數學教育 7.5.1

Genius 4.2.1 | A Fraction of Algebra, 2.1

這段改編自 2010 年 4 月 24 日的對話。

.

另外,他提的另一個,有關學習數學的要點是,即使假設你在大學中,學到的數學,在日常生活中沒有用,單單是為獲取,那些嶄新的元素概念本身,就已經能夠令你有超能力;令你有一些,常人沒有的思考工具、比喻語言。

.

(安:但是,這個講法可能有一個問題。

雖然,你剛才列舉了數個例子,來示範如何將高深數學,間接應用到人生處世,但是,一般人未必有那種能力。所以我想問,你又是如何去跨過這個難關呢?)

.

什麼難關?

.

(安:去翻譯那些抽象數學概念,到其他範疇,或者日常生活。)

.

那不是「難關」。你的意思是,一般人也沒有那個能力,而我有。所以,那是超能力;我當年一定是,用了一些秘技,才獲取之。

天才之道,點滴累積。其實並沒有所謂的「秘技」。只要一步一步地,學習數學,就自然建構出,一個相對接近完整的數學思考體系,生成「翻譯抽象數學概念到其他範疇」等能力。

所以,我猜想你的疑問是,其實我所講的「點滴累積」,或者「一步一步地」,雖然理想上是,基本的要求,但是現實中是,大部人也做不到。那就代表著,大部人可能也會遇到,一個共通的「難關」。那個「難關」究竟是什麼?我又是如何克服它,而做到「一步一步地」「點滴累積」的呢?

.

你問題的最簡化版本是:「如何學習數學,開創人生?」

有起碼以下三個先決條件:

1. 對數學(及其他學問人生),有極大興趣;

2. 遇到合理的老師和書籍:

重點是,數學概念或運算上的主要步驟,亳無違漏。支節可免,但主旨必須。細節可以無師自通,大節必靠前人指點。平地自己行,斜地靠梯級。平地可跳步,梯不可跳級。

3. 極超大量的背誦和練習:

數學是理科,所以其背誦方法,不是「死背」零碎隨機的資料,而是「生背」息息相關的訊息。融匯貫通地背誦的唯一方法是,極超大量的操練。

.

你所講的「難關」,就是以上的第二點。老師有分好老師和差老師。大部分也是差老師。而差老師再分兩類:不懂數學和不懂教學。

— Me@2022.05.02 11:48 PM

.

.

2022.05.03 Tuesday (c) All rights reserved by ACHK

Visualizing higher dimensions, 2

Geometry is global.

Space is what we can see at once.

Dynamics is local.

Time is what we cannot see at once.

— Me@2017-02-07 10:11:34 PM

.

If we could see, for example, several minutes at once, that several minutes would become a spatial dimension.

In other words, that dimension is visualized for us.

— Me@2017-02-03 07:31:25 AM

.

.

2021.08.23 Monday (c) All rights reserved by ACHK

Two dimensional time 5.2.3

The first time direction is uncontrollable; the second is controlled by making choices, traveling through different realities. Future is a set of parallel universes.

— Me@2017-12-15 10:59:49 AM

.

The first time direction, which is along the timeline, is uncontrollable, because one can only travel from the past to the future, not the opposite.

The second direction, which is across different timelines, is controlled by making choices, forming different realities.

— Me@2019-12-21 11:03:23 PM

.

.

2019.12.22 Sunday (c) All rights reserved by ACHK

Two dimensional time 5.2.2

time direction ~ direction of change

multiple time directions ~ multiple directions of change

— Me@2019-12-22 04:38:47 PM

.

the first dimension of time ~ direction of change

the second dimension of time ~ direction of change of changes

— Me@2019-12-22 04:46:47 PM

.

.

2019.12.22 Sunday (c) All rights reserved by ACHK

Visualizing higher dimensions

The trick of visualizing higher dimension is: not to visualize it.

— Wikipedia

— Me@2011.08.19

.

Besides trying to visualize, there are other methods to understand higher dimensions.

— Me@2018-10-28 04:28:01 PM

.

.

What is the meaning of visualization?

— Me@2018-09-02 4:35 pm

.

feel ~ receive all the data at once

(This definition is not totally correct, but is useful in the meantime.)

visual ~ feel at once through eyes

.

you can visualize a 3D object ~ you can see all of a 3D object at once

you cannot visualize a 4D object ~ you cannot see all of a 4D object at once

.

Actually, you can only visualize a 2D object, such as a square.

You cannot visualize a 3D object, such as a cube.

That’s why the screen of any computer monitor is 2 dimensional, not 3.

— Me@2018-10-28 04:32:41 PM

.

.

2018.10.28 Sunday (c) All rights reserved by ACHK

Juan Maldacena 2

What is the difference between 10 and 11?

The simplest string theory is ten dimensional. Strings can interact with each other. If the interaction among strings is large, the theory is hard to describe. It turns out that when strings interact very, very strongly, something surprising happens. A new dimension opens up and we have a theory in eleven dimensions, the ten we started with plus an extra circle. In eleven dimensions we do not have strings, we have membranes. Membranes wrapped along the 11th dimension give rise to strings.

— Who’s Counting? Is it 10 or 11? (dimensions, that is — M Theory is making me Manic!)

— Prof. Juan Maldacena

— Institute for Advanced Study

2013.09.01 Sunday ACHK

Juan Maldacena

We do not know yet whether a description in terms of 10 or 11 dimensions is more appropriate for the universe where we live in. But these two possibilities are continuously connected. They are simply different possibilities for the internal geometry. Since the geometry of the internal space is quantum mechanical, asking what its dimension is might not be the right question.

