Additional Mathematics Quotes

Infinite is a meaningless word: except – it states / The mind is capable of performing / an endless process of addition.

The sciences do not try to explain, they hardly even try to interpret, they mainly make models. By a model is meant a mathematical construct which, with the addition of certain verbal interpretations, describes observed phenomena. The justification of such a mathematical construct is solely and precisely that it is expected to work - that is correctly to describe phenomena from a reasonably wide area. Furthermore, it must satisfy certain esthetic criteria - that is, in relation to how much it describes, it must be rather simple.

Numbers should always add up and reconcile but the only thing that can be added up and reconciled with numbers are numbers.

...whatever we do whatever way we move forward there will be damage carried from all previous rows and columns in this mathematical computation we make, all the multi-configured additions and subtractions in language America, in all bases, particularly I'm thinking about at the moment 1492, 1776, 1861, 1867, 1980, 2016, 2020.

People enjoy inventing slogans which violate basic arithmetic but which illustrate “deeper” truths, such as “1 and 1 make 1” (for lovers), or “1 plus 1 plus 1 equals 1” (the Trinity). You can easily pick holes in those slogans, showing why, for instance, using the plus-sign is inappropriate in both cases. But such cases proliferate. Two raindrops running down a window-pane merge; does one plus one make one? A cloud breaks up into two clouds -more evidence of the same? It is not at all easy to draw a sharp line between cases where what is happening could be called “addition”, and where some other word is wanted. If you think about the question, you will probably come up with some criterion involving separation of the objects in space, and making sure each one is clearly distinguishable from all the others. But then how could one count ideas? Or the number of gases comprising the atmosphere? Somewhere, if you try to look it up, you can probably fin a statement such as, “There are 17 languages in India, and 462 dialects.” There is something strange about the precise statements like that, when the concepts “language” and “dialect” are themselves fuzzy.

It was mathematical, marriage. Not, as one might expect, additional. It was exponential.

So a)To what extent might human relationships be expressed in a mathematical or logical formula? And b) If so, what signs might be placed between the integers? Plus and minus, self- evidently; sometimes multiplication, and yes. division. But these signs are limited. Thus an entirely failed relationship might be expressed in terms of both loss/minus and division/ reduction, showing a total zero; whereas an entirely successful one can be represented by both addition and multiplication. But what of most relationships? Do they not require to be expressed in notations which are logically insoluble?

As to the need of improvement there can be no question whilst the reign of Euclid continues. My own idea of a useful course is to begin with arithmetic, and then not Euclid but algebra. Next, not Euclid, but practical geometry, solid as well as plane; not demonstration, but to make acquaintance. Then not Euclid, but elementary vectors, conjoined with algebra, and applied to geometry. Addition first; then the scalar product. Elementary calculus should go on simultaneously, and come into vector algebraic geometry after a bit. Euclid might be an extra course for learned men, like Homer...

The sciences do not try to explain, they hardly even try to interpret, they mainly make models. By a model is meant a mathematical construct which, with the addition of certain verbal interpretations, describes observed phenomena. The justification of such a mathematical construct is solely and precisely that it is expected to work.

Peano. He showed that the entire theory of the natural numbers could be derived from three primitive ideas and five primitive propositions in addition to those of pure logic.

A3 0 is an additive identity: 0 + a = a for any number a. That is all you need to know about 0. Not what it means – just a little rule that tells you what it does.

Success is the circle, hard work is the perimeter of it. the diameter of success is the addition of radius and radius of success is the failures of it

The impulse to all movement and all form is given by [the golden ratio], since it is the proportion that summarizes in itself the additive and the geometric, or logarithmic, series.

One must know addition and subtraction (material knowledge) before proceeding to higher mathematics (spiritual knowledge). Without knowledge, life does not blossom and a person remains in raja guna and tama guna.

There are two moments in the course of education where a lot of kids fall off the math train. The first comes in the elementary grades, when fractions are introduced. Until that moment, a number is a natural number, one of the figures 0, 1, 2, 3 . . . It is the answer to a question of the form “how many.”* To go from this notion, so primitive that many animals are said to understand it, to the radically broader idea that a number can mean “what portion of,” is a drastic philosophical shift. (“God made the natural numbers,” the nineteenth-century algebraist Leopold Kronecker famously said, “and all the rest is the work of man.”) The second dangerous twist in the track is algebra. Why is it so hard? Because, until algebra shows up, you’re doing numerical computations in a straightforwardly algorithmic way. You dump some numbers into the addition box, or the multiplication box, or even, in traditionally minded schools, the long-division box, you turn the crank, and you report what comes out the other side. Algebra is different. It’s computation backward. When you’re asked to solve

One can be enlightened about proofs as well as theorems. Without enlightenment, one is merely reduced to memorizing proofs. With enlightenment about a proof, its flow becomes clear and it can become an item of astonishing beauty. In addition, the need to memorize disappears because the proof has become part of your soul.

5.4 The question of accumulation. If life is a wager, what form does it take? At the racetrack, an accumulator is a bet which rolls on profits from the success of one of the horse to engross the stake on the next one. 5.5 So a) To what extent might human relationships be expressed in a mathematical or logical formula? And b) If so, what signs might be placed between the integers?Plus and minus, self-evidently; sometimes multiplication, and yes, division. But these sings are limited. Thus an entirely failed relationship might be expressed in terms of both loss/minus and division/ reduction, showing a total of zero; whereas an entirely successful one can be represented by both addition and multiplication. But what of most relationships? Do they not require to be expressed in notations which are logically improbable and mathematically insoluble? 5.6 Thus how might you express an accumulation containing the integers b, b, a (to the first), a (to the second), s, v? B = s - v (*/+) a (to the first) Or a (to the second) + v + a (to the first) x s = b 5.7 Or is that the wrong way to put the question and express the accumulation? Is the application of logic to the human condition in and of itself self-defeating? What becomes of a chain of argument when the links are made of different metals, each with a separate frangibility? 5.8 Or is "link" a false metaphor? 5.9 But allowing that is not, if a link breaks, wherein lies the responsibility for such breaking? On the links immediately on the other side, or on the whole chain? But what do you mean by "the whole chain"? How far do the limits of responsibility extend? 6.0 Or we might try to draw the responsibility more narrowly and apportion it more exactly. And not use equations and integers but instead express matters in the traditional narrative terminology. So, for instance, if...." - Adrian Finn

Eleven days ago I coined the expression [Solar Normalized Resonant Frequency of the GPG's Height] after I have demonstrated mathematically the valid link between time and space on the Giza Plateau. But now, the revelation is even more glorious after I have shown the physical link between time and space on that area; in addition to confirming the existence of the normalizing factor.

It as mathematical, marriage, not, as one might expect, additional; it was exponential. This one man, nervous in a suite a size too small for his long, lean self, this woman, in a green lace dress cut to the upper thigh, with a white rose behind her ear. Christ, so young. The woman before them was a unitarian minister, and on her buzzed scalp, the grey hairs shone in a swab of sun through the lace in the window. Outside, Poughkeepsie was waking. Behind them, a man in a custodian's uniform cried softly beside a man in pajamas with a Dachshund, their witnesses, a shine in everyone's eye. One could taste the love on the air, or maybe that was sex, or maybe that was all the same then. 'I do,' she said. 'I do,' he said. They did. They would. Our children will be so fucking beautiful, he thought, looking at her. Home, she thought, looking at him. 'You may kiss,' said the officiant. They did, would. Now they thanked everyone and laughed, and papers were signed and congratulations offered, and all stood for a moment, unwilling to leave this gentile living room where there was such softness. The newlyweds thanked everyone again, shyly, and went out the door into the cool morning. They laughed, rosy. In they'd come integers, out they came, squared. Her life, in the window, the parakeet, scrap of blue midday in the London dusk, ages away from what had been most deeply lived. Day on a rocky beach, creatures in the tide pool. All those ordinary afternoons, listening to footsteps in the beams of the house, and knowing the feeling behind them. Because it was so true, more than the highlights and the bright events, it was in the daily where she'd found life. The hundreds of time she'd dug in her garden, each time the satisfying chew of spade through soil, so often that this action, the pressure and release and rich dirt smell delineated the warmth she'd felt in the cherry orchard. Or this, each day they woke in the same place, her husband waking her with a cup of coffee, the cream still swirling into the black. Almost unremarked upon this kindness, he would kiss her on the crown of her head before leaving, and she'd feel something in her rising in her body to meet him. These silent intimacies made their marriage, not the ceremonies or parties or opening nights or occasions, or spectacular fucks. Anyway, that part was finished. A pity...

