Math class has two goals:
- Verify thata statement is true
- Understand whya statement is true
There's a tendency to put the goals in opposition, assuming concepts are either "easily understood but wrong" or "difficult to understand yet correct".
It's like a restaurant that believes in having tasteor营养，但不是两者都有。为什么选择?
Our goal is a deep intuition for correct things. And it's ok to start with an understood "sorta-true" concept and refine it to an understood "very-true" version:
正确性不是二进制。多年来，我们对有效证明的标准一直在演变，一个世纪以来，我们对什么是真实的概念似乎是令人尴尬的原始。没关系:让我们好好理解，并努力让它变得更好。
We have to balance two roles: the safety inspector who makes sure the food is safe, and the customer who wants to enjoy the meal. In my head I think about "inspection mode" and "tasting mode": the secret is inspecting things that already taste good.
Example: The Pythagorean Theorem
The Pythagorean theorem is usually introduced as a statement abouttriangles. A common proof is a visual rearrangement, like this:
This is nutritious and correct, but not tasty to me. It seems like a special case, an optical illusion: withjustthe right shape, things can be re-arranged.
A tastier proof is that the Pythagorean Theorem is really about the nature of 2d area. A big shape, when split, yields two smaller shapes. The total area must be the same:
The split-apart area can come from a triangle, circle, or cardboard cutout of Thomas Jefferson. It doesn't matter: two pieces, when cut from a larger one, must have the same total area.
Aha!
This intuition can then be refined into a moreformal statement.
Example: Euler's Formula
Here's a trickier example: Euler's Formula.
It's a baffling statement, and here's thecommon justification:
It's crisp and concise, but unsatisfying to even other math fans:
I agree: it's a bunch of symbols that happen to line up. Here's atastier version:
- $e^x$ represents continuous growth (interest earning interest, which earns interest…)
- $\sin(x)$ and $\cos(x)$ represent vertical and horizontal directions
- $i$ represents rotation
如果我们创建“连续旋转”($e^{ix}$)，那么我们在一个圆中移动，它可以被分离成水平和垂直分量($\cos(x)$和$i \sin(x)$)。
Again, this intuition can be sharpened further:
- $e^x$可以看作是一个无穷级数，从初始值($1$)开始，它赚取的利息($x$)，赚取的利息($\frac{x^2}{2!}$)，依此类推。
- Sine(and cosine) are infinite series based on an initial impulse, which creates a restoring force, which creates a restoring force, and so on. This is like interest that earns interest in the opposite direction (and why sine oscillates without going to infinity: its motion opposes itself.)
- 将$i$代入$e^x$意味着我们获得“虚利息”($i$)，它获得“虚利息”($-1$)，然后获得“虚利息”($-i$)，依此类推。有些兴趣与前面的术语相反，我们可以将它们集合成匹配正弦和余弦的模式。
Rather than staring at a dry proof and trying to understand it directly, get a rough intuition (世界杯2022赛程时间表最新 ) and then see if the proof makes sense. It's a bit of math inception, where we try tounderstandthe verification step, not simplyverify验证步骤。
Happy math.
Appendix: On Proof and Progress in Mathematics
William Thurston (Fields Medal Winner) wrote a great essay,On Proof and Progress in Mathematics. It's full of ideas I found interesting:
- 数学家的问题是:“数学家如何促进人类对数学的理解?”
例如，当Appel和Haken用大规模的自动计算完成了四色地图定理的证明时，引起了很大的争议。我认为这场争论与人们对定理的正确性或证明的正确性的怀疑没有多大关系。相反，它反映了人类对证明的持续渴望，除了知道定理是正确的以外……通过这种经历，他们发现他们真正想要的通常不是“答案”的集合，他们想要的是理解。
- We're never done explaining a concept:
We may think we know all there is to say about a certain subject, but new insights are around the corner. Furthermore, one person’s clear mental image is another person’s intimidation.
- On the role of intuition:
Personally, I put a lot of effort into “listening” to my intuitions and associations, and building them into metaphors and connections. This involves a kind of simultaneous quieting and focusing of my mind. Words, logic, and detailed pictures rattling around can inhibit intuitions and associations.
- The "emperor's clothes" problem in math happens even for professionals:
尽管如此，大多数参加普通讨论会的听众并没有从中获得什么价值。也许他们在开始的5分钟里迷失了方向，但在接下来的55分钟里却一直保持沉默。或者他们很快就会失去兴趣，因为说话者在没有给出任何调查理由的情况下就投入到技术细节中。在演讲结束时，少数几个与演讲者领域相近的数学家会问一两个问题，以避免尴尬。
更进一步的问题是，人们有时需要或想要一个被接受和验证的结果，不是为了学习它，而是为了引用和依赖它。
- On the difference between everyday explanations and and technical ones:
为什么从非正式讨论到谈话到报纸会有这么大的扩张?一对一，人们使用广泛的沟通渠道，远远超过正式的数学语言。他们使用手势，他们画图片和图表，他们制作声音效果，使用肢体语言……在报纸上，人们仍然更加正式。作者将他们的思想转化为符号和逻辑，而读者试图将其转化回来。
It’s like a new toaster that comes with a 16-page manual. If you already understand toasters and if the toaster looks like previous toasters you’ve encountered, you might just plug it in and see if it works, rather than first reading all the details in the manual.
- On what motivates us to do math:
是什么激励人们学习数学?做数学，学习解释、组织和简化的思维方式，是一种真正的乐趣。人们可以感受到这种发现新的数学，重新发现旧的数学，从一个人或文本中学习一种思维方式，或找到一种解释或观察旧数学结构的新方法的喜悦。
I love the "aha!" moments when a concept click. People willing seek out mysteries and puzzles (movies where we don't know the ending, games like Tetris). Math is an experience with similar emotional payoffs when approached correctly.