<the greatest weight, the greatest weight> 
Imagine an angel comes to you in the night, when you are feverish and in the midst of metaphysical reveries. The angel says that she has been taught metaphysics by God, and so she can answer truthfully any questions you might have. You are slow to react, and this is the first question that you think to ask: Do ordered pairs exist?

The angel boggles and, after an awkward pause, asks you to explain the question.

Well, you say, you encounter individuals like your armchair, your pajamas, your fireplace, and so on. (You pick these because they are close at hand, and you can point at them.) Making sense out of the semantics of relations leads to positing a further thing which is an ordered pair of two objects. For example, the relation "__ is wearing __" holds of a pair of things; e.g. <you, your pajamas>.

The angel laughs. The sound is a bit like a wind chime and a bit like a carnival.

Your mind is too small, says the angel. God considered making a world like that, built out of individuals which stood in pairwise relations to one another. This seemed like it would be a waste of infinite power and infinite intellect. What God did instead was to directly create all of what you think of as ordered pairs. To God, each of these has a separate and true name.

You blurt out, almost interrupting: That's absurd! Surely God thinks of individual things, rather than only of pairs.

In a way, the angel replies. God can think of you by using the name for what you, with your limited intellect, think of as the ordered pair <you, you>.

But... surely individuals are more fundamental than ordered pairs. If two things exist, like the angel and the fireplace, then there are necessarily ordered pairs <angel, angel>, <fireplace, fireplace>, <angel, fireplace>, and <fireplace, angel>. Having enumerated the four possibilities, you consider saying "QED" but instead thump your fist on the arm of your chair.

Of course God made all those, and what God makes must necessarily be so. Yet God, considering only infinite power, might have refrained from making <fireplace, fireplace> while still making <fireplace, angel> and <angel, fireplace> and all the rest. There would not be the thing which you think of as the individual fireplace.

You know, says the angel, you've been very sick and you're taking this lesson very seriously. I think we better stop now.

What would you make of this encounter, after your fever had passed?

[edited for clarity]

Is the angel claiming that God could've made <angel, fireplace> WITHOUT making <fireplace, fireplace>?

Greg: That's precisely what the angel claims.

Hmmm... I'm not sure what I would make of this encounter. At the very least, all of our current definitions of, or fundamental axioms governing, 'ordered pair' are wrong. I.e. ordered pairs are not what we thought they were. And more generally: it sounds like maybe we would start thinking of ordered pairs as individuals -- or at least, ordered pairs would play the cognitive role in our future thinking that individuals play in my current thinking. In this future mode of thought, what we now call 'individuals' would have to be something else, like 'sub-individuals' or something similar.

And also, that we perhaps misunderstand mathematical objects more generally. I don't think I can explain exactly and precisely what prompts that feeling, but this case seems similar to saying "God could've made 1, 2, 3, 6, and 7, without making 4 and 5."

(Sorry I took so long to respond.)

Administrator (P.D. Magnus) 
Greg: The analogy which I had in the back of my head was with non-Euclidean geometry. The geometry of non-Euclidean plane can be shown to be (relatively) consistent by showing it to be isomorphic to the structure of a curved surface in non-Euclidean space. Similarly, the world that the angel describes can be show to be consistent by mapping ordered pairs of N individuals onto a structure with N^2 individuals. Certain axioms would fail to hold, just as the parallel postulate fails to hold in non-Euclidean geometry.

Perhaps the numbers without 4 and 5 might be spelled out in a similar way.

What prompted the thought experiment for me was the suspicion that there is always a gap between knowing the metaphysical structure of things to some level of depth and knowing the fundamental structure of things. Any would-be fundamental structure could turn out to have something very different at a further level down.


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