Axioms in this article explains why the universe, as in other pursuit systems, has only one largest quantity, why natural laws are invariant in all situations, etc.


Axioms about Methodology and Advantage

For fixed mathematical expression of pursued quantity (purpose), methodologies and their advantage (expanding speed) have the following properties.

Axiom 1: A methodology is a one-one mapping between a set of methods and a set of applicable conditions.

A methodology must be both complete and consistent.

Axiom 2: For a methodology, in order to be complete, there must exist a method to be used under any condition; in order to be consistent, there is no more than one method under any condition.

For the following axioms, there is a common condition "when mathematical expression of pursued quantity is the same".

Axiom 3: For any two quantities in a close pursuit system, when both are pursued well enough, in order to pursue one better, the other must be pursued worse.

For any methodology, there is at most one quantity that is actually the best pursued among all possible methodologies. From axiom 3, a pursuit system cannot pursue equality and expansion speed the best together, cannot have the largest satisfaction at each moment, cannot pursue the largest satisfaction for each particle. This is called the incompatibility property of pursuit.

Axiom 4: Advantage of all methodologies forms an order, the order is invariant with dimension and unit of pursued quantity. Larger advantage leads to higher expanding velocity.

In science of pursuit, truth or fallacy is only decided by which makes system expanding faster (in infinite long time). One ultimate truth (the best methodology) will triumph over infinite bad methodologies just because of its advantage. From axiom 4, if we could find advantage order in one pursuit system, it will be valid in all pursuit systems with the same mathematical expression.

If unit of pursued quantity does not influence advantage, large (more developed) and small (less developed) systems have the same expanding velocity when following the same methodology. rent purposes have different measurement for advantage and lead to different best methodology. A small difference in expanding velocity will bring large difference in pursued quantity, so expanding velocity is the main factor influencing advantage.

Axiom 5: A(M) is continuous function of methodology.

Continuity of A(M) means: For any M, if dM is enough small,

|dA|=|A(M+dM)-A(M)|

can be smaller than any degree. So any methodology that is similar to a good methodology is good too.

Number of methodologies in unit advantage is called methodology density in advantage order, marked as N(A).

Axiom 6: N(A) decreases with advantage. There is only one methodology at the maximum value of advantage: the best methodology. The best methodology consists of natural laws only.

dN(A)/dA<0

From axiom 6, methodologies cannot be infinitely improved, so there is ultimate truth in science of pursuit. This is the reason why natural laws are time-invariant: because they are always the best.

One fact foundation for axiom 6 is that truth is scarce compared with fallacies. The larger advantage is, the fewer methodologies there are.

In social history, people of different countries and races made many decisions in choosing methodologies. Statistically, these decisions are toward the same direction--basic natural laws, including more exchange, free market, better desire equality and pursuer independence. This should not be explained as coincident, because its possibility is too small. So "there are many perfect methodologies" is not true. The best methodology is unique. This also shows a new way to discover truth in science of pursuit. The methodology improvement process in history provides us materials to establish causal relation from methodology to advantage.

In a perfect pursuit system, pursuing positive quantity S the best also means pursuing negetive quantity -S the worst. In the universe, pursuing action the worst is equivalent with pursuing negative action the best. In society, pursuing positive happiness the best also means pursuing negative happiness the worst.

So methodologies that are neither too good nor too bad for any pursuit are called "wicked".

Axiom 7: Around any M except BM, there exists an infinitesimal substitution dM at least, satisfying:

A(M+dM)>A(M)

There is also dm satisfying:

A(M+dm)<A(M)

Apart from BM, any methodology can be both improved and worsen a little. So BM are the only extreme value for A(M).

A(M) is measured to be larger when average expanding velocity in whole system is larger. ˇ°Averageˇ± refers to average value of expanding velocity for all time and all places. Exploiting the rest can make part of pursuit system expanding faster, but it cannot make whole system expanding faster, so advantage cannot be judged by expanding velocity in part of a pursuit system. In a pursuit system, truth or fallacy is decided by which makes whole system expanding faster.

One ultimate truth, best methodology, will triumph over infinite bad methodologies because of its advantage in expanding velocity.

Axioms about Purposes of Methodologies

Pursuing different purposes (different mathematical expressions for pursued quantities) will lead to different advantage for a given methodology. A better mathematical expression can make pursued quantity growing faster. If two purposes P and P' satisfy P=kP'+c, then A(M,P)=A(M,P'). Such purposes belong to the same set of purposes. So, even when two purposes are quite different they can still be very similar, even the same. When k is negative, pursuing largest will become pursuing least. For a given methodology, the purpose making pursued quantity growing the fastest is called best purpose (BP).

Axiom 8: Purpose and methodology are symmetrical: a purpose inludes a set of sub-purposes, each is a coefficient on a quantity, making the quantity have the dimension of the pursued quantity; the set of coefficients must be complete and harmonious (having one and only one coefficient on each quantity); after pursuing fine enough, sub-purposes are incompatible; for given methodology, advantage of purposes form an order A(M,P); for given M, A(M,P) is continuous relative to P; N(A,M) represents number of purposes in unit A, it decreases with A; apart from BP, there is dP for any P satisfying

A(P+dP)>A(P)

and:

A(P+dP)<A(P)

A sub-purpose is similar to a method and it condition.

In a pursuit system, pursued quantity is pursued, but whose and when does a methodology works for? Purpose can also be treated as a weight function stressing or discriminating some pursued quantity. For any weight function w(dP), there is one methodology at least, making weighted total pursued quantity ¨°w(dP)dP the largest.

In society, different social systems work for different w(dS). Human beings cannot establish a perfect society partially because they do not have a correct weight function to evaluate different social systems. For example, deficit is better than basic natural laws when dw/dt is negative, which works for present human beings better than future ones.

A feudal society works for the lords' happiness more than for peasants' happiness, w is larger for the lords' happiness than that for the peasants. In economy, when exchange rate changes, w between different countries will change, so currencies ought to be unified in a perfect pursuit system.

I think every pursuit system pursues its purpose very well, but they have many imperfect purposes, like in society, economy and research.

In a pursuit system, w can be negative (like w for enemies' pursued satisfaction) and discontinuous. For the basic natural laws, weight function is the same for all pursued quantity, including expected pursued quantity. No pursued quantity is discriminated. So it has real advantage.

Some discrimination will be very complex. For example, ability discrimination is very complex in society. Variables include culture, sex, timing, relation with leaders, distribution principle (according to ability, time, stock, position), etc. So, society cannot have a simple mathematical expression for profit and happiness.

Purpose and its BM, methodology and its BP, are continuous relative to each other.

A bad methodology for a mathematical expression will become a good methodology when there is a negative weight. By adjusting purpose, a methodology can change continuously from best methodology and worst methodology; by adjusting methodology, a purpose can change continuously from best purpose to worst purpose.

A methodology cannot make every purpose be realized the best. In society, there are many purposes, like happiness, equality and employment; in a research system, there are purposes like finding more problems, solving more problems, etc. A pursuit system must learn to establish equilibrium between various pursuit purposes, not to pursuing other purposes too much. Better to forget purposes other than the pursued quantity, this is the best way to find the best equilibrium.

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