Judgment and Decision Making

Jared and Todd talk about the study of judgment and decision making, the work of Kahneman and Tversky, System 1 and System 2, heuristics and biases, naturalistic decision making, prospect theory and value functions, loss aversion, anchoring, and nudge theory.

Judith Butler and Gender Trouble

Judith Butler’s Gender Trouble is a key text in the history of feminism and gender studies. This episode explores the significant points of Butler’s system of ideas, in particular the concept of gender performativity.

With this episode I’d like to talk about the philosophy of Judith Butler and the 1991 book Gender Trouble: Feminism and the Subversion of Identity. Gender Trouble is one of a series of books I’ve been reading related to the field of gender studies. Some of the other books I’ve read most recently on the subject are Simone de Beauvoir’s The Second Sex and Michel Foucault’s The History of Sexuality. I’ve known for a while that I would want to study one of these books in more depth and eventually do an episode on it. And I eventually chose Gender Trouble because I felt like it encompassed the broadest set of philosophical issues that interest me. The topics in Gender Trouble are all presented in the context of feminism but they also have implications for those foundational topics that philosophers have been talking about for centuries. Most significant is the question of where words and concepts get their meaning, whether our conceptual categories are ‘out there’ in nature or whether they are socially constructed; for example, categories like male and female.

These topics are necessarily political, minimally in the technical sense of pertaining to the ways that societies organize and conduct themselves, and also in the more popular, partisan sense of being controversial topics on which people have strong opinions. What I’d like to do here is give Butler’s views as fair a presentation as I can. And I actually don’t think I have any prejudice against these views; I find them quite interesting. So any errors in my overview would be due only to honest misunderstandings of admittedly highly sophisticated philosophy. But I’ve been trying to read this book quite carefully so hopefully any such errors would be minimal, or at least amount to reasonable interpretations. I’ll try to bracket the points on which I agree or disagree with their philosophy. The goal is, before anything else, to promote understanding of their ideas. And this is a matter of principle. I think it’s crucial, prior to any kind of serious criticism of philosophical ideas, to present those ideas fairly and accurately. Many treatments of things like postmodernism, gender studies, and critical race theory do not give fair or accurate presentations of them or give any understanding of how intelligent, well-intentioned people would find these ideas persuasive. And you have to do that. If you look at these ideas and can’t understand how anyone could possibly think this way then that’s an indication that your understanding of it was not adequate and that you have to go deeper. Not necessarily to be persuaded, but to at least have a fair understanding. So I’ll try to do that here with Butler’s philosophy.

Butler described Gender Trouble as a work of cultural translation engaging with the ideas of many French thinkers as their work relates to gender studies. So much of the book is Butler talking about the ideas of Claude Lévi-Strauss, Michel Foucault, Jacques Lacan, Julia Kristeva, and Monique Wittig. I found all that very informative and was happy to learn more about these thinkers, particularly Kristeva and Wittig. But I won’t get into any of that here because that would just be too much. The only exception is Foucault, who I will talk about a little on the subject of language and power. But for the most part I want to focus on Butler directly. I’ll break the ideas of the book down into five major topics that stood out to me.

Gender as Construct and Performance

Butler’s primary thesis in Gender Trouble is that gender is performative. This is very similar to saying that gender is culturally constructed. With the thesis that gender is performative Butler seems not to be saying precisely the same thing as that gender is culturally constructed. But the two ideas seem to be consistent and Butler also speaks in the book of gender being culturally constructed. By the performativity of gender Butler means that gender has its origins in our actions and performance. We perform gender in the ways that we dress, speak, sit, walk, and certainly in the interests that we pursue. These all operate within the context and expectations of our cultures. These things constitute or create gender. And crucially, no one has a gender prior to such performances. It’s in these performances that people consolidate an impression of being a man or being a woman.

Butler says in a preface to a later edition that, “The view that gender is performative sought to show that what we take to be an internal essence of gender is manufactured through a sustained set of acts, posited through the gendered stylization of the body.” One important philosophical issue they address repeatedly is ontology. Ontology is the philosophical study of the nature of being and one can also speak of having “an ontology” to refer to the things that are understood to exist in a worldview. A physicalist ontology, for instance, would not include any kind of spiritual element. Key for Butler’s philosophy is that they do not include gender in their fundamental ontology. Butler says, “That the gendered body is performative suggests that it has no ontological status apart from the various acts which constitute its reality.” Also: “There is no gender identity behind the expressions of gender; that identity is performatively constituted by the very ‘expressions’ that are said to be its results.” The alternative would be that gender has independent ontological status, in which case there would be gender identity behind the expressions of it. It would be this ontological gender that would constitute and produce the expressions of it. But Butler is saying that, to the contrary, it’s not the case that there is an ontologically self-sufficient gender identity behind the expressions and acts, that is producing them. Rather it’s the expressions and acts that entirely constitute gender.

Does that mean that gender is insignificant in Butler’s philosophy? Far from it. They say, for example: “To claim that gender is constructed is not to assert its illusoriness or artificiality.” One of Butler’s major points in the book is that gender constructs are extremely consequential and that’s why they merit attention. It’s not that gender, as a construct, is illusory or artificial but that it can be changed to better accommodate the needs of individuals for whom it’s currently not working so well.

Still much of that project involves undermining the apparent necessity and objectivity of gender. So they make a point of this. This makes allowance for thinking about gender in a different way. For example, they say: “If constructed gender is all there is, then there appears to be no ‘outside,’ no epistemic anchor in a precultural ‘before’ that might serve as an alternative epistemic point of departure for a critical assessment of existing gender relations.” I think this idea of an “outside” is quite illustrative. Something Butler has to do for their project is address what kind of standard any proposed change to the way we think about gender is to be evaluated. This kind of “outside” perspective is sometimes called an “Archimedean point”, a “view from nowhere” where we could see things as they are, without any kind of perceptual or conceptual filter. There doesn’t really seem to be a place for an Archimedean point in Butler’s outlook, but regardless gender wouldn’t factor in there anyway because gender is constituted by the actions of individuals and interpreted culturally. So there’s no standard of evaluation beyond that. “If gender attributes and acts, the various ways in which a body shows or produces its cultural signification, are performative, then there is no preexisting identity by which an act or attribute might be measured; there would be no true or false, real or distorted acts of gender, and the postulation of a true gender identity would be revealed as a regulatory fiction.” Also: “Genders can be neither true nor false, neither real nor apparent, neither original nor derived.”

One of the implications of this view of there not being an original nor a derived is that minority gender performances at the margins are no less original than what is perceived to be the norm. For example, on homosexuality Butler says: “The replication of heterosexual constructs in non-heterosexual frames brings into relief the utterly constructed status of the so-called heterosexual original. Thus, gay is to straight not as copy is to original, but, rather, as copy is to copy.” Or we might say, as construct is to construct. The idea being that heterosexuality and homosexuality are on equal ontological footing; neither is more original or real than the other. They are equally viable ways that individuals can perform gender.

The constructed and performative nature of gender will be important in Butler’s ultimate prescriptive project in the book, which will be to perform parodies of the norms in order to highlight their non-essential nature, to make room for other ways of performing gender.

Sex as Construct

That gender is a construct might seem controversial enough. But under the distinction between gender and sex, that’s been operative in intellectual circles for quite some time, it’s not necessarily all that radical. Under this distinction sex would be the physical, anatomical features distinctive to males and females. Whereas gender would be all the social roles and self-perceptions understood as male or female, or any other gender we might consider. And this actually isn’t just a distinction made by liberal theorists. Roger Scruton, a philosopher who pretty much anyone would place decidedly on the conservative end of the spectrum, said in his 1986 book Sexual Desire that, “Failure to distinguish sex and gender — to distinguish the material base from the intentional superstructure — is responsible for many interesting confusions.” Scruton even made use of gender as a construction: “I shall use the term ‘gender’ to denote both a way of perceiving things and a particular artificial feature of the thing perceived (it’s ‘gender construction’).” Scruton placed gender among a class of similar concepts, such as personhood, that “have this effect of changing the reality to which they are applied” with which we “reconstruct ourselves according to the requirements of a fundamental perception.” There’s nothing essentially radical about making this kind of distinction. Sex is the physical fact; gender is our psychological and social response to it.

Feminists have also made use of this distinction, notably Simone de Beavoir. Feminists have tended to use the sex/gender distinction to more reform-minded ends than conservative thinkers like Scruton. But Butler goes even farther. Butler actually refuses the sex/gender distinction by which sex is factually given and gender is culturally constructed. Instead, for Butler sex is also constructed. So Butler’s is a more radical position.

To make the case that sex is also constructed Butler needs to make use of important ideas from twentieth century developments in the philosophy of language. The philosophy of language is not Butler’s primary focus in the book but I’d like to give some of the background to make the case for the plausibility of this idea, regardless of whether or not it ends up being convincing. Butler talks about the way “language itself [produces] the fiction construction of ‘sex’ that supports these various regimes of power.” They are clearly influenced by Michel Foucault and his philosophy of the way scientific discourse is used to support certain power structures. And we’ll get to that. For my part though, I think Richard Rorty has a better philosophy to support Butler’s position.

Richard Rorty, in his book Philosophy and the Mirror of Nature, argued against what he called “representationalism”, a view that our words and concepts are representations of things as they are, as if we were holding up a mirror to nature and just reflecting an image of it. But in Rorty’s view our knowledge of and speech about things is so thoroughly mediated and filtered that it is hardly representational. And there’s a lot of history behind this kind of idea in the history of philosophy and particularly in the epistemology of folks like John Locke and Immanuel Kant. The radical upshot of this is that it becomes impossible for statements about the world to be conclusively true or false. The best statement of this I know of from Rorty on this subject is from his book Contingency, Irony, and Solidarity:

“We need to make a distinction between the claim that the world is out there and the claim that truth is out there. To say that the world is out there, that it is not our creation, is to say, with common sense, that most things in space and time are the effects of causes which do not include human mental states. To say that truth is not out there is simply to say that where there are no sentences there is no truth, that sentences are elements of human languages, and that human languages are human creations. Truth cannot be out there – cannot exist independently of the human mind – because sentences cannot so exist, or be out there. The world is out there, but descriptions of the world are not. Only descriptions of the world can be true or false. The world on its own – unaided by the describing activities of human beings – cannot.”

