irrelevance of contradictions

As everyone knows some contradictions are merely surface contradictions (like ‘jumbo shrimp’) and can be ignored (or joked about), and intermittent contradictions can often be shrugged off as ‘the exception that proves the rule’, or even ‘collateral damage’. Yet other contradictions arise from the failure to distinguish between meta-level and object-level. For example, “We’re not expecting any emergencies.” is not the oxymoron it might seem (and neither is “Never say ‘never’.”).

Three more takes on contradictions are worth highlighting:

“Do I contradict myself? Very well, then I contradict myself, I am large, I contain multitudes.”
— Walt Whitman

“contradictions, inherent in ideological theories, are never a serious embarrassment to them. For ideologies are not aimed at deepening cognition, but at determining the will.”
— Hans Kelsen (p. 286 of ‘Pure Theory of Law’)

“Men do not differ much about what things they will call evils; they differ enormously about what evils they will call excusable.”
— G. K. Chesterton

keywords: Logic, distinction


The long-standing rivalry between the incenter and the circumcenter was permanently weighted in favor of the circumcenter when the circumcenter had the perspicacity to set up a refreshment area (with free donuts and coffee) that it designated as ‘Drop-In Center’, whose subliminal message will surely have a telling effect in the long run.
keywords: Mathematics, Geometry, triangles, circles

relative strengths of mathematical results

The fact that a vertex angle of a triangle can be divided (bisected, trisected, etc.) in a specified manner before the opposite side is finally fixed (imagining the triangle to be under gradual construction), is consistent with the apparent situation that a result based on vertex-angle division is ‘stronger’ than the corresponding result for division of the opposite side. For example, the three vertex-angle bisectors meet at the incenter of the triangle, whereas the three side bisectors (medians) meet at the centroid (balance point). Arguably, the center of the inscribed circle is ‘more important’ than the balance point. Again, the vertex-angle trisectors give rise to an equilateral triangle, whereas the side trisectors give rise to a similar triangle of 1/5 the linear length of the original triangle. Here, there is even less doubt about the relative importance: the former is known as ‘Morley’s miracle’, whereas the latter seems merely awesome.
keywords: Logic, Mathematics, Geometry

lorem ipsum

Being compromised is the price of not moving fast enough.
More formally: the following reciprocity relationship obtains: the amount of compromise times the amount of speed is constant.
keywords: life, General Systems Theory