Learning to Learn: Math Abstraction

我们简化复杂的概念来理解它们。当数学运作良好时，它会让事情变得更简单。(有时药不如病。)

Let's change the generic "Math teaches you to think" to a more specific "Math helps us simplify ideas". We hide detail after detail to reveal an essential truth.

Is this style of thinking necessary? Required for survival? Usually not. But it's often interesting.

What's the simplest drawing you can recognize as a face? What's the simplest joke that's still funny? The simplest exercise that grows a muscle? Would knowing that improve your art, humor, or fitness?

Abstracting Counting

这幅画里有什么?

电脑说“百万像素”，你说“三只狮子”。在几秒钟内，你扔掉了无数的细节，以揭示更深刻的见解。

发生了什么事?We abstracted the scenario into something simpler.

Remove background from foreground
Remove differences between each animal
Remove "animal-ness" (treat lions as generic "lines")
Remove need to count objects with literal lines
Remove need to specify a fixed number ("n happens to be 3 today")

我们倾向于将我们明确意识到的步骤称为“数学”。一旦它变得自然，它就是“毫不费力的看到”。(把更多的概念归入“毫不费力”的范畴不是很好吗?)

Abstracting Learning

Let's try the "math simplification" on a bigger idea: learning.

What does learning involve? At its bare essence, what do we need?

Learning = Insight + Enthusiasm

Insight (for me) comes from Analogies, Diagrams, Examples, Plain English descriptions, and Technical definitions. (Read more about the世界杯2022赛程时间表最新 )

Enthusiasm comes from humor, warmth, empathy, and being treated like a human (not math robot).

好的教训两者兼而有之。但是等等:热情本身就足够了吗?人力资源管理。Maybe it's better written:

Learning = Insight * Enthusiasm

如果两者都是0，你就不是真正的学习。
"Negative insight" is learning something false.
"Negative enthusiasm" is hating something, even to the point of discouraging others.
"Negative insight with negative enthusiasm" could be discouraging others from learning something false (which is good, right?).

This is just playing with words and pseudo-equations. Sure. But seeing how enthusiasm impacts education reveals a truth: an educational experience can become negative when enthusiasm points the wrong way.

The equations above don't have to be "right". They're helping us work through an idea. The math approach is to isolate the key factors and figure out how they're related.

Abstracting Technology

For something like a car, the key elements seem to be:

Car = Propulsion * Control

Traditionally, the details of propulsion involve a gas engine, and the details of control require a human driver. But we're interested in abstraction: are these details we can hide?

也许推进可以是电力的。也许控制可以来自电脑。自动驾驶电动汽车以不同的细节满足了这个等式的本质。(就像1个苹果+ 2个苹果= 3个苹果一样，1头狮子+ 2头狮子= 3头狮子。)

Asking the right question is difficult, and critical. For this problem, what are the essential variables? What counts, and what can be thrown away?

Abstracting Programming

A lot of people argue that "math helps your programming". Yes, but not in the way you think.

大多数程序员除了代数和基本统计之外，什么也不使用。(是的，是的，如果你正在制作电子游戏物理引擎，你可以坐下来。)

从数学中得到的关键教训是，它是如何抽象出世界的巨大复杂性的。Here are a few fundamental types of "quantity":

integers (whole numbers)
floats (decimal numbers)
hexadecimal numbers (whole numbers with a simpler way to use powers of 2)
null (an unset number,different from zero)

程序员不需要数学技能，所以他们可以解决算术问题。他们需要数学来看到世界变得更简单的例子。

Any piece of data (text, images, video, etc.) can be expressed as a giant list of numbers, a combination of the above elements. That's pretty simple.

What other metaphors from math (functions, structure, change, chance) can help us simplify our code?

The Ladder of Abstraction

Bret Victor has a wonderful essay on theLadder Of Abstraction.

If a new concept is difficult for me, I wonder if I'm at the right level of detail. There's no all-purpose answer like "less detail is better". Sometimes you're staring at your feet and need to zoom out, sometimes you're in the clouds and need to zoom in.

Analogies, Diagrams, Examples, Plain English and Technical definitions, throw them at the wall and trial-and-error a way to better understanding. Like getting an eye exam, move closer or further from an idea until it snaps into focus.

快乐数学。

Abstracting Counting

Abstracting Learning

Abstracting Technology

Abstracting Programming

The Ladder of Abstraction

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