In this article, I discuss the relation between mathematical expressions(called "purpose") and advantage of methodology.

In a pursuit system, pursued quantity is pursued, but whose and when does a methodology works for? From axiom 5, purpose can also be treated as a weight function stressing or discriminating some pursued quantity. Different 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 1:

For any weight function w(S), there is one methodology at least, making weighted total satisfaction

the largest. A(M,w) is called the advantage order for purpose w(S).

In society, different social systems work for different w(dS). Human beings can not 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', w(dS) is larger for the lordsˇŻhappiness than that for the peasants. In economy, when exchange rate changes, w(dS) between different countries will change, so currencies ought to be unified in a perfect pursuit system.

I think every pursuit system pursues the given purpose the best, but there are many imperfect purposes, like in society, economy and research.

If, for any mathematical expression of negative action, there is a result not existing in universe, then universe is not a pursuit system.

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

Fact foundation for axiom 1 is that people can always find a methodology to pursue their weighted satisfaction well. Whether (1), converse proposition of axiom 1, is correct or not is still doubtful,

(1) for any methodology, there exists w(dS) making the methodology to be the best methodology for the w(dS).

Another problem is:

(2) Can the best methodology for a weight function be found in finite steps?

Axiom 2:

For two weight functions, w and u, if w=ku+c (k and c are constants), advantage order A(u) and A(w) are the same.

So when we say u and w are very different, it means ˇ°for any k and c, (u-kw-c)/u is largeˇ±. Difference between two weight functions is the least value of (u-kw-c)/u, for all possible k and c. So if dw can be expressed as ˇ°kw+cˇ±, it means w+dw is the same as w.

Axiom 3:

For a methodology M, a weight function w(dS) and its advantage A(M,w), if dM and dw is small, then dA is small.

Axiom 4:

Weight function w(dS) and its best methodology, BM(w), are continuous relative to each other.

A bad methodology for a mathematical expression will become a good methodology when there is a negative weight. If (1) is true, by adjusting purpose, a methodology can change continuously from best methodology and worst methodology; by adjusting methodology, a purpose can change continuously from expanding the fastest to shrinking the fastest. For fixed methodology, the purpose having the fastest expanding speed can be called "best purpose". Similar to advantage symmetry(axiom 7 in "axioms about advantage and methodology"), there is symmetry between good and bad purposes, which can be called purpose symmetry. This similar to that a vector can be projected to different directions. A methodology is the BM for one purpose at most; a purpose is the best purpose for one methodology at most.

If dw(dS) is small enough, then smaller dw will lead to monotonous decrease of dBM(w), or monotonous increase of similarity between BM(w) and BM(w+dw). If difference between BM(w) and BM(u) is small enough, difference between w and u can be smaller than any degree.

A purpose can be projected to different purposes too, some components are large and some are small.

Axiom 5:

One purpose can be resolved as the the sum of many expressions(partially pursued quantities). When all components are given, the expression is uniquely given.

So, all possible discrimination will cover all possible mathematical expressions.

A methodology can not 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 one purpose too much.