What’s it Like?

The Science of Scientific Analogies

by Chris Edwards

To fully understand the significance of the argument made by Douglas Hofstadter and Emmanuel Sander in their new book Surfaces and Essences: Analogy as the Fuel and Fire of Thinking, one must first be familiar with the fence (not a wall) separating mathematics and science from the humanities. Physicists, for example, are trained to see mathematics as the only “real” way of understanding scientific phenomena. Many great physicists explain their concepts to lay audiences using metaphors and analogies, but often regard this process as sharing Platonic shadows with those who could not comprehend the ideal. As a brief aside into the history of analogical thinking in philosophy will show, a good many serious thinkers have been widening the holes in this fence for a while now, but Hofstadter and Sander not only want to tear the fence down, but seek to have the humanities annex mathematics entirely. Central to their argument is that even Einstein’s insights came to him primarily in the form of analogy, and while the authors answer no significant philosophical or theoretical questions, their argument does open up new pathways for approaching these questions.1

As is often the case in philosophy, this analogical approach has intellectual ancestors. Perhaps the first statement involving the concept can be attributed to the Enlightenment philosopher Joshua Reynolds, who in 1776 wrote:

This search and study of the history of the mind, ought not to be confined to one art only. It is by the analogy that one art bears to another, that many things are ascertained, which either were but faintly seen, or, perhaps, would not have been discovered at all, if the inventor had not received the first hints from the practices of a sister art on a similar occasion.2

The counterintuitive idea here is that if a thinker wants to advance in an academic field, then she should study widely outside the field in search of analogies that can then be superimposed on problems from the original field. Here Reynolds is identifying the centrality of analogical thinking for the sciences.

Although Ludwig Wittgenstein’s famously impenetrable 1921 Tractatus Logico-Philosophicus, continues to spark controversy—largely because of its vagueness—modern forays into epistemology rescue the core of Wittgenstein’s philosophy. He wrote, for example:3

2.063 The total reality is the world

2.1 We make to ourselves pictures of facts

2.11 The picture presents the facts in logical space, the existence and nonexistence of atomic facts.

2.12 The picture is a model of reality

2.13 To the objects correspond in the picture the elements of the picture.

2.131 The elements of the picture stand, in the picture, for the objects

2.14 The picture is a fact.

2.15 That the elements of the picture are combined with one another in a definite way, represents that the things are so combined with one another. This connexion of the elements of the picture is called its structure, and the possibility of this structure is called the form of representation of the picture.

2.151 The form of representation is the possibility that the things are combined with one another as are the elements of the picture.

2.1211 The pictures is linked with reality; it reaches up to it.

2.1512 It is like a scale applied to reality.

In the above passage, Wittgenstein presages Leonard Mlodinow and Stephen Hawking’s theoretical position of Model-Dependent Realism (MDR) by noting that the mind makes models based upon “atomic facts.”4 Wittgenstein does not deign to explain his concepts and phrases, but by putting his words into a philosophical context, we may infer that “atomic facts” refers to the way that objects are perceived by human senses.

Wittgenstein’s assumption, then, is that the model of reality constructed by the mind acts in accordance with the “atomic” reality. Using analogical language, he compares the mind model to reality, saying our mind models of the “atomic facts” are, in fact, like a scale model. Much of his argument relies on simple faith. In fact, evolutionary biology likely gives us a representative model of the universe which is grounded in “atomic” reality, simply because it is difficult to see how a radically counterintuitive mind model would be useful for helping an organism survive or mate.

In 1980, George Lakoff and Mark Johnson published the importantly insightful work Metaphors We Live By, in which they explained how metaphorical thinking, such as “time is money,” lies at the heart of language and of thought.5 People tend to think through analogies and metaphors, and therefore the unattended mind can get caught in false metaphors. Prior to that, in his 1962 book The Structure of Scientific Revolutions, Thomas Kuhn, while avoiding the language of metaphors and analogies, coined the word paradigm to describe the totality of the scientific understanding of the universe at any given time.6 Kuhn theorized that paradigms often face crisis from new facts which, to borrow the language of Lakoff and Johnson, would threaten the existing set of metaphors and analogies.

The problem, from the point of view of physicists, is that the notion that analogy or metaphors might get thinkers closer to an understanding of how the universe functions than mathematics makes the scientific endeavor seem a little too much like, well, like the humanities. Hofstadter and Sander argue against this concept repeatedly in the book and are quite right to note that mathematicians and physicists are reluctant to stray from their numbers into the murky waters of epistemology or metaphysics.

