Latest Posts by mildlyramified - Page 3

Sleepy, confused puffins and dodo birds

A puffin rests while overlooking the ocean near Látrebjarg, Iceland.

Go World Travel Magazine

Isomorphism

everything IS like everything else

So, my spouse has been exploring his gender lately; he also just built himself a new laptop. Today he told me that he in an attempt to process some genderfeels through metaphor, he made a post on a trans forum along the lines of: "I'm a lifelong Windows user and I think I'm pretty good at it. I want to find out what Linux has to offer but I'm afraid I wouldn't be any good at it. And how do you choose the right Linux distro, anyway? Do you have to try them all?"

The responses, he said, were a mix of useful advice about feeling out your gender and useful advice about choosing a Linux distro.

I love trans people so much

The alphabet up to homotopy equivalence:

Upper case:

O 8 . O . . . . . . . . . . O O O O . . . . . . . .

Lower case:

O O . O O . O . : : . . . . O O O . . . . . . . . .

Tamil Linguistics thread (bc nobody cares but me)

but really, if you are interested in linguistics at all, give this post a read, because this shit really blew my mind ...

have been reading the following paper: https://ccat.sas.upenn.edu/~haroldfs/public/h_sch_9a.pdf

"The Tamil Case System" (2003) written by Harold F. Schiffman, Professor Emeritus of Dravidian Linguistics and Culture, University of Pennsylvania

Tamil is one of the oldest continuously-spoken languages in the world, dating back to at least 500 BCE, with nearly 80 million native speakers in South India and elsewhere, and possessed of several interesting characteristics:

a non-Indo-European language family (the Dravidian languages, which include other languages in South India - Malayalam being the most closely related major language - and one in Pakistan)

through the above, speculative ties to the Indus Valley Civilization, one of the first major human civilizations (you can read more about that here)

an agglutinative language, similar to German and others (so while German has Unabhängigkeitserklärungen, and Finnish has istahtaisinkohankaan, in Tamil you can say pōkamuṭiyātavarkaḷukkāka - "for the sake of those who cannot go")

an exclusively head-final language, like Japanese - the main element of a sentence always coming at the end.

a high degree of diglossia between its spoken variant (ST) and formal/literary variant (LT)

cool retroflex consonants (including the retroflex plosives ʈ and ɖ) and a variety of liquid consonants (three L's, two R's)

and a complex case system, similar to Latin, Finnish, or Russian. German has 4 cases, Russian has at least 6, Latin has 6-7, Finnish has 15, and Tamil has... well, that's the focus of Dr. Schiffman's paper.

per most scholars, Tamil has 7-8 cases - coincidentally the same number as Sanskrit. The French wikipedia page for "Tamoul" has 7:

Dr. Schiffman quotes another scholar (Arden 1942) giving 8 cases for modern LT, as in common in "native and missionary grammars", i.e. those written by native Tamil speakers or Christian missionaries. It's the list from above, plus the Vocative case (which is used to address people, think of the KJV Bible's O ye of little faith! for an English vocative)

... but hold on, the English wiki for "Tamil grammar" has 10 cases:

OK, so each page adds a few more. But hold on, why are there multiple suffix entries for each case? Why would you use -otu vs. -utan, or -il vs -ininru vs -ilirintu? How many cases are there actually?

Dr. Schiffman explains why it isn't that easy:

The problem with such a rigid classification is that it fails in a number of important ways ... it is neither an accurate description of the number and shape of the morphemes involved in the system, nor of the syntactic behavior of those morphemes ... It is based on an assumption that there is a clear and unerring way to distinguish between case and postpositional morphemes in the language, when in fact there is no clear distinction.

In other words, Tamil being an agglutinative language, you can stick a bunch of different sounds onto the end of a word, each shifting the meaning, and there is no clear way to call some of those sounds "cases" and other sounds "postpositions".

Schiffman asserts that this system of 7-8 cases was originally developed for Sanskrit (the literary language of North Indian civilizations, of similar antiquity to Tamil, and the liturgical language of Vedic Hinduism) but then tacked onto Tamil post-facto, despite the languages being from completely different families with different grammars.

