跟讀練習: Mathematician Collapses All Functions to One Weird Formula - 通過影片學習英語口語

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plus, minus, x times y, x to the power of y, sine and cosine.

Maths is full with complicated operations that you do to numbers.

An ambitious new paper now claims that actually you can do everything with only one operation.

That's quite a claim.

Maybe that finally explains why the Americans have only one math, where the Brits have maths.

I've had a look.

The new paper's a preprint that appeared on the archive the other week and comes from a mathematician in Poland.

The one operation

that he says can give everything is taking the exponential function of a number x minus the logarithm of another number y.

He calls that ML, short for exponential minus logarithm.

For example, if you put x and 1 into the ML function, you get e to the x minus log of 1.

Log of 1 is 0, so this gives you just the exponential function.

You can also get the logarithm itself back out.

For this, you first take ml of 1 and x, that gives you e minus log x.

Then you put that back into the first slot with another one.

So ml of ml 1 x 1, that gives e to the e over x.

And then you put that yet again into the second slot of ml with a 1 in the first.

If you work through the algebra, all the extra pieces cancel and you're left with just log of x.

So the same function gives you exponentials and logarithms indirectly.

But how do we get the standard operations, addition, subtraction, multiplication, etc.? They're all in there.

Take subtraction.

We just saw that you can express the exponential function and the logarithm through ML operations.

If you put those back into the ML function, what do you get?

You get x minus y.

You can also get a plus just by entering minus y.

And now you see you can put the sum of two logarithms into the exponent of the exponential function.

That'll give you multiplication.

And once you have multiplication, you can define powers and so on.

If you want to get trigonometric functions, you first need imaginary numbers.

You can get those from the logarithm at negative numbers, though there's a branch cut involved there.

I know that sounds like a gardening accident, but it just defines the logarithm for a negative number.

But once you have that, you get complex numbers, and you get sine and cosine and even the number pi.

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.

But the author doesn't make that claim.

He says one can get all mathematical operations.

The paper hasn't yet been peer -reviewed, but I'm reasonably sure it's correct.

I saw a lot of people online worry about the arbitrary branch cut of the logarithm, but I don't think this matters.

Yes, you could choose it differently, and that it'd give you a different definition of complex numbers, but it'd still work.

Now, you may be wondering whether this is actually useful.

The answer is, for normal people, probably not.

No one's going to replace a normal calculator with one

that has a single deranged button that expects you to rebuild cosine by nesting 15 different operations.

The interesting part is structural.

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.

This paper's trying to find the analog for ordinary continuous maths.

The claim is that EML is kind of a NAND gate for scientific calculators.

And there is a possible practical angle.

If every ordinary formula can be rewritten as a tree made from one repeated building block,

then that gives you a very uniform way to search for formulas with a computer.

The author suggests this could be useful for symbolic regression, which means trying to guess the equation behind a set of data.

So this may be less about helping humans do maths

and more about giving machines a simpler language in which to search for equations,

which is nice because if the machines are going to take over science, then at least they will do it in an orderly manner.

I give this paper a 0 out of 10 on the bullshit meter.

It's a cute paper.

I think that the point isn't super surprising to anyone, but it is surprising that no one has done it before.

There are two ways one can look at this result.

One is that mathematics appears more complicated than it is.

The other way to look at it is that what we think of as complicated is very subjective.

You could rightfully argue that maybe using 10 or

so different operations is easier than nesting just one operation 20 times in itself.

In the end, it depends on just how you quantify simplicity.

This is also an issue that haunts the foundations of physics, because what really do we mean when we say

that simple explanations are better if we don't agree on how to measure simplicity?

Or maybe it just tells us that mathematicians have too much time.

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本課的詞彙與口語要點

這段影片共有 81 個句子、1038 個單詞可供跟讀。講話部分時長為 6:24。說話人語速自然，每分鐘約 162 個詞，接近日常對話。只有 79% 的單詞屬於英語最常用的 3,000 詞，詞彙難度較高。

影片中的重點詞彙

影片中 15 個值得學習的單詞，附發音和釋義：

單詞	發音	釋義
minus	/ˈmaɪ.nəs/	減 /減
mathematics 名詞	/mæθ(.ə)ˈmæt.ɪks/	數學 /數學, 算學 /算學
complicated 形容詞	/ˈkɑm.plɪˌkeɪ.tɪd/	複雜 /複雜
earn 動詞	/ɝn/	掙 /掙, 賺 /賺
practical 形容詞	/ˈpɹæk.tɪ.kəl/	使用的, 實用 /實用
gate 名詞	/ˈɡeɪ̯t/	門口 /門口, 門 /門
formula 名詞	/ˈfɔː.mjʊ.lə/	公式
ordinary 形容詞	/ˈɔɹdɪnɛɹi/	普通, 平常
define 動詞	/dɪˈfaɪn/	界定
nest 名詞	/nɛst/	вә, 岫
equation 名詞	/ɪˈkweɪ.ʒən/	方程式, 方程
calculator 名詞	/kæl.kjə.leɪ.tɚ/	計算器 /計算器, 電子計算器 /電子計算器
argue 動詞	/ˈɑɹ.ɡju/	辯論 /辯論, 爭辯 /爭辯
explanation 名詞	/ˌɛkspləˈneɪʃən/	解釋 /解釋, 說明 /說明
physics 名詞	/ˈfɪz.ɪks/	物理學 /物理學

影片中出現的短語動詞

單詞	釋義
take over 動詞	接管, 接手

影片中的語法

說話人最常用的結構，並附上影片中的原話：

結構	影片中的用法
現在完成時 have/has + 過去分詞 — 過去發生但與現在仍有關聯的事	I've had · has done · has already built
定語從句 who / which + 從句 — 補充說明人或事物	NAND, which is · regression, which means · those who don
被動語態 be + 過去分詞 — 強調發生了什麼，而不是誰做的	can be rewritten · is rated

需要注意的發音

說話人用了 14 次縮略和弱讀形式，例如 don't, it'd, doesn't。請按聽到的簡短形式來說。

“th” 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” 和 “zh” 音: equation /ɪˈkweɪ.ʒən/, explanation /ˌɛkspləˈneɪʃən/, structural /ˈstɹʌk(t͡)ʃəɹəl/, ambitious /æmˈbɪʃ.əs/, compilation /ˌkɑm.pɪˈleɪ.ʃən/
長單詞——注意重音位置: mathematics /mæθ(.ə)ˈmæt.ɪks/, complicated /ˈkɑm.plɪˌkeɪ.tɪd/, ordinary /ˈɔɹdɪnɛɹi/, simplicity /sɪmˈplɪsɪti/, calculator /kæl.kjə.leɪ.tɚ/

如何用這段影片練習

先完整聽一遍影片，不要開口，記下不認識的單詞。
先用 0.75 倍速逐句跟讀，熟練之後再回到正常速度。
錄下自己的聲音並與原聲對比，特別注意 minus, mathematics, complicated 這類單詞。

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跟讀練習: Mathematician Collapses All Functions to One Weird Formula - 通過影片學習英語口語

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