Historical Perspectives on Models and Modeling

Michael S. Mahoney

Princeton University

13th DHS-DLMPS Joint Conference

Scientific Models: Their Historical and Philosophical Relevance

Historical Perspectives on Models and Modeling

Michael S. Mahoney

Princeton University

Colby College Science, Technology, and Society Lecture

Modeling the heavensgeometrically

Modeling the worldmechanically(analytically)

Modeling the worldas discrete systems(combinatorially)

Mechanicalmodel

geometry

algebra, calculus

Analyticalextension

Mathematicalmodel (structure

Extended modelreaching limits,e.g. non-linear DEs

Computational modeling ofmathematics

Computationalmodel

Plato’s Original Model

Motion ofheavensover time

Rotation of spheresthrough anglescorresponding to times

line of sight

Planet at time t1

Planet at time t2

Point P on sphere1

Point P on sphere2

line of sight

how model correspondsto the world

how model workshow world works

how model correspondsto the world

f W f M

SW

S’W

SM

S’M

= SW

f M f W

SWS’

M=

Corollary result

Motion ofheavensover time

Properties (symptomata) of the hyperbola

gnomon’s shadowat time t2

point P on hyperbola1

point P on hyperbola2

Direct measurementgnomon’s shadow

at time t 1

Direct measurement

Descartes’ Optics

reflection andrefraction of light

laws of impactand vectorial motion

rays -> lines of force -> motion of particle

Newton’s Model

Kepler’sLaws of PlanetaryMotion

Central Force

Mechanics

observationalmeasurement

observationalmeasurement

Newton, Principia (1687), I, 41Assuming any sort of centripetal force, and granting the quadrature of curvilinear figures, required are both the trajectories in which the bodies move and the times of motions in the trajectories found.

physical object

mathematicalmodel

Varignon (1700)

v = ds

dt

y = ds

dx

dds

dt2

x

r

zdx

ds

s

Lagrange -Analytic Mechanics

No drawings are to be found in this work. The methods I set out there require neither constructions nor geometric or mechanical arguments, but only algebraic operations subject to a regular and uniform process. Those who love analysis will take pleasure in seeing mechanics become a new branch of it and will be grateful to me for having thus extended its domain

ˆ)rr2(r̂)rr(r

ˆrr̂rr

r̂rr

2

radial acceleration

centrifugal acceleration Coriolis acceleration

angular acceleration

const.r

rrr2

rr2

2

2

Principia I,1 = Kepler, Law 2 (equal areas)

central force only, no torque

The Classical Model

PhysicalWorld

Central Force

Mechanics

instrumentalmeasurement

instrumentalmeasurement

The Classical Model Extended

PhysicalWorld

measurement

measurement

(P)DE

FE

Analyticmodel

Series expansionfinite differences

etc

Numericalmethods

Series summationsumsetc

221 mvmvdvmadSFdS

The Classical Model Extended

PhysicalWorld

How analytic modelcorresponds to the

world(P)DE

FE

How Analyticmodelworks

HowNumerical

modelworks

How numerical modelcorresponds to the

analytic model

Increasingly uncleartruncation, rounding,

critical values

Increasingly unclearcomplexity

John von Neumann on Models

To begin with, we must emphasize a statement which I am sure you have heard before, but which must be repeated again and again. It is that the sciences do not try to explain, they hardly even try to interpret, they mainly make models. By a model is meant a mathematical construct which, with the addition of certain verbal interpretations, describes observed phenomena. The justification of such a mathematical construct is solely and precisely that it is expected to work –that is, correctly to describe phenomena from a reasonably wide area. Furthermore, it must satisfy certain esthetic criteria –that is, in relation to how much it describes, it must be rather simple.

John von Neumann

computer as calculator

EDVAC Report

programming

numerical analysis

IAS MeterologicalProject

Computer as artificial organism

stabilityself-replication

evolution

Burks at Michigan

Ulam > cellular automata

ToffoliCA machine Holland

complex adaptive systemsgenetic algorithms

Langtonsynthetic biology

Artificial Life

WolframCAs in physics

SANTA FE

1980s: computer graphicslead to resurgence of CAs

1970s: supercomputers onJvN model and variants

finite automata

theoretical CSNumerical models

dynamical systems

chaos

General and Logical Theory of Automata

A New Kind of Science

Real World (Physical) System

Computational model of systemwithout referenceto specific

implementation

Computational model of systemin terms of specific implementation

Systems analysis

Computational model of systemin terms of finite-state machine

microprogramming

high-level language[intermediate language]