Schiffman goes through a variety of examples of the incoherence of this model, one of my favorites quoted from Arden 1942 again:

There is no rule as to which ending should be used ... Westerners are apt to use the wrong one. There are no rules but you can still break the rules. Make it make sense!!

Instead of sticking to this system of 7-8 cases which fails the slightest scrutiny, Dr. Schiffman instead proposes that we throw out the whole system and consider every single postposition in the language as a potential case ending:

Having made the claim that there is no clear cut distinction between case and postpositions in Tamil except for the criterion of bound vs. unbound morphology, we are forced to examine all the postpositions as possible candidates for membership in the system. Actually this is probably going too far in the other direction ... since then almost any verb in the language can be advanced to candidacy as a postposition. [!!]

What Schiffman does next is really cool, from a language nerd point of view. He sorts through the various postpositions of the language, and for each area of divergence, uses his understanding of LT and ST to attempt to describe what shades of meaning are being connoted by each suffix. I wouldn't blame you for skipping through this but it is pretty interesting to see him try to figure out the rules behind something that (eg. per Arden 1942) has "no rule".

On the "extended dative", which connotates something like "on the behalf of" or "for the sake of":

I especially find his analysis of the suffix -kitte fascinating, because Schiffman uncovers a potential case ending in Spoken Tamil that connotes something about the directness or indirectness of an action, separate from the politeness with which the person is speaking to their interlocutor.

Not to blather on but here's a direct comparison with Finnish, which as stated earlier has 15 cases and not the 7-8 commonly stated of Tamil:

What Schiffman seems to have discovered is that ST, and LT too for that matter, has used existing case endings and in some cases seemingly invented new ones to connote shades of meaning that are lost by the conventional scholar's understanding of Tamil cases. And rather than land on a specific number of cases, he instead says the following, which I find a fascinating concept:

The Tamil Case System is a kind of continuum or polarity, with the “true” case-like morphemes found at one end of the continuum, with less case-like but still bound morphemes next, followed by the commonly recognized postpositions, then finally nominal and verbal expressions that are synonymous with postpositions but not usually recognized as such at the other extreme. This results in a kind of “dendritic” system, with most, but not all, 8 of the basic case nodes capable of being extended in various directions, sometimes overlapping with others, to produce a thicket of branches. The overlap, of course, results from the fact that some postpositions can occur after more than one case, usually with a slight difference in meaning, so that an either-or taxonomy simply does not capture the whole picture.

How many cases does Tamil have? As many as its speakers want, I guess.

I need to stop thinking about my work days as "productive" days and my days off as "unproductive" days that I waste if I haven't built something or deep cleaned my house. What the fuck am I accomplishing at work. My job doesn't wash my dishes

mathematics/art

Laziness Does Not Exist

Psychological research is clear: when people procrastinate, there's usually a good reason

good read for teachers.

Just learnt about this dude Galois, absolutely crazy. so basically, he:

- tries to get into "Ecole Polytechnique", the French school for science at the time, and fails

-publishes several papers on polynomials

-his father dies

-tries again (and fails) to get into the Ecole

-publishes several papers basically founding group theory

-attends a lesser school, gets kicked out and eventually arrested for political beliefs

-died in a duel shortly after he got out of prison

AT THE AGE OF 20

This proof is left as an exercise for the grader

Prof after saying maybe we should be teaching these to first year undergrads, the calculation is really simple but they might struggle with understanding forms - stuck on said calculation for the next 40 minutes of class.

Why are mathematicians* so USELESS at simple arithmetic? You're telling me I possess knowledge that would get me hailed as a lord 2000 years ago, but if I had two bags of wheat in one hand and three in the other, I would say there's six bags? What the fuck

*it's me, I'm mathematicians

I'm running out of fonts to use and it is now blackbaords

Them : What do you even do in graduate-level math?

Me: Invent symbols no one understands, stare at whiteboards for hours, and occasionally cry into my coffee.

