gay-mathematician-portmanteau

[gay-mathematician-portmanteau]

it's kinda fucked up how omnipresent e is in discussions of exponential functions. today, i feel weird about the fact that f(e) is the absolute maximum of the function

$$f: \mathbb{R}^{+} \rightarrow \mathbb{R}^{+} | f(x) = x^{\frac{1}{x}}$$

i can't and won't explain further what seems so profound about this.

[gay-mathematician-portmanteau]

single thread math episode 23: systems of polynomial equations are just systems of linear equations with superscripts instead of subscripts. i don't know what algebraic geometers are doing studying them; that's the kind of thing they teach you in an undergraduate linear algebra course. you all should be doing real math instead (i.e. universal algebra).

[gay-mathematician-portmanteau]

suppose my lips are analytic. which parts of your body do you want to compose them with to determine analyticity?

[gay-mathematician-portmanteau]

single thread math episode 22: i was thinking about writing a soap opera about graph theory when i realized,,, soap operas *are* graphs

[gay-mathematician-portmanteau]

they told me (i told myself) that skeleton category jokes were only for halloween (what ind of person would say that? this is not a believable meta-narrative, g-m-p)!!! nonetheless, we have hit peak necrocategorist in the blessed month of december (i want to post this despite having been too depressed to maintain this blog for a while).

single thread all hallow’s math: on this, the most haunted of all nights, my research into skeleton categories can finally be completed! they called me evil and scorned my work. now,,, now, they’ll see that necrocategories are the way of the future!

[gay-mathematician-portmanteau]

single thread math episode 21: is determining whether a person is well-defined a well-defined process? if you said yes, then you must accept that we can divide the set of all people into the subsets "people who are well-defined" and "people who are not well-defined." bertrand russell would necessarily be sorted into one of these sets. since bertrand russell discovered a paradox, we have a contradiction. thus, there cannot exist a set of all people.

[gay-mathematician-portmanteau]

i can't tell if i genuinely believe there's not a fire in my apartment or if i just can't summon the willpower to care if there is one. anyway, the fire alarm is going off and i haven't moved from my bed.

[gay-mathematician-portmanteau]

someone on a dating app just sent me a .gif and i responded "eyeballs don't have .gif compatibility" so, yeah, i'm never hearing from that person again. honestly, that's the dorkiest thing i've ever written and i write this blog every day.

[gay-mathematician-portmanteau]

alone alone alone alone alone alone alone alone alone alone alone alone alone alone alone alone alone alone alone alone alone alone alone alone alone alone alone alone alone alone alone alone alone alone alone alone alone alone alone alone alone alone alone alone alone alone alone alone alone alone alone alone alone alone alone alone alone alone alone alone alone alone alone alone alone alone alone alone alone alone alone alone alone alone alone alone alone alone alone alone alone alone alone alone alone alone alone alone alone alone alone alone alone alone alone alone alone but like math or something

[gay-mathematician-portmanteau]

Single thread compendium episode 1: my single thread math series isn't gaining the traction i’d hoped for. rather than believing the series just isn’t good or that i’ve possibly failed in some way, we’re going to go ahead and explain the joke (because obviously, that will make it funnier).

the series might be beyond my skills as a writer, so we’ll see if this actually helps.

the series is obviously centered around math. it’s best defined as a loose collection of absurd math jokes, which may or may not be interconnected in some way. the hope is that people who aren’t necessarily interested in mathematics can still understand the jokes and find them funny. at the same time, i’m trying to include enough mathematical depth that those who are into math will get a little extra punchline to enjoy. i succeed more often at the latter than the former,  but i genuinely believe you don’t need math knowledge to enjoy the series. 

if this sounds interesting to you and you’d like to experience the series as intended, i’d encourage you to stop reading here. From this point forward, i’ll be discussing topics that are intentionally left vague in the series. if you do decide to keep reading, that’s totally okay—the speculation i’m hoping to encourage can be pretty energy-intensive, and it’s completely valid not to have the space to dedicate to such a ridiculous project. you can find the posts under the tag #single thread math on my blog.

the truest focus of the series is on unnamed characters. my hope is to encourage the reader to grow attached to the characters and assign to them personhood through their beliefs, actions, and interactions without my ever having to explicitly discuss the characters (i’ve clearly already failed at this goal).

in this post, I’m featuring the first two (and my favorite) characters. i usually refer to them as the algebraist and the analyst in my head.

for those less familiar with the politics of theoretical mathematics: there are algebraists and analysts who view their subjects as irreconcilable. while i don’t share this opinion, i thought it would be funny to highlight this “war” between the characters (whose personalities reflect stereotypes about real analysts and algebraists, along with the subjects themselves).

the algebraist is smug, arrogant, and clever. her problem is that she never stops to consider whether she might be wrong. she’s so fixated on meta-mathematics that she loses sight of the concrete. i felt this was an important trait to highlight right from the first story arc of the series (the posts i’ll feature here).