By our very nature, we humans are linear thinkers. We evolved to estimate a distance from the predator or to the prey, and advanced mathematics is only a recent evolutionary addition. This is why it’s so difficult even for a modern man to grasp the power of exponentials. 40 steps in linear progression is just 40 steps away; 40 steps in exponential progression is a cool trillion (with a T) – it will take you 3 times from Earth to the Sun and back to Earth.

Note that a rotation by 360 degrees is equivalent to doing nothing at all, or rotating by zero degrees. This is known as the identity transformation. Why bother to define such a transformation at all? As we shall see later in the book, the identity transformation plays a similar role to that of the number zero in the arithmetic operation of addition or the number one in multiplication-when you add zero to a number or multiply a number by one, the number remains unchanged.

More generally, we underestimate the share of randomness in about everything, a point that may not merit a book—except when it is the specialist who is the fool of all fools. Disturbingly, science has only recently been able to handle randomness (the growth in available information has been exceeded only by the expansion of noise). Probability theory is a young arrival in mathematics; probability applied to practice is almost nonexistent as a discipline. In addition we seem to have evidence that what is called “courage” comes from an underestimation of the share of randomness in things rather than the more noble ability to stick one’s neck out for a given belief. In my experience (and in the scientific literature), economic “risk takers” are rather the victims of delusions (leading to overoptimism and overconfidence with their underestimation of possible adverse outcomes) than the opposite. Their “risk taking” is frequently randomness foolishness.

Economics is haunted by more fallacies than any other study known to man. This is no accident. The inherent difficulties of the subject would be great enough in any case, but they are multiplied a thousandfold by a factor that is insignificant in, say, physics, mathematics or medicine - the special pleading of selfish interests. While every group has certain economic interests identical with those of all groups, every group has also, as we shall see, interests antagonistic to those of all other groups. While certain public policies would in the long run benefit everybody, other policies would benefit one group only at the expense of all other groups. The group that would benefit by such policies, having such a direct interest in them, will argue for them plausibly and persistently. It will hire the best buyable minds to devote their whole time to presenting its case. And it will finally either convince the general public that its case is sound, or so befuddle it that clear thinking on the subject becomes next to impossible. In addition to these endless pleadings of self-interest, there is a second main factor that spawns new economic fallacies every day. This is the persistent tendency of man to see only the immediate effects of a given policy, or its effects only on a special group, and to neglect to inquire what the long-run effects of that policy will be not only on that special group but on all groups. It is the fallacy of overlooking secondary consequences.

There is no doubt that Earth Central, the planetary and sector AIs, and even some ship and drone AIs are capable, without acquiring additional processing space, of setting up synergetic systems within themselves that result in an exponential climb in intelligence (mathematically defined as climbing beyond all known scales within minutes). So why not? Ask then why a human, capable of learning verbatim the complete works of Shakespeare, instead drinks a bottle of brandy, then giggles a lot and falls over.

Regular expressions are widely used for string matching. Although regular-expression systems are derived from a perfectly good mathematical formalism, the particular choices made by implementers to expand the formalism into useful software systems are often disastrous: the quotation conventions adopted are highly irregular; the egregious misuse of parentheses, both for grouping and for backward reference, is a miracle to behold. In addition, attempts to increase the expressive power and address shortcomings of earlier designs have led to a proliferation of incompatible derivative languages.

I don’t know to what extent ignorance of science and mathematics contributed to the decline of ancient Athens, but I know that the consequences of scientific illiteracy are far more dangerous in our time than in any that has come before. It’s perilous and foolhardy for the average citizen to remain ignorant about global warming, say, or ozone depletion, air pollution, toxic and radioactive wastes, acid rain, topsoil erosion, tropical deforestation, exponential population growth. Jobs and wages depend on science and technology. If our nation can’t manufacture, at high quality and low price, products people want to buy, then industries will continue to drift away and transfer a little more prosperity to other parts of the world. Consider the social ramifications of fission and fusion power, supercomputers, data “highways,” abortion, radon, massive reductions in strategic weapons, addiction, government eavesdropping on the lives of its citizens, high-resolution TV, airline and airport safety, fetal tissue transplants, health costs, food additives, drugs to ameliorate mania or depression or schizophrenia, animal rights, superconductivity, morning-after pills, alleged hereditary antisocial predispositions, space stations, going to Mars, finding cures for AIDS and cancer.

A critical step was made sometime before the ninth century AD, when a new partial script was invented, one that could store and process mathematical data with unprecedented efficiency. This partial script was composed of ten signs, representing the numbers from 0 to 9. Confusingly, these signs are known as Arabic numerals even though they were first invented by the Hindus (even more confusingly, modern Arabs use a set of digits that look quite different from Western ones). But the Arabs get the credit because when they invaded India they encountered the system, understood its usefulness, refined it, and spread it through the Middle East and then to Europe. When several other signs were later added to the Arab numerals (such as the signs for addition, subtraction and multiplication), the basis of modern mathematical notation came into being.

On my fourth day in the sick quarters I had just been detailed to the night shift when the chief doctor rushed in and asked me to volunteer for medical duties in another camp containing typhus patients. Against the urgent advice of my friends (and despite the fact that almost none of my colleagues offered their services), I decided to volunteer. I knew that in a working party I would die in a short time. But if I had to die there might at least be some sense in my death. I thought that it would doubtless be more to the purpose to try and help my comrades as a doctor than to vegetate or finally lose my life as the unproductive laborer that I was then. For me this was simple mathematics, not sacrifice. But secretly, the warrant officer from the sanitation squad had ordered that the two doctors who had volunteered for the typhus camp should be “taken care of” till they left. We looked so weak that he feared that he might have two additional corpses on his hands, rather than two doctors.

There are two moments in the course of education where a lot of kids fall off the math train. The first comes in the elementary grades, when fractions are introduced. Until that moment, a number is a natural number, one of the figures 0, 1, 2, 3 . . . It is the answer to a question of the form “how many.”* To go from this notion, so primitive that many animals are said to understand it, to the radically broader idea that a number can mean “what portion of,” is a drastic philosophical shift. (“God made the natural numbers,” the nineteenth-century algebraist Leopold Kronecker famously said, “and all the rest is the work of man.”) The second dangerous twist in the track is algebra. Why is it so hard? Because, until algebra shows up, you’re doing numerical computations in a straightforwardly algorithmic way. You dump some numbers into the addition box, or the multiplication box, or even, in traditionally minded schools, the long-division box, you turn the crank, and you report what comes out the other side. Algebra is different. It’s computation backward.

Now, if on the one hand it is very satisfactory to be able to give a common ground in the theory of knowledge for the many varieties of statements concerning space, spatial configurations, and spatial relations which, taken together, constitute geometry, it must on the other hand be emphasised that this demonstrates very clearly with what little right mathematics may claim to expose the intuitional nature of space. Geometry contains no trace of that which makes the space of intuition what it is in virtue of its own entirely distinctive qualities which are not shared by “states of addition-machines” and “gas-mixtures” and “systems of solutions of linear equations”. It is left to metaphysics to make this “comprehensible” or indeed to show why and in what sense it is incomprehensible. We as mathematicians have reason to be proud of the wonderful insight into the knowledge of space which we gain, but, at the same time, we must recognise with humility that our conceptual theories enable us to grasp only one aspect of the nature of space, that which, moreover, is most formal and superficial.

The properties that define a group are: 1. Closure. The offspring of any two members combined by the operation must itself be a member. In the group of integers, the sum of any two integers is also an integer (e.g., 3 + 5 = 8). 2. Associativity. The operation must be associative-when combining (by the operation) three ordered members, you may combine any two of them first, and the result is the same, unaffected by the way they are bracketed. Addition, for instance, is associative: (5 + 7) + 13 = 25 and 5 + (7 + 13) = 25, where the parentheses, the "punctuation marks" of mathematics, indicate which pair you add first. 3. Identity element. The group has to contain an identity element such that when combined with any member, it leaves the member unchanged. In the group of integers, the identity element is the number zero. For example, 0 + 3 = 3 + 0 = 3. 4. Inverse. For every member in the group there must exist an inverse. When a member is combined with its inverse, it gives the identity element. For the integers, the inverse of any number is the number of the same absolute value, but with the opposite sign: e.g., the inverse of 4 is -4 and the inverse of -4 is 4; 4 + (-4) = 0 and (-4) + 4 = 0. The fact that this simple definition can lead to a theory that embraces and unifies all the symmetries of our world continues to amaze even mathematicians.

