This is the most interesting and new of the three parts of the series, where I actually define mathematically the concepts of beauty, goodness and morality and their precise relations to love and talent. I have sensed (I mentioned it on usenet many years ago) for a decade at least that beauty can be thought of as a geometric sequence, but now that I distinguish both effective love from love and effective morality from morality, things are clearer, hopefully. Much of this I wrote last summer, but I made significant improvements and corrections today.
Beauty, goodness, and morality defined mathematically
As mentioned earlier, I define beauty as having two parts: a concrete part not involving love, which I call talent, and a more abstract part involving love, which I call goodness. Morality I define as love of beauty. As it turns out, there is good reason to believe goodness is the same thing as effective morality, but when making definitions it is useful and necessary by way of avoiding circularity to have separate terms—the concepts are not defined the same, they just happen to be the same, rather as 3 - 1 and 2 are the same even though they are defined differently.
Since the basic concepts are defined in terms of love, it is well that I elaborate on what I mean in this context by "love". Obviously some if not many concepts have to remain undefined, or one must choose between circularity or an infinite descending chain of definitions, neither situation being desirable, but at least I can give a better idea of what I mean. In particular, the expression "love of A", where A is some quality, will denote a desire to increase the amount of A. Even romantic and sexual love can be made to fit the definition since such love basically involves behaviors that tend to increase the progeny and likely descendants of the beloved, thereby increasing the representation of the qualities of the beloved in the world. Love is an unselfish ideal in the sense it involves wanting to increase the amount of A represented in the world and not (say) the amount of A in one's bank vault or in one's own descendants.
Though it is hard to be more precise about what is meant by love, we can be more clear about what we mean if we clarify what we mean for a person to love A n-times as much as B. Clearly what we do not mean is that as a result of love the person tends to increase A by an amount n-times that by which he tends to increase B, i.e., that he effectively loves A n-times as much as he loves B . For the amount a person is able to increase A depends on his skill in increasing A (which might differ from his skill in increasing B), and surely it would be grossly contrary to usage that his love for A should depend on his skill in increasing A. The appropriate definition for loving A r-times more than B is that his desire to increase A and B are such that given a choice between increasing A by a small amount a and increasing B by a small amount b = ra, he will be indifferent as to which choice he chooses, all things else equal. The concepts of effective love and love turn out to be related. If a person loves A r-times as much as B, then it seems entirely reasonable to assume that he loves effective love of A r-times as much as effective love of B; before explaining why, let me first be more clear about the concepts and the difference.
The first clarification to make is that effective love is to be taken as a long-term phenomenon (at least mostly--here might be a good place for further exactness for those willing to do things at a higher level). What matters in judging the effective love of A arising from a characteristic is not so much how effective the characteristic will likely be in the individual possessing it at enabling him to directly increase A, but rather how effective the characteristic tends to be in enabling individuals to directly increase A. For instance, if a moral person is nevertheless very able at deceiving others as to his character, this ability to thus deceive others is very likely a bad trait that tends in most individuals to cause them to behave so as to decrease the beauty in the world, and so effective-love-of-beauty would have a negative value (likely exceeding in magnitude the value of talent there) when evaluated at this deceptive characteristic, even if the person were so moral and hesitant to deceive as to character that the characteristic would not affect his behavior.
Effective love of A by definition is proportional to the amount love tends to be effective in increasing A. So loving effective love of A r-times as much as effective love of B amounts to loving an actual increase a' in A (a change arising because effective love causes change) the same amount as an actual increase ra' in B. But this is the same case with loving A and B directly. If I love A r-times more than B, then if by loving B directly (by say caring unselfishly for someone with much B) I can increase B an amount r-times greater than I can increase A by loving A directly (by say caring unselfishly for someone with much A), then the choice will be a matter of indifference to me. I.e., loving A r-times as much as B effectively means loving an increase a in A the same amount as an increase ra in B. The difference is how the changes a' and a are created. The change a' is caused because loving now people who have effective love of A does over the generations cause A to increase as more and more people with A are unselfishly rewarded by the increased unselfish effective love of A in the population. The change a on the other hand is more immediate, caused merely by rewarding directly with unselfish love someone possessed of much A. I see no reason whatsoever to think otherwise than that if I or anyone else love a direct immediate increase a in A (induced by loving A) the same as a similarly induced direct immediate increase ra in B, then I will love an indirect future increase a' in A (induced by loving effective love of A) the same as a similarly induced indirect future increase ra' in B, and vice versa. I shall assume as much for moral people. To reiterate, it shall be considered an axiom that for any qualities A and B and for all moral persons X, X loving A n-times as much as B is equivalent to X loving effective love of A n-times as much as effective love of B.
