Практика Shadowing: Mathematician Collapses All Functions to One Weird Formula - Изучайте разговорный английский по видео
Создание урока...
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plus, minus, x times y, x to the power of y, sine and cosine.
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Maths is full with complicated operations that you do to numbers.
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An ambitious new paper now claims that actually you can do everything with only one operation.
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That's quite a claim.
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Maybe that finally explains why the Americans have only one math, where the Brits have maths.
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I've had a look.
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The new paper's a preprint that appeared on the archive the other week and comes from a mathematician in Poland.
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The one operation
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that he says can give everything is taking the exponential function of a number x minus the logarithm of another number y.
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He calls that ML, short for exponential minus logarithm.
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For example, if you put x and 1 into the ML function, you get e to the x minus log of 1.
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Log of 1 is 0, so this gives you just the exponential function.
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You can also get the logarithm itself back out.
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For this, you first take ml of 1 and x, that gives you e minus log x.
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Then you put that back into the first slot with another one.
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So ml of ml 1 x 1, that gives e to the e over x.
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And then you put that yet again into the second slot of ml with a 1 in the first.
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If you work through the algebra, all the extra pieces cancel and you're left with just log of x.
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So the same function gives you exponentials and logarithms indirectly.
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But how do we get the standard operations, addition, subtraction, multiplication, etc.? They're all in there.
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Take subtraction.
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We just saw that you can express the exponential function and the logarithm through ML operations.
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If you put those back into the ML function, what do you get?
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You get x minus y.
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You can also get a plus just by entering minus y.
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And now you see you can put the sum of two logarithms into the exponent of the exponential function.
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That'll give you multiplication.
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And once you have multiplication, you can define powers and so on.
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If you want to get trigonometric functions, you first need imaginary numbers.
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You can get those from the logarithm at negative numbers, though there's a branch cut involved there.
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I know that sounds like a gardening accident, but it just defines the logarithm for a negative number.
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But once you have that, you get complex numbers, and you get sine and cosine and even the number pi.
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What you do not get, for all I can see, is all real numbers starting from just 1 and 0, not without taking infinite limits.
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But the author doesn't make that claim.
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He says one can get all mathematical operations.
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The paper hasn't yet been peer -reviewed, but I'm reasonably sure it's correct.
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I saw a lot of people online worry about the arbitrary branch cut of the logarithm, but I don't think this matters.
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Yes, you could choose it differently, and that it'd give you a different definition of complex numbers, but it'd still work.
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Now, you may be wondering whether this is actually useful.
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The answer is, for normal people, probably not.
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No one's going to replace a normal calculator with one
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that has a single deranged button that expects you to rebuild cosine by nesting 15 different operations.
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The interesting part is structural.
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In computer logic, there's a logical operation called NAND, which is enough on its own to build all the usual logical gates in a computer.
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This paper's trying to find the analog for ordinary continuous maths.
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The claim is that EML is kind of a NAND gate for scientific calculators.
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And there is a possible practical angle.
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If every ordinary formula can be rewritten as a tree made from one repeated building block,
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then that gives you a very uniform way to search for formulas with a computer.
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The author suggests this could be useful for symbolic regression, which means trying to guess the equation behind a set of data.
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So this may be less about helping humans do maths
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and more about giving machines a simpler language in which to search for equations,
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which is nice because if the machines are going to take over science, then at least they will do it in an orderly manner.
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I give this paper a 0 out of 10 on the bullshit meter.
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It's a cute paper.
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I think that the point isn't super surprising to anyone, but it is surprising that no one has done it before.
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There are two ways one can look at this result.
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One is that mathematics appears more complicated than it is.
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The other way to look at it is that what we think of as complicated is very subjective.
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You could rightfully argue that maybe using 10 or
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so different operations is easier than nesting just one operation 20 times in itself.
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In the end, it depends on just how you quantify simplicity.
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This is also an issue that haunts the foundations of physics, because what really do we mean when we say
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that simple explanations are better if we don't agree on how to measure simplicity?
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Or maybe it just tells us that mathematicians have too much time.
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Лексика и советы по произношению к этому уроку
В этом видео 81 предложений и 1038 слов для шедоуинга. Речь длится 6:24. Говорящий говорит в естественном темпе, около 162 слов в минуту, близком к повседневной речи. Только 79% слов входят в 3000 самых частых слов английского языка, поэтому лексика сложная.