instruction set

Machine as operational model of system

Specification and design

programming (direct and embedded)

hardware design

There exists today a very elaborate system of formal logic, and, specifically, of logic as applied to mathematics. This is a discipline with many good sides, but also with certain serious weaknesses. This is not the occasion to enlarge upon the good sides, which I certainly have no intention to belittle. About the inadequacies, however, this may be said: Everybody who has worked in formal logic will confirm that it is one of the technically most refractory parts of mathematics. The reason for this is that it deals with rigid, all-or-none concepts, and has very little contact with the continuous concept of the real or of the complex number, that is, with mathematical analysis. Yet analysis is the technically most successful and best-elaborated part of mathematics. Thus formal logic is, by the nature of its approach, cut off from the best cultivated portions of mathematics, and forced onto the most difficult part of the mathematical terrain, into combinatorics.

The theory of automata, of the digital, all-or-none type, as discussed up to now, is certainly a chapter in formal logic. It will have to be, from the mathematical point of view, combinatory rather than analytical.

John von Neumann, 1948

In a mathematical science, it is possible to deduce from the basic assumptions, the important properties of the entities treated by the science. Thus, from Newton's law of gravitation and his laws of motion, one can deduce that the planetary orbits obey Kepler's laws.

John McCarthy, “Towards a Mathematical Science of Computation”, 1962

Shannon

circuit design (1938) information theory (1948)

switching theory

sequential machines

coding

Schützenberger (1957)

monoid theory of automata

Rabin/Scott

von Neumann

Kleene

McCulloch/Pitts

finite automata

cellular automata

regular events

Chomsky (1956)

mathematical linguistics

finite automata as Boolean algebras

phrase structure grammars (1959)

formal power series

FS CF

Ginsburg/Rice

lattice of sets

IAL (1958)

Algol (1960)

BNF

CFL as fixpoint of BNF-defined CFG

“algebraic”

other classes

mechanical theorem proving(AI)

IPL

FORTRAN (1956)

LISP

McCarthy

formal semantics

CPL

Strachey

Landin

operational semantics

-calculus

Scott

denotational semantics basedon -calculus as contnuous lattice

category theory

Tarski (1955)fixpoint theorem

for latticesapplicative (function) languages

as mathematical systems

msm 98

Computational Model

PhysicalWorld

(ComputationalModel)

Programstatic

(ComputerProcess)

Simulationdynamic

Brian Cantwell Smith, On the Origin of Objects (MIT Press, 1996), 35

Computational Model

PhysicalWorld

(ComputationalModel)

Program(ComputerProcess)

Simulation

MathematicalModel

Robert Rosen on Simulation

These considerations show how dangerous it can be to extrapolate unrestrictedly from formal systems to material ones. The danger arises precisely from the fact that computation involves only simulation, which allows the establishment of no congruence between causal processes in material systems and inferential processes in the simulator. We therefore lack precisely those essential features of encoding and decoding which are required for such extrapolations. Thus, although formal simulators can be of great practical and heuristic value, their theoretical significance is very sharply circumscribed, and they must be used with the greatest caution.

(“Effective Processes and Natural Law”, 1995)

Lindenmayer Systems

Computational Model

PhysicalWorld

Howcomputational

modelworks

How computational modelcorresponds to world

unanalyzable

unspecified

Computational Model

PhysicalWorld

(ComputationalModel)

Programstatic

(ComputerProcess)

Simulationdynamic

How computational modelcorresponds to world

unspecified

[How static programcorresponds to dynamic process

Howcomputational

modelworks

unanalyzable

]

John Holland on MathematicsMathematics is our sine qua non on this part of the journey. Fortunately, we need not delve into the details to describe the form of the mathematics and what it can contribute; the details will probably change anyhow, as we close in on our destination. Mathematics has a critical role because it along enables us to formulate rigorous generalizations, or principles. Neither physical experiments nor computer-based experiments, on their own, can provide such generalizations. Physical experiments usually are limited to supplying input and constraints for rigorous models, because the experiments themselves are rarely described in a language that permits deductive exploration. Computer-based experiments have rigorous descriptions, but they deal only in specifics. A well-designed mathematical model, on the other hand, generalizes the particulars revealed by physical experiments, computer-based models, and interdisciplinary comparisons. Furthermore, the tools of mathematics provide rigorous derivations and predictions applicable to all cas. Only mathematics can take us the full distance. (Hidden Order, 1995)