The important difference between me and the pope is that he has an essentialist view of human nature (and hence human dignity): it is something given and fixed, so changing it is both impolite (since it was a gift) and impossible (since it is absolute). We might try, but we will become inauthentic and hence unhappy. This is very much the same argument Michael Sandel makes in a secular form. It also has the same problem: there are obviously many parts of the human condition that are bad and we can and ought to change (ignorance, cruelty), and it is not clear how to delineate which ones are in that category, in the optional category (hair color? circumcision?), and in the impossible and/or bad categories (better than healthy). One can make some arguments for what goes where, but they typically seem to be consequentialist arguments - in which case there is no need to invoke human nature. Deontological arguments run into problems since typically they make claims of the type "One should always do X, when it is possible", and these arguments then produce the "wrong" conclusions as the borders of the possible are shifted by new technology.
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It seems to me that the family obsession of Christianity is very much based on the transcendentalisation of certain family relations typical of humans - parent-child relations, strong altruist bonds to family members, social emotions such as gratitude, and strict gender roles. But the last one does not seem to have the same deep biological basis as the first three: we know a bit about their neuroscience already, but as far as I know there are no neural correlates of fixed gender roles. There are certainly biological gender differences in the brain, but the various meta analyses collected by J.S. Hydes suggest that few of them are very large, and even those that do exist do not have much moral implications. This later point was dealt with at length in Janet Radcliffe Richard's excellent 2012 Uehiro lectures, and she also argued that conservatives of all colors tend to think of the world as having a fixed underlying moral/natural order that must be preserved at all costs - and against all evidence (see here, here and here for a brief summary; the lectures will appear as a book sooner or later). Opposed to this "natural order" view is the view that humans must be active in both structuring the moral order, and changing the world to fit it. We are moral agents, so we better figure out a good morality and try to implement it - all based on empirical data as much as possible. You can guess where I stand.
The pope's model predicts that changed gender roles, such as female suffrage, gender equality and gay rights, produce bad psychological effects. Not just occasional or individual problems (no doubt there are some) but profound malaise since they supposedly interfere with the essential human nature. His model predicts that very equal places like Scandinavia would be hotbeds of mental trouble, while traditional societies should work very well. Playing around a bit with tools like Gapminder tends to dispel that view. Just like anybody predicting dire consequences from new technology the pope can of course claim that they are long-term and invisible so far, but I think we would have seen some effects by now from female suffrage.
Of course, being a dualist the pope could conveniently claim there are profound damage in some spiritual dimension not accessible to empirical study. We Swedes might be thoroughly spiritually corrupt, we just don't know it. Except that if there is no clear way of detecting it even from the inside we will just have to take it on faith, and it is going to be a hard sell to claim something that does provide real, observable benefits to people is actually very bad. It won't be the first time for the Catholic church, of course, but given the past results of opposing contraception the pope should not feel optimistic about doing much good.
Go, read the whole essay.