As mentioned earlier, as best as I can define it, beauty has a concrete talent component, and another component called goodness comprising the various components involving love. We shall define the components inductively. The talent component of beauty I shall denote by B0. For each n, Bn + 1 will be defined as rn E(Bn), where E is the "effective love operator", and rn is a coefficient that for each n gives an indication of the weight "effective love of Bn" is to have in the definition of beauty. Since beauty is what morality consists in loving (by definition of morality), one can determine how much a moral person loves an aggregate of qualities (e.g., those possessed by a person) by considering all the values of the Bn and summing them; letting B be the (infinite) sum, B gives a numerical indication of how beautiful the aggregate is, i.e., how much the moral person loves what is under consideration. This amounts to it being the case that for any natural numbers m and n, Bm and Bn are loved the same amount. Since in particular Bm + 1 is loved the same as B n + 1, it follows that for each m, n that rn times the amount a moral person loves E(Bm) equals rm times the amount a moral person loves E(Bn); i.e., that a moral person loves E(Bm) an amount that is rm/rn times the amount the same moral person loves E(Bn). But we also know by the remarks ending the last paragraph that, what since Bm and Bn are loved the same amount, E(Bm) and E(Bn) are loved the same amount. It follows that for all m, n, rm/rn = 1, i.e., that for all m, n, rm= rn. In other words, all the coefficients rm have the same value, the effective love ratio with respect to E, which we shall denote r or rE.
Clearly (easy proof by induction), to say that for each natural number n, Bn + 1 = r E(Bn) is the same as to say that for each n, Bn = rnEn(B0).
Note that hitherto, all that I have suggested about the definition of the effective-love operator is that its value on a quantity that corresponds to a kind of beauty B' is proportional to the amount the person possessing the love is able to increase that particular type of beauty; by redefining E by taking a constant multiple, one can make r whatever one wishes. At first glance, the most straightforward way of defining E would be to do so flatly, so that if a person's effective love of B' (where B' has the dimension of beauty) tends to increase B' by δ beauty units in people, the value of E(B') where B' has a value of one beauty unit is δ beauty units; I shall call E defined thus the flat love operator. Then B is the sum, over all natural numbers n (including n=0), of the terms Bn = rn En(B0). The coefficients rn form what is called a geometric sequence with ratio r. I shall call r with respect to an E thus defined flatly the effective-love ratio . In some respects, this is all quite cool, but why not just define effective love as L = rE so that rL = 1? Doing so, if in fact B' is a function that actually gives the beauty of something, L(B') gives the beauty arising from whatever effective love of B' is present, i.e., L(B') is the goodness involved with loving B'. To see this is the case, notice that B' must be a restriction of B (since B is what measures beauty). Furthermore, L(Bn) is nothing other than Bn + 1, and so L applied to the sum of all the terms Bn is just the sum of all the terms Bn except B0. Thus L(B) is just the sum of those components of B involving love, which is what goodness, the part of beauty that is effective love (of beauty), is defined as. This in fact is also what we need to ensure goodness is just effective morality. Indeed, since morality is defined as love of beauty, effective morality is nothing other than effective love of beauty, i.e., L(B). To reiterate, defining effective love as L ensures effective morality and goodness are the same thing. In situations where we are not comparing people with differing definitions of beauty (involving, say, people who define beauty with different effective-love ratios), L is probably the preferable effective love operator, and so from here on the effective love operator shall be defined as L. If I have occasion to consider the flat-love operator, I shall denote it as E.
That goodness is in fact the same as effective morality makes it especially reasonable that people would evolve to find things beautiful more-or-less as I have mathematically defined it. For if goodness (the non-talent part of what makes a moral person love you) is effective morality, in order for a person to be loved the best by a moral person, it behooves the former to weight the various components of love in beauty in the same way that a moral person would, or his love of beauty, i.e., his morality, will not be as effective, causing him to be less good and thus relatively less well-loved by her. Indeed, morality involves loving beauty, i.e., wanting to increase the sum of the beauty terms r n En(B0). If one instead tries to increase the sum of beauty terms involving other coefficients (e.g., those obtained by choosing a different value of r), one will not be as effective in increasing beauty; one instead will be maximizing something else determined by one's own preferences (but of course it would not make a difference if one merely changed the units in which one measured beauty, notwithstanding the numerical value of a beauty measurement and of the corresponding beauty terms depends on the units used), which would reduce the effectiveness of one's morality, i.e., one's goodness would be reduced, making one less loved by moral people. Beauty defined as mentioned causes moral people to possess the advantage of idealism. A moral person will be loved unselfishly by fellow moral people, just the way idealists are loved by those who share their ideals--what enables idealism to prosper to the extent it can be judged and appreciated. Also, the part of beauty that does not involve goodness, i.e., talent, seems among matters not involving unselfishness more-or-less what it is most useful to love. Indeed, love tends to be expressed through mutual children, and it is more rewarding to have children via talented mates, since the success and quantity of one's descendants is positively influenced by the talent of one's mates.