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A View of the World Based on Science: Determinism By Kevin Padden and Jake Kildoo

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An Overview Traditionally, science claims that it can explain all of nature. Need to make the world seem understandable. Science replacing religion. This worldview is often characterized as philosophically materialistic—monism vs. dualism. These all characterize a deterministic outlook.

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Historical Origins of Determinism Ionian Philosophers First to put anthropomorphism to rout Materialism—all substances derived from a single substrate (often the 4 elements). Thales (c. 624-c. 546 B.C.) Water composes all matter Based this claim off of the importance of moisture in organisms’ lives Introduced abstract geometry into Greek thinking

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Historical Origins (contd.) Heraclitus (c. 540-c. 480 B.C.) Fire composes all matter Called fire the “Logos” The universe is like a flowing river controlled by the Logos Anaxamenes (c. 585-c. 528 B.C.) Air composes all matter Justified this by noting air’s ability to change density Xenophanes (not in Kushing) (c. 570-c. 475 B.C.) Earth composes all matter Anaxagoras (c. 500-c. 428 B.C.) Did not believe that properties such as life, sentience and intelligence can be explained in terms of the 4 elements Claimed that there are “seeds” that compose all material substance A strange prelude to Aristotelian natural philosophy.

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Historical Origins (contd.) Anaximander (c. 610-c. 546 B.C.) The universe is infinite in extension and duration The first known philosopher to suggest the geocentric model The “indefinite” composes all matter Did not like that other philosophers used the 4 elements (because they have completely opposing qualities) The primary matter must be quality-less The last of the Ionian philosophers that we discuss in this presentation Democritus (c. 460-c. 370) and Leucippus (fl. c. 450 B.C.) First among the Atomists Atoms compose all matter and are indivisible, eternal and imperishable.

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Historical Origins (contd.) Pythagoras (c. 570-c. 495 B.C.) Wanted to use numbers and mathematics as the “fundamental entity for understanding the universe” (Kushing, p. 165). Ex. Tetratics Not a materialistic worldview—all things composed of form, not matter (“all is number”). William of Ockham (c. 1288- c. 1348) Scholastic philosopher Founded the nominalistic view of universals Ockham’s razor (exemplified perfectly by Copernicus’ objection to Ptolemy’s equants)

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Historical Origins (contd.) Francis Bacon (1561-1626) Sense data more important than preconceived theoretical constructs Dislikes the a priori Leibniz (1646-1716) God acts according to the principle of “sufficient reasoning” Nothing exists which is unnecessary Nature acts according to a principle of economy

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Historical Origins (contd.) Pierre-Louis de Maupertius (1698-1759) Nature always works so as to minimize “action” Where action is the product of mass, speed and distance Ernst Mach (1838-1916) Believed that science’s purpose is to represent the phenomena of nature in the most economical way possible. The general principle held between these two and many others (i.e. Leibniz and Ockham) is that of the simplicity and succinctness of science.

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How do these Trends Exemplify the Evolution of Determinism? Reliance upon a few basic, underlying principles of nature to describe all universal phenomena. The Ionians—strayed from anthropomorphism. Pythagoras—used mathematics. Ockham (et. al.)—used simplicity.

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Questions/Comments about these Historical Figures Consider this: The Ionian philosophers seem to have decided to reject anthropomorphic views so suddenly and arbitrarily— this appears to be a huge reform in belief. Why might you speculate this dramatic shift to happen?

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What is Determinism? “Every event is necessitated by antecedent events and conditions together with the laws of nature” ~Carl Hoefer (2008) After Newton, universe is seen as completely deterministic If we know the initial positions and velocities of all particles, Newton’s second law can predict future trajectories forever

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Propagation of Initial Uncertainty Possible Locations of Particle v0v0 Δv0Δv0 Initial location of particle Predictions based on precision of initial conditions Can our instruments deliver on precision?

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Newton and the Stable Universe Newton believed in an omnipresent and omniscient God Uncertain that mechanics could explain the deterministic evolution of a stable universe: “To what end are comets, and whence is it that planets move all one and the same way in orbs concentric, while comets move all manner of ways in orbs very eccentric; and what hinders the fix’d stars from falling upon one another?” Mechanical universe required God to create and maintain the universe as explained in Optics

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Laplace on Determinism He perfected perturbation-theory calculation and used them to argue for stability of solar system Stated in his Philosophical Essays on Probabilities : “Present events are connected with preceding ones by a tie based upon the evident principle that a thing cannot occur without a cause which produces it. This axiom, known by the name of the principle of sufficient reason, extends even to actions which are considered indifferent.” “We ought to regard the present state state of the universe as the effect of its anterior state and as the cause of the one which is to follow”

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Laplace’s Demon “If an intelligence, for one given instant, recognizes all the forces which animate Nature, and the respective positions of the things which compose it, and if that intelligence is also sufficiently vast to subject these data to analysis, it will comprehend in one formula the movements of the largest bodies of the universe as well as those of the minutest atom: nothing will be uncertain to it, and the future as well as the past will be present to its vision. The human mind offers in the perfection which it has been able to give to astronomy, a modest example of such an intelligence.” Claims that the future and past would be present to this “demon” with perfect information and computational abilities. Is this claim true?

