The idea that the Universe is a conscious mind that responds to value strikes us a ludicrously extravagant cartoon. But we must judge the view not on its cultural associations but on its explanatory power. Agentive cosmopsychism explains the fine-tuning without making false predictions; and it does so with a simplicity and elegance unmatched by its rivals. It is a view we should take seriously.
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In December of 2021 and January of 2022, two teams of physicists, one an international collaboration including researchers from the Institute for Quantum Optics and Quantum Information in Vienna and the Southern University of Science and Technology in China, and the other led by scientists at the University of Science and Technology of China (USTC), showed that a version of quantum mechanics devoid of imaginary numbers leads to a faulty description of nature. A month earlier, researchers at the University of California, Santa Barbara reconstructed a quantum wave function, another quantity that cannot be fully described by real numbers, from experimental data. In either case, physicists cajoled the very real world they study to reveal properties once so invisible as to be dubbed imaginary.
For most people the idea of a number has an association with counting. The number five may remind someone of fingers on their hand, which children often use as a counting aid, while 12 may make you think of buying eggs. For decades, scientists have held that some animals use numbers as well, exactly because many species, such as chimpanzees or dolphins, perform well in experiments that require them to count.
In their simplest mathematical formulation, imaginary numbers are square roots of negative numbers. This definition immediately leads to questioning their physical relevance: if it takes us an extra step to work out what negative numbers mean in the real world, how could we possibly visualise something that stays negative when multiplied by itself? Consider, for example, the number +4. It can be obtained by squaring either 2 or its negative counterpart -2. How could -4 ever be a square when 2 and -2 were both already determined to produce 4 when squared? Imaginary numbers offer a resolution by introducing the so-called imaginary unit i, which is the square root of -1. Now, -4 is the square of 2i or -2i, emulating the properties of +4. In this way, imaginary numbers are like a mirror image of real numbers: attaching i to any real number allows it to produce a square exactly the opposite of the one it was generating before.
Western mathematicians started grappling with imaginary numbers in earnest in the 1520s when Scipione del Ferro, a professor at the University of Bologna in Italy, set out to solve the so-called cubic equation. One version of the challenge, later referred to as the irreducible case, required taking the square root of a negative number. Going further, in his book Ars Magna (1545), meant to summarise all of algebraic knowledge of the time, the Italian astronomer Girolamo Cardano declared this variety of the cubic equation to be impossible to solve.
In physics, however, the oddness of imaginary numbers was disregarded in favour of their usefulness. For instance, imaginary numbers can be used to describe opposition to changes in current within an electrical circuit. They are also used to model some oscillations, such as those found in grandfather clocks, where pendulums swing back and forth despite friction. Imaginary numbers are necessary in many equations pertaining to waves, be they vibrations of a plucked guitar string or undulations of water along a coast. And these numbers hide within mathematical functions of sine and cosine, familiar to many high-school trigonometry students.
At the same time, in all these cases imaginary numbers are used as more of a bookkeeping device than a stand-in for some fundamental part of physical reality. Measurement devices such as clocks or scales have never been known to display imaginary values. Physicists typically separate equations that contain imaginary numbers from those that do not. Then, they draw some set of conclusions from each, treating the infamous i as no more than an index or an extra label that helps organise this deductive process. Unless the physicist in question is confronted with the tiny and cold world of quantum mechanics.
Quantum theory predicts the physical behaviour of objects that are either very small, such as electrons that make up electric currents in every wire in your home, or millions of times colder than the insides of your fridge. And it is chock-full of complex and imaginary numbers.
In stark contrast to theories concerning electricity and oscillations, in quantum mechanics a physicist cannot look at an equation that involves imaginary numbers, extract a useful punchline, then forget all about them. When you set out to try and capture a quantum state in the language of mathematics, these seemingly impossible square roots of negative numbers are an integral part of your vocabulary. Eliminating imaginary numbers would highly limit how accurate of a statement you could make.
Historically, real quantum mechanics has had not only proponents but also some successes in the realm of mathematical proofs and investigations. Theorists have been able to show that certain properties of quantum-mechanical systems can indeed be captured without resorting to imaginarity. Within the last year, however, a new crop of proofs and experiments proved that this line of reasoning can only go so far. Laboratory experiments involving quantum computers and quantised light now strongly indicate that imaginary and complex numbers are an indispensable part of the quantum, and therefore our own, world.