The perspective that Rorty is promoting here is, depending on how you look at it, either radical or trivial. Rorty understands truth in purely linguistic terms. So it would follow just definitionally, and trivially, that you wouldn’t have truth without sentences. But to limit truth in such a linguistic way is itself bound to be controversial. That’s another topic though. The point is that there is at least a way to make the case for this kind of talk.

Let’s apply this now to sex. Rorty says that “most things in space and time are the effects of causes which do not include human mental states”. The things that, in the following step, we will refer to in our vocabulary as sexual organs and sexual features, are in space and time and effects of causes which do not include human mental states. But in the Rortyian view there is no truth or falsehood about them until we make statements about them. But in that process we are constructing sex.

Butler has some similar ideas. “Is there a ‘physical’ body prior to the perceptually perceived body? An impossible question to decide. Not only is the gathering of attributes under the category of sex suspect, but so is the very discrimination of the ‘features’ themselves. That penis, vagina, breasts, and so forth, are named sexual parts is both a restriction of the erogenous body to those parts and a fragmentation of the body as a whole.” Butler doesn’t have to deny and can affirm, as Rorty affirms, that these features that will be marked as “named sexual parts” are effects of causes in space and time, causes which do not include human mental states. Presumably Butler can concede that much. But Butler denies that there is one and only one possible way to pick out features of the world as entities of significance, that only certain groups of tissues can be marked out as organs meriting special distinction. They say, “sex’ imposes an artificial unity on an otherwise discontinuous set of attributes.” And further that, “the body is not a ‘being,’ but a variable boundary, a surface whose permeability is politically regulated.”

I think this addresses a common point of criticism made against the kind of view Butler is proposing: the criticism that this all ignores biology. I think there are different ways to make this criticism that can be either too simplistic and unsatisfactory or sophisticated and satisfactory. Whether Butler’s view or the criticism is ultimately convincing is another matter that I’ll leave to the side for now. Certainly the point about biology needs to be addressed. But what’s crucial is that Butler hasn’t ignored biology. Butler’s points are meta-biological and get beneath the methods and interpretations of biology itself. So if the criticism via biology is going to be made it’s going to have to be at this level; it’s going to have to be meta-biological as well. It won’t be sufficient to just say, “What do you mean sex is constructed? Just look at biology.” We can’t just “look at biology”. We have to do a more second-order look at what it means to look at biology. And this is actually characteristic of the whole discipline of the philosophy of science.

As an example of the kind of meta-scientific critique Butler makes, in one part of the book they get into different scientific studies conducted on the genetics of sex determination. I won’t get into the details but the basic takeaway was that Butler argues that the conclusions of the experiments were constrained from the outset by the kinds of questions that the experimenters allowed themselves to ask: “The framework suggests a refusal from the outset to consider that these individuals implicitly challenge the descriptive force of the available categories of sex; the question he pursues is that of how the ‘binary switch’ gets started, not whether the description of bodies in terms of binary sex is adequate to the task at hand… The conclusion here is not that valid and demonstrable claims cannot be made about sex determination, but rather that cultural assumptions regarding the relative status of men and women and the binary relation of gender itself frame and focus the research into sex-determination.”

There’s some similarity here to the ideas of Thomas Kuhn and his classic book The Structure of Scientific Revolutions, which looks at the way certain “paradigms” of thinking constrain the kinds of questions scientists will think to ask, until a scientific revolution occurs to shift the paradigm. This kind of thinking isn’t necessarily anti-scientific. But it’s a reflection on the possibilities and limitations of scientific work.

Butler asks: “What is ‘sex’ anyway? Is it natural, anatomical, chromosomal, or hormonal, and how is a feminist critic to assess the scientific discourses which purport to establish such ‘facts’’ for us?… If the immutable character of sex is contested, perhaps this construct called ‘sex’ is as culturally constructed as gender; indeed, perhaps it was always already gender, with the consequence that the distinction between sex and gender turns out to be no distinction at all.” And so for Butler the distinction breaks down and sex is also a cultural construct.

The Power of Language and Norms

I mentioned Michel Foucault earlier. Butler definitely sees gender constructs in terms of power relations, which marks their thought as political and certainly on the Left. Michel Foucault’s work is foundational to much of this since his work highlighted the way discourse, both scientific and juridical, is used to enforce power structures. In his classic work The History of Sexuality Foucault argues that, far from being reticent and repressed in our willingness to talk about sex in modernity we actually talk about it a great deal. But our abundant discourse on sex has not liberated us sexually. Instead discourse on sexuality has functioned to regulate sex. We talk about sex in order to control it. We mark out certain sexual practices and sexual drives and give them names in order to regulate and control them in ways that wouldn’t have been possible without those linguistic tools. Butler picks this idea up and applies it to gender in general; saying that the concept of gender is regulative.

Butler says that, “Under conditions of normative heterosexuality, policing gender is sometimes used as a way of securing heterosexuality.” I think this is an interesting way of understanding the concept of “heteronormativity”. One might wonder, what’s wrong with heteronormativity? Is being heterosexual bad? No, it’s not that. The problem is not with heterosexuality but with using heterosexuality as a normative standard from which to stigmatize homosexuality and judge it defective.

For both Butler and Foucault the way we use language has important implications for the way we see the world and understand what is normal and what is just common sense. Butler says, “Language gains the power to create ‘the socially real’ through the locutionary acts of speaking subjects.” Why would that be? One of the interesting things about language is that it dramatically expands the range of what it is possible for a single individual to think, because it is shared across time and space by millions of people. We have words like “curious”, “dormant”, “dependent”, “receptive”, ideas that are at some levels removed from physical objects that would be most obvious to just point to and give words for. But those concepts are quite useful once we have them and we don’t usually even stop to think about how remarkable it is that we have words like this for such abstract ideas. I understand Butler and Foucault to understand also that terms and concepts like heterosexuality and homosexuality are similarly abstract but that, once we have these concepts, they take on a character of seeming obvious as features of reality.

I think this gives a bit of insight into what we might sometimes think of as language policing from the Left. There’s an intellectual history here in which thinkers on the Left have been looking at the ways language operates in ways that structure and limit our thinking. Butler has some interesting comments about the way language affects what seems to be just common sense or transparent. For example: “There is nothing radical about common sense.” Conservatives and radicals might both agree on that and only disagree with whether that’s a problem; whether the status quo is desirable or needs to be changed. Butler also asks, “What does ‘transparency’ keep obscure?” At a surface level, looking at the apparent ‘way things are’ leaves uninvestigated all the presuppositions and philosophical underpinnings of our worldview. So in a way, the transparent, ‘way things are’ view leaves obscure or unexamined everything that supports it.

For both Butler and Foucault it’s not possible to just drop the exercise of power altogether, even if we want to. Discourse just does wield power. Their aim rather is to disperse power so that no single view dominates. “If sexuality is culturally constructed within existing power relations, then the postulation of a normative sexuality that is ‘before,’ ‘outside,’ or ‘beyond’ power is a cultural impossibility and a politically impracticable dream, one that postpones the concrete and contemporary task of rethinking subversive possibilities for sexuality and identity within the terms of power itself. This critical task presumes, of course, that to operate within the matrix of power is not the same as to replicate uncritically relations of domination.” And for Butler they are going to affirm that it is indeed possible to operate within this matrix of power without replicating relations of domination. That’s an important point, I think, because one of the criticisms of this outlook is that it merely seeks to replace one power structure with another. And maybe that happens. Like, say university campuses and faculties being dominated by the Left. But in Butler’s view that kind of replication of relations of dominations is not a necessity and can be avoided, and presumably should be avoided.

Personal Identity and the Internal Psyche

The most interesting idea in Gender Trouble, for me, was Butler’s perspective on personal identity and the internal psyche. This isn’t one of the primary topics in the book but it’s one that really stood out to me and their treatment of it was one of the reasons I found this book the most interesting of the books I’ve read on gender studies. Butler says: “I continue to think that it is a significant theoretical mistake to take the ‘internality’ of the psychic world for granted.” This is quite fascinating. The internality of the psychic world is the kind of privileged knowledge we have of our own thoughts. This would seem to be the most basic, undeniable thing there is. And I think this might place Butler into conflict with some on the Left who would want to privilege “lived experience” and a person’s own account of it. But for Butler much of the internal psychic world is also constructed: “the ‘coherence’ and ‘continuity’ of ‘the person’ are not logical or analytic features of personhood, but, rather, socially instituted and maintained norms of intelligibility.” This is a significant claim and I don’t think Butler justifies it philosophically in the book. That would be a huge undertaking in the field of philosophy of mind. And that doesn’t seem to have been part of the project; not every presupposition needs to be justified; that chain of justification needs to stop at some point for practical purposes to just get on with the principle project. But I appreciate that they state this presupposition because it helps to show how many of the other ideas of the book fit together under this perspective.

By challenging the internality of the psychic world Butler is challenging some basic premises to much of feminist theory. Butler says: “For the most part, feminist theory has assumed that there is some existing identity, understood through the category of women, who not only initiates feminist interests and goals within discourse, but constitutes the subject for whom political representation is pursued… Within feminist political practice, a radical rethinking of the ontological constructions of identity appears to be necessary in order to formulate a representational politics that might revive feminism on other grounds.” By rethinking the ontological constructions of identity Butler is getting into what has been called a metaphysics of substance.

