on the practice [P] of using abstract variables [V]

an intermission

With apologies to [Hg], [P₀], [P₁], [P₂], [Ts], and my readers in general.

I have been distracted from quotient agents ii. Probably(?) we’ll be back to it tomorrow, sorry to anyone who was excited to read it tonight.

Anyways: a Human¹¹ presumably, if anyone is, maybe we’re all demons in the Abyss guy, gal, or gentleperson [Hg] with whom I am acquainted recently wrote a blog post [B] containing, if I am counting correctly, twelve different variables ([V₀] through [V₁₁]). Three of these (WLOG [V₀], [V₁], and [V₂]) referred to people [P₀], [P₁], and [P₂]; four (still WLOG, [V₃], [V₄], [V₁₀], and [V₁₁]) referred to various putative latent causes [C₃], [C₄], [C₁₀], and [C₁₁]; one (you get the idea, [V₅]) referred to a trait [T₅]; two ([V₆] and [V₇]) referred to problems [P₆] and [P₇]; one ([V₈]) referred, if I am understanding it correctly, to a law [L₈]; and one ([V₉]) referred to the act [A₉] of clearing variables.

Far be it from me to take issue with the act of introducing unnecessarily many variables into a blog post as a bit—that would be kind of hypocritical of me, it’s totally the sort of bit I would do. Nonetheless, I think I have thoughts on practice [P].

Why?

Oh no not another letter I have to keep track of—oh, you meant the word, okay that’s a good question.

As a mathematician [M], I think I do have some sort of obligation to defend [P], and in particular the use of [V]s that are letters to represent various objects [O].

Technically, my own previous post [Q] contains:

• X, a set

• a, an element

• b, an element

• A, an element

• B, an element

• 𝓟, the power set… functor… Set → Set

• Z, the integers

• R, a relation

• B_*, the function returning the *th Bell number

• π, half the circumference of a unit circle

• e, the base of the unique function of the form b^x which is its own derivative and isn’t 0^x

• 𝕴, the imaginary part function

• +²² The symbol + might(?) ultimately be derived from the Latin word “et” much like & is. Also, it looks kinda like a t. If I can count ∫ as a kind of s, I think I can get away with counting + as a kind of t., the product of an abelian group (especially C, Z, or Z/nZ)

• ∫, the… integral… operator³?

• θ, a real variable probably meant to represent an angle

• d, the… uh… differential… operator?

• a, a real number

• b, a real number

• i, a square root of -1

• P, a partition (of X)

• n, an integer

• a, an integer

• b, an integer

• A, a subset of Z (or more generally of any desired magma)

• B, a subset of Z (ditto)

• a, an element of A

• b, an element of B

• a, an integer

• n, an integer

• b, an integer

• n, an integer

• A, an [O] in a concrete category

• B, an [O] in a concrete category

• ι, a structure-preserving injective function from A to B

• q, a structure-preserving surjective function from B to A

• a, an element of A

• b, an element of B

• b₁, an element of B

• b₂, an element of B

• 𝕽, the real part function

• Q, the rationals

• R, the reals

• H, the quaternions⁴⁴ But not C. For some reason.

• O, the octonions

• X, a subset of B

• x, an element of X

Which, when you put it like that, maybe makes the [P] in [Q] seem rather less excusable than all of [Hg]’s [V_i]s in [B]. But let’s consider some of these examples in more detail:

• π, e, i, 𝕽, 𝕴, d, ∫, 𝓟, Z, Q, R, H, O, and I guess also B_* are all standard [V]s for specific [O]s.

  • The reader of [Q] is expected to be already familiar with the referents of at least π, e, i, d, ∫, Z, Q, and R—and mostly with the symbols for them, too.

  • Also, 57%⁵⁵ Do you remember why this is a round number? of these symbols are only included for a single one-off joke.

  • Another 21% (R, H, and O) are only used for one single long list of examples.

  • 𝕽 and 𝕴 are highly mnemonic choices of variable, though this might be unclear to anyone who cannot see at a glance that 𝕴 is obviously just a form of the letter I. Also, 𝕽 is a projection onto R.

  • 𝓟 is similarly mnemonic and also only occurs in a single parenthetical clarifying a technical point.

  • Who even thinks of ∫ as a letter anyways.

• Similarly, while +, ι, and q might not be specific [O]s exactly, these are the standard [V]s for abelian group operations, inclusion maps, and quotient maps. Like, come on, are you really going to [O] to +? Also, q is central to the point of [Q].

• θ is a variable of integration, and furthermore is a common choice of variable for representing an angle (or an input to sin). It also is only relevant to the aforementioned one-off joke that uses π, e, i, 𝕽, 𝕴, d, ∫, and B_*.

