Shadowing-Übung: Mathematician Collapses All Functions to One Weird Formula - Englisch Sprechen Lernen mit Video
Lektion wird erstellt...
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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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Wortschatz und Sprechhinweise zu dieser Lektion
Dieses Video enthält 81 Sätze und 1038 Wörter zum Nachsprechen. Der gesprochene Teil dauert 6:24. Der Sprecher spricht in natürlichem Tempo, etwa 162 Wörter pro Minute, ähnlich wie im Alltagsgespräch. Nur 79 % der Wörter gehören zu den 3.000 häufigsten im Englischen, der Wortschatz ist also anspruchsvoll.
Wichtiger Wortschatz in diesem Video
15 lernenswerte Wörter aus dem Video, mit Aussprache und Bedeutung:
| Wort | Aussprache | Bedeutung |
|---|---|---|
| minus | /ˈmaɪ.nəs/ | weniger, minus |
| mathematics Substantiv | /mæθ(.ə)ˈmæt.ɪks/ | Mathematik |
| complicated Adjektiv | /ˈkɑm.plɪˌkeɪ.tɪd/ | kompliziert, umständlich |
| earn Verb | /ɝn/ | verdienen |
| practical Adjektiv | /ˈpɹæk.tɪ.kəl/ | praktisch |
| gate Substantiv | /ˈɡeɪ̯t/ | Tor |
| formula Substantiv | /ˈfɔː.mjʊ.lə/ | Formel |
| ordinary Adjektiv | /ˈɔɹdɪnɛɹi/ | gewöhnlich, normal |
| define Verb | /dɪˈfaɪn/ | bestimmen |
| logical Adjektiv | /ˈlɑd͡ʒɪkəl/ | logisch |
| slot Substantiv | /slɑt/ | Schlitz, Nut |
| nest Substantiv | /nɛst/ | Nest |
| equation Substantiv | /ɪˈkweɪ.ʒən/ | Ausgleich, Gleichgewicht |
| simplicity Substantiv | /sɪmˈplɪsɪti/ | Einfachheit |
| calculator Substantiv | /kæl.kjə.leɪ.tɚ/ | Rechner, Taschenrechner |
Phrasal Verbs, die du hören wirst
| Wort | Bedeutung |
|---|---|
| back out Verb | sich zurückziehen, aussteigen |
| take over Verb | übernehmen |
Grammatik in diesem Video
Die Strukturen, die der Sprecher am häufigsten verwendet, mit den genauen Worten aus dem Video:
| Struktur | Im Video |
|---|---|
| Passiv be + Partizip Perfekt – wichtig ist, was geschieht, nicht wer es tut | is rated · be prepared · are limited |
| Relativsätze who / which + Satz – eine Zusatzinformation über eine Person oder Sache | NAND, which is · regression, which means · those who don |
| Present Perfect have/has + Partizip Perfekt – eine vergangene Handlung, die jetzt noch wichtig ist | has done · has already built |
Aussprache, auf die du achten solltest
Der Sprecher verwendet 14 Kurzformen und abgeschwächte Formen, zum Beispiel don't, it'd, doesn't. Sprich sie in der kurzen Form, so wie du sie hörst.
- Die „th“-Laute: mathematics /mæθ(.ə)ˈmæt.ɪks/, mathematical /ˌmæθ(.ə)ˈmæt.ɪ.kəl/, logarithm /ˈlɑ.ɡə.ɹɪ.ð(ə)m/, mathematician /ˌmæθ.(ə.)məˈtɪʃ.ən/
- Die Laute „sh“ und „zh“: equation /ɪˈkweɪ.ʒən/, explanation /ˌɛkspləˈneɪʃən/, structural /ˈstɹʌk(t͡)ʃəɹəl/, ambitious /æmˈbɪʃ.əs/, compilation /ˌkɑm.pɪˈleɪ.ʃən/
- Lange Wörter – auf die Betonung achten: mathematics /mæθ(.ə)ˈmæt.ɪks/, complicated /ˈkɑm.plɪˌkeɪ.tɪd/, ordinary /ˈɔɹdɪnɛɹi/, simplicity /sɪmˈplɪsɪti/, calculator /kæl.kjə.leɪ.tɚ/
So übst du mit diesem Video
- Höre dir das ganze Video einmal an, ohne zu sprechen, und notiere die Wörter, die du nicht kennst.
- Beginne mit 0,75-facher Geschwindigkeit, sprich Satz für Satz nach und wechsle zur normalen Geschwindigkeit, sobald es leichtfällt.
- Nimm dich auf und vergleiche mit dem Original; achte dabei auf Wörter wie minus, mathematics, complicated.
Was ist die Shadowing-Technik?
Shadowing ist eine wissenschaftlich fundierte Sprachlerntechnik, die ursprünglich für die professionelle Dolmetscherausbildung entwickelt und durch den Polyglotten Dr. Alexander Arguelles populär gemacht wurde. Die Methode ist einfach aber wirkungsvoll: Du hörst englisches Audio von Muttersprachlern und wiederholst es sofort laut — wie ein Schatten, der dem Sprecher mit nur 1–2 Sekunden Verzögerung folgt. Anders als passives Hören oder Grammatikübungen zwingt Shadowing dein Gehirn und deine Mundmuskulatur, gleichzeitig echte Sprachmuster zu verarbeiten und zu reproduzieren. Studien zeigen, dass es Aussprachegenauigkeit, Intonation, Rhythmus, verbundene Sprache, Hörverständnis und Sprechflüssigkeit signifikant verbessert — was es zu einer der effektivsten Methoden für die IELTS Speaking-Vorbereitung und reale englische Kommunikation macht.