Undergrad: This proof is so hard!

Grad student: My proof is 17 pages long, involves six lemmas, and the author just ‘leaves the rest to the reader.

Actually this is what diogenes was trying to say....

the default way for things to taste is good. we know this because "tasty" means something tastes good. conversely, from the words "smelly" and "noisy" we can conclude that the default way for things to smell and sound is bad. interestingly there are no corresponding adjectives for the senses of sight and touch. the inescapable conclusion is that the most ordinary object possible is invisible and intangible, produces a hideous cacophony, smells terrible, but tastes delicious. and yet this description matches no object or phenomenon known to science or human experience. so what the fuck

Ooohhhh that's quite something.

Fun scary fact: Let g: ℝ→ℝ be given by g(x)=x if x∈ℚ and g(x)=0 if x∉ℚ. Then the function f: ℝ->ℝ given by f(x)=g(x)tan(x) is differentiable at x=0. In fact it is only differentiable at x=0

REBLOG THIS POST IF YOU ARE A MATH ENJOYER

No it is not optional, I desperately need to follow y’all so that there is more math on my dashboard.

Zariski topologies

So if you take kⁿ, the n-dimensional coordinate space over some field k, the Zariski topology on kⁿ is the topology whose closed sets are of the form

Z(S) = { x ∈ kⁿ : f(x) = 0 for all f ∈ S }

for some subset S ⊆ k[x₁,...,xₙ]. That is, the closed sets are the common zero loci of some set of polynomials over k in n variables, i.e. they are the solution sets for some system of algebraic equations. Such sets are called algebraic sets. If I is the ideal generated by S, then Z(S) = Z(I), so we can restrict ourselves to ideals.

Now if you take a commutative unital ring R, we let Spec R denote its prime spectrum, the set of prime ideals of R. We let Max R ⊆ Spec R be the subset consisting of the maximal ideals, the maximal spectrum. The Zariski topology on Spec R is the topology whose closed sets are of the form

Z(S) = { P ∈ Spec R : P ⊇ S }

for some subset S ⊆ R. A prime ideal P contains S if and only if it contains the ideal generated by S, so again we can restrict to ideals. What's the common idea here? Classically, if k is algebraically closed, then Hilbert's Nullstellensatz (meaning Zero Locus Theorem) allows us to identify the points of kⁿ with those of Max k[x₁,...,xₙ], by mapping a point (a₁,...,aₙ) to the maximal ideal (x₁ - a₁,...,xₙ - aₙ), and the Zariski topologies will agree along this identification. There's nothing very special about these algebraic sets though.

Let X be any (pre-)ordered set with at least one bottom element. For a subset Y ⊆ X, define the lower and upper sets associated to Y as

L(Y) = { x ∈ X : x ≤ y for all y ∈ Y }, U(Y) = { x ∈ X : x ≥ y for all y ∈ Y }.

We call a lower [upper] set principal if it is of the form L(x)= L({x}) [U(x) = U({x})] for some x ∈ X. If X is complete (any subset has at least one least upper bound and greatest lower bound), then any lower or upper set is principal. Note that ⋂ᵢ L(Yᵢ) = L(⋃ᵢ Yᵢ), so lower sets are closed under arbitrary intersections; they provide what's called a closure system on the power set of X. The lower closure of a set Y is the intersection of all lower sets containing Y. We have that Y ⊆ L(x) if and only if x ∈ U(Y), so the lower closure of Y is given by L(U(Y)). If the lower sets were furthermore closed under finite unions (including empty unions), then they would form the closed sets of a topology on X.

This is not generally true; first of all, note that any lower set contains the bottom elements of X, of which there is at least one, so the empty set is not a lower set. As for binary unions, generally we have L(Y₁) ∪ L(Y₂) ⊆ L(Y₁ ∩ Y₂), but this inclusion might be strict. This is something we can fix by restricting to a subset of X.

