FOM: Re: "Relativistic" mathematics?
csilver at sophia.smith.edu
Mon Oct 12 07:17:14 EDT 1998
> > Vladimir Sazonov:
> > > > > But there is nothing here (except possibly of my
> > > > > education) what forces me to think that the structure I am working
> > > > > with is mysteriously unique one, even up to isomorphism, because
> > > > > I even do not know WHAT DOES IT MEAN *any* "isomorphism" and *any*
> > > > > structure in this general context.
C. Silver said:
> > Take these second-order axioms:
> > 1) (Ax)Sx <> 0
> > 2) (Ax)(Ay)(Sx = Sy --> x = y)
> > 3) (AX)(X0 & (Ay)(Xy -> XSy) --> (Ay)Xy)
> > What is wrong with the proof that any structure satisfying these
> > axioms must be isomorphic to <N,0,S>, the standard structure? Or is it
> > that you accept this proof, but want to raise the question: Just what *is*
> > this standard structure? Or are you asking just what is it to *be* a
> > structure in the first place? Or,...?
> Any (mathematical) proof assumes (at least implicitly) a formal
> system where it is written.
Already I disagree. Maybe this is at the heart of our
disagreement. I think mathematics proceeds intuitively first, on the
basis of agreed-upon structures. For example, suppose I were to say to
you: I'm thinking of a Tree that starts out with two elements. To make
this pseudo-Biblical, let these elements be Adam and Eve. I'm supposing
that each of them alone can beget "children". And, in fact--continuing
with this silly example--let us suppose each of them has two children the
first generation, three the second, five the third, and so forth (i.e.,
the children of each generation arrive in terms of the sequence of prime
numbers). And, the children can have children too, in accordance with the
same procedure of having 2,3,5,7,... in each generation. I'd like to make
this a little more complicated, but I'm not sure how. Let's see.
Supposing that some of them die according to the following pattern....
I'm not going to continue with this. It's probably rather a dumb example.
My only point is that you and everyone else can follow this without
benefit of some formal system. I think when I started out saying that
this whole thing begins with Adam and Eve, you pictured two entities.
Then, when I said they each begat two children, I think you also pictured
that. And, I think you also pictured three children for the next
generation, and so forth. Of course, when it gets complicated, you may
need to draw some lines or something. My only very simple-minded point
here is that there is no formalism involved. Just you following along,
picturing my silly made-up structure.
If you were to say that this *could* be formalized, I'd agree with
you. But, the thinking, the understanding, the theorem-proving, and so
forth is *prior* to the formalization of it all. Again, I'm not saying
that it doesn't help to formalize things. I agree that it does. But, the
fact that something ultimately is given a formal explanation does not, to
my mind anyway, indicate that the essence of the mathematics involved is
that it could later on be formalized. I know many people disagree with
this. And, perhaps you do too. Perhaps, our disagreement hinges on your
thinking that all of mathematics makes sense only within a formal system.
Do you? I don't believe this at all.
I wanted to say some more things, but I think I'll stop and wait
to hear what you have to say about this.
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