Latenight Decaf

  • Issues with Partitions

    This’ll be a short post. I screwed up with the partition stuff. I tried to share the project with a few mathematicians in the field who I thought might respond. Total silence. After a while, I decided to try and model it myself, see if I could find anything interesting. So I learned the basics of a mathematical programming tool called Sage.

    Anyhow, it turns out that my alternative definition of the crank only almost works. The core issue is that, while the crank is determined correctly by my setup in most cases, it gets the sign wrong for the cranks of canonical partitions. The absolute value is perfect, but the whole thing is not quite there.

    This is very dispiriting, but I still intend to look into it, see if these concepts still have any interesting properties that I haven’t noticed yet. Anyhow, that’s all.

  • Coming Back to Partitions

    I won’t go into too much detail today, because it’s my first time back to the site in quite a while and I just want to get the information online. On the off-chance that this is a meaningful thing to think of, I want some kind of record that I figured at least the basics out.

    As has probably been mentioned in previous posts, the inspiration for this line of inquiry came as early as late 2021, soon after I first learned of the crank of an integer partition as a concept. What was the issue? The properties of the crank are very nice, but the definition is incredibly ugly. I suspect that this has been “allowed” because most combinatorics happens in the realm of generating functions, and the generating functions surrounding the crank are perfectly fine-looking. So what did I want? I wanted a way to consider the crank, visually or just in relation to individual partitions, which made it actually seem like it might have meaning. In some sense, I wanted the explanation for why this ugly formula is special to be better than “if you work out the generating functions, they’re pretty nice.”

    For context, the usual definition of the crank is as follows. For a partition P, let w(P) be the number of ones in the partition. If w(P) = 0, the crank is equal to the largest part of the partition. Otherwise, the crank is equal to the number of parts greater than w(P), minus w(P).

    Very messy. I mean, it is impressive that it was even found. That said, I am not surprised that it was found primarily by working backwards from generating functions (at least, I suspect this, heaving read the paper that introduced the formula). It mixes the sizes of different parts with the numbers of other parts in multiple ways. It’s just messy. So here we start my new definitions. I apologize for the lack of LaTeX.

    We define partitions of n to be equivalence classes of all permutations of n characterized by cyclic decomposition.

    Definition 1: For a permutation R of n, a nonempty set of points X \subseteq {1, 2, …, n} is called cyclic if R(X) = X and \emptyset \subsetneq X’ \subsetneq X implies R(X’) /neq X’.

    Definition 2: For a permutation R of n, a nonempty set of points X \subseteq {1, 2, …, n} is called fixed if R(X) = X and \emptyset \subsetneq X’ \subsetneq X implies R(X’) = X’.

    In plain language, a cyclic set of points make up a single cycle, whereas a fixed set is made up exclusively of fixed points. These have been phrased in such a way to show that they are, in some sense, complementary concepts.

    Definition 3: A partition P on n is canonical if, for any nonempty C \subseteq {1, 2, …, n}, (C is cyclic in some R in P) implies (C is fixed in no R in P).

    Definition 3: A partition P on n is anti-canonical if, for any nonempty C \subseteq {1, 2, …, n}, (C is cyclic in some R in P) implies (C is fixed in some R in P).

    These definitions are phrased to show their complementarity. In fact, one can show that for n >= 1, there are exactly as many canonical partitions of n as there are anti-canonical partitions of n.

    However, if one wants a geometric understanding of these partitions, they can be phrased as: (a) a partition is canonical if it has no ones, and (b) a partition is anticanonical if no part is greater than the number of ones.

  • Introducing Canonical and Anti-Canonical Partitions

    This is the first post in a series about integer partitions (in which older posts will be continuously updated to mesh better with newer posts). My previous post on the topic had some issues, both mathematically and conceptually, and after realizing that I decided to start from scratch. The goal: figure out an alternate definition of the Crank invariant of a partition that feels motivated, unlike the original.

    I think I’ve accomplished that, though I feel that my terminology could be better, since I accidentally wandered across a few fields of math I’ve never touched before, making up my own terminology along the way. Wherever possible, I will use what I believe to be the correct terminology.

    A final note before actual content: don’t spoil me on where the math might go. I’m slowly trying to work out any further implications, and I’d much rather feel any accomplishment that will come with doing so by myself. Okay… actual content time.

