Introduction
At its present state, mathematics has advanced to a point where two randomly chosen mathematicians, or graduate students, may have never heard of what each other is doing. The reason is not that mathematics is hard (it is hard as always!); given enough time and devotion, a competent student can understand something about "almost anything". The main reason is that there is too much mathematics to learn in a lifetime. Just to get a feeling, you may click this link to see the current classification of mathematical fields (and sub-fields, sub-sub-fields, etc). Of course, we must not miss the forest for the trees, but the popular criticism about "knowing more and more about less and less" is not unjustifiable.

Another problem is the highly inefficient way of storing mathematical results. Theorems are scattered in textbooks, monographs and journal articles, and it is often difficult to find them. To make the matter worse, different notations and conventions are adopted by different authors. In more advanced work, authors tend to skip steps in order to leave space (it is easy to see that...), and even experts (not to mention the average author) make occasional errors. All these make the life of beginners quite difficult. An example is the delicate measurability problem of stochastic processes from which I am now suffering. Sometimes I wonder whether this can be avoided.

The information age
The rise of computer and internet has revolutionized our lifestyles. To give a simple example, international business is not possible without a highly sophisticated computerized system. (Think about the stock market, foreign exchanges, and Bloomberg!) As a matter of fact, computer has inspired mathematical theories such as optimization, statistics and numerical analysis. I believe that there will be another revolution that change our way of doing mathematics.

Mathematical logic
The basis of this potential revolution is mathematical logic, which, since the time of Boole, Russel, and Godel, has developed greatly and is now widely regarded as the foundation of all mathematics. The basic idea is that every mathematical theory can be expressed as (1) a collection of axioms about some primitive, undefined objects, and (2) theorems which are derived from these axioms by logical deduction.

For example, in probability theory, the basic object is that of a "probability space". A probability space is a triple (S, F, P), where S is a set, F is a Borel field of subsets of S, and P is a measure on F with total mass 1. The actual names of these objects (sample space, event space, probability) are not important. What matters is the properties being assumed. From these axioms (with further definitions and appropriate hypotheses) we can derive properties of probability models.

The main achievement of mathematical logic is that the "axiomatic method" described above can be expressed in terms of a formal language, and that virtually all mathematics can be reduced to statements of set theory. This makes the computerized storage and management of mathematical results possible.

A database of theorems
The title of this subsection may remind you of Wikipedia. It is indeed a very comprehensive database, but it is nothing more than a very huge digital book.

Imagine that a database of theorems is built. From the Zermelo–Fraenkel axioms of set theory to the classification of finite groups, all mathematical propositions are stored in the system. In the system, the statement and proof of each proposition are translated into the formal language of logic, and the validity of each step is verifed by a sophisticated system so that no mistakes are then possible. We can also imagine that the system has reasonable translation ability. For example, if we enter "A is a set with two points", the computer will know that we mean

我之前沒有看　youtube 的習慣。但因為宿舍的電腦什麼都沒有，想聽歌的時候，便會去 youtube 找來聽。無意間找到一套電影的原聲碟，一邊聽，一邊回想電影的情節。我不懂音樂，可是能泛起心中漣漪的，應該是好音樂吧。

G. H. Hardy 說過：

Exposition, criticism, appreciation, is work for second-rate minds.

換言之，創作是第一流的人做的。(這裡，我想不到怎樣翻譯 mind 這個字。例如 "He has a great mind" 譯作 "他有好的思想" 並不恰當。mind 是活的，不斷思考的，但 "思想" 卻好像是靜止的東西。) 創作不為什麼，只為留下一些痕跡：

Here, on the level sand,
Between the sea and land,
What shall I build or write
Against the fall of night?

Tell me of runes to grave
That holds the bursting wave,
Or bastions to design
For longer date than mine.

雖然 Hardy 的話很刻薄，但我很認同。我就不能創作這些能感染人的音樂，所以我很羨慕音樂家。我知道很多很漂亮的數學，它們都令人廢寢忘餐。將來我也想創造新的東西，但這些和音樂畢竟不同。尤其是漏斷人靜的夜晚，一段琴音，不用語言，便使人神往。

拓樸學最基本的概念是開集 (open set) 與閉集 (closed set)。在尺度空間 (metric space) (X, d) 中，子集 A 是開當且僅當對 A 的每一點 x，都有 r > 0 使得 B(x, r) = {y: d(x, y) < r} 是 A 的子集。而子集 A 是閉當且僅當 A (相對於 X) 的餘集 (complement) 是開的。問題是，為什麼開集叫開集？為什麼閉集叫閉集？