[FOM] First-order arithmetical truth
V.Sazonov@csc.liv.ac.uk
V.Sazonov at csc.liv.ac.uk
Fri Oct 27 07:12:23 EDT 2006
Quoting "Timothy Y. Chow" <tchow at alum.mit.edu> Wed, 25 Oct 2006:
> Apparently you also choose to ignore my explanation of why I chose to
> ignore certain parts of your message.
Sorry, I do not remember any explanations concerning your choice to
ignore, except presented here.
>
> However, I believe that I've said all I can say on the original topic
> (namely, the point that Arnon Avron was trying to make to Francis Davey).
> You and I also appear to be in agreement on that point.
I understand that style of reasoning and can follow it, but cannot
agree with it because it is based on a vicious circle inherent to any
reasoning based on a belief.
Therefore I'll
> say a couple of things about the new topic that you repeatedly insist on
> bringing up.
>
>> Naive formal systems are something from our real life and (for example,
>> computer) practice. They are quite rigid and reliable and even can be
>> represented physically by computer systems. The main point is that they
>> are REAL
Let me clarify myself if it is still unclear: REAL in the sense to be
REPRESENTED physically, say, by computer systems or in any other
reasonable way. It is also a specific kind of real (not imaginary)
human activity with finite physically presented objects of the discrete
character.
>> unlike ABSTRACT mathematical numbers and abstract
>> (meta)mathematical formal systems (quite similar by the nature to
>> mathematical numbers).
>
> I don't see how (naive) formal systems are more real than (naive) numbers.
NAIVE formal systems are NOT more real than NAIVE numbers. The main
difference is that naive (I actually assume unary) numbers have much
simpler real representation (as strings like |||||) than general formal
systems which are more complex "devices".
NAIVE formal systems (something similar to really written computer
programs) are real (JUST REALLY and ADEQUATELY PRESENTED) in comparison
with ABSTRACT numbers or any other abstract, imaginary mathematical
objects, including abstract formal system as objects of metamathematics.
Any abstract mathematical objects such as numbers and sets and whatever
else are represented by REAL and NAIVE formal system VIA our
imagination.
> Say you take a pencil and move it in a certain fashion relative to a piece
> of paper. Is the resulting physical object a formal system?
... Then some discussion follows on physical objects, whether they are
the same (formal systems/sheets of papers, etc) or not.
Timothy continues:
> But I am baffled when you say that the section of your
> disk drive is the same physical object as the paper. What is the relevant
> *physical* property that they have in common?
>
You unnecessarily overcomplicate things. Naive formal systems are like
lego, only a bit more complex. Like with lego we can construct formal
derivations, even computer having no human intellect is able to do
this. This is something "mechanical" provided appropriate physical
peaces are given. This is about quite simple style of behaviour of
people (even small children) with some discrete physical objects.
Of course, I agree that working with naive finite objects (numbers or
formal systems) assumes some naive level of abstraction. Say, quite
small children are able to recognize that two pieces of lego of
different colours have the same shape - just can use them practically
in the same "mechanical" way connecting them with other pieces. They
absolutely do not need to have any scientific and philosophical
considerations about this activity. They just do this in some way, and
that is all. Given the simple (or not very simple, but does it matter?)
ability to distinguish and identify letters in an alphabet written on a
sheet of paper, we can manipulate with strings of letters or more
complicated figures (of logical proof rules). It is still highly naive
ability to operate with finite objects - too far from the ability of
the abstract mathematical thinking. Also no need to come to the general
idea of arbitrary (HOW MUCH arbitrary?) finite string of symbols, to
the idea of quantification over these strings, nothing to say about
alternation of quantifiers. Only simple mechanical manipulations with
some kind of figures. Just take it as it is.
> I cannot see that the process of abstraction that allows you to equate the
> two (naive) formal systems is any more "concrete" or "real" than the
> process of abstraction that allows most people to work with (naive)
> numbers.
I did not asssert that. It seems this is again some misunderstanding
about naive formal system vs. naive numbers which I have already
commented above.
The real mathematics begins with using these naively understood formal
systems to describe/restrict/regulate/govern, that is formalise our
imagination about abstract objects such as numbers, sets, non-Euclidean
geometry, and whatever else. This way our unruly fantasies and
imaginations become something serious (mathematical).
That is, mathematics is a (specific) interplay between naive formal
systems and our imaginations and fantasies ABOUT what is this formal
play about. Fom this moment formal logical quantifier rules, induction
axiom, etc. start playing a serious, nontrivial role - not just a
childish game.
From the point of view on mathematics and foundations of mathematics
there is seemingly no real need to mathematise the above naive level of
our formal abilities. It is naive, comparatively simple and
fundamental. This is a good and sufficiently solid point and ground to
start. (Even if we would try to mathematise, for example, naively
understood numbers as a theory of feasibe numbers (e.g., as I suggested
in other posts) this will become something abstract and different from
the naive numbers because any abstraction adds something new to the
initial intuition.)
Any attempt to equate this naive ability with abstract mathematical
thought such as
(i) naively understood numbers with abstract numbers as the subject of
mathematics or just of first order Peano Arithmetic,
(ii) naively understood formal systems with abstract formal systems as
the subject of metamathematics
leads to vicious circle. What I mean is just a simple way to break off
this circle: 1. Do not mix the (naive finite) reality with the
imaginary world of mathematics 2. Take the naive finite reality in the
form of naive formal systems as the ground for formalisation of
mathematics.
Is not this so evident and clear?
Vladimir Sazonov
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