
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.
Clearly mathematics can be reasoned about as being about Rule 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
The rule "is" is about "identity" which is about replaceability and 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 metatheory 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 counterintuitive while the axioms of Z themselves are intuitive.
So the statement for example: IF Con(ZFC) > BanachTarski 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) Metatheory 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 metatheory 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 (fictional or real) is made. So this answers the Metatheory 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 uninteresting 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 interpersonal 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 metatheories 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

