mycroft, on 2015-March-12, 09:25, said:

At one point I started working through my then partner's Real Analysis book.

This was showing how math works (from the inside).

I am an Engineer - our math was copious, but included many places where the professor said "there's a hole here. You'll never see it so we aren't worrying about it, but..."

I lasted 85 pages before I was happy to go back to the Engineering view of math as "it works, don't worry about it".

This is the step between *my* "math is fundamental" and the real "math is fundamental".

One of my Profs from the 1960s,

Larry Markus , was at the board whipping through an argument with various tensors. he remarked "The whole purpose of tensor notation is to allow you to do things far faster than you can understand what it is that you are doing". As with most memorable jokes, there is some truth to it.

My own observations about the utility of mathematics runs like this: "Most mathematicians I have known went into math because they enjoyed it. People pay us because it is useful. It's a good arrangement. People who do not enjoy it will pay us to do something that we do enjoy".

Is a deep understanding of math useful? I guess Norbert Wiener would say yes, Or Alan Turing. Or John von Neumann, or Sergei Brin. Etc. It is my understanding that when von Neumann first began employing operators in hilbert space to study quantum mechanics, a fair portion of the physics community though he was nuts. So you wait a few years for people to catch up.

It's a big wide world with room for different approaches.

Getting back to the more elementary level with fractions, or maybe geometry and algebra, I think that one practical reason for young people to become comfortable with the basic ideas is that it substantially increases their options for making a living. I was at a store in a mall once when I saw the following (numbers made up, but it was along these lines): A shirt is marked at $40, with 30% off, discount taken at the register. A customer asks what the actual price will be. The store clerk has no idea. OK, the customer has enough money to buy a decent shirt, and the store clerk has a job. But maybe he would like to move beyond being a store clerk. I think it would help (yes I know there are other considerations) if he wasn't completely thrown for a loop by the simplest of calculations.

But mostly, for me, I think a person should be able to work out the price with the discount simply because he lives in the world. Iwatched, not for the first time, Born Yesterday with Judy Holliday as Billie. The Supreme Court is mentioned and Billie says "What is it?". You can get through life without knowing, but it seems like it might be intersting to learn a few things here and there. If I have it right, this is Janet's intention. She just got tired of always having to say "well, I don't understand any math stuff".