“The metaphysics of substance is a phrase that is associated with Nietzsche within the contemporary criticism of philosophical discourse. In a commentary on Nietzsche, Michel Haar argues that a number of philosophical ontologies have been trapped within certain illusions of ‘Being’ and ‘Substance’ that are fostered by the belief that the grammatical formulation of subject and predicate reflects the prior ontological reality of substance and attribute. These constructs, argues Haar, constitute the artificial philosophical means by which simplicity, order, and identity are effectively instituted. In no sense, however, do they reveal or represent some true order of things.”

The metaphysics of substance, so described, is very much at odds with notions of construction, especially of the most fundamental concepts like personhood and gender. Butler asks: “What is the metaphysics of substance, and how does it inform thinking about the categories of sex? In the first instance, humanist conceptions of the subject tend to assume a substantive person who is the bearer of various essential and nonessential attributes. A humanist feminist position might understand gender as an attribute of a person who is characterized essentially as a pregendered substance or ‘core,’ called the person, denoting a universal capacity for reason, moral deliberation, or language… According to Haar, the critique of the metaphysics of substance implies a critique of the very notion of the psychological person as a substantive thing.”

Butler appropriates an idea from Nietzsche for their project of placing the origins of gender in performativity. “The challenge for rethinking gender categories outside of the metaphysics of substance will have to consider the relevance of Nietzsche’s claim in On the Genealogy of Morals that ‘there is no ‘being’ behind doing, effecting, becoming; ‘the doer’ is merely a fiction added to the deed—the deed is everything.’” “The deed is everything” is a succinct restatement of Butler’s theory of gender. For Butler gender is not so much a thing as it is activity; a verb rather than a noun.

This puts Butler in an interesting position relative to both identity politics and transgenderism, both of which Butler is in many ways sympathetic toward, so the difference would seem to be on technical theoretical grounds rather than partisan affiliation. Butler says: “The foundationalist reasoning of identity politics tends to assume that an identity must first be in place in order for political interests to be elaborated and, subsequently, political action to be taken. My argument is that there need not be a ‘doer behind the deed,’ but that the ‘doer’ is variably constructed in and through the deed.”

Trouble and Opening Up Possibilities

The final topic I’d like to go over is the prescriptive, call to action, element of Butler’s book. What do they propose that we do with all of this? The title Gender Trouble indicates that Butler is going to propose making trouble for gender. And they want to do this through parody of prevailing assumptions about gender in order to expose their constructed, non-essential nature and so open up space for more ways of performing gender than those that currently dominate to the exclusion of others. Butler says: “The prevailing law threatened one with trouble, even put one in trouble, all to keep one out of trouble. Hence, I concluded that trouble is inevitable and the task, how best to make it, what best way to be in it.”

For Butler the best way to make trouble and be in trouble is through parodic practices. They give drag as an example of consummate parody. “I describe and propose a set of parodic practices based in a performative theory of gender acts that disrupt the categories of the body, sex, gender, and sexuality and occasion their subversive resignificationand proliferation beyond the binary frame.” “Is drag the imitation of gender, or does it dramatize the signifying gestures through which gender itself is established?” “I would suggest as well that drag fully subverts the distinction between inner and outer psychic space and effectively mocks both the expressive model of gender and the notion of a true gender identity.” This is similar to the idea quoted earlier where Butler says that “gay is to straight not as copy is to original, but, rather, as copy is to copy. The parodic repetition of ‘the original’”. Butler says: “This perpetual displacement constitutes a fluidity of identities that suggests an openness to resignification and recontextualization; parodic proliferation deprives hegemonic culture and its critics of the claim to naturalized or essentialist gender identities. Although the gender meanings taken up in these parodic styles are clearly part of hegemonic, misogynist culture, they are nevertheless denaturalized and mobilized through their parodic recontextualization. As imitations which effectively displace the meaning of the original, they imitate the myth of originality itself.”

Butler is particularly concerned or wary of anything that has the potential reinstitute exclusionary boundaries. This will necessarily mean that they are wary of many of the categories that group people together, such as identities. “The mobilization of identity categories for the purposes of politicization always remain threatened by the prospect of identity becoming an instrument of the power one opposes. That is no reason not to use, and be used, by identity.” Since woman is one such category that means Butler’s feminism is going to be quite different from traditional feminism. “If a stable notion of gender no longer proves to be the foundational premise of feminist politics, perhaps a new sort of feminist politics is now desirable to contest the very reifications of gender and identity, one that will take the variable construction of identity as both a methodological and normative prerequisite, if not a political goal.” The variable construction of identity; that’s a key notion for Butler.

One of the ways they talk about this that I found interesting was the idea of the “illimitable et cetera”. “The theories of feminist identity that elaborate predicates of color, sexuality, ethnicity, class, and able-bodiedness invariably close with an embarrassed ‘etc.’ at the end of the list… This illimitable et cetera, however, offers itself as a new departure for feminist political theorizing.” Multiplicity is the word of the day. Butler wants to see the redeployment of power through parodic practices by which “Cultural configurations of sex and gender might then proliferate.”

Concluding Thoughts

A few final thoughts here. One is that I don’t think we can downplay or even want to downplay the radical nature of Butler’s ideas. For most people these ideas will seem quite radical. And Butler seems to embrace that. But what I am interested in is at least shining some light on the reasoning and assumptions behind these views, to see that they’re not merely assertions, but that there is some serious and sophisticated thought behind them. I’ve wanted principally to give Butler’s ideas a fair and hopefully accurate presentation. So where do I come down on all this? To the extent that I have opinions on Butler’s ideas I’d like to remain charmingly coy about them. But for the most part I don’t even really have any conclusive evaluations and don’t feel much of a need to. One of things I enjoy about the life of the mind, a life of intellectual study and contemplation is being able to explore different ideas and systems of ideas without necessarily having to adopt or reject them, but just explore them. And I think Butler’s system of ideas is kind of interesting. And I’ll wrap it up there. Thank you for reading.

Slavoj Žižek and the Sublime Object

Slavoj Žižek is a uniquely entertaining and engaging philosopher with an animated oratorical style. But this can often make it difficult to follow his train of thought and pick out key ideas from it. This episode gives an overview of his most important book, The Sublime Object of Ideology, which presents many of the ideas persisting in all his work. In particular, this overview focuses on Žižek’s theory of the Sublime. Žižek’s ideas bring him into close contact with the German philosophers Immanuel Kant and G.W.F. Hegel.

If we were to put philosophers on a scale from dry, boring presentation styles on one end to engaging, entertaining presentation styles on the other, Slovenian philosopher Slavoj Žižek would be off the chart with his exuberant and oratorical approach. Half of his philosophy seems to consist of jokes. He has a charming (or maybe frustrating) tendency to evade direct argument and debate, opting instead for a kind of friendly banter that opens up a space for people to make up their own minds. For all that, it can be difficult to follow his train of thought or identify some kind of thesis from it all. That difficulty would certainly not be unique among philosophers, especially those philosophers with whom Žižek tends to engage. For example, most of Žižek’s philosophy is a kind of interpretation of G.W.F. Hegel through the ideas of Jacque Lacan. Hegel and Lacan may be two of the most difficult philosophers in the entire tradition of European Continental philosophy. So it’s not surprising that Žižek would be somewhat difficult to follow as well. Be that as it may, Žižek does have some core ideas that recur in his books and talks. Many of these core ideas are present in what has been called his magnum opus, his 1989 book, The Sublime Object of Ideology.

I haven’t found much out there by way of detailed summaries for this book, so after I read it I thought I’d make one myself. I won’t give a summary of all the ideas in the book. There are a lot of them. Instead I’d like to focus on the subject I found most interesting. And that is Žižek’s theory of the Sublime. Žižek is perhaps best known for his purported and self-attributed communism. A provocative claim, to be sure. And that provocation might even be its most significant aspect. Žižek’s communism and leftist politics are interesting features of his thought but for the purposes of this summary I’m not going to focus as much on the political aspects of the book, in spite of the word “ideology” being right there in the title. Instead I want to focus on the concept of the Sublime and look at how Žižek fits into the history of the philosophical treatment of the Sublime.

I’ll introduce his basic position here first and then fill in the details and background information to explain it. Žižek says that there are certain objects that we elevate and believe have or act as if they had greater qualities behind their surface level. But, there is actually nothing greater behind this surface level. Nevertheless, for Žižek there is a real experience of sublimity. It is not of some greater thing, in or behind the object. Rather we experience the Sublime in the very process of discovering that there is nothing, no greater thing behind the object. This realization is Žižek’s sublime object, the realization that there is nothing being concealed.

This will all require some explanation and background. First let’s look at the concept of the Sublime and its philosophical history.

Žižek’s theory of the Sublime is in dialogue with the aesthetic theory of Immanuel Kant. In Kant’s thought aesthetic responses include the Beautiful and the Sublime. The Beautiful is well-ordered and harmonious. The Sublime is awesome and even frightening. The Sublime is found in those things that transcend our ability to grasp them; powers of nature like thunder and lightning. Žižek’s underlying theory of the Sublime is quite different from Kant’s but it also has a quality of not being directly approachable. He says in The Sublime Object of Ideology, that “The sublime object is an object which cannot be approached too closely”. But for Žižek the reason for this is very different from Kant’s. For Žižek, “if we get too near it, it loses its sublime features and becomes an ordinary vulgar object”. Kant’s sublime object would not be something that loses its sublimity. But for Žižek the object of the Sublime is not anything special in itself. And this is an important difference. For Žižek it’s not the object itself that is special. The object is elevated to a special position through a formal fiction. And the sublimity is found in the very process of recognizing that there is nothing greater behind the ordinary, vulgar object.

