“Few terms are used in popular and scientific discourse more promiscuously than ‘model’.” (Goodman 1976, 171).
Abstract
In this essay, I first consider a popular view of models and modeling, the similarity view. Second, I contend that arguments for it fail and it suffers from what I call “Hughes’ worry.” Third, I offer a deflationary approach to models and modeling that avoids Hughes’ worry and shows how scientific representations are of apiece with other types of representations. Finally, I consider an objection that the similarity view can deal with approximations better than the deflationary view and show that this is not so.
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Notes
Notoriously, Nelson Goodman argued that this proposal was “useless” (Goodman 1972). For any two objects, there is at least one set of which they are both members. Hence, the claim that similarity can be understood in terms of shared properties is universal and thus useless. The same is true if we say that a and b are more similar than c and d if, and only if, the former have more properties in common than the latter. For any two things, they will have exactly the same number properties in common. If the number of objects is n, then the number of shared properties is \(2^{n-2}\) and if the number of objects is infinite, the number of properties shared is infinite. But, Goodman assumes properties simply are sets. Many would argue this is false because there are sets for which there is no property (or as Lewis puts it, no “natural” property for every set) (Lewis 1983).
Weisberg includes weights given to terms in the equation, but I ignore those for simplicity; i.e. including them would not make a difference to the arguments presented below.
I am not making a historical claim of influence (though I think there is such a chain of influence). Rather, the type of view articulated by Hesse, Giere, and Weisberg are all developing similar thoughts on the matter. By emphasizing similarity, we are locating the model-world relation as one of analogy with positive (the intersections), negative (the differences), and neutral analogies. Additionally, scientists select the respects in which a model and target are thought to be similar.
Chakravartty (2001) provides similar worries to the ones Hughes provided.
As we will see, my own view is that concrete objects like inscriptions and utterances do represent the world. However, they don’t do so, or least don’t generally do so, by being similar to it. My name represents me but doesn’t do so by being similar to me.
This is true even if we restrict our mathematics to that utilized in scientific theories insofar as they employ the real numbers. Hartry Field proposes a very large number of spacetime points and their relations as truthmakers for Newtonian classical mechanics. But these seem as recondite as pure mathematical objects. Likewise, Kitcher understands mathematical claims as made true by idealized constructors who group and permute. These constructors are also as recondite as the objects they replace.
Of course, if mathematical realism is correct, one might correctly claim there are “Cambridge properties” they share; I am thinking about \(\pi \) and a beer right now. Both share the property being thought of by Jay. But those are not relevant to our purposes.
Incidentally, I am inclined to think that Weisberg could reformulate his view to avoid Hughes’ worry. First, Weisberg’s models and target systems can be construed as relational structures. Second, we can formulate whatever morphism we like between an abstract object and a concrete one construed as relational structures. For example, sets A and B have the same cardinality, if there is a bijection from A to B. Two sets having the same cardinality is a property they can share regardless of the ontology of their members of the respective sets. This presumes that the model and target system have both been construed as mathematical objects. But now we have a problem of how a mathematical object can denote a concrete one. Bas van Fraassen (2010) has argued that we model phenomena of the world with data models or what he calls appearances. We then determine the fit of our theory to the appearances. That is, we evaluate how one model fits another another model. But you ask, how does something abstract like a mathematical structure represent something concrete? van Fraassen suggests we ignore this question since it engages in metaphysics. As a naturalist, I do not think we can reasonably avoid this question.
Giere (1999, Chap. 6) utilizes psychological work on categorization involving prototypes and exemplars to understand how models form families with great insight. However, I would argue that the similarity view per se is not driven by findings in cognitive science.
To be fair, one might ask what my own view of mathematics is. If pressed, I am inclined to adopt a structuralist philosophy of mathematics (Resnik 1997; Shapiro 1997). Mathematics describes patterns with positions. For example, the natural number system is the pattern shared by any system of objects that has a specific initial object and a successor relation that satisfies the induction principle. However, I am inclined to accept in rem rather than ante rem structuralism. Ante rem structuralists claim that mathematical structures exist independently of their exemplifications whereas the in re structuralist thinks that the structures exist in virtue of their exemplifications. A different way of putting the view is that there are no mathematical objects but only mathematical properties.
The logistic model assumes a constant carrying capacity, linear density-dependence, no time lags, no migration, no genetic variation, or age structure in the population.
My view is not that all successful models are approximately true. Rather, it is that models which are accurate representations are approximately true. Models can be successful and not approximately true provided that satisfy other scientifically relevant aims (Odenbaugh 2005).
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I thank Steve Downes, Catherine Elgin, Melissa Vergara Fernández, Jim Griesemer, Andoni Ibarra, Iñaki San Pedro, and Chris Pinnock for their help with this essay. Additionally, I thank two anonymous referees for the very helpful feedback.
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Odenbaugh, J. Models, models, models: a deflationary view. Synthese 198 (Suppl 21), 1–16 (2021). https://doi.org/10.1007/s11229-017-1665-8
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DOI: https://doi.org/10.1007/s11229-017-1665-8