In either case, the outcome of the game was impossible to predict accurately by any version of quantum physics that renounced complex numbers. Not only did physicists infer that imaginary numbers can indeed show up in experiments, but that, even more strikingly, they had to be considered in order for experiments in the quantum realm to be understood correctly at all.
Barbara acknowledges that the independent personal drive to learn creates a desire to fulfill the need of knowledge. Armstrong complements this though by asserting that the gift of memory is very key to fulfilling these needs. He further explains that the desire to learn and understand your environment constitute the reasons for all education that are perception, thought, and communication. These reasons can be effectively achieved by use of the three basic skills in education which are: finding what you want, fixing it in your mind, and organizing it for use (Armstrong 19). Barbara then explains the importance of a teacher in aiding a learner to refine these skills and focus them for effective use. To further affirm the importance of teachers in a students learning, Barbara says, No one has the expertise to be all things to their children on issues relating to learning She further explains that this is the reason for the existence of piano, dance, soccer, and teachers to help learners. "Getting good at math," she adds involves a little bit of practice on an everyday basis. It is this daily practice that characterizes the learners with a self-driven desire to learn.
'Tis hard to say, if greater want of skillAppear in writing or in judging ill;But, of the two, less dang'rous is th' offenceTo tire our patience, than mislead our sense.Some few in that, but numbers err in this,Ten censure wrong for one who writes amiss;A fool might once himself alone expose,Now one in verse makes many more in prose. 'Tis with our judgments as our watches, noneGo just alike, yet each believes his own.In poets as true genius is but rare,True taste as seldom is the critic's share;Both must alike from Heav'n derive their light,These born to judge, as well as those to write.Let such teach others who themselves excel,And censure freely who have written well.Authors are partial to their wit, 'tis true,But are not critics to their judgment too? Yet if we look more closely we shall findMost have the seeds of judgment in their mind;Nature affords at least a glimm'ring light;The lines, tho' touch'd but faintly, are drawn right.But as the slightest sketch, if justly trac'd,Is by ill colouring but the more disgrac'd,So by false learning is good sense defac'd;Some are bewilder'd in the maze of schools,And some made coxcombs Nature meant but fools.In search of wit these lose their common sense,And then turn critics in their own defence:Each burns alike, who can, or cannot write,Or with a rival's, or an eunuch's spite.All fools have still an itching to deride,And fain would be upon the laughing side.If Mævius scribble in Apollo's spite,There are, who judge still worse than he can write. Some have at first for wits, then poets pass'd,Turn'd critics next, and prov'd plain fools at last;Some neither can for wits nor critics pass,As heavy mules are neither horse nor ass.Those half-learn'd witlings, num'rous in our isleAs half-form'd insects on the banks of Nile;Unfinish'd things, one knows not what to call,Their generation's so equivocal:To tell 'em, would a hundred tongues require,Or one vain wit's, that might a hundred tire. But you who seek to give and merit fame,And justly bear a critic's noble name,Be sure your self and your own reach to know,How far your genius, taste, and learning go;Launch not beyond your depth, but be discreet,And mark that point where sense and dulness meet. Nature to all things fix'd the limits fit,And wisely curb'd proud man's pretending wit:As on the land while here the ocean gains,In other parts it leaves wide sandy plains;Thus in the soul while memory prevails,The solid pow'r of understanding fails;Where beams of warm imagination play,The memory's soft figures melt away.One science only will one genius fit;So vast is art, so narrow human wit:Not only bounded to peculiar arts,But oft in those, confin'd to single parts.Like kings we lose the conquests gain'd before,By vain ambition still to make them more;Each might his sev'ral province well command,Would all but stoop to what they understand. First follow NATURE, and your judgment frameBy her just standard, which is still the same:Unerring Nature, still divinely bright,One clear, unchang'd, and universal light,Life, force, and beauty, must to all impart,At once the source, and end, and test of art.Art from that fund each just supply provides,Works without show, and without pomp presides:In some fair body thus th' informing soulWith spirits feeds, with vigour fills the whole,Each motion guides, and ev'ry nerve sustains;Itself unseen, but in th' effects, remains.Some, to whom Heav'n in wit has been profuse,Want as much more, to turn it to its use;For wit and judgment often are at strife,Though meant each other's aid, like man and wife.'Tis