Abstract
The chapter articulates weak and strong versions of the problem of representation. It also argues that enhancing the expressive power of logical systems does not result in a conclusive solution to the problem of representation. I examine structural realists’ reliance on ‘partial isomorphisms’ and ‘pragmatic truth’ to substantiate the message of this chapter. The message is that the set/model-theoretic framework cannot account for the biologically explicable relationship between the structure of the scientific theories and reality.
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- 1.
This works by assigning cube, fourth, fifth, sixth, etc., powers of prime numbers greater than 2 to symbols of Carnap’s system of Language I, i.e., to its variable, positive integers, undefined and defined predicates, and functions (Carnap 1937, 55).
- 2.
For example, Halvorson argued that:
Let L(T) be the language with a countable infinity of 1-place predicate symbols P1, P2, P3, …, and let T have a single axiom ∃1(x = x) (there is exactly one thing). Let L(T′) be the language with a countable infinity of 1-place predicate symbols Q0, Q1, Q2, …, and let T′ have axioms ∃1(x = x) as well as Q0x⊢ xQix for each i∈ℕ.
Every model of T is isomorphic to a model of T′ and vice versa. Indeed, a model of T has a domain with one object that has a countable infinity of monadic properties, and model of T′ also has a domain with one object that has a countable infinity of monadic properties. Therefore, T and T′are equivalent according to criterion…. And yet, T and T′ are intuitively inequivalent. We might reason as follows: the first theory tells us nothing about the relations between the predicates, but the second theory stipulates a nontrivial relation between one of the predicates and the rest of them. Again, our intuition is backed up by the syntactic account of equivalence: the theories T and T′ are not definitionally equivalent. (Halvorson 2012, 192–93).
- 3.
As van Fraassen has pointed out, actually there are some proofs (e.g., De Bouvere’s theorem) which demonstrate that model theory is not deprived of the necessary resources for identifying theories that are equivalent or distinguishing those that are not (van Fraassen 2014). For a textbook version of De Bouvere’s theorem, see (Hodges 1993, section 2.6). Then again, it could be argued that the SVT had regrettably deprived itself even of model theory’s linguistic resources, such as what is offered by the De Bouvere’s theorem.
- 4.
As I have remarked several times, if the structural realist give up the dichotomy, SR will collapse into a Platonic or Pythagorean view. This is the foremost problem. But I overlook this eventuality for the sake of argument.
- 5.
Notice that there are approaches that do not need to assume that only abstract mathematical structures play this role. For example, Giere’s approach allows for invoking physical models (as well as mapping, comparison, analogies, etc.) for fulfilling the representational job. This conception is in line with actual scientific practice, and instances of actual scientific model making support it. For example, consider concrete, physical model of the double helix made by Watson and Crick. This was intended to represent many diverse chemical systems that are similar in the relevant ways (what counts as relevant is presumably determined by the intentions of Watson and Crick). CMSA recognises such instances of representation. However, within the context of SR, which is at issue here, only mathematical structures can play the representational role.
- 6.
OSR is a metaphysically and epistemologically audacious theory. According to OSR, epistemology is tailored to the results of scientific theories which are supposed to reveal the real features of the external world. Ontology is based on epistemology, and it assumes that the structures that are discovered by scientific theories are all that there is in the world. Therefore, OSR wants to dispense with quiddities and hidden intrinsic natures which could not be exhausted by the scientific theories (and our epistemology and ontology).
- 7.
It assumes that the ground state is stable, and therefore the electron in such a state would not radiate energy and spiral into the nucleus in a manner determined by classical physics.
- 8.
It should be noted that, in Wittgenstein’s context, the problem could not be dissolved by invoking a meta-language, or by suggesting that one can represent the relation between a picture and reality in another picture. This is because, for Wittgenstein, there is only one (correct) language/logic and one cannot shift to another language (or to a meta-language, for that matter) to represent the relation between the language and reality. For Wittgenstein’s problem and Carnap’s attempt at dissolving it see (Beni 2015). In this book, too, I presume that enhancing the expressive power of an abstract formal system does not help with providing a naturalistically plausible account of the relationship between the system and reality.
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Beni, M.D. (2019). The Problem of Representation and a Formal Solution. In: Cognitive Structural Realism. Studies in Brain and Mind, vol 14. Springer, Cham. https://doi.org/10.1007/978-3-030-05114-3_3
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