Wait a minute! Naturally, you may raise an objection as such: the intention that defines one’s action (e.g. fulfilling a shopping list) remains unconscious to that person until he retrieves a line labeled “intention” from somewhere in the brain through “introspection”. This objection serves as a perfect beginning of a Socratic dialogue: Me: If so, knowing what oneself is doing would cause many surprises, just like suddenly becoming aware that he is daydreaming or having tripped on a pebble. Interlocutor: We “introspect” our own intentions all the time and in real time, and that is the reason why our actions are unsurprising to us. Me: But how about shifting to a new action, i.e. forming a new intention that defines a new action? Does the shift surprise you like tripping on a pebble? Interlocutor: No, it does not. Me: It should surprise you like tripping on a pebble, because each action has a beginning at a definite moment in time, which is, in this sense, “sudden” or “abrupt”. Unless... unless you are also aware of something else that makes it unsurprising. Interlocutor: Yes! Let us say that we are also aware of something going before it – the decision-making that leads to the intention or at least a judgment about its appropriate nature given the circumstances facing us. Me: What if the decision-making is too fast? Interlocutor: Be it fast or slow. It must be there to make the action unsurprising. Me: Now you also need to assume that we “introspect” our decision-making or judgment about the action’s appropriateness all the time and in real time to eliminate the surprise. Interlocutor: Let us assume that we can “introspect” many mental states at the same time, as many as needed. To tell you the truth, besides “introspection” I cannot think of any other access through which we can be aware of any mental state. Me: But the problem of too many surprises is merely delayed rather than solved by resorting to “something going before the action”. For example, why aren’t you surprised by the abrupt beginning of the decision-making itself? Interlocutor: Perhaps I am also aware of the decision to begin that decision-making! Me: But then you also need to be aware of the decision to begin the decision to begin that decision-making, and so on and so forth. It will form an infinite regress that burns out the best computer in the world. Interlocutor: That indeed sounds too much for our brains... but who knows? Me: Perhaps what we need in order to eliminate the too many surprises is not “something going before the action” but something else. Perhaps the surprise brought about by suddenly becoming aware of daydreaming or tripping on a pebble is not due to the “suddenness” of those events. Interlocutor: That sounds refreshing to me. At least “suddenness” here cannot simply mean “having a beginning at a definite moment in time”. Any event has a beginning and an end, but it does not necessarily feels sudden. Me: Suppose you are driving on a highway and suddenly notice a McDonald’s sign. You suddenly feel hungry and take the exit to buy a burger. Does that surprise you? Interlocutor: I may be surprised by the sign or the hunger, but definitely not my taking the exit. Me: And why? Interlocutor: Because I see the sign and I feel hungry! Me: Yes! That constitutes a reason for your action. Perhaps being suddenly aware of daydreaming or tripping on a pebble constitutes a surprise not due to its “suddenness” but due to its occurring without a “reason”. Interlocutor: But I have already said that through “introspection” I am aware of my decision-making or my judgment about the appropriate nature of my action. Me: But then you also need to be aware of your judgment about the appropriateness of your judgment about the appropriateness of your action and so on and so forth. Another infinite regress. Interlocutor: There must be a way to stop the regress. Me: The unsurprising nature of what we are doing in contrast with the surprising nature of what happens to us can only be accounted for by our direct awareness of the reasonable nature of the action prior to and during the action, instead of the awareness of any judgment we make about its reasonableness. Interlocutor: I can give you that. But we still need an “intention” to define the action! The awareness of the action’s reasonable nature cannot be the “intention” itself, can it? Me: Why not? If an “intention” is a mental state that makes our bodily movement unsurprising, that is, makes it an “action” rather than a mere happening, we can find nothing other than that to be the “intention”. Interlocutor: I am sort of satisfied with your answer now, except that we also need “introspection” to be aware of an action’s reasonableness. Me: Here comes the punchline. “Introspection” offers us access to an existing mental or brain state, yet the reasonableness of an action is NOT a mental or brain state. It means that our awareness of it cannot be attributed to “introspection”. Interlocutor: Then to what shall we attribute it? Me: ... You can give a name to it. The important thing is that we now see our awareness of our intention or action cannot simply be attributed to “introspection”.