What are some examples of this kind of formal fiction? Žižek finds theoretical similarities of this process in the work of Marx and Freud. Both Marx and Freud theorize that certain formal processes endow ordinary objects with fictionally extraordinary qualities. This is also called a fetishization. For Marx this happens with the commodity and for Freud this happens with dreams. In both cases people act as if the commodity or the dream have some greater significance. They don’t. But the process of acting as if they did still has real effects in the social structures of which they are a part. Fiat currency is a pretty clear case of this. Money only has value because people act as if it did. And so, in a way, it does. Žižek says:

“There is a fundamental homology between the interpretative procedure of Marx and Freud – more precisely, between their analysis of commodity and of dreams. In both cases the point is to avoid the properly fetishistic fascination of the ‘content’ supposedly hidden behind the form: the ‘secret’ to be unveiled through analysis is not the content hidden by the form (the form of commodities, the form of dreams) but, on the contrary, the ‘secret’ of this form itself.”

Žižek says the key is not the content, like what a dream means for instance. The key is that there is a formal process of apparent concealment in the first place. We come to understand this process not by figuring out what is being concealed but by realizing that the only thing being concealed is that nothing is concealed. There’s a pattern that will recur here, the kind of pattern to which Žižek will apply this sort of deflation. There’s an object and a presumed concealed object behind it. For all such patterns Žižek will locate the Sublime in the process of dispelling the notion of concealment to reveal that there is nothing being concealed. 

Another example where Žižek applies this pattern is to Kant’s philosophy of phenomena and noumena. That is (1) things as we conceive of them and (2) what we posit to be the inaccessible things in themselves. In Kant’s philosophy we understand the world around us using mental structures called categories that organize the raw perceptions of our senses. We take raw sense data and impose our categories onto them like unity, plurality, causality, dependence, necessity, contingency, etc. We use these categories to order our sense perceptions and make them intelligible. But what’s crucial for Kant is that the product of any such mental construction is just that: a construction. It is not the thing in itself. This is the Kantian distinction between noumena and phenomena. A noumenon is the “thing in itself” as it is, independent of our conception of it. A phenomenon is the thing as it is to us after it has passed through our mental processing. For Kant the phenomenon is not the same thing as the noumenon. In fact, the noumenon, the thing in itself, is forever inaccessible to us.

Žižek compares this to there being something hidden behind a curtain. Sort of like in the Wizard of Oz. The phenomenon is what we see, but we imagine that there’s something deeper. Something behind the curtain. Žižek calls this “the Kantian gap dividing forever the Thing from the world of phenomena”. And naturally, Žižek is going to challenge this idea. For Žižek the Sublime arises in the realization that there is nothing behind the curtain: “The illusion that there is something hidden behind the curtain is thus a reflexive one: what is hidden behind the appearance is the possibility of this very illusion – behind the curtain is the fact that the subject thinks something must be behind it.” What the Kantian gap conceals is “a foreboding that perhaps this Thing is itself nothing but a lack”. For Žižek this introduces a paradox. For Kant the noumenon is forever inaccessible. But for Žižek (and for Hegel) it’s this very failure that enables us to access – not the thing in itself, since that’s a fiction – but the Sublime: “The paradox of the Sublime is as follows: in principle, the gap separating phenomenal, empirical objects of experience from the Thing-in-itself is insurmountable – that is, no empirical object, no representation (Vorstellung) of it can adequately present (darstellen) the Thing (the suprasensible Idea); but the Sublime is an object in which we can experience this very impossibility, this permanent failure of the representation to reach after the Thing.”

Žižek traces this insight back to Hegel and sees it as a Hegelian corrective to Kant. A few illustrative quotes on this point:

“Hegel’s reproach to Kant… is… that it is Kant himself who still remains a prisoner of the field of representation… Kant still presupposes that the Thing-in-itself exists as something positively given beyond the field of representation, of phenomenality… Hegel’s position is, in contrast, that there is nothing beyond phenomenality, beyond the field of representation.”

One of the most important ideas in the interpretation of Hegel’s philosophy is of the dialectic. A dialectic is, in its most basic form, a dialogue, a conversation in which ideas are exchanged. Often the ideas being exchanged are in tension and the hope is that in this interchange of competing ideas there can be some kind of productive insight, that this dialectic will be philosophically generative. This was an idea going around among German philosophers at the time. We also see this in Johann Fichte. The pattern is sometimes expressed as a triad of a thesis, a competing antithesis, and a resulting synthesis. We also find a materialist, economic version of this in Marx.

Žižek has a somewhat different and interesting perspective on the Hegelian dialectic, one that he understands to be more true to Hegel’s own. For Žižek the result of the dialectic is fundamentally a change in perspective that reveals the Nothingness, that dispels the illusion of concealment. It’s the pattern we’ve been seeing where there’s an object, a supposed concealed object, and then a revelation that there’s nothing being concealed. Žižek also sees this in the Hegelian dialectic.

“The only philosophical counterpart here is Hegelian dialectics: at the very beginning of his Logic, Being and Nothingness are not complementary, neither is Hegel’s point that each of them obtains its identity through its difference from the other. The point is that Being in itself, when we try to grasp it “as it is’, in its pure abstraction and indeterminacy, without further specification, reveals itself to be Nothingness.”

I’ve gone far too long talking about Žižek without sharing any of his jokes so I’ll share one here that he uses to highlight this idea:

“Here we have a kind of dialogic economy: we articulate a proposition denying the subject, our attempt fails, we experience the absolute contradiction, the extreme negative relationship between the subject and the predicate – and this absolute discordance is the subject as absolute negativity. It is like a well-known Soviet joke about Rabinovitch, a Jew who wants to emigrate. The bureaucrat at the emigration office asks him why; Rabinovitch answers: ‘There are two reasons why. The first is that I’m afraid that in the Soviet Union the Communists will lose power, there will be a counter-revolution and the new power will put all the blame for the Communist crimes on us, Jews – there will again be anti­-Jewish pogroms… ‘But’, interrupts the bureaucrat, ‘this is pure nonsense, nothing can change in the Soviet Union, the power of the Communists will last forever!’ ‘Well,’ responds Rabinovitch calmly, ‘that’s my second reason’. The logic is the same here as in the Hegelian proposition ‘the spirit is a bone’: the very failure of the first reading gives us the true meaning… The Rabinovitch joke also exemplifies the logic of the ill-famed Hegelian triad: if the first reason for emigrating is the ‘thesis’ and the bureaucrat’s objection the ‘anti-thesis’, then the ‘synthesis’ is not any kind of return to the thesis, some kind of healing of the wound made by the anti-thesis – the ‘synthesis’ is exactly the same as the ‘anti-thesis’, the only difference lies in a certain change of perspective.”

So is the problem that the Communists will lose power or that they won’t? That, for Rabinovitch, is precisely the problem. It doesn’t matter. It’s bad either way. This is a humorous, sort of parabolic illustration of the kind of dialectic where Žižek sees discord as fundamental; “absolute discordance” as he calls it. Early in the book Žižek proposes that Hegelian dialectics don’t magically resolve antagonisms. Instead, Žižek asks: “What if, for Hegel, the point, precisely, is to not ‘resolve’ antagonisms ‘in reality’, but simply to enact a parallax shift by means of which antagonisms are recognized ‘as such’ and thereby perceived in their ‘positive’ role?”

This kind of recognition of the fundamentality of discordance is instrumental in Hegel’s system for how the mind progresses to higher levels of knowledge. Arriving at such discordance is not a failure but a successful insight into the nature of things. “We pass from Understanding to Reason not when this analysis, or tearing apart, is overcome in a synthesis that brings us back to the wealth of reality, but when this power of ‘tearing apart’ is displaced from being ‘merely in our mind’ into things themselves, as their inherent power of negativity.” This “tearing apart” is intrinsic to the nature of things. Later in the book Žižek says in a gesture of “decisive emphasis” that “it is not only that the appearance, the fissure between appearance and essence, is a fissure internal to the essence itself; the crucial point is that, inversely, ‘essence’ itself is nothing but the self rupture, the self fissure of the appearance.” Contradiction is “an internal condition of every identity.”

What’s crucial is that in this realization of negativity, negativity itself takes on a positive role. When an illusion is dispelled and we realize that there’s nothing behind it this is a negativity. But it’s not just a negativity. There’s a positive aspect to it in the realization of the negativity, in seeing the negativity as such, where before it was unrealized and unnoticed. 

“So ‘we’ (who have already ‘gone through the fantasy’) can see that there is nothing where the

consciousness thought that it saw something, but our knowledge is already mediated by this ‘illusion’ in so far as it aims at the empty space which makes the illusion possible. In other words, if we subtract from the illusion the illusion itself (its positive content) what remains is not simply nothing but a determinate nothing, the void in the structure which opened the space for the ‘illusion’. To ‘unmask the illusion’ does not mean that ‘there is nothing to see behind it’: what we must be able to see is precisely this nothing as such – beyond the phenomena, there is nothing but this nothing itself, ‘nothing’ which is the subject.”

This idea of a “determinate nothing” is interesting. It reminds me of another joke that Žižek often tells, not in this book but elsewhere. “‘Waiter! A cup of coffee without cream, please!’ ‘I’m sorry, sir, we have no cream, only milk, so can it be a coffee without milk?’” There’s a difference to us between just simply nothing and a determinate nothing. The determinate nothing is the nothing where before, because of the illusion, there was thought to be something.

Žižek believes that this kind of stripping away toward Nothingness is instrumental for Hegel. Hegel’s most famous and arguably most important book was his Phenomenology of Mind. The Phenomenology is a challenging and intimidating book, but I might summarize it as a philosophy of the way that the mind comes to understand things. This is basically the project Hegel outlines in his Preface to the book. Process is fundamental for all of this. Hegel outlines how the mind progresses along successive stages toward fuller understanding.

