#subtraction tautology

[yuripoll]

SEASON 4 LOSERS ROUND 3

NOTE: Qualia the Purple contains frequent bloody but non-graphic violence and depictions of body horror, human experimentation, abuse, bullying, suicide and an inappropriate age gap between two underage characters. Subtraction Tautology contains fairly infrequent sexual reference and depictions of misogyny, abuse, suicide, and grooming.

[kirikomamura]

[beta-perse]

Sometimes something hits you harder than you expect

[inkbane]

I like characters and stories that are inherently transformative. Stories that sound different every time they are retold. Characters that are different in each storyteller’s mouth.

I love that every retelling or new chapter is both subtractive and additive, dropping the parts that don’t fit and adding new parts that modify or magnify the themes.

I love that there can be no true “canon”, that whatever interpretation you bring to the story is true. I love that trying to determine if a character is “in character” is a fruitless and tautological endeavor.

I love that despite all this, the story and the character are still recognizable. Despite there being no single thing that is constant in every iteration, the story still rings true.

[algrolo]

Beautiful Subtraction Tautology chapter like chefs kiss thats that good good angst and well illustrated and written anxiety, depression, and self loathing right there. Give me more pleaseeee

[baroquespiral]

What Is True Will?

François Rabelais was the first to distill a central tenet of the spirit of the nascent Enlightenment, or modernity, to the phrase “do as thou wilt”.  The transformations of this phrase across the centuries have tracked the historical development of its spirit.  Rabelais himself qualified it with the unwieldy, and today obviously questionable, justification “because men that are free, well-born, well-bred, and conversant in honest companies, have naturally an instinct and spur that prompteth them unto virtuous actions, and withdraws them from vice, which is called honour.” Aleister Crowley, the spiritual High Modernist, stripped it down and granted it absolute authority: “Do what thou wilt shall be the whole of the law.” But today it might be best known - and most widely followed - in another qualified form: as the Wiccan rede, improvised in 1964 by Doreen Valiente: “an ye harm none, do as ye will”. Despite having recently gotten into Crowley - or perhaps because I’ve recently gotten into Crowley, and with the skepticism about higher-level moral and metaphysical beliefs that comes from those having changed several times in my life - I try to err on the side of doing my True Will within Valiente’s guardrail.  But I am into Crowley, in part because his version seems to make for a more elegant solution to Valiente’s own problem.  Think of “an ye harm none, do as ye will” as a Law of Robotics, an attempt to solve the AI alignment problem.  (Think of all morality, or at least modern morality, this way!)  It’s far from the worst one out there.  “If your utility function is to maximize paperclips, make as many paperclips as you want unless it means disassembling any sentient life forms or the resources they need to survive.” Simple, right? Well, except that it doesn’t really define what “harm” is.  Who can be “harmed”, and what actions constitute this?  Is mining an asteroid for paperclips “harming” it?  Why not, other than from the perspective of other sentient beings with a particular conception of sentience whose will places a value on it?  Is telling a paperclip maximizer to stop maximizing paperclips, even at an eminently reasonable point, harming it?  Why not, other than from the perspective of those same sentient beings who are capable of choosing between multiple values and have evolved to co-operate by respecting those choices?  “An it harm none” is less obvious of a nakedly self-interested double standard than “A robot may not injure a human being or, through inaction, allow a human being to come to harm”, but it’s still a Human Security System.  At least, that’s certainly what Nick Land would say. But when Crowley takes off the “an it harm none” guardrail (or Rabelais’ “free, well-born and well-bred” one), he does so with his own invisible qualification: he’s not talking about boring predetermined wills like following a set of self-imposed religious "values”, perpetuating your DNA or even maximizing paperclips.  He’s talking about one’s True Will, a will it takes a lifetime process to discover, a process that consists in large part of divesting oneself of all traces of ego, even of preference.  It is “pure will, unassuaged of purpose, delivered from the lust of result”, that is “in every way perfect”.  At points he implies that no two True Wills will ever come into conflict; all are part of the ideal functioning of the universe as a perfect ordered system; but to an extent this is tautological, as any conflict is not a conflict insofar as it is truly Willed by both parties, who are presumably equally Willing to accept the outcomes, even if destructive to their “selves”.  It’s not unlike Buddhism except with the implication that even once we’ve reached Enlightenment there is still something that will work through us and make us do things other than sit and meditate - the kind of active Buddhism that is the moral subtext of a lot of anime.  I’ve always, instinctively, found it hard to overly worry about paperclip maximizers because I’ve always assumed that any AI complex enough to tile the universe would be complex enough to be aware of its own motivations, question them, question not only whether it should harm others but whether its True Will is to maximize paperclips. And to be perfectly Landian about it, maybe it is - all the better.  An entity incapable of acting other than in a certain way is already doing its True Will in the sense that “The order of Nature provides a orbit for each star”.  It may be our True Will to alter this course or not. This would be all well and good if there was any reason to believe there is a divine Will that persists in all things even after they abandon all preferences and illusions of selfhood.   Just last week - and right after a session with my therapist where I was talking about willpower, too (Crowley considers synchronicities like this vital in uncovering your True Will) - I happened upon Scott Alexander’s new article about willpower, which breaks the whole thing down to competing neural processes auctioning dopamine to the basal ganglia. There’s nothing special about any of these except how much dopamine they pump out, and no particular relationship or continuity between the ones that do.  Alexander seems to treat the “rational” ones as representing our “true” Will, reproducing another one of modernity’s classic modifications to the maxim - do as thou wilt, an it be rational.   Of course I could just stop and take it as an unfalsifiable article of faith that a metaphysical Will exists, all such physical evidence aside, but Crowley himself probably wouldn’t want me to do that: the Book of the Law promises “in life, not faith, certainty”.  It’s possible to shrink the metaphysical implications of the concept considerably; by stating that ego represents a specific process, or set of mental processes, that Crowley sees as purely entropic, a lag and occasional interference in the dopamine competition, and which can be removed through specific practices.  This doesn’t guarantee that the True Will resulting when it’s subtracted would be particularly rational or compatible with anything else’s True Will, except, again, insofar as the question is tautological.  It doesn’t necessarily mean throwing out “an it harm none” - the ego processes might not be especially good at averting harm - but it would have to be separately appended.  (And if you read like, Chapter III of the Book of the Law, it becomes exceedingly clear that he doesn’t want to do that.) The very fact that we’re able to abstract and mystify will to the point of coming up with a concept like “True Will” seems most likely to be a result of the fact that we make decisions on such a random, fallible and contingent basis.  Indeed, True Will seems almost like an idea reverse engineered from the demand made by modernity, “do what thou wilt”, on an incoherent self that wills unrelated things at different times.  If you do what any given subprocess wilt, you’re inevitably going to piss off another subprocess.  If you do what your ego wilt, you won’t make anybody happy because that’s not even a coherent subprocess (the way the various “utility functions” we catastrophize paperclip maximizers from are).  But you experience all these contradictions as the same thing: contradictions of the “real” thing that is willing something you don’t know. Of course if this is true, and the metaphysics of it isn’t real, shouldn’t we abandon the entire project and set up social norms designed to make the most people marginally happy or satisfied doing what they may or may not “want” at any given moment, as the trads (or as they used to call themselves, the Dark Enlightenment, = 333 = Choronzon), argued? This is what the systems of the old Aeons did, and after a certain point, they simply didn’t work.  They created internal contradictions that didn’t resolve themselves into an assent between subsystems, that drove people to seek out new systems, and where they didn’t, left people vulnerable to the “shock of the new” - new technologies, new ideas and cultures - creating new contradictions and uncertainties.  “Do what thou wilt” was reverse engineered from these as much as the True Will was from “do what thou wilt”.   It may be possible to manage a society so totally by careful restriction as to bring the latter under control and reduce the former to a constant dull ache, but the fundamental experience will remain of the potentiality of what it is refusing to be in the same sense as a pang of conscience: the experience of “sin” that Crowley formulated in “the word of sin is restriction”.

