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This article was submitted to Frontiers in Plant Evolution and Development, a specialty of Frontiers in Plant Science.

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Philosophy can shed light on mathematical modeling and the juxtaposition of modeling and empirical data. This paper explores three philosophical traditions of the structure of scientific theory – Syntactic, Semantic, and Pragmatic – to show that each illuminates mathematical modeling. The Pragmatic View identifies four critical

Three distinct philosophical perspectives on the nature and dynamics of scientific theory were sequentially developed in the twentieth century. Each was a critique of the previous perspective. Each illuminates scientific modeling.

The SYNTACTIC VIEW (advocated by the Vienna Circle “Logical Positivists”) took issue with nineteenth century German idealism and argued that a scientific theory was actually a set of sentences expressed in a logical language

In the late 1950s and early 1960s, some philosophers of science critiqued the SYNTACTIC VIEW and focused on the history and sociology of scientific practice (e.g., Thomas Kuhn, Paul Feyerabend, N.R. Hanson). Another group suggested that a philosophy of science should focus on

Models were “always a mathematical structure” (Van Fraassen,

The perspectives on syntactic structures and semantic relations offered by the SYNTACTIC AND SEMANTIC VIEWS are insufficient to describe scientific modeling because

Practices, instruments, and experiments interweave with mathematical modeling (e.g., Hacking,

There are a variety of modeling syntaxes – mathematics, diagrams, narratives, simulations, and programs, etc. (e.g., Morgan and Morrison,

Models have implicit functions and assumptions (e.g., Wimsatt,

The PRAGMATIC VIEW focuses on contextual factors, the agents, and use. For example, philosopher Leo Apostel wrote:

Let then R(S,P,M,T) indicate the main variables of the modelling relationship. The subject S takes, in view of the purpose P, the entity M as a model for the prototype T.

The PRAGMATIC VIEW includes syntax and semantics, and explores assumptions and functions of models.

The PRAGMATIC VIEW helps us focus on the functions of mathematical models, including (1)

Unification involves the integration and synthesis of disparate types of evidence and models. For instance, Darwin unified hybridization, developmental, paleontological, and biogeographical data under a single theory. In mathematical modeling, unification often entails

Model fitting employs statistical procedures such as regression analysis and hypothesis testing. Although fitting is essential for model verification, models can be over fitted to accommodate all of the data, a practice called “fudging” by Lipton (

Mechanism identification uses various strategies, including (i) analysis (i.e., decomposition into constituent parts and processes, given a theoretical perspective), (ii) causal surgery (Pearl,

The relevance of an observation or datum for the testing, confirmation, and choice of mathematical models and theories is significantly greater if it is a novel prediction (e.g., Popper,

The foregoing is illustrated with a debate between two camps of modelers, each focusing on different modeling functions. The possibility of collaboration rather than conflict is explored and advocated.

Prusinkiewicz et al. (_{A} and _{B}. Since lateral branches do not usually produce flowers (p. 1453), the model assumes that meristems in

Álvarez-Buylla et al. (

Prusinkiewicz et al. (

Now Prusinkiewicz et al. (

In contrast, Álvarez-Buylla et al. (

The preceding illustrates that no model can fully satisfy all the functions and virtues of modeling. For instance, every model is limited in scope and identifies only some mechanisms. There are trade-offs among functions (Wimsatt,

Modelers should not become prisoners of their own abstractions (Levins and Lewontin,

Mariana Benítez, Karl Niklas, Michael Titcomb, and Michael J. Wade provided comments. Support was provided by the Academic Senate Committee on Research, UC Santa Cruz, and the Biocomplexity Center, Niels Bohr Institute, University of Copenhagen.