Book recommendation

[Nine Divines]

Can anyone recommend a good book on non standard analysis using the hyperreal numbers? I've been going over Elementary Calculus - An Infinitesimal Approach by Keisler and he constructs the calculus by means of the hyperreals. I've done some homework on these numbers and they were formally proved around about the 1960s.

Further, does anybody know what the latest research in maths is at the moment?

 

[pka]

Can anyone recommend a good book on non standard analysis using the hyperreal numbers? I've been going over Elementary Calculus - An Infinitesimal Approach by Keisler and he constructs the calculus by means of the hyperreals. I've done some homework on these numbers and they were formally proved around about the 1960s. Further, does anybody know what the latest research in maths is at the moment?

Well Keisler is the very best. He also has a small paper-back Foundations of Infinitesimal Calculus. The book that started it all is Non-standard Analysis by A. Robinson. I always liked Martin Davis's book Applied Nonstandard Analysis. So far as I know there is no active research program going the field.
The coming of computer algebra systems killed the need.

 

[Deleted member 4993]

Well Keisler is the very best. He also has a small paper-back Foundations of Infinitesimal Calculus. The book that started it all is Non-standard Analysis by A. Robinson. I always liked Martin Davis's book Applied Nonstandard Analysis. So far as I know there is no active research program going the field.
The coming of computer algebra systems killed the need.

I detect a sense of loss in your response here..........

 

[Nine Divines]

Well Keisler is the very best. He also has a small paper-back Foundations of Infinitesimal Calculus. The book that started it all is Non-standard Analysis by A. Robinson. I always liked Martin Davis's book Applied Nonstandard Analysis. So far as I know there is no active research program going the field.
The coming of computer algebra systems killed the need.

Yes, I'm aware Keisler has another book as a follow on. Is it a formal approach on the hyperreals?

As for the computer aspect of your argument, I believe a computer can only achieve what it's programmed to do.

 

[Deleted member 4993]

Yes, I'm aware Keisler has another book as a follow on. Is it a formal approach on the hyperreals?

As for the computer aspect of your argument, I believe a computer can only achieve what it's programmed to do.

But HAL is coming to overtake IBM......

 

[pka]

Yes, I'm aware Keisler has another book as a follow on. Is it a formal approach on the hyperreals?
As for the computer aspect of your argument, I believe a computer can only achieve what it's programmed to do.

Well you completely misunderstood my point. It had nothing to with computers themselves. The remark was about a class of software know as computer-algebra systems(CAS). From the late 1960's through the 1980's there was a great effort to reform calculus teaching. One of the leaders was a group at Harvard, the Harvard Calculus project, and Keisler's group both had many off shoots. The National meeting was held in New Orleans in mid 80's which was more-or-less devoted to the topic Calculus Reform. What none of us realized at the was the impact CAS would have on mathematics teaching. I believe its main impact was the power of visualization: seeing how things work in real time. Relatively cheap computer software for a fast growing field of technology.