Re: Definition of real number ℝ --infinitesimal--

On 02/04/2024 12:05, Ben Bacarisse wrote:

[... I]t's true that any new theory
has a up-hill struggle.  And the hill will be steep if the theory is
motivated by common sense since most mathematicians follow Russell and
are sceptical of common sense.

Um.  Of course, most new theories are simply complete bunkum.
Of those that are generally accepted or are refuted only by refined
experiment/argument, there are lots both ways.  I started to produce
a list, but it soon got too long and involved [and debatable].

  After all, we (as animals) have no
physical experience of the infinite so what value can our common sense
expectations of it have?

If you insist on Cantor as the only meaning of infinite/infinity
then you're probably right.  But there have been other experiences and
theories, some of which are more accessible.  For example, there is the
[surreal] "blank cheque" view.  A [mathematical] blank cheque trumps any
finite amount of money;  if you have $1234567, then I write $1234568 on
my cheque and can outbid you.  If we both have blank cheques, then
whoever bids first loses.  In this PoV, "infinity" is not a huge distance
away, but rather "as far as you choose".  Thus, "0.999..." is not an
"infinite" sequence of "9"s after the decimal point, but rather "as many
as you want".  When you decide how many you want, the numbers become
"crystallised", and everything thereafter is finite.  Making that decision
is always disadvantageous, which is why numbers make boring games.  The
surreals give an operational version of maths rather than the traditional
static version.
[There are interesting and playable games with infinite and/or
infinitesimal values, so it is possible to gain physical experience of
these.  There are, for example, some relatively simple chess positions
that are most simply analysed in terms of (surreal) infinitesimals.]
--
Andy Walker, Nottingham.
    Andy's music pages: www.cuboid.me.uk/andy/Music
    Composer of the day: www.cuboid.me.uk/andy/Music/Composers/Dvorak