very interesting post.

I have several questions, but here I will pose only one, I hope not too confused.

Suppose I have NON separable preferences across time, as in the rational addiction model of Becker and Murphy (1988). Preferences in that model are also stationary.

Suppose I can somehow test Time Invariance in preferences.

Can I say that if Time Invariance is satisfied, than those preference are also time consistent? And if Time Invariance is NOT satisfied preference are Time Inconsistent?

Thank you for your reply.]]>

Many of the reviewers seemed to be either mathematicians or cognitive scientists who (1) didn't appreciate any of the wider philosophical content and (2) complained about minor details that are completely irrelevant to my main points. For example, one reviewer commented on my use of the term "doxastic space": "Ordinary mathematical use of ‘space’ suggests that the au wants to have some structure on the extended doxastic space beyond what comes from it being based on a Cartesian product (most likely a topology); but we are told nothing about that structure, leaving the au’s use of ‘space’ mysterious."

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1. I haven't looked at Nodelman and Zalta carefully enough (I should!), but the view doesn't look like the classical kind of structuralism I had in mind. The structuralism I had in mind postulates only concrete individuals and abstract properties. Since a "place in a structure" can hardly be a concrete individual, it would have to be a property. Nodelman and Zalta don't seem to think of structures as properties; in any case, they also say that 'i' and '-i' are not referring terms at all, but have to be quantified away. So they agree that e.g. 'i ≠ -i' doesn't express the non-identity of two definite things.

2. I wasn't assuming any indiscernibility principle. Instead, I was assuming that complex numbers are properties, defined by the conjunction of all structural predicates which they satisfy in the complex field. It then follows immediately that i = -i. You're right that if we have an ontology in which property-like "structures" are in some mysterious sense related to individual-like platonic "places", then we could say that i and -i are primitively distinct "places" even though they share their structural properties.

(And yes, Button's hybrid solution amounts to a kind of eliminative structuralism for constructed structures.)

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Interesting read! I'm certainly no expert and not necessarily a friend of structures myself, but here are some loose thoughts:

1. One thing that perplexes me a bit is why do you presume that a stucturalist would want (or need) to identify the number 2 with some property. Isn't it quite customary for a structuralist to say instead that numbers are not properties but objects of a special kind - namely, positions within the structure? For this to be convincing, of course, you'd have to take seriously the structuralist's talk of positions and it looks like you dismiss it as being too contrived and metaphorical. What would convince you otherwise? (In particular, I wonder what's your take on an axiomatic approach of the sort provided by Nodelman and Zalta.)

2. The argument leading to $latex i = -i$ seems to hinge on an indiscernibility principle, and one strong enough to guarantee that two elements (or places/positions) of a given structure are identical whenever they are related by an automorphism of the structure in question (cf. Keranen's paper). But why think that a structuralist must commit herself to such principle in the first place? Why can't she, say, treat the identity of positions as primitive and proceed from there, thus avoiding the need for an explicit identity criterion? I suppose that some epistemological considerations might be at work here, but as I said, I'm not overly familiar with this.

P.S. Oh, this could be a red herring, but what you say about $latex i \neq -i$ being a universally quantified statement bears a prima facie resemblance to Tim Button's hybrid solution and his distinction between basic and constructed structures. ]]>