Is There Anything to be Learned from Very Long Computations? While I understand that doing computations can help us gain a greater appreciation for the underlying concepts of our subject matter, it seems that sometimes textbook problems can become more an exercise in tedium than in understanding. For example, in Deisenroth's Mathematics for Machine Learning, I was working through the following problem.

I understand that the intended purpose of this question is to impress upon the reader the technique by which we calculate the projection of a vector onto a subspace using the Moore-Penrose pseudo-inverse of the matrix composed of the basis vectors of that subspace. What I don't understand is why it is necessary to complete such a tedious computation to gain this understanding when the same algorithm can be executed on shorter examples. What do we gain from this?
 A: One of the most extraordinary pieces of mathematical work in the history of humankind is Gauss's Disquisitiones arithmeticae. In that book he sets the groundwork for much of modern number theory — for example, in the first few pages he invents the idea of congruences.
In many ways, lots of what he does there is the result of his trying do compute things. He famously spent a lot of time building tables of prime numbers --- which he had to compute by hand, of course. He must also have spent a huge amount of time finding quadratic residues, and what not… Doing those calculations with a critical mind — and being Gauss! — led him to notice patterns and from that to conjectures and to theorems.
That book is a testament to the most human of all enterprises: that of trying to do less work. Somewhat paradoxically, learning how to do less work very often requires a lot of work.
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Theorems and conjectures and math does not come to people in dreams sent by the gods: they are the result of observation. And you cannot go out into the woods and observe math like you'd observe birds. You have to build a lab out of examples, and calculations, to build sensible prejudices about what should be true and what should be false, and so on. That is done by calculation — in a very general sense of the term.

As for large examples… Dealing with vectors in $R^5$ is not a large example by any means, but it will help you exercise the muscle that lets you deal with big examples. A big example forces you to have to find ways to organize what you are doing, to come up with strategies to check your work, to notice what is exactly routine and what is not, and so on.