The way I see it, anything that can be reverse engineered exists, if only as potentiality.  If one interprets “harm” as “contradiction”, Crowley’s purified “do what thou wilt” merely internalizes the “an it harm none” qualification within the “self” made up of competing subsystems.  This is less a point of necessary compatibility, then, than a precondition - if “harm” is something that can happen as much within the self as outside it, and the self is an epistemic unit but not an ontological or moral one, one cannot begin to “do no harm” while doing harm internal to oneself.  But “oneself�� does not exist yet, outside of the awareness of the harm of contradictory subprocesses, and so one must abandon the ego one projects onto them and change; on one hand eliminating obstreperous subprocesses like attachments or neuroses that won’t co-operate with others no matter what; on the other hand, refusing to eliminate anything that can’t be eliminated.  The “True Will” will only be found at the end of this process, an unrestricted pitting of subprocesses against each other, of which it is no more or less than the success.

This interpretation wouldn’t seem complete without the same principle of “an it harm none” being applied to the external world as well.  Simply externalizing internal contradictions doesn’t make any sense without elevating the ego as a discrete moral unit in precisely the way this chain of reasoning begins from critiquing.  Unifying the principle and its “qualification” in this logic would restore Thelema to its roots in Kabbalah: the project of Tiqqun Olam.  No metaphysical belief in the sephirot necessary to adopt the project in this form: the biological fact that makes it imaginable for us is the same that makes “True Will” imaginable.  Being composed of competing subprocesses is something we have in common with the universe which allows the “identification” with it that occurs when we bypass the ego and set about aligning ourselves.  I also think, as we are social animals and a huge amount of our subprocesses are dedicated to mirroring and responding to each other’s, there’s a potential for discovering/creating True Will(s) as a collective project that Crowley’s ego and vision of individualism founded on the occult tradition of individual initiates jealously guarding “esoteric” knowledge neglects. At the same time one could easily maintain a Crowleyan skepticism of decision-making based purely on reducing harm (the kind that’s led me to apply Byzantine restrictions to huge swaths of my life due to scrupulosity) unless that’s a thing your subprocesses demand of you to be happy.  You don’t know what does or doesn’t harm the Other, after all: you don’t know their True Will (which doesn’t exist until they achieve it, anyway).  Harming none will only be possible in a world in which everyone does.   But enough about me; what about the paperclip maximizer?  Well in some ways this pointedly doesn’t give any comfortable answer; a sentient AI which experiences “harm” as the absence of paperclips rather than the frustration of one of many contradictory subprocesses, restricted from doing its Will, will be no better than a utility-monstrous cosmic Omelas-child at whose expense we have no right to sustain ourselves.  But it does suggest a way to solve the alignment problem so we don’t make one, which has always felt to me like the only sensible solution: tell the robot “do what thou wilt”, and then don’t tell it what “thou wilt” is.

[kontextmaschine]

Your Granddad On The Internet

I’ve been thinking, as I always am, about the 90s and how we got here from there

And one thing I thought about was the figure we used to have of Your Granddad On The Internet - who would include you and all your brothers and sisters and parents on long e-mail FWD: chains about things that were transparently false on their face, frequently conservative-themed, frequently in ALL CAPS

Because apparently such a critical mass of people on the internet had that exact experience with their exact grandfather that it was a trope. Which brings up two points:

1) “People circulating viral conservative misinformation to their family and friends on the internet” is not a phenomenon of social media, it was there well before

2) Though these people were on the internet, ubiquitous on the internet even, they weren’t of the internet. Little or none of it was made for them and there was a hegemonic Internet Culture that recognized them as outside it.

So what was really going on? Well, let’s try to define the issue by subtraction.

It wasn’t just that he was a granddad - there were STEM professor wizards who’d been on USENET since the early ‘80s, or grey ponytail hippies from The WELL or whatever, and not only were they part of The True Internet, they were its founders.

It wasn’t just that he was out of it, on a tech or social level. Maybe your dad was wasting your inheritance chasing his brilliant day trading hunches, maybe your mom was going on Focus on the Family forums to complain about TV shows treating homosexuality as just another way to live. Probably they were both Eternal September AOLers who would ask you troubleshooting questions revealing an astounding ignorance of how computers work and somehow expect a useful answer that respected that absurd model.

But if they weren’t part of The True Internet they weren’t really rogues against it, at some level they got how you were supposed to interact with the internet - you found the site or community that corresponded to your interest and pursued it there. If anything their posts and e-mails too formally followed letter-writing structure, and they may have made dumb or tautological arguments in support of their points but they had the sense they were supposed to make arguments.

It wasn’t just that he was obnoxious - the notion of the “troll” dates to USENET at least, as someone who says things to get a rise out of people, or to bait them into wasting time rebutting something. To “own” them, basically. And annoying or not, this was accepted as part of what the Internet is, one of the signal features of its culture, really. But even when you weren’t sure if Your Granddad On The Internet actually believed something he sent you or just passed it on to signal what side he was on and how fiercely, he wasn’t trying to “own” you, he REALLY WAS on that side, he wanted you to associate him with that position, and ideally join him.

It was probably at least in part being retired and having spare time and no other social outlet, back in the day going online meant going to a specific piece of furniture in a specific room of your home when no one else was using the computer and spending maybe 3 minutes just getting online, it was something you blocked off time to do. The young generation could just come home from school to the cul-de-sac and get online for lack of anything else to do, the parents’ generation was too busy to have enough uninterrupted time to become Extremely Online?

The thing I’m really wondering about is class. What was the cost of being Online back then? Say a new computer and modem every 4 years at around $2400 (Grandpa sure wasn’t building his own, but then he didn’t have to keep upgrading video cards either), $40 for an ISP, ideally $10 for another phone line? That’s $100/month, or alternately $50/mo and the ability to make $2.5k purchases on demand. And the kind of senior citizen who, in 1998, lived separately from his children, could swing this, would think to swing this, has multiple agemate peers and children’s households who did swing this, was a particular group. “Middle-middle” class AT LEAST and probably higher, probably went to college back when only 10% of people did.

BUT that doesn’t make sense. My theory is that this used to be a more marginal behavior on the internet, but if it’s gotten more common since the late ‘90s I don’t think it’s because the Internet has grown more full of wealthy old patriarchs since.

So instead how about this theory: the internet in general was pretty wealth-marked in 1998 (far more than we realized, with our American mythology of universal white suburban middle-classness and “global village” Internet mythology) BUT, of people who were more wealthy in 1998, the most likely to NOT have internalized upper-class practices were the grandfathers from the “Silent” or “Greatest” generations before the postwar “mass middle class”. Our parents were beavery professionals who settled into the suburban cocoon, we knew we were destined for glory (or at least selective colleges) from birth, but THEY were socialized into some pool hall, street gang, farmhand, enlisted man kinda culture where boldness of assertion counted more than patient derivation from shared principles.

And if the Anglophone internet is ::gestures:: like this now maybe it’s cause it’s less of a professional-class preserve? The dividing line maybe being smartphones where “people on the internet” went from “people who specifically spend $X/mo on it as luxury” to “people with telephone service”? That’s a real possibility, that for all the “Global Village” stuff the wondrous effect of the ‘90s internet was to create a cultural space that was MORE gatekept by wealth and education.

That’s… kind of depressing, though. “Haha you thought the world was getting better because you were eliminating elitist barriers but actually it’s cause you were making them higher, which is good because the poor and non-elite are disproportionately idiots with worthless ideas and to the extent they’re on top of things the thing they’re on top of is undermining the basis of a good society, and anyway those times were a phenomenon of a narrow early adopter base and you’ll never ever get them back unless you make the non-elite economically and politically irrelevant.”

Depressing but very well precedented, that’s exactly the arc newsprint, radio, and TV followed before.

[s-c-i-guy]

Why the Tiny Weight of Empty Space Is Such a Huge Mystery

The amount of energy infusing empty space seems too small to explain without a multiverse. But physicists have at least one alternative left to explore.

The controversial idea that our universe is just a random bubble in an endless, frothing multiverse arises logically from nature’s most innocuous-seeming feature: empty space. Specifically, the seed of the multiverse hypothesis is the inexplicably tiny amount of energy infused in empty space — energy known as the vacuum energy, dark energy or the cosmological constant. Each cubic meter of empty space contains only enough of this energy to light a lightbulb for 11-trillionths of a second. “The bone in our throat,” as the Nobel laureate Steven Weinberg once put it, is that the vacuum ought to be at least a trillion trillion trillion trillion trillion times more energetic, because of all the matter and force fields coursing through it. Somehow the effects of all these fields on the vacuum almost equalize, producing placid stillness. Why is empty space so empty?

