There’s a model of philosophical rigor that denounces all analogies, metaphors, and examples.  It is an ancient tradition.  Plato, of course, denounces the use of examples tirelessly throughout his writings.  “I did not ask you for an example of piety, Euthyphro, but to articulate what piety itself is or what is common to all instances of piety!”  In the Republic, dianoia– mathematical reasoning –is not yet the highest form of knowledge because it relies on writing and the use of diagrams.  The symbols I use in my notation can get in the way of recognizing relationships as in the case of the near impossibility of doing even the simplest arithmetic using Roman numerals, and the diagrams I use to help me to think of a conic section or a triangle can lead me astray insofar as no matter how hard I try I can’t draw triangleness, only a triangle.  Writing, images, and examples in all of their forms, this tradition claims, are doomed to lead us astray.  Thus, much later in the Science of Logic, Hegel will attempt a pure logic that eschews any examples, metaphors, or analogies altogether.  These will be deemed beneath the dignity of the concept.

While I understand these arguments and the dangers of writing, the image, the example, the metaphor, and the analogy, I nonetheless believe that we will never do better than examples, metaphors, and analogies.  The attempt to purify thought, to produce pure thought, by chasing away all examples, metaphors, and analogies is, in my view, despite appearances to the contrary, a lack of rigor.  Lurking somewhere in the thought, as formal and free of examples as it might appear, is always a privileged example that governs ones inquiry, a master metaphor or analogy.  Yet in that writing that strives for pure formalism, the privileged example, master metaphor, and analogy are hidden from view…  They hidden from both the author and the reader alike.  As a result, they govern and structure their thought like a gravitational attractor without author or reader realizing it.  Rigorous thought involves taking responsibility for our examples, metaphors, and analogies so as to explore how they might influence our thinking while also recognizing that there is no escaping them…  Even in mathematics.