• a, b, A, B, a, b, n, a, b, A, B, a, b, a, n, b, n, A, B, a, b, b₁, b₂, X, and x are subsets or elements of specific sets. There is rarely any particular further detail that must be tracked about these [O]s, other than what [O] exactly they are a subset or element of. Other than the elements A and B from the example set X = {a, b, A, B}, we consistently have lowercase letters represent elements and uppercase letters represent subsets.

  • The set an element is from can often be immediately inferred from the symbol:

    • If there’s both an a and an A in consideration, you can bet that a is an element of A.

    • If you see “a+bi” it’s easy to infer that this is a complex number with real part a and imaginary part b.

    • The symbol n representing an integer is unsurprising, though it is also often used for a natural.

    • If you’re working with Z and see something like “a≡₂b” introduced as part of the process of defining ≡₂, then it is clear that two arbitrary integers were needed to represent potentially related elements of Z.

    • The variables n, a, b, n, and x occur in set-builder notation. This means that their scope is limited to a singular formula.

  • Furthermore, even those among these variables which don’t occur inside set-builder notation have a scope which is very limited, hopefully predictably so. Several of them occur in only one clause of one sentence, and many of the others are limited to a single paragraph.

• While X, R, and P do not exactly fit in with the above grouping, they are morally similar. While X is explicitly defined as {a, A, b, B}, it is often actually used to mean something more like “an arbitrary set, such as (for example) {a, A, b, B}”. This is perhaps a sloppy use of language, but the important point is that the details of X (or P) beyond being an arbitrary set or being an arbitrary partition of P can be totally ignored except insofar as having a concrete example is helpful. R, on the other hand, is an arbitrary relation rather than a set or element—though I suppose it could be viewed as an arbitrary element of 2^{X^2}.

While it is kind of funny to list every single variable in a post like [Q] out explicitly—and while I do think that there can be a learning curve to becoming comfortable with the level of abstract symbolic manipulation that can occur in university-level mathematics, to the extent that it makes total sense for even a considerably intelligent person to balk at them a little—I do not think it is actually exceptionally difficult to track these things, once you have practice, at least if you’re the sort of person to learn to program. [M]s tend to choose symbols for sensible enough reasons, and generally one is used to variables representing numbers if they have learned any mathematics beyond high school algebra, which the target audience of my blog has⁶⁶ If I am wrong about this—look, a not inconsiderable portion of the math I write is basically me trolling by discussing things in intentionally obtuse manners, and I’m probably not going to stop doing that, because it’s funny, but I would appreciate feedback if people who want to understand my posts are confused. I think explaining funny obtuse mathematics often does not ruin it, and also I would ideally prefer it to not inhibit understanding of those things I am saying which are not themselves funny obtuse mathematics..

In contrast, in [Hg]’s [B]:

• [V₀] is the title of a section that uses topic [T] as a primary example, as well as a variable representing person [P₀], a central figure in [B].

  • [V₀] appears fifteen times, all inside [V₀]. It seems to me like a fine choice of variable name; there is a particular word which [V₀] can be interpreted as standing in for in order to produce a legible if bizarrely worded version of [B]—what I mean is that it is a workable mnemonic. It also probably helps that [V₀] is the first variable introduced and that [P₀] is so central to [B].

  • Just the first paragraph of [V₀] introduces four different variables [V₃], [V₄], [V₁₀], and [V₅], all representing various putative traits of [P₀].

    • Only [T₅] is actually described as being a “trait”—the rest are not described any more concretely than as arbitrary putative causes of [T₅] or of [P₀]'s beliefs and statements regarding [P₀]’s possession of [T₅].

    • The symbol [V₁₀] is chosen in a way that suggests a particular other variable, [V₁₁]. [V₁₁] indeed appears in a later paragraph, but the relationship between [C₁₀] and [C₁₁] within [B] does not end up being all that different in character from the relationship between [C₃] and [C₁₀].

    • The symbol [V₄] is, arguably, loosely related to the symbols [V₃] and [V₁₀].

    • Three distinct beliefs of [P₀] regarding [C₃], [C₄], and [C₁₀] are described, all of which are related to each other in specific manners.

    • Essentially, at least three different propositions about [P₀] end up needing to be tracked, all of which are phrased in terms of abstract variables [V₃], [V₄], [V₁₀], and [V₅].

  • The second paragraph introduces two new characters, [P₁] and [P₂].

    • [V₁] has a mnemonic provided, but it’s considerably worse than [V₀]'s. In my opinion.

    • [V₂] has no mnemonic provided.

    • [P₁] and [P₂] each express complicated opinions about the relationships between [P₀], [T₅], [C₃], [C₄], and [C₁₀].