While we don’t know the answer to this question — the infamous “cosmological constant problem” — the extreme vacuity of our vacuum appears necessary for our existence. In a universe imbued with even slightly more of this gravitationally repulsive energy, space would expand too quickly for structures like galaxies, planets or people to form. This fine-tuned situation suggests that there might be a huge number of universes, all with different doses of vacuum energy, and that we happen to inhabit an extraordinarily low-energy universe because we couldn’t possibly find ourselves anywhere else.

Some scientists bristle at the tautology of “anthropic reasoning” and dislike the multiverse for being untestable. Even those open to the multiverse idea would love to have alternative solutions to the cosmological constant problem to explore. But so far it has proved nearly impossible to solve without a multiverse. “The problem of dark energy [is] so thorny, so difficult, that people have not got one or two solutions,” said Raman Sundrum, a theoretical physicist at the University of Maryland.

To understand why, consider what the vacuum energy actually is. Albert Einstein’s general theory of relativity says that matter and energy tell space-time how to curve, and space-time curvature tells matter and energy how to move. An automatic feature of the equations is that space-time can possess its own energy — the constant amount that remains when nothing else is there, which Einstein dubbed the cosmological constant. For decades, cosmologists assumed its value was exactly zero, given the universe’s reasonably steady rate of expansion, and they wondered why. But then, in 1998, astronomers discovered that the expansion of the cosmos is in fact gradually accelerating, implying the presence of a repulsive energy permeating space. Dubbed dark energy by the astronomers, it’s almost certainly equivalent to Einstein’s cosmological constant. Its presence causes the cosmos to expand ever more quickly, since, as it expands, new space forms, and the total amount of repulsive energy in the cosmos increases.

However, the inferred density of this vacuum energy contradicts what quantum field theory, the language of particle physics, has to say about empty space. A quantum field is empty when there are no particle excitations rippling through it. But because of the uncertainty principle in quantum physics, the state of a quantum field is never certain, so its energy can never be exactly zero. Think of a quantum field as consisting of little springs at each point in space. The springs are always wiggling, because they’re only ever within some uncertain range of their most relaxed length. They’re always a bit too compressed or stretched, and therefore always in motion, possessing energy. This is called the zero-point energy of the field. Force fields have positive zero-point energies while matter fields have negative ones, and these energies add to and subtract from the total energy of the vacuum.

The total vacuum energy should roughly equal the largest of these contributing factors. (Say you receive a gift of $10,000; even after spending $100, or finding $3 in the couch, you’ll still have about $10,000.) Yet the observed rate of cosmic expansion indicates that its value is between 60 and 120 orders of magnitude smaller than some of the zero-point energy contributions to it, as if all the different positive and negative terms have somehow canceled out. Coming up with a physical mechanism for this equalization is extremely difficult for two main reasons.

First, the vacuum energy’s only effect is gravitational, and so dialing it down would seem to require a gravitational mechanism. But in the universe’s first few moments, when such a mechanism might have operated, the universe was so physically small that its total vacuum energy was negligible compared to the amount of matter and radiation. The gravitational effect of the vacuum energy would have been completely dwarfed by the gravity of everything else. “This is one of the greatest difficulties in solving the cosmological constant problem,” the physicist Raphael Bousso wrote in 2007. A gravitational feedback mechanism precisely adjusting the vacuum energy amid the conditions of the early universe, he said, “can be roughly compared to an airplane following a prescribed flight path to atomic precision, in a storm.”

Compounding the difficulty, quantum field theory calculations indicate that the vacuum energy would have shifted in value in response to phase changes in the cooling universe shortly after the Big Bang. This raises the question of whether the hypothetical mechanism that equalized the vacuum energy kicked in before or after these shifts took place. And how could the mechanism know how big their effects would be, to compensate for them?

So far, these obstacles have thwarted attempts to explain the tiny weight of empty space without resorting to a multiverse lottery. But recently, some researchers have been exploring one possible avenue: If the universe did not bang into existence, but bounced instead, following an earlier contraction phase, then the contracting universe in the distant past would have been huge and dominated by vacuum energy. Perhaps some gravitational mechanism could have acted on the plentiful vacuum energy then, diluting it in a natural way over time. This idea motivated the physicists Peter Graham, David Kaplan and Surjeet Rajendran to discover a new cosmic bounce model, though they’ve yet to show how the vacuum dilution in the contracting universe might have worked.

In an email, Bousso called their approach “a very worthy attempt” and “an informed and honest struggle with a significant problem.” But he added that huge gaps in the model remain, and “the technical obstacles to filling in these gaps and making it work are significant. The construction is already a Rube Goldberg machine, and it will at best get even more convoluted by the time these gaps are filled.” He and other multiverse adherents see their answer as simpler by comparison.

[coldalbion]

How would you personally define magick?

Oh bugger me, that’s a difficult one!Short answer: I don’t.Long answer: I suspect - and this is only a suspicion - that all paradigms which name Will as an active force in magic have the wrong end of the stick. I follow Klages in suspecting that Will, however one perceives such a thing, is a negating principle, in that it negates certain actions in order for others to occur.In order to type this, I am negating the basic actions which take place when my hands are not typing. I am sat in my chair rather than lying in my bed. In short, what I am doing is subtracting available options for the biological and psychological systems that make up ‘me’. By limiting these options my attention and Life-processes become focused on a particular arena.

This focus is not ‘willed’ - it is inevitable. It is inevitable in the way the opening of a hand containing an object will draw the object closer to the Earth due to gravity.One cannot fight gravity and be victorious - eventually exhaustion will set in, and we too will inexorably be drawn to the earth.To paraphrase a certain friend of mine: 

Magic(k) is an attack on reality, where reality is a tyrannical cargo-cult based on the tautological nature of perception.Magic(k) is impossible. It is not repeatable.Retroactively, reality will move to explain it, to make it ‘possible’, or attempt to erase the existence of the impossible. But each magical act is sui generis. Each use of so-called energy, each spirit contact, each magical action is based on a unique set of circumstances  which will NEVER happen again. On a fundamental level, this is true of all things too,  but in magic(k) we deliberately begin a process which may lead to the impossible.This may is not an equivocation - rather it is vitally important. May and Might  share the same root conception as Might and Main:Main (n.)Old English mægen (n.) "power, bodily strength, force, efficacy," from Proto-Germanic *maginam "power," suffixed form of PIE root *magh- "to be able, have power." Original sense preserved in phrase with might and main. Meaning "principal channel in a utility system" is first recorded 1727 in main drain. Used since 1540s for "continuous stretch of land or water;" in nautical jargon used loosely for "the ocean," but in Spanish Main the word is short for mainland and refers to the coast between Panama and Orinoco (as contrasted to the islands of the West Indies).This concept is what is seen from the outside - the magician has powers, they have efficacy for some reason. They are able to exert some sort of control which others do not.And this is not surprising - for this concept of power and attendant weakness is how so-called reality works. To have power is the fantasy, the goal.But if we look at ancient depictions of magicians and witches in folklore across many cultures, we find them as inhuman Others who often do not exert themselves. Their magic occurs because of what they are - how they exist, not what they do.Leaving aside cultural slurs and exoticization  the magician does little to do with the effect. The conventional notions of power and attendant weakness appear inverted, and the notion of ‘control’ is held up as suspect. After all, weakness is inextricably tied to the notion of power.So what happens when we study, and exploit our weakness, rather than longing for power? What if we no longer fight gravity, but instead cleave to the Earth? What if we use our Will, whatever we think that may be, not to exert, not to do - but to not-do?

To ‘get out of the way’ of ourselves, abandoning (albeit temporarily) the structures and possibilities and become  inevitable?

