PDF A Subject With No Object: Strategies for Nominalistic Interpretation of Mathematics

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Hellman , p. The introduction of isomorphism in this context comes, of course, from the need to accommodate the preservation of structure between the applied mathematical part of the domain under study and the non-mathematical part. Recall that Hellman started with an applied mathematical theory T. This can be illustrated with a simple example.

Suppose that finitely many physical objects display a linear order. We can describe this by defining a function from those objects to an initial segment of the natural numbers. What is required by the modal structuralist's synthetic determination condition is that the physical ordering among the objects alone captures this function and the description it offers of the objects.


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It is not claimed that the full natural number structure is thus captured. This example also provides an illustration of the applied mathematical statement mentioned above. The Urelemente objects that are not sets are the physical objects in question, the relevant mathematical relation is isomorphism, and the mathematical structure is a segment of natural numbers with their usual linear order. On the modal structural conception, mathematics is applied by establishing an appropriate isomorphism between parts of mathematical structures and those structures that represent the material situation.

This procedure is justified, since such isomorphism establishes the structural equivalence between the relevant parts of the mathematical and the non-mathematical levels. However, this proposal faces two difficulties. The first concerns the ontological status of the structural equivalence between the applied mathematical and the non-mathematical domains. Of course, given that the structural equivalence is established by an isomorphism the material objects are already formulated in structural terms—this means that some mathematics has already been applied to the domain in question.

In other words, in order to be able to represent the applicability of mathematics, Hellman assumes that some mathematics has already been applied.

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This means that a purely mathematical characterization of the applicability of mathematics via structure preservation is inherently incomplete. The first step in the application, namely the mathematical modeling of the material domain, is not, and cannot, be accommodated, since no isomorphism is involved there. Indeed, given that by hypothesis the domain is not articulated in mathematical terms, no isomorphism is defined there. It may be argued that the modal structural account does not require an isomorphism between applied mathematical structures and those describing the material situation.

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In reply, note that this only moves the difficulty one level up. In this way, an isomorphism between structures describing the material situation and those arising from applied mathematics is still required. The second difficulty addresses the epistemological status of the claim that there is a structural equivalence between the mathematical and the non-mathematical domains. On what grounds do we know that such equivalence holds?

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Someone may say that the equivalence is normatively imposed in order for the application process to get off the ground. But this suggestion leads to a dilemma. Either it is just assumed that we know that the equivalence holds, and the epistemological question is begged given that the grounds for this are in question , or it is assumed that we do not know that the equivalence holds—and that is why we have to impose the condition—in which case the latter is clearly groundless.

However, it may be argued that there is no problem here, since we establish the isomorphism by examining the physical theories of the material objects under consideration. But the problem is that in order to formulate these physical theories we typically use mathematics. And the issue is precisely to explain this use, that is, to provide some understanding of the grounds in terms of which we come to know that the relevant mathematical structures are isomorphic to the physical ones.

The main point underlying these considerations has been stressed often enough although in a different context : isomorphism does not seem to be an appropriate condition for capturing the relation between mathematical structures and the world see, e.

There is, of course, a correct intuition underlying the use of isomorphism at this level, and this relates to the idea of justifying the application of mathematics: the isomorphism does guarantee that applied mathematical structures S and the structures M which represent the material situation are mathematically the same. The problem is that isomorphism-based characterizations tend to be unrealistically strong. They require that some mathematics has already been applied to the material situation, and that we have knowledge of the structural equivalence between S and M.

What is needed is a framework in which the relation between the relevant structures is weaker than isomorphism, but which still supports the applicability, albeit in a less demanding way e. The modal structuralist solves partially the epistemological problem for mathematics. Assuming that the modal-structural translation scheme works for set theory, modal structuralists need not explain how we can have knowledge of the existence of mathematical objects, relations or structures—given the lack of commitment to these entities.

However, they still need to explain our knowledge of the possibility of the relevant structures, since the translation scheme commits them to such possibility. One worry that emerges here is that, in the case of substantive mathematical structures such as those invoked in set theory , knowledge of the possibility of such structures may require knowledge of substantial parts of mathematics.