    We will be alternately be using two different sets of terminology when it comes to partitions. The first is considering partitions as finite non-increasing sequences of natural numbers, with each element in the sequence being called a part. The second is considering partitions as equivalence classes of permutations within symmetric groups, where the equivalence relation maintains cyclic decomposition. I will give a gentler introduction to each of these in a future post. For now, I just want to get to the first part of the point. I apologize in advance for the lack of LaTeX.

    Definition 1: Given a permutation rho in S_n, a non-strict subset C of {1, 2, …, n} is called cyclic if: (1) rho(C) = C; and (2) for any non-empty, strict subset C’ of C, rho(C’) != C’.

    Definition 2: Given a permutation rho in S_n, a non-strict subset C of {1, 2, …, n} is called fixed if: (1) rho(C) = C; and (2) for any non-empty, strict subset C’ of C, rho(C’) = C’.

    Note first the fundamental relatedness of these two definitions. Note second some less opaque interpretations of these definitions. A cyclic subset of points in a permutation is simply the set of points that make up a single cycle in the cyclic decomposition. A fixed subset of points in a permutation is composed entirely of fixed points, or points that are, in themselves, an entire cycle.

    Now for the important definitions. Again, don’t steal these, I’m still working on them.

    Definition 4: A partition mu (on n points) is canonical if, for any subset C of {1, 2, …, n} such that C is cyclic for some permutation rho in mu, there exists no rho’ in mu for which C is fixed. That is, (C cyclic for some permutation in mu) => (C fixed for no permutation in mu).

    Definition 5: A partition mu (on n points) is anti-canonical if, for any subset C of {1, 2, …, n} such that C is cyclic for some permutation rho in mu, there exists some rho’ in mu for which C is fixed. That is, (C cyclic for some permutation in mu) => (C fixed for some permutation in mu).

    Again note first the symmetry between these two definitions. Again note second some simplified versions. A partition is canonical if every part is greater than the number of ones. A partition is anti-canonical if no part is greater than the number of ones. In fact, the canonical definition in particular we can make even simpler: a partition is canonical if it has no ones. I say this form last because it hides the connection to anti-canonical partitions.

    That’s where I’ll end this post, though I do have more (and it involves actual results). In the meantime, I again apologize for both the lack of LaTeX as well as the lack of diagrams. I am quickly reaching the point where I may just pay up to allow them. Thanks for reading- and don’t spoil me!

  • What is a Good Friend?

    It’s not that they care; everyone should care about everyone. Instead, it’s because they know more about a person that they already care about.
    You can look toward a stranger, care about their life, sacrifice your life for it, even. That’s because you know that when a person is falling, they’ll only ever be fulfilled if you catch them. That’s true of anyone.

    But if you know more than that, a lot more, things get messy, since you get a much more complex idea of what their fulfilment could look like. If you see your friend on a path to fulfillment, their fulfillment, you support them. If they’re on a path that you don’t think leads to their fulfillment, it’s more complicated, but you still support them.

    You support them, because if they continue, they’ll need it, and if they change paths, they’ll need it. But that support, if you don’t think they’re on an in-the-end-fulfilling path, won’t be a pair of hands pushing them forward.
    It’ll be a pair of hands holding them up, with a slight backward pull to keep their progress on that path from getting away from them. But your hands are mostly just holding them up.

    Do you think you know their fulfillment better than them? The direction of your hands’ force depends on that. Do you push forward strongly? Do you hold up and slightly push? Do you hold up and slightly pull? Do you yank back from an edge and shake their shoulders, snap them out of it?

    This all relies on your being self-aware, knowing what you truly know about your friend’s path to fulfillment. The better the friend, the closer the friend, the more you know, the more you need to be self-aware. Yet also the more you can push before your hands will be shaken off, and perhaps rightly so. No matter how close you are, sometimes, not always, but sometimes, a person knows their own fulfillment like no other.

    A good friend always supports but, knowing what they truly know, rarely shoves. A good friend is a constant, gentle force. Perhaps it’s not in the direction you yourself want or expect, but it’s gentle, and it’s for you.

  • Natural Law as a Conditional

    Note: This is Part 5 of the “Justice as Conditionals” series.

    This idea is the last of those from them period about a year ago when I came up with the core idea of the theory, meaning that all posts after this one will be recent ideas, at most a month old and quite possibly come up with since the last update to the series. This also means that updates will be less reliable and potentially less frequent. We’ll have to see. More important to this post is that this part of the theory is not foundation for further components. What I mean is that while this is something I believe, it is not necessary for the functioning of the rest of the theory. I stress this mostly because this is the only religious segment of the theory and I don’t want people thinking that the theory itself is a religious one.