Žižek challenges one interpretation of Hegel in which he is understood to say that the process of progressive understanding is one of continual accretion, constantly taking on more and more information. “Is not the Hegelian Idea effectively a voracious devourer which ‘swallows up’ every object it comes upon?” Žižek says No! That this is not a correct understanding of Hegel. “The standard reading constructs the Hegelian absolute Substance-Subject as thoroughly constipated [classic Žižek right there] – retaining within itself the swallowed content.” Continuing with the digestive metaphor Žižek says the Hegelian system is a much more healthy process of the usual consumption and excretion, integrating and using conceptual nutrients for sustenance and development, then discarding what’s left over. He calls this process one of “notional deployment”. The idea is not just to accumulate and retain everything, but rather to process and use. “The Idea, in its resolve/decision, ‘freely releases itself’ into Nature, lets Nature go, leaves it off, discards it, pushes it away from itself, and thus liberates it.”

Thus for Žižek abstraction, which is a crucial process of understanding, is also a process of subtraction. “For Hegel the true problem is… the fact that, when we observe a thing, we see too much in it, we fall under the spell of the wealth of empirical detail which prevents us from clearly perceiving the notional determination which forms the core of the thing.” We might think of the difference between someone enjoying nature while out on a hike and a biologist seeking to identify some property of a particular species in that setting. The hiker wants to experience the totality without reduction. But the biologist can’t do that. The biologist needs to isolate, dissect, and analyze.

What is the endpoint of the process of abstraction and subtraction? Žižek says: “If we make an abstraction, if we subtract all the richness of the different modes of subjectivation, all the fullness of experience present in the way the individuals are ‘living’ their subject-positions, what remains is an empty place which was filled out with this richness; this original void, this lack of symbolic structure, is the subject, the subject of the signifier. The subject is therefore to be strictly opposed to the effect of subjectivation: what the subjectivation masks is not a pre- or trans-subjective process of writing but a lack in the structure, a lack which is the subject.”

So here again we see Nothingness behind objects. The idea being that this would be the eventual result of repeated abstraction. But this idea that the end point of the process of abstraction is a void is not nihilistic. At least I don’t understand it to be nihilistic. We’re often interested in more general principles than in the particulars to which they refer. Scientists use data sets to find general patterns. The patterns, as in the form of equations, strip away many of the particulars and abstract from them. If there is some Theory of Everything it will certainly be highly abstract and several levels removed from almost all the particulars that populate our immediate reality. This is just a consequence of the kind of process that abstraction is.

That being said however, non-abstracted, un-subtracted particulars are also useful. What are we to do with the Sublime object in those settings where we have grasped the negativity behind the illusion? Interestingly enough, Žižek says we ought to “come to terms with it”.

“The point is not just that we must unmask the structural mechanism which is producing the effect of subject as ideological misrecognition, but that we must at the same time fully acknowledge this misrecognition as unavoidable – that is, we must accept a certain delusion as a condition of our historical activity, of assuming a role as agent of the historical process… There is no solution, no escape from it; the thing to do is not to ‘overcome’, to ‘abolish’ it, but to come to terms with it, to learn to recognize it in its terrifying dimension and then, on the basis of this fundamental recognition, to try to articulate a modus vivendi with it.”

Why is that? Why find some way to come to terms with the fundamental discordance? The reason is that the alternative is unacceptable. And here we see some of political implications of Žižek’s ideas:

“All ‘culture’ is in a way a reaction-formation, an attempt to limit, canalize – to cultivate this imbalance, this traumatic kernel, this radical antagonism through which man cuts his umbilical cord with nature, with animal homeostasis. It is not only that the aim is no longer to abolish this drive antagonism, but the aspiration to abolish it is precisely the source of totalitarian temptation: the greatest mass murders and holocausts have always been perpetrated in the name of man as harmonious being, of a New Man without antagonistic tension…”

“We have the same logic with democracy: it is – to use the worn-out phrase attributed to Churchill – the worst of all possible systems; the only problem is that there is no other which would be better. That is to say, democracy always entails the possibility of corruption, of the rule of dull mediocrity, the only problem is that every attempt to elude this inherent risk and to restore ‘real’ democracy necessarily brings about its opposite – it ends in the abolition of democracy itself.”

There’s a bit of similarity here to Richard Rorty’s liberal ironism. One of Žižek’s favorite phrases is: “They know very well what they are doing, but still, they are doing it.” Whether it’s in the canalizing aspects of culture or in a reverence for democracy we persist in acting out our cultural and democratic roles as if the ideologies underlying them were not discordant and illusory. Far worse would be to try to iron out the wrinkles, as totalitarian regimes strive to do.

I’ll wrap all this up with two parting thoughts. Or rather, two parting questions. First is: How accurate is Žižek’s interpretation of Hegel? I guess that presumes that there are accurate and inaccurate interpretations of Hegel, something that the logical positivists might have taken issue with. But whatever. Hegel is one of those philosophers who has so many commentators he almost lives entirely through them. The Hegel of Alexandre Kojève, for example, being one very significant incarnation of Hegel. The second question is what to make of Žižek’s ideas as such. In particular the idea we experience the Sublime in the very process of discovering that there is nothing, no greater thing behind the object; the realization that there is nothing being concealed. In how many domains does this idea apply? Are all apparent concealments false concealments? Are there any true “things in themselves” that exist in reality, even if beyond our reach? I’m inclined to think that there are.

So, if for whatever reason you found yourself interested in Žižek and wondering what some of his ideas were about I hope this was useful to you and not too far off base. Thanks for listening.

Philosophy of Structure, Part 3: Chemistry

Part 3 in this series on the philosophy of structure looks at examples and general principles of structure in chemistry. Subjects covered include quantum mechanics, the Schrödinger equation, wave functions, orbitals, molecules, functional groups, and the multiple levels of structure in proteins. General principles discussed include the nature of functions, the embedding of lower-level structures into higher-level structures, and the repeated use of a limited number of lower-level structures in the formation of higher-level structures.

In this third episode on the philosophy of structure I’d like to look at examples of structure in the field of chemistry. I’d like to see how some of the general principles of structure discussed in previous episodes apply to chemistry and to see what new general principles we can pick up from the examples of chemical structures. I’ll proceed along different scales, from the smallest and conceptually most fundamental components of chemical structure up to the larger, multicomponent chemical structures. For basic principles I’ll start with quantum mechanics, the Schrödinger  equation, and its wave function solutions, which constitute atomic and molecular orbitals. From there I’ll look at different functional groups that occur repeatedly in molecules. And lastly I’ll look at the multiple levels of structure of proteins, the embedding of chemical structures, and the use of repeatable units in the formation of multicomponent chemical structures.

One aspect from previous discussions that won’t really show up in chemistry is an aesthetic dimension of structure. That’s not to say that chemical structures lack beauty. I find them quite beautiful and the study and application of chemical structures has actually been the primary subject of my academic and professional work. In other words, I probably find chemistry more aesthetically satisfying than most people commonly would. But what I’m coming to think of as the philosophical problem of systematizing the aesthetic dimension of structure, in fields like music, art, and literature, isn’t so directly applicable here. I’ll get back to that problem in future episodes. The aesthetic dimension is not so intrinsic to the nature of the subject in the case of chemistry.

So let’s start getting into chemical structures by looking at the smallest and most conceptually fundamental scale.

Matter Waves

There is, interestingly enough, an intriguing point of commonality between music and chemistry at the most fundamental level; and that is in the importance of waveforms. Recall that the fundamental building block of a musical composition is a sound wave, a propagation of variations in the local pressure in which parts of the air are compacted and parts of the air are rarified. Sound waves are governed by the wave equation, a second order partial differential equation, and its solutions, in which a series of multiple terms are added together in a superposition, with each individual term in that summation representing a particular harmonic or overtone. There are going to be a lot of similarities to this in the basic building of chemical structures.

One of the key insights and discoveries of twentieth century science was that matter also takes the form of waves. This is foundational to quantum mechanics and it is known as the de Broglie hypothesis. This was a surprising and strange realization but it goes a long way in explaining much of what we see in chemistry. Because a particle is a wave it also has a wavelength. 

Recall that in acoustics, with mechanical waves propagating through a medium, wavelength is related to the frequency and speed of the wave’s propagation. That relation is:

λ = v/f

Where λ, is the wavelength, f is frequency, and v is the wave propagation velocity. With this kind of mechanical wave the wave is not a material “thing” but a process, a disturbance, occurring in a material medium.

But with a matter wave the wave is the matter itself. And the wavelength of the matter wave is related to the particle’s momentum, a decidedly material property. A particle’s wavelength is inversely proportional to its momentum. This relation is stated in the de Broglie equation:

λ = h/p

In which λ is the wavelength, h is a constant called Planck’s constant (6.626×10−34 J/s), and p is momentum. Momentum is a product of mass and velocity:

p = mv

Because the wavelength of a matter wave is inversely proportional to momentum the wavelength for the matter waves of macroscopic particles, the kinds of objects we see and interact with in our normal experience, is going to be very, very short, so as to be completely negligible. But for subatomic particles their wavelengths are going to be comparable to the scale of the atom itself, which will make their wave nature very significant to their behavior.

One interesting consequence of the wave nature of matter is that the precision of simultaneous values for momentum and position of a matter wave is limited. This is known as the Uncertainty Principle. There’s actually a similar limit to the precise specification of both wavelength and position for waves in general, i.e. for any and all waves. But because wavelength is related to momentum in matter waves this limitation gets applied to momentum as well.

Recall that with sound waves a musical pitch can be a superposition of multiple frequencies or wavelengths. This superposition is expressed by the multiple terms in a Fourier Series. Any function can be approximated using a Fourier Series, expressed in terms of added sinusoidal (oscillating) waves. A function that is already sinusoidal can be matched quite easily. The Fourier Series can converge on more complicated functions as well but they will require more terms (that’s important). In the case of musical pitches the resulting functions were periodic waves that repeated endlessly. But a Fourier Series can also describe pulses that are localized to specific regions. The catch is that more localized pulses, confined to tighter regions, require progressively more terms in the series, which means a higher number of wavelengths.