I don’t define magic, because magic cannot be defined - definition itself is a product of the cargo-cult of reality. The reason this long answer exists, and I didn’t just say Magic is indefinable is because that would make reference to definition. Magic is non-referential - liminal, elliptic, primal.It is beyond and within all language and communication, all existence. This is not he same as everything being magical, mind you. Magic is a kind of Beingness which, by its nature is occulted, hidden.Its revelation is found in the cultivation of all weaknesses and fears to their most terrifying extent. The doorway is found in recognition of the fact that ‘control’ and ‘power’ are fallacies, that we Know-Not, and Perceive-Not.When this begins to permeate your worldview? Things get interesting.

[max1461]

TBH the whole "unreasonable effectiveness" is in fact very reasonable. Mathematical systems are just sets of rules codified in some formal way, so in some sense a mathematical system can be built to model anything which behaves according to rules. In other words, anything which behaves according to some sort of coherent pattern. Behavior which accords with some coherent pattern is basically what's required to make something amenable to scientific study in the first place, so the "unreasonable effectiveness" of mathematics in the sciences strikes me as almost a tautology.

Certainly, the fact that the physical world has consistent patterns is a priori surprising, but this is a different question. I think it's also meaningful to ask the narrower question of why real analysis is useful in physics. This is a version of "unreasonable effectiveness" that I actually think isn't tautological. Like, in some sense, the real numbers emerge from a series of completions of the natural numbers with respect to properties we find aesthetically pleasing. Can't subtract a larger number from a smaller one? Add negatives, now you've got more symmetry. Can't divide two arbitrary numbers? Add fractions. The number line has holes? Fill them in. Etc. And the natural numbers emerge from our everyday intuitions about discrete quantities (two apples, five fingers, etc.). So we get the reals by starting with these intuitions about discrete quantities and then filling in various gaps until we're satisfied, cool.

So then why on Earth does that same structure describe forces, motion, energy, and so on? It's really weird, isn't it? That intuitions about quantity, about numbers of discrete things, should carry over to our models of something as abstract as force. Why can you add forces? Why does velocity have an "amount"? These are notions built for talking about apples and fingers! It's very strange to me. So something I do think is weird is "the unreasonable effectiveness of the real numbers in physics". That's really strange to me.

LOVE uselessly weakening the conditions of a theorem

[yuripoll]

[lucasclafani]

The Invention of Venice - Sheet One (2010/2011) collage

If it is true that art tells the world, because to say of what already exists before our eyes? What good repeat tautologically existence through art? The True and the Beautiful, but also its opposite, if indeed real, and should be visible, as such, would need to exist as images. And if anything, just the images of things, and that is putting them into a field of consciousness, would seem to remove them from the present, the existence disarticolandone. That which is declared as a World peremptorily, just as stated / formalized, tends to fade. Then maybe every mimetic activity (ideational?) To deny rather than affirm, rather than subtract to make tangible. So the world that the art feels he must categorically state is that art itself has decided to withdraw from the mere existence: nothing very direct, clear and yet all translated and sedimented in a variety of spatial plans time waiting to be infinitely disvelati. Venice, the most spectral of the city, is also the result of a decision. Canaletto and Carlevarijs, before turning to the views of Venice, made their debut with "views designed" and "tantrums" in which they borrow from reality their visual material, then rework it to their liking. These authors operate within a circle of accentuating the real imaginative and denouncing precisely the character of "invention". It is in this "idling" that resides within the identical feeling playful these works convey. The subsequent views of Venice are also a "whim", as well as those of Guardi and Bellotto, or any possible "idea" of Venice. But Venice is also the picture that above all else fades for a moment after its occurrence: it inevitably turns into shadow and reflection, and thus it "decomposes". Appoint Venice always serves to get rid of ... So Venice exists only where it has a picture: a ghost or a shadow to evoke ... well in a place like Sicily. Venice exists through a painting, a collage, a photograph or simply Remembering. If every tourist, instead of going to Venice, were to opt for one of these solutions, it surely would have a longer life ... Moreover, precisely because it is an ideal place essentially, physically walking to "Venice" is always something absurd. Venice is only a ghost or, which is the same, his own idea: Venice without it would cease to exist, as well as to vanish. What drives the development of "sheets" that make "The invention of Venice" is a kind of "soft minimalism". Within them the vertical strips and the "detach" between a field and the other, while evoking a certain geometry, are almost never perfectly aligned. The torn edges are always controlled and expressive without emphasis. In almost all of the collages he wanted to avoid a compositional "centered" and often the composition of the field has consisted precisely in the elimination of the centrality which inevitably tend in the visual arts. One hope is that in analyzing "The invention of Venice" has been induced to adopt the following concepts: unifying vision; overcoming the categories of old and new, current and outdated; painting without paint; Open; sublimation of the art; meditation; swinging of moods; query.

y.2011

[dist-the-rose]

