The Toronto Star, July 30, 2006
The poet Jan Zwicky once wrote, “Those who think metaphorically are enabled to think truly because the shape of their thinking echoes the shape of the world.”
Zwicky, whose day job includes teaching philosophy at the University of Victoria in British Columbia and authoring books of lyric philosophy such as Metaphor & Wisdom, from which the above quotation was taken, has lately directed considerable attention to contemplating the intersection of “Mathematical Analogy and Metaphorical Insight,” giving numerous talks on the subject, including one scheduled at the European Graduate School in Switzerland next week.
Casual inquiry reveals that metaphor, and its more common cousin analogy, are tools that are just as important to scientists investigating truths of the physical world as they are to poets explaining existential conundrums through verse. A scientist, one might liken, is an empirical poet; and a poet a scientist of more notional hypotheses. Neither, of course, holding a monopoly on those methodologies.
A balloon eventually pops… whereas a universe does not. At least not yet.
Both are seeking “the truth of the matter,” says Zwicky. “As a species we are attempting to articulate how our lives go and what our environment is like, and mathematics is one part of that and poetry is another.”
Analogies, whether in science or poetry, she says, are not arbitrary and meaningless, not merely “airy nothings, loose types of things, fond and idle names.”
To bolster her thesis, Zwicky cites Austrian ethologist and evolutionary epistemologist Konrad Lorenz: “Lorenz has argued that, ok, yeah, we are subject to evolutionary pressure, selection of the fittest, but that means what we perceive about the truth of the world has to be pretty damn close to what the truth of the world actually is, or the world would have eliminated us. There are selection pressures on our epistemological choices.”
Analogy appearing in scientific methodology, then, is no accident.It is fundamental to the way scientists think and the way they whittle their thinking down to truth. Zwicky, not being a mathematician (though she teaches elementary mathematical proofs in her philosophy courses), relies on historical testimony from mathematicians such as Henri Poincaré and Johannes Kepler.
“I love analogies most of all, my most reliable masters who know in particular all secrets of nature,” Kepler wrote in 1604. “We have to look at them especially in geometry, when, though by means of very absurd designations, they unify infinitely many cases in the middle between two extremes, and place the total Essence of a thing splendidly before the eyes.”
The University of Toronto’s late and great classical geometer Donald Coxeter, for instance, investigated the abstract and seemingly visually inaccessible geometric objects that reside in higher dimensions (objects known as polytopes) through a process he called “dimensional analogy.” Starting with his knowledge of our concrete three-dimensional space, he extrapolated by analogy and thus was able to investigate and intuit properties of shapes in higher dimensions.
“Mathematicians don’t talk a lot about analogy in mathematics,” says Simon Kochen, Henry Burchard Fine professor of mathematics at Princeton. “Not because it isn’t there, but just the opposite. It permeates all mathematics. It is pervasive. It’s a powerful engine for new mathematical advances.”According to Kochen, the modern mathematical method is that of axiomatics — rooted abstraction and analogy. Indeed, mathematics has been called “the science of analogy.”
“Mathematics is often called abstract,” Kochen says. “People usually mean that it’s not concrete, it’s about abstract objects.But it is abstract in another related way. The whole mathematical method is to abstract from particular situations that might be analogous or similar (to another situation). That is the method of analog.”
It is an age-old method, originating with the Greeks, with the axiomatic method applied in geometry. It entailed abstracting from situations in the real world, such as farming, and deriving mathematical principles that were put to use elsewhere. Eratosthenes used geometry to measure the circumference of the Earth in 276 BC, and with impressive accuracy.
In the lexicon of cognitive science, this process of transferring knowledge from a known to unknown is called “mapping” from the “source” to the “target.” Keith Holyoak, a professor of cognitive psychology at UCLA, has dedicated much of his work to parsing this process. He discussed it in a recent essay, “Analogy,” published last year in The Cambridge Handbook of Thinking and Reasoning.
“The source,” Holyoak says, providing a synopsis, “is what you know already — familiar and well understood. The target is the new thing, the problem you’re working on or the new theory you are trying to develop. But the first big step in analogy is actually finding a source that is worth using at all. A lot of our research showed that that is the hard step. The big creative insight is figuring out what is it that’s analogous to this problem. Which of course depends on the person actually knowing such a thing, but also being able to find it in memory when it may not be that obviously related with any kind of superficial features.”