The conception of substance, which is fundamental in the philosophies of Descartes, Spinoza, and Leibniz, is derived from the logical category of subject and predicate. Some words can be either subjects or predicates; e.g. I can say 'the sky is blue' and 'blue is a colour'. Other words—of which proper names are the most obvious instances—can never occur as predicates, but only as subjects, or as one of the terms of a relation. Such words are held to designate substances. Substances, in addition to this logical characteristic, persist through time, unless destroyed by God's omnipotence (which, one gathers, never happens). Every true proposition is either general, like 'all men are mortal', in which case it states that one predicate implies another, or particular, like 'Socrates is mortal', in which case the predicate is contained in the subject, and the quality denoted by the predicate is part of the notion of the substance denoted by the subject. Whatever happens to Socrates can be asserted in a sentence in which 'Socrates' is the subject and the words describing the happening in question are the predicate. All these predicates put together make up the 'notion' of Socrates. All belong to him necessarily, in this sense, that a substance of which they could not be truly asserted would not be Socrates, but some one else. Leibniz was a firm believer in the importance of logic, not only in its own sphere, but as the basis of metaphysics. He did work on mathematical logic which would have been

Underlying the doctrines which disregard the radical novelty of each moment of evolution there are many misunderstandings, many errors. But there is especially the idea that the possible is less than the real, and that, for this reason, the possibility of things precedes their existence. They would thus be capable of representation beforehand; they could be thought of before being realised. But it is the reverse that is true. If we leave aside the closed systems, subjected to purely mathematical laws, isolable because duration does not act upon them, if we consider the totality of concrete reality or simply the world of life, and still more that of consciousness, we find there is more and not less in the possibility of each of the successive states than in their reality. For the possible is only the real with the addition of an act of mind which throws its image back into the past, once it has been enacted. But that is what our intellectual habits prevent us from seeing.

One thing that we conclude from all this is that the 'learning robot' procedure for doing mathematics is not the procedure that actually underlies human understanding of mathematics. In any case, such bottom-up-dominated procedure would appear to be hopelessly bad for any practical proposal for the construction of a mathematics-performing robot, even one having no pretensions whatever for simulating the actual understandings possessed by a human mathematician. As stated earlier, bottom-up learning procedures by themselves are not effective for the unassailable establishing of mathematical truths. If one is to envisage some computational system for producing unassailable mathematical results, it would be far more efficient to have the system constructed according to top-down principles (at least as regards the 'unassailable' aspects of its assertions; for exploratory purposes, bottom-up procedures might well be appropriate). The soundness and effectiveness of these top-down procedures would have to be part of the initial human input, where human understanding an insight provide the necesssary additional ingredients that pure computation is unable to achieve. In fact, computers are not infrequently employed in mathematical arguments, nowadays, in this kind of way. The most famous example was the computer-assisted proof, by Kenneth Appel and Wolfgang Haken, of the four-colour theorem, as referred to above. The role of the computer, in this case, was to carry out a clearly specified computation that ran through a very large but finite number of alternative possibilities, the elimination of which had been shown (by the human mathematicians) to lead to a general proof of the needed result. There are other examples of such computer-assisted proofs and nowadays complicated algebra, in addition to numerical computation, is frequently carried out by computer. Again it is human understanding that has supplied the rules and it is a strictly top-down action that governs the computer's activity.

This partial script was composed of ten signs, representing the numbers from 0 to 9. Confusingly, these signs are known as Arabic numerals even though they were first invented by the Hindus (even more confusingly, modern Arabs use a set of digits that look quite different from Western ones). But the Arabs get the credit because when they invaded India they encountered the system, understood its usefulness, refined it, and spread it through the Middle East and then to Europe. When several other signs were later added to the Arab numerals (such as the signs for addition, subtraction and multiplication), the basis of modern mathematical notation came into being.

Strictly used, concepts are (a) acquired dispositions to recognize perceived objects as being of this kind or of that kind, and at the same time (b) to understand what this kind or that kind of object is like, and consequently (c) to perceive a number of perceived particulars as being the same in kind and to discriminate between them and other sensible particulars that are different in kind. In addition, concepts are acquired dispositions to understand what certain kinds of objects are like both (a) when the objects, though perceptible, are not actually perceived, and (b) also when they are not perceptible at all, as is the case with all the conceptual constructs we employ in physics, mathematics, and metaphysics.

Mother nature is a master mathematician specialising in addition and subtraction. Our lives run linearly, a series of integer numbers alternating between two poles to opposite ends of the spectrum through our lives, making a journal of our time. Mankind tries to quantify these events for easier understanding, study for future prevention or record keeping. Oftentimes they're events that mother nature throws at us which cannot be enumerated or fit on mathematical scales. These are events that cause big shifts but are still incomprehensible. They remain an enigma to us and requires an inner understanding that's different in each and every person. True human grit is to soak ourselves in each moment on separate points of the spectrum either for good or worse and knowing there's no other way except through the centre of every singular moment Real strength comes as we accept the chapters as they're and keeping the long-term outlook of our feelings constant to the extreme right pole despite fluctuations from events. So until subtractions exceeds the left pole we'll meet.

Wealth is where history shows up in your wallet, where your financial freedom is determined by compounding interest on decisions made long before you were born. That is why the Black-white wealth gap is growing despite gains in Black education and earnings, and why the typical Black household owns only $17,600 in assets. Still, having little to no intergenerational wealth and facing massive systemic barriers, descendants of a stolen people have given America the touch-tone telephone, the carbon filament in the lightbulb, the gas mask, the modern traffic light, blood banks, the gas furnace, open-heart surgery, and the mathematics to enable the moon landing. Just imagine the possibilities if—in addition to rebuilding the pathways for all aspirants to the American Dream—we gave millions more Black Americans the life-changing freedom that a modest amount of wealth affords. A 2020 Citigroup report calculated that “if racial gaps for Blacks had been closed 20 years ago, U.S. GDP could have benefitted by an estimated $16 trillion.

The process by which [a logical theorem] is deduced shows us that it does not differ essentially from mathematical theorems; it is only more general, e.g. in the same sense that “addition of integers is commutative” is a more general statement than “2 + 3 = 3 + 2”. This is the case for every logical theorem: it is but a mathematical theorem of extreme generality; that is to say, logic is a part of mathematics, and can by no means serve as a foundation for it.

Brouwer, the founder of intuitionist mathematics, has shown that in certain mathematical problems dealing with infinite sets of numbers the elementary rule of the excluded middle is not admissible, without an additional arbitrary assumption. Statements like: there is a number . . . and: there is no number . . . , in this case only seemingly, by virtue of their abbreviated linguistic formulations, have the form of contradictory opposites.

The Brouwerian believed that this conception was wholly wrong from the beginning. They accused it of misunderstanding the nature of mathematics and of unjustifiedly transferring to the realm of infinity methods of reasoning that are valid only in the realm of the finite. By regaining the right perspective, mathematics could be constructed on a basis whose intuitive soundness could not be doubted. The antinomies were only the symptoms of a disease by which mathematics was infected. Once this disease was cured, one need worry no longer about the symptoms. All Russellians thought that our naiveness consisted in taking for granted that every grammatically correct indicative sentence expresses something which either is or is not the case, and some — among them Russell himself — believed, in addition, that through some carelessness a certain type of viciously circular concept formation had been allowed to enter logico-mathematical thinking. By restricting the language — and proscribing the dangerous types of concept formation— the known antinomies could be made to disappear. Their faith in the consistency of the resulting, somewhat mutilated, systems was less strong than that of the Brouwerians, since certain intuitively not too well founded devices had to be used in order to restore at least part of the lost strength and maneuverability. Zermelians, finally, thought that our blunder consisted in naively assuming that to every condition there must correspond a certain entity, namely the set of all those objects that satisfy this condition. By suitable restriction of the axiom of comprehension, in which this assumption is formulated, they tried to construct systems which were free of the known antinomies yet strong enough to allow for the reconstruction of a sufficient part of classical mathematics.