Bringing this back to matter waves, these same principles apply. Under the de Broglie formula wavelength is related to momentum. A pure sine wave that repeats endlessly has only one wavelength. But it also covers an infinite region. As a matter wave this would be a perfect specification of momentum with no specification of position. A highly localized pulse is confined to a small region but requires multiple terms and wavelengths in its Fourier Series. So its position is highly precise but its momentum is much less precise.

The limit of the simultaneous specification of momentum and position for matter waves is given by the equation:

σxσp ≥ h/(4π)

Where σx is the standard deviation of position, σp is the standard deviation of momentum, and h is Planck’s constant. The product of these two standard deviations has a lower limit. At this lower limit it’s only possible to decrease the standard deviation of one by increasing the standard deviation of the other. And this is a consequence of the wave nature of matter.

The most important application of these wave properties and quantum mechanical principles in chemistry is with the electron. Protons and neutrons are also important particles in chemistry, and significantly more massive than electrons. But it’s with the electrons where most the action happens. Changes to protons and electrons are the subject of nuclear chemistry, which is interesting but not something we’ll get into this time around. In non-nuclear chemical reactions it’s the electrons that are being arranged into the various formations that make up chemical structures. The behavior of an electron is described by a wave function and the wave equation is governed by the Schrödinger equation.

The Schrödinger equation is quite similar to the classical wave equation that governs sound waves. Recall that the classical wave equation is:

d2u/dx2 = (1/v2) * d2u/dt2 

Where u is the wave displacement from the mean value, x is distance, t is time, and v is velocity. A solution to this equation can be found using a method of separation of variables. The solution u(x,t) can be written as the product of a function of x and a sinusoidal function of time. We can write this solution as:

u(x,t) = ψ(x) * cos (2πft)

Where f is the frequency of the wave in cycles per unit time and ψ(x) is the spatial factor of the amplitude of u(x,t), the spatial amplitude of the wave. Substituting ψ(x) * cos (2πft) into the differential wave equation gives the following equation for the spatial amplitude ψ(x).

d2ψ/dx2 +2f2/v2 * ψ(x) = 0

And since frequency multiplied by wavelength is equal to velocity (fλ = v) we can rewrite this in terms of wavelength, λ:

d2ψ/dx2 +2/λ2 * ψ(x) = 0

So far this is just applicable to waves generally. But where things get especially interesting is the application to matter waves, particularly to electrons. Recall from the de Broglie formula that:

λ = h/p

In which h is a constant called Planck’s constant (6.626×10−34 J/s) and p is momentum. We can express the total energy of a particle in terms of momentum by the equation:

E = p2/2m + V(x)

Where E is total energy, m is mass, and V(x) is potential energy as a function of distance. Using this equation we can also express momentum in these terms:

p = {2m[E – V(x)]]1/2

And since,

λ = h/p

The differential equation becomes

d2ψ/dx2 + 2m/ħ2 * [E – V(x)] ψ(x) = 0


ħ = h/(2π)

This can also be written as

2/2m * d2ψ/dx2 + V(x) ψ(x) = E ψ(x)

This is the Schrödinger equation. Specifically, it’s the time-independent Schrödinger equation. So what do we have here? There’s a similar relationship between the classical wave equation (a differential equation) and its solution u(x,t), which characterizes a mechanical wave. The Schrödinger equation is also a differential equation and its solution, ψ(x), is a wave function that characterizes a matter wave. It describes a particle of mass m moving in a potential field described by V(x). Of special interest to chemistry is the description of an electron moving in the potential field around an atomic nucleus.

Let’s rewrite the Schrödinger equation using a special expression called an operator. An operator is a symbol that tells you to do something to whatever follows the symbol. The operator we’ll use here is called a Hamiltonian operator, which has the form:

H = -ħ2/2m * d2/dx2 + V(x)

Where H is the Hamiltonian operator. It corresponds to the total energy of a system, including terms for both the kinetic and potential energy. We can express the Schrödinger equation much more concisely in terms of the Hamiltonian operator, in the following form:

H ψ(x) = E ψ(x)

There are some special advantages to expressing the Schrödinger equation in this form. One is that this takes the form of what is called an eigenvalue problem. An eigenvalue problem is one in which an operator is applied to an eigenfunction and the result returns the same eigenfunction, multiplied by some constant called the eigenvalue. In this case the operator is the Hamiltonian, H. The eigenfunction is the wave function, ψ(x). And the eigenvalue is the observable energy, E. These are all useful pieces of information to have that relate to each other very nicely, when expressed in this form.


In chemistry the wave functions of electrons in atoms and molecules are called atomic or molecular orbitals. And these are also found using the Schrödinger equation; they are solutions to the Schrödinger equation. The inputs to these wave functions are coordinates for points in space. The output from these wave functions, ψ, is some value, whose meaning is a matter of interpretation. The prevailing interpretation is the Born Rule, which gives a probabilistic interpretation. Under the Born Rule the value of ψ is a probability amplitude and the square modulus of the probability amplitude, |ψ|2, is called a probability density. The probability density defines for each point in space the probability of finding an electron at that point, if measured. So it has a kind of conditional, operational definition. More particularly, we could say, reducing the space to a single dimension, x, that |ψ(x)|2 gives the probability of finding the electron between x and x + dx. Going back to 3 dimensions, the wave function assigns a probability amplitude value, ψ, and a probability density value, |ψ|2, to each point in space. Informally, we might think of the regions of an orbital with the highest probability density as the regions where an electron “spends most of its time”.

Solutions to the Schrödinger equation, electron wavefunctions, can be solved exactly for the hydrogen atom. Other solutions cannot be solved analytically but can be approximated to high precision using methods like the variational method and perturbation theory. And again, we call these wave functions orbitals. I won’t get into the specifics of the methods for finding the exact solutions for the hydrogen atom but I’ll make some general comments. For an atom the Cartesian (x,y,z) coordinates for the three dimensions of space aren’t so convenient so we convert everything to spherical coordinates (r,θ,φ) in which r is a radial distance and θ and φ are angles with respect to Cartesian axes. The term for potential, V(r) in the Hamiltonian operator will be defined by the relation between a proton and an electron. And the mass of the electron also gets plugged into the Hamiltonian. Solving for the wave function makes use of various mathematical tools like spherical harmonics and radial wave functions. Radial wave functions in turn make use of Laguerre polynomials. Then solutions for the hydrogen atom will be expressed in terms of spherical harmonic functions and radial wave functions, with the overall wave function being a function of the variables (r,θ,φ).

Because the orbitals are functions of r, θ,and φ they can be difficult to visualize and represent. But partial representations can still give an idea of their structure. An orbital is often represented as a kind of cloud taking some kind of shape in space; a probability density cloud. The intensity of the cloud’s shading or color represents varying degrees of probability density.

The shapes of these clouds vary by the type of orbital. Classes of orbitals include s-orbitals, p-orbitals, d-orbitals, and f-orbitals. These different kinds of orbitals are grouped by their orbital angular momentum. s-orbitals are sphere shaped, nested shells. p-orbitals have a kind of “dumbbell” shape with lobes running along the x, y, and z axes. d-orbitals are even more unusual, with lobes running along two axes, and one orbital even having a kind of “donut” torus shape. Although we sometimes imagine atoms as microscopic solar systems with electrons orbiting in circles around the nucleus their structure is much more unusual, with these oddly shaped probability clouds all superimposed over each other. The structure of atoms into these orbitals has important implications for the nature of the elements and their arrangements into molecules. But before getting into that let’s pause a moment to reflect on the nature of the structure discussed up to this point.

Reflection on the Structure of the Wave Function

As with a sound wave, the function for an electron wave function is a solution to a differential equation, in this case the Schrödinger equation. This wave function ψ, is a function of position. In spherical coordinates of r, θ, and φ this function is ψ(r,θ,φ). In the most basic terms a function is a rule that assigns elements in a set, or a combination of elements from multiple sets, to a single element in another set. This rule imposes additional structure on relations between these sets. So in our case we have a set for all r values, a set for all θ values, a set for all φ values, and a set for all ψ values. Prior to the imposition of structure by any function we could combine elements from these sets in any way we like. In a four-dimensional (abstract) phase space or state space with axes r, θ, φ, and ψ all points are available, any ordered quadruple (r,θ,φ,ψ) is an option. That’s because an ordered triplet (r,θ,φ) can be associated with any value of ψ. There’s no rule in place limiting which values of ψ the ordered triplet (r,θ,φ) can associate with. The entire phase space is available; all states are available. But with the imposition of the function ψ(r,θ,φ) the region of permissible states conforming to this rule is significantly smaller. An ordered triplet (r,θ,φ) can be assigned to one and only one value of ψ.

It’s useful here to distinguish between logical possibility and physical possibility. In what sense are all ordered quadruples (r,θ,φ,ψ) in the state space “possible”? Most of them are not really physically possible for the electron in an atom because they would violate the laws of physics, the laws of quantum mechanics. That’s because the function, the wave function in fact is imposed. But in the theoretical case that it were not imposed, any ordered quadruple (r,θ,φ,ψ) would be logically possible; there’s no contradiction in such a combination. At least, not until we start to develop the assumptions that lead to the Schrödinger equation and its solutions. But since the actual, physical world follows physical laws only the states satisfying the function ψ(r,θ,φ) are physically possible.

This distinction between logical possibility and physical possibility highlights one, very basic source of structure: structure that arises from physical laws. Atomic orbitals are not man-made structures. There certainly are such things as man-made structures as well. But atomic orbitals are not an example of that. I say all this to justify including atomic orbitals as examples of structure in the first place, since in a physical sense they seem “already there” anyway, or as something that couldn’t be otherwise. But in light of the much more vast state space of logically possible states I think it makes sense to think of even these physically given states as highly structured when compared to the logically limitless states from which they stand apart.

