[FOM] RE:2^1000

[Matt Insall]

Dear Vladimir,
Your original posts, if I recall correctly, indicated that you consider
``infinity'' to be less than or equal to 2^1000.  Now, it is known that no
natural number is infinite, so this indicates to me that you do not believe
that numbers larger than 2^1000 exist.  You seemed to give ``feasibility''
as a justification for this conclusion.  I do not agree that feasibility of
notation should be used as the basis for determining what exists and what
does not exist, but if we momentarily adopt your position, that feasibilty
of _notation_ should be used in this metaphysical way, to determine what we
consider to exist, then the main question becomes not just feasibility of
the notation you prefer, but also why should one prefer such notation.  The
notation you proposed is woefully inefficient, as is the one I described
(but, I might add, I do not _prefer_ it).  So why do you prefer your
notation over the one I described?  Better yet, can you explain to me why
_I_ should prefer your notation?  If you can convince me that your notation
is the superior one, and if you can convince me that the only numbers which
exist are those denoted by some expressions in your notational system, then
we can discuss what is meant by ``infinity'', and where it lies.  But I am
not currently convinced that you are using the term ``infinity'' in a
precise, meaningful way.

Now, you state that
``To "measure" feasibility of numbers (how big?)
the unary notation is, evidently, most adequate one.''

I think I disagree with this, but first, I must parse it a bit, and see if I
understand it.  You seem here to be equating size of a number with
feasibility of writing down an expression that describes it in the unary
notation you described.  I gather that you are only interested in unary
notations in which it is feasible to write the number 1 with a single
hashmark.  There are hidden assumptions in your statement, I think, that
bear more careful scrutiny.  For instance, in an extreme case, one might
require each hashmark to be 99 light-years long.  Personally, I would in
such a (unary) notation, consider even the number 1 to be infeasible.  In
another extreme case, one might allow the hashmarks to vary in size as one
writes them.  For instance, if the first hashmark one writes is 4 cm in
length, and each additional hashmark is half the length of the one that
immediately precedes it, then very large numbers are quite feasible in this
``unary'' notation.  Clearly, even the idea of a ``unary'' notation does not
completely prescribe what you mean by ``feasible''.  What does?  I propose
that it is the general concept of number (without a specific notation).  For
you seem to be _actually_ basing your notion of feasibility on the _number_
of hashmarks, and you seem to be _ignoring_ the variety of physical ways to
_implement_ a (unary) notation.  Having said all this, I still do not agree
with your metaphysical identification between ``feasible'' and ``existent''.

You wrote:
``By which miracle will you avoid considering what is feasible unary
(and then also binary, decimal) finite string in discussion of
feasibility? Sorry, what are you talking about? Is it about
feasibility or about what? Do not you substitute one question by
some other (I do not even understand which one)?''

First, no miracles here.  By a simple application of logic, one may of
course deduce that if I am discussing with you feasibility, then I will not
avoid discussion of feasibility.  But I am not only discussing feasibility,
but also your metaphysical conclusions based upon a discussion of
feasibility.  I apologize for not making that clear before.  But I think I
have now made it clear.  I do not substitute one question for another, but I
think I exposed some of the concerns one needs to consider in discussing
feasibility and its _meaning_ in the context of many other notions already
available in mathematics, its foundations, and its applications.  You seemed
to identify ``feasible'' with ``existent'', and I suggest that these are
distinct.  (In fact, while some people may debate it, I consider
``existence'' to be a much less ``fluid'' concept than ``feasibility''.  Put
another way, a number that exists exists, but a number that is feasible in
one context is infeasible in another, as demonstrated by my discussion of
the effects of notational preferences in determining which numbers are
``feasible''.)

You wrote:
``If you want, consider my question as about feasible strings (say,
unary ones). However, for me, identifying unary string with natural
numbers is the most natural thing.''

The question is what are your ``feasible strings'' made of?  Must we use as
our basic unary symbol the hashmark that appears on my screen when I press
the key that is located on my keyboard right above my ``enter key'', along
with the shift key, which produces ``|'', or can the basic unary symbol be
smaller?  Can it be larger?  Can the basic unary symbol vary in size, as it
probably does when I write it by hand?  All these things affect what numbers
you will consider to be feasible and what numbers you will consider to be
infeasible.  If you do not agree with me, then I submit that you are not
_actually_ identifying numbers with (physical) unary strings, but with some
sort of ``ideal unary strings'', and you are therefore making another
metaphysical assumption about the existence of these ideal unary strings,
since you seem to consider their corresponding feasible class to determine
which numbers exist.


Best regards,
Matt Insall

PS:  Have a happy holiday, if this is to you a holiday season, as it is for
us in the USA.  (Have a happy season, in any case.)



Dr. Matt Insall
Associate Professor of Mathematics
Department of Mathematics and Statistics
University of Missouri - Rolla
Rolla MO 65409-0020

insall at umr.edu
(573)341-4901