[K.B.]

The 4 most popular major philosophies of mathematics are Formalism, Inventionism, Intuitionism, and Platonism.

Brower was an intuitionist (as it appears you are.) He was also Dutch (which I am) He made many important discoveries using intuitionism. I certainly bow to his mathematical prowess. However Errett Bishop produced a systematic treatment showing that almost all of the important intuitionist material can be dealt with by methods of classical approach. In other words, there are different ways to reach the same mathematical conclusions.

I see why you are not supportive of using Boolean logic, true false, binary, computers. Brower likely supported 3 state logic, fuzzy logic, yes no maybe.

"Is pure mathematics simply a game which we play or do our theorems describe an external reality?"

Did we discover counting or did we invent it?

The slow development of artificial intelligence shows how complex the human brain is. We can’t duplicate it with computers very well. Computers have a difficult time understanding simple language, something that comes easy to us. Computers have an easy time solving math problems; something that often times seems difficult for us. Math is not difficult, it is merely contrary to our normal way of thinking .

If we better understood how the brain handles math, we could find approaches to teach math more effectively. I agree with your statement that the smart kids will make it no matter what. Even if they have a few bad teachers they seem to bounce back in no time. The cream rises to the top.

I did not say the educational system that brought us to where we are, was bad. I am saying our society is changing and so are our educational needs. Education has been changing continuously anyway. If it didn’t change it would become stagnant. Current educational literature and brain research contains a body of knowledge (or opinion) that differs from your opinion. It's not about math its about the human brain. I grew up in a “Leave it to Beaver” household. That doesn’t exist for the average kid any more. The undiscovered math that is out there will need new and different ways to discover it. It is not that math is changing; it’s our world, our people, our knowledge, and the tools at our disposal.

Our educational system appears to lag behind other countries systems for many reasons.
A. We educate everybody. When we compare tests we include the low along with the high. Other countries only promote the best. We are comparing our average with their top (see the Timms report)
B. Other countries have a much higher respect for education and teachers. It is valued more when it is not free for everybody.

The days are gone when a single person could understand all of math. For hundreds of years a person could study all of the known body of math. This body has become too large for any single person to learn. People will need to work collectively in hives to conquer larger problems. A collective conscious can solve more than a single mind. The famous mathematicians of the past are going to be replaced by groups, corporations, communities, working together to solve larger problems. We will need new methods to solve new problems. Our educational system is adapting to new knowledge of the human brain. If this is BS to you then so be it. I agree to disagree. If you don't think kids are different today then visit your local schools.

Snowman, this conversation has probably stretched beyond OT capacity. Thanks for the oportunity to discuss. You are welcome to have the last word (or last 2 letters if you wish)