[FOM] real numbers

[Robbie Lindauer]

> -> Is this the same sense in which Santa Claus can be said to be real?
> There is NOT!  There is *huge* disagreement, both in detail and in 
> basics.
>
> Also, he doesn't "kick back", (short of parental conjuring tricks).

To say that mathematical objects and their ontological foundations are 
well understood and agreed upon is to obviate this list.  CF Hartry 
Field.

There is clearly a sense in which "Santa Claus" "kicks back".  Though 
his borders are not perfectly well-defined ("Is Santa Claus identical 
with Sinter Klas?")

There are disagreements, for instance, about whether infinity is a 
number and whether or not the expression "-1 things" ever refers.

> ->Isn't this the same sort of "reality" which
> ->mathematical objects have, a "reality" dependent on a human cultural 
> context?
>
> You are speaking of Popper's "third world" here - the world of 
> cultural objects.
> Those could be said to have an "emergent reality", perhaps, but not a 
> basic one.
> Their outlines are vague and unclear; not like math whose chief 
> characteristics
> are clarity and precision.   Compare a dream and waking life.

There's great value in unclarity and imprecision, even in mathematics 
when applied.  Being able to formally "deal" with vagueness and chaos 
is very valuable.

If I have three apples and you take away two or three or four, how many 
do I have left?

> But you can't adjust math to suit yourself - that's utterly crucial - 
> it
> kicks back.  Often and again we grind our teeth in frustration and 
> have to
> abandon a cherished hope concerning a conjecture or whatever, because 
> we
> can't just do it the way we want - we have to do it the way IT wants!

Are there necessary rules about how any language is to be interpreted?  
"You can only interpret this remark as a truism" is almost certainly 
false.

> Sure we can change the emphasis of our enquiry, re-define terms, etc, 
> but
> the original unpalatable fact remains to bug us however much we dodge 
> it.

Which are those?  Which terms are we not allowed to redefine?  I don't 
think they're mathematical facts, more likely logical ones "every 
statement is either true or not true" and such.

There may be a physical fact, that when you take an apple away from 
someone who has two, they only have one left.  But this would be very 
different in a world in which apples somehow only came in pairs (for 
instance, if there were an evil demon who once you only had one apple 
left, took the other away so it was impossible to have only one apple). 
  In that world, we might say that "2-1=0 for apples."

> Well that's a relief I suppose.  You may call it MY basic view, but I 
> suspect
> it's merely what almost all practising mathies, and a great majority of
> math logicians/philosophers actually think.  So-called "Platonism"; or 
> realism.

The case is parallel with the interpretation of Quantum Mechanics.  
There are plenty of working physicists who just don't care whether or 
not the underlying objects correspond in any strong way to there being 
actual objects and properties as described in the formalism.

Most likely, there is a "moderate realism", for instance the natural 
numbers are real, the rest are fabrications or something like that.

This would be parallel to physics as well.  We might have a theory of 
Cows, for instance.  And we might ask "What happens if you take 
half-a-cow from three hundred farms and bred them."  While we know 
perfectly well that we can't breed half-cows, we can still build a 
formalism about half-cows so we could answer this question.  But if 
there weren't any cows at all, then we'd say our formalism was useless. 
  It's the cows that are real, in this case, the half-cows are creations 
for simplicity sake.

Best,

Robbie Lindauer