von Neuman and understanding

So the joke goes something like this: A student asked John Von Neumann when he would start understanding things. And van Neuman replied "Young man, in mathematics you don't understand things. You just get used to them." 

 
I like this way of putting things. I can recall vividly all the ways things that I now hold dear in mathematics were once completely impenetrable. 

From the time I was first taught in high school about matrices and I kept asking myself and the teacher, but `what difference does it make if I write numbers on shapes or not?' It shouldn't make any difference how we lay out the numbers! To when I finally got the hang of Linear Algebra, after months of resisting it: the notion that a vector space could be defined by the properties it has, instead of by what it is. This was, mind-boggling and it still is, a little. 

More disturbing still was when I first learned about Natural Deduction from my first Logic teacher (Luiz Carlos Pereira) and I had the bad idea of telling him that I couldn't see the point of having axioms, sequents and natural deduction to define the "same" system.  It seemed to me that logicians hadn't gotten their formalizations sorted out, yet. Ah, the foolishness of youth.  Now I can see that some of this Bourbakianism of believing in "the most" perfect formalization of mathematical concepts is not only silly, it's bad for mathematics and for science in general. (also the most embarrassing detail is that Luiz Carlos told Prof Prawitz about my stupid remarks.) 

oh well, everyone knows I have very strong convictions, that is --translating-- that I am as stubborn as a mule. Mostly my stupidity is harmless, like, it took me forever to learn how to bike, because I was convinced that it was not possible for bikes to stay upright. or when I decided that people could not swim, as otherwise, why would anyone drown? Again it took me forever to learn how to swim, since I believed it was impossible to float.

But despite all my blunders, I still think that it's important to have your own ideas on the mathematical concepts you're taught and what they mean and don't mean. and whether they `have legs' (will go far or not). So I hope to re-ignite the more abstract side of this blog and to discuss a few more mathematics, while I can. The worst thing that can happen is that I am wrong. I have plenty of experience in this department.

ps: Check out the Wikipedia page on "The Martians", I had no idea all these guys were from Budapest!