[FOM] real numbers

[Bill Taylor]

->What does "real" mean here?  

Well, I already gave an (admittedly ultra-brief) outline, when I said it was
intersubjective and licked back.  Without getting overly undergraduately
philosophical, that seems to be a good rough and ready guide to what is
*objective* - i.e. "real".

Of course huge expansion is possible, but it's too early in the thread to write
a whole essay!   I would hope the basic idea is clear, even if objected to.

-> Is this the same sense in which Santa Claus can be said to be real?

Oh puh-LEEZ!

->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.

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

->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.

-> 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! 

That, surely, is the essence of objecthood, or reality, or objectivity.

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.


->[By the way, I don't think I'm disagreeing with your basic view. 

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.

-> But I'd be inclined to avoid
->the use of the word "real" to describe the concept.

As you see I disagree.  I feel the distinction between real and physical
is a very helpful one to make.  And getting back to Popper's "worlds",
those here who know my penchant for "bi-dichotomies", may recall this 
next one, which I find covers the situation very well!

Anyway; it is just in fun, but I suppose there is a serious grain of meme:


             objective    subjective
           ,---------------------------.
           |            |              |
  abstract |   World 4  |   Popper-3   |    World 4 is the new idea here.
           |            |              |    
           |------------+--------------|
           |            |              |    The Popper-2 thingies are
  physical |  Popper-1  |   Popper-2   |    not *concrete* of course,
           |            |              |    but they may be termed physical
           `---------------------------'

These worlds talk about, as their subject matter,

             objective    subjective
           ,---------------------------.
           |            |              |
  abstract |   numbers  |    memes     |     *coins* are physical of course,
           | structures |    money     |     but money is abstract
           |            |              |
           |------------+--------------|
           |            |              |    "feelings" here is both literal
           |   objects  |   thoughts   |     AND metaphorical; that is, both
  physical |   energy   |   feelings   |     sense-data impressions & emotions
           |            |              |
           `---------------------------'

And the standard university departments that study these are...

             objective    subjective
           ,---------------------------.
           |            |              |
  abstract |    math    |  humanities  |
           |  comp sci  |  economics   |
           |            |              |
           |------------+--------------|
           |            |              |
           |  science   |  psychology  |
  physical | engineering|  psychiatry  |
           |            |              |
           `---------------------------'

I often note that when the "diagonal pairs" of such bi-dichotomies can
be seen to have some coherence, then we are really rolling! (I wouldn't
go so far as to pontificate "on the brink of an insight").

And so it is here.   The top left has barely  1.5  departments devoted
to its study, and the bottom right barely even 1.   Whereas the other two
are both enormous.   The main diagonal is "very thin" and the co-diagonal
"very thick".   So I guess that shows a very strong correlation between
the physical and the objective, i.e. the abstract and the subjective.

That seems reasonable, but shows a notable modification of mere Cartesian
dualism.  It looks very good to me.   I know many others think the top left
is actually empty, that its stuff belongs to the top right.
I think most mathemticians and many logicians would agree with me, 
and most of everybody else would agree with those many others.

But at least now we know what we're arguing about!


------------------------------------------------------------------------------
        Bill Taylor                  W.Taylor at math.canterbury.ac.nz
------------------------------------------------------------------------------

              a priori   a posteriori
           .--------------------------.
           |            |             |
 synthetic |   math     |  science    |
           |            |             |
           |------------+-------------|   The Kant/Taylor bidichotomy.
           |            |             |
 analytic  |  logic     |  sophistry! |
           |            |             |
           `--------------------------'