FOM: naive or brainwashed? (objective vs. subjective in math.)

Vladimir Sazonov sazonov at
Fri Mar 13 14:23:42 EST 1998

Reuben Hersh wrote in responce to Martin Davis:

> > As you mentioned in your posting
> >about Lagrange's theorem, it's clear to you that such a theorem about the
> >natural numbers was true before there were people,
> >and I suppose before there was an Earth or a Sun.  Anyway,
> >I said that to me, as a materialist, such a view was hard to
> >connect or relate to my overall materialist philosophy.  You
> >said that you also didn't know how to make such a connection.
> >To me, that's a reason why Platonism is untenable.  To
> >you, evidently, this gap does not prevent you from continuing
> >as a Platonist.

Martin Davis replied:

> One consequence of Lagrange's theorem is that any bunch of pebbles can be
> rearranged into four piles each of which is arranged as a square array. As a
> materialist do you really believe that it is conceivable that bunches of
> pebbles on a jurasic beach were immune to this law just because there were
> no mathematicians around to formulate it?

Thus, Martin Davis uses *experimental arithmetic* in his 
argumentation.  I have presented in this FOM list another 
*experimental arithmetical low* that base two logarithm function 
of arbitrary integer argument in unary notation is *bounded* by 
the number one thousand, and that logarithm of logarithm is 
bounded by ten. However, Peano arithmetic proves the contrary, 
namely, that logarithm is *unbounded* function.

What is true, what is false, and what is just *our*
(social or individual) imagination and agreement here?

> >It's not a matter of finding a physical location for 7.  It's
> >a matter of explaining the sense in which we can say it
> >exists.  Its objectivity is not an explanation of that.
> I believe that for mathematics the fact that an entity's property's are
> objective (there for us to find out about, not for us to fix) is what we
> need to know to say that that entity EXISTS. No more and no less. I'm happy
> to leave to philosophers whatever deeper ontological issues there may be.
> But I dispute that "locating math in social interactions" sheds the least
> light on WHY there is that objectivity.

First, my question: WHAT may shed the light on WHY there is that 
objectivity? I guess that this should be something like a belief 
in some kind of objective existence of "the" standard model of 
Peano Arithmetic? 

By the way, can anybody here explain what is this fully 
"determinate" "standard model"? I am not asking what is, say, 7, 
but what are "all" the natural numbers, altogether. I never seen 
such a clear explanation and do not believe that it exists at 
all (of course, except the formal definition of this model in 
set theoretic terms as the ordinal omega; I have no problem with 
*this* definition).  Everybody uses this term in a general 
context but nobody explains its meaning as if it is trivial or 
indecent question. I ask to forgive me my ignorance. 

I believe that ignoring the subjective character of essential 
part of mathematical activity is also nonproductive. (Compare 
this with the subjective character of creating by a master, say, 
a table or some new instrument, etc. The form and the 
construction of it may heavily depend on the master's 
imagination. Nevertheless, and even because of this! it may be 
objectively a very useful thing.) It is important to realize 
how, by which lows of thought mathematical and logical "truths" 
(like induction axiom, choice axiom, and even transitivity of 
implication, etc., etc.) are consciously or unconsciously 
*created* and may be *manipulated* by us, and, of course, also 
how and why such a subjective activity can be purposeful successful 
in achieving the proper relation to experimental objective truth in 
the material world (the applied role and "unreasonable 
effectiveness" of mathematics). 

Vladimir Sazonov
Computer Logic Laboratory,
Program Systems Institute,  	| Tel. +7-08535-98945 (Inst.),
Russian Acad. of Sci.		| Fax. +7-08535-20566
Pereslavl-Zalessky,		| e-mail: sazonov at
152140, RUSSIA			|

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