Ключевая лексика этого видео
15 слов из видео, которые стоит выучить, с произношением и значением:
| Слово | Произношение | Значение |
|---|---|---|
| minus | /ˈmaɪ.nəs/ | ми́нус |
| mathematics существительное | /mæθ(.ə)ˈmæt.ɪks/ | матема́тика |
| complicated прилагательное | /ˈkɑm.plɪˌkeɪ.tɪd/ | сло́жный, запу́танный |
| earn глагол | /ɝn/ | зараба́тывать, зарабо́тать |
| practical прилагательное | /ˈpɹæk.tɪ.kəl/ | практи́ческий |
| gate существительное | /ˈɡeɪ̯t/ | воро́та, врата́ |
| formula существительное | /ˈfɔː.mjʊ.lə/ | фо́рмула |
| ordinary прилагательное | /ˈɔɹdɪnɛɹi/ | обыкнове́нный, обы́чный |
| define глагол | /dɪˈfaɪn/ | определя́ть, определи́ть |
| logical прилагательное | /ˈlɑd͡ʒɪkəl/ | логи́ческий |
| slot существительное | /slɑt/ | щель, отве́рстие |
| nest существительное | /nɛst/ | гнездо́ |
| equation существительное | /ɪˈkweɪ.ʒən/ | уравне́ние |
| simplicity существительное | /sɪmˈplɪsɪti/ | простота́ |
| calculator существительное | /kæl.kjə.leɪ.tɚ/ | калькуля́тор, вычисли́тельная маши́на |
Фразовые глаголы, которые вы услышите
| Слово | Значение |
|---|---|
| back out глагол | сдавать назад |
| take over глагол | брать на себя́, взять на себя́ |
Грамматика в этом видео
Конструкции, которые говорящий использует чаще всего, с точными словами из видео:
| Конструкция | В видео |
|---|---|
| Пассивный залог be + причастие прошедшего времени — важно, что происходит, а не кто это делает | is rated · be prepared · are limited |
| Придаточные определительные who / which + предложение — уточнение о человеке или предмете | NAND, which is · regression, which means · those who don |
| Present Perfect have/has + причастие прошедшего времени — прошлое действие, важное сейчас | has done · has already built |
Произношение, на которое стоит обратить внимание
Говорящий использует 14 сокращённых и редуцированных форм, например don't, it'd, doesn't. Произносите их коротко, так, как слышите.
- Звуки «th»: mathematics /mæθ(.ə)ˈmæt.ɪks/, mathematical /ˌmæθ(.ə)ˈmæt.ɪ.kəl/, logarithm /ˈlɑ.ɡə.ɹɪ.ð(ə)m/, mathematician /ˌmæθ.(ə.)məˈtɪʃ.ən/
- Звуки «sh» и «zh»: equation /ɪˈkweɪ.ʒən/, explanation /ˌɛkspləˈneɪʃən/, structural /ˈstɹʌk(t͡)ʃəɹəl/, ambitious /æmˈbɪʃ.əs/, compilation /ˌkɑm.pɪˈleɪ.ʃən/
- Длинные слова — следите за ударением: mathematics /mæθ(.ə)ˈmæt.ɪks/, complicated /ˈkɑm.plɪˌkeɪ.tɪd/, ordinary /ˈɔɹdɪnɛɹi/, simplicity /sɪmˈplɪsɪti/, calculator /kæl.kjə.leɪ.tɚ/
Как заниматься с этим видео
- Прослушайте всё видео один раз молча и выпишите незнакомые слова.
- Начните со скорости 0,75×, повторяйте предложение за предложением и вернитесь к обычной скорости, когда станет легко.
- Запишите себя и сравните с оригиналом, обращая внимание на такие слова, как minus, mathematics, complicated.
Что такое техника Shadowing?
Shadowing — это научно обоснованная техника изучения языка, изначально разработанная для подготовки профессиональных переводчиков и популяризированная полиглотом доктором Александром Аргуэльесом. Метод прост, но эффективен: вы слушаете аудио на английском от носителей языка и немедленно повторяете вслух — как тень, следующая за говорящим с задержкой в 1–2 секунды. В отличие от пассивного прослушивания или грамматических упражнений, Shadowing заставляет мозг и мышцы рта одновременно обрабатывать и воспроизводить реальные речевые паттерны. Исследования показывают, что это значительно улучшает точность произношения, интонацию, ритм, связную речь, понимание на слух и беглость речи — что делает его одним из самых эффективных методов для подготовки к IELTS Speaking и реального общения на английском.