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Topic: Then answer to Frege's two objections to formalism.
Replies: 17   Last Post: Apr 9, 2013 7:56 AM

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 Zaljohar@gmail.com Posts: 2,665 Registered: 6/29/07
Then answer to Frege's two objections to formalism.
Posted: Apr 5, 2013 7:25 AM

I just want to argue that

"Mathematics is analytic processing fictional or real"

First I'll try to explain what is meant by "analytic".

Analytic statements are those that can be known to be true or false
just from examining their syntax, i.e. their written or spoken
components. So there is no need whatsoever to seek external
observation to known their truth status. An obvious example is the
statement "All bald men are bald", now this statement is true and this
can be known on grammatical basis from analyzing the language in which
that statement was made, we don't need at all to go and select a
random sample of bald men in the population and check if they were
bald or not? The above statement is True by virtue of the meaning
given to the individual components of it. To make a nice contrast take
the statement "Napoleon weighed 80 kg when he was dead", now this is a
statement lets say referring to the body weight of Napoleon the known
emperor of France, however there is no way to know whether it is true
or false without a record of a measurement of his body weight at his
death, so external observation is needed. Those kinds of propositions
are said to be synthetic because in addition to understanding them one
needs other requirements to reach at their truth so understanding them
is not enough to determine their truth.

So Analytic statement are those whose understanding their meaning is
enough to know their truth status.

Anyhow analytic reasoning is not that simple, it can be made too
complex that one cannot even know the answer to them!

Analytic is much of a Rule following game really.

An example of that is the following:

If we say that the following rule applies to R for whatever x,y,z are

If x R y and y R z Then x R z.

Then for fixed objects A,B,C it is true that

If A R B and B R C then A R C

This is a simple substitution instance of the above rule, and the
above reasoning is purely analytic.

Now lets try take some simple mathematical examples and see how can
they fit in to analytics:

We may say that "1+1" which is a string of symbols Is replaceable by
the symbol 2 and vice verse whenever those occur without any
alternation in truth. So this is definitional rule. Also we may also
defined the string of symbols 1+1+1 to be symbolically interchangeable
(replaceable) with the symbol 3.
Then it analytically follows that 2+1=3 and also it analytically
follows that 1+2=3
and it analytically follows that 2+1=1+2. All those are simple
analytic consequences of the above rule.

So analytic processing is about RULE FOLLOWING.

following processes.

Now what I mean by a process is nothing but the fellowship of a rule.
It is to be understood that this can be highly complex, especially if
we start from separate rules, and then stipulate that those are
consistent, and then examine what sentences could be proved in them by
rule following only.

Those rule following scenarios can be REAL or can be simply FICTIONAL.

A real scenario occurs if Concrete objects happens to follow those
rules, or
to be more practical 'can be reasonably viewed to follow such rules'.
On the
other hand Fictional scenarios occur when imaginary entities are the
main stuff
of those analytic arguments. To Address that we give the following two
examples:

Julius Caesar is Julius Caesar

Oliver Twist is Oliver Twist

it holds analytically of real objects as well as of fictional ones.

Actually we may regard the question "Was Oliver Twist an orphan?" as a
fictionally synthetic question, the answer to which can only be
"observed Within" the fiction itself.

Now obviously known formal theories uses fictional syntax (for we
don't have
enough ink nor space at our disposal to write all logical theorems of
first order logic) and the rules by then is governing those fictional
entities. However the derivation of a theorem from the supposed axioms
is fully Analytic and is absolutely true from the analytic standpoint.
Also the meta-theory of the known formal theories is definitely
written in fictional realm. However fictional those may be the
important matter is that the complex analytic reasoning that they
convey is absolutely referable to them by strict rule following, and
that's the piece of knowledge Mathematics is concerned with, i.e. non
trivial analytic processing whether of real or fictional objects. It
needs to be emphasized that there is no claim here made of those
fictional realms being real or can be played by real objects or the
alike just because significant mathematical rules are uncovered,
nothing of that is true, however what is important is the non trivial
complex rule following. So we are not saying that Z for example is
true, what we are concerned with is that IF Z is consistent then t
follows, where t is some theorem of Z. especially if t is not that
expected to follow or is counter-intuitive while the axioms of Z
themselves are intuitive.