一加一 (记录于2016年9月21日,关于儿童对算术认知性质的认知,亦即“meta-cognition of arithmetic cognition”)


“妈妈,你非常粗心,连一加一都会做成三。” 他又开玩笑。知道我丢三落四,现在经常提醒我看看对他来说重要的东西(比如ipad)有没有忘在哪里,路上没事干可以看的后备书有没有放在包里。


看他将信将疑,我继续解释,“六十九加七十三和一加一用到的数学能力是不同的。至少一到九的加法要靠对math facts的记忆,大数字加法就需要操作这些facts了,容易出现计算错误。这些数学事实没记好,很难说这个人会加法而只是粗心。”



“什么?那你还会做加法吗?” 我以为他又回到了刚才的问题。

“我忘了一加一等于几,但我还记得一加二等于几,我还记得二加二等于几,我还记得一加三等于几......” 他更认真了。


他的表情忽然轻松起来,让我意识到我上钩了。原来他是在质疑记忆math facts跟数学能力的关系,以至于“两种数学能力”的说法。


“哪里看来这个词的呢?” 这个词听上去年代好久远。

“十万个为什么里面的.....” 家里还真有好几版十万个为什么,有我小时候最初看的那一版。我的母亲每出一版都会收藏,哪怕在我长大以后。

“Math facts不是‘僵死教条’呀。” 他终于彻底暴露了意图。



A More Honest Picture of Life

One thing that can be said about life is that it is eventful in the short term, and uneventful in the long term. Events that are said to be “life-changing” often turn out to be only minor upheavals in the history of an individual. Although many books portray an action-packed plot filled with twists and turns leading to an uplifting resolution, this structure is not very common in the real world.

Earlier this year, I read the book Sag Harbor by Colson Whitehead. I found the plot hard to follow, because there was not much plot. Almost all the ideas conveyed in the novel were details. It was a very unconventional way to organize a book. And this kind of structure made me feel, “hey, this is what life is like.”

In Sag Harbor, every chapter forms its own little compartment, with its own characters, events, and everything, save for the occasional recurring facet of everyday life. Randomly pick up a chapter and read from there, and probably you can still understand it without difficulty. The fact that the chapters are named like short stories, such as “if I could pay you less, I would”, without numbers, reinforces the mutual independence between chapters. You will notice that Whitehead, while writing Sag Harbor, decided to take the concept of Chekhov’s gun and throw it right out of the window. Many details appear once, only to have its minor effect on the plot a few pages later, and are wrapped up, never to be used again. On page 145 of Sag, a literal gun (a BB gun) was introduced to us as a toy gun that was supposed to be harmless, and that Benji and the gang wanted to fight with. Unfortunately, during the BB gunfight, it turned out that Benji was in fact (fortunately, lightly) wounded by a bullet. The chapter is wrapped up simply with sentences such as “the BB guns didn’t come out again that summer”. And the BB guns did not come out for the rest of the book, having lasted for a single chapter. Life is just like this. Although I keep track of the order of time, days, weeks and months, together with certain long-term plan, what feels important if not decisive is always the interests and conflicts of the moment. For example, “I need to finish this essay.” I do, but it’s due in a week, and it only affects a transient period of my life. Last year, for a few months, the MATHCOUNTS math competition become my life. I competed in multiple rounds, traveling to other cities for knockouts, practicing and practicing and having “Eureka!” moments, even making it to the nationals, granting me “prestige forever”. However, a year later, it has been put away in the drawer as a fun fact about me, while I navigate much of the same work and anxiety that came the year before.

Although it could be argued that Sag Harbor features recurring themes, it appears as if most of them are simply byproducts of the growth patterns of an average adolescent. For example, many times throughout the book, love is involved in some event or another. These events, however, are told individually and don’t really build on each other to make “a milestone in life”, or to convey “the meaning of life”, or something like that. In the modern day, it is a truth universally acknowledged that a boy Benji’s age would likely be exploring love, and so naturally many parts of his life would revolve around it.

So, is life a flat, meaningless void after all? Probably not. There are many moments when Benji is experiencing what is indisputably happiness. At the start of the final chapter, a Labor Day party appears, distracting the characters from the impending school year. “But forget all those city intimations. Today was the Sag Harbor Hills Labor Day Party.” This portrays the feeling of a carefree moment regardless of what happens around it, a time to have fun, elevated above the void. Sometimes, such moments can serve as a distraction from the stubborn course of life. For me, moments like this might be taking a break from homework and visiting the newly hatched Canada geese in the central park, or when a friend says hello to me halfway through an exhausting school day.

Many people believe that life is full of drama and turning points that make up a breathtaking journey of progress (e.g. in the personal essays we write for college admission). Unfortunately, that mentality is a subtle misrepresentation of life. If you expect huge permanent changes or progresses to happen to your life every year, you’ll be disappointed in the end. True, there are life-changing events, but they usually only change specific aspects of life, and many of them turn out to be temporary. In the book, Benji decides that he would rather be called Ben, since he felt Benji was childish. The new name never stuck, and eventually he went back on his decision.

Continuing to go down the path of such beliefs, you’ll end up being blind to the true face of life. If you take a closer look at what you have experienced, you will see that it hasn’t changed that much from a few years ago. Even though the circumstances of your life have changed, your personality, motivations and values haven’t. And they are, so to speak, the backbone of your life.

The underlying fabric of life simply does not change that much. Most of the changes in life are just ripples on the surface of a lake. The deeper waters are eternal like heaven and firm like stones, only moving slowly with the flow of time.