Shadowing Practice: Mathematician Collapses All Functions to One Weird Formula - Learn English Speaking with Video
Creating lesson...
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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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Vocabulary and speaking notes for this lesson
This video has 81 sentences and 1038 words to shadow. The speech runs for 6:24. The speaker talks at a natural 162 words per minute, close to everyday conversation. Only 79% of the words are among the 3,000 most common in English, so the vocabulary is demanding.
Key vocabulary in this video
15 words from the video worth learning, with pronunciation and meaning:
| Word | Pronunciation | Meaning |
|---|---|---|
| minus | /ˈmaɪ.nəs/ | Made less or reduced by (followed by an expression of number or quantity). |
| mathematics noun | /mæθ(.ə)ˈmæt.ɪks/ | An abstract representational system studying numbers, shapes, structures, quantitative change and relationships between them. |
| complicated adjective | /ˈkɑm.plɪˌkeɪ.tɪd/ | Difficult or convoluted. |
| earn verb | /ɝn/ | To gain (success, reward, recognition) through applied effort or work. |
| practical adjective | /ˈpɹæk.tɪ.kəl/ | Relating to, or based on, practice or action rather than theory or hypothesis. |
| gate noun | /ˈɡeɪ̯t/ | A doorlike structure outside a house. |
| formula noun | /ˈfɔː.mjʊ.lə/ | Any mathematical rule expressed symbolically. |
| ordinary adjective | /ˈɔɹdɪnɛɹi/ | Having regular jurisdiction; now only used in certain phrases. |
| define verb | /dɪˈfaɪn/ | To determine with precision; to mark out with distinctness; to ascertain or exhibit clearly. |
| logical adjective | /ˈlɑd͡ʒɪkəl/ | In agreement with the principles of logic. |
| slot noun | /slɑt/ | A narrow depression, perforation, or aperture; especially, one for the reception of a piece fitting or sliding into it. |
| nest noun | /nɛst/ | A structure built by a bird as a place to incubate eggs and rear young. |
| equation noun | /ɪˈkweɪ.ʒən/ | The act or process of equating two or more things, or the state of those things being equal (that is, identical). |
| simplicity noun | /sɪmˈplɪsɪti/ | The state or quality of being simple |
| calculator noun | /kæl.kjə.leɪ.tɚ/ | A mechanical or electronic device that performs mathematical calculations; (now usually) an electronic one specifically. |
Phrasal verbs you will hear
| Word | Meaning |
|---|---|
| back out verb | To reverse (a vehicle) from a confined space. |
| take over verb | To assume control of something, such as a business or enterprise, and sometimes by force. |
| work through verb | To get past a difficult or stressful situation by thinking or talking about it. |
Grammar in this video
The structures the speaker uses most, with the exact words from the video:
| Structure | In the video |
|---|---|
| Passive voice be + past participle — the focus is on what happens, not who does it | is rated · be prepared · are limited |
| Relative clauses who / which + clause — extra information about a person or thing | NAND, which is · regression, which means · those who don |
| Present perfect have/has + past participle — a past action that still matters now | has done · has already built |
Pronunciation to watch
The speaker uses 14 contractions and reduced forms, such as don't, it'd, doesn't. Say them the short way, as you hear them.
- The “th” sounds: mathematics /mæθ(.ə)ˈmæt.ɪks/, mathematical /ˌmæθ(.ə)ˈmæt.ɪ.kəl/, logarithm /ˈlɑ.ɡə.ɹɪ.ð(ə)m/, mathematician /ˌmæθ.(ə.)məˈtɪʃ.ən/
- The “sh” and “zh” sounds: equation /ɪˈkweɪ.ʒən/, explanation /ˌɛkspləˈneɪʃən/, structural /ˈstɹʌk(t͡)ʃəɹəl/, ambitious /æmˈbɪʃ.əs/, compilation /ˌkɑm.pɪˈleɪ.ʃən/
- Long words — get the stress right: mathematics /mæθ(.ə)ˈmæt.ɪks/, complicated /ˈkɑm.plɪˌkeɪ.tɪd/, ordinary /ˈɔɹdɪnɛɹi/, simplicity /sɪmˈplɪsɪti/, calculator /kæl.kjə.leɪ.tɚ/
How to practise with this video
- Listen to the whole video once without speaking and note the words you do not know.
- Start at 0.75× speed, shadow it sentence by sentence, then go back to normal speed once it feels easy.
- Record yourself and compare with the original, paying attention to words like minus, mathematics, complicated.
What is the Shadowing Technique?
Shadowing is a science-backed language learning technique originally developed for professional interpreter training and popularized by polyglot Dr. Alexander Arguelles. The method is simple but powerful: you listen to native English audio and immediately repeat it out loud — like a shadow following the speaker with just a 1–2 second delay. Unlike passive listening or grammar drills, shadowing forces your brain and mouth muscles to simultaneously process and reproduce real speech patterns. Research shows it significantly improves pronunciation accuracy, intonation, rhythm, connected speech, listening comprehension, and speaking fluency — making it one of the most effective methods for IELTS Speaking preparation and real-world English communication.