Models and Metaphors

I’m reading two books about models and metaphors in science. One is straightforward with many interesting vignettes, aimed at highlighting the role of models in describing abstract concepts in science. The other is difficult, aimed at telling a single story, and uses abstraction to conceptualize what scientists do when creating a model. Both highlight the role of metaphor, in different ways, to parse inherent problems and uncertainties in science and our understanding of the natural world.

 

The first book, published in 2003, is Making Truth: Metaphor in Science by Theodore Brown, a retired chemistry professor at the University of Illinois (Urbana-Champaign). His name is familiar to many students from his chemistry textbook, now in its fourteenth edition. You don’t have to be a chemist to enjoy and learn from Making Truth. As an educator, he builds his case step-by-step with many examples. As a chemistry instructor, I particular enjoyed the two chapters devoted to the concept of the atom, both classical and modern.

 

Atomic theory is the foundation of chemistry. This seems obvious to students today, steeped in images and metaphors that assume all matter is made up of tiny entities called atoms, and that molecules (the combination of two or more atoms), give rise to all the interesting stuff you can see and touch and some which you can’t see. Atoms are tiny, tiny, tiny. Nanoscopic tiny. And they’re actually rather strange. My first semester general chemistry course has been themed around the idea of making visible the invisible; discovering the molecular basis for matter in its unity and diversity!

 

I am familiar with most of the examples used by Brown, several of which also make an appearance in my classes. I emphasize the use of models and their limitations, and the important role they play. We use various visual aids, and discuss their pros and cons. Brown also talks about these. But I have not emphasized the metaphors that I use except when I tell the Happy Atoms story. More recently I’ve also used teleportation (Apparition in Harry Potter) to describe electrons changing energy levels in an atom. Brown’s book reminded me to pay attention to metaphors; especially since a metaphor that’s clear to me may not be as clear to my students, a generation removed from my own touchstones and experiences. I’ve now encountered students who’ve never read the Harry Potter books or seen the movies, believe it or not, and would not understand why my blog is called Potions for Muggles. (This proved interesting in one class session.)

 

Brown devotes a chapter to molecular models in chemistry and biology. The story of Watson and Crick’s elucidation of DNA structure is the crowning vignette; no surprises there. But what I enjoyed most from that chapter was the story of van’t Hoff trying to make sense of optical activity, and coming up with the idea of a tetrahedral carbon. Optical isomerism isn’t usually in our G-Chem syllabus although I’ve included it more often in recent years. Instead the tetrahedral shape is discussed through VSEPR theory. I like the van’t Hoff story, and will try to incorporate it into my G-Chem classes next semester. Brown also reminded me of the power of visuals. While I already use such visuals in my classes, I feel motivated to think more carefully about how I can include more interesting and relevant images, in the context of the metaphors I use, when discussing chemistry.

 

There’s a chapter on the protein folding problem (how the ‘primary’ polypeptide structure turns into its ‘tertiary’ functional structure) and its associated models and metaphors. Brown uses the metaphor of language, employing the distinction between syntax and semantics, something I’ve been thinking about recently. If you misspell a word, sometimes you change the entire meaning of a sentence, and other times the reader recognizes it as a typo and the meaning is unchanged. Using this example, Brown gets to the crux of the matter, and I’ll quote him here.

 

The gestalt that consists of the complex of associations and ideas that make up our understanding and use of the written language maps onto the molecular domain of protein sequence. Notice that this mapping does not involve a directly emergent physical experience but rather a human artifact in the social domain. This is an early example of an important and interesting aspect of metaphors in science: As the scientist attempts to understand systems of increasing complexity, metaphors based solely on embodied physical experiences no longer suffice.

 

Brown goes on to illuminate the metaphors we use regularly in our classrooms, sometimes not giving a second thought to our use of them. Energy is one of those nebulous things; we use metaphors such as the waterfall with its attendant directionality of a downward flow. Higher energy is UP. Lower energy is DOWN. Like Hermione, I’ve even had an epiphany about Pipes, although my metaphor was linked to a research problem rather than to finding a fantastic beast. Brown examines the word ‘folding’ as a metaphor that “evokes the notion of bringing into contact various parts of the object, as in folding clothing or card table chairs.” My example would be origami. Extending this image to ‘solving’ the Levinthal Paradox, creasing or pre-folding your origami is key to obtaining the end goal of a beautiful intricate structure.

 

But let’s not forget Brown’s point that the metaphor is a mapping that is also an artifact. He splits the physical domain from the human social domain, but how one imposes this artificial separation is a conundrum. That’s why we have trouble answering the question “What is Life?” as my students have encountered.

 

This brings us to the second book, Life Itself by Robert Rosen, published in 1991. Rosen, a theoretical biologist, was a student of Nicholas Rashevsky at the University of Chicago. Both names are likely unfamiliar to most biologists. The book’s subtitle, A Comprehensive Inquiry into the Nature, Origin, and Fabrication of Life, sounds grandiose. Rosen will attempt to answer the question “What is Life?” by distinguishing machines and organisms. It’s a very challenging book and not for the faint-of-heart. Background knowledge is assumed, and I especially struggled through the mathematical abstractions. Who would have thought that set theory and algebraic topology should be employed for such an endeavor?

 

I will honestly say that I don’t understand large chunks of the book. I have a glimpse of where Rosen is going; I suspect he’s on the right track; I can partially follow the arguments made; but there’s this feeling of swirling in a fog as I grope my way through, partially blind. Perhaps this is how my students feel in P-Chem. Rosen, now deceased, isn’t available to answer my questions, and he doesn’t have any disciples that I know of – although I have alluded to Mikulecky’s work, and to Rosen’s follow-up book. I’ve also mentioned Noble’s concept of Biological Relativity, that no level in biology is privileged when discussing causation – Rosen and Rashevsky’s version is called Relational Biology. But where Noble stays with qualitative examples that illustrate the problems with reductionism in biology, Rosen delves deep into the source of reductionism. And it has to do with models and metaphors.