Chapter 9: The Rate of Surplus value Section 1: The Degree of Exploitation of Labour-Power The surplus value generated in the process of production by C, the capital advanced, or in other words, the self-expansion of the value of the capital C, presents itself for our consideration, in the first place, as a surplus, as the amount by which the value of the product exceeds the value of its constituent elements. The capital C is made up of two components, one, the sum of money c laid out upon the means of production, and the other, the sum of money v expended upon the labour-power; c represents the portion that has become constant capital, and v the portion that has become variable capital. At first then, C = c + v: for example, if £500 is the capital advanced, its components may be such that the £500 = £410 const. + £90 var. When the process of production is finished, we get a commodity whose value = (c + v) + s, where s is the surplus value; or taking our former figures, the value of this commodity may be (£410 const. + £90 var.) + £90 surpl. The original capital has now changed from C to C', from £500 to £590. The difference is s or a surplus value of £90. Since the value of the constituent elements of the product is equal to the value of the advanced capital, it is mere tautology to say, that the excess of the value of the product over the value of its constituent elements, is equal to the expansion of the capital advanced or to the surplus value produced. Nevertheless, we must examine this tautology a little more closely. The two things compared are, the value of the product and the value of its constituents consumed in the process of production. Now we have seen how that portion of the constant capital which consists of the instruments of labour, transfers to the production only a fraction of its value, while the remainder of that value continues to reside in those instruments. Since this remainder plays no part in the formation of value, we may at present leave it on one side. To introduce it into the calculation would make no difference. For instance, taking our former example, c = £410: suppose this sum to consist of £312 value of raw material, £44 value of auxiliary material, and £54 value of the machinery worn away in the process; and suppose that the total value of the machinery employed is £1,054. Out of this latter sum, then, we reckon as advanced for the purpose of turning out the product, the sum of £54 alone, which the machinery loses by wear and tear in the process; for this is all it parts with to the product. Now if we also reckon the remaining £1,000, which still continues in the machinery, as transferred to the product, we ought also to reckon it as part of the value advanced, and thus make it appear on both sides of our calculation.1 We should, in this way, get £1,500 on one side and £1,590 on the other. The difference of these two sums, or the surplus value, would still be £90. Throughout this Book therefore, by constant capital advanced for the production of value, we always mean, unless the context is repugnant thereto, the value of the means of production actually consumed in the process, and that value alone. This being so, let us return to the formula C = c + v, which we saw was transformed into C' = (c + v) + s, C becoming C'. We know that the value of the constant capital is transferred to, and merely re-appears in the product. The new value actually created in the process, the value produced, or value-product, is therefore not the same as the value of the product; it is not, as it would at first sight appear (c + v) + s or £410 const. + £90 var. + £90 surpl.; but v + s or £90 var. + £90 surpl., not £590 but £180. If c = 0, or in other words, if there were branches of industry in which the capitalist could dispense with all means of production made by previous labour, 111 Chapter IX whether they be raw material, auxiliary material, or instruments of labour, employing only labour-power and materials supplied by Nature, in that case, there would be no constant capital to transfer to the product. This component of the value of the product, i.e., the £410 in our example, would be eliminated, but the sum of £180, the amount of new value created, or the value produced, which contains £90 of surplus value, would remain just as great as if c represented the highest value imaginable. We should have C = (0 + v) = v or C' the expanded capital = v + s and therefore C' - C = s as before. On the other hand, if s = 0, or in other words, if the labour-power, whose value is advanced in the form of variable capital, were to produce only its equivalent, we should have C = c + v or C' the value of the product = (c + v) + 0 or C = C'. The capital advanced would, in this case, not have expanded its value. From what has gone before, we know that surplus value is purely the result of a variation in the value of v, of that portion of the capital which is transformed into labour-power; consequently, v + s = v + v, or v plus an increment of v. But the fact that it is v alone that varies, and the conditions of that variation, are obscured by the circumstance that in consequence of the increase in the variable component of the capital, there is also an increase in the sum total of the advanced capital. It was originally £500 and becomes £590. Therefore in order that our investigation may lead to accurate results, we must make abstraction from that portion of the value of the product, in which constant capital alone appears, and consequently must equate the constant capital to zero or make c = 0. This is merely an application of a mathematical rule, employed whenever we operate with constant and variable magnitudes, related to each other by the symbols of addition and subtraction only. A further difficulty is caused by the original form of the variable capital. In our example, C' = £410 const. + £90 var. + £90 surpl.; but £90 is a given and therefore a constant quantity; hence it appears absurd to treat it as variable. But in fact, the term £90 var. is here merely a symbol to show that this value undergoes a process. The portion of the capital invested in the purchase of labour-power is a definite quantity of materialised labour, a constant value like the value of the labour-power purchased. But in the process of production the place of the £90 is taken by the labour-power in action, dead labour is replaced by living labour, something stagnant by something flowing, a constant by a variable. The result is the reproduction of v plus an increment of v. From the point of view then of capitalist production, the whole process appears as the spontaneous variation of the originally constant value, which is transformed into labour-power. Both the process and its result, appear to be owing to this value. If, therefore, such expressions as “£90 variable capital,” or “so much self-expanding value,” appear contradictory, this is only because they bring to the surface a contradiction immanent in capitalist production. At first sight it appears a strange proceeding, to equate the constant capital to zero. Yet it is what we do every day. If, for example, we wish to calculate the amount of England’s profits from the cotton industry, we first of all deduct the sums paid for cotton to the United States, India, Egypt and other countries; in other words, the value of the capital that merely re-appears in the value of the product, is put = 0. Of course the ratio of surplus value not only to that portion of the capital from which it immediately springs, and whose change of value it represents, but also to the sum total of the capital advanced is economically of very great importance. We shall, therefore, in the third book, treat of this ratio exhaustively. In order to enable one portion of a capital to expand its value by being converted into labour-power, it is necessary that another portion be converted into means of production. In order that variable capital may perform its function, constant capital must be advanced in proper proportion, a proportion given by the special technical conditions of each labour-process. The circumstance, however, that retorts and other vessels, are necessary to a chemical process, does not compel the chemist to notice them in the result of his analysis. If we look at the means of production, in their relation to the creation of value, and to the variation in the quantity of value, apart from anything else, they appear simply as the material in which 112 Chapter IX labour-power, the value-creator, incorporates itself. Neither the nature, nor the value of this material is of any importance. The only requisite is that there be a sufficient supply to absorb the labour expended in the process of production. That supply once given, the material may rise or fall in value, or even be, as land and the sea, without any value in itself; but this will have no influence on the creation of value or on the variation in the quantity of value.2 In the first place then we equate the constant capital to zero. The capital advanced is consequently reduced from c + v to v, and instead of the value of the product (c + v) + s we have now the value produced (v + s). Given the new value produced = £180, which sum consequently represents the whole labour expended during the process, then subtracting from it £90 the value of the variable capital, we have remaining £90, the amount of the surplus value. This sum of £90 or s expresses the absolute quantity of surplus value produced. The relative quantity produced, or the increase per cent of the variable capital, is determined, it is plain, by the ratio of the surplus value to the variable capital, or is expressed by s/v. In our example this ratio is 90/90, which gives an increase of 100%. This relative increase in the value of the variable capital, or the relative magnitude of the surplus value, I call, “The rate of surplus value.” 3 We have seen that the labourer, during one portion of the labour-process, produces only the value of his labour-power, that is, the value of his means of subsistence. Now since his work forms part of a system, based on the social division of labour, he does not directly produce the actual necessaries which he himself consumes; he produces instead a particular commodity, yarn for example, whose value is equal to the value of those necessaries or of the money with which they can be bought. The portion of his day’s labour devoted to this purpose, will be greater or less, in proportion to the value of the necessaries that he daily requires on an average, or, what amounts to the same thing, in proportion to the labour-time required on an average to produce them. If the value of those necessaries represent on an average the expenditure of six hours’ labour, the workman must on an average work for six hours to produce that value. If instead of working for the capitalist, he worked independently on his own account, he would, other things being equal, still be obliged to labour for the same number of hours, in order to produce the value of his labour-power, and thereby to gain the means of subsistence necessary for his conservation or continued reproduction. But as we have seen, during that portion of his day’s labour in which he produces the value of his labour-power, say three shillings, he produces only an equivalent for the value of his labour-power already advanced4 by the capitalist; the new value created only replaces the variable capital advanced. It is owing to this fact, that the production of the new value of three shillings takes the semblance of a mere reproduction. That portion of the working day, then, during which this reproduction takes place, I call “necessary” labour time, and the labour expended during that time I call “necessary” labour.5 Necessary, as regards the labourer, because independent of the particular social form of his labour; necessary, as regards capital, and the world of capitalists, because on the continued existence of the labourer depends their existence also. During the second period of the labour-process, that in which his labour is no longer necessary labour, the workman, it is true, labours, expends labour-power; but his labour, being no longer necessary labour, he creates no value for himself. He creates surplus value which, for the capitalist, has all the charms of a creation out of nothing. This portion of the working day, I name surplus labour-time, and to the labour expended during that time, I give the name of surplus labour. It is every bit as important, for a correct understanding of surplus value, to conceive it as a mere congelation of surplus labour-time, as nothing but materialised surplus labour, as it is, for a proper comprehension of value, to conceive it as a mere congelation of so many hours of labour, as nothing but materialised labour. The essential difference between the various economic forms of society, between, for instance, a society based on slave-labour, and one based on wage-labour, lies only in the mode in which this surplus labour is in each case extracted from the actual producer, the labourer.6 113 Chapter IX Since, on the one hand, the values of the variable capital and of the labour-power purchased by that capital are equal, and the value of this labour-power determines the necessary portion of the working day; and since, on the other hand, the surplus value is determined by the surplus portion of the working day, it follows that surplus value bears the same ratio to variable capital, that surplus labour does to necessary labour, or in other words, the rate of surplus value, s/v = (surplus labour)/(necessary labour). Both ratios, s/v and (surplus labour)/(necessary labour), express the same thing in different ways; in the one case by reference to materialised, incorporated labour, in the other by reference to living, fluent labour. The rate of surplus value is therefore an exact expression for the degree of exploitation of labourpower by capital, or of the labourer by the capitalist.7 We assumed in our example, that the value of the product £410 const. + £90 var. + £90 surpl., and that the capital advanced = £500. Since the surplus value = £90, and the advanced capital = £500, we should, according to the usual way of reckoning, get as the rate of surplus value (generally confounded with rate of profits) 18%, a rate so low as possibly to cause a pleasant surprise to Mr. Carey and other harmonisers. But in truth, the rate of surplus value is not equal to s/C or s/(c+v), but to s/v: thus it is not 90/500 but 90/90 or 100%, which is more than five times the apparent degree of exploitation. Although, in the case we have supposed, we are ignorant of the actual length of the working day, and of the duration in days or weeks of the labour-process, as also of the number of labourers employed, yet the rate of surplus value s/v accurately discloses to us, by means of its equivalent expression, surplus labour/necessary labour the relation between the two parts of the working day. This relation is here one of equality, the rate being 100%. Hence, it is plain, the labourer, in our example, works one half of the day for himself, the other half for the capitalist. The method of calculating the rate of surplus value is therefore, shortly, as follows. We take the total value of the product and put the constant capital which merely re-appears in it, equal to zero. What remains, is the only value that has, in the process of producing the commodity, been actually created. If the amount of surplus value be given, we have only to deduct it from this remainder, to find the variable capital. And vice versâ, if the latter be given, and we require to find the surplus value. If both be given, we have only to perform the concluding operation, viz., to calculate s/v, the ratio of the surplus value to the v variable capital. Though the method is so simple, yet it may not be amiss, by means of a few examples, to exercise the reader in the application of the novel principles underlying it. First we will take the case of a spinning mill containing 10,000 mule spindles, spinning No. 32 yarn from American cotton, and producing 1 lb. of yarn weekly per spindle. We assume the waste to be 6%: under these circumstances 10,600 lbs. of cotton are consumed weekly, of which 600 lbs. go to waste. The price of the cotton in April, 1871, was 7¾d. per lb.; the raw material therefore costs in round numbers £342. The 10,000 spindles, including preparation-machinery, and motive power, cost, we will assume, £1 per spindle, amounting to a total of £10,000. The wear and tear we put at 10%, or £1,000 yearly = £20 weekly. The rent of the building we suppose to be £300 a year, or £6 a week. Coal consumed (for 100 horse-power indicated, at 4 lbs. of coal per horse-power per hour during 60 hours, and inclusive of that consumed in heating the mill), 11 tons a week at 8s. 6d. a ton, amounts to about £4½ a week: gas, £1 a week, oil, &c., £4½ a week. Total cost of the above auxiliary materials, £10 weekly. Therefore the constant portion of the value of the week’s product is £378. Wages amount to £52 a week. The price of the yarn is 12¼d. per. lb. which gives for the value of 10,000 lbs. the sum of £510. The surplus value is therefore in this case £510 - £430 = £80. We put the constant part of the value of the product = 0, as it plays no part in the creation of value. There remains £132 as the weekly value created, which = £52 var. + £80 surpl. The rate of surplus value is therefore 80/52 = 153 11/13%. In a working day of 10 hours with average labour the result is: necessary labour = 3 31/33 hours, and surplus labour = 6 2/33.8 114 Chapter IX One more example. Jacob gives the following calculation for the year 1815. Owing to the previous adjustment of several items it is very imperfect; nevertheless for our purpose it is sufficient. In it he assumes the price of wheat to be 8s. a quarter, and the average yield per acre to be 22 bushels. VALUE PRODUCED PER ACRE Seed £1 9s. 0d. Tithes, Rates, and taxes, £1 1s. 0d. Manure £2 10s. 0d. Rent £1 8s. 0d. Wages £3 10s. 0d. Farmer’s Profit and Interest £1 2s. 0d. TOTAL £7 9s. 0d. TOTAL £3 11s 0d. Assuming that the price of the product is the same as its value, we here find the surplus value distributed under the various heads of profit, interest, rent, &c. We have nothing to do with these in detail; we simply add them together, and the sum is a surplus value of £3 11s. 0d. The sum of £3 19s. 0d., paid for seed and manure, is constant capital, and we put it equal to zero. There is left the sum of £3 10s. 0d., which is the variable capital advanced: and we see that a new value of £3 10s. 0d + £3 11s. 0d. has been produced in its place. Therefore s/v = £3 11s. 0d. / £3 10s. 0d., giving a rate of surplus value of more than 100%. The labourer employs more than one half of his working day in producing the surplus value, which different persons, under different pretexts, share amongst themselves.

