쉐도잉 연습: Mathematician Collapses All Functions to One Weird Formula - 영상으로 영어 말하기 배우기
레슨 만드는 중...
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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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이 영상에는 섀도잉할 문장 81개와 단어 1038개가 있습니다. 말하는 구간의 길이는 6:24입니다. 화자는 분당 약 162단어로, 일상 대화에 가까운 자연스러운 속도로 말합니다. 영어에서 가장 많이 쓰이는 3,000단어에 속하는 단어가 79%뿐이라 어휘가 어려운 편입니다.
이 영상의 핵심 어휘
영상에 나오는 익혀 둘 만한 단어 15개를 발음, 뜻과 함께 정리했습니다.
| 단어 | 발음 | 뜻 |
|---|---|---|
| minus | /ˈmaɪ.nəs/ | 빼기 |
| mathematics 명사 | /mæθ(.ə)ˈmæt.ɪks/ | 수학 |
| complicated 형용사 | /ˈkɑm.plɪˌkeɪ.tɪd/ | 복잡하다 |
| earn 동사 | /ɝn/ | 벌다 |
| practical 형용사 | /ˈpɹæk.tɪ.kəl/ | 실질적인 |
| gate 명사 | /ˈɡeɪ̯t/ | 문, 대문 |
| formula 명사 | /ˈfɔː.mjʊ.lə/ | 공식 |
| ordinary 형용사 | /ˈɔɹdɪnɛɹi/ | 정규의, 보통의 |
| nest 명사 | /nɛst/ | 둥지, 새집 |
| equation 명사 | /ɪˈkweɪ.ʒən/ | 방정식 |
| calculator 명사 | /kæl.kjə.leɪ.tɚ/ | 계산기 |
| argue 동사 | /ˈɑɹ.ɡju/ | 논하다, 의논하다 |
| explanation 명사 | /ˌɛkspləˈneɪʃən/ | 설명 |
| physics 명사 | /ˈfɪz.ɪks/ | 물리학, 물리 |
| angle 명사 | /ˈæŋ.ɡəl/ | 각, 각도 |
이 영상의 문법
화자가 가장 많이 쓰는 문형을 영상 속 실제 표현과 함께 정리했습니다.
| 문형 | 영상 속 표현 |
|---|---|
| 수동태 be + 과거분사 — 누가 하는지보다 무슨 일이 일어나는지에 초점 | is rated · be prepared · are limited |
| 관계절 who / which + 절 — 사람이나 사물에 대한 추가 정보 | NAND, which is · regression, which means · those who don |
| 현재완료 have/has + 과거분사 — 과거의 일이 지금도 관련이 있을 때 | has done · has already built |
주의할 발음
화자는 don't, it'd, doesn't 같은 축약형과 약화된 형태를 14번 사용합니다. 들리는 대로 짧게 발음하세요.
- “th” 소리: mathematics /mæθ(.ə)ˈmæt.ɪks/, mathematical /ˌmæθ(.ə)ˈmæt.ɪ.kəl/, logarithm /ˈlɑ.ɡə.ɹɪ.ð(ə)m/, mathematician /ˌmæθ.(ə.)məˈtɪʃ.ən/
- “sh”와 “zh” 소리: equation /ɪˈkweɪ.ʒən/, explanation /ˌɛkspləˈneɪʃən/, structural /ˈstɹʌk(t͡)ʃəɹəl/, ambitious /æmˈbɪʃ.əs/, compilation /ˌkɑm.pɪˈleɪ.ʃən/
- 긴 단어 — 강세 위치에 주의: mathematics /mæθ(.ə)ˈmæt.ɪks/, complicated /ˈkɑm.plɪˌkeɪ.tɪd/, ordinary /ˈɔɹdɪnɛɹi/, simplicity /sɪmˈplɪsɪti/, calculator /kæl.kjə.leɪ.tɚ/
이 영상으로 연습하는 방법
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쉐도잉이란? 영어 실력을 빠르게 키우는 과학적 방법
쉐도잉(Shadowing)은 원래 전문 통역사 훈련을 위해 개발된 언어 학습 기법으로, 다언어 학자인 Dr. Alexander Arguelles에 의해 대중화된 방법입니다. 핵심 원리는 간단하지만 매우 강력합니다: 원어민의 영어를 들으면서 1~2초의 짧은 지연으로 즉시 소리 내어 따라 말하는 것——마치 '그림자(shadow)'처럼 화자를 따라가는 것입니다. 문법 공부나 수동적인 청취와 달리, 쉐도잉은 뇌와 입 근육이 동시에 실시간으로 영어를 처리하고 재현하도록 훈련합니다. 연구에 따르면 이 방법은 발음 정확도, 억양, 리듬, 연음, 청취력, 말하기 유창성을 크게 향상시킵니다. IELTS 스피킹 준비와 자연스러운 영어 소통을 원하는 분들에게 특히 효과적입니다.