シャドーイング練習: Roger Penrose - Is Mathematics Invented or Discovered? - 動画で英語スピーキングを学ぶ

Roger, I have been fascinated by mathematics my entire life, and it is a pleasure to come to you to discuss two of its fundamental aspects.

One, its incredible capacity to describe reality, and then what mathematics really is.

So let's start with the first.

How accurately does math describe the physical world?

Well, it is extraordinarily precise.

And in different areas, more precise.

In some areas, we know less about it.

But I think people often find it puzzling that something abstract like mathematics could really describe reality as we understand it.

I mean, reality, you think of something like a chair or something, something made of solid stuff.

And then you say, well, what's our best scientific understanding of what that is?

Well, you say it's made of fibers and cells and so on.

And these are made of molecules, and those molecules are made of atoms.

Those atoms are made out of nuclei and electrons going around.

And then you say, well, what's a nucleus?

And you say, well, it's protons and neutrons.

And they're held together by things called gluons.

And then neutrons and protons are made of things called quarks and so on.

And then you say, well, what is an electron and what's a quark?

And at that stage, the best you can do is to describe some mathematical structure.

You say, they're things that satisfy the Dirac equation or something like that, which you can't understand what that means without mathematics.

I mean, the mathematical description of reality is where we're always led.

And these equations are fantastically accurate.

The Dirac equation, which describes the electron or quarks, is a very precise equation.

And for example, there's a calculation which describes the magnetic moment, that is, electrons behave like little magnets.

And the magnetic moment, the strength of that magnet can be described in terms of other parameters.

And there's a calculation which gets the accuracy of that.

Well, Feynman had a very good description.

He said it describes the distance between New York

and Los Angeles to an accuracy of less than the thickness of human hair.

So that's pretty precise.

That's unbelievable.

And that's describing the microstructure of atoms and...

Yeah, but these are the particles...

The electron and the gluons.

The electrons...

This is specifically electrons and quarks, the things which are called spin-half particles, but don't worry about that.

Gluons are slightly different.

Now mathematics can also describe things in the ordinary physical world, the gravitational attract, electromagnetic attraction, and with the same kind of descriptive accuracy?

Well, there's another, even more in a certain sense, because gravity, according to Einstein's theory, I mean Newton's theory already,

had a precision of something like one part in 10 to the 7, so that's 10 million.

Wow.

One part in that.

And then there was discrepancies seen in the behavior of mercury and so on,

And that's where you start to see differences with Newton's scheme.

And then Einstein comes along and produces a theory

which is now known to have a precision something like 10 to the power 14.

And that precision is a measure of how accurate.

There's a particular system of two stars going around, special kinds of stars, called neutron stars, very dense objects.

And these stars, one of them is what's called a pulsar.

It emits pulses of signals, which can be timed extremely precisely.

And over a period of, well, I suppose it's maybe more than 30 years now, I can't remember, they've been observing this thing.

And in that period of time, the accuracy over that length of time is known to something like one part in 10 to the 14.

And the agreement between Einstein's theory and the observations.

So it's telling you these are very, very precise theories.

So whether we're talking about the structure of very large entities, neutron stars over great distances in the universe with gravity,

or the structure of an electron, mathematics in both cases is able to describe it with that kind of incredible precision.

Exactly.

And these are small equations.

I mean, they're not giant.

That's right.

They're relatively small equations.

I mean, they're a little difficult to understand and Einstein's theory is certainly subtle.

It's not complicated in the sense that the ideas, okay, you have to understand about curved space and that sort of thing,

which is not an easy thing to get your mind around.

But once you get over that.

Once you get over that, it's about the simplest thing you could write down.

Wow.

In that kind of term.

So we have this extraordinary precision between mathematics on the macroscopic level

with neutron stars and at the microscopic level with the nature of the electron.

And mathematics is incredibly precise in both cases.

So what does that now begin to tell us about what mathematics really is?

Yes, well, in a sense, this is telling us that our picture of physical reality depends on something in a sense which is more precise, at least in our understanding of it,

than how we think about the world.

And this precision really dates back to the ancient Greeks, the time of Pythagoras and later,

where they developed the mathematical ideas as a field of study, stimulated to some degree by physical reality.

Because the geometry of Euclid, which was very much part of the mathematics that was being studied then,

which we know now isn't extraordinarily precise, I mean it is extraordinarily precise, but it's not as precise as Einstein's theory.

So one has to go a little bit beyond the geometry that they had.

I don't think they quite appreciated that they were doing physics, because they didn't realize that the geometry of the world could have been anything else.

But they developed this mathematical scheme purely as a study on its own.

And so mathematics was studied as a pure intellectual activity,

And without necessarily it being related to the structure of the physical world, although geometry clearly was a big input.

But then the properties of numbers and how you add and multiply and the notions of prime numbers, the fact there are infinitely many prime numbers, that goes back to Euclid and earlier.

And so these things just about numbers were developed very much from the time of the Greeks.

And ever since then, mathematics has been a subject which you can study for its own sake.

It has its own life, in a sense.

And certainly mathematicians view it this way.

It's something out there which seems to have a reality independent of the reality, the ordinary kind of reality like things like chairs and

so on which which are what we normally think of as real

but uh okay the mathematical reality is something different it's sometimes referred to as a platonic world a platonic reality

and sometimes people have a lot of trouble thinking

that is real i mean philosophers uh worry about that and

so on what does that what would

that mean a platonic reality well i think uh it's a different kind of reality from the reality of the physical world.