For instance, in order to know that the structures formulated in Zermelo set theory are possible, presumably we need to know that the theory itself is consistent. But the consistency of the theory can only be established in another theory, whose consistency, in turn, also needs to be established—and we face a regress.

It would be arbitrary simply to assume the consistency of the theories in question, given that if such theories turn out to be in fact inconsistent, given classical logic, everything could be proved in them. Of course, these considerations do not establish that the modal structuralist cannot develop an epistemology for mathematics.

They just suggest that further developments on the epistemological front seem to be called for in order to address more fully the epistemological problem for mathematics. Similarly, the problem of the application of mathematics is partially solved by the modal structuralist. After all, a framework to interpret the use of mathematics in science is provided, and in terms of this framework the application of mathematics can be accommodated without the commitment to the existence of the corresponding objects.

One concern that emerges besides those already mentioned at the end of section 4. Rather than explaining how mathematics is in fact applied in scientific practice, the modal-structural framework is advanced in order to regiment that practice and dispense with the commitment to mathematical entities.

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But even if the framework succeeds at the latter task, thus allowing the modal structuralist to avoid the relevant commitment, the issue of how to make sense of the way mathematics is actually used in scientific contexts still remains. Providing a translation scheme into a nominalistic language does not address this issue. A significant aspect of mathematical practice is then left unaccounted for. The status of the indispensability argument within the modal-structural interpretation is quite unique.

On the one hand, the conclusion of the argument is undermined if the proposed translation scheme goes through , since commitment to the existence of mathematical objects can be avoided. On the other hand, a revised version of the indispensability argument can be used to motivate the translation into the modal language, thus emphasizing the indispensable role played by the primitive modal notions introduced by the modal structuralist.


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  7. The idea is to change the argument's second premise, insisting that modal-structural translations of mathematical theories are indispensable to our best theories of the world, and concluding that we ought to be ontologically committed to the possibility of the corresponding structures. In this sense, modal structuralists can invoke the indispensability argument in support of the translation scheme they favor and, hence, the possibility of the relevant structures, which are referred to in the conclusion of the revised argument.

    But rather than supporting the existence of mathematical objects, the argument would only support commitment to modal-structural translations of mathematical theories and the possibility of mathematical structures. With the introduction of modal operators and the proposed translation scheme, the modal structuralist is unable to provide a uniform semantics for scientific and mathematical theories.

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    Only the latter, as opposed to the former, requires such operators. In fact, Field has argued that if modal operators were invoked in the formulation of scientific theories, not only their mathematical content, but also their physical content would be nominalized Field After all, in that case, instead of asserting that some physical situation is actually the case, the theory would only state the possibility that this is so. One strategy to avoid this difficulty of losing the physical content of a scientific theory due to the use of modal operators is to employ an actuality operator.

    By properly placing this operator within the scope of the modal operators, it is possible to undo the nominalization of the physical content in question Friedman Without the introduction of the actuality operator, or some related maneuver, it is unclear that the modal structuralist would be in a position to preserve the physical content of the scientific theory in question.

    But the introduction of an actuality operator in this context requires the distinction between nominalist and mathematical content. That such a distinction cannot be drawn at all is argued in Azzouni Otherwise, there is no guarantee that the application of the actuality operator will not yield more than what is physically real. However, even with the introduction of such an operator, there would still be a significant difference, on the modal-structural translation scheme, between the semantics for mathematical and scientific discourse.

    For the former, as opposed to the latter, does not invoke such an operator. The result is that modal structuralism does not seem to be able to provide a uniform semantics for mathematical and scientific language.

    Given the need for introducing modal operators, the modal structuralist does not take mathematical discourse literally. In fact, it may be argued, this is the whole point of the view! Taken literally, mathematical discourse seems to be committed to abstract objects and structures—a commitment that the modal structuralist clearly aims to avoid. However, the point still stands that, in order to block such commitment, a parallel discourse to actual mathematical practice is offered.