Essay Psychology Invention Mathematical Field

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I was very interested and impressed by the quality of the reasoning, but it was quite hard to stay alert and focused.

After a few experiences of reading a few pages only to discover that I really had no idea what I'd just read, I learned to drink lots of coffee, slow way down, and accept that I needed to read these books at 1/10th or 1/50th standard reading speed, pay attention to every single word and backtrack to look up all the obscure numbers of equations and theorems in order to follow the arguments.

However, in formalized expositions, this psychological difference vanishes.

In the same way, any idea in mathematics can be thought about in many different ways, with competing advantages.

The closest explanation [for this mathematical universe] is Einstein’s statement that “the only physical theories which we are willing to accept are the beautiful ones” …

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the concepts of mathematics have this quality of beauty. I assumed that such an elaborate buildup must be leading to a fantastic denouement, which I eagerly awaited -- and waited, and waited.It was only much later, after much of the mathematics I had studied had come alive for me that I came to appreciate how ineffective and denatured the standard ((definition theorem proof)^n remark)^m style is for communicating mathematics.Our minds depend on many interconnected special-purpose but powerful modules.We allocate everyday tasks to these various modules instinctively and subconsciously.The term `geometry', for instance, refers to a pattern of processing within our brains related to our spatial and visual senses, more than it refers to a separate content area of mathematics.One illustration of this is the concept of correlation between two measurements on a set, which is formally nearly identical with the concept of cosine of the angle between two vectors.Mathematics is not something we sense directly: it lives in our imagination and we sense it only indirectly.The choices of how it flows in our brains are not standard and automatic, and can be very sensitive to cues and context.In exploring this phenomenon, Hadamard produced one of the most famous and cogent cases for the existence of unconscious mental processes in mathematical invention and other forms of creativity.Written before the explosion of research in computers and cognitive science, his book, originally titled The Psychology of Invention in the Mathematical Field , remains an important tool for exploring the increasingly complex problem of mental life.

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