PH201/400 – Week 16 Scientific Modelling

1. PH201/400 – Week 16 Scientific Modelling

2. Introduction Starting point: Models matter! Examples: The Schelling model of social segregation The Fibonacci model of population growth The billiard ball model of a gas The Bohr model of the atom The Lorenz model of the atmosphere The Lotka-Volterra model of predator and prey ...

3. ‘Few terms are used in popular and scientific discourse more promiscuously than “model”. A model is something to be admired or emulated, a pattern, a case in point, a type, a prototype, a specimen, a mock-up, a mathematical description – almost anything from a naked blonde to a quadratic equation […]’ (Goodman)

4. What Models (here) are not: • Jane the model student (example to follow) • Claudia Schiffer (Someone presenting fashion) • The French model of democracy (an institutional structure) • Little Jimmy’s model railway (a toy) • The Ford T Model (A certain product) • ‘This is just a model’ (Model as a tentative hypothesis ) • …

5. Example 1: Population Growth Question: If you get pair of now-born rabbits now, how many rabbits will you have at a later point in time?

6. Fibonacci Model Fibonacci aka Leonardo of Pisa Liber Abaci (Book of Calculation), 1202

7. Modelling Assumptions: • The rabbit pair is one male and one female, and both are heterosexual. • Rabbits are able to reproduce for the first time six months after birth. • They they reproduce every six month. • The mate again immediately after the female gives birth.

8. 5. Every female gives birth to exactly one male-female pair of rabbits which satisfy assumptions 1 and 2 • There are no limitations on living space. • There are no limitations on food supplies. • Rabbits don’t die. • Rabbits keep up their 6-months birth rhythm indefinitely.

9. Rabbit Pairs A Rabbits born: 6 months later 12 months later 18 months later 24 months later …

10. Rabbit Pairs A A Rabbits born: 6 months later 12 months later 18 months later 24 months later …

11. Rabbit Pairs A A A, B Rabbits born: 6 months later 12 months later 18 months later 24 months later …

12. Rabbit Pairs A A A, B A, B, C Rabbits born: 6 months later 12 months later 18 months later 24 months later …

13. Rabbit Pairs A A A, B A, B, C A, B, C, D, E, … Rabbits born: 6 months later 12 months later 18 months later 24 months later …

14. Rabbit Pairs A A A, B A, B, C A, B, C, D, E, … Number 1 1 2 3 5 … Rabbits born: 6 months later 12 months later 18 months later 24 months later …

15. Generalisation: Rabbits born: 1 6 months later 1 12 months later 2 18 months later 3 24 months later 5 … …

16. Generalisation: Rabbits born: 1 6 months later 1 12 months later 2 18 months later 3 24 months later 5 … … Sequence of numbers

17. Generalisation: t0 t1 t2 t3 t4 … Rabbits born: 1 6 months later 1 12 months later 2 18 months later 3 24 months later 5 … … Sequence of numbers

18. Generalisation: t0 t1 t2 t3 t4 … Rabbits born: 1 6 months later 1 12 months later 2 18 months later 3 24 months later 5 … … = n(t0) = n(t1) = n(t2) = n(t3) = n(t4) … Sequence of numbers

19. Generalisation: t0 t1 t2 t3 t4 … Rabbits born: 1 6 months later 1 12 months later 2 18 months later 3 24 months later 5 … … = n(t0) = n(t1) = n(t2) = n(t3) = n(t4) … Sequence of numbers

20. Generalisation: t0 t1 t2 t3 t4 … Rabbits born: 1 6 months later 1 12 months later 2 18 months later 3 24 months later 5 … … = n(t0) = n(t1) = n(t2) = n(t3) = n(t4) … Sequence of numbers

21. Generalisation: t0 t1 t2 t3 t4 … Rabbits born: 1 6 months later 1 12 months later 2 18 months later 3 24 months later 5 … … = n(t0) = n(t1) = n(t2) = n(t3) = n(t4) … Sequence of numbers

22. Generalisation: t0 t1 t2 t3 t4 … Rabbits born: 1 6 months later 1 12 months later 2 18 months later 3 24 months later 5 … … = n(t0) = n(t1) = n(t2) = n(t3) = n(t4) … Sequence of numbers General Law:

23. Generalisation: t0 t1 t2 t3 t4 … Rabbits born: 1 6 months later 1 12 months later 2 18 months later 3 24 months later 5 … … = n(t0) = n(t1) = n(t2) = n(t3) = n(t4) … Sequence of numbers General Law: Sequence: 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, …

24. Properties: • After 5 years you have 89 Rabbit pairs. • The population grows monotonically. • There is no bound to the population size. • The speed of growth increases.

25. Questions: • Representation • Notice that most of the model assumptions are false, and some of them dramatically (there are no immortal rabbits!) • So models are not just a description of facts. • Yet the model tells us something about the world. • What, if anything, does this model represent and how does it do so?

26. 2. Ontology • Is it the equation ? • Is it the sequence 1, 1, 2, 3, …? • Is the set of assumptions we made? • Is it a fictional population of rabbits? • Is it combination of some (all?) of these? • …

27. 3. Truth in the Model • A model has an ‘internal structure’ or ‘mechanism’ that generates results. • Some claims are true in the model and some are not. • This is pressing issue because in a good model there are true claims that are not explicitly written into the model.