IRCC Announces Eligible Programs for PGWPs Immigration, Refugees and Citizenship Canada (IRCC) has updated its guidelines regarding the programs eligible for a Post-Graduation Work Permit (PGWP). As of November 1, international graduates applying for a PGWP must meet additional field of study requirements to qualify for this essential work permit. Eligible Fields of Study for PGWPs The eligible fields of study for the PGWP correspond to the occupation-based Express Entry categories introduced by IRCC in 2023. These categories are aligned with national labor market demands and include the following: • Agriculture and Agri-Food • Healthcare • Science, Technology, Engineering, and Mathematics (STEM) • Trade • Transport Eligible programs in these fields are classified using the Classification of Instructional Programs (CIP), a systematic approach to describing and categorizing educational programs in Canada, akin to the National Occupation Classification (NOC) system used for job classification. Below is a summary of selected instructional programs eligible for the PGWP, along with their respective CIP codes: CIP 2021 Title CIP 2021 Code Field of Study Category Agricultural business and management, general 01.0101 Agriculture and agri-food Animal/livestock husbandry and production 01.0302 Agriculture and agri-food Plant nursery operations and management 01.0606 Agriculture and agri-food Animal health 01.0903 Agriculture and agri-food Agronomy and crop science 01.1102 Agriculture and agri-food Special education and teaching, general 13.1001 Healthcare Exercise physiology 26.0908 Healthcare Physical therapy assistant 51.0806 Healthcare Polysomnography 51.0917 Healthcare Cytotechnology/cytotechnologist 51.1002 Healthcare Computer programming/programmer, general 11.0201 STEM Chemical engineering 14.0701 STEM Engineering mechanics 14.1101 STEM Water, wetlands and marine resources management 03.0205 STEM Computer graphics 11.0803 STEM Electrician 46.0302 Trade Heating, air conditioning, ventilation and refrigeration maintenance technology/technician 47.0201 Trade Machine tool technology/machinist 48.0501 Trade Insulator 46.0414 Trade Plumbing technology/plumber 46.0503 Trade Heavy equipment maintenance technology/technician 47.0302 Transport Air traffic controller 49.0105 Transport Truck and bus driver/commercial vehicle operator and instructor 49.0205 Transport Flight instructor 49.0108 Transport Transportation and materials moving, other 49.9999 Transport

Since scientists have no answer to dark matter (except dark matter filament), the most logical explanation is that it is invisible. Some questions remain: 1. Can matter be invisible (or imperceptible by our senses and instruments)? 2. Even if matter could be theoretically invisible, is it possible that such a vast amount of matter, like dark matter, would escape all our knowledge and existing laws of physics and be unidentified until recently but wholly invisible and beyond our reach? 3. If dark matter is imperceptible, what makes it imperceptible? 4. Is it potentially perceptible but not perceptible to us as human beings? 5. Is there anything that would still avoid perception even if we possessed the absolute perceptive ability or technology with these abilities? 6. Or, is dark matter our way of explaining the unexplainable and offering a linguistic form to unknown phenomena? 7. Or, is it our inability to go beyond the spectrum, outside the existing frames, and try to decipher the unknown beyond the known frame of reality or what we see and understand as reality and the Universe? The answer to the first question is known; even atoms are invisible not only to the eyes but to microscopes. It is, therefore, theoretically possible that matter can be hidden and imperceptible. Still, it is hard to imagine that vast amounts of the mass of the Universe would stay unaccounted for within the realm of already advanced understanding of the laws of physics, instruments, and experiments. It would be possible to prove mathematically, based on what we already know about the Universe, the mass, the dispersion of energy and mass, and by these comparisons to conclude, without the CERN accelerator, that this is, most likely, impossible. This was a short answer to the second question. The third question is important because it would lead scientists in the right direction by avoiding the possible net of perplexed ideas. If we have already established that something exists, it would be better to define it as precisely as possible to avoid guessing only. In addition, how do we guess? We do not know anything about its nature, origin, or how it came into existence except that we came to this discovery almost accidentally by pure and relatively simple measurements and experiments. But what about us? How do we think? What methods do we use in experiments and the way we think? The answer to these questions could lead to better discoveries than only focusing on something we do not know and, even worse if we do not know where to look for it. Based on an accidental discovery, it is a good start to conclude that there is more mass in the Universe than can be detected. Still, it would be better and more productive to go beyond the Universe as we see it, beyond our existing knowledge and perception, not toward the stars we already know but toward another bottomless sky of darkness and the unknown. Although light is the source of life, darkness is also the source of light and life. Maybe the brightest “star” sleeps in the darkness and feeds the world from darkness. Is there only one Universe? If we start from the premise of the Big Bang theory, it would be logical to ask why there is only one Big Bang. It is easy to conclude that if there is a Big Bang at one point in “space” (nothingness), there can be another one at another “point,” past or future, although this may sound strange.

But the concepts of truth and falsity are only easy to apply in cases where a representation is in the form of language. In addition to linguistic representations, science often uses mathematical models, and other kinds of models, to describe phenomena. A scientific claim might also be expressed using a diagram. So I use the term "accurate representation" in a broad way to include true linguistic descriptions, pictures and diagrams that resemble reality in the way they are supposed to, models that have the right structural similarity to aspects of the world, and so on.

violations of regression assumptions, and strategies for examining and remedying such assumptions. Then we extend the preceding discussion and will be able to conclude whether the above results are valid. Again, this model is not the only model that can be constructed but rather is one among a family of plausible models. Indeed, from a theoretical perspective, other variables might have been included, too. From an empirical perspective, perhaps other variables might explain more variance. Model specification is a judicious effort, requiring a balance between theoretical and statistical integrity. Statistical software programs can also automatically select independent variables based on their statistical significance, hence, adding to R-square.2 However, models with high R-square values are not necessarily better; theoretical reasons must exist for selecting independent variables, explaining why and how they might be related to the dependent variable. Knowing which variables are related empirically to the dependent variable can help narrow the selection, but such knowledge should not wholly determine it. We now turn to a discussion of the other statistics shown in Table 15.1. Getting Started Find examples of multiple regression in the research literature. Figure 15.1 Dependent Variable: Productivity FURTHER STATISTICS Goodness of Fit for Multiple Regression The model R-square in Table 15.1 is greatly increased over that shown in Table 14.1: R-square has gone from 0.074 in the simple regression model to 0.274. However, R-square has the undesirable mathematical property of increasing with the number of independent variables in the model. R-square increases regardless of whether an additional independent variable adds further explanation of the dependent variable. The adjusted R-square (or ) controls for the number of independent variables. is always equal to or less than R2. The above increase in explanation of the dependent variable is due to variables identified as statistically significant in Table 15.1. Key Point R-square is the variation in the dependent variable that is explained by all the independent variables. Adjusted R-square is often used to evaluate model explanation (or fit). Analogous with simple regression, values of below 0.20 are considered to suggest weak model fit, those between 0.20 and 0.40 indicate moderate fit, those above 0.40 indicate strong fit, and those above 0.65 indicate very strong model fit. Analysts should remember that choices of model specification are driven foremost by theory, not statistical model fit; strong model fit is desirable only when the variables, and their relationships, are meaningful in some real-life sense. Adjusted R-square can assist in the variable selection process. Low values of adjusted R-square prompt analysts to ask whether they inadvertently excluded important variables from their models; if included, these variables might affect the statistical significance of those already in a model.3 Adjusted R-square also helps analysts to choose among alternative variable specifications (for example, different measures of student isolation), when such choices are no longer meaningfully informed by theory. Empirical issues of model fit then usefully guide the selection process further. Researchers typically report adjusted R-square with their

If by “system” is meant the logical arrangement of certain laws which have been derived and deduced from one fundamental principle by pure thought, in disregard of the actual facts (speculative system), then theology certainly is not a system. Systems of that kind are in place only where one is not dealing with things as they actually exist but only with ideal things, as is the case in “pure” mathematics, as distinguished from “applied” mathematics. They are not in place in the field of natural science and history, much less in theology. To apply the methods of the speculative systematizer to science and history is not only unscientific, but also downright nonsensical, because it proceeds on the assumption that facts are obedient to human thinking. Philosophical idealism has been justly called a derangement of the human mind, in which man labors under the obsession that his thoughts (ideas) are the rule and measure of things. Edm. Hoppe remarks in Der Alte Glaube: “Nature” [and he might have added history] “is not so obliging as to follow the diagram of the textbook.”195 Much less may the speculative systematizer intrude into the field of theology, because the Christian doctrine is fixed by Scripture. It is a finished product, which no human thinking may or can change in the least. Here every addition and every subtraction is absolutely and expressly forbidden (Joshua 23:6; Matt. 5:17-19; John 10:35; 8:31; Gal. 1:6-9). It is not the business of the theologian to deduce, through a process of thought, the Christian doctrine from some one fundamental principle or from some one fact, e. g., from the fact of regeneration, nor to construct it from the so-called “whole of Scripture,” which is a logical monstrosity. His work is limited to drawing the Christian doctrine in all its parts directly from those Scripture statements that treat the respective doctrine (sedes doctrinae). When we arrange all the Scriptural statements concerning the various doctrines under the proper heads, we have that well-ordered system of Christian doctrine which we need in this life; and God will not have us ask for something better.