Analogies roll off his tongue with the effortless precision of a Michael Jordan lay-up.
In an earlier book, Mental Leaps: Analogy in Creative Thought, Holyoak and co-author Paul Thagard, a professor of philosophy and director of the Cognitive Science Program at the University of Waterloo, argued that the cognitive mechanics underlying analogy and abstraction is what sets humans apart from all the other species, even the great apes. They touch upon the use of analogy in politics and law but focus a chapter on the “analogical scientist” and present a list of “greatest hits” science analogies.
The ancient Greeks used water waves to suggest the nature of the modern wave theory of sound. A millennia and a half later, the same analogical abstraction yielded the wave theory of light.
Charles Darwin formed his evolutionary theory of natural selection by drawing a parallel to the artificial selection performed by breeders, an analogy he cited in his 1859 classic The Origin of Species.
Velcro, invented in 1948 by Georges de Mestral, is an example of technological design based on visual analogy — Mestral recalled how the tiny hooks of burrs stuck to his dog’s fur. Velcro later became a “source” for further analogical designs with “targets” in medicine, biology, and chemistry. According to Mental Leaps, these new domains for analogical transfer include abdominal closure in surgery, epidermal structure, molecular bonding, antigen recognition, and hydrogen bonding.
Physicists currently find themselves toying with analogies in trying to unravel the puzzle of string theory, which holds promise as a grand unified theory of everything in the universe. Here the tool of analogy is useful in various contexts — not only in the discovery, development, and evaluation of an idea, but also in the exposition of esoteric hypotheses, in communicating them both among physicists and to the layperson.
Brian Greene, a Columbia University professor cum pop-culture physicist, has successfully translated the foreign realm of string theory for the general public with his best-selling book The Elegant Universe and an accompanying NOVA documentary, both replete with analogies to garden hoses, string symphonies, and sliced loaves of bread. As one profile of Greene observed, “analogies roll off his tongue with the effortless precision of a Michael Jordan lay-up.”
Yet at a public lecture at the Strings05 conference in Toronto, a member of the audience politely berated physicists for their bewildering smorgasbord of analogies, asking why the scientists couldn’t reach consensus on a few key analogies so as to convey a more coherent and unified message to the public.
The answer came as a disappointment. Robbert Dijkgraaf, a mathematical physicist at the University of Amsterdam, bluntly stated that the plethora of analogies is an indication that string theorists themselves are grappling with the mysteries of their work; they are groping in the dark and thus need every glimmering of analogical input they can get.
“What makes our field work, particularly in the present climate of not having very much in the way of newer experimental information, is the diversity of analogy, the diversity of thinking,” says Leonard Susskind, the Felix Bloch professor of theoretical physics at Stanford, and the discoverer of string theory.
“Every really good physicist I know has their own absolutely unique way of thinking,” says Susskind. “No two of them think alike. And I would say it’s that diversity that makes the whole subject progress. I have a very idiosyncratic way of thinking. My friend Ed Witten at Princeton’s Institute for Advanced Study has a very idiosyncratic way of thinking. We think so differently, it’s amazing that we can ever interact with each other. We learn how. And one of the ways we learn how is by using analogy.”
Susskind considers analogy particularly important in the current era because physics is almost going beyond the ken of human intelligence.
“Physicists have gone through many generations of rewiring themselves, to learn how to think about things in a way which initially was very counterintuitive and very far beyond what nature wired us for,” he says. Physicists compensate for their evolutionary shortcomings, he says, either by learning how to use abstract mathematics or by building analogies.
For his own part, Susskind deploys more of the latter. Analogy is one of his most reliable tools (visual thinking is the other). And Susskind has a few favourites that he always returns to, especially when he is stuck or confused.
He thinks of black holes as an infinite lake with boats swirling toward a drain at the bottom, and he envisions the expanding universe as an inflating balloon.
However, the real art of analogy, he says, “is not just making them up and using them, but knowing when they’re defective, knowing their limitations. All analogies are defective at some level.”
A balloon eventually pops, for example, whereas a universe does not. At least not yet.