A quote from the eminent physicist Leonard Susskind, in his book about his disagreement with Stephen Hawking over black holes, summarizes the attitude of most physicists. Susskind wrote that, after having finished Thomas Kuhn’s classic work that the ideas of the book sparked his thinking prior to a lecture he gave in Santa Barbara in 1993:

I would have liked to add some philosophical remarks about how evolution has created a mental picture that guides our actions when it comes to caves, tents, houses, and doors, but that it misleads when it comes to black holes and horizons. Yet those remarks would have been ignored. Physicists want facts, equations, and data—not philosophy and evolutionary pop psychology.7

Susskind’s disdain is an echo of Richard Feynman’s famous put down that the philosophy of science is about as useful to scientists as ornithology is to birds. Yet, Edward O. Wilson, one of the most prolific and influential scientists of the last half century, recently has written that mathematics really is not all that crucial for science. In a Wall Street Journal opinion editorial, Wilson wrote that most mathematical theorems are actually translations from already existing scientific theories. Charles Darwin, according to Wilson, bordered on mathematical illiteracy. When Both Hofstadter and Wilson argue that mathematical thinking is secondary to analogical thinking, both philosophers and mathematicians should pay attention.8

Furthermore, many serious physicists have started to recognize the importance of epistemology, especially when it comes to the search for the mirage of a “Theory of Everything” which would unify classical and quantum mechanics. In his 1995 book Schrödinger’s Kittens and the Search for Reality, John Gribbin attacked the concept of the “Theory of Everything” as a myth:

[If] scientific knowledge really is a product of culture then scientific communities that exist in different worlds…would regard different natural phenomena as important, and would explain those phenomena in different theoretical ways (using different analogies). The theories from different scientific communities—the different worlds—could not be tested against one another, and would be, in philosopher’s jargon, “incommensurable.”9

Here, Gribbin united an epistemological position from the humanities with hard science, and later in his book he compared mathematics with language. (Mathematics represents a rich language indeed). Once one accepts the analogy of mathematics and language, the next step is simply to note that languages cannot be right or wrong but more or less descriptive, and the same is true of mathematical theorems and scientific theories.

In his 2010 book A Tear at the Edge of Creation, the astrophysicist Marcelo Gleiser, also critiqued the search for TOE:

There is no final “right to be arrived at, only a sequence of improved descriptions of the cosmos. Each era, each generation even, will describe the Universe in ways that may be radically different from the preceding one.10

A few months after that book’s publication, Stephen Hawking and Leonard Mlodinow published The Grand Design (Order the lecture), and echoed Gleiser and Gribbin while giving this theoretical position a title:

According to model-dependent realism, it is pointless to ask whether a model is real, only whether it agrees with observation. If there are two models that both agree with observation… then one cannot say that one is more real than another. One can use whichever model is more convenient in the situation under consideration….11

So why is this new book by Hofstadter and Sander even necessary if the concepts they wrote about seem so well-developed by other thinkers? Is there anything new here? Yes. Hofstadter and Sander go further in their arguments than any previous physicist or philosopher by claiming that all thinking, from the superficial to the most profound, is based upon analogy. The authors devote an entire chapter called “Analogies that Shook the World” to arguing that Einstein’s insights into physics were, in fact, the elaborate creation of abstract analogies. This argument is fascinating, because it represents a total annexation of mathematics by the humanities. They write:

To some people, it might seem far-fetched to imagine that any role at all might be played by analogical thinking in the professional activity of a mathematician. After all, of all intellectual domains, mathematics is generally thought of as the one where rigor and logic reach their apogee. A mathematical paper can seem like an invincible fortress with ramparts built from sheer logic, and if it gives that impression it is not accident, because that is how most mathematicians wish to present themselves. The standard idea is that in mathematics, there is less place for intuitions, presentiments, vague resemblances, and imprecise instincts than in any other discipline. And yet this is just a prejudice, no more valid for mathematics than for any other human activity.12

Or, as Gribbin wrote earlier and more succinctly, “[T]he discovery that mathematics is a good language for describing the Universe is about as significant as the discovery that English is a good language for writing plays in.”13 The conceit that Hofstadter and Sander put forth here is not necessarily new for those interested in the history and philosophy of science, but the authors further the argument to a greater degree than anyone before by claiming that Einstein thought primarily through analogy.

The authors turn Einstein—typically seen as a logical and mathematical genius—into an exemplar for thinking analogically. For mathematicians this is not quite, perhaps, the equivalent of arguing that Jesus was a Hindu, but it’s not far from it. To use but one example of the authors’ argument, Einstein found an analogy between light and electromagnetic radiation. They wrote: “Although not in the least controversial today, Einstein’s bold suggestion in 1905 that light must consist of particles was harshly and unanimously dismissed by his colleagues. Later in life, he declared this hypothesis, based upon the shakiest of analogies, to be the most daring of his entire career.”14 The rest of the argument about Einstein is deeply thought out and, ultimately convincing. It does seem, however, that the authors stopped short of spelling out the implications and applications of their thesis. In other words, for those of us who buy their argument, what does this all mean?

