So still my answer would be the same. Metaphysical-strength axioms seem self-evident when they result from dichotomous reasoning. — apokrisis

No, the point is that such axioms result from a description of what is, reality, not from dichotomous reasoning. The dichotomous reasoning follows the description. The description is "objects are bounded", what follows from the dichotomous reason is that it is impossible that objects are not bounded. So if we assume the existence of something which is not bounded, it is impossible that this is an object.

But then as I say, my own take is that dichotomies only do produce ideal limits. — apokrisis

See, contrary to what you say here, the dichotomy produced is between the ideal (not bounded), and the practical, the description of objects. The description is not an ideal, it is a representation, a model.

So now we could talk the same way about your own proposed dichotomy here - objects and boundaries. — apokrisis

It's not a dichotomy which I proposed, it is a description, which is proposed as an axiom. It is not proposed as a dichotomy between objects and boundaries, but as a description of objects. Therefore "boundary" is to be read as a property of objects, not as dichotomous to objects.

So the idea of a bounded object is a crisp metaphysical ideal that in reality only really exists in this fashion. — apokrisis

Correct, in reality boundaries are porous, as you describe. So the question is where do we get this idea of a continuous boundary. Boundaries, as we know them, are as you describe, yet we also want to assume an ideal boundary, the continuous one. If we cannot describe how this boundary could exist, in reality, what it could be bounding, this supposed ideal is nonsense.

So on the one hand, we can easily imagine a world of bounded objects. We can axiomatise a metaphysical dichotomy in that fashion - one that is built up from ancient debates about the continuous and the discrete, the one and the many, to arrive at an atomistic conception of bounded objects. — apokrisis

I don't see where you get the axiomatic dichotomy from. We have an axiom concerning the nature of an object, that it has a boundary, and we have an ideal which is "boundless". The ideal of boundless must be described in a self-evident way to become an axiom. I believe that this was the ancient trick of the theologians, to demonstrate that the boundless (God) is self-evident.

No, the point is that such axioms result from a description of what is, reality, not from dichotomous reasoning. — Metaphysician Undercover

Empirical claims about "what is" - the kinds of things people say as a result of common experience of the world - were the departure point for Ancient Greek metaphysical inquiry.

So in the world, we see all kinds of objects and non-objects. Is a cloud an object? Is the wind an object? Is a river an object?

Reason is then applied to the question - the unexamined assumption. So the starting point is only self-evident in the sense no one has really thought to question it systematically. It is only axiomatic in being acted upon without being philosophically considered.

Therefore "boundary" is to be read as a property of objects, not as dichotomous to objects. — Metaphysician Undercover

And I accounted for the conditions under which it can be considered a property of an object - if the object has the semiotic power to define its own boundaries. Otherwise the boundary is probably an idea that we ourselves impose on an unbounded nature. It is only us who might be concerned about identifying the true source of the nile or deciding whether some bump on a landscape is a hill or a mountain.

So the question is where do we get this idea of a continuous boundary. — Metaphysician Undercover

Well what bounded objects did you have in mind as an example? Let's see how necessary continuity might be to that idea of it being an object.

The ideal of boundless must be described in a self-evident way to become an axiom. — Metaphysician Undercover

Like the axiom of vagueness you mean? Surely you can see how it arises automatically via a dichotomy with the ideal of the crisp. To be absolutely crisp would be to absolutely lacking in vagueness. And thus, transitively, the same must apply in the other direction.

So if you can tell me about boundedness in any absolute fashion, you will be also telling me about absolute unboundedness as its logical corollary.

And if you can't give that kind of crisp definition of a boundary, then - again logically - your idea of a boundary is rather vague and lacking in metaphysical-strength axiomisation.

So let me see if I have this straight, the position you're arguing. It is useless to seek self-evident axioms, as there is no such thing, because meaning is context dependent. Therefore we should only use mathematical axioms, as apokrisis suggests, which have crisply defined, and fixed meaning within a mathematical system. This entails that anything which is logically possible is also true. — Metaphysician Undercover

What I'm saying is that the "atoms" in these axioms are not so atomistic. They are nodes in a network, or rather they are nodes in billions of similar but differing networks. Math works by fixing meanings more or less exactly. I know the definition of a continuous function, for instance. I can enlarge what I know about continuous functions in terms of other defined objects via a normalized method (a formalized logic, although used informally ). So the meaning evolves as relationships are deduced, but there's is no disagreement about this evolving meaning. It's the meta-law that proof is the law.

But for me philosophy is the supreme example of an abnormal discourse, even if one of its central fantasies is exactly the normalization of discourse --to define itself or science or rationality, etc. It's a permanent revolution, though. One doesn't play by the rules of the epistemology that one is trying to replace. Aren't "great" philosophers those who reinvent philosophy's self-image and method? This is largely done in terms of seduction by metaphors and narratives ("showing the fly the way out of the bottle") and not really so much in terms of refutations in a "word-math." Rhetoric partly succeeds by appeals to logic -- I won't deny that. But language seems too soft for the sort of "word math" that I associate with lots of traditional metaphysics. We can argue from shared investments and assumptions, but this mass of investments and assumptions is a mess. I'm not saying we can't do it at all in a useful way, but I do look at utility as a epistemological principle. "The smallest unit of meaning is a personality as a whole."