Hence, the energy for independent thoughts is additive except for a term log[B(n1,n2)], the log of a binomial coefficient. Since binomial coefficients are always bigger than (or equal to) one, it follows that energy is super-additive. Combining thoughts demand more and more mental power as the sizes increase: the MIND is limited in the complexity of thoughts.

seemed almost certain to the mathematicians that since the general first, second, third, and fourth degree equations can be solved by means of the usual algebraic operations such as addition, subtraction, and roots, then the general fifth degree equation and still higher degree equations could also be solved. For three hundred years this problem was a classic one. Hundreds of mature and expert mathematicians sought the solution, but a little boy found the full answer. The Frenchman Évariste Galois (1811— 1832), who refused to conform to school examinations but worked brilliantly and furiously on his own, showed that general equations of degree higher than the fourth cannot be solved by algebraic operations. To establish this result Galois created the theory of groups, a subject that is now at the base of modern abstract algebra and that transformed algebra from a a series of elementary techniques to a broad, abstract, and basic branch of mathematics.

As an example, let us choose base six. To write the quantities from zero to five we would use the symbols 0, 1, 2, 3, 4, 5, as in base ten. The first essential difference comes up when we wish to denote six objects. Since six is to be the base we indicate this larger quantity by the symbols 10, the 1 denoting one times the base, just as in base ten the 1 in 10 denotes one times the base, or the quantity ten. Thus, the symbols 10 can mean different quantities, depending upon the base being employed. To write seven in base six we would write 11, because in base six these symbols mean 1.6+ 1, just as 11 in base ten means 1. 10 + 1. Similarly, to denote twenty in base six we write 32 because these symbols now mean 3 · 6 + 2. To indicate the quantity forty in base six we write 104, because these symbols mean 1 . 62 + 0 . 6 + 4, just as in base ten 104 means 1 · 102 + 0 · 10 + 4 or one hundred and four. It is clear that we can express quantity in base six. Moreover, we can perform the usual arithmetic operations in this base. We would, however, have to learn new addition and multiplication tables. For example, in base ten 4 + 5 = 9, but in base six 9 would be written 13.

Hence, the energy for independent thoughts is additive except for a term log[B(n1,n2)], the log of a binomial coefficient. Since binomial coefficients are always bigger than (or equal to) one, it follows that energy is super-additive. Combining thoughts demand more and more mental power as the sizes increase:

With the coming of Nazi-inspired racial laws, many promising Jewish graduate students were also dismissed from the universities. Castelnuovo organized special courses in his home, and in the homes of other Jewish former professors, to enable the graduate students to continue their studies. In addition to writing books on the history of mathematics, Castelnuovo spent the last of his eighty-seven years examine the philosophical relationship between determinism and chance and trying to interpret the concept of cause and effect.

If times or ages are mentioned in any activity, please take that as a guideline only. Children will learn the activity as it comes naturally to them. A thirty minute activity may take your child forty minutes. If the suggested age is four, but you have a three year old that can grasp the concepts do not let the recommendations hinder you. This information has not been provided for all activities. This is for a couple of reasons. One being that some of these activities span broader age ranges. The other is that while I find that sometimes it is useful to have an idea of where to start your child, it is better to look at your child in terms of ability and readiness rather than number of years. If times are included, they are meant for assistance in planning your day only. They are not in any way intended to be a marker for your child’s success. The activities have been divided up into the instructional areas of language, mathematics, sensory development and practical life skills. Many of these activities can serve as crossovers, allowing you to introduce multiple concepts at once. Some subjects such as cultural studies or science are included in practical life skills, since at this age that is primarily what those subjects encompass. Where applicable, additional skill areas have been included

One might think that math aptitude would apply across the code-writing world, but this was not so. The logical operations of a computer, while abstract, were abstract in a familiar way. When computers added two numbers together, for instance, they did so in much the same way as people do: One plus one equals two. A computer performed this addition faster than a person, but the route to the answer was the same. This was not so with graphics. When an ordinary person (as opposed to a civil engineer) wanted to draw a circle, he did not apply a special algorithm to his choice of a diameter. He just drew a circle with his hand. Having no hands, computers drew circles—and all the other shapes—by applying mathematical formulas. The better the formulas, the more accurate and versatile the shapes.

The remaining part of the first description consist of low-energy open strings moving on the three-branes. We recall from Chapter 4 that low-energy strings are well described by point particle quantum field theory, and that is the case here. The particular kind of quantum field theory involves a number of sophisticated mathematical ingredients (and it has an ungainly characterization: conformally invariant supersymmetric quantum gauge field theory), but two vital characteristics are readily understood. The absence of closed strings ensures the absence of the gravitational field. And, because the strings can move only on the tightly sandwiched three-dimensional branes, the quantum field theory lives in three spatial dimensions (in addition to the one dimension of time, for a total of four spacetime dimensions). The remaining part of the second description consists of closed strings, executing any vibrational pattern, as long as they are close enough to the black branes' event horizon to appear lethargic-that is, to appear to have low energy. Such strings, although limited in how far they stray from the black stack, still vibrate and move through nine dimensions of space (in addition to one dimension of time, for a total of ten spacetime dimensions). And because this sector is built from closed strings, it contains the force of gravity. However different the two perspectives might seem, they're describing one and the same physical situation, so they must agree. This leads to a thoroughly bizarre conclusion. A particular nongravitational, point particle quantum field theory in four spacetime dimensions (the first perspective) describes the same physics as strings, including gravity, moving through a particular swath of ten spacetime dimensions (the second perspective). This would seem as far-fetched as claiming...Well, honestly, I've tried, and I can't come up with any two things int he real world more dissimilar than these two theories. But Maldacena followed the math, in the manner we've outlined, and ran smack into this conclusion. The sheer strangeness of the result-and the audacity of the claim-isn't lessened by the fact that it takes but a moment to place it within the line of thought developed earlier in this chapter. As schematically illustrated in Figure 9.5, the gravity of the black brane slab imparts a curved shape to the ten-dimensional spacetime swath in its vicinity (the details are secondary, but the curved spacetime is called anti-de Sitter five-space times the five sphere); the black brane is itself the boundary of this space. And so, Maldacena's result is that string theory within the bulf of this spacetime shape is identical to a quantum field theory living on its boundary. This is holography come to life.

NOTHING CAN GO FASTER THAN LIGHT Of course the idea that there is an ultimate speed limit seems absurd. While the speed of light is very high by earthly standards, the magnitude is not the point; any kind of speed limit in nature doesn't make sense. Suppose, for example, that a spaceship is traveling at almost the speed of light. Why can't you fire the engine again and make it go faster-or if necessary, build another ship with a more powerful engine? Or if a proton is whirling around in a cyclotron at close to the speed of light, why can't you give it additional energy boosts and make it go faster? Intuitive explanation. When we think of the spaceship and the proton as made of fields, not as solid objects, the idea is no longer ridiculous. Fields can't move infinitely fast. Changes in a field propagate in a "laborious" manner, with a change in intensity at one point causing a change at nearby points, in accordance with the field equations. Consider the wave created when you drop a stone in water: The stone generates a disturbance that moves outward as the water level at one point affects the level at another point, and there is nothing we can do to speed it up. Or consider a sound wave traveling through air: The disturbance in air pressure propagates as the pressure at one point affects the pressure at an adjacent point, and we can't do anything to speed it up. In both cases the speed of travel is determined by properties of the transmitting medium- air and water, and there are mathematical equations that describe those properties. Fields are also described by mathematical equations, based on the properties of space. It is the constant c in those equations that determines the maximum speed of propagation. If the field has mass, there is also a mass term that slows down the propagation speed further. Since everything is made of fields - including protons and rocketships - it is clear that nothing can go faster than light. As Frank Wilczek wrote, One of the most basic results of special relativity, that the speed of light is a limiting velocity for the propagation of any physical influence, makes the field concept almost inevitable. - F. Wilczek ("The persistence of Ether", p. 11, Physics Today, Jan. 1999) David Bodanis tried to make this point in the following way: Light will always be a quick leapfrogging of electricity out from magnetism, and then of magnetism leaping out from electricity, all swiftly shooting away from anything trying to catch up to it. That's why it's speed can be an upper limit - D. Bodanis However, Bodanis only told part of the story. It is only when we recognize that everything, not just light, is made of fields that we can conclude that there is a universal speed limit.

