January 14th, 2016, 04:09 PM  #11 
Math Team Joined: Dec 2013 From: Colombia Posts: 7,615 Thanks: 2604 Math Focus: Mainly analysis and algebra 
Oh yes, I agree with that. My main problem with the hyperreals is that they don't seem to add any value. The only "benefit" I recall hearing about being that they can make calculus accessible to people who don't understand limits. To which my response is: well teach them limits in a clearer fashion then! Or, don't bother teaching the rigorous foundations at all.

February 23rd, 2016, 04:57 AM  #12 
Math Team Joined: Jan 2015 From: Alabama Posts: 3,264 Thanks: 902 
It has a historical value When Sir Isaac Newton developed the Calculus, he used 'infinitesmals' without properly defining them. Hyperreal numbers allow one to rigorously define those 'infinitesmals'.

February 23rd, 2016, 06:44 AM  #13 
Math Team Joined: Dec 2013 From: Colombia Posts: 7,615 Thanks: 2604 Math Focus: Mainly analysis and algebra 
Indeed so (and Leibnitz used them too). Infinitesimals show that their intuition that infinitesimals could be used to penetrate the subject was well founded. But unfortunately the modern theory retains the core objection to those infinitesimals by validating the idea that you just ignore them when you are ready to transport your result to the reals. It's all perfectly rigorous but just as unsatisfying now as it was 300 years ago.


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difference, hyperreal, numbers, real 
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