Hysteresis and Commutativity

I don't have a pithy sounding introduction at the moment so let's just get into it.

Averages and Essences

When processing information for the purposes of feedback it's a given that there will be some degree of error. The most well-known and simple way of dealing with this is to measure multiple times until the law of large numbers smooths these errors out; for a short time a coin might seem loaded if it comes up heads twice, maybe even thrice in a row, but over hundreds of flips you can be sure that if it's fair, it'll come up heads more or less 50% of those flips, with the amount of plausible variation from that number getting less and less as you flip more and more times. By this logic, one could repeatedly make a bet on a loaded coin, and in doing so more or less guarantee a profit, unless of course one has to bet a lot of their initial money on those first couple of flips, in which case you could lose whatever money you have to bet with before you rake in a profit; in other words, the order of heads and tails in the sequence can change how much money you make in the end and even bankrupt you. Should one bet a percentage of their money, on the other hand, well, then you'll never lose it all, and not only that, but the order of heads and tails won't even change how much money you make in the end. In this latter case, we have a match between what one expects over time and what one would expect if all bets were made at the same time, which is a manifestation of the property of ergodicity.

In loose informal terms, ergodicity means that the ensemble average is the same as the time average: that is, your final expected outcome isn't sensitive to the order of your outcomes but only how they add up in the end. In other words, it's not path-dependent, and one can represent this same idea diagrammatically. If we were to represent these different sequences as paths from the top-left corner of the below diagram to the bottom-right corner, these paths would all be effectively the same:

This is known in mathematics as a commutative diagram: one in which all paths from one node to another are equivalent. In more technical terms (for those interested), if we think of p as the percentage you gain or lose on each bet and every node as the set of real numbers and each arrow H as a function h(x) = x*(1+p) and each arrow T as t(x) = x*(1-p), and each path as a chain that composes these functions, then every composite function that takes you from one node of this diagram to another is the same insofar that it has the same output for each input, even if one can at the very least suppose a different internal process.

While ergodicity is a concept from statistical physics that can be a way of (at least loosely) talking about path-independence, the diagram above hints at a way of generalizing this concept: paths can still be constructed in distinct ways, but upon composing them into a set of paths from a common source to a common destination their distinctions are negated. This more generalized notion goes beyond probability, however, and straight into how one can denote the kernel of a phenomenon. Consider, for example, this diagram from my earlier essay, Narrative Engineering about what constitutes a successful model:

Here, it does not matter whether you first encode your observations into the model and then change the model to represent a different set of observations, or if you first observe the change and then encode it into the model: in both cases, you get the same result. And this is what makes something a representation as opposed to a symbol that simply purports to stand for an underlying reality: unlike a mere symbol, a true representation preserves some structure of the phenomenon in question, and in doing so can be reliably used as something that "stands for" it. That is, you could plug it into some other model or computer program or scientific protocol as a substitute for the original thing and not have to worry about anything changing so long as your representation encodes the information specifically needed by whatever is using it. In other words, a representation does something: it enables composition between processes.

This also gives away what the distinction is in language between pragmatics and semantics. Usually the answer is the extremely useless "distinction" between what a speech act "does" versus what the "content" of such speech "means", but what's really at play here is that language can always do all sorts of stuff, even if it's agreed to be outright drivel, but the semantics of language is a question of what structures can be inferred by seeing what remains invariant, i.e. what commutes.

A Question of Value

These ideas, when put together, can solve the question of what "value" means, at least in an economic sense. This is doubtless a contentious question, but these concepts vindicate Marx by putting him in an entirely different light from conventional interpretations, that for something to have economic "value" in some concrete sense requires a numeraire, and numeraires are not merely denoted but constructed. This requires a little bit more elaboration before proceeding:

A numeraire is simply a way of ranking things: i.e. the relationships between its elements are transitive. Now one may imagine that this can be seen as an arbitrary matter of "preference", but this kind of mentalism is facile: it's a tautology that if somebody does X and not Y then they must have "preferred " X, and this could be applied even if they "preferred" Y a nanosecond ago; more importantly for the purposes of this chapter, there's no notion of any kind of structure being preserved, which means there's nothing being represented about the subject and their desires. So instead, take on a stricter idea of what must make something "at least as good as" something else: it would have to as a bare minimum do everything that the other thing can do. This can be represented as a partial order, and indeed a very popular kind of partial order to look at is one defined on sets: a set is "at least as big as" another set if and only if it contains all of the elements of that other set. Simply having more elements doesn't suffice.

