[FOM] internal/external distinction

[José Manuel Rodríguez Caballero]

Dear  Mikhail,
In the paper that you cited [1], the author wrote (I changed the variables
for better readability):

Fix an extension of the reals, and suppose Alice assigns an infinitesimal x
> [...] to the hypothesis that a fair coin flipped countably infinitely often
> always landed heads, while Bob assigns 2x to that hypothesis. Both assign
> consistent finitely additive probabilities. On what grounds could we say
> that Alice’s assignment of x rather than 2x was rationally right--or even
> better reflective of Alice's doxastic state?


The problem is that Alice cannot assign a nonzero infinitesimal, because an
infinitesimal, say an infinitesimal segment, is something that, using the
language of Newton [3], may be assumed of less length than any assignable.

In his lecture 16 on nonstandard analysis [2], Alexander Shnirelman shows a
case where the arbitrariness on the choice of infinitesimals has an effect
on the standard real numbers and he explains how to solve this apparent
paradox.

In my personal experience, I used nonstandard analysis as a tool in the
proof assistant Isabelle/HOL [4], before knowing its theoretical
foundations. So, it is possible to prove a theorem in functional analysis
using the transfer principle without knowing all the details about the
formulation and meaning of this principle, because anytime when there was
something wrong, the software pointed out the mistake.

Kind Regards,
Jose M.

References:
[1] Pruss, A.R. Synthese (2018). https://doi.org/10.1007/s11229-018-02064-x
[2]  Nonstandard Analysis, Lecture 16, Alexander Shnirelman, Concordia
University, Montreal, Canada, Winter 2019. https://youtu.be/cw-hPFo_9Xk
[3] Newton, Isaac. The Principia: mathematical principles of natural
philosophy. Univ of California Press, 1999.
[4] Fleuriot, Jacques D., and Lawrence C. Paulson. Mechanizing nonstandard
real analysis. LMS Journal of Computation and Mathematics 3 (2000):
140-190.
https://www.cambridge.org/core/journals/lms-journal-of-computation-and-mathematics/article/mechanizing-nonstandard-real-analysis/A1A4325EE3C91A18EF68D55407D184FB



--

José Manuel Rodríguez Caballero

arvutiteaduse instituut / Institute of Computer Science
Tartu Ülikool / University of Tartu

Personal Research Page: https://josephcmac.github.io/
-------------- next part --------------
An HTML attachment was scrubbed...
URL: </pipermail/fom/attachments/20190809/1fc532e2/attachment.html>