[FOM] General Foundations Discussion

[Ayan Mahalanobis]

Harvey Friedman wrote:

>Mahalanobis wrote:
>
>  
>
>>>I like nonstandard analysis a lot, but I don't see it as measuring up
>>>to the epochal work of Goedel and Turing.  In particular, one can't
>>>use nonstandard analysis as the basis of an exposition of mathematics
>>>from the ground up.  Part of the difficulty is that, in a precise
>>>sense, one can't give an example of an infinitesimal.
>>> 
>>>
>>>      
>>>
>>I don't know much about non-standard analysis. I will dare ask a
>>question here in hope that at the end I might end up wiser. Why is the
>>non-existence of infinitesimals an issue? I have tried to understand the
>>concept of continuity in classical mathematics and it seems to me that
>>the use of epsilon-delta is highly conceptual. So what is wrong with
>>conceptual infinitesimals?
>>
>>
>>    
>>
>One can easily define the real numbers explicitly, and explicitly define
>what a continuous function is. In fact, THE field of real numbers is the
>UNIQUE complete ordered field, up to isomorphism.
>
>However, apparently one cannot define THE nonstandard real numbers
>explicitly, and explicitly define what a nonstandard continuous function is.
>THE field of nonstandard real numbers makes no sense, at least currently.
>
>In fact, various undefinability issues have been discussed here on the FOM
>some years ago. I got involved in the discussion, and proved some negative
>results. But I remember that there were still some interesting relevant
>questions that remained open. Anybody care to restart this productive
>discussion where it left off?
>
>The apparent lack of THE nonstandard real numbers is a DECISIVE drawback
>against the use of nonstandard analysis in its most obvious foundational
>role (although there may be other roles). This is what is behind the
>categorical rejection by the mathematics community of teaching calculus this
>way. Keisler did write a big calculus book that promptly went out of print.
>
>Mathematicians wish to concentrate their attention on mathematical
>structures that are not only explicitly defined, but also are canoncial in
>various hard nosed senses.
>  
>
I don't think you tried to answer my question. Though I am thankful for 
your comments. Since you bought up the topic that in classical 
mathematics THE real numbers are defined explicitly and you put an 
emphasis to the fact that real numbers should be explicitly defined and 
be unique. Do the phrase "explicit definition" has any foundational 
value without trying to understand what it means or what it should mean? 
The constructions of sets which depend on the axiom of choice might not 
be explicitly defined for some and probably they are right in the 
context of Banach Taraski paradox and others.

My point being that if we go in the cauchy definition and use 
convergence to define the complete field of Reals then we are already in 
the trap of epsilon-delta to which I already raised an objection.

In Errett Bishop's constructive mathematics cauchy reals are not 
isomorphic to Dedekind reals but they seem to be doing fine. So I have 
little difficulty in believing that the uniqueness is a big issue. I 
once tried to read through the work of Bell , "Infinitesimal analysis" 
and yes I also thought that some form of existence of those real numbers 
would be nice. Could some one more knowledgeable in this subject please 
contribute to this.

Best Regards
Ayan.