For math, abstraction is both a blessing and a curse. It is a blessing, because the mighty of math comes from abstraction—his ubiquity rests on its transcending specific physical forms of the matter. It is a curse, because abstraction runs against the nature way of human thinking.

Although natural languages are a form of abstraction. Learning them involves trials and errors, the practice, the frustration, and the desire to express ourselves. When practice occurs daily, picking up a natural language comes naturally. For most people, however, that learning process stops once he master the natural language. But , like any other art form, math requires long, boring practice. Just as exercise strengthens one’s muscle, doing math thickens one’s logic fiber.  But because practice requires effort(let’s face it, we human are lazy), we won’t do it, if we can get by.

Then there is the misconception. Most people believe, wrongly, math is all about numbers and formulas. This misconception is largely our education’s fault. Indeed, before college, math hardly moves beyond numbers and formulas. But they are just a small part of the big math enterprise. If anything, math is about relations. The trouble is, they are about abstract objects, not sexy ones.


[Point Loma, San Diego,  CA, 12/2009]



They say math is a universal language, for the Nature mother. This analogy seems odd with our experience: as most people can testify, we learn to speak the mother tongue effortlessly. But math never comes naturally. Well, ask an Englishman whether math or Chinese is more difficult. It is the environment we living in that makes natural languages natural: we PRACTICE it on a daily basis.

One may also argue that we tend to think intuitively, not abstractly. But by nature math is abstract. This argument, however, cannot fully explain why we can learn natural languages, a form of abstraction. For example, “blue” means so many different shades of blue (ask Russians how many different blues they can distinguish). We acquire them by living through it. Thus, like beer, abstraction is an acquired taste. It is not abstraction itself that prevents one from learning math; it is practice.


[Muir Woods Redwood Forest Near San Francisco, CA, 2009]


Math is everywhere. From the first sound in the morning  (alarm) till the last touch at night (electric toothbrush), hardly a day goes by without math’s touch. Indeed, math powers all kinds of gadgets,  from cell phone,  to TV, car, train,  and computer; it also dictates how modern society carries out daily activities, from sales, trading, auction, forecasting, to sports (Moneyball). Indeed, if we put a label “Math Inside” on everything that uses math, we would be amazed by its ubiquity.

Its might aside, math is also genuinely beautiful. Like writing, math values simplicity, coherence, consistency, and precision. Often more so than writing. For example, my favorite symbol, the integral \displaystyle \int, is just visually gorgeous. It is also extremely powerful to express all sorts of relationships—e.g., between speed and distance, between density and mass, between stock and flow. It does so by capturing their common core in such a simple, unified, elegant way:

\displaystyle F(x ) = F(a ) + \int_a^x f(t ) dt.

Despite its ubiquity and beauty, people seem to dislike math. Few joke about their deficiency in literature or music, but many are eager to admit their illiteracy of math.


2011-10-12 157


Samantha has an interesting post. It reminds me of Gilda Radner:

“Some stories don’t have a clear beginning, middle, and end. Life is about not knowing, having to change, taking the moment and making the best of it, without knowing what’s going to happen next. Delicious Ambiguity.”

Gilda must be thinking about the relative importance of human actions and sheer luck in life. She argues that we cannot completely control our future,  but we can partially influence it by taking proper action now.

Now let’s try to make her idea more precise.

In current period t, I have a reward f_t(a_t, X_t) of happiness. a_t is the action I take in period t, X_t is a random element that summarize the external, environmental impact on me, say luck; X_t follows probability distribution F_t(X_t|h_t), where h_t=\{(a_s, X_s)\}_{s=0}^t is the history of actions I’ve taken and external factors till now. Hence, the future may not be completely chaos: although I cannot know future X_s precisely, by taking actions a_t now, I can influence how future will unfold via F_t(X_t|a_t, X_t).

Suppose I am a rational hedonist,  in a disciplined pursuit of happiness. In the current moment t, I take the best action a_t^* to maximize the expected, accumulative happiness from current time t till the end of my life T (a random variable), given the history h_t. Let V_t(X_t) = \sum_{s=t}^T f_t(a_s^*, X_s) be the best accumulative happiness in expectation from now onward.

In this model, luck is captured by random variable X_t. My karma h_t will influence how future state X_t plays out by partially controlling F_t(X_t|h_t), which is influenced by relative important of action a_t and luck X_t. Given current history h_t, I make the best decision a_t^*, without knowing X_{t+1}, X_{t+2}, ... precisely (although I may know the distribution F_s):

\displaystyle\Large \max_{a_t} f_t(a_t, X_t) + \mathbb{E}[ V_{t+1}(X_{t+1})|a_t].

That is,
“taking the moment and making the best of it, without knowing what’s going to happen next.”

Solving this model requires backward induction. The irony is:

“Life can only be understood backwards; but it must be lived forward.”


[Atlanta, June, 2006]