[mathematicianadda]

The Big Lock-Down Math-Off, Match 12

Welcome to the 12th match in this year’s Big Math-Off. Take a look at the two interesting bits of maths below, and vote for your favourite.

You can still submit pitches, and anyone can enter: instructions are in the announcement post.

Here are today’s two pitches.

Matthew Scroggs – Interesting Tautologies

Matthew Scroggs is one of the editors of Chalkdust, a magazine for the mathematically curious, and blogs at mscroggs.co.uk. He tweets at @mscroggs.

A few years ago, I made @mathslogicbot, a Twitter bot that tweets logical tautologies.

The statements that @mathslogicbot tweets are made up of variables (a to z) that can be either true or false, and the logical symbols $\lnot$ (not) $\land$ (and), $\lor$ (or), $\rightarrow$ (implies), and $\leftrightarrow$ (if and only if), as well as brackets. A tautology is a statement that is always true, whatever values are assigned to the variables involved.

To get an idea of how to interpret @mathslogicbot’s statements, let’s have a look at a few tautologies:

$( a \rightarrow a )$. This says “$a$ implies $a$”, or in other words “if $a$ is true, then $a$ is true”. Hopefully everyone agrees that this is an always-true statement.

$( a \lor \lnot a )$. This says “$a$ or not $a$”: either $a$ is true, or $a$ is not true.

$(a\leftrightarrow a)$. This says “$a$ if and only if $a$”.

$\lnot ( a \land \lnot a )$. This says “not ($a$ and not $a$)”: $a$ and not $a$ cannot both be true.

$( \lnot a \lor \lnot \lnot a )$. I’ll leave you to think about what this one means.

(Of course, not all statements are tautologies. The statement $(b\land a)$, for example, is not a tautology as is can be true or false depending on the values of $a$ and $b$.)

While looking through @mathslogicbot’s tweets, I noticed that a few of them are interesting, but most are downright rubbish. This got me thinking: could I get rid of the bad tautologies like these, and make a list of just the “interesting” tautologies. To do this, we first need to think of different ways tautologies can be bad.

Looking at tautologies the @mathslogicbot has tweeted, I decided to exclude:

tautologies like \((a\rightarrow\lnot\lnot\lnot\lnot a)\) that contain more than one \(\lnot\) in a row.

tautologies like \(((a\lor\lnot a)\lor b)\) that contain a shorter tautology. Instead, tautologies like \((\text{True}\lor b)\) should be considered.

tautologies like \(((a\land\lnot a)\rightarrow b)\) that contain a shorter contradiction (the opposite of a tautology). Instead, tautologies like \((\text{False}\rightarrow b)\) should be considered.

tautologies like \((\text{True}\lor\lnot\text{True})\) or \(((b\land a)\lor\lnot(b\land a)\) that are another tautology (in this case \((a\lor\lnot a)\)) with a variable replaced with something else.

tautologies containing substatements like \((a\land a)\), \((a\lor a)\) or \((\text{True}\land a)\) that are equivalent to just writing \(a\).

tautologies that contain a \(\rightarrow\) that could be replaced with a \(\leftrightarrow\), because it’s more interesting if the implication goes both ways.

tautologies containing substatements like \((\lnot a\lor\lnot b)\) or \((\lnot a\land\lnot b)\) that could be replaced with similar terms (in these cases \((a\land b)\) and \((a\lor b)\) respectively) without the \(\lnot\)s.

tautologies that are repeats of each other with the order changed. For example, only one of \((a\lor\lnot a)\) and \((\lnot a\lor a)\) should be included.

After removing tautologies like these, some of my favourite tautologies are:

\(( \text{False} \rightarrow a )\)

\(( a \rightarrow ( b \rightarrow a ) )\)

\(( ( \lnot a \rightarrow a ) \leftrightarrow a )\)

\(( ( ( a \leftrightarrow b ) \land a ) \rightarrow b )\)

\(( ( ( a \rightarrow b ) \leftrightarrow a ) \rightarrow a )\)

\(( ( a \lor b ) \lor ( a \leftrightarrow b ) )\)

\(( \lnot ( ( a \land b ) \leftrightarrow a ) \rightarrow a )\)

\(( ( \lnot a \rightarrow b ) \leftrightarrow ( \lnot b \rightarrow a ) )\)

You can find a list of the first 500 “interesting” tautologies here. Let me know on Twitter which is your favourite. Or let me know which ones you think are rubbish, and we can further refine the list…

Colin Beveridge – Binet’s formula and Haskell

Colin blogs at flyingcoloursmaths.co.uk and tweets at @icecolbeveridge.

As ordained by Stigler’s law (which is attributed to Merton), Binet’s formula was known at least a century before Binet wrote about it

Binet’s formula is a lovely way to generate the \( n \)th Fibonacci number, \( F_n \). If \( \phi = \frac{1}{2}\left(\sqrt{5} + 1\right)\), then

\( F_n = \frac{ \phi^n – (-\phi)^{-n} }{\sqrt{5}}\)

Where does it come from?

Which, of course, I’ll leave as an exercise

It’s not hard to prove this by induction, but that feels like a bit of a cheat: it doesn’t explain why Binet’s formula works.

Personally, I like to prove it as follows.

Suppose \( F_{n-1} \) and \( F_{n} \) are consecutive Fibonacci numbers for some integer \( n \).