  • The remaining paragraphs all seem largely inoffensive, especially paragraph five (the final one), which very clearly and helpfully states the point of [V₀]. Possibly this is partially because [V₁₁] is the only variable newly introduced during the final three paragraphs, and it occurs only once.

  • At least two of the symbols, though I will not say which in order to not risk exposing [Hg] to alloyed hearts, have a secret bonus interpretation [SBI] if you know [T] well enough to deduce the identity of the [O] they represent⁷⁷ [V₈] is an arguable third [SBI], in that the corresponding mnemonic term does not explicitly occur in [B], but my interpretation of [V₈] relies only on information conveyed within [B] and not on side-channel attacking [Hg]’s writing process.. This is exciting, I love when posts do this, people should do it more. However, this suggests the possibility of some additional spurious [SBI]s, under which [V₃] would represent [C₁₀] and [V₄] would represent [P₂]. While I do not expect these pseudo-[SBI]s to actually confuse anyone at all, I think it is very disappointing that they aren’t valid.

• [V₉] is also the title of a section. It only uses two variables, [V₆] and [V₇].

  • [V₆] and [V₇] do not have very good mnemonics, though [V₆] sort of has one.

  • The use of [V₆] and [V₇] is perhaps unnecessary, but it does not cause many problems because they are each only used in two sentences, one of which is the one that defines them—so there is very little information to track.

  • Unlike every single variable used in [V₀], [V₆] and [V₇] represent [P₆] and [P₇], which are explicitly-specified concrete [O]s. This is much less confusing than simultaneously tracking three different unspecified people and five different unspecified putative traits of [P₀].

• The final section, [V₈], contains no variables.

I found [B] mildly confusing and intimidating, and I had personally witnessed many of the events described, though I haven’t engaged with any of [P₀]’s claims regarding [T₅] to any substantial level of depth (which might have contributed to my difficulties tracking [C₃], [C₄], [C₁₀], and [C₁₁]). If I imagine a version of myself [A’] who has no idea what any of the events described in [V₀] could possibly have been, I think [A’] gets almost nothing out of most paragraphs in [V₀].

Maybe it doesn’t really matter if [A’] is confused by [P]. [A’] probably isn’t confused by

If someone uses a process to produce one answer, and then uses the same process to get another, mutually exclusive answer, you know that the process sometimes produces wrong answers. You should doubt the second answer. If the same process produces a third answer, you should doubt the process even more.

which is arguably the only important thing to get out of the [V]-heavy part of [B]. This is too simple a point for the details of the example to especially matter. But then, why [P]?

The [O] of [P]

• Obfuscating [T] especially hard

• Intimidating the reader

  • For fun

  • To seem impressive

  • To make it less likely for anyone of alloyed heart to determine [T]

• To include enough structural details that the pure of heart⁸⁸ Or, for example, those and only those who already know [T], and therefore would not benefit from being spared from knowledge about [T] but might benefit from the use of it as an example. can deduce [T] while being confusing enough / excluding enough details that only the pure of heart can

• [P] is really funny

• No really, [P] is funny

• If you think it isn’t funny, have you ever tried writing a blog post that does [P]? No? Try it

• Abstracting away details

  • To focus attention only on the details of relevance to the argument of [B]

  • To allow readers to read the post while filling in their own example [X] isomorphic to [T]

• To turn your post about [P] into an example of [P] that is conveniently available to all readers of your post explaining [P]

I believe that almost all of these are pretty good reasons, at least under the presumption that they are being intentionally pursued—obviously, if you don’t want to intimidate readers, then “to intimidate readers” is, at best, a not very good reason for [P].

The “focusing attention” one could definitely be a good reason for [P], but if it counterfactually were the primary reason for [Hg] to [P] in [B] then I would think that the execution of [P] in [B] was poor. Cutting down the number of [V]s as much as possible is a good idea if one aims to narrow the focus of [B], and multiple [V]s could’ve been excluded at little cost, such as [V₁] and [V₁₁] (which were each used only once). I also might have placed [V₃] in series with [V₁₀] and [V₁₁] (or just with [V₁₀], if [V₁₁] were cut.) However, at least the first of those two interventions would be actively counterproductive if even humor was among [Hg]’s goals—the cost in clarity of introducing [V₁] and [V₁₁] is marginal, especially if one is already committed to the inclusion of [V₃], [V₄], [V₁₀], and [V₅], so they are worth it for the humor value (not to mention the intimidation and obfuscation value).

Personally, though, I am most interested by the idea of a post where readers can fill in an [X] isomorphic to [T]. I aspire to someday write a post where multiple people think they understand exactly what I’m talking about, but have actually ended up with entirely incompatible interpretations of my statements. [P] (especially the sort of [P] featuring [SBI]s) is probably useful, maybe even indispensable, for this goal.