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2014 05 unibuc optimization and minimization


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A talk in Philosophy of Physics, about minimization in classical mechanics and a project in computational mechanics

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2014 05 unibuc optimization and minimization

  2. 2. Scientific modalities? What I am concerned with?  the interplay between natural science and philosophy (both metaphysics and epistemology)  In metaphysics, I argue for a form of scientism, called “scientific metaphysics”: inspired by the practice and advancement of science  How do we build modalities from a scientific theory?  This is a new naturalism for metaphysics (esp. analytic metaphysics)  In this present case I argue that scientific epistemology guides scientific metaphysics
  3. 3. Modality and science in general  Standard answer: science describes “reality as actuality” and not “as possibility”  Hence, a “division of labor” between science and metaphysics  Science is all about actuality, philosophy is about possible or impossible entities  The non-standard answer: natural science (at least) is involved with modalities  B. van Fraassen: the modeling process is a process of dealing with possible outcomes of experiments  Any model is un to its neck in modalities  Quantum mechanics is up to its neck in modality esp. many worlds interpretations of QM (see: D. Wallace (2012), Saunders, Barrett, Kent Wallace (eds.) 2011  Ditto about cosmology and multiverse (Tegmark 2013)  String dualities are “modally involved” (Muntean 2014)  Even biology adds a new type of modality  Statistical mechanics relates to modality (viz. ensembles)  The involvement of classical mechanics with modality is rarely discussed in philosophy of science. Exceptions: Stoltzner, Butterfield, Pulte.
  4. 4. Minimization and philosophy What has been done?  relate minimization (and related concepts) to the work of David Lewis (Butterfield 2002, 2004)  Relate minimization to simplicity and unification in physics (cf. M. Planck, D. Hilbert in the 1900s)  The anti-metaphysics of the minimization of action and logical empiricism (esp. Ph. Frank, M Schlick)  Historical approach to the minimization of action (Pulte 1989, Stoltzner 2003)  Relate minimization to dispositions (Katzav 2008)
  5. 5. Prospects  Integrate better minimization with optimality and computational methods.  This is how science is practice nowadays.  This will relate better epistemology to modality (in this context!)  Relate the CM minimization to the standard approaches in QM and probably to QFT  Relate it to statistical mechanics
  6. 6. Metaphysical voltages of minimization  Teleology  “the perfectly acting being… can be compared to a clever engineer who obtains his effect in the simplest manner one can choose” (Malebranche)  Simplicity:  Nature has “the greatest simplicity in its premises and the greatest wealth in its phenomena” (Leibniz)  Unification  of particle mechanics with wave optics (Hamilton)  A gate to modality (in conjunction with D. Lewis’ work on counterfactuals)
  7. 7. Physical voltage of minimization  With PLA mechanics is freed of forces and becomes “non- Newtonian”.  Forces are replaced with “virtual motions” and “virtual work” in statics  Universality and generality  One can infer conservation of energy of different quantities and invariants  Can be used in conjunction with symmetries  One can infer Newton’s equations of motion from minimization  Hence a leap towards special relativity is possible through minimization
  8. 8. What makes the actual special?  At least in one interpretation, minimization involves a calculation over an infinite number of possible evolutions.  Ostwald 1893: “from an infinite number of possibilities of one process, the one that actually happens is distinguished among the possible cases.”
  9. 9. Deflationism about minimization  It is a convenient alternative to standard mechanics, but with no metaphysical implications  Euler,  Mach  Ph. Frank  It is just a mathematical tool with no philosophical implications.  It is more or less like a clever coordinate transformation, or a Fourier transformation or a Laplace transformation, with no metaphysical or philosophical implication.  Mach: as a mathematical principle, “it is an economically ordered experience of counting”  Butterfield mentions a eliminativism attitude towards modality in minimization.
  10. 10. The tale of the 18th century  Huygens used Fermat’s principle to infer the laws of optics  Euler and Maupertuis used the principle of least action to infer the laws of collision of bodies  “Nature makes some quantities a minimum or a maximum”  Euler (1743): “all processes in nature obey certain min or max laws.
  11. 11. Hamilton’s equation   Ldt  0  A system moves from one configuration to another such that the variation of the integral between the actual and the virtual path, co-terminus in space and time, is zero
  12. 12. Hamilton’s principle (in Butterfield)  For any one-parameter family, parametrized by say α, of kinematically possible histories of the mechanical system, that may deviate from the actual history between t0 and t1, but must match the actual history as regards the configurations q0, q1 at times t0, t1: the action as a function of: I    Ldt  with the integral taken along the history labelled by parameter-value , has zero gradient at the value of corresponding to the actual history. 2 1 ( ) t t
  13. 13. 19th century and the Hamilton Jacobi formalism  The H-J equation S S   H q t  ( , , )  0 t q  
  14. 14. Philosophical stances towards minimization in the 20th century  Realism about minimization and action: Planck, Hilbert. Minimization points to a deeper structure in nature:  Pure heuristics: E. Mach, Ph. Frank. It is an useful alternative to PDE
  15. 15. The metaphysics of minimization Butterfield postulates three levels of involvement with modality:  1: keep the problem (i.e. the L) keep the laws of motion, but change initial conditions.  2: alter the problem (change the L), alter the initial condition but keep the laws  3: alter the laws of nature.