Applications For and Problems with the Analogical Approach

The universe sometimes presents us with areas of time or space that defy explanation—mathematical, analogical, or otherwise—these are called singularities. Philosophy, also, has its singularities such as the mystery of consciousness. The reason these singularities defy explanation for so long is that certain phenomena don’t act like anything we come into contact with in our everyday lives. Analogies tend to evolve slowly and often in tandem with technology. Galileo understood planetary motion only after new cannon technology helped to evolve mathematical techniques to explain how airborne spheres move in relation to gravity. Newton developed a theory of a clockwork universe only after the clock proliferated. When Leonard Susskind theorizes about a holographic universe, he falls into storied tradition of using technological analogies as explanatory tools.

Douglas Hofstadter has devoted much of his philosophical career to exploring the nooks and crannies of physics and philosophy while looking for connections or analogies that might be applied to deep questions. His thinking seems to be that abstract phenomena will require abstract analogies in order to be understood. For example, in his book I Am a Strange Loop he seemed to argue (readers of Hofstadter must sift through a lot of sand to get the gold) that consciousness can be understood as through the analogy of an information loop.

In Surfaces and Essences, Hofstadter and Sander argued that all thinking is analogical and that naïve analogies poison a lot of thoughts. The authors stated that when schoolchildren are taught to view division as sharing rather than as measurement, they make it harder for students to complete more complex mathematics later on. The ability to create proper analogies is therefore central to higher thought. The authors wrote:

To choose one analogy over another is to favor one viewpoint over another. It amounts to looking at things from a particular angle, to taking a specific perspective on a situation. An insightful analogical take on a situation gives you confidence in your beliefs about the situation while also revealing new facts about it. A teacher, a lecturer, a lawyer, a politician, a writer, a poet, a translator, or a lover may pass hours or days in search of the most convincing analogy….15

Before going any further, it’s important to note that the authors developed a philosophy of analogical thinking that is deeper than a glib “this is like that” type that most of us associate with the concept. The philosophy, which Hofstadter and Sander argue is central to mathematics, is that analogies reach greater levels of abstraction (at some level, the ideas become difficult to convert into ordinary language). One must think about this abstraction until it becomes solid, and then look to find points of connections between complex analogies in other fields. They wrote that “In short, the secret of making good analogies involves making good but more abstract analogies—analogies between encodings, or conceptual skeletons.”16

Now, Hofstadter and Sander note in their book that people can only make analogies based upon what they know. Individuals who have not studied deeply in a topic can only understand the topic through superficial analogies that lead to incomplete thoughts. (I found myself wondering how many economic policies are driven by the naïve analogy that the government’s budget is like a household budget.) Bear in mind that someone could, if she wanted, make a decent explanation to a six-year old of how photosynthesis works using boxes of apple juice. The six year old might think it pretty cool that apple juice boxes and photosynthesis are closely related, but then would be disappointed later by how arbitrary and incomplete this analogy is.

Hofstadter and Sander further argue that, historically, only in cases where a thinker became obsessed with a topic did it become possible for that person to see the deep analogies in that area. When dealing with abstract questions, one might expect to find the best analogies in other areas of abstraction. This is where the Hofstadter and Sander make a crucial point:

A passion for horses or dogs does not instantly turn these animals into sources for analogies that yield insights into triangle geometry, quilt design, fly-fishing, or who knows what else. On the other hand, a double obsession could surely give rise to such analogies. That is, a simultaneous fanatic for, say, Euclidean geometry and for fly-fishing would doubtless find plenty of phenomena in these two domains on which to found analogies, for in this case the search would be intense on both sides.17

And here we have returned to the concept first stated in 1776 by Joshua Reynolds. An analogical approach would require thinkers to study deeply in several fields, looking for connections between, and analogies from, each field for application to abstract problems. The future of philosophy, and the future of thought, belongs as it always has, to those individuals who can think deeply in various disciplines.