NOTHING CAN GO FASTER THAN LIGHT Of course the idea that there is an ultimate speed of light is very high by earthly standards, the magnitude is not the point; any kind of speed limit in nature doesn't make sense. Suppose, for example, that a spaceship is traveling at almost the speed of light. Why can't you fire the engine again and make it go faster-or if necessary, build another ship with a more powerful engine? Or if a proton is whirling around in a cyclotron at close to the speed of light, why can't you give it additional energy boosts and make it go faster? Intuitive explanation. When we think of the spaceship and the proton as made of fields, not as solid objects, the idea is no longer ridiculous. Fields can't move infinitely fast. Changes in a field propagate in a "laborious" manner, with a change in intensity at one point causing a change at nearby points, in accordance with the field equations. Consider the wave created when you drop a stone in water: The stone generates a disturbance that moves outward as the water level at one point affects the level at another point, and there is nothing we can do to speed it up. Or consider a sound wave traveling through air: The disturbance in air pressure propagates as the pressure at one point affects the pressure at an adjacent point, and we can't do anything to speed it up. In both cases the speed of travel is determined by properties of the transmitting medium- air and water, and there are mathematical equations that describe those properties. Fields are also described by mathematical equations, based on the properties of space. It is the constant c in those equations that determines the maximum speed of propagation. If the field has mass, there is also a mass term that slows down the propagation speed further. Since everything is made of fields - including protons and rocketships - it is clear that nothing can go faster than light. As Frank Wilczek wrote, One of the most basic results of special relativity, that the speed of light is a limiting velocity for the propagation of any physical influence, makes the field concept almost inevitable. - F. Wilczek ("The persistence of Ether", p. 11, Physics Today, Jan. 1999) David Bodanis tried to make this point in the following way: Light will always be a quick leapfrogging of electricity out from magnetism, and then of magnetism leaping out from electricity, all swiftly shooting away from anything trying to catch up to it. That's why it's speed can be an upper limit - D. Bodanis However, Bodanis only told part of the story. It is only when we recognize that everything, not just light, is made of fields that we can conclude that there is a universal speed limit.

the kernel of a homomorphism is closed under addition, and also under multiplication by any element of the ring. These two properties define the notion of an ideal.

Modules are to rings as vector spaces are to fields. In other words, they are algebraic structures where the basic operations are addition and scalar multiplication, but now the scalars are allowed to come from a ring rather than a field.

The sciences do not try to explain, they hardly even try to interpret, they mainly make models. By a model is meant a mathematical construct, which with the addition of certain verbal interpretations, describes observed phenomena. The justification of such a mathematical construct is solely and precisely that it is expected to work. —JOHN VON NEUMANN

More important, there is a revealing logical lacuna in the positivist's argument, one that will reintroduce us immediately to the nature of revolutionary change. Can Newtonian dynamics really be derived from relativistic dynamics? What would such a derivation look like? Imagine a set of statements, E1, E2,...,En, which together embody the laws of relativity theory. These statements contain variables and parameters representing spatial position, time, rest mass, etc. From them, together with the apparatus of logic and mathematics, is deducible a whole set of further statements including some that can be checked by observation. To prove the adequacy of Newtonian dynamics as a special case, we must add to the E1's additional statements, like (v/c)^2<<1, restricting the range of the parameters and variables. This enlarged set of statements is then manipulated to yield a new set, N1,N2,....,Nm, which is identical in form with Newton's laws of motion, the law of gravity, and so on. Apparently, Newtonian dynamics has been derived from Einsteinian, subject to a few limiting conditions.

Although "What's the meaning of life?" can be interpreted in many different ways, some of which may be too vague to have a well-defined answer, one interpretation is very practical and down-to-Earth: "Why should I want to go on living?" The people I know who feel that their lives are meaningful usually feel happy to wake up in the morning and look forward to the day ahead. When I think about these people, it strikes me that they split into two broad groups based on where they find their happiness and meaning. In other words, the problem of meaning seems to have two separate solutions, each of which works quite well for at least some people. I think of these solutions as "top-down" and "bottom-up." In the top-down approach, the fulfillment comes from the top, from the big picture. Although life here and now may be unfulfilling, it has meaning by virtue of being part of something greater and more meaningful. Many religions embody such a message, as do families, organizations and societies where individuals are made to feel part of something grander and more meaningful that transcends individuality. In the bottom-up approach, the fulfillment comes from the little things here and now. If we seize the moment and get the fulfillment we need from the beauty of those little flowers by the roadside, from helping a friend or from meeting the gaze of a newborn child, then we can feel grateful to be alive even if the big picture involves less-cheerful elements such as Earth getting vaporized by our dying Sun and our Universe ultimately getting destroyed. For me personally, the bottom-up approach provides more than enough of a raison d'etre, and the top-down elements I'm about to argue for simply feel like an additional bonus. For starters, I find it utterly remarkable that it's possible for a bunch of particles to be self-aware, and that this particular bunch that's Max Tegmark has had the fortune to get the food, shelter and leisure time to marvel at the surrounding universe leaves me grateful beyond words.

The collection of all real or complex numbers that are integral linear combinations of 1 and τd is closed under addition, subtraction, and multiplication, and is therefore a ring, which we denote by Rd. That is, Rd is the set of all numbers of the form a + bτd where a and b are ordinary integers. These rings Rd are our first, basic, examples of rings of algebraic integers beyond that prototype, , and they are the most important rings that are receptacles for quadratic irrationalities. Every quadratic irrational algebraic integer is contained in exactly one Rd.

The American Works Progress (later Projects) Administration, founded in 1935 to provide jobs for “employable workers” during the Great Depression, established the Mathematical Tables Project in 1938 as one of its “small useful projects.” Useful it was, but hardly small: it was one of the largest-scale computing operations in the pre-ENIAC age, headed by a Polish-born mathematician, Gertrude Blanch, who supervised 450 clerks.18 Just as de Prony had learned a lesson from Adam Smith, Blanch took her cue from Henry Ford—she gave each group of workers a single task: some did only addition, some only subtraction. The best were trusted with long division. The resulting tables of logarithms and other functions were published in twenty-eight volumes; in some of them, no one to this day has discovered a single error.

Algebra is another branch of mathematics; it studies sets on which there have been defined things called “operations”. An operation on a set is a rule whereby two or more elements of the set can be combined to form another element of the set. High school algebra is the algebra of one specific set, the set of real numbers, and four specific operations defined on that set, addition, subtraction, multiplication, and division. High school algebra is only the tip of the algebraic iceberg.

...although confederations are meant to attract and collaborate with allies, the word confederation, although correlated with uniting allies, seems to be more so connected with making enemies. It’s just an elementary math and logic problem---7.9 billion people are in the world population, so it is mathematically supported that it would be improbable for all 7.9 billion people to agree with each other. In addition, “confederation” actually seems to add to division, because confederations are alliances where people with same/similar interests are united and organized, and in order for there to be an alliance of people with same/similar interests, there has to be people with interests that differ from theirs. Otherwise, they should not have made an alliance dedicated to one particular interest. If everyone agreed with them, they wouldn’t need to create an alliance.

Difficulties of technical translation: features, problems, rules Technical translation is one of the most important areas of written translation in modern translation practice. Like the interpretation technique, it has its own characteristics and requirements. The need for this type of work is due to economic and scientific and technical progress, as well as the development of international relations. Thanks to technical translation, people share experience, knowledge and developments in various fields. What are the features of this type of translation? What pitfalls can be encountered on the translator's path? You will learn about this and much more from our article. ________________________________________ Technical translation is one of the most difficult types of legal translation. This is due to the large number of requirements for such work. Technical translation includes all scientific and technical texts, documents, instructions, reports, reference books and dictionaries. The texts of this plan contain a lot of specific terminology, which is the main difficulty of technical translation. A term is a word or a combination of words that accurately names a phenomenon, subject or scientific concept, revealing its meaning as much as possible. The most common technical texts in the following areas: • engineering; • defense; • physics and mathematics; • aircraft construction; • oil industry; • shipbuilding, etc. The main feature of technical translation is the requirement for its high accuracy (equivalence). The task of the translator is to convey information as close as possible to the original. Otherwise, distortions may appear in the text, leading to a misunderstanding of important information. Vocabulary selection is carried out carefully and carefully. The construction of phrases should be logical and meaningful. Other technical translation requirements include adequacy and informativeness. It is equally important to maintain the style of such texts. This includes not only vocabulary, but also the grammatical structure of the text, as well as the way the material is presented. Most often, this is a formal and logical style. Unlike artistic translation, where the main task is to convey the content, and the translator can use his imagination, include fancy turns and various figures of speech, the presence of emotionality and subjectivity is unacceptable in technical translation. Let's consider the peculiarities of technical translation in English. According to the well-known linguist and translator Y. Y. Retsker, English technical literature is characterized by the predominant use of complex or complex sentences, which include adjectives, nouns, as well as impersonal forms of verbs (infinitives, gerundial inflections, etc.). Passive constructions are also often found. In this direction, it is permissible to use only generally accepted grammatical structures. Another feature of such texts may be the absence of a predicate or subject and a large number of enumerations. In addition, the finished text should have an appropriate layout equivalent to the original. Let's consider the basic rules of technical translation for a specialist: • knowledge of the vocabulary, grammar and word structure of the foreign language from which the translation is performed (at the level required for understanding the source text); • knowledge of the language into which the translation is performed (at a level sufficient for a competent presentation of the material); • excellent knowledge of the specifics of texts and terminology; • ability to use linguistic and technical sources of information; • familiarity with the specifics of the field