I tend to think of there being different ways of looking at reality.

There's the reality of our mental experience, which interrelates with the physical reality,

but so then does the mathematical reality of this platonic world, which gives reality to these notions.

So if you like mathematical facts, like there is no largest prime number, it's something independent of ourselves.

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It's always been true.

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It doesn't somehow become true as soon as somebody saw how to prove it.

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It's always been true.

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There are wonderful examples like the...

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And it would have been true if nobody ever...

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Exactly.

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Yes.

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And in a sense that had to be so because if the physical world depended so precisely on these mathematical laws,

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I couldn't have known what to do in a certain sense if the mathematics hadn't already been there.

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It's not us that imposes this on the world.

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It's out there.

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Sometimes people think

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that maybe the reason we have good mathematical laws of physics

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is that's the best way we can come to understand the world.

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But it's something more than that.

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It really is out there in the world.

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Well, that's the argument.

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Whether mathematics is invented by us, by human beings, trying to impose our way of thinking on the physical world,

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or whether it is discovered because it's already out there and we're finding it because it's already there.

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Those are the two polar views.

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Sometimes people do argue.

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They say, well, you know, it's just our way of organizing what we see about us.

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But I really don't think that's good enough, because Newton, for example, the observations probably had about three figures,

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three decimal places, and he produced this theory which kept on working until about seven figures, you see.

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Then there were discrepancies seen.

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Einstein produced his theory mostly out of his head, appealing to things that were known to Galileo and so on.

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But apart from that, it was not much more empirical evidence.

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But he produced this theory which extended far beyond anything that the observations at that time told us about.

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And they keep on agreeing with the observations.

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So that theory, which is, if you like, a platonic absolute thing,

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it's a mathematical thing, seems to be inbuilt into the way the world operates.

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It's not as though you see a new effect and say, OK, we're going to think of a better theory to accommodate that one.

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Sometimes science is like that.

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But these really good physical theories are not like that.

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You're revealing something in the way the world operates, which is there all the time.

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And I don't think there's any way of understanding

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that just in terms of our trying to understand what we see around us.

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A critical fact really seems to be what you said, that when these mathematical theories were discovered,

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the accuracy that the observation that they had at the time was small compared to the accuracy that those theories then produced.

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That's right, that's right.

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I mean, Einstein's case, okay, seven figures of decimal were known perhaps in the planetary motions, but there's another seven out there.

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The precision is over and above that.

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It's as much as, yes.

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10 million, 10 millions.

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It's 10 million, 10 million, yes, that's right.

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Yes, yes.

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I mean, that's unbelievable.

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Yes, no, it's incredible.

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So push it further.

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What does that mean?

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Because mathematics is almost infinite in terms of all the different relationships and expressions and things that we already know.

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Yes.

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What does that mean in terms of how much mathematics is sitting out there?

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Well, that's a good point, because there's an awful lot of mathematics which doesn't seem to have any clear relation to the physical world.

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The way I like to picture it is there is this world of mathematics, and only a small part of that, and it's a very fruitful part, it's an extraordinarily fruitful part,

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has relevance to the physical world.

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There's an awful lot out there which, as far as we know, has no relation to physical behavior.

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Well, of course, people said some of that in the past, and then we've been surprised to find some other things that later have.

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But still, there's so much math out there, and so much bizarre, I don't know how else to put it,

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structures, that it would seem impossible that that could relate to the physical world.

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But what does that mean about, if it is out there in some platonic world, what is out there?

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In other words, all these infinite ideas and structures and possibilities?

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Yes.

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Sometimes people think of these as mental creations, you see, but it doesn't really explain...and there's this wonderful example of the Mandelbrot set, extraordinarily complicated.

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The fractional...

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Yes, and you can magnify little bits of it and you see all this incredible detail.

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And that's all there in a very simple mathematical idea, and it's encompassed by this very simple piece of mathematics.

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How does that give your own sense of what mathematics really is?

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Well, I think there are two aspects to mathematics, at least how I look at it.

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Some people are just exploring the mathematics, and that's their real interest.

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And it's the beauty in the subject often, and that's why they're doing it, because they find it exhilarating, something they find really wonderful to do.

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But there's the other side of it, which is how it relates to the physical world.

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And there is this extraordinary precision that we find when you get the mathematics right.

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It really mirrors the behavior of the physical world to an unbelievable degree.

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And so there's these two sides to mathematics.

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It has this reality which you can study quite independently of its role in physics.

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And the other side, which is how it really does seem to reveal how the real world operates, in a certain sense, what the world is, as far as we can understand it.

シャドーイングとは？英語上達に効果的な理由

シャドーイング（Shadowing）は、もともとプロの通訳者養成プログラムで開発された言語学習法で、多言語習得者として知られるDr. Alexander Arguelles によって広く普及されました。方法はシンプルですが非常に効果的：ネイティブスピーカーの英語を聞きながら、1〜2秒の遅延で声に出してすぐに繰り返す——まるで「影（shadow）」のように話者を追いかけます。文法ドリルや受動的なリスニングと異なり、シャドーイングは脳と口の筋肉が同時にリアルタイムで英語を処理・再現することを強制します。研究により、発音精度、抑揚、リズム、連音、リスニング力、そして会話の流暢さが大幅に向上することが確認されています。IELTSスピーキング対策や自然な英語コミュニケーションを目指す方に特におすすめです。

シャドーイング練習: Roger Penrose - Is Mathematics Invented or Discovered? - 動画で英語スピーキングを学ぶ

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