 

If I had any reasonable grasp of Rosen’s work, I would try to explain it, but I don’t – and so you’ll have to make do with some quotes from the book and some ill-defined rambling from me. In the first half of the book, Rosen defines something called the Modeling Relation (see diagram below). In a nutshell: The real world is complex. To discover something about nature, we simplify by utilizing a formal system. This requires two additional steps: encoding and decoding, loosely associated with the activities of observation and prediction in science. And if the process 1 commutes with the sum of the processes 2+3+4, we might have a reasonably good model for what is going on in nature. I’m significantly oversimplifying things here.

 

Here’s a quote from Rosen on models and their role in causation (what he calls entailment).

 

Modeling… Is the art of bringing entailment structures into congruence. It is indeed an art, just as surely as poetry, music, and painting are. Indeed, the essence of art is that, at root, it rests on the unentailed, on the intuitive leap. I have stressed repeatedly that the encodings and decodings on which modeling relations depend are themselves unentailed. Thus theoretical scientists must be more artist than craftsmen; Natural Law assures them only that their art is not in vain, but it in itself provides not the slightest clue how to go about it.

 

Um, that’re reassuring. But Rosen has good reasons for this. So let me quote him some more.

 

Modeling relations can be thought of as transductions, in which one kind of entailment can be replaced by, or converted into, another, in an invariant way. We convert causal entailments to inferential ones for a very simple and basic reason; we can hope to understand what goes on in a formal system. The concrete manifestation of such understanding lies mainly in our capacity to predict and (though this is a quite different order of question) perhaps to control. It would be nice if could pull the modeling process itself inside a formal system, where we can see it whole. We cannot do this directly, but we can do it metaphorically…

 

In probing the natural world, we are rather limited (yes, even with state-of-the-art technology) with what we can actually pull out as actionable data, i.e., Step 2 is very difficult and often incomplete. Nevertheless, we try. Then using some inferential rules (Step 3), we eventually use Metaphor to decode (Step 4). In a sense, science is always speaking in metaphors. Brown’s examples of how we probed the atom and drew ‘conclusions’ expressing them in metaphors are aplenty.

 

Rosen will subject his own framework to intense scrutiny, at least that’s what if feels like to me, the foggy reader. He will carefully define a simulation and how that differs from a model. There will be a distinction between analytic and synthetic models, the former deriving from (mathematical) Cartesian products (and not easy to decompose into its parts) and the latter from direct summands which will correspond to the reductionist approach in science championed by physics. Reductionism is very useful, and Rosen acknowledges this. But it is limiting. Much too limiting when it comes to the wider world of biology. Synthetic models can give us machines with hardware and software, but they can’t give us organisms. Rosen has a solution, but I don’t understand it well enough to summarize.

 

With reductionism, one is stuck with the “Cartesian metaphor of organism as machine”, at least formally, and with as much rigor as the physicist can muster. When we talk about the limits of reductionism and hint at emergence, it’s mostly hand-waving in the fog. Rosen’s goal is to give it rigor through abstraction in the world of mathematics and graph theory. Since I’m still in the fog, I’ll simply quote Rosen again on the (philosophical) Cartesian metaphor.

 

It succeeds in likening organisms to machines, to the extent that both classes of systems admit relational descriptions. But beyond that, it is fundamentally incorrect; it inherently inverts the notions of what is general and what is special. On balance, [it] has proved to be a good idea. Ideas do not have to be correct to be good; it is only necessary that, if they do fail, they do so in an interesting way.

 

Rosen will use the same analogy as Brown, comparing the machine-organism distinction to the syntax-semantic distinction. I’ll leave this to the astute reader of his book to figure out how he does it. In fact, maddeningly, Rosen does this all over the book with “I leave it to the reader…” statements where something is clearly obvious to him and so he skips over steps. I try very, very hard not to do this when teaching P-Chem. Like Brown, Rosen also uses protein folding as an example, although writing in 1991 he is not sanguine about the prospects of solving the problem. Interestingly, in 2021 we’ve gotten much better at the prediction part of the problem, but with a significant loss in understanding, as the number-crunching parts have moved to an A.I. black box. We don’t know the details of Step 3 any longer. What does that tell us about Step 1? I’m no longer sure.

 

Why do proteins fold into their specific structures to carry out their specific functions? Rosen uses the metaphor of a scaffold.

 

Sequence pertains to scaffolding… held together, not by any direct intersymbol bonds, but by being suspended in a larger structure. Conversely, any larger structure that maintains their configuration would create the sequence; its exact nature, its chemistry, if you will, is otherwise irrelevant. If we perchance interpret the elements of such configurations to be… chemical groups… then such scaffolded configurations may themselves act like conventional chemical species. If so, they are in fact much more general than conventional molecules… only “exist” when scaffolded together… If the scaffolding as a whole is perturbed, or disrupted, they disappear, they cease to exist, they denature. But they do not “decompose” in any conventional sense, and they reappear when the scaffolding is restored.

 

This is a metaphor I find very helpful as I’m puzzling over my present origin-of-life projects. I might even come up with a model to go with the metaphor. Perhaps that will help concretize my foggy understanding of this whole business. It’s no wonder many folks think science is difficult. Rosen certainly hasn’t made it any easier. But he might have made it more profound.