remains that the lines look that way. (The Muller-Lyer lines are the ones placed parallel to each other, but with arrow-heads pointing in opposite directions)...."
To see a graphic of Muller-Lyer lines, see the Epistemology of Perception at The Internet Encyclopedia of Philosophy.
*** "Website version 2004. A first version appeared, under the same title, in a 1993 issue of Aspects of Education (University of Hull, England), number 49, pages 76 - 80. A second version, again under the same title, appeared in S.Appel, editor, Psychoanalysis and Pedagogy (Bergin and Garvey: Westport, Connecticut), pages 45 - 51. Copyright material used by permission."
The basic problem with "hand" proofs as opposed to "mechanical" proofs is that, as noted by Neeraj Suri, Michelle M. Hugue, and Chris J. Walter in Synchronization Issues in Real-Time Systems:
"As hand proofs are sensitive to the skills of the prover, mechanical proofs are sensitive to the correctness of the theorem prover and its underlying logic. "
In other words, to employ a phrase used by Thomas Kuhn (The Structure of Scientific Revolutions), if the paradigms (viz. "mechanical" proofs) underlying a given view of "shared knowledge" are wrong, then that knowledge is likely also wrong. What this means is that someone along the way has done an intuitive "hand proof" which does not conform to the judgmental mechanical proofs in vogue. A hand proof made by a skilled prover is thus always at the root of progress, in any field.
Another example of "hand proofs" is the method by which our legal system relies upon the opinions of judges, rather than on computer-produced verdicts applying fixed mechanical theorems. Here, "skilled" provers are viewed as superior to a computer theorem.As concerns the progress of science (and law), Thomas Kuhn's theory of scientific revolutions is thus in our view more accurate than Karl Popper's ideas concerning scientific thought as the falsifiability of mainstream statements because "hand proofs" - also in fields other than mathematics - are generally made to conform to the observations at hand, often initially ignorning completely any presumed attempt to "falsify" existing mainstream ideas.
In law, precedents may in fact have to be overturned, but that is not the main purpose of an opinion which overrules prior judicial decisionmaking. Rather, new rules are being made to conform to new observations and events.
Only after an observation-fitting hand proof is made and then compared with the mechanical proofs in vogue does the battle with the inertia of existing paradigms begin.
Mainstream scientists want the ensuing discussion to proceed under their terms and thus demand that their theories be proven false. This, however, does not accurately describe the process of scientific discovery, nor does it describe the primary motivation for overturning precedents in law - and this constitutes Popper's main error in analysis. Popper, by concentrating on mainstream science, does not actually describe the actual process of scientific (or legal) advance - rather, he describes the process of mainstream resistance to advance and the inefficient mechanisms by which that resistance is or can be broken.
The true pioneers in science (or law), on the other hand, and this is where Kuhn's analysis is the more accuracte, have no interest to waste their time on developing proofs to falsify the erroneous theories in vogue, but rather, prefer to be busy building up their own systems which correspond to the evidence at hand. The falsification process of the erroneous prevailing theories is then later carried out by others, i.e. the innovators and early adopters of new theories.
Good examples here are the "hand proof" works of Isaac Newton, which presented new interpretations of observed phenomena and spent as little time as possible wasting time in disproving the erroneous ideas of others.
Another example of new paradigms and hand proofs is the Constitution of the United States, which is a "new discovery" that concentrates on new things to be achieved, rather than on old things to be "disproven". This in fact is still the genius of America, several hundred years later. America is a "Kuhnian" world of new paradigms and "hand proofs", whereas the Old World (Europe, Middle East) is in part still caught in a maelstrom of Popperian inertia of resistance to change, functioning by antiquated and long outdated mechanical solutions (unreformed social systems, entrenched social classes, overemphasis on tradition, no longer state-of-the-art customs, deference to nobility at the cost of modern social innovation, etc.)
As the "hand proof" says, "go for it". That's the American way which is sorely lacking in the Old World.