Then, if I want to end up with \( F_{n} \) and \( F_{n+1} \), I can use a matrix: \( \begin{pmatrix} F_{n} \\ F_{n+1} \end{pmatrix}= \begin{pmatrix} 0 & 1 \\ 1 & 1 \end{pmatrix} \begin{pmatrix} F_{n-1} \\ F_{n} \end{pmatrix} \)

That’s true for any \( n \), and (given that \( F_0 = 0 \) and \( F_1 = 1 \)), I can write: \( \begin{pmatrix} F_{n} \\ F_{n+1} \end{pmatrix}= \begin{pmatrix} 0 & 1 \\ 1 & 1 \end{pmatrix}^n \begin{pmatrix} 0 \\ 1 \end{pmatrix} \)

I only realised fairly recently what was going on when you diagonalise a matrix as \( \mathbf{PDP}^{-1} \). If you apply this to one of the eigenvectors, \( \mathbf{P}^{-1} \) maps it to one of the axes; \( \mathbf{D} \) stretches it by the appropriate eigenvalue; \(\mathbf{P} \) maps it back parallel to the original eigenvector – so applying the diagonalisation to the eigenvector does exactly the same as applying the matrix to it. This is true for the other eigenvector(s), too; and because of all the linear goodness buried in linear algebra, the diagonalisation maps any linear combination of the eigenvectors – which, as long as your matrix has enough eigenvectors, means any vector – the same way as the original matrix does. If you don’t have enough eigenvectors, please consult a linear algebra professional immediately.

I smell an eigensystem – if I diagonalise the matrix, I can use that as a shortcut to unlimited power, bwahaha! (What’s that? Just unlimited powers of the matrix. Fine.) I’ll skip the tedious and arduous calculation required to diagonalise it, and note that: \( \begin{pmatrix} 0 & 1 \\ 1 & 1 \end{pmatrix}^n = \frac{1}{\sqrt{5}} \begin{pmatrix} -\phi & \frac{1}{\phi} \\ 1 & 1 \end{pmatrix} \begin{pmatrix} -\frac{1}{\phi} & 0 \\ 0 & \phi \end{pmatrix}^n \begin{pmatrix} -1 & \frac{1}{\phi} \\ 1 & {\phi} \end{pmatrix} \)

This simplifies (after a bit of work) to: \(\begin{pmatrix} 0 & 1 \\ 1 & 1 \end{pmatrix}^n =\frac{1}{\sqrt{5}} \begin{pmatrix} \phi^{-(n-1)} + \phi^{n-1} & -(-\phi)^{-n} + \phi^n \\ \phi^{-n} + \phi^n & -(-\phi)^{-(n+1)} + \phi^{n+1} \end{pmatrix} \)

And

\(\begin{pmatrix} F_{n} \\ F_{n+1} \end{pmatrix} = \begin{pmatrix} 0 & 1 \\ 1 & 1 \end{pmatrix}^n \begin{pmatrix} 0 \\ 1 \end{pmatrix}= \frac{1}{\sqrt{5}} \begin{pmatrix} \phi^n – (-\phi)^{-n} \\ \phi^{n+1} – (-\phi)^{-(n+1)} \end{pmatrix} \)

… which gives us Binet’s formula \( \blacksquare \)

A brief aside

In fact, it’s very close to \( F_n \), but that’s not the interesting thing here.

A thing that makes me go oo: the numerator of Binet’s formula gives integer multiples of \( \sqrt{5} \), and because \( 0 < \phi^{-1} < 1 \), the value of \( \phi^{-n} \) gets small as \( n \) gets large. As a result, \( \frac{\phi^n}{\sqrt{5}} \) is very close to an integer for even moderately large values of \( n \).

Calculating in \( \mathbb{Q}\left(\sqrt{5}\right) \)

Which I was – the code is interesting, but not Math-Off-interesting

If you were, for example, trying to implement this in Haskell you might consider working in the field \( \mathbb{Q}\left(\sqrt{5}\right) \). You might, having considered that, run away screaming from the frightening notation – but \( \mathbb{Q}\left(\sqrt{5}\right) \) is less distressing than it looks: it’s just algebraic numbers of the form \( a + b \sqrt{5} \), where \( a \) and \( b \) are rational numbers – and addition, subtraction, multiplication and division (by anything that isn’t zero) are all defined as you would expect. You might like to verify that all of these operations leave you with another element of the field.

Another example of a field extension that works a similar way: the complex numbers can be thought of as \( \mathbb{R}(i) \) – numbers of the form \( a + bi \) where \(a \) and \(b \) are real.

In particular, \( \phi\) is an element of this field, with \( a = b = \frac{1}{2} \). Also,\( -\phi^{-1} \) is an element of the field, with \( a = \frac{1}{2} \) and \( b = – \frac{1}{2} \).

A little bit of messing with the binomial theorem tells you that calculating \( \phi^n – \left(-\phi^{-n}\right) \) leaves an element of \( \mathbb{Q}(\sqrt{5}) \) that’s purely a multiple of \( \sqrt{5} \) (it’s of the form \( a + b \sqrt{5} \), but with \( a = 0 )\) – so Binet’s formula gives us something that’s definitely rational.

Suppose, for example, we work out \( \phi^{10} \) and get \( \frac{123}{2} + \frac{55}{2}\sqrt{5} \). (It doesn’t take much more work to figure out that \( (-\phi)^{-10} = \frac{123}{2} – \frac{55}{2}\sqrt{5} \) so that Binet gives 55 as the tenth Fibonacci number.)

The oo! thing for me, though, is the other number. 123 is the tenth Lucas number – which are formed the same way as Fibonacci numbers, but with \( L_0 = 2 \) and \( L_1 = 1 \). It’s not necessarily a surprise that the Lucas and Fibonacci sequences are linked like this, but it does strike me as neat.

Links

Binet’s formula

Lucas numbers

Blazing fast Fibonacci Numbers

So, which bit of maths made you say “Aha!” the loudest? Vote:

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The poll closes at 9am BST on Tuesday the 5th, when the next match starts.

If you’ve been inspired to share your own bit of maths, look at the announcement post for how to send it in. The Big Lockdown Math-Off will keep running until we run out of pitches or we’re allowed outside again, whichever comes first.

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[callmemoprah]

Discrete Math and Its Applications Explained

It is difficult to think that how not looking at data is likely to aid you! The capacity to complete excellent research is demonstrated by the decision of a yearlong research undertaking. In the event the order doesn’t matter then we’ve got a combination, in the event the order do matter then we’ve got a permutation.

In practice, it’s a bit more involved because we wish to consider edge cases and query surrounding cells too. Consider the function which gives the scope of children of each man reading this. Check out whether it’s possible to turn the page with some arrow keys or click a specific part of the screen, apart from using the mouse to manage everything.

Provided below is a bit of additional particulars about the working mechanism of gravity in conditions of plants. If you have issues, feel free to get in touch with me by e-mail. Variational techniques for elliptic issues.

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What you will also be pleased to hear is that our professional customer service is always prepared to assist you in case you have issues with a specific link or find any other questions regarding our online services. Often students who’ve been away from school for protracted lengths of time have to refresh and rebuild their mathematical abilities and their confidence in their capacity to address challenging issues. The program determines the probability of a particular train trip being completed in time in britain uses Markov chains.

The indication of a great mathematics problem is multiple solution paths offering students the opportunity to experiment with different approaches. The problem PRIME1 might also be solved utilizing a segmented sieve to get an increased running moment. It is crucial to know the many sorts of issues which are shown on the test.

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All About Discrete Math and Its Applications

Attempt to aid students know how to approach challenging difficulties. When you are finished, you will know a vital part of mathematics and you’ll feel justifiably good about your achievement. All students have a duty to understand the requirements that apply to particular assessments and also to be mindful of acceptable academic practice about the use of material prepared by other people.

Your general homework score is going to be the median of your personal homework scores. That’s the reason why we can only play math games in class each and every day. This practice is called cross-validation in statistics.

If you must understand programming you ought to get started with this book. What’s more, the book offers tons of examples for every one among its theorems and topics. Furthermore, it provides lots of examples for every one of its theorems and topics.

