前沿讲座要写讲座报告，给我们看了一个讨论数学的 TED，短短十多分钟，然鹅要写 1000 字的报告。很悲催的是我走了 hard 模式：用英语写，写了一半发现我要是用中文写字数会多很多…这篇演讲还挺不错，喜欢里面的各种视觉效果，总体来说就是学到了三点：一数学是一门表达结构的学科，二数学教会我们从不同角度来理解，三数学需要我们充分发挥想象力。以下是我的废话，我都不知道自己在写什么了，为凑字数而凑字数。

In some people’s eyes, mathematics can be a headache. But we have to deal with it in many aspects of our daily life, such as paying money and calculating the time. We would rather not understand mathematics until we need it. However, from Roger Antonsen’s perspective, mathematics is beautiful and wonderful. It is mathematics who teaches us the essence of understanding, and how can we understand the world.

Mathematics is about to find patterns, and everything can be attributed to a specific pattern, that is, the connection between different things. Discovering and recognizing such connections and structures is very important in mathematics and computer science. The connection is inseparable from the structure, and it results from observing the structure from a certain perspective. The observers may have different ideas and understanding when viewing from a different perspective. Just like the example the lecturer showed, we can understand a number by vision, hearing and other human sensory system.

Then there will definitely be some permutations and combinations of structures, and the connections presented cannot be understood and perceived. Observing the same structure, like the octahedron shown in the video, from different angles (rotate around different axes) will also lead to different views and understanding.

How do we express, describe, and transfer these connections? The lecturer said people will make up a language for it. Just like a simple equation, “x + x = 2 • x”. On the left, it’s a sum of the same number. While on the other hand, it’s a multiplication. It is a language specialized for mathematical operations. I bet most of us haven’t thought about that equation before, and we just take it for granted. But it is a very nice pattern which teaches us an analogy between two things. Just like the lecturer said, if you are viewing something and taking two different points of view, you express that in a language. A very subtle feeling, isn’t it?

When it comes to describing the connections between different things, I think about a good structure - graph. Well, as a computer science student, we know it contains nodes and edges from our data structure course. We can model many practical problems as a graph, like the Internet, the route planning and the social networks. Generally, graph is more like an abstract structure and we often adopt the concept of network in the real world. From my perspective, the network conveys more about the concept of information flow while the graph is static. Each node contains its own information, at the same time, each node also transfers some information to other nodes connected with it. Therefore, in a big picture, information describes the relationship, that is, describes the structure. From the perspective of the observer, what the structure is, naturally, is the permutation and combination of information.

The carrier of information is also a structure, so it can be described by other information, so information is a descriptive structure. Just like a piece of text data that conveys information, this information may describe another structure, and the translation or interpretation of this text is the information describing the information, which is also a structure.

It can be seen that the structure of the data itself, and the relationships presented are also information. If you analyze the data from different angles, you will see different relationships, and thus get different information. The main function of the data structure is to convey information. Its carrier and form are not important. Instead, what is important is the arrangement and combination of its structures, the formed connections, that is, information.

Therefore, transmitting information is transmitting structure. Structures can absorb information, in fact, it is absorbing structures, so that it can form a new structure and transmit new information. Take the neural network for example. Recently, I read a paper named Graph-Based Global Reasoning Networks, which is accepted by CVPR 2019. This paper impressed me a lot with its handling of relationships. First, how to define the relationships of each pixel in an image? Previous work may use a time-consuming object detector, but this work use convolution layers for projection to form a binary combination. As we can imagine, this is a clever way because with the training of the whole network and the information flow between different nodes, each pixel in the image can naturally learn which regions it belongs to. Besides, turning an image with numerous pixels to a group of pixels can highly reduce the dimension, which is also conducive to the training. After projecting to the new interaction space, they can do reasoning with graph convolution, that is, learning the node features by information flow between different nodes. I think this step can perfectly describe the process that structures absorb information to form a new structure.

As I’ve said over and over again, which is also an essential part of this talk, mathematics also teaches us to think in a different way. A impressing example in the lecture is 4⁄3. We can see it, and also we can hear it. If we limit the perception to vision, we still have a different type of views. Think it just a fraction, a traditional arabic numeral or a new expressing way of numbers. If we follow some rules convey the information of 4⁄3 to draw some graphs, then you won’t know how many ways you can express. Even if you use 4⁄3 to draw a graph, you can rotate and stitch it, making it a three-dimensional graph that has different feelings from different angles. In a sense, mathematics is full of philosophy. “When I view the world from your perspective, I have empathy with you. If I truly understand what the world looks like from your perspective, I am empathetic.” Maybe this is another way of saying put yourself in others’ shoes and it requires imagination and creativity. Therefore, as the lecturer said, have a mind like water, open your mind and use your creative imagination, understand the world with different perspective, you’ll experience a wonderful feeling.