We say that p ∈ X is prime if p is not a bottom element and for all x, y such that for all z such that x ≤ z and y ≤ z we have p ≤ z, we have that p ≤ x or p ≤ y. That is, if p is smaller than every upper bound of x and y, then p is smaller than x or y. Furthermore, we say that p is a prime atom if it is a minimal prime element. Let P(X) and A(X) denote the sets of primes and prime atoms of X, respectively. For a subset Y ⊆ X, let the Zariski closed set associated to Y be given by

Z(Y) = L(Y) ∩ P(X) = { p ∈ P(X) : p ≤ y for all y ∈ Y }.

We again have ⋂ᵢ Z(Yᵢ) = Z(⋃ᵢ Yᵢ), so the Zariski closed sets are closed under arbitrary intersections. Note also that Z(X) = ∅, so the empty set is closed. Now let Y₁, Y₂ be subsets of X. We find that Z(Y₁) ∪ Z(Y₂) = Z(U(Y₁ ∪ Y₂)). Clearly if p is smaller than all of the elements of one Yᵢ, then it is smaller than every upper bound; the interesting part is the other containment.

Assume that p ∈ Z(U(Y₁ ∪ Y₂)), so p is smaller than every upper bound of Y₁ ∪ Y₂. If p is smaller than every element of Y₁ then we are done, so assume that there is some y ∈ Y₁ with p ≰ y. For every y' ∈ Y₂ we have that p is smaller than every upper bound of y and y', so because p is prime we get that it is smaller than y or y'. It is not smaller than y, so p ≤ y'. We conclude that p ∈ Z(Y₂), and we're done.

As before, the Zariski closure of a set of primes Q ⊆ P(X) is given by Z(U(Q)). Note however that for a point x ∈ X we have L(x) = L(U(x)), so the Zariski closure of a prime p is Z(p). It follows that A(X) is exactly the subspace of closed points of P(X).

So we have defined the Zariski topology on P(X). How can we recover the classical examples?

If X is the collection of algebraic subsets of kⁿ ordered by inclusion, then P(X) consists of the irreducible algebraic subsets, and we can identify kⁿ itself with A(X). Our Zariski topology coincides with the standard definition.

If X = R is a unital commutative ring, ordered by divisibility, then being prime for the ordering coincides with being either prime for the ring structure, or being equal to 0 if R is an integral domain. Note that this ordering is not generally antisymmetric; consider 1 and -1 in a ring of characteristic not equal to 2.

A more well-behaved version of the previous example has X = { ideals I ⊴ R }, ordered by reverse inclusion. Note that for principal ideals (r), (s) we have (r) ⊇ (s) if and only if r divides s. We have P(X) = Spec R and A(X) = Max R, and our Zariski topology coincides with the standard definition.

You can play the same game if X is the lattice of subobjects of any structure H. If H is a set (or a topological space) and X is its power set, then the primes and prime atoms are the same; the points. The Zariski topology is the discrete topology on H. If H is a vector space, then P(X) is empty, because any non-zero subspace V can be contained in the span of two subspaces that don't contain V. It seems that the sweet spot for 'interesting' Zariski topologies is somewhere in between the rigidity of vector spaces and the flexibility of sets.

If H is an affine space, then again the prime elements are exactly the points. The resulting Zariski topology has as closed sets the finite unions of affine subspaces of H.

An interesting one is if X is the set of closed sets of some topological space S (generalizing the first example). The prime elements are the irreducible closed sets, and if S is T1 (meaning all points are closed), then the points of A(X) can be identified with those of S. Then the Zariski topology on A(X) is the same as the topology on S, and the Zariski closure of an irreducible closed set is the set of all irreducible closed sets contained in it.

Mathematics is taught very rigidly. When I'm independently working and studying math, it feels like art - like I'm making something and it tickles the creative side of my brain. In class it feels like the structured STEM course I initially signed up for.

It's a world of rules and structures people have carefully built over the millennium and you can add to it (if you can) or just walk around and observe and learn.

Analogously, learning mathematics, especially higher mathematics and even more so Algebra and Category Theory also feels like learning a new language. Working with it feels like writing poetry. Mathematics literature has a lot of the characteristic features of literature. There are many rules, but if you can break them, you are a mad genius!