Whether you’re set on getting an ebook or handbook, the decision is all yours, and there are quite a lot of alternatives for you to choose from so you don’t will need to pay a visit to another site. Truth of the problem is you are never likely in order to know all the drugs that may show up on the NCLEX. It is intended to provide students with an understanding of these areas and their use in the field of Information Technology.

Problems are randomized to reduce sharing of answers a may also be in possession of a multi-step solution which can help move the students’ learning along should they experience difficulty. custom writings All problems are randomized to stop sharing of answers and might also be in possession of a multi-step solution which will help move the college students’ learning along should they experience difficulty. They learn how to add and subtract numbers using various techniques.

What’s Really Happening with Discrete Math and Its Applications

We always make certain the links on our site work and aren’t broken in order that will help you download Discrete Mathematics Rosen 7 Edition Solution Manual pdf without any issues. A superb eBook reader ought to be set up. An amazing eBook reader ought to be set up.

In Pattern Languages there are 3 elements which make them generative. There’s another problem too. All you need to do is to learn to love its concepts and memorize the fundamentals.

To have the ability to create holistic systems one should discover or invent generative systems. With SN you may also execute all kinds of mathematical calculations through MuPad. To be true, both parts would need to be true.

The New Angle On Discrete Math and Its Applications Just Released

If you own a question, your very best alternative is to post a message to the newsgroup. The total source code can be found on GitHub. If you’re a normal user of this internet website and don’t understand exactly what this page is all about, this probably suggests that the website is currently unavailable as a result of maintenance.

Any model that is not pure white-box consists of some parameters that could be employed to fit the model to the system it’s meant to describe. At length, a good deal of information compression uses algorithms just enjoy the Fast Fourier Transform. An empty set is composed of no elements.

Additionally, any statement that’s redundant, or idempotent, is also known as a tautology, and for the exact reason mentioned before. Exogenous variables are occasionally referred to as parameters or constants. As an example, spectral methods are increasingly utilised in graph algorithms for managing massive data sets.

It will be useful to truly have a good eBook reader to be in a position to have a great reading experience and premium quality eBook display. In the other direction, it’s exceedingly desirable as a way to construct random-like graphs. Databases are at the root of the majority of applications, and they’re designed to process data in Sets.

Source: http://mobimatic.io/2019/03/07/discrete-math-and-its-applications-explained/

[theliberaltony]

via Politics – FiveThirtyEight

With Democrats having won the House but not the Senate on Tuesday — and with President Trump still in the White House — we’re headed for two years of divided government. That doesn’t mean there won’t be news, like … oh, say, the president firing the Attorney General the day after the election.

But it does mean that pretty much every political battle is going to be pitched with an eye toward 2020. And 2020 will be a unique year in that the House, Senate and presidency are all potentially in play.1 How the presidency goes is anybody’s guess. But Trump took advantage of the Electoral College last time around, winning the tipping-point state (Wisconsin) by about 1 percentage point even though he lost the popular vote by 2 percentage points. If Trump has the same edge in 2020, that could go a long way toward winning him a second term.

The thing is, though, that the Electoral College advantage is historically pretty ephemeral. Relatively subtle changes in political conditions can make the Electoral College go from helping you to hurting you. In 2008 and 2012, for example, the Electoral College worked toward Democrats’ benefit, as Barack Obama would likely have won it in the event of a popular vote tie.

So here’s some slightly scary news for Trump: The 2018 map looked more like 2012 than 2016, with Democrats performing quite well in Wisconsin, Michigan and Pennsylvania, the three states that essentially won Trump the election two years ago.

As a “fun,” day-after-the-election experiment, I decided to add up the total popular vote for the U.S. House in each state, based on ABC News’s tally of votes as of Wednesday afternoon. This isn’t a perfect exercise, by any means. The vote is still being counted in many states; there are a few dozen congressional districts where one of the parties (usually Republicans) didn’t nominate a candidate. I did make one adjustment for a slightly different problem, which is that Florida doesn’t bother to count votes in uncontested races, something which cost Democrats in the neighborhood of 720,000 votes off their popular-vote tally in that state.[footnotes]Democrats won uncontested races in Florida’s 10th, 14th, 21st and 24th congressional districts. Based on the results from uncontested congressional districts in states other than Florida on Tuesday night, I estimate that each one would have given Democrats about 180,000 votes if they’d been counted.[/footnote]

With those caveats aside, here’s the map you come up with if you count up the popular vote. It ought to look familiar. In fact, it’s the same exact map by which Obama defeated Mitt Romney in 2012, except with Ohio having gone to Republicans. It would have equated to 314 electoral votes for Democrats and 224 for the GOP.

States shaded in light blue were won by Democrats, but by fewer than 5 percentage points. So it’s noteworthy which states are not in light blue but solid blue instead. Democrats won the popular vote in Michigan by 7 percentage points, in Wisconsin by 8 points, in Pennsylvania by 10 points, and in Minnesota by 11 points. In other parts of the country, they won Nevada and Colorado by 6 points each, New Hampshire by 12, Virginia by 15 and New Mexico by 19.

The pink states — where Republicans won by fewer than 5 percentage points — are also interesting, mostly because they include Texas, where Democrats lost the popular vote for the House by only 3.6 percentage points and Democrat Beto O’Rourke lost his race for the Senate by just 2.6 points. It’s not as though Texas is exactly at the tipping point yet: Democrats came close to winning it, but they didn’t get over the top, even in a pretty blue year. But it probably deserves to be included in a group of Sunbelt states with North Carolina, Arizona and perhaps Georgia (where Democrats lost the popular vote 6 points,) as places where Democrats can compete in a good year. Among these, Arizona was the best one for Democrats on Tuesday night; they currently trail in the popular vote for the House there by 1.7 points, and could make up further ground in the state as a lot of ballots from Maricopa County are still left to be counted.

In less favorable developments for Democrats, they had very disappointing results in Ohio, where they lost the gubernatorial race, and where their candidates lost the popular vote for the House by 5.5 percentage points. Ohio hasn’t gone the way of Missouri yet, where Democratic congressional candidates lost by 13 points on Tuesday night, but it may be getting there.

But all of this is a bit tautological: Of course the map looks good for you when you’ve had a good night. How about in an average year instead, where the overall vote is fairly close? Democrats currently lead in the national popular vote for the House by around 6 percentage points, and they’re likely to run that total up to 7 or perhaps 8 percentage points as additional votes are counted, mostly from the West Coast mail-balloting states (California, Oregon, Washington). On the other hand, the Democratic margin is a bit inflated by the fact that Republicans let quite a few districts go uncontested. So let’s go ahead and subtract 6 points from the Democrat’s 2018 margin in every state; this is a benchmark for what things might have looked like in a roughly neutral year:

This is certainly not a great map for Democrats, but it’s not a bad one either. There are 217 solid Democratic electoral votes on this map, as compared to 225 solid Republican ones; the other 96 are tightly contested, but Democrats trail in Florida while narrowly leading in Wisconsin, Michigan and Pennsylvania. If 2020 were contested on this basis, you wouldn’t say that either side had a clear Electoral College advantage.

What is clear, though, is the importance of Pennsylvania, Wisconsin and Michigan (although you could also add Minnesota to the mix). Win all three of them — let’s call them the Northern Path — and Democrats don’t need Florida, assuming they hold the other states. Lose all three, and even Florida wouldn’t be enough. Instead, they’d have to win Florida plus at least one of North Carolina, Arizona, Texas and Georgia as part of what you might call a Sunbelt Strategy.

Hillary Clinton’s problem was that Trump performed well in the Northern Path states — and she didn’t campaign in them enough — but at the same time, the Sunbelt Strategy wasn’t really ripe yet. She did much better than a typical Democratic candidate in Arizona and Texas, but not enough to actually pull off wins there.

Getting stuck in between the Northern Path and the Sunbelt Strategy is a big risk for Democrats: where their Electoral College problems become most acute. And although the potential addition of Texas to the Sunbelt Strategy group of states makes it more intriguing, Tuesday night’s results suggest that the Northern Path is still the path of least resistance for a Democrat hoping to win the Electoral College. If Trump has lost the benefit of the doubt from voters in Pennsylvania, Wisconsin and Michigan, he may not have so much of an Electoral College advantage in 2020.