If one peruses the mathematical literature for the last one hundred years one will notice that in most cases no diagrams or pictures appear. The level of rigor in all cases is impressive though, but unfortunately this makes the understanding of the results much more difficult. There seems to be an inverse relationship between rigor and understanding in mathematics, at least for those who are new to the subject at hand. In order to gain this understanding, the drawing of pictures and diagrams is useful, along with a certain amount of experimentation with the concepts at hand. Intuition, how mysterious, and however ill defined, plays a role in both the understanding of mathematical results and in their discovery. Many mathematicians do not want to acknowledge this, as a visitation to a typical conference will readily verify. The attitude has been expressed that mathematics has "always been abstract" and therefore that pictures or diagrams violate its spirit. Even a well-known geometry center whose goal was to use sophisticated computer graphics to visualize complex mathematical objects lost its funding, to the consternation of a few but with glee to most.
Thus the way to discovery of mathematics, i.e. the heavy use of intuition, the disorganized shuffling of concepts, and the experimental doodling, has been masked by the final product of this process: a superb example of logical rigor and organization called modern mathematics. The authors of this book however think otherwise, and they give the best apology for the role of experimental mathematics than anyone else in the literature. The book is packed with highly interesting examples and challenging exercises, all of which are ample proof of the need for doing experimentation in mathematics.
In addition to these considerations, the book is just plain fun to read, and even though time constraints may prohibit the working out of every exercise, the book could be used profitably in a graduate course in mathematics or even possibly in an undergraduate course at the senior level. Hopefully this approach to scholarship in mathematics will take hold in this century, and mathematicians will not only write down their final results with all their splendid rigor, but also how they got there. This would serve to educate younger generations of mathematicians in just how discovery in mathematics is done and increase their efficacy in the same. The book will also assist those who are trying to build machines capable of discovering novel results in mathematics. Machine proofs of difficult theorems and conjectures are now a reality, and in the twenty-first century we will no doubt see many more of these.
This book therefore contains a lot of hints about how to proceed in mathematics. Its acceptance will depend on how well it does its job in the creation of new mathematical results and in the teaching of them. Results in mathematics that seem plausible serve to make conjectures and motivate the construction of rigorous proofs. This book is a first step in a hopefully larger work.
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