more to guide, than spur the Muse's steed;Restrain his fury, than provoke his speed;The winged courser, like a gen'rous horse,Shows most true mettle when you check his course. Those RULES of old discover'd, not devis'd,Are Nature still, but Nature methodis'd;Nature, like liberty, is but restrain'dBy the same laws which first herself ordain'd. Hear how learn'd Greece her useful rules indites,When to repress, and when indulge our flights:High on Parnassus' top her sons she show'd,And pointed out those arduous paths they trod;Held from afar, aloft, th' immortal prize,And urg'd the rest by equal steps to rise.Just precepts thus from great examples giv'n,She drew from them what they deriv'd from Heav'n.The gen'rous critic fann'd the poet's fire,And taught the world with reason to admire.Then criticism the Muse's handmaid prov'd,To dress her charms, and make her more belov'd;But following wits from that intention stray'd;Who could not win the mistress, woo'd the maid;Against the poets their own arms they turn'd,Sure to hate most the men from whom they learn'd.So modern 'pothecaries, taught the artBy doctor's bills to play the doctor's part,Bold in the practice of mistaken rules,Prescribe, apply, and call their masters fools.Some on the leaves of ancient authors prey,Nor time nor moths e'er spoil'd so much as they:Some drily plain, without invention's aid,Write dull receipts how poems may be made:These leave the sense, their learning to display,And those explain the meaning quite away. You then whose judgment the right course would steer,Know well each ANCIENT'S proper character;His fable, subject, scope in ev'ry page;Religion, country, genius of his age:Without all these at once before your eyes,Cavil you may, but never criticise.Be Homer's works your study and delight,Read them by day, and meditate by night;Thence form your judgment, thence your maxims bring,And trace the Muses upward to their spring;Still with itself compar'd, his text peruse;And let your comment be the Mantuan Muse. When first young Maro in his boundless mindA work t' outlast immortal Rome design'd,Perhaps he seem'd above the critic's law,And but from Nature's fountains scorn'd to draw:But when t' examine ev'ry part he came,Nature and Homer were, he found, the same.Convinc'd, amaz'd, he checks the bold design,And rules as strict his labour'd work confine,As if the Stagirite o'erlook'd each line.Learn hence for ancient rules a just esteem;To copy nature is to copy them. Some beauties yet, no precepts can declare,For there's a happiness as well as care.Music resembles poetry, in eachAre nameless graces which no methods teach,And which a master-hand alone can reach.If, where the rules not far enough extend,(Since rules were made but to promote their end)Some lucky LICENCE answers to the fullTh' intent propos'd, that licence is a rule.Thus Pegasus, a nearer way to take,May boldly deviate from the common track.Great wits sometimes may gloriously offend,And rise to faults true critics dare not mend;From vulgar bounds with brave disorder part,And snatch a grace beyond the reach of art,Which, without passing through the judgment, gainsThe heart, and all its end at once attains.In prospects, thus, some objects please our eyes,Which out of nature's common order rise,The shapeless rock, or hanging precipice.But tho' the ancients thus their rules invade,(As kings dispense with laws themselves have made)Moderns, beware! or if you must offendAgainst the precept, ne'er transgress its end;Let it be seldom, and compell'd by need,And have, at least, their precedent to plead.The critic else proceeds without remorse,Seizes your fame, and puts his laws in force. I know there are, to whose presumptuous thoughtsThose freer beauties, ev'n in them, seem faults.Some figures monstrous and misshap'd appear,Consider'd singly, or beheld too near,Which, but proportion'd to their light, or place,Due distance reconciles to form and grace.A prudent chief not always must displayHis pow'rs in equal ranks, and fair array,But with th' occasion and the place comply,Conceal his force, nay seem sometimes to fly.Those oft are stratagems which errors seem,Nor is it Homer nods, but we that dream. Still green with bays each ancient altar stands,Above the reach of sacrilegious hands,Secure from flames, from envy's fiercer rage,Destructive war, and all-involving age.See, from each clime the learn'd their incense bring!Hear, in all tongues consenting pæans ring!In praise so just let ev'ry voice be join'd,And fill the gen'ral chorus of mankind!Hail, bards triumphant! born in happier days;Immortal heirs of universal praise!Whose honours with increase of ages grow,As streams roll down, enlarging as they flow!Nations unborn your mighty names shall sound,And worlds applaud that must not yet be found!Oh may some spark of your celestial fireThe last, the meanest of your sons inspire,(That on weak wings, from far, pursues your flights;Glows while he reads, but trembles as he writes)To teach vain wits a science little known,T' admire superior sense, and doubt their own! 2ff7e9595c
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