analogies vs metaphors — Tarik Yildirim

“The existence of analogies between central features of various theories implies the existence of a general abstract theory which underlies the particular theories and unifies them with respect to those central features.”
- Eliakim Hastings Moore

Conceptual similarities manifest themselves as analogies, where one recognizes that two structures X and Y have a common meaningful core, say A, which can be pulled up to a higher level. The resulting relationship is symmetric in the sense that the structure A specializes to both X and Y. In other words, one can say either “X is like Y via A” or “Y is like X via A”.

The analogy get codified in the more general structure A which in turn is mapped back onto X and Y. (I say “onto” because A represents a bigger set than both X and Y.) Discovering A is revelatory in the sense that one recognizes that X and Y are special instances of a more general phenomenon, not disparate structures.

Metaphors play a similar role as analogies. They too increase our total understanding, but unlike analogies, they are not symmetric in nature.

Say there are two structures X and Y where is X is more complex but also more familiar than Y. (In practice, X often happens to be an object we have an intuitive grasp of due to repeated daily interaction.) Discovering a metaphor, say M, involves finding a way of mapping X onto Y. (I say “onto” because X - via M - ends up subsuming Y inside its greater complexity.)

The explanatory effect comes from M pulling Y up to the familiar territory of X. All of a sudden, in an almost magical fashion, Y too starts to feel intuitive. Many paradigm shifts in the history of science were due to such discrete jumps. (e.g. Maxwell characterizing the electromagnetic field as a collection of wheels, pulleys and fluids.)

Notice that you want your analogy A to be as faithful as possible, capturing as many essential features of X and Y. If you generalize too much, you will end up with a useless A with no substance. Similarly, for each given Y, you want your metaphor pair (X,M) to be as tight as possible, while not letting X stray away from the domain of the familiar.

You may be wondering what happens if we dualize our approaches in the above two schemes.

Analogies. Instead of trying to rise above the pair (X,Y), why not try to go below it? In other words, why not consider specializations that both X and Y map onto, rather than focus on generalizations that map onto X and Y?
Metaphors. Instead of trying to approach Y from above, why not try approach it from below? In other words, why not consider metaphors that map the simple into the complex rather than focus on those that map the complex onto the simple?

The answer to both questions is the same: We do not, because the dual constructions do not require any ingenuity, and even if they turn out to be very fruitful, the outcomes do not illuminate the original inputs.

Let me expand on what I mean.

Analogies enhance our analytic understanding of the world of ideas. They are tools of the consciousness, which can not deal with the concrete (specialized) concepts head on. For instance, since it is insanely hard to study integers directly, we abstract and study more general concepts such as commutative rings instead. (Even then the challenge is huge. You could devote your whole life to ring theory and still die as confused as a beginner.)
In the world of ideas, one can easily create more specialized concepts by taking conjunctions of various X’s and Y’s. Studying such concepts may turn out to be very fruitful indeed, but it does not further our understanding of the original X’s and Y’s. For instance, study of Lie Groups is exceptionally interesting, but it does not further our understanding of manifolds or groups.
Metaphors enhance our intuitive understanding of the world of things. They are tools of the unconsciousness, which is familiar with what is more immediate, and what is more immediate also happens to be what is more complex. Instruments allow us to probe what is remote from experience, namely the small and the big, and both turn out to be stranger but also simpler than the familiar stuff we encounter in our immediate daily lives.
- What is smaller than us is simpler because it emerged earlier in the evolutionary history. (Compare atoms and cells to humans.)
- What is bigger than us is simpler because it is an inanimate aggregate rather than an emergent life. (Those galaxies may be impressive, but their complexity pales in comparison to ours.)
In the world of things, it is easy to come up with metaphors that map the simple into the complex. For instance, with every new technological paradigm shift, we go back to biology (whose complexity is way beyond anything else) and attack it with the brand new metaphor of the emerging Zeitgeist. During the industrial revolution we conceived the brain as a hydraulic system, which in retrospect sounds extremely naive. Now, during the digital revolution, we are conceiving it as - surprise, surprise - a computational system. These may be productive endeavors, but the discovery of the trigger metaphors itself is a no-brainer.

Now is a good time to make a few remarks on a perennial mystery, namely the mystery of why metaphors work at all.

It is easy to understand why analogies work since we start off with a pair of concepts (X,Y) and use it as a control while moving methodically upwards towards a general A. In the case of metaphors, however, we start off with a single object Y, and then look for a pair (X,M). Why should such a pair exist at all? I believe the answer lies in a combination of the following two quotes.

"We can so seldom declare what a thing is, except by saying it is something else."
- George Eliot
“Subtle is the Lord, but malicious He is not.”
- Albert Einstein

Remember, when Einstein characterized gravitation as curvature, he did not really tell us what gravity is. He just stated something unfamiliar in terms of something familiar. This is how all understanding works. Yes, science is progressing, but all we are doing is just making a bunch of restatements with no end in sight. Absolute truth is not accessible to us mere mortals.