Consider, however, that when things cost money, they can be bought and they can be sold, so if you could reliably sell something for at least the cost of something else, then it is more valuable than that something else because you can in fact do everything the other thing you can do if you sell your current thing for enough money to afford that other thing. Of course, in the real world, reselling something takes time and you rarely get much in the way of resale value as things go bad. Not only that, but prices can vary in different regions or amongst different sellers. On the other hand, consider the deed to an ounce of gold stored electronically: this is an asset where you can sell just about any fine-grained fraction of it and where you can buy and sell instantaneously. So if you're offered an ounce of gold or an ounce of silver, you take the gold even if you like silver better or think silver will one day be more valuable, because in that case you just sell the gold, buy the equivalent amount of silver, and pocket the difference. A more painstaking elaboration of all this can be found in my polemic Imagine A Utility Function Stamping On A Human Face Forever, but hopefully what's being said here is intuitive enough.

The important thing about the example with gold and silver is that these are liquid assets, insofar that they can be bought and sold instantaneously for a single uniform quoted price. Now even financial assets exchanged electronically are not always liquid: if you have to sell too much at once, you may have to sell at lower and lower prices, or your claim to the asset may be valid but the institution that backs it up can't immediately provide you the means to sell it, and so on. The more liquid, however, the more fungible, and the more fungible, the less-path dependent your outcome: that is, to the degree that you can instantaneously buy and sell things for the listed price, the less it matters what order you do all of these financial operations in when tallying up the final outcome. The "value" of these assets then becomes exactly one thing since they're interchangeable and not time-sensitive: the profit you expect to make.

And whether or not he intended it (I can't read minds and don't care what people intend), Marx demonstrated that capitalism is the construction of value as a precise operator of comparison. Everything else flows from this: under the right circumstances, the output of a factory worker can be measured as how many copies of a uniform good they produced from which can be derived how much they contribute to the bottom line. The labor theory of value (LTV), while contentious and existing before Marx, becomes an empirical and material hypothesis rather than a moral conjecture: if the task is sufficiently rote and the produced goods sufficiently uniform, then the amount of hours worked will correlate to how much is made which will correlate to the company's bottom line. Now of course there will be fluctuations around this, but the fluctuations will be gaussian in nature: you'll see plenty of workers who are 10% more efficient than others, maybe even some who are twice or triple as efficient, but you'll never see a worker who is 100 times more efficient. Compare this, by contrast, to a field like software engineering, where a single misguided line of code could potentially set the company back by months or where a developer with sufficient insight could lead the team down a path that cuts total work on an assignment for everyone in half. The theory of surplus value is then very simply measuring the discrepancy between these precise levels of output and how much the workers are getting compensated.

But one doesn't even need to suppose the labor theory of value to get the point about surplus value across: what matters is that to the extent a job and its outputs can be standardized, notions of surplus and deficit in terms of compensation become a tenable concept. But I digress, as this is not a chapter about Marxism.

The key point to take away here is that a concrete notion of the "value" of something emerges from establishing sufficient interchangeability between different components. Standardization, mechanization, commodification, financialization, and many other forms of containerization are different facets of this phenomenon, and what all of them have in common is that they create value by making distinct paths, each representing different orders of operations and different specific inputs and outputs, commute. To the extent that the process commutes, all particulars seen in the nodes and arrows between them become irrelevant.

I should stress that this is not an inherently good or inherently bad thing. Streamlining things reduces all kinds of overhead and makes possible processes that weren't before, but on the other hand there is an erasure of particulars.

Asymmetry and Knowledge

Commutativity is the concept that underlies all notions of constancy and composability, but this property deals specifically with what is not novel, and without novelty nothing can progress. For example, physicists have been distraught for years now that their predictions on particle accelerator experiments have been succeeding, since that means there's no discrepancy from their current "standard model" to investigate. To refer to an earlier diagram, if your diagram that maps your observations to your models always commutes, then that means your scientific paradigm can only "see" the things already encoded in the model. Yes, there may be particulars that you can refer to in some sense that exist outside the model, but those particulars have to be meaningful insofar that they are encoded somewhere in the wider paradigm, even if not in the model itself.