In summary, in a quantum spacetime the dimension might not be a well defined notion. When the space in question is small, it can interpolate continuously between different dimensions.

— Who’s Counting? Is it 10 or 11? (dimensions, that is — M Theory is making me Manic!)

— Prof. Juan Maldacena

— Institute for Advanced Study

2013.08.29 Thursday ACHK

Looper, 5.4

Paradox 5.5 | Meta-time 4.5 | Cumulative concept of time, 13.5 | Two dimensional time 4.4 | 二次元時間 4.4

To be logically consistent WITHIN the movie’s story, Young Joe (in the year 2044) should not be able to influence Old Joe, who had time-travelled to the year 2044 from the year 2074,because that Old Joe is from another timeline. The proof is that Young Joe’s experience in the year 2044 is different from Old Joe’s experience in the year 2044 when he was young.

They are not the same person, nor the same person at different ages within the same timeline. At most, they are different versions of the “same” person from two different timelines (aka “parallel universes” or “histories”).

Young Joe’s changes should affect the same-timeline-Old-Joe, but not any Old Joe’s from any other timelines. So the Old Joe within the movie should not have been affected when Young Joe hurt himself.

Also, the changes of the same-timeline-Old-Joe due to the actions of Young Joe should be seen only by the author (meta-time), but not by Young Joe until he has become that Old Joe 30 years later. 

The author unintentionally, or intentionally, has confused two story timelines. Moreover, the author unintentionally, or intentionally, has confused the story-time and its meta-time.

— Me@2013-07-05 10:32 PM

2013.07.11 Thursday (c) All rights reserved by ACHK

Looper, 5.3

Paradox 5.4 | Meta-time 4.4 | Cumulative concept of time, 13.4 | Two dimensional time 4.3 | 二次元時間 4.3

In a single-mutable-timeline time travel story, the two dimensional time is not due to the internal causal structure of the story. Instead, it is due to the author’s timeline (aka meta-time). The author’s timeline is the second time dimension (aka independent direction).

The single-mutable-timeline model of time travel is not logically consistent within the story. If it is “mutable”, it is not “single”.

The single-mutable-timeline model of time travel is logically consistent only outside the story, from the perspective of the story’s author.

— Me@2013-07-02 3:47 PM 

2013.07.09 Tuesday (c) All rights reserved by ACHK

Looper, 5.2

Paradox 5.3 | Meta-time 4.3 | Cumulative concept of time, 13.3 | Two dimensional time 4.2 | 二次元時間 4.2

In the movie Looper, Young Joe (in the year 2044) influences Old Joe (in the year 2074) in the sense that Young Joe’s every action affects the state of Old Joe, because Old Joe is Young Joe’s future self.

For example, after Young Joe had hurt his own arm, the corresponding wound also appeared on Old Joe’s arm, even though Old Joe had already time-travelled back to the year 2044. 

All of Young Joe’s actions are the causes of Old Joe’s state. Young Joe is in the past of Old Joe.

Old Joe (2074-Joe) = [ …, Young Joe (2044-Joe), … ]

B = [ …, A, … ]

However, Old Joe (2074-Joe) had time-travelled back to the year 2044, meeting the Young Joe.

So, some of Old Joe’s actions would affect Young Joe’s decisions on his own actions. In this sense, Old Joe also influences Young Joe indirectly. Some of Old Joe’s actions are the causes of Young Joe’s state. Part of Old Joe is also in the past of Young Joe.

Young Joe (2044-Joe) = [ …, Old Joe (2074-Joe), … ]

A = [ …, B, … ]

However, it is logically impossible to have both

B is in the past of A 

and

A is in the past of B

just as it is logically almost impossible to have both

D is a part of C

and

C is a part of D

If you insist that it is the case, the only possibility is that

C = D

In this analogy, neither C nor D is really a “part” of another. In the time travel case, neither A nor B is really in the past of another. In other words, A (Young Joe) and B (Old Joe) have no time relationship. Neither’s actions are the causes of the state of another.

The real causes of Young Joe or Old Joe’s states are actually not within the movie story’s timeline. The real causes are the decisions of the author of the story.

— Me@2013-07-03 6:19 PM

2013.07.08 Monday (c) All rights reserved by ACHK

Looper, 5

Paradox 5.2 | Meta-time 4.2 | Cumulative concept of time, 13.2 | Two dimensional time 4.1 | 二次元時間 4.1

In a “if and only if” case, there is no time. 

If A is a necessary condition of B, we say “A is a cause of B“. In other words, A is in the past of B.

However, in some time travel story, it is “possible” to have both

A is a cause of B

(A is a necessary condition of B)

(B -> A)

and

B is a cause of A

(B is also a necessary condition of A)

(A -> B)

In this case, A and B are just equivalent.

(A B)

Neither is in the past of another. A and B have no causal relationship. In this sense, there is no time.

— Me@2013-07-03 6:19 PM

2013.07.05 Friday (c) All rights reserved by ACHK

Factors

因子

factors = independent causes

因素 = 原因元素 = 原因粒子

product = the results of factors multiplying together

產品 = 積 = 因素成果

multiply

乘 = 騎 = 順勢 = 利用

— Me@2013-02-28 8:16 pm

2013.02.28 Thursday (c) All rights reserved by ACHK

Over

over ~ finished ~ transcended

— Me@2013-02-01 01:54:02 PM

Finishing is one of the two methods of transcending. For example, once you have earned enough money, you would never have to worry about money anymore.

Finishing is more time-consuming and should be avoided if possible. But sometimes, it is necessary.

— Me@2013-02-03 02:02:07 PM

2013.02.04 Monday (c) All rights reserved by ACHK