Re: Formalisation

On Jun 5, 8:34 am, Aatu Koskensilta <aatu.koskensi...@xxxxxxxxx>
wrote:
An important note about terminology: in the following 'statement' will
always refer to a mathematical statement in ordinary mathematical English
while 'sentence' or 'formal sentence' will usually refer to a string of
symbols in the mathematically defined language.

Why should we want to formalise arithmetical
statements and notions in the first place?

You cannot put the genie back in the bottle.
If you stand on the shoulders of giants, you cannot
get back down, not easily enough to make bothering-with-it
worthwhile, anyway. Analogy:
http://www.amazon.com/Wild-Trees-Story-Passion-Daring/dp/product-description/0812975596
It turns out that the tallest trees in the world
(northern California coast redwoods)
are so tall that
The deep redwood canopy is a vertical Eden filled with mosses,
lichens, spotted salamanders, hanging gardens of ferns, and thickets
of huckleberry bushes, all growing out of massive trunk systems that
have fused and formed flying buttresses, sometimes carved into blackened
chambers, hollowed out by fire, called "fire caves."
Thick layers of soil sitting on limbs
^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^
harbor animal and plant life that is unknown to science.
Humans move through the deep canopy suspended on ropes,
far out of sight of the ground, knowing that the price of a small
mistake can be a plunge to one's death.

My point is, if you want to view modern mathematical knowledge
as growing up from some soil and forming an ecosystem, I think
my opponents are mis-identifying the soil it is growing in. It is NOT
growing in the GROUND. It is NOT growing from basic simple things
that mathematically unlearned people think about. IT IS RATHER
growing
in the soil matted in the redwood canopy, the soil accumulating ON
THE BRANCHES that are ALREADY 300 FEET UP OFF the ground.
Everything WE think about math, we think in an intellectual context
brought to us AFTER Godel, AFTER Cantor, AFTER Russell, AFTER
Frege and Gauss. Piagetian innocence IS NOT re-capturable.
It WOULD NOT be reasonable to try to re-plant this ecosystem
300 feet down on the same ground that the redwood trunk is growing
out of! It is higher and better and JUST RIGHT JUST WHERE it IS!

One possible reason is that we'd like to know, with
mathematical certainty, whether some statement can be proved

No. Statements cannot be proved. END of story.
You yourself will say this later. But before we agree
with your point (modulo one single misused word "about"),
I want to make MY point.
My point here is that alleging that
we want to formalise arithmetical
statements and notions
is preposterous. Don't parse that adjective as "ridiculous" or some
other simplistic blanket condemnation. I chose it carefully; I meant
it literally: "preposterous" as in putting what ought to be
"posterior"
in the "pre" or "before" position, i.e., it is ASS-BACKWARDS.
Because:
NO notion *is* ARITHMETICAL to begin with
UNLESS it is FORMAL to BEGIN with!
You CAN'T be talking about FORMALIZING something that
NECESSARILY AND BY DEFINITION STARTED OUT formal!
This is NOT a question of the language you phrase it in!
Its MEANING/CONTENT IS INHERENTLY formal!
Numbers and all other mathematical entities are, precisely to the
extent
that they are mathematical, ABSTRACT AND THEREfore
formal TO BEGIN with! IT IS NOT as though we BEGAN
with some NATURAL language notion that was born correct and
meant exactly what we wanted it to mean, NOT if we are talking
about THIS stuff! IT IS RATHER that we began with some knowledge
about a realm that was INHERENTLY ABSTRACT AND FORMAL TO
BEGIN WITH and then ATTEMPTED TO APPROXIMATELY DESCRIBE
it using natural language! To the extent that that began to seem
inconvenient,
we then tried with various formal languages or linguistic paradigms,
but
because they were formal, we were able to PROVE that THEY were
not merely inconvenient to use, but actually inaccurate in certain
ways.
Of course, to say of anything that it is inaccurate is to claim
knowledge
on some level of ideal that it was supposed to match, but my point
is, natural language IS NOT so perfect that you can claim that our
natural-linguistic practice in trying to point at and describe that
ideal
is itself necessarily successful, because:
Fundamentally, natural languages are simply inconsistent.
You can ameliorate some of that by making them partial,
but that, too, takes A LOT of work.

In any case, natural languages do not even NEED to be RELEVANT
in the FIRST place, GIVEN that the WHOLE REALM here IS FORMAL
to begin with!


using these or those principles.
Mathematical statements aren't mathematically defined
objects

OH, SHUT UP.
You can have a mathematical theory of natural language just as easily
as you can have a mathematical theory of anything else.
LETTERS, TOO, are abstractions. Sounds are not but they're close;
just because all the tokens have to be concrete doesn't mean the class
is.

and thus there can be no question of mathematically proving stuff
about them;
Almost right. This is the agreement I was speaking of above.
This is you and me AGREEING that natural language statements
canNOT be PROVED. ALMOST: because you said
proving stuff about them.
Bzzzzt. INcorrect use of ABOUT.
There can be no question of mathematically proving THEM, the
statements,
THEMSELVES, BECAUSE THEY are in natural language, while proof,
like numbers, is, again, INHERENTLY abstract and formal.
But proving things ABOUT natlang statements (mathematical or
otherwise;
THAT doesn't even MATTER) is entirely possible; again, you just
postulate
some axioms for a formal theory about the natural language.

thus we need to provide some mathematical model of our
mathematical reasoning and mathematical language, a model that captures
faithfully those aspects of our mathematical reasoning and language we're
interested in, if we are to investigate these things mathematically.

Completely false.
Again, this assumes that all the true mathematical statements and
therefore all the reasoning proving new ones from old ones WAS going
on in natural language, and that we are just trying to formalize it.
THAT IS NOT THE CASE. That cannot POSSIBLY be the case
EXCEPT to the extent that the natural language HAS INCORPORATED
DEFINED PARTS OF some formalism, IS APING a formalism (it is
plastic enough to do that).

Hence: formalisation.

No. Hence, naturalization, for purposes of shared communication,
of a process that was formal but perhaps idiosyncratic (the way I
reason about addition may not be the way you do). People inherently
have a faculty for formal reasoning and dealing with abstractions.
It is not necessarily properly expressed or encumbered with the
separate
faculty for dealing with natural language (though of course there is
some
intersection; my point is simply that it IS intersection at the
SUPERstructural
level and NATURAL language IS NOT the FOUNDATION or substructure
for reasoning WITH or about ABSTRACTIONS).

To get started we define

Oh, shut up.
Before you can issue/utter a definition, you have to have
already defined some criteria for how THAT'S permissible.
How are you supposed to define anything if you don't even
know what a definition is (unless you are Phil)? How are you
supposed to learn what a definition is before somebody TELLS
you the DEFINITION of "definition"? You can't be simplistic
and blase' about how we get started. You have to DEFEND
your choice of starting point.

.

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