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God’s Role in Determinism Newton believed that God actively maintained the universe in order to maintain stability Laplace, when asked by Napoleon Bonaparte if he had mentioned God in his Celestial Mechanics replied, “I did not need that hypothesis.” Laplace found no place for God in keeping the universe running To this point, who would you side with?

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The Clockwork Universe Kushing’s “rough and ready” definition of determinism: “the requirement that the present state of the universe (or system) plus the laws of mechanics uniquely determine the future state of the universe.” What is the basis for this belief? In Newton’s view God runs the universe in orderly fashion. Science is simply a discovery of God’s method. He did not believe his mathematics in the Principia were sufficient to explain the universe’s evolution, long-term. It was not until after Newton that determinism based on his laws of mechanics were accepted.

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The Clockwork Universe (contd.) How would one use Newton’s laws to justify absolute determinism? The question boils down to this: do classical systems governed by Newton’s laws evolve deterministically and are they stable against small perturbations? Let us take an example: the motion of a planet around a fixed force center. This system works according to Newton’s second law and his law of universal gravitation. When worked out mathematically, this turns into a differential equation that can be solved exactly to obtain conic sections as orbits. Question: Does this example represent all mechanical systems?

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The Clockwork Universe (contd.) Science is believable precisely because of the predictive power it gives us. Some physical phenomena are regularly unpredictable. Ex. The weather The traditional view says that systems consisting only of a few parts are easier to predict. Does this really make a difference with regards to determinism, though? It appears not… The assumption was, for a long time, that classical mathematical analysis can account for all physical systems. This is not the case…

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The Clockwork Universe (contd.) The problem is that we assumed, for nearly 300 years that we completely understood classical mechanics. Chaotic behavior inheres in many even simple mechanical systems. This chaos confounds the project of determinism considerably The question thus takes on a new angle: Is the world fundamentally deterministic with only a few atypical indeterminate exceptions, or is it fundamentally indeterministic with only a few atypical determinable irregularities?

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Questions What are the theological implications of this sort of determinism? How well does this work as a proof and/or justification of atheism? Good counterarguments? Are we warranted in going beyond a mere instrumentalist view of scientific laws and attributing actual reality to them?

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Unwarranted Optimism Deterministic chaos is a reality Exhibits itself in: turbulent flow, heat convection, mechanical vibrations, population dynamics, and possibly the stock market Classical dynamic chaos is caused by rapid separation of system trajectories during dynamic evolution Trajectories that begin close together at some initial time grow apart in a short period of time Extreme sensitivity to the initial conditions

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The Butterfly Effect The notion that chaotic behavior is so unpredictable that the slightest change in initial conditions can rapidly change the trajectory of nearly identical systems or particles, with unexpected and seemingly impossible to determine results Coined by Edward Lorenz (1917-2008) A hurricane’s formation may be contingent on whether or not a distant butterfly had flapped its wings weeks before

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Intro to Chaos Theory General form of chaotic behavior λ is an external parameter Temperature or amplitude of disturbance x is a vector with many components The function need not depend explicitly on time Function must be at least non-linear for chaotic behavior, however some non-linear systems can be approximated as linear

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Chaos Theory (cont.) One cannot easily determine chaotic behavior from a non-linear differential equation Discrete mapping problems can help one to understand the long-term behavior of a system Makes it possible to study universal features of chaos for nonlinear differential equations by analyzing generic properties of much simpler discrete maps

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The Logistic Map Driving equation: Results along parabola: Value of next iteration is equivalent to a series of reflections between the parabola and the line y=x This makes for maps that clearly depict the amount of chaos in the behavior based on the parameter size and initial conditions

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Key Takeaways from Logistic Map

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Summary of Logistic Map a<1; x =0 is a stable fixed point a>1; x=0 is an unstable fixed point and a new one arises at the intersection of the parabola and 45° line at x = (a-1)/a 1<a<3; population moves towards this point a>3; populations cycles between two limit points a>~3.57; limit points bifurcate (double) to infinity a~4; population is chaotic and jumps around all possible values For x 0 =0 or x 0 =1, x n remain 0 forever for all a

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The Kicked Rotor x is the angular displacement y is the angular velocity k is a parameter controlling the “strength” of the kicks Only for k = 0 is a closed-form solution known At k = 0.971635 the system becomes a combination of regular and chaotic

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K = 0.5K = 0.971635K = 1.3 K = 2.1K = 5K = 10

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Key Takeaways from Kicked Rotor Even amongst in chaotic systems there are smooth and predictable regions The chaotic space is seemingly random and simply “fills an area” These are simple systems which exhibit extremely complex deterministic chaos The features of these examples are generic ones of most mechanical systems Both exhibit a loss of predictive power which is of utmost importance to determinism

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Summary Two populations beginning close together can be widely separated after only a few iterations Chaotic systems rapidly magnify small differences into big ones Extreme sensitivity to initial conditions renders prediction of future behavior impossible Output of deterministic chaotic systems can be so irregular that they model simply random systems Although one individual systems cannot be predicted, the average behavior of a large set of them can be predicted by probability In this sense, determinism reigns almost nowhere and chaos everywhere. Determinism remains only an unrealized theory.

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