In the Introduction, the word 'isomorphism' was defined as an information preserving transformation. We can now go into that notion a little more deeply, and see it from another perspective. The word 'isomorphism' applies when two complex structures can be mapped onto each other, in such a way that to each part of one structure there is a corresponding part in the other structure, where 'corresponding' means that the two part play similar roles in their respective structures. This usage of the word 'isomorphism' is derived from a more precise notion in mathematics. Meaning and Form in Mathematics 57 It is cause for joy when a mathematician discovers an isomorphism between two structures which he knows. It is often a 'bolt from the blue', and a source of wonderment. The perception of an isomorphism between two known structures is a significant advance in knowledge-and I claim that it is such perceptions of isomorphism which create meanings in the minds of people. A final word on the perception of isomorphisms: since they come in many shapes and sizes, figuratively speaking, it is not always totally clear when you really have found an isomorphism. Thus, 'isomorphism' is a word with all the usual vagueness of words-which is a defect but an advantage as well.

• meaning
• roles
• knowledge
• perspective
• structure
• joy
• perception
• mathematics
• transformation
• other

If there is a test for theoremhood, a test which does always terminate in a finite The MU-puzzle 48 amount of time, then that test is called a decision procedure for the given formal system. When you have a decision procedure, then you have a very concrete characterization of the nature of all theorems in the system.

• time
• nature

There seems to he one common culprit in these paradoxes, namely self-reference, or 'Strange Loopiness'. So if the goal is to ban all paradoxes, why not try banning selfreference and anything that allows it to arise? This is not so easy as it might seem, because it can be hard to figure out just where self-reference is occurring. It may be spread out over a whole Strange Loop with several steps, as in this 'expanded' version of Epimenides, reminiscent of Drawing Hands: The following sentence is false. The preceding sentence is true. Taken together, these sentences have the same effect as the original Epimenides paradox: yet separately, they are harmless and even potentially useful sentences. The 'blame' for this Strange Loop can't he pinned on either sentence-only on the way they 'point' at each other. In the same way, each local region of Ascending and Descending is quite legitimate; it is only the way they are globally put together that creates an impossibility.

• self
• other

The 'Strange Loop' phenomenon occurs whenever, by moving upwards (or downwards) through the levels of some hierarchical system, we unexpectedly find ourselves right back where we started. (Here, the system is that of musical keys.) Sometimes I use the term Tangled Hierarchy to describe a system in which a Strange Loop occurs. As we go on, the theme of Strange Loops will recur again and again. Sometimes it will be hidden, other times it will be out in the open; sometimes it will be right side up, other times it will be upside down, or backwards. 'Quaerendo invenietis' is my advice to the reader.

• will
• other

A similar usage survives in English today: the word 'recherche' means, literally, 'sought out', but carries the same kind of implication, namely of esoteric or highbrow cleverness.

• etymology
• english