Shadowing Practice: Mathematician Collapses All Functions to One Weird Formula - Learn English Speaking with Video
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81 משפטים
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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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See you tomorrow.
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אוצר מילים והערות הגייה לשיעור הזה
שיעור הדיבור הזה ברמה C1 מבוסס על הסרטון “Mathematician Collapses All Functions to One Weird Formula”. המילים שחוזרות הכי הרבה: operation, logarithm, function, minus, exponential. בסרטון הזה יש 81 משפטים ו-1038 מילים לתרגול Shadowing. הדיבור נמשך 6:24. הדובר מדבר בקצב טבעי של כ-162 מילים בדקה, קרוב לשיחה יומיומית. רק 79% מהמילים נמצאות בין 3,000 המילים הנפוצות ביותר באנגלית, כך שאוצר המילים מאתגר.
אוצר מילים מרכזי בסרטון הזה
15 המילים המתקדמות ביותר בסרטון, עם ההגייה והמשמעות שלהן:
| מילה | הגייה | משמעות |
|---|---|---|
| minus | /ˈmaɪ.nəs/ | פחות, מינוס |
| exponential שם תואר | /ˌɛk.spəˈnɛn.t͡ʃəl/ | מעריכי, אקספוננציאלי |
| cosine שם עצם | /ˈkoʊ.saɪn/ | קוסינוס |
| mathematician שם עצם | /ˌmæθ.(ə.)məˈtɪʃ.ən/ | מָתֶמָטִיקַאי |
| sine שם עצם | /saɪn/ | סינוס |
| calculator שם עצם | /kæl.kjə.leɪ.tɚ/ | מַחְשְׁבוֹן, מונה |
| simplicity שם עצם | /sɪmˈplɪsɪti/ | פשטות |
| equation שם עצם | /ɪˈkweɪ.ʒən/ | משוואה \ מִשְׁוָאָה |
| nest שם עצם | /nɛst/ | קֵן |
| exponent שם עצם | /ɛkˈspəʊ.nənt/ | מַעֲרִיך |
| haunt פועל | /hɔnt/ | רָדַף |
| accelerate פועל | /ɪkˈsɛl.əˌɹeɪt/ | האיץ |
| analog שם תואר | /ˈænəlɔɡ/ | אָנָלוֹגִי |
| metre שם עצם | /ˈmiːtəɹ/ | מֶטֶר |
| algebra שם עצם | /ˈæl.d͡ʒɪ.bɹə/ | אַלגֶבּרָה |
דקדוק בסרטון הזה
המבנים שהדובר משתמש בהם הכי הרבה, עם המילים המדויקות מהסרטון:
| מבנה | בסרטון |
|---|---|
| הווה מושלם (Present perfect) have/has + past participle — פעולה שקרתה בעבר ועדיין יש לה חשיבות או השפעה עכשיו | I've had · has done · has already built |
| משפטי זיקה (Relative clauses) who / which + clause — מידע נוסף על אדם או דבר | NAND, which is · regression, which means · those who don |
| סביל (Passive voice) be + past participle — הדגש הוא על מה שקורה, לא על מי שעושה את הפעולה | can be rewritten · is rated |
הגייה ששווה לשים לב אליה
הדובר משתמש ב-14 קיצורים וצורות מקוצרות, כמו don't, it'd, doesn't. אמרו אותם בצורה הקצרה, כפי שאתם שומעים אותם.
- הצלילים של “th”: logarithm /ˈlɑ.ɡə.ɹɪ.ð(ə)m/, mathematician /ˌmæθ.(ə.)məˈtɪʃ.ən/, mathematical /ˌmæθ(.ə)ˈmæt.ɪ.kəl/
- הצלילים של “sh” ו-“zh”: multiplication /ˌmʌltɪplɪˈkeɪʃən/, mathematician /ˌmæθ.(ə.)məˈtɪʃ.ən/, subtraction /səbˈtɹækʃən/, equation /ɪˈkweɪ.ʒən/, regression /ɹiːˈɡɹɛʃ.ən/
- מילים ארוכות — שימו לב להטעמה: logarithm /ˈlɑ.ɡə.ɹɪ.ð(ə)m/, exponential /ˌɛk.spəˈnɛn.t͡ʃəl/, multiplication /ˌmʌltɪplɪˈkeɪʃən/, mathematician /ˌmæθ.(ə.)məˈtɪʃ.ən/, calculator /kæl.kjə.leɪ.tɚ/
צלילים שדוברי עברית מתקשים בהם:
- /w/ — שפתיים מעוגלות, לא /v/: equation /ɪˈkweɪ.ʒən/, quantify /ˈkwɑn.tɪˌfaɪ/
- /æ/ — פתוח יותר מה-“אֶ” העברית, לא כמו “a” ב-“cup”: mathematician /ˌmæθ.(ə.)məˈtɪʃ.ən/, subtraction /səbˈtɹækʃən/, calculator /kæl.kjə.leɪ.tɚ/, analog /ˈænəlɔɡ/, algebra /ˈæl.d͡ʒɪ.bɹə/
- /r/ — הלשון לא נוגעת בשום מקום, לא ה-“ר” הגרונית: logarithm /ˈlɑ.ɡə.ɹɪ.ð(ə)m/, subtraction /səbˈtɹækʃən/, deranged /dɪˈɹeɪnd͡ʒd/, orderly /ˈɔɹdɚli/, regression /ɹiːˈɡɹɛʃ.ən/
איך לתרגל עם הסרטון הזה
- הקשיבו לכל הסרטון פעם אחת בלי לדבר, ורשמו את המילים שאתם לא מכירים.
- התחילו במהירות 0.75×, תרגלו Shadowing משפט אחר משפט, וחזרו למהירות רגילה כשזה מרגיש קל.
- הקליטו את עצמכם והשוו למקור, תוך תשומת לב למילים כמו minus, exponential, cosine.
מהי טכניקת השאדואינג?
שאדואינג היא טכניקת למידת שפה מבוססת מדע שפותחה במקור להכשרת מתורגמנים מקצועיים ופופולרית על ידי הפוליגלוט ד"ר אלכסנדר ארגואלס. השיטה פשוטה אך עוצמתית: אתה מקשיב לאודיו באנגלית ילידית וחוזר עליו בקול רם מיד — כמו צל העוקב אחרי הדובר עם עיכוב של 1–2 שניות בלבד. בניגוד להאזנה פסיבית או תרגילי דקדוק, שאדואינג מכריח את המוח והשרירים בפה לעבד ולשחזר דפוסי דיבור אמיתיים בו זמנית. מחקרים מראים שזה משפר משמעותית את דיוק ההגייה, האינטונציה, הקצב, הדיבור המחובר, הבנת השמיעה ושטף הדיבור — מה שהופך אותו לאחת השיטות היעילות ביותר להכנה לIELTS Speaking ולתקשורת באנגלית בעולם האמיתי.