Prática de Shadowing: Mathematician Collapses All Functions to One Weird Formula - Aprenda a falar inglês com vídeo
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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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Vocabulário e dicas de fala para esta lição
Este vídeo tem 81 frases e 1038 palavras para praticar shadowing. A fala dura 6:24. O falante fala em ritmo natural, cerca de 162 palavras por minuto, próximo de uma conversa do dia a dia. Apenas 79% das palavras estão entre as 3.000 mais comuns do inglês, por isso o vocabulário é exigente.
Vocabulário principal deste vídeo
15 palavras do vídeo que vale a pena aprender, com pronúncia e significado:
| Palavra | Pronúncia | Significado |
|---|---|---|
| minus | /ˈmaɪ.nəs/ | menos |
| mathematics substantivo | /mæθ(.ə)ˈmæt.ɪks/ | matemática |
| complicated adjetivo | /ˈkɑm.plɪˌkeɪ.tɪd/ | complicado |
| earn verbo | /ɝn/ | receber |
| practical adjetivo | /ˈpɹæk.tɪ.kəl/ | prático |
| gate substantivo | /ˈɡeɪ̯t/ | portão |
| formula substantivo | /ˈfɔː.mjʊ.lə/ | fórmula |
| ordinary adjetivo | /ˈɔɹdɪnɛɹi/ | ordinário, normal |
| define verbo | /dɪˈfaɪn/ | definir |
| logical adjetivo | /ˈlɑd͡ʒɪkəl/ | lógico |
| slot substantivo | /slɑt/ | fenda, ranhura |
| nest substantivo | /nɛst/ | ninho |
| equation substantivo | /ɪˈkweɪ.ʒən/ | equação |
| simplicity substantivo | /sɪmˈplɪsɪti/ | simplicidade |
| calculator substantivo | /kæl.kjə.leɪ.tɚ/ | calculadora |
Phrasal verbs que você vai ouvir
| Palavra | Significado |
|---|---|
| take over verbo | tomar posse de, apoderar-se de |
Gramática neste vídeo
As estruturas que o falante mais usa, com as palavras exatas do vídeo:
| Estrutura | No vídeo |
|---|---|
| Voz passiva be + particípio passado — o foco está no que acontece, não em quem faz | is rated · be prepared · are limited |
| Orações relativas who / which + oração — informação extra sobre uma pessoa ou coisa | NAND, which is · regression, which means · those who don |
| Present perfect have/has + particípio passado — uma ação passada que ainda importa agora | has done · has already built |
Pronúncia para ficar de olho
O falante usa 14 contrações e formas reduzidas, como don't, it'd, doesn't. Diga-as na forma curta, do jeito que você ouve.
- Os sons de “th”: mathematics /mæθ(.ə)ˈmæt.ɪks/, mathematical /ˌmæθ(.ə)ˈmæt.ɪ.kəl/, logarithm /ˈlɑ.ɡə.ɹɪ.ð(ə)m/, mathematician /ˌmæθ.(ə.)məˈtɪʃ.ən/
- Os sons de “sh” e “zh”: equation /ɪˈkweɪ.ʒən/, explanation /ˌɛkspləˈneɪʃən/, structural /ˈstɹʌk(t͡)ʃəɹəl/, ambitious /æmˈbɪʃ.əs/, compilation /ˌkɑm.pɪˈleɪ.ʃən/
- Palavras longas — acerte a sílaba tônica: mathematics /mæθ(.ə)ˈmæt.ɪks/, complicated /ˈkɑm.plɪˌkeɪ.tɪd/, ordinary /ˈɔɹdɪnɛɹi/, simplicity /sɪmˈplɪsɪti/, calculator /kæl.kjə.leɪ.tɚ/
Como praticar com este vídeo
- Ouça o vídeo inteiro uma vez sem falar e anote as palavras que você não conhece.
- Comece na velocidade 0,75×, faça shadowing frase por frase e volte à velocidade normal quando ficar fácil.
- Grave a sua voz e compare com o original, prestando atenção a palavras como minus, mathematics, complicated.
O que é a Técnica de Shadowing?
Shadowing é uma técnica de aprendizado de idiomas com base científica, originalmente desenvolvida para o treinamento de intérpretes profissionais. O método é simples, mas poderoso: você ouve áudio em inglês nativo e repete imediatamente em voz alta — como uma sombra seguindo o falante com 1-2 segundos de atraso. Pesquisas mostram melhora significativa na precisão da pronúncia, entonação, ritmo, sons conectados, compreensão auditiva e fluência na fala.