    St. Thomas Aquinas is generally considered the most important theologian of the Catholic Church since the end of the second millennium. This is largely because, though he was certainly not the first to consider reconciling Aristotle’s philosophy with that of Christianity, he was by far the most successful figure in doing so. His Summa Theologica is a masterwork of philosophy, bringing together with impressive unity the mainstream theology of Christianity and “The Philosopher” who the philosophers of the Middle Ages respected so much. Though many of the exact arguments given by Aquinas are no longer those held as true by the Church, it is safe so say that the bedrock supporting modern theology is still that laid by St. Aquinas. (As a side note, he has the title “Doctor of the Church,” which is only given to critical theologians who were also saints. It’s a pretty slick title.)

    One of the concepts developed heavily by Aquinas is that of the different types of Law. He discusses four types: Eternal Law, Natural Law, Human Law, and Divine Law. The gist is that Eternal Law is the overarching plan of existence, Natural Law is the inclinations and moralities built into human nature, Human Law is the set of laws put in place by humans to govern humans, and Divine Law is the scripture. All of this serve different roles and have significant interplay, but diving into that is a job for some other blog post. Let’s get to how this fits into the theory of Justice as Conditionals.

    Remember that a Conditional is a function from behaviors of the recipient to behaviors of the originator. In this digression, we might instead generalize and say that a Conditional is any intentional transmission of information which results in preferences/inclinations in the recipient. We consider this alternate definition because Natural Law is built into human nature, so considerations of phrasing and imprecision are not very meaningful like they are in the analysis of human-generated Conditionals. This generalization exceeds the name “Conditional,” but we’ll use the term anyway for consistency.

    In short, I propose that Natural Law itself is a Conditional. This set of unconscious inclinations and a conscience in a person may be considered transmitted information regarding the nature of God and/or morality as the two are related to the actions of the person. Precise information about how the two may behave retributively is not necessarily included, but that’s not an issue. The point of such indications is to transfer the related preferences onto the recipient’s perceptions of their own actions. As Natural Law imprints these inclinations directly, such indirect retributive threat is unneeded. A Conditional is expressing your morality and preferences onto another person in such a way that it impacts their own morality and preferences.

    If Natural Law is taken to be a special case of Conditional, we may learn something from the definition of Justice as being the study of Conditionals: the dictates of conscience, away from the more particular moralities we may develop alongside them, are the most fundamental decrees of Justice. In other words, if this interpretation is to be believed, the preferences imposed by conscience and given to some extent to almost all people are an expression of the morality of God. That is, the dictates of your conscience are the Conditionals that reflect True Morality.

    Divine Law is probably also interpretable in part as a collection of Conditionals, but that’s a discussion for another post.

    ,

  • Integer Partition Conventions

    The purpose of this post is mostly to point out that this is not exclusively a philosophy blog. That said, the site doesn’t currently have a LaTeX plugin, so the math is gonna be a bit janky to communicate. If you feel you need it, an introduction to the topic can be found at https://en.wikipedia.org/wiki/Partition_(number_theory).

    The definition of a partition used in George Andrews’ “The Theory of Partitions” is “a finite nonincreasing sequence of positive integers.” This definition is perfectly fine and I intend to build on it here, not truly replace it. The motivation for my altered definition is the definition of the Crank invariant of a partition. Its standard definition is as follows: if the partition contains no ones, the Crank is equal to the largest element of the partition. Otherwise, it equals the number of elements larger than the number of ones in the partition.

    I very much wanted to get rid of the conditional, which seemed to me to indicate something imperfect about the definition. Further, the Crank behaves similarly in many respects to another, simpler invariant called the Rank, which is defined as the largest element in the partition minus the number of parts in the partition. The considerable difference in the form of their definitions also bothered me. Note that my alternate Crank definition isn’t fundamentally different from the original; it provides a value equal to the standard Crank of the partition conjugate to the one it considers. As conjugation is well-defined and bijective on all partitions of a given positive integer, this switching maintains all of the partition-organizing properties of the original definition while being more aesthetically pleasing. Because of my lack of LaTeX, we will call N the standard natural numbers and N_0 the natural numbers including zero.

