1. On a natural construal, propositional attitude ascriptions, as in (1)–(3), characterise or describe relations between subjects and propositions (or proposition-like elements, including facts).
(1) Jill knows that crisps are delicious
(2) Flo believes that Jill is a glutton
(3) Eliza wishes that Flo would be less judgemental
For a variety of reasons, philosophers typically think of propositions and proposition-like elements as abstract—e.g. as not located in space-time. For example, Flo is in London. Bill, in Australia, also believes that Jill is a glutton. On the natural construal, Jill and Bill stand in the same relation to the same proposition: that Jill is a glutton. If the proposition is located in space-time, there appear to be four options: (i) it’s located where Flo is; (ii) it’s located where Bill is; (iii) it’s located where Jill is; (iv) it’s located where both Flo and Bill are, or where all three are (e.g. it’s omnipresent). Any argument in favour of (i) is liable to make problematic how Bill can stand in the same relation to the proposition; mutatis mutandis for (ii) and Flo. (iii) may avoid the immediate difficulties with (i) and (ii), but begins to seem awkward when one considers propositions about relations: is the proposition that Jill is within five light years of Alpha Centauri located with Jill, or Alpha Centauri, or both, or spread over a space containing both? (iv) is difficult to distinguish from the view that the proposition is not located in space-time at all. So, there is at least one type of reason for thinking that propositions are not located in space-time.
2. Suppose that propositions are not located in space-time. That supposition has induced some philosophers to find puzzling the apparent fact that appeal to propositions appears to figure essentially in characterisations of human psychology. How, such philosophers wonder, can humans—located in space-time—be in psychologically efficient contact with things that are outside space-time? One response has been to compare ascriptions of propositional attitudes with quantitative measurements. For in the latter, we allow that the properties of spatio-temporally located things can be characterised by appeal to relations to numbers, as when we say that Jill weighs 72kg. On the plausible assumption that numbers, like propositions, are not located in space-time, such measurements therefore share features with propositional attitude ascriptions. To the extent that we find un-mysterious the assignment of numbers to spatio-temporally located things in measurements, the analogy might be helpful in rendering un-mysterious the role of propositions in propositional attitude ascriptions. (For recent development and discussion, see Robert J. Matthews The Measure of Mind: Propositional Attitudes and their Attribution, 2010, Oxford: OUP.)
3. Not everyone is convinced of the probity of the analogy. Tyler Burge writes:
“Sometimes representational contents have been compared to numbers in measurement. This metaphor is misleading in two ways. First, the numbers in measurement can, according to standard representation theorems, be eliminated. By contrast, there is absolutely no solid reason to believe that representational content can be eliminated from explanations of propositional attitudes (or logic, or semantics). Second, the numbers in measurement are relative to a unit of measure, which is relatively arbitrary. By contrast, representational contents are fixed by natural, objective relations to a subject matter. They are constituent elements in natural psychological, or other representational, kinds.” (Truth, Thought, Reason: Essays on Frege, Oxford: OUP, 2005: 29–30.)
4. The main thing I don’t understand here is Burge’s claim about the elimination of numbers. I’m no expert on measurement theory, so this could easily be a reflex of my ignorance. But as I understand the standard representational theorems, they establish homomorphisms between the represented domain and the numerical measures. (Homomorphisms are a type of structural correlation. Roughly, the aim is to set up correlations between, on one hand, the target domain of things and its structure and, on the other hand, the system of numbers used in the measurements and their structure, so that one can use the structure of the number system to characterise the structure of the domain to be measured.) Presumably, the establishment of such homomorphisms typically supports the possibility of distinct, but still homomorphic, assignments of objects in structures, which may or may not be numbers, depending on the availability of suitable non-mathematical structures. This is generalisation of the fact that, where one can measure something by the use of one system of numbers, one can also measure it by the use of a structurally similar system of numbers. This happens, for example, when one switches from measuring weight in pounds to measuring it in kilos, or from measuring temperature on the Fahrenheit scale to measuring on the Celsius scale. It may be that the theorems deliver the result, then, that any specific assignment of numbers may be eliminated, in favour of some other specific assignment. Or, it may be that they deliver the stronger result, that any assignment of numbers may be eliminated in favour of a different type of structure, which may or may not be mathematical, but need not include numbers. Or, perhaps Burge has in mind an even stronger result. Anyway, the three main questions I have here are:
Q1. Which theorems does Burge have in mind, and which of their immediate consequences does he mean to appeal to here?
Q2. What precisely does Burge intend by the claim that “the numbers in measurement can be eliminated”?
(One sub-question here concerns whether there are any explanatory losses in eliminating numbers through the theorems. Presumably, self-representation could be set up to meet conditions on homomorphism, but it’s plausible that eliminating numbers in favour of self-representation would involve a loss of explanatory power.)
Q3. Is Burge right that the standard representation theorems that he has in mind deliver the result that numbers can be eliminated?