What I Mean by Reduction…

A couple of days ago, the HBL Web site provided an excerpt on concretization and reduction. It was from Jean Moroney Binswanger. [Update on Sept. 16, 2009: I am taking the liberty to copy-paste the excerpt in toto here.]

I have thought about the issues Damon Cole raises, and have a somewhat different perspective. Some ideas:

1) Concretization and reduction have two different purposes.

The purpose of concretization is to clarify the meaning of an idea (a concept, phrase, or proposition) by using perceptual concretes. Although examples are the archetypical form of concretization, there are many other forms. These include descriptions, diagrams, models, and analogies. Note that it is possible to concretize a false or invalid idea.

The purpose of reduction is to validate the idea by tracing it back through the chain of reasoning to its basis in perceptual reality. It is not enough that it connect to perceptual reality “somehow” or even that it be concretizable. Every step in the process has to be validated.
—  Jean Moroney Binswanger

Since I seem to differ from her view of reduction, I would like to write a bit about it.

(BTW, the reason I have not joined HBL, even if there is a free trial subscription on offer currently, is that many topics on the list wouldn’t be of direct relevance to me, and yet, since the quality of the discussions appears to be fairly good, subscribing might have a distracting effect on me. As it is, I fall so woefully short of time… Several books and articles to be finished, immediately…)

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On the issue of concretization, I think I agree with JMB. Concretization is the process of identifying what kind of concretes are subsumed under a given abstraction. This may involve examples, diagrams, etc.

However, reduction is where I seem to differ from JMB’s view/description (as judged by the excerpt that was given).

Given below are my thoughts (without being able to find any time to make any references).

Reduction is a process of simplifying a given abstraction. Consider the following examples of reduction. (In other words, let us concretize the process of reduction.)

  • (A) Simplification of Form (the easiest form of reduction)
    • e.g. 42/8 reduces to 21/4.
    • e.g. \sqrt{ 1 - cos^2 \theta} = \sin\theta
  • (B) Componentization (my word!), i.e. simplification via an identification of a “has-a” relationship (i.e. a Whole-to-Part relationship)
    • e.g. Upon dissociation, H_2O reduces to H_2 and O_2
  • (C) Specialization, i.e. simplification via an identification of an “is-a” relationship
    • e.g. Effecting the separation of variables, the partial differential wave equation reduces to the Helmholtz equation

A Few Notes:
(i) The “is-a” and “has-a” relations are often nicely explained in the OO programming books.
(ii) There is no suggestion that the above (A) through (C) categorization gives an exhaustive scheme.
(iii) Science provides conceptually easy examples, but examples from humanities are not as neat to classify. For example, consider this usage: “During those summer vacations, her life had been reduced to eating, reading and sleeping.” (Technically, this would be an example of componentization. But it is not as clear a case as reducing all water exclusively to hydrogen and oxygen. The list of the parts which comprise “life” is incomplete here, as often is the case.)

Notice, we did not have to go down to the level of perceptual concretes in all our examples. When it comes to reduction, the relation to perceptual concretes is important only in the ultimate sense. It is directly relevant only if the concept being reduced itself is a fairly low-level one so that the components or specialized cases that it refers to must turn out to be perceptual concretes. So, it is to be emphasized that, as a process, reduction is not primarily about identifying meaning of a concept; it is about establishing conceptual relationships from the more complex to the more simple.

Speaking metaphorically, the focus in concretization is on the final destination itself; in reduction, it is on the journey—the particular path-way. In either case, the final destination need not be perceptual concretes. (Given Ayn Rand’s works, one assumes that the “ultimate destination” is going to consist of some or the other perceptually evident concretes.)

Of the two, reduction does tend to appear more formal and rigorous because (i) the easy examples of reduction do seem to involve deduction, and (ii) of the two, it is reduction which is amenable to being summarized. Concretization, by its nature, involves elaboration or expansion of a sort. People often associate rigour with terse formulations, deduction, symbols. Therefore, reduction seems to involve rigour whereas concretization doesn’t seem to. However, do notice, deduction is not the only process involved in reduction. And, concretization does take real work

In the fallacy of “reductio ad absurdum” the real error occurs not in reduction but in the major premise. This much is true. But this does not make reduction any more error-proof than is concretization. Errors are possible in either case.

Observe that as a general rule, in the process of reduction, there occurs a concomitant reduction of scope. Forgetting this can easily lead to errors, and so can making it out as if there is no such a reduction. For example, asserting that Indian culture equals Hindu culture involves an error of reduction. (The error involved in reducing India to Indira/Sonia was/is another blatant error.)

An aside about “Western reductionism.” I do not at all like that term; it’s an anti-concept. (Has anyone ever offered a rigorously valid definition of it?) At the same time, also remember that the mind-body dichotomy (MBD) has always been, unfortunately, a very prominent part of the Western culture throughout its history. The MBD involves nothing but an erroneous reduction of Man to either the mind without the body or the body without the mind.

Exercises (if you care for one or two):
(i) Find examples of improper reduction in the context of today’s American cultural context
(ii) Identify what kind of error is involved in the fallacy of frozen abstraction (see Ayn Rand, “Philosophy: Who Needs It.”) Is it an error of concretization? of reduction? Something else?

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The reason I decided to write this post is because I wanted to clarify what I mean when I say something like: “…[under certain circumstances] FAQ reduces to the Monte Carlo method.” In all such cases, I certainly don’t mean to say that FAQ is based on the Monte Carlo (—a position seemingly implied by JMB’s piece). What I mean to say that the Monte Carlo may be considered a special case of the FAQ. … In mechanics and physics (not to mention engineering), this seems to be an acceptable usage… But if there is a correction to be made, I would like to know about it.

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Three Songs I Like…

1. (Hindi) “khoyaa khoyaa chaand, khulaa aasmaan…”
Singer: Mohammad Rafi
Music: S. D. Burman

2. (Hindi) “chandaa o chandaa (2), kisine churaayee…”
Singers: Kishore Kumar, Lata Mangeshkar
Music: R. D. Burman

3. (Marathi) “punavechaa chandram aalaa ghari, chaandaachi kirran_ daryyaavari…”
Lyrics: shanta sheLke
Singer: krishna kalle
Music: baL parTe

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