The theorem can be likened to a pearl, and the method of proof to an oyster. The pearl is prized for its luster and simplicity; the oyster is a complex living beast whose innards give rise to this mysteriously simple gem.
Context thus changes categorization and can modify how we perceive even the most familiar of items.
No thought can be formed that isn’t informed by the past; or, more precisely, we think only thanks to analogies that link our present to our past.
The repetition of the orphanage drama year after year, echoing the Nietzschean idea of eternal recurrence – that everything that has happened will happen again and again – seems to rob the little world of any real meaning. Why should the repetition of the fire inspector’s lament make it sound so hollow?
Often compound words have drifted so far from their etymological roots that native speakers can easily miss what is right in front of their eyes.
Although dictionaries give the impression of analyzing words all the way down to their very atoms, all they do in fact is graze their surfaces.
When a system of “meaningless” symbols has patterns in it that accurately track, or mirror, various phenomena in the world, then that tracking or mirroring imbues the symbols with some degree of meaning-indeed, such tracking or mirroring is no less and no more than what meaning is. Depending on how complex and subtle and reliable the tracking is, different degrees of meaningfulness arise.
Just as we need to hide the massively complex details inside our fancy gadgets by elegant and user-friendly packaging, so we need to hide the details of many ideas in order to talk about them in a sufficiently compact way that we won’t get lost in a mountain of details.
Any situation permits a host of diverse categorizations.
At the outset, there is a concrete situation with concrete components, and thus it is perceived as something unique and cleanly separable from the rest of the world. After a while, though – perhaps a day later, perhaps a year – one runs into another situation that one finds to be similar, and a link is made.
Poised midway between the unvisualizable cosmic vastness of curved spacetime and the dubious shadowy flickerings of charged quanta, we human beings, more like rainbows and mirages than like raindrops or boulders, are unpredictable self-writing poems – vague, metaphorical, ambiguous, sometimes exceedingly beautiful.
There can be conceptual skeletons on several different levels of abstraction. For instance, the “isomorphism” between Bongard problems 70 and 71, already pointed out, involves higher-level conceptual skeleton than that needed to solve either problem in isolation.
This was a tall order, and one could criticize it on the grounds that it was somewhat circular: how can you justify your methods of reasoning on the basis of those same methods of reasoning? It is like lifting yourself up by your own bootstraps.
Why is it that we would use the very same word to denote two different levels on a ladder of abstraction?
We mortals are condemned not to speak at that level of no information loss. We necessarily simplify, and indeed, vastly so. But that sacrifice is also our glory. Drastic simplification is what allows us to reduce situations to their bare bones, to discover abstract essences, to put our fingers on what matters, to understand phenomena at amazingly high levels, to survive reliably in this world, and to formulate literature, art, music, and science.
Categorization thus helps one to draw conclusions and to guess about how a situation is likely to evolve.
In short, nonstop categorization is every bit as indispensable to our survival in the world as is the nonstop beating of our hearts. Without the ceaseless pulsating heartbeat of our “categorization engine”, we would understand nothing around us, could not reason in any form whatever, could not communicate with anyone else, and would have no basis on which to take any action.
If categorization is central to thinking, then what mechanism carries it out? Analogy is the answer.
Such concepts, be they concrete or abstract, are selectively mobilized instant by instant, and nearly always without any awareness on our part, and it is this ceaseless activity that allows us to build up mental representations of situations we are in, to have complex feelings about them, and to have run-of-the-mill as well as more exalted thoughts. No thought can be formed that isn’t informed by the past; or, more precisely, we think only thanks to analogies that link our present to our past.
For Kant, analogy was the wellspring of all creativity, and Nietzsche gave a famous definition of truth as “a mobile army of metaphors”.
Where there’s a pattern, there’s a reason.