This has already begun. Werner R. Lowenstein’s 2013 Physics in Mind: A Quantum View of the Brain, represents just such an attempt to use insights from quantum physics to explain consciousness. (He never mentions Hawking and Mlodinow’s Model Dependent Realism, but spells out the logic of the position.) Lowenstein aptly describes the actual physical makeup of the senses from the cellular level, but then explains the problem of consciousness in language very similar to the way in which Hawking has presented the mathematical issues surrounding the Big Bang, but then issues a call of a dual-field approach to the problem:

Regarding the nature of the very foundation of mind, consciousness, we know as much as the Romans did: nothing. We don’t even have a valid paradigm. Painful as it is, I say this at the start, so as not to raise false hopes (there has been enough of that in recent years). But it would be unfair to lay all of the blame at the door of biologist. The gap of knowledge, as I see it, falls as much into the field of physics as it does in that of biology. Consciousness provides a natural meeting ground for biologists and physicists—it is the ultimate frontier.18

The problem with this notion of studying in abstract fields for abstract analogies reveals itself immediately. Can we really expect anyone to become deeply informed in both biology and quantum physics? It takes half a lifetime of study to really understand many of the concepts inherent in each field. Even if we did produce these types of thinkers, then how exactly are we to find a community of similar scholars who could peer-review the work? And where would this type of work be published?

Furthermore, Hofstadter and Sander point out that most analogies are false matches. Someone could spend a lifetime searching for analogies in metaphors from multiple fields but not find anything necessarily meaningful. Or, even worse, someone studying in fields far separate from their own risks becoming a dilettante and creating naïve analogies that might actually do intellectual damage. And it would require someone else to follow in the same steps across multiple fields to check or correct the theory.

One point upon which the authors breeze by too quickly involves the lessons that can be learned from historical geniuses. The authors state that creating a work of genius is not a deliberate process but something that comes as a natural result of an obsession with problems in multiple fields. Historically, this is true, however as I’ve argued in my work on educational theory, there’s no reason to leave this process up to serendipity. Why not capture this historical force and put it to work as an educational philosophy? The authors do not call for this, but perhaps a new approach is needed in higher education. Why not make the process of analogy hunting a new field of knowledge in itself? Would any modern university president be bold enough to introduce a Department of Consilience and Analogy?

Hofstadter and Sander make a convincing argument, but nothing in the book indicates that they fully understood the implications of what they’ve done to science, or in particular, to Einstein’s theories. Their intent appears to have been to show the centrality of analogical thinking even to one of history’s greatest geniuses based upon the assumption that Einstein intuited great truths. Yet, the full implications of their argument would alter our perception of what Einstein did.

Follow me here; if Einstein thought analogically by equating mass with energy and gravity with acceleration, and most importantly, space with time, then his set of analogies are just that—analogies that are applicable to real world phenomena. Einstein famously conjectured that the speed of light is the universe’s only constant, but he merely pulled this from his own formidable mind. We might note that if Einstein had begun by conjecturing that the speed of sound was the universe’s only constant he could have made a descriptive theory with all of the same elements. Only it would have described every particle and wave that moved more slowly than sound. His genius was to begin with the fastest speed that anyone knew about, maybe the fastest speed we can know about, and then created a theory that described all the particles we come into contact with.

Now, physicists assume that certain rules hold true for different levels of phenomena. To borrow the language of Hofstadter and Sander, a physicist who throws a pebble into a mud puddle can, based upon that analogy, extrapolate those same forces outward to calculate what would happen if a meteor struck the Atlantic Ocean. The same forces apply only to greater degrees. The same is true of slingshots and rocket ships. Yet, it could be that at the particle level our analogies of propulsion, derived from the Newtonian level, do not hold up. Our confusions about the consciousness, the big bang, and other singularities all stem from the fact that there is nothing in our experience that provides an adequate analogy for understanding.

If this was the case, Einstein’s brilliant theory would hold true for everything in the universe except for things travelling faster than what his theory forbids. If faster than light particles are ever conclusively found, then we will have found out that Einstein’s genius was in the creation of a brilliant theoretical model, but not in the intuition of a universal truth. It could simply be that at certain speeds the analogies about propulsion that we derive from engines or muscles simply do not apply any longer.

While the ideas that Hofstadter and Sander put forth may sound abstract, in his book Loewenstein says that 30 percent of the U.S. Gross National Product is now based upon quantum physics, something which just a century ago existed only in the minds of a few obscure thinkers and in the largely unopened pages of a handful of academic journals. Furthermore, the uncovering of potentially apt analogies to consciousness might yield new insights that could have benefits we cannot currently conceive of. It’s not too far-fetched to imagine that someone well-trained in lattice theory might be able to understand the way in which neurons are packed into the brain better than a neuroscientist who lacked such insights. An epistemological understanding that encouraged thinkers to be polymaths and analogy hunters could be as useful as anything since the development of quantum theory.

Information is now readily available, but the human mind must be able to make sense of that information for it to be coherent and useable, and this can only be done through minds that are educated to think in a certain way. Hofstadter and Sander, by annexing all fields of knowledge under the banner of analogies, have presented philosophers and scientists with a challenge that carries deep implications of change for each field of knowledge. After finishing the book, this reader is left with the impression that we still have the same old problems but perhaps we have a new way forward.