“Truths are illusions which we have forgotten are illusions — they are metaphors that have become worn out and have been drained of sensuous force, coins which have lost their embossing and are now considered as metal and no longer as coins.”
- Friedrich Nietzsche

The reason why we can come up with metaphors of any practical significance is because nature subtly keeps recycling the same types of patterns in different places and at different scales. This is what Einstein means when he says that the Lord is not malicious, and is why nature is open to rational inquiry in the first place.

Unsurprisingly, Descartes himself, the founder of rationalism, was also a big believer in the universality of patterns.

Descartes followed this precept by liberal use of scaled-up models of microscopic physical events. He even used dripping wine vats, tennis balls, and walking-sticks to build up his model of how light undergoes refraction. His statement should perhaps also be taken as evidence of his belief in the universality of certain design principles in the machinery of Nature which he expects to reappear in different contexts. A world in which everything is novel would require the invention of a new science to study every phenomenon. It would possess no general laws of Nature; everything would be a law unto itself.
John D. Barrow - Universe That Discovered Itself (Page 107)

Of course, universality does not make it any easier to discover a great metaphor. It still requires a special talent and a trained mind to intuit one out of the vast number of possibilities.

Finding a good metaphor is still more of an art than a science. (Constructing a good analogy, on the other hand, is more of a science than an art.) Perhaps one day computers will be able to completely automate the search process. (Currently, as I pointed out in a previous blog post, they are horrible at horizontal type of thinking, the type of thinking required for spotting metaphors.) This will result in a disintermediation of mathematical models. In other words, computers will simply map reality back onto itself and push us out of the loop altogether.

Let us wrap up all the key observations we made so far in a single table:

Now let us take a brief detour in metaphysics before we have a one last look at the above dichotomy.

Recall the epistemology-ontology duality:

An idea is said to be true when every body obeys to it.
A thing is said to be real when every mind agrees to it.

This is a slightly different formulation of the good old mind-body duality.

Minds are bodies experienced from inside.
Bodies are minds experienced from outside.

While minds and bodies are dynamic entities evolving in time, true ideas and real things reside inside a static Platonic world.

Minds continuously shuffle through ideas, looking for the true ones, unable to hold onto any for a long time. Nevertheless truth always seems to be within reach, like a carrot dangling in the front.
Minds desperately attach names to phenomena, seeking permanency within the constant flux. Whatever they refer to as a real thing eventually turns out to be unstable and ceases to be.

Hence, the dichotomy between true ideas and real things can be thought of as the (static) Being counterpart of the mind-body duality which resides in (dynamic) Becoming. In fact, it would not be inappropriate to call the totality of all true ideas as God-mind and the totality of all real things as God-body.

Anyway, enough metaphysics. Let us now go back to our original discussion.

In order to find a good metaphor, our minds scan through the X’s that we are already experientially familiar with. The hope is to be able to pump up our intuition about a thing through another thing. Analogies on the other hand help us probe the darkness, and bring into light the previously unseen. Finding a good A is like pulling a rabbit out of a hat, pulling something that was out-of-experience into experience. The process looks as follows.

First you encounter a pair of concepts (X,Y) in the shared public domain, literally composed of ink printed upon a paper or pixels lighting up on a screen.
Your mind internalizes (X,Y) by turning it back to an idea form, hopefully in the fashion that was intended by its originator mind.
You generalize (X,Y) to A within the world of ideas through careful reasoning and aesthetic guidance.
You share A with other minds by turning it into a thing, expressed in a certain language, on a certain medium. (An idea put in a communicable form is essentially a thing that can be experienced by all minds.)
End result is a one more useful concept in the shared public domain.

Analogies lift the iceberg, so to speak, by bringing completely novel ideas into existence and revealing more of the God-mind. In fact, the entirety of our technology, including the technology of reasoning via analogies, can be viewed as a tool for accelerating the transformation of ideas into things. We, and other intermediary minds like us, are the means through which God is becoming more and more aware of itself.

Remember, as time progresses, the evolutionary entities (i.e. minds) decrease in number and increase in size and complexity. Eventually, they get

so good at modeling the environment that their ideas start to resemble more and more the true ideas of the God-mind, and
so good at controlling the environment that they become increasingly indistinguishable from it and the world of things start to acquire a thoroughly mental character.

In the limit, when the revelation of the God-mind is complete, the number of minds finally dwindles down to one, and the One, now synonymous with the God-mind, dispenses with analogies or metaphors altogether.

As nothing seems special any more, the need to project the general onto the special ceases.
As nothing feels unfamiliar any more, the need to project the familiar onto the unfamiliar ceases.

Of course, this comes at the expense of time stopping altogether. Weird, right? My personal belief is that revelation will never reach actual completion. Life will hover over the freezing edge of permanency for as long as it can, and at some point, will shatter in such a spectacular fashion that it will have to begin from scratch all over again, just as it had done so last time around.