シャドーイング練習: Mathematician Collapses All Functions to One Weird Formula - 動画で英語スピーキングを学ぶ
レッスンを作成中...
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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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このレッスンの語彙とスピーキングのポイント
この動画には、シャドーイング用の文が81文、単語が1038語あります。 音声の長さは6:24です。 話す速さは1分あたり約162語で、日常会話に近い自然なペースです。 英語の頻出3,000語に含まれる単語は79%だけなので、語彙は難しめです。
この動画の重要語彙
動画に出てくる覚えておきたい単語15語を、発音と意味つきで紹介します。
| 単語 | 発音 | 意味 |
|---|---|---|
| minus | /ˈmaɪ.nəs/ | 引く |
| mathematics 名詞 | /mæθ(.ə)ˈmæt.ɪks/ | 数学 |
| complicated 形容詞 | /ˈkɑm.plɪˌkeɪ.tɪd/ | 複雑な |
| earn 動詞 | /ɝn/ | 稼ぐ, 得る |
| practical 形容詞 | /ˈpɹæk.tɪ.kəl/ | 実践的な |
| gate 名詞 | /ˈɡeɪ̯t/ | 門, 大門 |
| formula 名詞 | /ˈfɔː.mjʊ.lə/ | 公式, 定式 |
| ordinary 形容詞 | /ˈɔɹdɪnɛɹi/ | 通常な, 普通な |
| nest 名詞 | /nɛst/ | 巣, ネスト |
| equation 名詞 | /ɪˈkweɪ.ʒən/ | 方程式 |
| calculator 名詞 | /kæl.kjə.leɪ.tɚ/ | 電卓, 計算機 |
| argue 動詞 | /ˈɑɹ.ɡju/ | 議論する, 論ずる |
| explanation 名詞 | /ˌɛkspləˈneɪʃən/ | 説明 |
| physics 名詞 | /ˈfɪz.ɪks/ | 物理学 |
| angle 名詞 | /ˈæŋ.ɡəl/ | 角度 |
動画に出てくる句動詞
| 単語 | 意味 |
|---|---|
| take over 動詞 | 引き受ける |
この動画の文法
話し手がよく使っている文型を、動画の実際の表現とともに紹介します。
| 文型 | 動画での表現 |
|---|---|
| 受動態 be + 過去分詞 — 誰がするかより、何が起きるかに焦点を当てる | is rated · be prepared · are limited |
| 関係詞節 who / which + 節 — 人や物について情報を加える | NAND, which is · regression, which means · those who don |
| 現在完了形 have/has + 過去分詞 — 過去の出来事が今も関係している | has done · has already built |
注意したい発音
話し手はdon't, it'd, doesn'tなど、短縮形や弱形を14回使っています。聞こえたとおりの短い形で発音しましょう。
- 「th」の音: mathematics /mæθ(.ə)ˈmæt.ɪks/, mathematical /ˌmæθ(.ə)ˈmæt.ɪ.kəl/, logarithm /ˈlɑ.ɡə.ɹɪ.ð(ə)m/, mathematician /ˌmæθ.(ə.)məˈtɪʃ.ən/
- 「sh」と「zh」の音: equation /ɪˈkweɪ.ʒən/, explanation /ˌɛkspləˈneɪʃən/, structural /ˈstɹʌk(t͡)ʃəɹəl/, ambitious /æmˈbɪʃ.əs/, compilation /ˌkɑm.pɪˈleɪ.ʃən/
- 長い単語（アクセントの位置に注意）: mathematics /mæθ(.ə)ˈmæt.ɪks/, complicated /ˈkɑm.plɪˌkeɪ.tɪd/, ordinary /ˈɔɹdɪnɛɹi/, simplicity /sɪmˈplɪsɪti/, calculator /kæl.kjə.leɪ.tɚ/
この動画での練習方法
- まず声を出さずに動画を最後まで聞き、知らない単語をメモします。
- まず0.75倍速で一文ずつシャドーイングし、慣れてきたら通常の速度に戻します。
- 自分の声を録音して元の音声と比べます。minus, mathematics, complicatedなどの単語に特に注意しましょう。
シャドーイングとは？英語上達に効果的な理由
シャドーイング（Shadowing）は、もともとプロの通訳者養成プログラムで開発された言語学習法で、多言語習得者として知られるDr. Alexander Arguelles によって広く普及されました。方法はシンプルですが非常に効果的：ネイティブスピーカーの英語を聞きながら、1〜2秒の遅延で声に出してすぐに繰り返す——まるで「影（shadow）」のように話者を追いかけます。文法ドリルや受動的なリスニングと異なり、シャドーイングは脳と口の筋肉が同時にリアルタイムで英語を処理・再現することを強制します。研究により、発音精度、抑揚、リズム、連音、リスニング力、そして会話の流暢さが大幅に向上することが確認されています。IELTSスピーキング対策や自然な英語コミュニケーションを目指す方に特におすすめです。