In my recent article "The Sacrilege of Schism", I argued that the Protestant notion of "the visible Church" is equivalent to the set of all believers, or the set of all congregations of believers. I pointed out that sets per se are not actual, but merely conceptual., and thus that the Protestant notion of the visible Church is such that it [per se] cannot be affected by schism. My argument is important because it cuts away the illusion of a middle position between institutional unity and the notion that the Church (per se) is entirely invisible. If my argument is a good argument, it entails that we must choose between finding out which institution is the institution that Christ founded, and giving up the notion that there is such an actual entity as the "visible Church".
One possible objection to my argument is that I am setting up a false dilemma between the conception of the visible Church as a mere plurality of believers (or mere plurality of congregations) and the conception of the visible Church as an actual unity of the institutional sort. According to the objection, the existence of the various congregations and their agreement on basic essentials of the faith creates a kind of quasi-institution, or something with 'institutional dimensions', or 'institutional structure', a thing with a kind of unity between that of a mere plurality and that of a concrete institution such as the Catholic Church.
I think that this objection fails, and I want to use an illustration to explain why. Imagine a plurality of twelve wooden children's blocks, scattered around the living room floor. We ask the question: Is there one thing composed of these twelve blocks? The answer is "No". Now I pull four of the blocks together and stack them into a tower. Then I pull four more together and stack them into a different tower. And then I pull the remaining four together and stack them into a third tower. So now there are three separate towers on the living room floor, each composed of four blocks. Now we ask the same question again: Is there one thing composed of these twelve blocks? The correct answer is still "No". That is because unifying subsets of the members of a plurality into subset unities does not unify all the members of the plurality into one unity. Such an action does not turn the mere plurality into a unity. The mere plurality remains a mere plurality, even if the members of the mere plurality become subset unities.
That is why whether the visible Church is the "plurality of believers" or is the "plurality of congregations", it is still a mere plurality, and thus only a conceptual unity. Unifying subsets of believers into congregations does not unify all believers into one actual unity. And we should not treat what is a merely conceptual unity as if it is an actual unity.
But doesn't the fact that all believers hold to certain basic essentials of the faith make them one actual unity, i.e. the visible Church? No. Imagine that each of the twelve blocks in our three towers had the letter 'A' on it. That would not make the twelve blocks into one thing composed of these twelve blocks. Formal unity is not ontological unity. Otherwise photocopy machines would only keep spitting out the very same piece of paper that you were trying to photocopy.
If my argument is correct, does that mean that schism is impossible, given a Protestant conception of the visible Church? No. It means that the Protestant has to define (or redefine) schism as something that does nothing to the visible Church. If there is no actual visible Church, then whatever schism does, it doesn't do anything to the visible Church. Putting it a different way, if there is no actual visible Church, then schism cannot possibly reduce the unity of the visible Church, because what is not actual cannot lose unity. But a Protestant could easily redefine the term 'schism' as something that can occur between individual believers, between individual congregations and between individual denominations.
The Catholic position does not have the problem of treating a merely conceptual unity as an actual unity because the Catholic Church teaches that through baptism, each person is incorporated in some respect into the institution of the Catholic Church. Those "who believe in Christ and have been properly baptized are put in a certain, although imperfect, communion with the Catholic Church." (CCC 838 ) So when the Catholic Church talks about the "visible Church", it is talking about an actual unity, not a mere conceptual unity. But the Protestant conception of the visible Church is such that the visible Church is a mere plurality, i.e only a conceptual unity, not an actual unity. The Protestant talk of an actual "visible Church" depends [though Protestants do not realize this] upon there being an actual institution into which all believers are, in some way, incorporated. In other words, it depends upon there being something like the Catholic Church. But what other institution is claiming that all believers are in some way incorporated into itself? I am not aware of any.
The objection that because the visible Church has 'institutional dimensions' or an 'institutional structure' therefore there is a middle position between the two horns of my dilemma, is an attempt [through the use of such metaphors] to point out the institutional unities at the subset level and the formal unity (i.e. unity of doctrine on certain basic essentials) across the entire set of believers. And in reply I'm pointing out that (1) unity at the subset level does not make unity at the set level, and that (2) formal unity at the set level is not sufficient for ontological unity. So I am pointing out that those two types of unities do not do the sort of unitive work that is needed in order for the visible Church to be an actual entity and not a mere plurality (i.e. conceptual unity).
"Let unity, the greatest good of all goods, be your preoccupation." - St. Ignatius of Antioch (Letter to St. Polycarp)