Should one find such a codifiable discrepancy, however, there becomes an opening for inventing and testing new models, and in some cases entirely new concepts. This opening, by its very definition, cannot be given a positive definition, at least not with respect to the paradigm in question, which is why Hegel referred to this idea as the negative. For Hegel, "negation" is not the layman's sense of "this proposition is false", but rather the part of reality that is not merely invisible but in fact inconceivable to a given system of logic (note: I am not necessarily using the term "logic" in the sense that Hegel uses it, as I do not know enough about how he defines "logic".) Deleuze takes a very similar concept (in fact so similar that I don't really know if there's a fundamental difference, feel free to reach out if you can enlighten me) called the virtual, which, unlike Kant's noumenal realm, which is effectively an ethereal mirror of the empirical realm of the senses, is not such a mirror but instead is a source of new predicates.

These predicates are not magically "generated" but instead exist as a result of the inherent differences between every repetition of the "same" act. Each such repetition carries its unique signature by virtue of happening at a different time. But instead of thinking of a different time as a mere difference in chronological timestamp, consider instead the idea that time, according to Kant, is the concept that signifies the possibility of change. Therefore, insofar that two paths in a diagram don't commute, one can consider those two paths as representing distinct potential timelines, and the fact that there is a difference between them that is identifiable by the diagram's failure to commute turns this difference into a new concept signified by the diagram.

Now, it should be stated that one does not need to ever think of any predicates ever being explicitly encoded in the diagram. The nodes need not have any specific anatomies, the arrows need not have any specific extensional identifiers: all that matters is the relationships between all of these things. Insofar that the diagram fails to commute then, one derives backwards from there that there exists something that one could identify as "why" the diagram didn't commute, so the failure to commute denotes the intensional difference, which implies that there is a "reason" for such a difference. You could say, in both the layman's definition and Deleuze's definition, that this reason virtually exists, and the next step would be to find a way to derive some consistent account, a kernel, of this difference, thus turning it into a persistent concept.

That last sentence was a bit convoluted so a breakdown of how Deleuze sees the genesis of difference is worth noting, since it is entirely in line with what's been said so far: difference is not (necessarily) a difference between overt properties but something that exists in its own right, and new concepts are born by deriving the "properties" that would differ by virtue of such a difference (I say "derive" to avoid the debate between "invent" and "discover".)

But where do differences come from then? This is a bit unsatisfactory, so let's get back to the idea of the commuting diagram: insofar that these different paths don't commute and therefore represent distinct timelines if one is to take a Kantian defintion of time, then what you're seeing is two timelines that are out of sync: that is, you cannot, for whatever reason, take those two paths and put them back together again like you'd be able to with synchronized processes. Hysteresis, which is defined by changes in a phenomenon lagging behind changes in what's causing those changes, is itself doubtlessly something that makes things fall out of sync, since it'll now be affected differently depending on when it gets that input and will in turn feed a different value into that input, making it no longer predictable in the way that it would be if it were just a discrete input-output-input-output sequence where each "input" is based on the last "output".

To even draw a diagram at all of a process subject to hysteresis is itself problematic, but doable, but you now can't expect them to commute unless the inputs and outputs are not sensitive to exact timing: for example, in the case in the beginning of betting a specific percentage of money on coin flips, if the inputs and outputs for betting on coin flips were out of sync, you'd still actually not be path-dependent. In that case, however, the lag is something with no actual consequence, and there is hysteresis only in a pedantic sense. And so, in the absence of the kind of convergence on an outcome that the property of ergodicity guarantees, hysteresis creates a condition by which one's model does not commute and a difference is found within the model without any external reference to any properties. For Deleuze, such differences are known as intensive differences, with the virutal being itself the unsunbstantiated "property" of which there is ostensibly a difference. That is, difference comes first, but a difference must by definition be between two things, so those two things must in turn be specified in order to incorporate that difference into our vocabulary.

Whereas I must confess here that I still have not finished Difference and Repetition and cannot speak for where Deleuze says the true source of these intensive differences is, I can nonetheless sum up my own thesis here: that difference is logically entailed from the failure to commute that is itself our formal way of denoting hysteresis; this failure exists not as a mere abstraction but as an actual real-life breakdown of what we're actually doing with such a diagram, even if that diagram has never been explicitly posited but is merely implied by the way in which a process (for example, a scientific project) is done. From here, knowledge proceeds by finding a kernel elsewhere that formalizes that difference, and in turn expands our lexicon for the next round of invention and discovery. In other words, while commutativity inheres in that which stabilizes life, any kind of élan vital that one may speak of is a question of hysteresis.