Mathematical analogy and metaphorical insight

Abstract

Chandler Davis. M y interest in this topic grows out of a long-standing, handson engagement with the making of metaphors. As a poet, and as an editor and reader of poetry, I have often been struck by the power of good metaphors to change my stance in the world, to alter in a profound and, it seems, permanent way how I look at things. Whence this power? And further, how is it that we are able to distinguish such 'good', world-altering, metaphors from metaphors that are merely outr6 or arcane--surprising linguistic constructions that lack, or seem to lack, genuine ontological depth? It is difficult to offer examples without quoting whole poems or paragraphs. A metaphor is like a depth charge: if you know nothing about the material in which it is embedded, it can be difficult to evaluate its force. Some very powerful metaphors can speak to us sans context; but often metaphors in the shallow and mid-ranges won' t yield up their full meaning on their own. As a consequence, almost any candidate for a shallow metaphor that I might offer without context can appear as a challenge to invent a context in which it would appear effective. With this caveat, let me offer without further commentary the following: "the eyes are the windows of the soul" (a good or strong metaphor); "the table fizzed like a platypus" (a weak or shallow metaphor); "the river/Is a strong brown god" (good); "the luggage resembled/godly" (weak); "the road was a ribbon of highway, perfect for Pekinese" (weak); "If I have exhausted the justifications I have reached bedrock, and my spade is turned" (good). My interest, as I say, is in what appears to be an intuitive capacity many of us have for being struck by certain metaphors, and for being left cold by others. What is involved in the comprehension of good metaphors? How do metaphors mean? (It will be clear from the preceding that I am using the term 'metaphor ' broadly to cover any linguistic expression of focussed analogical thinking. Thus what we would strictly regard as a simile is also 'metaphorical' in the sense I am concerned with.) In reflecting on these issues, three things struck me more or less simultaneously. The first was that metaphors involve what Wittgenstein called "seeingas" [1], a seeing of one thing in terms of another. The second was that, as a working poet, I find that understanding a metaphor feels like understanding certain kinds of mathematical demonstrations: I am aware of features of various figures or expressions, or various images or ideas, being pulled into revealing alignment with one another by the demonstration or the metaphor. The final observation---or, in this case, i dea was that seeing-as involves a kind of re-cognition, and, as such, is what we mean when we say we understand something. This last claim is essentially empirical in nature. If we ask "When do people say that they 'understand' or 'get' or 'see' or 'grasp' something?", it turns out that the experience of 'getting it' seems to involve a reconfiguration of an initially problematic array or scenario---a redirection of emphasis that somehow affects the overall shape of the problem. And the emergence of this new way of looking at things is often accompanied by a feeling of astonishment, or of things falling into place, of their coming home. In sum, I began my investigations with the intuition that both metaphors and certain kinds of mathematical demonstrations are species of analogical reasoning: both say, in effect, "Look at things like this, if you want to understand them". But how close is the connexion between mathematical analogy and metaphor? (And here I should perhaps emphasize that my use of the phrase 'mathematical analogy' is intended to be at least as broad as my use of 'metaphor'---it embraces everything from certain visual proofs of the Pythagorean theorem to Euler's result about the sum of the reciprocals of the squares.) Mathematics clearly involves reasoning--but poetry? Aren't literary metaphors simply inventions---airy noth-