Axiomatic Consistency

The pre-eminent exemplar of diagrammatic thinking is, of course, Euclid's geometry.  Unlike those diagrams relying on a constant conjunction with some particular object, the object of Euclidean geometry is axiomatically defined.

1. "To draw a straight line from any point to any point.
2. To produce a finite straight line continuously in a straight line.
3. To describe a circle with any centre and distance.
4. That all right angles are equal to one another.
5. That, if a straight line falling on two straight lines make the interior angles on the same side less than two right angles, the two, if produced indefinitely, meet on that side on which are the angles less than the two right angles." [Heath's translation]

Euclid's axioms set the constraints for what can and cannot be done in geometry, but they also specify the planar and three-dimensional space where the geometry can be applied.  The result is an implicative universality.  An axiomatically defined diagram can go anywhere, not because it applies everywhere, but because it specifies just where and when it will apply.  The axioms underpin an internally consistent unity, cut off from any particular object, that can also self-direct its applications throughout the universe.  It is a serious, if common, short-changing of axiomatic thought to limit the axioms to maintaining an internal consistency.

Objective Consistency

There's a tendency with diagrammatic thinking, one Peirce succumbed to as well, to discuss it in terms of science.  Mark Twain [in Life on the Mississippi] describes what he calls the "science" of being a steamboat pilot on the  Mississippi River in the late 1800s.  The amount of knowledge was immense, one way up the river, another going down, and constantly changing as the river changed. What's more, as with all non-tacit diagrams, the pilot had to "see" it. As Mr. Bixby put it:

NO! you only learn THE shape of the river, and you learn it with such absolute certainty that you can always steer by the shape that's IN YOUR HEAD, and never mind the one that's before your eyes. [p. 103]

Twain called what he was learning a "science", yet, unlike modern science, it concerned itself with only one particular object, the Mississippi River.  In fact, as the object, the river played the crucial role of underpinning the internal consistency of the diagram as well as the inferences being made using it.  It provided an objective assurance that a conflict resolution procedure would not be needed in applying the rules.  Rather than saying such knowledge is not really "science", that it is not diagrammatic because it is not generalizable, it might be better to realize that the modern sciences, and other more general kinds of diagrams, must maintain the same kind of internal consistency without that anchor in a particular object.

Thinking Participants (cont)

The social sciences may mistreat thinking participants (my last post) in their theories, but the reality around us all the time is the avid willingness of thinking participants to take up residence in a single diagram. Many high-powered organizations have come to expect their employees/members to take up the diagrammatic underpinning of the organization as their own. Partly a matter complexity and partly a matter of loyalty, success within an organization often hinges on participant's ability to do that.

In the continuing tell-alls regarding Goldman Sachs, Jacki Zehner writes:

"I witnessed people getting promoted who were not positive 'culture carriers', … I sat and listened to arguments about how commercial people HAD to be promoted despite being poor team players, downright jerks or much more.

I also heard business leaders fight passionately for their people who were amazing positive culture carriers and less strong commercially.…  For both categories of people, there was frequently a fight and the more powerful leaders’ candidate often won.… What made the firm GREAT for so long is that one held the other in check. You need people who are very commercial but they cannot dominate or you risk the outcome that Mr. Smith described."

Two gotcha's present themselves here.

1. The corporate diagram in its dominance tends to blind the participant to other diagrams. It becomes increasingly difficult to imagine other diagrams, with differently defined inputs, different inferences, and different conclusions all more or less consistent but guided by different purposes and motives. Whether the zealous participant wants to scam a client or help them, he or she will increasingly run the risk of being unable to discern what makes them tick.
2. A total concentration on playing the game within the diagram results in the participant tending to take the game, the continued existence of the organization and it's diagram, for granted. Prudence dictates the ideal participant keep one eye on the "commercial" aspect of working the diagram for immediate profit and the other on the "cultural" aspect of maintaining the organization and its diagram over time. Yet such a balanced outlook possible? And how long can it last with onus of being "less strong commercially" hanging on it?

From the luxury of being retired, and having been a public employee, I have to say this looks like a Faustian bargain to me.

The Natural and the Social Sciences

BookTV aired a panel discussion tonight — "Panel Discussion on the Global Economy" — that took place on February 16, 2012 at the Metropolitan Museum of Art. During that discussion George Soros said:

"The natural sciences deal with facts that unfold independently of the scientific statements that are made about them. The facts, therefore, serve as an independent criteria by which the truth or validity of the statements can be judged. And that is what has enabled natural science to produce such amazing results. By contrast, the facts that form the subject matter of the social sciences are produced by thinking participants, and these agents think very differently from natural scientists. They don't have an independent criteria by which they can judge the validity of their views. Therefore, they base their decisions, not on knowledge, but on an inherently imperfect understanding of the facts."

