counterexamples

Real Analysis Gothic
  • For every epsilon there is a delta. Where is it? Epsilon does not know where delta is. Zero does not know where delta is, it is too far away. Nobody knows where delta is; every number is so far from the other numbers. They are getting further apart. Epsilon is so very small.
  • You are walking up a staircase. You try to take a step, but another step emerges in the middle. A hooded figure appears at the top. You scream and try to run downstairs, but this staircase has so many steps. It is getting closer, closer.
  • You compute a Fourier series at a point; it is different than the value of the function. That’s normal, you say, unconvinced, as you are overwhelmed by a sense of dread. You start to apply the inversion formula, but think better of it. You still have scars from the last time.
  • You cut yourself into three pieces and eat the middle one. It hurts, yes, but you are so tender. You do it again and again and again and again… Your mouth eats your mouth. You are perfect.
  • A year ago today you were teaching calculus. “So you see, it’s easier to take derivatives than integrals” you said. You coughed, the blood splattered out of your mouth, it formed Thomae’s function on the floor. Now you are at the same point in your lesson plan. The stain is still there. It looks up at you, expectantly.
  • You are talking to a complex analyst. Differentiable functions are smooth, he says. Bounded differentiable functions are constant, he says. What are you doing, he says. I’ll call the police, he says. He doesn’t say anything now.

People always talk about musical prodigies, I’m way more interested in late bloomers. And that’s partially because I am one - I did start learning piano around 5 or 6, but didn’t take it seriously and basically had to be forced to practice for 20 minutes a day by my mother. I only started putting any effort in around 19, which is supposed to be after the cut-off point but clearly isn’t since I got into one of the most competitive music programs in the country. But, again, 19 wasn’t anywhere near my first exposure to learning and playing music.

Now I have heard about other people like me, who were exposed to music at a young age but weren’t serious players until they were older - Wes Montgomery didn’t play the guitar until age 20 for instance. But are there any masters who didn’t have any musical training until their teens or later? Traditional wisdom says the cutoff point is very young but I’m curious if anyone knows any counterexamples.

anonymous asked:

*Curtsies* you said you rarely find equality in early modern relationships, but what about Beatrice and Benedick?

*Curtsies* I said rare, not nonexistent. It wasn’t like a challenge to find counterexamples, just a comment on the usual sexual power dynamics of the time. Beatrice and Benedick? Sure, if you play it the right way. But you could also argue not, and that doesn’t necessarily mean the man’s the one in control. Beatrice demanding Benedick murder his best friend? Yeah. That’s a lot to ask. So maybe she’s wearing the pants, because he does it, or at least plans to. Again, as with almost anything in Shakespeare, it’s open to interpretation.

R2-D2, the Heart of a Machine

In memory of Kenny Baker (1934-2016) 

One of the many things I love about Star Wars is that droids (and machines in general) are depicted as not inherently bad, but rather as tools that, just like the Force, and just like living beings themselves, can be used for both good and evil.

For even as we are shown us the mindlessness of the battle droids and the ruthless heartlessness of cyborgs like General Grievous, so too are we shown, from the very beginning, the shining counterexamples of C-P30 and R2-D2 – a pair of ‘machines’ that not only exhibit an extremely admirable sense of loyalty, but also possess a great capacity for Doing Good™.

Anakin Skywalker is the character perhaps most (in)famously known for having a great affinity for all things mechanical. To the point that some even see his need to ‘fix things’ as a manifestation of something deeply flawed or broken (or even inhuman) within himself. But though there is indeed an undeniable symbolism in Anakin becoming less and less physically human, losing pieces of his flesh and bone until he himself is fully imprisoned within a machine, I would argue that his affinity for fixing and modifying machines not intended to be viewed as solely a bad thing.

If anything, it originates as an expression of his own humanity. 

Even without taking into consideration the fact that, as a child, Anakin built C-3P0 from scratch as a gift for his mother simply in order to make her life easier, or all the various times he fixed broken starships using nothing but parts scavenged from crashes, saving countless lives in the process, there is one thing that always stands out to me: 

Anakin Skywalker programmed (or at the very least, encouraged) R2-D2 to have feelings.

It’s safe to say that Anakin loved that droid….and that R2-D2 ‘loved’ his master in his turn (much to the astonishment and consternation of many in their midst). And as General Skywalker’s most favoured astromech during the Clone Wars, R2-D2 saved his master’s life countless times in battle. But perhaps even more fascinatingly, even after his master was long-gone, lost within the “twisted and evil” machine of Darth Vader, this same droid went on to save the lives of the generations that came after.

While Vader is marching around being Supremely Evil™ and generally fucking shit up for everyone, this fiercely loyal little droid, this piece of Anakin’s goodness, is still out there, flying around the galaxy, keeping his beloved master’s kids company, and generally saving the day.

And that’s just beautiful. :’)

huffingtonpost.com
Some 450 Former EPA Officials Want The Senate To Reject Trump's Pick | The Huffington Post

In a letter, released Monday, 447 former EPA employees say Pruitt’s record raises “serious questions” about his ability “to act in the public’s interest based on current law and the best available science.”

The letter argues that during his six years serving as Oklahoma attorney general, Pruitt “has shown no interest in enforcing environmental laws” — which it describes as a “critically important function” of the EPA. Instead, the letter notes, Pruitt has repeatedly sued the EPA in an effort to overturn regulations and standards, such as those intended to reduce smog levels in cities and mercury emissions from power plants.