I’d like to make one point of clarification here, especially considering the reputation quantum mechanics has for being something especially inexact or even anti-realist. What is it that the wave function specifies at each point in space, for each ordered triplet (r,θ,φ)? It’s certainly not the position of the electron. That indeed isn’t specified. But what is specified is the amplitude, ψ. And the square modulus of the amplitude, |ψ|2 is the probability of finding the electron at that position, (r,θ,φ). The wave function doesn’t specify the electron’s exact position. Does this mean that chaos reigns for the electron? The electron could, after all, be anywhere in the universe (with the exception of certain nodes). But that infinite extension of possible positions doesn’t mean that chaos reigns or that the electron isn’t bound by structure. The probability density of the electron’s position in space is very precisely defined and governs the way the electron behaves. It’s not the case that just anything goes. Certain regions of space are highly probable and most regions of space are highly improbable.

This is something of a matter of perspective and it’s a philosophical rather than scientific matter. But still just as interesting, for me at least. It pertains to the kinds of properties we should expect to see in different kinds of systems. What kinds of properties should we expect quantum systems to have? What are quantum properties? Do quantum systems have definite properties? I’ve addressed this in another episode on the podcast, drawing on the thought of Sunny Auyang. In her view there’s an important distinction to be made between classical properties and quantum properties. Even if quantum systems don’t have definite classical properties that’s not to say they don’t have properties at all. They just have properties of a different kind, properties that are more removed from the kinds of classical properties we interact with on a daily basis. We’re used to interacting with definite positions and definite momenta at our macroscopic scale of experience. At the quantum level such definite predicates are not found for position and momentum, but they are found for the position representation and momentum representation of a system’s wave function. Quoting Auyang:

“Are there predicates such that we can definitely say of a quantum system, it is such and so? Yes, the wavefunction is one. The wavefunction of a system is a definite predicate for it in the position representation. It is not the unique predicate; a predicate in the momentum representation does equally well. Quantum properties are none other than what the wavefunctions and predicates in other representations describe.” (How Is Quantum Field Theory Possible?)

I think of this as moving our perspective “up a level”, looking not at position itself but at the wave function that gives the probability amplitude, ψ, and probability density, |ψ|2, of position. That is where we find definite values governed by the laws of physics. It’s appropriate to look at this level for these kinds of quantum systems, because of the kind of things that they are. Expecting something else from them would be to expect something from a thing that is not appropriate to expect from the kind of thing that it is.

Molecular Orbitals

Let’s move now to molecules. Molecules are groups of atoms held together by chemical bonds. This makes use of a concept discussed in the last episode that is pertinent to structure generally: that of embedding. Lower-level structures get embedded, as kinds of modules, into higher-level structures. The lower-level structures remain but their combinations make possible a huge proliferation of new kinds of structures. As we move from the level of atoms to molecules the number of possible entities will expand dramatically. There are many more kinds of molecules than there are kinds of atoms. As of 2021 there are 118 different kinds of atoms called elements. That’s impressive. But this is miniscule compared to the number of molecules that can be made from combinations and arrangements of these elements. To give an idea, the Chemical Abstracts Service, which assigns a unique CAS registry number to different chemicals, currently has a database of 177 million different chemical substances. These are just molecules that we’ve found or made. There are many more that will be made and could be made.

Electrons are again key players in the formation of molecules as well. The behavior of electrons, their location probability densities, and wave-like behavior, continue to be defined by mathematical wave functions and abide by the Schrödinger equation. A wave function, ψ, gives a probability amplitude and its square modulus, |ψ|2, gives the probability of finding an electron in a given region. So many of the same principles apply. But the nature of these functions at the molecular level is more complex. In molecules the wave functions take new orbital forms. Orbitals in molecules take two new important forms: hybridized orbitals and molecular orbitals.

Hybridized orbitals are combinations of regular atomic orbitals that combine to form hybrids. So where before we had regular s-type and p-type orbitals these can combine to form hybrids such as sp3, sp2, and sp orbitals. With a carbon atom for instance, in the formation of various organic molecules, the orbitals of the valence electrons will hybridize.

Molecular orbitals are the wave functions for electrons in the chemical bonds between the atoms that make up a molecule. Molecular orbitals are formed by combining atomic orbitals or hybrid atomic orbitals from the atoms in the molecule. The wave functions for molecular orbitals don’t have analytic solutions to the Shrõdinger equation so they are calculated approximately.

A methane molecule is a good example to look at. A methane molecule consists of 5 atoms: 1 carbon atom and 4 hydrogen atoms. It’s chemical formula is CH4.  A carbon atom has 6 electrons with 4 valence electrons that are available to participate in chemical bonds. In the case of a methane molecule these 4 valence electrons will participate in 4 bonds with 4 hydrogen atoms. In its ground state the 4 valence electrons occupy one 2s orbital and two 2p orbitals. In order to form 4 bonds there need to be 4 identical orbitals available. So the one 2s orbital and three 2p orbitals hybridize to form 4 sp3 hybrid orbitals. An sp3 orbital, as a hybrid, is a kind of mixture of an s-type and p-type orbital. The dumbbell shape of an p-orbital combines with the spherical shape of an s-orbital to form a kind of lopsided dumbbell. It’s these hybrid sp3 orbitals that then combine with the 1s orbitals of the hydrogen atoms to form molecular orbitals. In this case the type of molecular orbitals that form are called σ-bonds.

The 2s and 2p orbitals in the carbon atom can also hybridize in other ways to form two or three bonds. For example, a carbon atom can bond with 2 hydrogen atoms and 1 other carbon atom. When it does this the 2s orbital hybridizes with just 2 of the 2p orbitals to form 3 sp2 orbitals, which bond with the 2 hydrogens and the other carbon. The remaining 2p orbital combines with the other carbon atom again, to its corresponding 2p orbital. This makes two sets of orbitals combining into two molecular bonds, a σ-bond and what is called a π-bond. When a σ-bond and a π-bond form between atoms it is called a double bond. Carbon atoms can also form triple bonds in which two sp orbitals are formed from the 2s orbital and one 2p orbital. This leaves two 2p orbitals to combine with their counterparts in another carbon atom to form a triple bond, composed of 1 σ-bond and 2 π-bonds. Single bonds, double bonds, and triple bonds all have their own geometrical properties like bond angles and freedom of rotation. This has effects on the properties of the resulting molecule.

Functional Groups

σ-bonds, π-bonds, single bonds, double bonds, and triple bonds make possible several basic molecular structures called functional groups. Functional groups are specific groupings of atoms within molecules that have their own characteristic properties. What’s useful about functional groups is that they occur in larger molecules and contribute to the overall properties of the parent molecule to which they belong. There are functional groups containing just carbon, but also functional groups containing halogens, oxygen, nitrogen, sulfur, phosphorus, boron, and various metals. Some of the most common functional groups include: alkyls, alkenyls, akynyls, and phenyls (which contain just carbon); fluoros, chloros, and bromos (which contain halogens); hydroxyls, carbonyls, carboxyls, and ethers (which contain oxygen); carboxamides and amines (which contain nitrogen); Sulfhydryls and sulfides (which contain sulfur); phosphates (which contain phosphorus); and so forth.

Repeatable Units

The last subject I’d like to address with all this is the role of repeatable units in the formation of complex chemical structures. Let’s come at this from a different direction, starting at the scale of a complex molecule and work our way down. One of the most complex, sophisticated kinds of molecules is a protein. Proteins are huge by molecular standards. Cytochrome c, for example, has a molecular weight of about 12,000 daltons. (For comparison, methane, discussed previously, has a molecular weight of 16 daltons). What we find with such molecules is that they are highly reducible to a limited number of repeatable units. But we could imagine it being otherwise; a macromolecule being irreducible from its overall macrostructure and not having any discernible repeating components. Let’s imagine a hypothetical, counterfactual case in which a macromolecule of that size is just a chaotic lump. Imagine going to a landfill and gathering a bunch of trash from a heap with all sorts of stuff in it, gathering it all together, rolling it into a ball, and binding it with hundreds of types of unmixed adhesives. Any spatial region or voxel of that lump would have different things in it. You might find some cans and wrappers in one part, computer components in another, shredded office papers in another, etc. We could imagine a macromolecule looking like that; a completely heterogeneous assembly. We could imagine further such a heterogeneous macromolecule being able to perform the kinds of functions that proteins perform. Proteins can in fact be functionally redundant; there’s more than one way to make a protein that performs a given function. So we might imagine a maximally heterogeneous macromolecule that is able to perform all the functions that actually existing proteins perform. But this kind of maximal heterogeneity is not what we see in proteins.

Instead, proteins are composed of just 20 repeatable units, a kind of protein-forming alphabet. These are amino acids. All the diversity we see in protein structure and function comes from different arrangements of these 20 amino acids. Why would proteins be limited to such a small number of basic components? The main reason is that proteins have to be put together and before that they have to be encoded. And it’s much more tractable to build from and encode a smaller number of basic units, as long as it gives you the structural functionality that you’ll need in the final macrostructure. It might be possible in principle to build a macromolecule without such a limited number of repeatable units. But it would never happen. The process to build such a macromolecule would be intractable.

This is an example of a general principle I’d like to highlight that we find in chemistry and in structure generally. And it’s related to embedding. But it’s a slightly different aspect of it. Complex, high-level structures are composed by the embedding of lower-level structures. And the higher-level structures make use of a limited number of lower-level structures that get embedded repeatedly.

In the case of a protein, the protein is the higher-level structure. Amino acids are the lower-level structures. The structures of the amino acids are embedded into the structure of the protein. And the higher-level structure of the protein uses only a limited number of lower-level amino acid structures.