So the statement for example: IF Con(ZFC) -> Banach-Tarski theorem is
true.
This statement is "absolutely" true! as far as the analytic aspect is
concerned.

Now Frege had raised two objects to formalism, namely:

(1) Meta-theory objection
(2) Applicability objection

The first object was raised against the claim that Game formalism
avoided commitment to an infinite abstract realm of objects. The meta-
theory as well as the symbols, proofs, etc.. used in the theory are
all abstract entities.

The answer that I have to this is that Analytic processing have no
commitment so to say to abstract entities although it does utilize
them on fictional assumptive basis. So all what we are saying is that
IF was assume a fictional realm in which the meta-theory is carried in
according to the rules of the game, then the theory in that fictional
world produce the so and so ... theorems; as seen there is no
commitment to existence of such fictional world, the statement
analytic and thus mathematics is concerned with is consequence made in
that fictional world that strictly follows rules. It is that
fellowship of consequences from premises that is absolute and what
mathematics is concerned with, and not the truth of existence of those
objects nor fellowship of those rules by real objects. So under this
fictional assumption realm no commitment to abstract entities
objection.

As to the applicability objection, the answer relies in Psychology! It
is clear that no theory would be of common interest to many different
people if it was personally based, so there theories that were
investigated by mathematicians are of course about something that they
can agree upon or imagine the same fictional world for it, and those
things can only be about NATURAL relations and properties, and also
because mathematics had been and is continually ever since it was born
about applying those game following rules to matters around us helping
us in understand our surrounding and ourselves and thus enabling us to
interact with it. This of course confers some shared realm between
rule following and reality around us, otherwise mathematics couldn't
ever be applicable. So all in all mathematics has be practiced in the
sense of what I call as PROXIMAL ANALYTICS, i.e. analytic stuff that
somehow some real concrete objects happens to share some aspects or
can be understood in a proximity to it. For example a Circle as how
mathematically defined is too idea for any matter in our physical
universe to really behave in abidance by or be shaped exactly as, but
nevertheless we know for sure that concrete physical object can be
shaped and move to NEARLY or APPROXIMATION circular way.
So of course the INTERESTING MATHEMAICS would be about Analytic stuff
in proximity to the real physical realm. But this subject would more
fit what I call as Applied mathematics and not mathematics per se, we
can have much mathematics that have no applications whatsoever. Also
I'll present another clear and definite analytic example in which no
mathematician would be concerned with albeit it IS mathematical.

Take for example a game like any of the electronic games present
nowadays. Take the responses made by a specific person A at specific
times he played the game at, now lets take his responses in which he
made a choice as Axioms of the game, the rest would be consequences
that followed by the rules of the game, now the collection of ALL
consequences to the primary acts of person A are pure analytic
consequences that have absolute truth in being followed from the
actions of persons A at those specific times and the rules of the
game, those consequences are dame mathematical but yet un-interesting
since them reflect just the mathematics of choices of person A at that
time, that it might happen that person A itself cannot repeat them
again or any other person can memorize and do, so such analytic stuff
is not interesting because it is too personal. However it is
mathematics but not interesting kind of mathematics. So again it is
expected that during the history of mathematical development only rule
following issues that can be shared between different persons and
between a person itself at separate times would be investigated
otherwise who would mess his head about a multitude of personal rule
following styles that have no impact on inter-personal achievement. No
only that it is also as I said above to be expected that Shared issues
between different persons would be about clear and simple natural
relations and this would really increase our understanding of the
surrounding, all of that lead to Proximal analytics being subject to
study and thus explain what contemporary mathematics have all of that
wide applicability, because it is directed at matters that can be said
to behave or possess properties that are in proximity to those kinds
of rule following games. It is interesting of course to investigate
this nature of proximity and how did that occur, of course some of it
is obvious take for example a theory about parts and boundaries like
in Mereotopology this is clearly motivated by natural relations, truly
the theories and the meta-theories themselves are defined
instrumentally in a fictional world, but that fictional world is a
kind of mimicking what's occurring at the real world, and so it would
be expected that it would share something with it and consequences
made in that fictional world would be of importance to understanding
the real world since the chances are that the reals world would have
concrete objects that possess properties that approximate those
investigated at that fictional virtual world.

So Frege's second object has been answered.

So:

Mathematics is about analytics fictional or real, most interestingly
those in proximity with reality.

Zuhair