  16. 16. Metaphysics of Modality 1  Butterlfield thinks that we can identify a world with either:  a configuration (qi,t)  a state (qi,pi,t)
  17. 17.  A statement defended by Butterfield: “any actually true proposition (not only: any law of nature) should be made true by actual facts, i.e. goings-on in the actual world. (So the threat does not depend on the evolutions mentioned by the law being contralegal: what matters is that they are not actual.)”
  18. 18. Three strategies about modality in CM  (Vindicate): The role of possible (indeed, contralegal) histories can be vindicated—it is not problematic  (Eliminate): This role of possible histories can be eliminated—the laws can be formulated without invoking it;  (Useful): The variational formulation of the laws is nevertheless useful, or even advantageous compared with formulations that do not mention possible histories.
  19. 19. What is missing?  Here I am interested in the epistemology associated with minimization and optimization  Minimization is a search procedure  The practice of science is conducive to its metaphysics and its epistemology.  Analytical mechanics is a scheme for solving problems; therefore, a heuristics tool.  I concur with Butterfield: one can interpret Jacobi-Hamilton formalism metaphysically: you can read off some metaphysics from the formalism, but not that much
  20. 20. Problem solving and modality  Butterfield 2004: “aspects to do with problem-solving (the use of separation of variables, leading on to action-angle variables and Liouville’s theorem) […] are not illuminating about modality”.  I dissent: epistemology and metaphysics work hand in hand.
  21. 21. The forth strategy: the computational turn of the 20th century We need to think more pragmatically about minimization Minimization replaced with optimization, in a computational was
  22. 22. Enters the computational method  We need to remember the practice of current analytic mechanics which is deeply engaged with numerical simulations.  Minimization is replaced with optimization. The absolute minima of functions is rarely reached by numerical methods  So we discover minima, but good enough minima
  23. 23. Scientific reasoning and computation  two types of reasoning in science:  rule-based reasoning RBR: new theories or models are inferred based on a set of rules  Case-based reasoning CBR: exemplars used to solve problems in science (Kuhn, Nickles)  Kuhn argued that CBR is more frequently used in science than RBR  CBR computation applied to science (Bod 2006)
  24. 24. CBR and scientific reasoning  “scientists explain new phenomena by maximizing derivational similarity between the new phenomenon and previously derived phenomena. And the shortest derivation provides a possible way to attain this goal. The rationale behind maximizing derivational similarity is that it favors derivation trees which maximally overlap with previous derivation trees, such that only minimal recourse to additional derivational steps needs to be made.”  what is followed is previous patterns of derivations, not phenomena in itself.  Reminiscent of Kitcher’s unificatory explanation.  Possibly of consilience too (computational consilience?)
  25. 25. Scientific discovery and computation (III)  This is my contribution  A blatant counterexample to computational deflationism, and a negative answer to A1 is evolutionary computation.
  26. 26. Formalist and biomimetic strategies in computation  Three similarities: A. computation and reason/logic B. computation and mind C. computation and life  Witness that B and C are both biomimetic strategies. A is more or less a formalist strategy  The real novelty for a philosophy of science is, I argue, analyzing C
  27. 27. C: Computation and life “the oldest and the most fundamental of all questions about simulation” (Von Neumann, Keller 2003):  Q2: How closely can a mechanical simulacrum be made to resemble an organism? St. Ulam (Monte Carlo method): the right question when relating mathematics and computer science to biology is not: “What mathematics can do for biology?”, but:  “What biology can do for mathematics” (Ulam 1972).