28. For example: • It is true that the population in Fibonacci’s model never decreases • It is wrong that the earth in Newton’s model moves on parabolic orbit. • On what basis are claims about a model-system qualified as true or false, in particular if the claims concern issues about which the description of the system remains silent? • What notion of truth is at work here?

29. 4. Epistemology • How do we learn about models? If there are truths in the model that are not explicitly written into the description, how do we find out about them? • Models are in many ways like systems we can investigate. How do we explore a model?

30. 5. Models and Theory • How do models relate to theory? • Fibonacci’s model is independent from theory. Are all models like this? • (Answer ‘no’: many models are related to theory in a number of ways. The question is: what are these ways?)

31. 6. Other Topics in Philosophy of Science • Explanation: Does Fibonacci’s model explain how a population grows? • Realism: what is the implication of the use of model for the scientific realism debate? • Laws of nature: What, if anything, do models tell us about laws of nature? • …

32. 7. Definition of a model How, if at all, can a model be defined? ‘An object, real or imagined, is not a model in itself. But it functions as a model it is views as being in a certain relationships to other things. So the classification of model is ultimately a of the ways things and processes can function as models.” (Harré, 2004, underlining added) The question is what functions turn something into a model.

33. Three Kinds of Models • There are three different kinds of models: • Models of phenomena • Models of theory • Models of data • It is imperative not to conflate these concepts! • Yet a given model can be a model in several of these senses at once.

34. 1. Models of Phenomena • They are a model of a selected part or aspect of the world, a phenomenon. • This part is also called the target system. • Examples: • Fibonacci and Schelling models. • The billiard ball model of a gas • The Bohr model of the atom • The Lotka-Volterra model of predator-prey interaction • The Lorenz model of the atmosphere • Modern climate models • …

35. ‘Serious’ example: Global Climate Model Grid ≈ 300km 35

36. The central issue: representation. • When models are qualified as follows, this indicates that they are models of phenomena: • Replica • Scale model • Analogue model • Idealised model • … •  For more on this see the Stanford Ecyclopedia entry ‘Models in Science’.

37. 2. Models of Theory • These are models in the sense of logic. • Main idea: • A model is a structure that makes all sentences of a theory true. • In this a theory is taken to be a set of sentences in a formal language.

38. Simple Example: • Theory T: • Model: • The set S={S1, …,S100} consisting of all objects in this room. • All objects have mass. • All objects are subject to gravity. • Let F be the predicate “has mass”. • Let G be the predicate “is subject to gravity”. • The sentence is true in S. • Therefore S is a model of the theory T.

39. ‘Serious’ Examples: • Euclidean geometry consists of axioms such as: ‘any two points can be joined by a straight line’ and the theorems that can be derived from these axioms. Any structure of which all these statements are true is a model of Euclidean geometry. • Any solution to an equation (e.g. the Schrödinger equation) is model of that equation (and the equation is the theory).

40. Notice: • The structure is a ‘model’ in the sense that the model is what the theory is about. • But: the model S is not itself about anything! It’s just a set of objects. • Hence models in the logical sense are not ipso facto models of phenomena. • Sometimes it is said that logical models are an ‘interpretation’ of the theory, or that they ‘satisfy’ the theory.

41. 3. Models of Data Empirical observation sometimes provides evidence in the form of data points. A model of data is a corrected, rectified, regimented, and in many instances idealized version of the data we gain from immediate observation, the so-called raw data.

42. Simple Example: Evidence For instance: p and V of a gas at constant temperature

43. Simple Example: Evidence Data model For instance: p and V of a gas at constant temperature

44. Simple Example: Evidence Different data models are possible! For instance: p and V of a gas at constant temperature

45. ‘Serious’ Example: Venetian Sea Levels The city of Venice regularly is subject to intense flooding. Is there a pattern in the flooding which would allow us to predict the next flood and its magnitude?

46. Given: data of the annual maximum sea levels from 1931 to 2015: (1931, 103cm), …, (2015, 138cm). Linear regression: we assume that the curve through the data is an inclined straight line: This equation is the statistical model of the sea levels, which is a special kind of data model. The parameters m, b, are determined with the data, the error term is assumed to be Gaussian. Notice: no ‘physical modelling’ has gone into this!

47. 4. ‘Multi-functional’ models Mary Hesse (1967) pointed out, that many models in science are models in both senses: They are at once interpretation of a general formal theory and represent something in the word. But: this does not blur the boundary between the two concepts; this only shows that the same think can be at once a model in two different senses.

48. Example: The Newtonian Model of the Sun-Earth System Newtonian Model: Two ideal spheres with homogeneous mass distribution gravitationally interacting only with each other Newton’s Theory of motion: F=ma satisfies represents

49. Models and Theory A traditional view is that models typically function like the Newton model: satisfies represents Theory Model Target This is untenable as a general account. General message: relation between models and theories is a difficult one and not all models are the same.

50. 1. “Theory-Free” Models • Some models are constructed outside a theoretical framework. This is because there simply are no theories. • Examples: • Fibonacci’s model of population growth • The logistic model of population growth • Schelling’s model of social segregation • Self-organising criticality models of granular media • …