His “disturbing” record offers few counterexamples of genuine efforts to reduce pollution, according to the letter. Although Pruitt has cited his work to protect the Illinois River from phosphorous runoff, the letter notes that Pruitt’s action in that matter actually delayed enforcement of phosphorous limits.

The signers are also concerned that Pruitt has expressed “reluctance to accept and act on the strong scientific consensus on climate change,” an issue they say requires urgent action.


Pruitt is “a strange person” to put in charge of the EPA, said Eric Schaeffer, the Environmental Integrity Project’s executive director and co-founder who also signed the letter.

The letter also raises a transparency issue: Pruitt has acknowledged that some 3,000 emails and documents pertaining to his close ties with fossil fuel companies exist but have still not been produced in response to a Freedom of Information Act request filed two years ago.

While the letter is strongly worded, its perspective is described as nonpartisan.

The letter’s signers are hardly “flame-throwing radicals,” Schaeffer said. Among the group are scientists, administrators, lawyers and others who have served under both Democratic and Republican presidents.

The full Senate’s confirmation vote on Pruitt, which should take place this week, will likely succeed along party lines.


In his official state biography, Pruitt has described himself as “a leading advocate against the EPA’s activist agenda.” As Oklahoma attorney general, he has sued the agency more than a dozen times.

As EPA administrator, Pruitt is expected to reduce the agency’s budget and power. News outlet Axios reports that the “absolute hammering” considered by Trump’s transition team includes cuts to state and tribal assistance grants, the termination of climate programs, and a rollback of regulations under the Clean Air Act, the Clean Power Plan and the Clean Water Rule.

hello all, this is hilary (formerly evilregalcaptainswan) (yes i did change my url twice in one year faint) (i think it’s fitting tho)

i just thought i would briefly pop in and let you know that yes, i am still on tumblr, but i don’t anticipate being back blogging for quite a while. i need a lot of time away and have a lot of things going on right now, and the break from fandom and drama has been SO refreshing. i still love you, but yeah, i need a break. so a break it will be.

HOWEVER, something happened this morning that i’m just kinda gonna have to share with you, so:

apparently one of my readers emailed the north remembers to GRRM himself. and GRRM actually responded. not to sue me, thankfully – but to say that he found it utterly captivating from the first chapter and couldn’t put it down. also that, i quote, “this is a perfect counterexample to my dislike of fanfiction in general.” (he did add that the author/me needed to work on original stuff if i wanted to actually become an author – GRRM, way ahead of you there, but yeah). so i don’t know if he read the whole damn thing or just some, but in fact, he has read it and likes it very much. i honestly don’t know how i feel about someone sending it to him because i have known all along that he doesn’t like fanfic, but there is that. i also don’t know if the reason he likes it is because i’ve guessed the approximate shape of the story and my ideas are on target for what he himself has planned to happen at the end of asoiaf – but there you have it. thought i would pass it along. he read it and it kinda changed his views on fanfiction.

so um. yeah. that happened.

So, this morning I was thinking about how having debates / arguments in sociology is very similar to talking about things in mathematics in the sense that it is particularly helpful to have a bag full of counterexamples ready to go at all times. For example, if you’re talking about differentiable functions, it helps to remember the absolute value function is continuous everywhere, but not differentiable everywhere (so, continuity doesn’t imply differentiability). Similarly, if you’re facing off with someone who conflates genitals with gender, it helps to know about people who identify as transgender (so, genitals don’t imply gender).

Just having these counterexamples not only help you take apart arguments fairly quickly, but they help you remember the facts as well (does continuity imply differentiability? well no, since I remember abs(x). does having a penis imply being a man? well no, since I know there are trans women)

So now that we’re getting another, more intimate method of communication, I want to reiterate my (currently, at least) position on the subject:

Please feel perfectly, 100% free to take control of your social landscape.
And please don’t make anyone feel wrong for doing so.

Please feel free to block people for any of: spamming you, pressuring you, bugging you, attacking you, annoying you, being politically opposed to you, being politically neutral on a topic you care strongly about, being politically agressive on a topic you feel neutral about, have a URL you would really rather not see, having an avvie you would really rather not see, or basically any other reason.

That’s not to say that I wouldn’t be sad if someone whom I thought was a really interesting person blocked me for no apparent reason. Of course I’d be disappointed, and probably really want to know what it was that made them block me.

But I’d also want, even more than that, for anyone else whom I asked to leave me alone to just leave me alone.

And there are so many people here. There will be something.

In addition to the previous post: take a set with five elements and consider all possible 2-element subsets. There are exactly 10 such subsets. Now, we can define an interesting graph with the 10 subsets as vertices, by declaring two subsets adjacent precisely when they are disjoint.

The resulting graph is the Petersen graph, a very regular graph with lots of nice (and sometimes unexpected) properties. Donald Knuth said the Petersen graph is “a remarkable configuration that serves as a counterexample to many optimistic predictions about what might be true for graphs in general.”

Giuseppe Peano, born on 27th August 1858, was an Italian mathematician. The author of over 200 books and papers, he was a founder of mathematical logic and set theory, to which he contributed much notation ( in his first book dealing with mathematical logic, the modern symbols for the union and intersection of sets appeared for the first time.). The standard axiomatization of the natural numbers is named the Peano axioms in his honor. As part of this effort, he made key contributions to the modern rigorous and systematic treatment of the method of mathematical induction.

Moreover, Peano’s famous space-filling curve appeared in 1890 as a counterexample. He used it to show that a continuous curve cannot always be enclosed in an arbitrarily small region. This was an early example of what came to be known as a fractal.

He has done so many things ^_^