  28. 28. It’s Turing Year!  1950: “Computing Machinery and Intelligence”: “We cannot expect to find a good child-machine at the first attempt. One must experiment with searching one such machine and see how well it learns. One can then try another and see if it is better or worse. There is an obvious connection between this process and evolution, by the identifications:  Structure of the child machine = Hereditary material  Changes of the child machine = Mutations  Natural selection = Judgment of the experimenter”
  29. 29. Metabiology  G. Chaitin (the father of the algorithmic information theory):  Metabiology: a field parallel to biology that studies the random evolution of artificial software (computer programs) rather than natural software (DNA), and that is sufficiently simple to permit rigorous proofs or at least heuristic arguments as convincing as those that are employed in theoretical physics. (my emphasis). (Chaitin 2011, 100)
  30. 30. The biomimetic assumption  life = search for optimality  computation = search for optimality  In the 1930s S. Wright interpreted a biological species as a system that evolves in time by exploring a multi-peaked landscape heuristic of optimal solutions to a “fitness problem” (Wright 1932).  GA literature: (Tomassini 1995; Koza et al. 1999, 20sqq.; Koza et al. 2003; Affenzeller 2009; Olariu and Zomaya 2006).  More foundational approach (Jong 2006)
  31. 31. The idea  Start with a number of algorithms (individuals), mostly randomly chosen = initial population  Let them live in an environment (let thme output solutions to a problem)  Generate new algorithms by:  Reproduction (not perfect though!)  Mutation  Sexual crossover  Define a fitness function  Decide a termination condition
  32. 32. The genetic algorithm produce an initial population of individuals (1) while `termination condition’ not met do (2) evaluate the fitness of all individuals (3) select fitter individuals for reproduction (4) produce new individuals (5) generate a new population (by inserting some new ‘good’ individuals and by discarding some ‘bad’ individuals) (6) mutate some individuals (7) endwhile (8) Call the individual(s) who satisfy the `termination condition’ the “best-fit-so-far” (9)
  33. 33. Humans are here!  Pick the initial population  Pick the reproduction mechanisms: the mutation factor, the crossover breeding  Pick the termination condition  Pick the resources available
  34. 34. Eureqa=GA in science   Concrete results by Schmidt and Lipson who “re-discovered” not only analytical functions from empirical data, but structures which are highly relevant to physical sciences:  Hamiltonians, Lagrangians, laws of conservation, symmetries, and other invariants (Schmidt and Lipson 2009)  “Optimal forms” and meaningful invariants for the chaotic double pendulum  Schmidt&Lipson’s Termination condition: the decomposability (similar to mechanisms)
  35. 35. Complexity and predictability
  36. 36. An epistemic claim “These terms may make up an ‘emergent alphabet’ for describing a range of systems, which could accelerate their modeling and simplify their conceptual understanding. […] The concise analytical expressions that we found are amenable to human interpretation and help to reveal the physics underlying the observed phenomenon. Many applications exist for this approach, in fields ranging from systems biology to cosmology, where theoretical gaps exist despite abundance in data. Might this process diminish the role of future scientists? Quite the contrary: Scientists may use processes such as this to help focus on interesting phenomena more rapidly and to interpret their meaning” (Schmidt and Lipson 2009, 82)
  37. 37. Epistemology of Evolutionary Computing  Upward Epistemology  Rationality and Evolution  A new solution to the Meno’s Paradox?
  38. 38. Genetic algorithms and Numerical Simulations  They are not Numerical Simulations per se.  But they simulate the search process in science  They are a combination of:  Brute search  Guided Search  They are epistemic enhancers with an upward epistemology
  39. 39. Rationality and Evolution  GA are random  GA are inscrutable  Do they provide justification?  Are they irrational?  Can truth be reached by a set of stochastic procedures?  In principle, it can.  See the connection between rationality and evolution  (a) evolution produces individuals which are good approximations to an optimally well designed system and  (b) optimally well-designed systems are rational agents
  40. 40. Philosophical finale: Meno’s paradox  Meno: How will you look for it, Socrates, when you do not know at all what it is? How will you aim to search for something you do not know at all? If you should meet with it, how will you know that this is the thing you did not know? Socrates: […] Do you realize what a debater’s argument you are bringing up, that a man cannot search either for what he knows or for what he does not know? He cannot search for what he knows since he knows it, there is no need to search—nor for what he does not know, for he does not know what to look for. Meno, 80d4-e5, G.M.A. Grube in (Plato 1997).
  41. 41. Meno’s dilemma 1. How do we inquire into things we you know? 2. How do we inquire into things of which we know nothing?  Keyword: “searching” (ζητέω).
  42. 42. The Evolutionary Computation and Meno’s Paradox  Problem in PhilSci: “given the vast and noisy field of possible options, how do scientists identify which problems, techniques, and other resources are more likely to be fruitful to pursue” Nickles 2003  The EC answers:  There are no rules of searching  There are meta-rules of searching, i.e. the rules of evolution.  Search1 can combine with search2  Combine successful search with unsuccessful search too.
  43. 43. Conclusion  Science does inspire new ways of thinking about modality  But the practice of science, i.e. in its numerical age, inspires different kind of modalities.  The old debate about minimization of action, PLA, etc. finds new inspiration from the concrete application of optimization through numerical methods  When exact analytical methods are not available (and they usually are not) numerical solutions optimized at least locally and find minima.  See the concrete case of Schmidt and Lipson and GA
  44. 44.  Butterfield, Jeremy. “David Lewis Meets Hamilton and Jacobi.” Philosophy of Science 71, no. 5 (December 2004): 1095–1106.  ———. “Some Aspects of Modality in Analytical Mechanics.” arXiv:physics/0210081, October 19, 2002.  Ellis, Brian. “Katzav on the Limitations of Dispositionalism.” Analysis 65, no. 285 (January 1, 2005): 90–92.  Katzav, J. “Dispositions and the Principle of Least Action.” Analysis 64, no. 3 (July 1, 2004): 206–14. doi:10.1093/analys/64.3.206.  Pulte, Helmut. Das Prinzip der kleinsten Wirkung und die Kraftkonzeptionen der rationalen Mechanik. F. Steiner Verlag, 1989.  Stöltzner, Michael. “The Principle of Least Action as the Logical Empiricist’s Shibboleth.” Studies in History and Philosophy of Modern Physics 34, pt. B, no. 2 (June 2003): 285–318.  Yourgrau, Wolfgang., and Stanley. Mandelstam. Variational Principles in Dynamics and Quantum Theory. New York: Pitman, 1960.