[Eternal Student (OP)]

Hi.

    Calculus  ----> 

You've seen this stuff.  It involves small changes in y  divided by the corresponding small change in x.
You were probably taught Calculus in school with this concept.
Some time later there was a procedure to formally consider Limits.   You may have seen the ε , N  definition for limits.

   The thing is, why didn't we just stick with the intuitive idea of infinitesimals?

Disclaimer    This is not a new theory, it's an old theory.  It's also not mine.  It's not even all that new to suggest that we should teach Calculus with Non-standard Anlaysis.  However, there isn't an "old theory but ready for revival" section in this forum so I've placed the thread here. 

References
    If you have a week to spare:
    H. Jerome Keisler -  Foundations of infinitessimal calculus.
    Otherwise, use Wikipedia with  "hyperreal number"    and   "Nonstandard Analysis"  as search terms.   

New Theory    Should we teach Calculus with Nonstandard Analysis?
   (I know it's traditional to add a 20-page document to a New Theories post, so I've written something at the bottom of the post).
 
- - - - - - - -
Attached file:     NonStandard-thinking.zip               [Downloaded  0 times].     

[Origin]

Attached file:     NonStandard-thinking.zip               [Downloaded  0 times]. 

It looks like the file didn't link properly.

[Eternal Student (OP)]

Hi @Origin
   Just checking that you know there is no file.  However, the topic is open to discussion.
- - - - - -
Attached Files:       NONE      .   Sorry, I was being humorous when I added this line.           

[Origin]

Sorry, I was being humorous when I added this line.

Well that went completely over my head.😊

[Eternal Student (OP)]

Hi again.

   Another tradition of the New Theories section is to continue replying even if you're talking to yourself.  I'm going to do some of that now.   However, let's try and encourage some discussion since I rapidly tire of talking to myself and I only have housework to pass my time if I'm not here using this forum.

   There are at least two fields (mathematical fields, not magnetic fields or any other "field" as defined by physicists) that you are familiar with.
   You are familiar with (Q , +, x)   and  (ℜ, +, x)   which are the field of Rational numbers and the field of Real numbers respectively.
   These things are sets of objects that we will call numbers together with some binary operations that can be performed which we will call  addition and multiplication.    We'll just take the usual addition and multiplication.   You should then be able to recognise that the addition of two real numbers does give us another real number.  Similarly the multiplication of two reals gives us another real number.   A moments work will convince you that the Rational numbers behave the same way - the addition or multiplication of two rationals gives us another rational number.   The rational numbers are a smaller set of numbers but have the nice property that they remain closed under addition and multiplication (we can't add or multiply rational numbers to generate an irrational number, we always stay in the rationals).
   You can find more information about mathematical structures called Fields elsewhere if you're interested, I'm not going to bore everone by saying more about them here.

Question 1:    Space is commonly modelled as the Cartesian product   ℜ³
   This is especially useful in Physics since that structure can be treated as a vector space over the scalar field ℜ.  To say that another way, we can add two vectors and we can also multiply vectors by some number and it's useful - it describes another vector in space.
    Why don't we model space as  Q³ over the scalar field Q ?  This would also work as a perfectly good vector space.
    To say this another way, assume you have defined a unit of distance, 1 metre.  What makes you so sure that two objects can be placed an arbitrary distance apart in our real universe?   Perhaps they can only be a rational fraction of a metre apart.  Additionally, how are you defining "an arbitrary distance apart"?  Can two objects be infinitesimally close but not in exactly the same place?

Question 2   I mean you can probably see where I would be going with this and how it connects with Hyperreal numbers and Nonstandard Analysis:
  What if we find a field, ℜ^* that is an extension of the Reals instead of being a sub-field (a smaller set) of the Reals?  Can we model space with this?

Best Wishes.