the analyst, by contrast, is neurotic, devoted, and meticulous. she has a near-fanatical obsession with propriety and rigor, often rejecting conclusions if she finds the proofs invalid for any reason. i also decided early on that she would hate cantor. For those unfamiliar, cantor is most famous for proving that there are more real numbers than natural numbers (yes, some infinities are larger than others—very original reference). his work is foundational to analysis, so the analyst’s hatred of him is a bit of an issue for her prospects.

i’m not going to explain the math the posts contain here, but i hope anyone who wants such explanations will be willing to reach out to me via dm or ask. i’m truly always willing to answer such questions!

that’s all i have right now! the following entries in the series make up the first story arc, introducing the dynamic between the algebraist and the analyst. if you don’t enjoy math, i hope you’re able to find humor in the characters themselves. if you do enjoy math, i hope you appreciate the balance I have attempted to strike between absurdity and depth—i worked very hard on it.

episode 2: i’m pretty sure that if i write the derivative of the maclaurin expansion of \e^x\ out far enough, i’ll find a term that doesn’t perfectly match the original. this is how a real analyst proves something. take notes, cantor.

episode 7: if you’re close enough to the equator, polynomials are actually just piecewise-defined lines.

episode 8: okay, the derivative thing didn't take me quite as long as i expected. admittedly, i was wrong. nonetheless, i'm absolutely certain that the integral of the maclaurin expansion of e^x will have mismatched terms and i'll finally be vindicated in my support for cantor's institutionalization. 

episode 9: there's been a lot of talk on here about these taylor polynomial things, so i went to the equator to prove that they're nonsense.

claim: the maclaurin expansion for \e^x\ is not equal to the function \e^x.\ proof: consider the maclaurin expansion given by $$1+x+(1/2)x^2+…+(1/n!)x^n$$. this is trivially equivalent to $$1 \land x \land ((1/2)+2x) \land … \land (1/n!!)+nx$$. this can be rewritten as the piecewise function $$\begin{cases} 1&\text{if} x \le 0\\lim_{n \rightarrow \infty} \frac{1}{n!}+nx&\text{if} 0 < x \end{cases}$$ examine also $$e^x = \prod_{1}^{x}(e) = \sum_{1}^{x}e = ex$$.

it is easy to see that these functions are not equal to one another.

i don't know why anyone studies anything other than algebra. you're welcome, analysts.

episode 12: i just learned about this nifty induction thing. if only i had known about it a couple of weeks ago, then i wouldn't have needed to spend eternity writing out every single term of the derivative and integral of the maclaurin expansion of e^x. now, though, no cyclic set is safe from my proofs.

claim: the maclaurin expansion of \e^x\ is its own derivative.

proof: consider the maclaurin expansion of the function \e^x\ given by $$\sum_{n = 1}^{\infty}\frac{1}{n!}x^{n}$$ the derivative of sums is the sum of derivatives, allowing us to focus on terms of the form $$\frac{1}{j!}x^{j} | j \in \mathbb{N}$$

consider now the base case n=1. $$\frac{1}{1!}x^{1} = x$$ examine the derivative of this function. $$\frac{d}{dx}\left[x\right] = 1 = \frac{1}{0!}x^{0}$$ since this is a term of our original maclaurin expansion, the base case holds. now, examine the case $$n = k | k \ge 1$$ this term is given by $$\frac{1}{k!}x^{k}$$ suppose it has a derivative in the original maclaurin expansion given by $$\frac{1}{(k-1)!}x^{k-1}$$ lastly, examine the case n=k+1. $$\frac{d}{dx}\left[\frac{1}{(k+1)!}x^{k+1}\right] = \frac{d}{dx}\left[\frac{1}{k+1}x\frac{1}{k!}x^{k}\right]$$ which allows us to apply our assumption with the product rule. $$\frac{1}{k+1}\frac{1}{k!}x^{k}+\frac{1}{k}x\frac{1}{(k-1)!}x^{k-1}$$ we can now simplify the products in each term $$\frac{1}{(k+1)k!}x^{k}+\frac{1}{(k+1)(k-1)!}x^{k}$$ adding the fractions normally gives us $$\frac{k+1}{(k+1)k!}x^{k} = \frac{1}{k!}x^{k}$$ by induction, since this step holds for n=k+1, this holds for all natural numbers n. as a result, the derivative of any term of the maclaurin expansion is a term of the maclaurin expansion, demonstrating that the two functions are equal.

[gay-mathematician-portmanteau]

what do you want from me?

a joke?

tell me honestly, when was the last time i said anything you found funny?

never? well, that's a little harsh.