I was talking to a professor and he told me about realising that he could read mathematics, granted it's not the same as picking up a story book, but there is this entire new world out there when you start reading mathematics. He also pulled up the linguistics definition of a language and said that perhaps mathematics is the only language with no exceptions in class once.

I just watched a beutifull and and emotionall video by Patricia Taxxon on youtube and it reminded me of something I used to be very close to but have since stopped doing, and that is to just look at math and play! Math is inherently arbitrary! By definition math is art! It's a way to grasp at the world, yes, but it's also a vehicle into creating entierly new worlds! New dimentions! New, novel kinds of logic and causality! You can take any single operation, map it to an arbitrary unused symbol, and just! Use it! In different eqasions! In different contexts! You can stretch you conceptions of what is a number, a value, space itself, and it's all so fun and inventive, and you can draw fractals! So so so so many fractals!

And it can be hard to see it that way. Especially after going through public education, especially after being beaten into the ground with incomprehensible notation of lond dead professors, after being offered only the most optimised and abstracted of formulas with no basis on which to place them but! Math is a jurney! To discovery of not only reality but yourself! What compells you; what you see with ease and what just does not compute, what is the rjeason for all of that notation, what would happen if you just assumed something stupid! And that stupid thing can sometimes be revolutionary to your understanding of numbers! Math and notation is an art form and I'm heartbroken people aren't seeing it for that! Because it's too esoteric or intimidating or because they've been made to feel like they could never succseed at it! Becouse someone hurt you in an attempt to teach you how to fill out a standardised test.

My plea is, to give math a try, look for something that resonated with you, or just draw some triangles and sqares; make a fractal; define how it's built and try to see what comes of it, there's so many things to discover. Try to proove a simple formula. Visualise! Solve a simple problem! It's intimidating; but it doesn't have to be! It's impossible to fail at mathematics, you can only learn from your mistakes, I beg of you, try.

This should have been my thesis

there should be a flat-earth-like conspiracy theory for every 2-manifold. show me the real projective plane earthers

Consult with dark powers to raise Paul Erdős from the dead to co-author your paper.

Just elementary topology will give you a good enough idea. The empty set is both open and closed and the universal set is both open and closed in a topology for instance.

Sometimes open balls are closed :)

where can I read more about this?

Fiber arts is just Math in sheep's clothing

she abstract my nonsense till my diagram commutes

she graph my theory til i form an edge set

thinking today about how much I love literally all fiber arts. I am hopeless at almost every other kind of art, but as soon as there is thread, yarn, or string I can figure it out fairly quickly.

I learned how to knit when i was eight, started sewing at nine, my dad taught me rock climbing knots around that age, I figured out from a book how to make friendship bracelets, I've made my own drop spindle to make yarn with, and more recently I've picked up visible mending. I've learned embroidery through fixing my overalls, and this year I've learned how to darn and how to do sashiko (which I did for the first time today). After years of being unable to crochet I finally figured it out last night and made seven granny squares in just a few hours.

I want to learn every fiber art that I can. I want to quilt, I want to use a spinning wheel, I want to weave, I want to learn tatting, I want to learn how to weave a basket, I want to learn them all. If I could travel through time and meet anyone in the Bible, high on my list are the craftsmen who made the Tabernacle.

I want to travel the world and learn the fiber arts of every culture, from the gorgeous Mayan weaving in Guatemala, to the stunning batik of Java, to Kente in Ghana. I want to sit at the feet of experienced men and women and watch them do their craft expertly and learn from them.

Of every art form I've seen, it's fiber arts that tug most at my heartstrings.

Hi what's this?

I wish I knew tbh. Usually there's more context for -this-

Old animation had such animated expressive emotions, the newer ones have been trying so hard to imitate and make it look as real as possible that we've lost the charm.

Love math textbooks so much <333

But like what's the fun if you don't get carpel tunnel

they should invent a hobby that doesn't require potentially destroying your wrists