When we say that we possess a table, then we have a tangible idea of what this means: I own an x, such that x is a table, and that specific table present to me now is the referent for the name “table” where x = table. But what about something non-physical, such as love? Or numbers? Or freedom? Or boredom? And so on.
What an abstract number is can be thought of as a set X, such that the elements of X are all of the things equinumerous to one another. Thus, the number 2 is the set X such that all the elements of X are all the pairs in existence and anything that is not equinumerous to these pairs is not an element of the set. We might expand this further and say that X contains all of the possible things that are equinumerous pairs: not only is my holding up of my index finger and middle finger an element of X, but so is any logical possibility of my doing so.
Could we then apply this to something like love?
Let’s say that love is a set L, such that the elements of L are all of things experiencing love. We might even add the same extension as before: L also contains all of the logically possible experiences of love (i.e. the possible love I could feel for every combination of things which could possibly exist). Thus, my love for my mom, for my dad, for a good nap, for a delicious meal, and so on, are all elements of L, along with all possible love I (and all possible love-experiencers) could feel for all other possible things.
I think most people would agree that love can come in degrees (as well as types, such as platonic love, erotic love, the love of non-sentient things, and so on, but I will assume all these types of love describe elements of L). For instance, I might love pepperoni pizza more than mushroom pizza, even though I love both; I might love a friend I’ve known for 20 years more than one I’ve known for 2 years; and so on. And so, we could then generate a series out of the elements of L based on the love-magnitude relationship the elements have with objects inside and outside the group. For instance, lets say we have
xm ∈ L
yn = yv + yu ≔ yv ∈ L ∧ yu∉ L
Rxmyn = Mmn
Where R is the love relationship between xm and yn and Mmn is some magnitude of the love that xm feels for yn. And so, if Rx1y1 is some love relationship between elements x1 and y1 with magnitude M11 and Rx1y2 is some love relationship between x1 and y2 with magnitude M12 where M11 < M12 then we could order Rxmyn based on the relative magnitude Mmn such that every R combination of xm with yn is ordered in increasing Mmn in the series S. This would satisfy transitivity (M11 < M12 ∧ M12 < M13 ⊢ M11 < M13), asymmetry (M11 < M12 ≠ M12 < M11), and connectivity (M11 < M12 ⊢ M11 precedes M12 in S). And thus we get the full well-ordered series S composed of the elements (M11 , M12 , M13 , … , M1n , M11 , M21 , M31 , … , Mm1) into a series of increasing magnitude.
Presumably Mmn would be some finite value, due to their being finite elements in L with finite magnitudes, and so it would not have infinite cardinality. But what is the nature of this finite value? Is there some finite maximum value that Mmn can take, or could it conceivably be infinite? The ontological argument for the existence of God would say that Mmn belongs to Anselm’s “the thing than which no greater can be thought.” Would it also be the case, then that
Mmn > SM
SM = Σm-1i=1 Σn-1j=1 (Rxiyj) = the sum of all the magnitudes in S except for Mmn
But what reason is there for leaving Mmn out of SM? L is the set of all love relationships, and so if SM is the serialization of all the elements of L, then SM must also include Mmn. But this would mean that
L > Mmn
And if we said that Mmn is defined as Anselm’s God, then this is contradictory, since L would be greater than “the thing than which no greater can be thought” and so would the pairing of Mmn and L (just in case one argued that Mmn and L are separate sets). And so, in order for God’s love to be a member of L, we would have to define L as a set containing a subset greater than the set into which the subset is contained:
Mmn ⊆ L ∧ Mmn > L
Which does not satisfy the property of reflexivity, since the finite cardinality of the subset Mmn is larger than the finite cardinality of the set L. This means 1) that there is no 1-to-1 mapping of points in the subset Mmn to points in the set L and 2) the axiom of regularity is violated if we say that Mmn = L, which would make:
(Mmn ⊆ L ∧ Mmn = L) → (Mmn ⊆ L ∧ Mmn = Mmn) → (Mmn ∈ Mmn)
Thus, this is one more reason why the ontological argument doesn’t work.