    We redefine a partition to be any function p: N_0 -> N_0 such that it has the two following properties for 0 < j < k:
    (1) p(j) >= p(k), and
    (2) [p(j) = 0] if and only if [j > p(0)].

    Note that if I had LaTeX, I would phrase it as “p(j) is not greater than 0 exactly when j is greater than p(0)” or something like that. The differences between this and the standard definition are twofold. First, the sequence is no longer finite, but instead features infinite trailing zeroes. Second, the sequence has a zeroth term equal to the number of parts of the partition.

    A nice side effect of this definition is that the Zero Partition is well-defined in this scheme, letting the typical generating sequences for partition-counting make a little more sense. I still need to work on the phrasing a bit more, to make this even clearer.

    The primary benefits, however, are how it allows us to redefine the Rank and Crank invariants into clear and clearly parallel expressions:
    We redefine the the Rank of a partition p to be [p(0) – p(1)];
    We redefine the Crank of a partition p to be [p(p(1) – p(2))].

    I’m not particularly well-versed in the study of integer partitions beyond a basic level, so I don’t know if this convention is useful in other places, but I intend to check, so nobody spoil it for me. Thanks!

  • Small Contracts

    Note: This is Part 4 of the “Justice as Conditionals” series.

    The conditionals, as exposited so far, are one-sided affairs. They are a single person imposing their preferences, what I’m calling their moralities, upon another who may not share them. Such a conception, however, is not the entirety of how Conditionals may appear. In particular, Conditionals can appear in sets.

                   Again, for simplicity, we will consider the two-person case with A and B. When A produces a Conditional to influence the behavior of B, A is taking a preference that B has on A’s behavior and is artificially adhering those preferences onto B’s perceptions of their own behavior. So far, we have tacitly assumed that the Conditionals are absolutely true and that B would not refrain to accept the adjustments to their preferences.

    Consider, however, what would happen if A provided B with too strict a Conditional: “if you don’t move to Ohio, I’m not going to pay for dinner.” A is taking the fact that B would like A to pay for dinner and applying it to B’s preferences regarding moving to Ohio. However, even if this is a true statement, it is not enough to meaningfully sway B’s behavior; B very much does not want to move to Ohio. What this Conditional does manage to usefully convey is that A has some meaningful preference regarding B’s moving to Ohio. This solid information about A’s preferences regarding B’s behavior allows B to construct their own, almost certainly more properly-balanced, Conditional: “if you give me $1M, I will move to Ohio.”

    This back and forth can continue on, each time providing more and more information about each’s preferences for the other, or at least whatever each can convince the other of. If this process converges to a situation where A and B each have their own behavioral restriction and each Conditional is “I will disobey my restriction if and only if you disobey yours,” what I will call a Small (or unenforced) Contract is formed. The only incentives which hold a Small Contract in place are the parties’ preferences regarding each others’ behavior and the sub-goal of maintaining the reputation of their own Conditionals. In this case, the deal might converge to “B will move to Ohio and A will buy them a nice house there.”

                   To clarify things slightly, a Small Contract is not something distinct from Conditionals. Rather, it is a set of Conditionals, one from every agent in question to every other, all of which agree on the behavioral restrictions placed upon each party as well as the behaviors which would result from any given violation.

    Lastly, we note that such agreements make the concept of Conditionals’ Acceptable Ranges (Domains and Codomains) all the more important. Small Contracts would be virtually impossible otherwise, as the back-and-forth Conditionals would not then have the capacity to vary enough to allow the convergence to a Contract. Each agent may care more to avoid restricting their own behavior than to actually restrict the other’s, and no convergence would happen. In these cases, the preferences of the parties regarding their own behavior would not significantly change.

  • Acceptable Ranges

    Note: This is Part 3 of the “Justice as Conditionals” series. This post’s concept is actually a recent addition to the theory but is crucial enough to the theory’s rigor that I’m inserting it before we move on to more of the older ideas.     

    As it stands, plain Conditionals cannot properly support things like governments without assuming some aspects of morality. The issue arises because not all moralities are compatible. In fact, when defined solely as strict preference sets, almost no moralities are compatible. We need to allow them to be compatible if the theory is to continue. Fortunately, simply looking at the world as it exists provides us a solution: that people demand not specific actions in Conditionals, but a range of actions, whether that range be thin or wide.