The final, two-way arrow indicates the most subtle and nefarious stage of this neurological programming, the feedback between the incoming energy (plus additions and minus subtractions) and the language system (including symbolic, abstract languages like mathematics) which the brain happens to use habitually. The final precept in humans is always verbal or symbolic and hence coded into the pre-existing structure of whatever languages or systems the brain has been taught. The process is not one of linear reaction but of synergetic transaction. This finished product is thus a neurosemantic construct, a kind of metaphor.

In the domain of mathematical assertions the property of absurdity, like the property of truth, is a universally additive property, that is to say, if it holds for each element α of a species of assertions, it also holds for the assertion which is the union of the assertions α. This property of universal additivity does not obtain for the property of non-contradictority. However, non-contradictority does possess the weaker property of finite additivity, that is to say, if the assertions ρ and σ are non-contradictory, the assertion τ, which is the union of ρ and σ, is also non-contradictory.

For Brouwer, and his followers (the intuitionists) , the constructive real numbers described above do hot constitute all of the real number system. In addition there are incompletely determined real numbers, corresponding to sequences of rational numbers whose terms are not specified by a master algorithm. Such sequences are called "free-choice sequences", because the creating subject, who defines the sequence, does not completely commit himself in advance but allows himself some freedom of choice along the way in defining the individual terms of the sequence.

Brouwer, the founder of intuitionist mathematics, has shown that in certain mathematical problems dealing with infinite sets of numbers the elementary rule of the excluded middle is not admissible, without an additional arbitrary assumption. Statements like: there is a number . . and: there is no number . . . , in this case only seemingly, by virtue of their abbreviated linguistic formulations, hm·e the forrn of contradictory opposites.

I added up their opinions, divided them by my truths, subtracted my expectations, and tallied them by the proof. Took a fraction of my fears and rounded them to a fifth, which equalled out to my freedom and summed up the way I live. And I ain't good at mathematics, know, I ain't no ancient Greek. I'm just the square root of a factor tree, and the answer is always…me.

In the traditional (biologically inspired) setup each neuron effectively has a certain set of “incoming connections” from the neurons on the previous layer, with each connection being assigned a certain “weight” (which can be a positive or negative number). The value of a given neuron is determined by multiplying the values of “previous neurons” by their corresponding weights, then adding these up and adding a constant—and finally applying a “thresholding” (or “activation”) function. In mathematical terms, if a neuron has inputs x = {x1, x2 ...} then we compute f[w . x + b], where the weights w and constant b are generally chosen differently for each neuron in the network; the function f is usually the same. Computing w . x + b is just a matter of matrix multiplication and addition. The “activation function” f introduces nonlinearity (and ultimately is what leads to nontrivial behavior). Various activation functions commonly get used; here we’ll just use Ramp (or ReLU):

Do you see?” asked Renee. “I’ve just disproved most of mathematics: it’s all meaningless now.” She was getting agitated, almost distraught; Carl chose his words carefully. “How can you say that? Math still works. The scientific and economic worlds aren’t suddenly going to collapse from this realization.” “That’s because the mathematics they’re using is just a gimmick. It’s a mnemonic trick, like counting on your knuckles to figure out which months have thirty-one days.” “That’s not the same.” “Why isn’t it? Now mathematics has absolutely nothing to do with reality. Never mind concepts like imaginaries or infinitesimals. Now goddamn integer addition has nothing to do with counting on your fingers. One and one will always get you two on your fingers, but on paper I can give you an infinite number of answers, and they’re all equally valid, which means they’re all equally invalid. I can write the most elegant theorem you’ve ever seen, and it won’t mean any more than a nonsense equation.” She gave a bitter laugh. “The positivists used to say all mathematics is a tautology. They had it all wrong: it’s a contradiction.” Carl tried a different approach. “Hold on. You just mentioned imaginary numbers. Why is this any worse than what went on with those? Mathematicians once believed they were meaningless, but now they’re accepted as basic. This is the same situation.” “It’s not the same. The solution there was to simply expand the context, and that won’t do any good here. Imaginary numbers added something new to mathematics, but my formalism is redefining what’s already there.” “But if you change the context, put it in a different light—” She rolled her eyes. “No! This follows from the axioms as surely as addition does; there’s no way around it. You can take my word for it.” 7

Nowhere in all this elaborate brain circuitry, alas, is there the equivalent of the chip found in a five-dollar calculator. This deficiency can make learning that terrible quartet—“Ambition, Distraction, Uglification, and Derision,” as Lewis Carroll burlesqued them—a chore. It’s not so bad at first. Our number sense endows us with a crude feel for addition, so that, even before schooling, children can find simple recipes for adding numbers. If asked to compute 2 + 4, for example, a child might start with the first number and then count upward by the second number: “two, three is one, four is two, five is three, six is four, six.” But multiplication is another matter. It is an “unnatural practice,” Dehaene is fond of saying, and the reason is that our brains are wired the wrong way. Neither intuition nor counting is of much use, and multiplication facts must be stored in the brain verbally, as strings of words. The list of arithmetical facts to be memorized may be short, but it is fiendishly tricky: the same numbers occur over and over, in different orders, with partial overlaps and irrelevant rhymes. (Bilinguals, it has been found, revert to the language they used in school when doing multiplication.) The human memory, unlike that of a computer, has evolved to be associative, which makes it ill-suited to arithmetic, where bits of knowledge must be kept from interfering with one another: if you’re trying to retrieve the result of multiplying 7 X 6, the reflex activation of 7 + 6 and 7 X 5 can be disastrous. So multiplication is a double terror: not only is it remote from our intuitive sense of number; it has to be internalized in a form that clashes with the evolved organization of our memory. The result is that when adults multiply single-digit numbers they make mistakes ten to fifteen per cent of the time. For the hardest problems, like 7 X 8, the error rate can exceed twenty-five per cent. Our inbuilt ineptness when it comes to more complex mathematical processes has led Dehaene to question why we insist on drilling procedures like long division into our children at all. There is, after all, an alternative: the electronic calculator. “Give a calculator to a five-year-old, and you will teach him how to make friends with numbers instead of despising them,” he has written. By removing the need to spend hundreds of hours memorizing boring procedures, he says, calculators can free children to concentrate on the meaning of these procedures, which is neglected under the educational status quo.

Mathematical modelling is a world-renowned field of research in mathematics education. The International Conference on the Teaching and Learning of Mathematical Modelling and Applications (ICTMA), for example, presents the current state of the international debate on mathematical modelling every two years. Contributions made at these conferences are published in Springer’s International Perspectives on the Teaching and Learning of Mathematical Modelling series. In addition, the ICMI study Modelling and Applications in Mathematics Education

The Banach-Tarski Theorem is an astonishing result. We have decomposed a ball into finitely many pieces, moved around the pieces without changing their size or shape, and then reassembled them into two balls of the same size as the original. I think the theorem teaches us something important about the notion of volume. As noted earlier, it is an immediate consequence of the theorem that some of the Banach-Tarski pieces must lack definite volumes and, therefore, that not every subset of the unit ball can have a well-defined volume. A little more precisely, the theorem teaches us that there is no way of assigning volumes to the Banach-Tarski pieces while preserving three-dimensional versions of the principles we called Uniformity and (finite) Additivity in chapter 7. (Proof: Suppose that each of the (finitely many) Banach-Tarski pieces has a definite finite volume. Since the pieces are disjoint, and since their union is the original ball, Additivity entails that the sum of the volumes of the pieces must equal the volume of the original ball. But Uniformity ensures that the volume of each piece is unchanged as we move it around. Since the reassembled pieces are disjoint, and since their union is two balls, Additivity entails that the sum of their volumes must be twice the volume of the original ball. But since the volume of the original ball is finite and greater than zero, it is impossible for the sum of the pieces to equal both the volume of the original ball and twice the volume of the original ball.) If I were to assign the Banach-Tarski Theorem a paradoxicality grade of the kind we used in chapter 3, I would assign it an 8. The theorem teaches us that although the notion of volume is well-behaved when we focus on ordinary objects, there are limits to how far it can be extended when we consider certain extraordinary objects - objects that can only be shown to exist by assuming the Axiom of Choice.

