[FOM] real numbers

[Mitchell Spector]

On Monday, May 12, 2003, at 11:54  PM, Bill Taylor wrote:

> ->What does "real" mean here?
> ...
> -> Is this the same sense in which Santa Claus can be said to be real?
>
> Oh puh-LEEZ!

Well, it's not such a bad example.  After
all, as is apparent from the rest of your
post, you knew what I meant!


> ->After all, there is general agreement within our
> ->culture on the characteristics of Santa Claus, on
> ->his appearance, on his behavior and motivations,
>
> There is NOT!  There is *huge* disagreement, both in detail and in 
> basics.

Maybe.  One could pick a different example, of course.

But, more to the point, there appears to be some
disagreement also on the nature of mathematical
objects.  Maybe the disagreement is similar.



> Also, he doesn't "kick back", (short of parental conjuring tricks).

Indeed not.  But neither do mathematical objects.

Can you do an experiment which will test whether
the continuum hypothesis is true or false?  CH
doesn't kick back.

But the problem is more fundamental than that
suggests.  Even simple number-theoretic facts
(1 = 1, say, or 2 + 2 = 4) do not kick back;
these are statements about abstract numbers,
not about physical objects in the real world.
(To see that this is true, suppose we suddenly
happened to observe one day that 2 rocks piled
on top of 2 other rocks now yielded only 3 rocks.
This would not change our unshakeable belief that
2 + 2 = 4, as a statement about numbers.)

Maybe you're suggesting that the objects
themselves (numbers, in this case) are what
kick back, rather than statements about those
numbers, but that seems even harder to justify.

There is no physical experiment that can be
conducted that would change a mathematician's
view of the mathematical universe.

On the other hand, a change in one's mental
model _can_ cause a change in one's perspective
on the mathematical universe.  This change
could be precipitated by philosophical
discussion (a conversion to intuitionism
or finitism, or from intuitionism or finitism
to a more traditional approach).  A change
could also conceivably be caused by brain
injury or even mental illness.


> ->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.

Mathematical objects live in what one could
call a fantasy world (maybe "mental world" is
a better phrase) that has been created by many
people over time.  Presumably it is constructed
in our minds in a fashion somewhat the same (only
more disciplined) as the way our mental model of
the real world is constructed.

Strictly speaking, we each have our own mental
model, our own version of this mathematical
world.  But the logically disciplined nature
of the subject, together with the feedback of
communication among mathematicians, tends
to keep our various models quite similar to
one another.  This is what takes the place
of what you're calling "kick back," keeping
our various models more or less in line with
one another.

> -> Reality in the
> -> physical world has no such dependence (or much less such, anyway).
>
> As in the abstract world of math, I claim.  We can adjust cultural 
> objects
> the way we please, to suit ourselves and others.  If your religion bugs 
> you,
> you can alter it a bit and no-one can gainsay you (since the end of the
> inquisition, anyway).
>
> 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!

A mathematical object doesn't kick back.  You can imagine
yourself in the mental world and _imagine_ that it's kicking
back, just like in a dream.  But that doesn't mean that it
has really kicked you; you're just wrapped up in a very
engrossing story.

In fact, I'm sure we've all seen students who do not
share the same model of the mathematical world that
we have.  And what is the consequence in the physical
world?  Maybe they get a D or an F in a course.
So who kicked back?  Not some fictional mathematical
object.  It's the math professor who kicked back.

It's a social, cultural "kick back" that keeps our
mental models aligned.

Of course, it's also true that the disciplined
structure underlying mathematics ensures that two
people with the same basic view (I'd say the same
axiomatization, except I don't want to presuppose
a particular method of organization) have extremely
similar developments, identical in many respects.
And this shouldn't really be an afterthought,
although I've presented it that way.  It's perhaps
the central features that distinguishes mathematics
from other cultural phenomena.



> 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.

I agree, and, as I said, I share it.  But I don't
think realism is a good name for it.  The approach
involves an immersion in a world with such internal
consistency that we can successfully maintain the
pretense that it's real.

It's not unlike virtual reality.

As for Platonism, I never understood the willingness
to identify realism and Platonism.  Plato, in the allegory
of the cave, seems to go well beyond a belief in the
reality of abstract objects; doesn't Plato (or Socrates)
propose that the physical world is less than real, a mere
set of shadows cast by the objects of eternal perfection
in the abstract world?

Mitchell
--
Mitchell Spector
Seattle University
E-mail: spector at seattleu.edu