That’s right. It’s time for another installment of “What has Josh been writing for class?” This week I responded mostly to an old article by Richard Paul—who, I think, bears a striking resemblance to Walker Texas Ranger: hold on to that.

He differentiates mainly between two types of styles of problem evaluation: weak-sense and strong-sense critical thinking. To paraphrase, perhaps unfairly, weak-sense is marred by an overly narrow subproblem formulation. It’s atomistic. First you take a big problem, chop it up into smaller problems, and then solve each of the bite-sized pieces one at a time. Paul rightly notes that oftentimes this method misses the larger problem that arrise from the interplay of the otherwise well-behaved subproblems. The mathematician in me has to note that the local-behavior-does-not-imply-global-behavior phenomenon has been a central theme in differential geometry from about its beginning. The same problem creeps up just about everywhere else you look for it. I’ve tried to talk about this before in vague terms relating to urban planning and chaos theory. Maybe I should try again sometime. But for now:

Journal 3: Weak-sense, Strong-sense, and Probabilities

I agree with Paul. Strong-sense thinking is more appropriate for lots modern problems. International conflict, curricular design, and global warming all require strong-sense critical thinking, for example. (Ordering dinner at a restaurant typically does not.) While I like Paul’s network approach to problem solving, I think the primary weakness of weak-sense thinking lies in its absolutist view of truth, not necessarily its divide-and-conquer methodology. Truth, when viewed as a certainty, is rigid and fragile. Today’s demanding social and business landscape calls for something more adaptive, fluid, and functional. (Yes, you were supposed to read that last line with an announcer’s voice.) So how do I amend his framework? With probabilities of course. Really dedicated readers will see that I’ve mostly recycled my blog entry about assumptions. But to keep things fresh, I had to add something. And you knew it would happen eventually. I couldn’t resist.

I center my discussion around a theorem from linear algebra. Gleason’s Theorem tells you exactly what the probabilistic measures on the closed subspaces of a Hilbert space are (basically they’re projection operators). And according to some, it’s central to future research in information retrieval. I use it to show the usefulness of multiple points-of-view with some scientific flare. Of course, my treatment is clumsy—but technically I’m only allowed one page per entry. How thorough could I have been? Maybe later I’ll clean this up and expand it a little. For now, it’s probably okay.

**References**

Paul, Richard. “Teaching critical thinking in the ‘strong’ sense: A focus on self-deception, world views, and a dialectical mode of analysis.” Informal Logic Newsletter, 1982.

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