[FOM] real numbers

[Bill Taylor]

Robbie Lindauer <robblin at thetip.org> writes:

->There is clearly a sense in which "Santa Claus" "kicks back". 

We should probably be very wary of claims involving the phrase "in a sense".


->There are disagreements, for instance, about whether infinity is a number

That is merely a matter of definition of an English word, thus inessential.

->and whether or not the expression "-1 things" ever refers.

I don't understand debate about "refers".  But certainly the concept can
be *applicable* to the external world, e.g. in bank accounts.  However
matters of applicability are not crucial to mathematical ontology, I feel.


->There's great value in unclarity and imprecision, even in mathematics 
->when applied.  

Undoubtedly.  But the whole thrust of pure mathematics and math logic is to
REMOVE these from mathematical ontology.  It always has been, from classical
Greek times at least.  In the last century-and-a-half we have seen astonishing
advances, though doubtless more are to come.  Let us not discard these
advances lightly.  Application is one thing, ontology is another.


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

I'm not sure what the point is here, but it sounds similar to noting that
if I add one puddle or one pile of sand to another, I still only get one
of them.  I hope no-one here is going to say this may cast doubt on
our idea that 1 + 1 = 2.  The latter is a mathematical fact, (world 4),
the former a matter of applicability to the physical world (world 1).


->Which are those?  Which terms are we not allowed to redefine?  

You may define anything you like.  But 3 + 4 = 7 is a fact however you
define anything (note my lack of mention-quotes BTW).  If you want it to
become 8 for whatever reason, you are doomed to failure.


->  In that world, we might say that "2-1=0 for apples."

You can say whatever you like about apples, but it won't change that  2 - 1 = 1.


->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.

That may be so, and certainly QM is an interesting lesson for those who say
the physical world simply MUST be such-and-such a way.      But it fails as 
a helpful parallel because math ontology is NOT CONCERNED with the physical
world.  Sometimes it applies, sometimes it doesn't.  Sometimes even 1 + 1 = 2
doesn't apply!  But that has no effect on its truth in math.


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

I have considerable sympathy with this view, in fact.  But it still has
little to do with the physical world, and science.


All this is MHO, naturally.  But I fancy it would also be core thinking for
most pure mathematicians, and many applied ones.  I DO know, though, that
a great many physicists would disagree, and support your (assumed?) view
that the physical world is directly relevant to mathematical truth.

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           Bill Taylor              W.Taylor at math.canterbury.ac.nz
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                They travel as waves but arrive as particles.
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