This clearly attributes Peirce's scientific or experimental method of fixing beliefs to the natural sciences, but denies that method to the social sciences. What's more, when it comes to the social sciences, the other methods of fixing beliefs don't look that good either. The method of authority would make it difficult to refer to the social sciences as "sciences," but the a priori method would also render that designation suspect.

Soros goes on to say:

"Economics, which became the most influential of the social sciences, sought to escape this inferior status by taking an axiomatic approach similar to Euclid's geometry. But where Euclid's axioms resembled reality very closely, the axioms of the official market hypotheses distorted reality quite substantially, because they eliminated the uncertainty caused by imperfect understanding."

using precisely the standard Friedman (my last post) was arguing against — trying to match the diagrammatic assumptions with reality — when it comes to economics. Peirce and pragmatism would not offer Soros much support in this regard either. Peirce doesn't rely on the analogy between experience and a diagram, between the relationships of our experience and the relationships being diagrammed, to substantiate a diagram. Diagrams, even Euclid's, are judged by means of their results, by how the consequences diagrammatically inferred play out in regard to subsequent kinds of experience. Paraphrasing one of my undergraduate philosophy professors: "You build railroads with the parallel postulate; you send men to the moon without it."

The operative phrase in Soros' distinction between the social sciences and the natural sciences is not "external facts" but rather "thinking participants." Human beings, as the 3-dimensional creators and users of diagrams, with minds that are themselves diagrams, inevitably chafe at the constraints of being rendered into the 2-dimensional indices and relationships of a single diagram. Such social sciences are, indeed, "inherently imperfect understanding[s]" of their subject matter.

Maps: Assumptions Made

Maps are one of the most common, and intuitively understandable, instances of diagrams.  And, if you compare the aerial photograph of the region around my boyhood home on the left with a road map of the same region below, you can see a clear example of the abstraction involved with a diagram. As Peirce put it: Diagrams are "a skeleton representation of the relations concerned in the problem before the mind's eye in a schematic shape." [CP3.556]  The map, in this case, has abstracted, selected and simplified, the relationships (roads) between the indices (towns) for the purpose driving in the region. And, doing this requires certain assumptions; assumptions as to what is "important" if nothing else.

The tendency, however, is to take these assumptions for granted. Milton Friedman [in "The Methodology of Positive Economics"] writes:

"… to suppose that hypotheses have not only 'implications' but also 'assumptions' and that the conformity of these 'assumptions' to 'reality' is a test of the validity of the hypothesis different from or additional to the test by implications. This widely held view is fundamentally wrong and productive of much mischief. [p. 14]

We rarely can afford the luxury of a "test by implications," that is of actually applying the diagram and then finding out it didn't work. More often we need to make a judgment prior to using the diagram as to whether it is appropriate for the situation and our purposes. Even the road map above, as functional as it may be for driving, is missing important information and perhaps making other inappropriate assumptions for, say, hiking. The hiker using this diagram and leaving it to be tested solely by the inferences that can be made from it may find him or herself in a very difficult situation.

More generally, though, I think saying we should not be concerned with assumptions is to say we should no longer be concerned with philosophy. For philosophy — before it became the chauvinistic and sectarian thing it is today — was concerned with just that: the assumptions a diagram, a conceptual system, or a set of beliefs might make; what others assumptions could be made; and what alternative systems might then be possible. Philosophy at the turn of the century did not consider questions within a minute domain of specialization, limiting itself to calculations within a given diagram. Philosophy was the discipline that considered questions regarding diagrams in general.

Deduction

Charles S. Peirce defines "deduction" as:

"… [D]eduction consists in constructing an icon or diagram the relations of whose parts shall present a complete analogy with those of the parts of the object of reasoning, of experimenting upon this image in the imagination, and of observing the result so as to discover unnoticed and hidden relations among the parts." [CP 3.363]

However, while Peirce is specifically describing "deduction" as it functions within a certain kind of diagram here, deduction can also be understood syllogistically as it functions in the mediating aspect of diagrams more generally. For example:

All men are mortal
Socrates is a man
Therefore, Socrates is mortal.

The major premise — "All men are mortal" — is part of an understanding of human beings, a diagram consisting of a set of beliefs regarding humans. The minor premise — "Socrates is a man" — is a factual statement applying that diagram to this aspect of experience, and the conclusion — "Socrates is mortal" — is a consequence, an expectation, deduced from the diagram. In this sense, the deductive aspect of a diagram lies in applying it to experience and using it, as the "middle term", to infer what can then be expected from experience.