In the beginning, everything was void, and J. H. W. H. Conway began to create numbers. Conway said, "Let there be two rules which bring forth all numbers large and small. This shall be the first rule: Every number corresponds to two sets of previously created numbers, such that no member of the left set is greater than or equal to any member of the right set. And the second rule shall be this: One number is less than or equal to another number if and only if no member of the first number's left set is greater than or equal to the second number, and no member of the second number's right set is less than or equal to the first number." And Conway examined these two rules he had made, and behold! They were very good. … And Conway said, "Let the numbers be added to each other in this wise: The left set of the sum of two numbers shall be the sums of all left parts of each number with the other; and in like manner the right set shall be from the right parts, each according to its kind." Conway proved that every number plus zero is unchanged, and he saw that addition was good. And the evening and the morning were the third day. And Conway said, "Let the negative of a number have as its sets the negatives of the number's opposite sets; and let subtraction be addition of the negative." And it was so. Conway proved that subtraction was the inverse of addition, and this was very good. And the evening and the morning were the fourth day. And Conway said to the numbers, "Be fruitful and multiply. Let part of one number be multiplied by another and added to the product of the first number by part of the other, and let the product of the parts be subtracted. This shall be done in all possible ways, yielding a number in the left set of the product when the parts are of the same kind, but in the right set when they are of opposite kinds." Conway proved that every number times one is unchanged. And the evening and the morning were the fifth day. And behold! When the numbers had been created for infinitely many days, the universe itself appeared. And the evening and the morning were N day. And Conway looked over all the rules he had made for numbers, and saw that they were very, very good.

Mathematical models are particularly important in the study of dynamics, because dynamic phenomena are typically characterized by nonlinear feedbacks, often acting with various time lags. Informal verbal models are adequate for generating predictions in cases where assumed mechanisms act in a linear and additive fashion (as in trend extrapolation), but they can be very misleading when we deal with a system characterized by nonlinearities and lags. In general, nonlinear dynamical systems have a much wider spectrum of behaviors than could be imagined by informal reasoning (for example, see Hanneman et al. 1995). Thus,

We do not give programs in any specific programming language; instead, algorithms are presented in pseudocode, a structured format using a combination of natural language, mathematics and programming structures. Algorithm 1.1 is a simple algorithm for integer multiplication using repeated addition.

A like-minded group coalesced with the common mission of bringing the study of living organisms in line with existing research into the inanimate world. In modern terms, they wanted to show that living organisms obeyed the same mathematical, physical, and chemical laws as everything else. However, this approach put Helmholtz and his network in conflict with a large section of the European scientific community who felt such a synthesis of the animate and inanimate worlds was not possible. Many scientists of the day believed in vitalism, the idea that living organisms, in addition to the sustenance they received from food, water, air, and so on, also possessed a “vital,” life-giving force. While an organism was alive, this vital force controlled the physical and chemical processes that took place within it. Logically, therefore, when it died, that vital force disappeared, leaving the dead organism to decay as if it were inanimate. Helmholtz and his friends opposed this “vitalist” view and felt disproving it was a crucial step to putting biology on the same footing as physics and chemistry.

Thus, calculus proceeds in two phases: cutting and rebuilding. In mathematical terms, the cutting process always involves infinitely fine subtraction, which is used to quantify the differences between the parts. Accordingly, this half of the subject is called differential calculus. The reassembly process always involves infinite addition, which integrates the parts back into the original whole. This half of the subject is called integral calculus.

Fasting insulin is a blood test for the hormone insulin. Your result should be less than 8 mIU/L (55 pmol/L). Fasting insulin can pick up severe insulin resistance. To detect milder insulin resistance, you’ll need the more sensitive insulin glucose challenge test. HOMA-IR index is a mathematical calculation using the ratio of glucose to insulin. For healthy insulin sensitivity, your HOMA-IR index should be less than 1.5. Insulin glucose challenge test is a blood test that is similar to the two-hour oral glucose tolerance test, in which several blood samples are taken during the two hours following a sweet drink. The difference with this test is that insulin is tested in addition to glucose.

The structure of de Prony’s computing office cannot be easily seen in Smith’s example. His computing staff had two distinct classes of workers. The larger of these was a staff of nearly ninety computers. These workers were quite different from Smith’s pin makers or even from the computers at the British Nautical Almanac and the Connaissance des Temps. Many of de Prony’s computers were former servants or wig dressers, who had lost their jobs when the Revolution rendered the elegant styles of Louis XVI unfashionable or even treasonous.35 They were not trained in mathematics and held no special interest in science. De Prony reported that most of them “had no knowledge of arithmetic beyond the two first rules [of addition and subtraction].”36 They were little different from manual workers and could not discern whether they were computing trigonometric functions, logarithms, or the orbit of Halley’s comet. One labor historian has described them as intellectual machines, “grasping and releasing a single piece of ‘data’ over and over again.”37 The second class of workers prepared instructions for the computation and oversaw the actual calculations. De Prony had no special title for this group of workers, but subsequent computing organizations came to use the term “planning committee” or merely “planners,” as they were the ones who actually planned the calculations. There were eight planners in de Prony’s organization. Most of them were experienced computers who had worked for either the Bureau du Cadastre or the Paris Observatory. A few had made interesting contributions to mathematical theory, but the majority had dealt only with the problems of practical mathematics.38 They took the basic equations for the trigonometric functions and reduced them to the fundamental operations of addition and subtraction. From this reduction, they prepared worksheets for the computers. Unlike Nevil Maskelyne’s worksheets, which gave general equations to the computers, these sheets identified every operation of the calculation and left nothing for the workers to interpret. Each step of the calculation was followed by a blank space for the computers to fill with a number. Each table required hundreds of these sheets, all identical except for a single unique starting value at the top of the page. Once the computers had completed their sheets, they returned their results to the planners. The planners assembled the tables and checked the final values. The task of checking the results was a substantial burden in itself. The group did not double-compute, as that would have obviously doubled the workload. Instead the planners checked the final values by taking differences between adjacent values in order to identify miscalculated numbers. This procedure, known as “differencing,” was an important innovation for human computers. As one observer noted, differencing removed the “necessity of repeating, or even of examining, the whole of the work done by the [computing] section.”39 The entire operation was overseen by a handful of accomplished scientists, who “had little or nothing to do with the actual numerical work.” This group included some of France’s most accomplished mathematicians, such as Adrien-Marie Legendre (1752–1833) and Lazare-Nicolas-Marguerite Carnot (1753–1823).40 These scientists researched the appropriate formulas for the calculations and identified potential problems. Each formula was an approximation, as no trigonometric function can be written as an exact combination of additions and subtractions. The mathematicians analyzed the quality of the approximations and verified that all the formulas produced values adequately close to the true values of the trigonometric functions.

In other words, how can we use the posterior distribution —which, after all, represents everything that we know about from the old set—to predict the number of correct responses out of the new set of questions? The mathematical solution is to integrate over the posterior, , where is the predicted number of correct responses out of the additional set of 5 questions. Computationally, you can think of this procedure as repeatedly drawing a random value from the posterior, and using that value to every time determine a single . The end result is , the posterior predictive distribution of the possible number of correct responses in the additional set of 5 questions. The important point is that by integrating over the posterior, all predictive uncertainty is taken into account.

Ghalib and Gerard translated no fewer than eighty-eight works of astronomy, mathematics, medicine, philosophy and logic, the very branches of learning which underpinned the great revival of scholarship in Europe referred to as the Twelfth-Century Renaissance.16 They focused especially on translating the greatest works of the Greek scientists of antiquity, particularly Ptolemy and Aristotle, but they also worked to bring original Indian and Arab works into Latin. Among these was Khwarizmi’s Book of Addition and Subtraction According to the Hindu Calculation, and the book which went on to define algebra for Europeans, the Kitab al-Jabr.17