Shadowing Practice: Mathematician Collapses All Functions to One Weird Formula - Learn English Speaking with Video
Ders oluşturuluyor...
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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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Bu dersin kelimeleri ve konuşma notları
Bu videoda gölgeleme çalışması için 81 cümle ve 1038 kelime var. Konuşma bölümü 6:24 sürüyor. Konuşmacı dakikada yaklaşık 162 kelimeyle, günlük konuşmaya yakın doğal bir hızda konuşuyor. Kelimelerin yalnızca %79’i İngilizcede en sık kullanılan 3.000 kelime arasında, bu yüzden kelime dağarcığı zorlayıcı.
Bu videodaki önemli kelimeler
Videodan öğrenmeye değer 15 kelime, telaffuzu ve anlamıyla:
| Kelime | Telaffuz | Anlam |
|---|---|---|
| minus | /ˈmaɪ.nəs/ | eksi |
| mathematics isim | /mæθ(.ə)ˈmæt.ɪks/ | matematik |
| complicated sıfat | /ˈkɑm.plɪˌkeɪ.tɪd/ | karışık, komplike |
| earn fiil | /ɝn/ | hak etmek |
| gate isim | /ˈɡeɪ̯t/ | kapı |
| formula isim | /ˈfɔː.mjʊ.lə/ | formül |
| ordinary sıfat | /ˈɔɹdɪnɛɹi/ | sıradan, olağan |
| nest isim | /nɛst/ | yuva |
| equation isim | /ɪˈkweɪ.ʒən/ | denklem |
| simplicity isim | /sɪmˈplɪsɪti/ | basitlik |
| calculator isim | /kæl.kjə.leɪ.tɚ/ | hesap makinesi |
| argue fiil | /ˈɑɹ.ɡju/ | münakaşa etmek, tartışmak |
| explanation isim | /ˌɛkspləˈneɪʃən/ | açıklama |
| physics isim | /ˈfɪz.ɪks/ | fizik, doğam (doğa kuramı) |
| angle isim | /ˈæŋ.ɡəl/ | açı |
Bu videodaki dil bilgisi
Konuşmacının en çok kullandığı yapılar, videodaki sözcüklerin aynısıyla:
| Yapı | Videoda |
|---|---|
| Edilgen yapı be + fiilin üçüncü hâli — kimin yaptığı değil, ne olduğu önemli | is rated · be prepared · are limited |
| İlgi cümlecikleri who / which + cümle — bir kişi ya da şey hakkında ek bilgi | NAND, which is · regression, which means · those who don |
| Present perfect have/has + fiilin üçüncü hâli — geçmişte olan ama şimdi de önemli olan bir eylem | has done · has already built |
Dikkat edilecek telaffuzlar
Konuşmacı don't, it'd, doesn't gibi kısaltılmış biçimleri 14 kez kullanıyor. Bunları duyduğunuz gibi kısa söyleyin.
- “th” sesleri: mathematics /mæθ(.ə)ˈmæt.ɪks/, mathematical /ˌmæθ(.ə)ˈmæt.ɪ.kəl/, logarithm /ˈlɑ.ɡə.ɹɪ.ð(ə)m/, mathematician /ˌmæθ.(ə.)məˈtɪʃ.ən/
- “sh” ve “zh” sesleri: equation /ɪˈkweɪ.ʒən/, explanation /ˌɛkspləˈneɪʃən/, structural /ˈstɹʌk(t͡)ʃəɹəl/, ambitious /æmˈbɪʃ.əs/, compilation /ˌkɑm.pɪˈleɪ.ʃən/
- Uzun kelimeler — vurguyu doğru yere koyun: mathematics /mæθ(.ə)ˈmæt.ɪks/, complicated /ˈkɑm.plɪˌkeɪ.tɪd/, ordinary /ˈɔɹdɪnɛɹi/, simplicity /sɪmˈplɪsɪti/, calculator /kæl.kjə.leɪ.tɚ/
Bu videoyla nasıl çalışılır
- Videonun tamamını konuşmadan bir kez dinleyin ve bilmediğiniz kelimeleri not edin.
- 0,75× hızla başlayın, cümle cümle tekrar edin ve kolaylaşınca normal hıza dönün.
- Kendinizi kaydedin ve orijinaliyle karşılaştırın; minus, mathematics, complicated gibi kelimelere özellikle dikkat edin.
Gölgeleme Tekniği Nedir?
Gölgeleme, başlangıçta profesyonel tercüman eğitimi için geliştirilen ve çok dilli Dr. Alexander Arguelles tarafından popüler hale getirilen, bilim destekli bir dil öğrenme tekniğidir. Yöntem basit ama güçlüdür: ana dili İngilizce olan bir sesi dinler ve hemen yüksek sesle tekrar edersiniz — konuşmacıyı 1-2 saniye gecikmeyle takip eden bir gölge gibi. Pasif dinleme veya dilbilgisi alıştırmalarının aksine, gölgeleme beyninizi ve ağız kaslarınızı gerçek konuşma kalıplarını eşzamanlı olarak işlemeye ve yeniden üretmeye zorlar. Araştırmalar, telaffuz doğruluğu, tonlama, ritim, bağlı konuşma, dinleme anlama ve konuşma akıcılığını önemli ölçüde geliştirdiğini göstermektedir — bu da onu IELTS Konuşma hazırlığı ve gerçek dünya İngilizce iletişimi için en etkili yöntemlerden biri yapar.