A comparison to writing systems comes to mind here. It’s possible to represent spoken words in written form in various ways. For example, we can give each word its own character. That would take a lot of characters, several hundred and into the thousands. And such a writing system takes several years to be able to use with any competence. But it’s also possible to limit the number of characters used in a writing system by using the same characters for phonemic properties common to all words, like syllables or phonemes. Many alphabets, for example, only have between 20 and 30 characters. And it’s possible to learn to use an alphabet fairly quickly. And here’s the key. There’s no functional representational loss by using such a limited number of characters. The representational “space” is the same. It’s just represented using a much smaller set of basic components.

Biochemists mark out four orders of biomolecular structure: primary, secondary, tertiary, and quaternary. And this is a perfect illustration of structural embedding.

The primary structure of a protein is its amino acid sequence. The primary structure is conceptually linear since there’s no branching. So you can “spell” out a protein’s primary structure using an amino acid alphabet, one amino acid after another. Like, MGDVEK: methionine, glycine, aspartic acid, valine, glutamic acid, and lysine. Those are the first 6 amino acids in the sequence for human Cytochrome c. What’s interesting about amino acids is that they have different functional groups that give them properties that will contribute to the functionality of the protein. We might think of this as a zeroth-level protein structure (though I don’t know of anyone calling it that). Every amino acid has a carboxyl group and an amino group. That’s the same in all of them. But they each have their own side chain or R-group in addition to that. And these can be classified by properties like polarity, charge, and other functional groups they contain. For example, methionine is sulfuric, nonpolar, and neutral; asparagine is an amide, polar, and neutral; phenylalanine is aromatic, nonpolar, and neutral; lysine is basic, polar, and positively charged. These are important properties that contribute to a protein’s higher-level structure.

The secondary structure of a protein consists of three-dimensional, local structural elements. The interesting thing about secondary structures in the context of embedding and repeatable units is that these local structures take common shapes that occur all the time in protein formation. The two most important structural elements are alpha helices and beta sheets. True chemical bonds only occur between the amino acid units of the primary structure but in the higher level structures the electrostatic forces arising from differences in charge distribution throughout the primary structure make certain regions of the primary structure attracted to each other. These kinds of attractions are called hydrogen bonds, in which a hydrogen atom bound to a more electronegative atom or group is attracted to another electronegative atom bearing a lone pair of electrons. In the case of amino acids such hydrogen bonding occurs between the amino hydrogen and carboxyl oxygen atoms in the peptide backbone.

In an alpha helix these hydrogen bonds form in a way that makes the amino acids wrap around in a helical shape. In a beta sheet strands of amino acids will extend linearly for some length and then turn back onto themselves, with the new strand segment extending backward and forming hydrogen bonds with the previous strand. These hydrogen bound strands of amino acids then form planar sheet-like structures. What’s interesting is that these kinds of secondary structures are very common and get used repeatedly, much like amino acids get used repeatedly in primary structures. Secondary structures, like alpha helices and beta sheets (among others), then get embedded in even higher-level structures.

The tertiary structure of a protein is its full three-dimensional structure that incorporates all the lower-level structures. Tertiary structures are often represented using coils for the alpha helix components and thick arrows for the beta sheet components. The way a protein is oriented in three-dimensional space is determined by the properties of its lower level structures all the way down to the functional groups of the amino acids. Recall that the different amino acids can be polar or nonpolar. This is really important because proteins reside in aqueous environments with highly polar water molecules. Nonpolar groups are said to be hydrophobic because conditions in which the surface area of exposure, the contact, between nonpolar groups and polar water molecules is minimized are entropically favored. Because of this polar and nonpolar molecules will appear to repel each other, a hydro-phobic effect. Think of the separation of oil and water as an example. Water is polar and oil is nonpolar. This is the same effect occurring at the scale of individual functional group units in the protein. Proteins can fold in such a way as to minimize the surface area of nonpolar functional groups exposed to water molecules. One way this can happen is that nonpolar amino acid sections fold over onto each other so that they interact with each other, rather than with water molecules and so that water molecules can interact with each other rather than with the nonpolar amino acid sections. These same kinds of effects driven by the properties of functional groups were also the ones bringing about the secondary structures of alpha helices and beta sheets.

Some proteins also have a quaternary structure in which multiple folded protein subunits come together to form a multi-subunit complex. Hemoglobin is an example of this. Hemoglobin is made up of four subunits; two alpha subunits and two beta subunits.

There’s a pattern here of structure building upon structure. But it does so with a limited set of repeatable structures. I’d like to address this again. Why should proteins be built out of only 20 amino acid building blocks. Certainly there could be (at least in theory) a macromolecule that has similar functionality and global structure, using the same functional group properties to get it to fold and form in the needed way, without the use of repeatable lower-level structural units. But that’s not what we see. Why? One important reason is that proteins need to be encoded.

Proteins are made from genes. Genes are sections of DNA that get translated into RNA and then transcribed in proteins. That’s a gene’s primary function: to encode proteins. DNA and RNA have further simplified components: only four types of nucleotides in each: guanine, adenine, cytosine, and thymine in DNA and guanine, adenine, cytosine, and uracil in RNA. These nucleotides have to match up with the proteins that they encode and it’s going to be very difficult to do that without dividing up the protein into units that can be encoded in a systematic way. There’s a complex biochemical process bringing about the process of transcribing an RNA nucleotide sequence into a protein. But since these are, at bottom, automatic chemical processes they have to proceed in systematic, repeatable ways. An entire macromolecule can’t have an entire intracellular biochemical system dedicated to just that macromolecule alone. For one thing, there are too many proteins for that. The same biochemical machinery for transcription has to be able to make any protein. So all proteins have to be made up of the same basic units.

The way this works in transcription is that molecules called transfer RNA (tRNA) are dedicated to specific combinations of the 4 basic RNA nucleotides. These combinations are called codons. A codon is some combination of 3 nucleotides. Since there are 4 kinds of nucleotides and each codon has 3 there are 43, or 64 possible combinations. Different codons correspond to different amino acids. Since they only code for 20 amino acids there is obviously some redundancy, also called degeneracy (which isn’t meant to be insulting by the way). The way that codons get transcribed into an amino acid is that the tRNA molecules that match the nucleotide sequences of the various codons in the RNA also convey their encoded amino acids. These tRNA molecules come together at the point of transcription, called ribosomes, and link the amino acids together into the chains that form the primary structure of the protein. This is just a part of the biochemical machinery of the process. What’s important to note here is that although there are a number of tRNA types it’s not unmanageable. There are at most 64 possible codon sequences. So there doesn’t have to be a unique set of transcription machinery dedicated to each and every kind of protein, which would be insane. The components only have to be dedicated to codon sequences and amino acids, which are much more manageable.

Key Takeaways

I’d like to summarize the foregoing with 4 key takeaways from this analysis of structure in chemistry that I think apply to a general philosophy of structure.

1. Structure can be modeled using functions

Recall that a function is a relation between sets that associates an element in one set or combination of elements from multiple sets to exactly one element in another set. The source sets are called domains and the target sets they map onto are called codomains. One example of a function we’ve looked at in both the previous episode on music and in this episode on chemistry is the waveform function. In chemistry mathematical functions called orbitals assign to each point in space (the domain) an amplitude value (the codomain).

2. Functions occupy only a small portion of a phase space

Functions, by nature, impose limitations. A relation that associates an element in a domain to more than one element in a codomain would not be a function. A function associates the domain element to only one codomain element. In this way functions are very orderly. To give an example, in an orbital a given point in space (a domain element) can have only one amplitude value (the codomain element). This is highly limited. To illustrate this, imagine a phase space of all possible combinations of domain and codomain values. Or to give a simpler comparison, imagine a linear function on an x-y plane; for example, the function y = x. This is a straight line at a 45 degree angle to the x and y axes. The straight line is the function. But the phase space is the entire plane. The plane contains all possible combinations of x and y values. But the function is restricted to only those points where y = x. A similar principle applies to orbitals. The corresponding phase space would be, not a plane, but a 4-dimensional hyperspace with axes r,θ,φ, and ψ. The phase space is the entire hyperspace. But the wave function, or orbital, is restricted to a 3-dimensional space in this 4-dimensional hyperspace. This kind of restriction of functions to small portions of phase spaces is a characteristic feature of structure generally.

3. Structural embedding

Embedding is a feature of structure that came up in music and has come up again in even more obvious form in chemistry. Just looking at proteins the different orders of structure is quite obvious and well known to biochemists, with their conceptual division of proteins into primary, secondary, tertiary, and quaternary protein structures, with each level of structure incorporating the lower level structures embedded into them. Using proteins as an example, even primary structures have embedded into them several layers of additional structure such as functional groups, molecular orbitals, atomic orbitals, and the general structure of the wave function itself. One key feature of such embedding is that properties and functionality of the lower-level structures are taken up and integrated into the higher-level structures into which they are embedded. We saw, for example, how the three-dimensional tertiary structure of a protein takes the form that it does because of the properties of functional groups in the side chains of individual amino acids, in particular polarity and nonpolarity.

4. Repeatable units

A final key takeaway is the use of repeatable units in the process of structural embedding. In retrospect this is certainly something that is applicable to music as well. We see repeatable units in the form of pitches and notes. In chemistry we see repeatable units in macromolecules like polymers and proteins. Polymers, like polyethylene, PVC, ABS, polyester, etc. certainly use repeatable units; in some cases a single repeating unit, or sometimes two or three. Proteins make use of more repeatable units but even there they make use of a limited number: 20 amino acids. We see here an important general principle of structure: that high-level structures tend to be composed through the repeated use of a limited number of lower-level structures rather than by forming as a single, bulk, irreducible macrostructure. The use of lower-level repeatable units in the higher-level structure facilitates the encoding and construction of high-level structures.

And that wraps up this study of structure in chemistry. Thank you for listening.