                   To show what I mean, here’s an example. The Conditional “If you rat me out, I’ll beat the crap out of you” seems fairly specific. We’re obviously missing some context, like who the speaker is being ratted out to, but we can make sense of what is being said overall. The “acceptable range” of the Conditional consists, to the receiver, of the set of actions which do not involve ratting out the speaker. The Conditional has nothing to say about where the receiver can go to dinner, nor does it care the tone of voice in which the receiver is speaking when ratting them out. Any Conditional is going to have an acceptable range, if only because language itself doesn’t allow for the the stated restrictions to be absolutely precise; one can’t meaningfully include the provision that “that atom of iron in that one red blood cell can’t twitch at 12:37 PM tomorrow!”

                   There is one more way to phrase this concept that will be even clearer to anyone of the right background. Basically, a Conditional said by A to B is a function from the set of B’s future behaviors to the set of A’s future behaviors. The set of future behaviors n-dimensional for a very large n and the function is discontinuous only on a finite number of (n-1)-dimensional surfaces. Just to be clear, this is still an imperfect analogy. I doubt that language or preferences can be properly modelled by Euclidean space. It’s a fairly good way to phrase it, though. Because of this, I except that rather than using “acceptable ranges” as the term for them, I’ll generally call them spaces within the Conditional’s domain and codomain (the receiver’s future behaviors and the speaker’s future behaviors, respectively).

  • Conditionals

    Note: This is Part 2 of the “Justice as Conditionals” series.

    For clarity, we will begin by considering a two-person case, the people named A and B. Suppose A has a preference regarding the behavior of person B. Such a preference is rational only if the behavior of person B impacts person A in a negative way or, more generally, if person B’s future behavior has the potential to violate person A’s morality on a level which the morality ascribes an allowance or obligation for person A’s intervention.

    Person A may either physically control person B’s behavior or socially control it. The first option is a topic for another day, clearly offering complications for a verbally-oriented theory. The form of person A’s social control is a very broad subset of communication I will call a Conditional. This is not A’s only option for social control, but is the one which doesn’t rely on some kind of prior relationship between the two, aside from the capacity to communicate.

    A Conditional consists of a description of person A’s future behavior conditional upon person B’s future behavior. Tacitly implied by the Conditional is the concept that the contents of the enumeration are backed by A’s goals or sub-goals (coming from the field of AI safety, sub-goals are the concept that many large-scale goals an agent may hold share sub-goals necessary to accomplish any of them; an example of a goal is becoming an astronaut and an example of a sub-goal is surviving until then). That is, making a Conditional implies that the giver of it will actually follow the actions they are prescribing themselves in each case; preserving one’s reputation in this way is a sub-goal of incentivizing their own morality using Conditionals.

    The preferences that person A is inducing in person B are not actually new to person B’s reasoning. Rather, person A is simply providing them with carefully-constructed information that allows them to control the future behavior of person A. By presuming preferences that person B has for the behavior of person A, such as preventing person A from punching them in the face, person A is able to associate those preferences with courses of action person B themselves may take.

    This series of posts argues that Justice may be considered the study of conditionals, the forms they may take, and the way in which they fit into the rest of human life. (First Drafted 10/2021)

  • An Introduction to the “Justice as Conditionals” Series

    Note: This is Part 1 of the “Justice as Conditionals” series.

    This post essentially replaces the previous “Justice as Open Conditional Behaviors” as the preferred beginning or the first elaboration of the political philosophy I’m developing. I initially thought of the concept around October of 2021 and the idea progressed steadily for a couple weeks. It then fell into stagnation, with a couple of exceptions, between then and now. Because of this history, the first few posts regarding the theory are likely to be revisions of what I wrote down back then. I’ll include an indication with the first posts truly new concepts begin appearing.

    Ahead of all that, however, I wish to state what I believe to be the greatest features of the theory. First, it overlaps with social contract theory in a number of places. Among these is that it does not make significant assumptions about the morality of the people it considers. Second, it seems obvious. In particular, one might hear it and say “Yeah, that is my main motivation.” This is unlike much of social contract theory, which relies on a historical account of the government’s formation. While I expect this theory to be capable of making such an account, it also accounts for people born into an existing system.

    Lastly, I wish to make clear my influences, primarily Aquinas and Nozick. Aquinas contributes most through his conceptions of the four types of Law. Nozick contributes to the more flaw-exposing, casual nature of these posts as well as what I expect to be a fairly libertarian bent once government begins being discussed.