...

ugh, fine, if you're not going to leave.

umm,,,

a countable set of mathematicians walks into a bar. the first element orders a beer. the second element orders half a beer. the third element orders a quarter of a beer. the fourth element orders an eighth of a beer. the fifth element orders a sixteenth of a beer.

exasperated, the bartender holds up their hand. "know your limits," they tell it and pour two beers.

are you happy now? actually, i don't care. good bye.

[gay-mathematician-portmanteau]

single thread math episode 20: so, we have the natural log of the natural log of the natural log of the... i'm sorry, i seem to have lost count. i'm sure it's fine, let's just move on with the proof. you see, this represents the current best result for the maximum gap between two prime numbers. it is vastly superior to the last result because it contains more nested natural logarithms.

[gay-mathematician-portmanteau]

being good at math stage 1: i like to count!

being good at math stage 2: add more often than i count.

being good at math stage 3: i'm starting to forget how to add because i do algebra so often.

being good at math stage 4: i can do trig subs but i can't quite remember what arithmetic is.

being good at math stage 5: it's just philosophy now.

being good at math stage 6: i like to count!

i think that i'm in some sort of stage 6 analog where i mostly do arithmetic but i have gone through all the other stages at some point. also stage 5 never really ends. what about you?

[gay-mathematician-portmanteau]

cw: reference to potentially triggering treatment of a child by a caregiver.

single thread math episode 19: i made a game to help reinforce the process of creating and analyzing closed n-ary operations on finite sets. here are the rules: my basement child will receive some large rocks onto which i have carved thorough definitions for various operations. they will then need to wander through the local woods, hunting for small rocks onto which i have carved elements of sets. they will need to construct enough sets from the scattered rocks that each n-ary operation they were provided at the beginning is part of a well-defined algebra. when they finish, i'll reward them by letting them come back indoors! the fun nature of the task should help reinforce the critical ideas we're developing. this represents a major breakthrough in educational psychology and early childhood development research.

[gay-mathematician-portmanteau]

i hate when people talk to me about how hard math is. like, yeah, it is pretty difficult, but it's also the easiest thing in my life right now. what does that say about my ability to manage myself? *shudders*

[gay-mathematician-portmanteau]

single thread math episode 18: it has come to my attention that people perceive me as behaving smugly or as though i'm superior to others (analysts). though i stand by my assertion that analysis is useless in the face of algebra, i decided to make you some alphabet soup as an apology. claim: the dot product of any two finite-dimensional vectors is distributive over vector addition. proof: consider the vectors given by $$\mathbf{a} = \begin{bmatrix} a_1\a_2\ \vdots \a_n \end{bmatrix}$$ $$\mathbf{b} = \begin{bmatrix} b_1\b_2\ \vdots \b_n \end{bmatrix}$$ $$\mathbf{c} = \begin{bmatrix} c_1\c_2\ \vdots \c_n \end{bmatrix}$$ notice first that each of these is n-dimensional, meaning that the relevant operations are defined. $$\mathbf{a} \cdot (\mathbf{b}+\mathbf{c}) = \begin{bmatrix} a_1\a_2\ \vdots \a_n \end{bmatrix} \cdot (\begin{bmatrix} b_1\b_2\ \vdots \b_n \end{bmatrix}+\begin{bmatrix} c_1\c_2\ \vdots \c_n \end{bmatrix})$$ which simplifies to $$\begin{bmatrix} a_1\a_2\ \vdots \a_n \end{bmatrix} \cdot \begin{bmatrix} b_1+c_1\b_2+c_2\ \vdots \b_n+c_n \end{bmatrix} = a_1(b_1+c_1)+a_2(b_2+c_2)+ \dots +a_n(b_n+c_n)$$ by the typical properties of fields, we may distribute multiplication over addition as normal. $$a_1b_1+a_1c_1+a_2b_2+a_2c_2+ \dots +a_nb_n+a_nc_n$$ and the commutative nature of addition allows us to write the following: $$a_1b_1+a_2b_2+ \dots +a_nb_n+a_1c_1+a_2b_2+ \dots +a_nc_n$$ we then notice that this very much looks like the definition of the dot products, and write our statement once more. $$\begin{bmatrix} a_1\a_2\ \vdots \a_n \end{bmatrix} \cdot \begin{bmatrix} b_1\b_2\ \vdots \b_n \end{bmatrix}+\begin{bmatrix} a_1\a_2\ \vdots \a_n \end{bmatrix} \cdot \begin{bmatrix} c_1\c_2\ \vdots \c_n \end{bmatrix}$$ each step of our proof was definitionally valid for vector spaces. therefore: $$\mathbf{a} \cdot (\mathbf{b}+\mathbf{c}) = \mathbf{a} \cdot \mathbf{b}+\mathbf{a} \cdot \mathbf{c}$$

[gay-mathematician-portmanteau]

wow, another math dork falling off the dominion deep end, obsessing over strategy wikis/discussion threads, and randomly generating dozens of kingdoms to play by herself? who'd have thought she'd be such a giant dork.