Brian Hayes offers the following delightful quote from John Aubrey’s *Brief Lives*, concerning the 17th century philosopher Thomas Hobbes:

He was 40 yeares old before he looked on geometry; which happened accidentally. Being in a gentleman’s library in…, Euclid’s Elements lay open, and ’twas the 47

El. libriI. He read the proposition. “By G—,” sayd he (he would now and then sweare, by way of emphasis), “this is impossible!” So he reads the demonstration of it, which referred him back to such a proposition; which proposition he read. That referred him back to another, which he also read.Et sic deinceps,that at last he was demonstratively convinced of that trueth. This made him in love with geometry.

Hobbes is of course most famous, and of continuing relevance, for his thoughts on what life is like in the absence of strong central government, i.e., “solitary, nasty, poor, brutish and short”, which seems an increasingly accurate picture of Iraq. But Hobbes made interesting observations on many other matters, including the nature of thought.

Hayes claims that

What’s most remarkable about this tale—whether or not there’s any trueth in it—is the way Hobbes is persuaded against his own will. He starts out incredulous, but he can’t resist the force of deductive logic. From proposition 47 (which happens to be the Pythagorean theorem), he is swept backward through the book, from conclusions to their premises and eventually to axioms. Though he searches for a flaw, each step of the argument compels assent. This is the power of pure reason.

“Remarkable” is to some extent in the mind of the marker. Hobbes’ inability to resist the force of deductive logic seems to me commonplace; every one of us, every day, is forced to accept unpalatable conclusions when presented with deductive, or even just strong, arguments.

Rather, the *most *remarkable feature of this situation may be the fact that Euclid’s propositions were presented in such a systematic and transparent way that Hobbes was able to follow, without confusion or misdirection, a long chain of inferences. In other words, the special feature of Euclid’s Elements – the feature of the greatest intellectual significance – is that a very complex set of arguments had been laid out in such a way that the inferential connections among propositions were unambiguously detectable.

In other words, Euclid’s arguments had been properly mapped.

This is quite remarkable because most of the time, in Euclid’s era and today, arguments of the greatest significance are *not *properly mapped out. They are not articulated to the point where even the authors are fully clear about the inferential relationships among their propositions; and their readers are presented with the hopeless task, at which they generally fail miserably, of attempting to figure out what those propositions and relationships are.

Euclid sets a standard to which we ought to be continually aspiring, even in non-mathematical domains. These days we have even less excuse for falling short, because we have better tools for rapidly mapping complex arguments and presenting the results.

Our problem today is largely a function of how we were taught. Arithmetic and math in grade schools in the US is not taught as a logical progression, but rather as a smattering of topics. Civics isn’t taught from a structure, such as the Constitution, but rather as unrelated current events. History itself is taught entirely without self-criticism. And so with other subjects. The reason, I think, is that the second-rate authors of elementary and high school textbooks, who make a great deal of money for their “work,” simply restate the obvious without giving it any thought.

Anthony,

This ramshackle approach to teaching that you mention is popular not only in elementary schools, but in universities and colleges as well.

Personally, my political economy classes in university were are real treat. They usually consisted of a long rant, a.k.a. a lecture, followed by an even longer emotionally super-charged and circular classroom discussion.

One of the great appeals of conceptual mapping is that the very act of visually recognizing the complexity of an idea can magically remove any political sticking points that may otherwise prevent a clear discussion.

The end result is hopefully that students leave the classroom feeling intellectually satisfied, rather than emotionally frustrated.