Modeling in the Natural and Social Sciences Haverford Science and Society seminar Fall 2002

Schedule Week 1: Philosophy, examples What is modeling? (seminar participants) What is modeling? (Max Black) Examples (Giorgio Israel, Mathematica) complex systems Week 2: Modeling complex systems in the natural sciences: Global Warming (Guest: Charles Miller) Week 3: Modeling complex systems in the social sciences (Guests: Suzanne Amador, Amy Slaton)

What is modeling? Give a two-sentence definition of modeling Give a two-sentence definition of modeling Describe an example of modeling in your discipline/experience. Be specific, give some details of what is included in the model, what it is used for Describe an example of modeling in your discipline/experience. Be specific, give some details of what is included in the model, what it is used for

According to Max Black Scale models: “likenesses of material objects, systems, or processes…that preserve relative proportions” Scale models: “likenesses of material objects, systems, or processes…that preserve relative proportions”

According to Max Black Analogue models: physical object “designed to reproduce as faithfully as possible in some new medium the structure or web of relationships in an original” Analogue models: physical object “designed to reproduce as faithfully as possible in some new medium the structure or web of relationships in an original” Hydraulic model of economics (Bill Phillips, 1949)

According to Max Black Mathematical models: “original field is thought of as `projected` upon… sets, functions, and the like that is the subject matter of the correlated mathematical theory” Mathematical models: “original field is thought of as `projected` upon… sets, functions, and the like that is the subject matter of the correlated mathematical theory” Differential Equation model of Predator-Prey System

According to Max Black Theoretical Models: describe original system as a simile/analogy/metaphor with a completely different system (like an analogue model, but without the actual construction) Theoretical Models: describe original system as a simile/analogy/metaphor with a completely different system (like an analogue model, but without the actual construction) Maxwell’s Ether

Key properties of modeling (echoing/refuting Black) Model is simpler than original system (“Simplifications, often drastic, are introduced for the sake of facilitating mathematical formulation and manipulation of the variables” [p. 224]). “Facilitating” is not quite right: more like “allowing” Model is simpler than original system (“Simplifications, often drastic, are introduced for the sake of facilitating mathematical formulation and manipulation of the variables” [p. 224]). “Facilitating” is not quite right: more like “allowing”

Key properties of modeling (echoing/refuting Black) Solve (or simulate) with model mathematically or computationally Solve (or simulate) with model mathematically or computationally Decide whether goal is quantitative predictions or global features/“plausible topology”; distinction is crucial, since there is a “serious risk of confusing accuracy of the mathematics with strength of the empirical verification in the original field”, especially when the model is drastically simplified, or the input data or parameters into the model are known only imprecisely. Decide whether goal is quantitative predictions or global features/“plausible topology”; distinction is crucial, since there is a “serious risk of confusing accuracy of the mathematics with strength of the empirical verification in the original field”, especially when the model is drastically simplified, or the input data or parameters into the model are known only imprecisely.

Key properties of modeling (echoing/refuting Black) Black’s distinction between mathematical and theoretical models seems spurious today: both involve mapping a real-world phenomenon onto a “heuristic fiction”; both involve “talking in a certain way” differently than previous approaches about that phenomenon; both involve introducing a new “language or dialect” into the study of the phenomenon. Black’s distinction between mathematical and theoretical models seems spurious today: both involve mapping a real-world phenomenon onto a “heuristic fiction”; both involve “talking in a certain way” differently than previous approaches about that phenomenon; both involve introducing a new “language or dialect” into the study of the phenomenon.

Question about causality Black says “Especially important is it to remember that the mathematical treatment furnishes no explanations” (p. 225) and that “causal explanations must be sought elsewhere” (p. 225). Is that really true? Doesn’t a mathematical model sometimes suggest a possible causation to be investigated much as any theory (cf. Israel’s discussion of the emergence of nonlinear dynamics and chaos)? Do theoretical models really supply more of an underlying cause than mathematical models as Black claims (p.226)? Black says “Especially important is it to remember that the mathematical treatment furnishes no explanations” (p. 225) and that “causal explanations must be sought elsewhere” (p. 225). Is that really true? Doesn’t a mathematical model sometimes suggest a possible causation to be investigated much as any theory (cf. Israel’s discussion of the emergence of nonlinear dynamics and chaos)? Do theoretical models really supply more of an underlying cause than mathematical models as Black claims (p.226)?

Examples (Mathematica, ODE Architect)

Modern Evolution of Mathematical Modeling (Giorgio Israel) Does science seek to “explain” (Newton) or merely “work” (von Neumann, modeling)? Does science seek to “explain” (Newton) or merely “work” (von Neumann, modeling)? Even within modeling, do we have descriptive or prescriptive goals? Even within modeling, do we have descriptive or prescriptive goals? 20 th century evolution of physical theories (quantum mechanics, nonlinear dynamics) causes us to re-evaluate our philosophy of modeling, from quantitative to qualitative, descriptive to prescriptive, simple to complex? 20 th century evolution of physical theories (quantum mechanics, nonlinear dynamics) causes us to re-evaluate our philosophy of modeling, from quantitative to qualitative, descriptive to prescriptive, simple to complex?

Modeling Complex Phenomena (Israel) Entry of mathematical modeling into biological and social sciences heightens these philosophical quandries (Guckenheimer & Oster, Levin) Entry of mathematical modeling into biological and social sciences heightens these philosophical quandries (Guckenheimer & Oster, Levin) Ambitious modeling in complex systems “opens the way to arbitrariness, and comes closer to the very practice of those verbal and informal sciences of which the uncertain, questionable, unverifiable natural has so often been deplored. Ambitious modeling in complex systems “opens the way to arbitrariness, and comes closer to the very practice of those verbal and informal sciences of which the uncertain, questionable, unverifiable natural has so often been deplored.