Shadowing Practice: Mathematician Collapses All Functions to One Weird Formula - Learn English Speaking with Video
Les maken...
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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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Woordenschat en spreektips bij deze les
Deze video bevat 81 zinnen en 1038 woorden om na te spreken. Het gesproken deel duurt 6:24. De spreker praat in een natuurlijk tempo van ongeveer 162 woorden per minuut, dicht bij een alledaags gesprek. Slechts 79% van de woorden hoort bij de 3.000 meest gebruikte Engelse woorden, dus de woordenschat is pittig.
Belangrijke woorden in deze video
15 woorden uit de video die het leren waard zijn, met uitspraak en betekenis:
| Woord | Uitspraak | Betekenis |
|---|---|---|
| minus | /ˈmaɪ.nəs/ | min |
| mathematics zelfstandig naamwoord | /mæθ(.ə)ˈmæt.ɪks/ | wiskunde, mathematiek |
| complicated bijvoeglijk naamwoord | /ˈkɑm.plɪˌkeɪ.tɪd/ | ingewikkeld, gecompliceerd |
| earn werkwoord | /ɝn/ | verdienen |
| practical bijvoeglijk naamwoord | /ˈpɹæk.tɪ.kəl/ | praktisch |
| gate zelfstandig naamwoord | /ˈɡeɪ̯t/ | poort |
| formula zelfstandig naamwoord | /ˈfɔː.mjʊ.lə/ | formule |
| ordinary bijvoeglijk naamwoord | /ˈɔɹdɪnɛɹi/ | gewoon, alledaags |
| define werkwoord | /dɪˈfaɪn/ | bepalen, determineren |
| logical bijvoeglijk naamwoord | /ˈlɑd͡ʒɪkəl/ | logisch |
| nest zelfstandig naamwoord | /nɛst/ | nest |
| equation zelfstandig naamwoord | /ɪˈkweɪ.ʒən/ | vergelijking |
| calculator zelfstandig naamwoord | /kæl.kjə.leɪ.tɚ/ | rekenmachine, zakrekenmachine |
| argue werkwoord | /ˈɑɹ.ɡju/ | redetwisten |
| explanation zelfstandig naamwoord | /ˌɛkspləˈneɪʃən/ | verklaring, toelichting |
Phrasal verbs die je zult horen
| Woord | Betekenis |
|---|---|
| back out werkwoord | zich terugtrekken |
| take over werkwoord | overnemen |
Grammatica in deze video
De structuren die de spreker het meest gebruikt, met de exacte woorden uit de video:
| Structuur | In de video |
|---|---|
| Lijdende vorm be + voltooid deelwoord — het gaat om wat er gebeurt, niet om wie het doet | is rated · be prepared · are limited |
| Betrekkelijke bijzinnen who / which + zin — extra informatie over een persoon of ding | NAND, which is · regression, which means · those who don |
| Present perfect have/has + voltooid deelwoord — iets uit het verleden dat nu nog telt | has done · has already built |
Uitspraak om op te letten
De spreker gebruikt 14 samentrekkingen en verkorte vormen, zoals don't, it'd, doesn't. Spreek ze kort uit, zoals je ze hoort.
- De “th”-klanken: mathematics /mæθ(.ə)ˈmæt.ɪks/, mathematical /ˌmæθ(.ə)ˈmæt.ɪ.kəl/, logarithm /ˈlɑ.ɡə.ɹɪ.ð(ə)m/, mathematician /ˌmæθ.(ə.)məˈtɪʃ.ən/
- De klanken “sh” en “zh”: equation /ɪˈkweɪ.ʒən/, explanation /ˌɛkspləˈneɪʃən/, structural /ˈstɹʌk(t͡)ʃəɹəl/, ambitious /æmˈbɪʃ.əs/, compilation /ˌkɑm.pɪˈleɪ.ʃən/
- Lange woorden — let op de klemtoon: mathematics /mæθ(.ə)ˈmæt.ɪks/, complicated /ˈkɑm.plɪˌkeɪ.tɪd/, ordinary /ˈɔɹdɪnɛɹi/, simplicity /sɪmˈplɪsɪti/, calculator /kæl.kjə.leɪ.tɚ/
Zo oefen je met deze video
- Luister de hele video één keer zonder te spreken en noteer de woorden die je niet kent.
- Begin op 0,75× snelheid, spreek zin voor zin na en ga terug naar normale snelheid zodra het makkelijk gaat.
- Neem jezelf op en vergelijk met het origineel; let daarbij op woorden als minus, mathematics, complicated.
Wat is de Shadowing-techniek?
Shadowing is een wetenschappelijk onderbouwde taalleermethode die oorspronkelijk is ontwikkeld voor professionele tolkentraining en gepopulariseerd door polyglot Dr. Alexander Arguelles. De methode is eenvoudig maar krachtig: je luistert naar native Engelse audio en herhaalt het onmiddellijk hardop — als een schaduw die de spreker volgt met slechts 1–2 seconden vertraging. In tegenstelling tot passief luisteren of grammaticadrills, dwingt shadowing je hersenen en mondspieren om echte spraakpatronen tegelijkertijd te verwerken en te reproduceren. Onderzoek toont aan dat het de uitspraaknauwkeurigheid, intonatie, ritme, verbonden spraak, luisterbegrip en spreekvaardigheid aanzienlijk verbetert — waardoor het een van de meest effectieve methoden is voor IELTS Speaking-voorbereiding en echte Engelse communicatie.