*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 B_{0}. For
each n, B_{n + 1} will be defined as r_{n}
E(B_{n}), where E is the "effective love operator",
and r_{n} is a coefficient that for each n gives an
indication of the weight "effective love of B_{n}"
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 B_{n}
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, B_{m}
and B_{n} are loved the same amount. Since in
particular B_{m + 1} is loved the same as B _{n
+ 1}, it follows that for each m, n that r_{n}
times the amount a moral person loves E(B_{m})
equals r_{m} times the amount a moral person loves
E(B_{n}); i.e., that a moral person loves E(B_{m})
an amount that is r_{m}/r_{n}
times the amount the same moral person loves E(B_{n}).
But we also know by the remarks ending the last paragraph that, what
since B_{m} and B_{n} are loved
the same amount, E(B_{m}) and E(B_{n})
are loved the same amount. It follows that for all m, n, r_{m}/r_{n}
= 1,
i.e., that for all m, n, r_{m}= r_{n}.
In other words, all the coefficients
r_{m} have the same value, the *effective
love ratio* with respect to E, which we shall
denote r or r_{E}.

Clearly (easy proof by
induction), to say that for each
natural number n, B_{n
+ 1} =
r E(B_{n}) is the same as
to say that for each n, B_{n} =
r^{n}E^{n}(B_{0}).

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 B_{n}
= r^{n}
E^{n}(B_{0}). The
coefficients r^{n}
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 r_{L}
= 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(B_{n})
is nothing other than B_{n + 1}, and so L applied
to the sum of all the terms B_{n} is just the sum
of all the terms B_{n}
except B_{0}. 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}
E^{n}(B_{0}).
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.