Historical Perspectives on Models and Modeling Michael S. Mahoney Princeton University 13th...
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Transcript of Historical Perspectives on Models and Modeling Michael S. Mahoney Princeton University 13th...
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)