Simplicity as a Surrogate

1. Simplicity as a Surrogate John D. Norton Department of History and Philosophy of Science Center for Philosophy of Science University of Pittsburgh Center for Philosophy of Science University of Pittsburgh, November 27, 2012.

2. The Claim of this Talk Simplicity is a surrogate for background facts or assumptions that warrant the relevant inductive inference. In so far as it has any epistemic power… Application of the material theory of induction to simplicity. Elliot Sober has been defending this view of simplicity for decades.

3. How it works

4. Bird Tracks What caused these tracks? One bird walking? Two coordinated one-legged birds hopping? Many one-legged birds touching down just once?

5. Simplicity as an Epistemic Criterion Rule I. We are to admit no more causes of natural things than such as are both true and sufficient to explain their appearances. To this purpose the philosophers say that Nature does nothing in vain, and more is in vain when less will serve; for Nature is pleased with simplicity, and affects not the pomp of superfluous causes. Rule II. Therefore to the same natural effects we must, as far as possible, assign the same causes. As to respiration in a man and in a beast; the descent of stones in Europe and in America; the light of our culinary fire and of the sun; the reflection of light in the earth, and in the planets. Isaac Newton, Rules of reasoning in philosophy “Nature is pleased with simplicity, and affects not the pomp of superfluous causes.” …ONE bird.

6. Bird Tracks again What caused these tracks? One bird walking a lot? Many birds birds each walking a little? …ONE bird? Rule II. Therefore to the same natural effects we must, as far as possible, assign the same causes. …MANY birds? (in ONE flock).

7. Background knowledge… …is what really decides, but we use simplicity talk to avoid having to explain lots of little details. ONE bird since we know that coordinated one-legged birds hopping are very rare. MANY birds in a flock since we know that single birds do not like to walk about a lot.

8. A Brief Farewell to theMetaphysics of Simplicity

9. Nature is Simple “…I would like to state a proposition that at present cannot be based upon anything more than upon a faith in the simplicity, i.e., intelligibility, of nature: there are no arbitrary constants of this kind…” Autobiographical Notes. Our experience hitherto justifies us in believing that nature is the realization of the simplest conceivable mathematical ideas.” On the Methods of Theoretical Physics, 1933.

10. Nature is NOT Simple. The term “simple” is vague. No single meaning broad enough to support a universal metaphysics of simplicity. Ontic simplicity: fewest entities. continuum gas molecular gas one entity 1023 entities infinitely many parts finitely many parts Aesthetic judgments of simplicity are made post hoc and reflect the achievement of comfort with a new theory. Descriptive simplicity. General relativity in 1920 General relativity in 1973 "...the complications of the theory of relativity are altogether too much...I fear it will always remain beyond my grasp..." Hale, 1920 “Einstein’s theory of gravity is simple; Newton’s is complex.” Misner, Thorne and Wheeler, 1973 Nature is NOT NOT Simple, either.

11. Curve Fitting

12. Hierarchy of Functions quartic cubic quadratic linear constant Choose the simplest that works. Real least squares fit to the data.

13. Data Compression Distinct projects The mark of truth My concern here. versus Present experimental data in a compact usable form. Most engineering uses of curve fitting. Search More efficient to check the simpler hypotheses first, independently of whether the truth is simple or not. Simplicity strips away confounding error noise to reveal truth. Simplicity is pragmatic, not an indicator of truth.

14. Background Assumptions make simplicity is a mark of truth. I.Error model holds Fails in data compression in engineering applications. There may be no true curve. error laden data true curve = + error density of primes in 0 to x y = Fails for density of primes error laden curve true data = + error

15. Background Assumptions make simplicity a mark of truth. Reparametrize 1, x, x2, x3, x4, x5, x6, x7, … II. The right parameterization is used. rescale z = x3 1, z, z2, … The right parametrization well-adapted to the true processes. III. Order hierarchy matches the strength, likelihood of processes, causes. For cyclic processes, first fit periodic function sin (t) = x – (1/3!) t3 + (1/5!) t5 - … before any finite order polynomial in t. Simplicity in curve fitting is a surrogate for these background assumptions.

16. II. and III. Combined. Reparameterize same data with z = sin-1x True curve y = sin z = z – (1/3!)z3 + (1/5!)z5 - … cannot be found in finite ascent of polynomial hierarchy. Data generated by true curve y=x

17. Curve FittingIllustrated

18. Fitting trajectories to planets, comets… Background assumptions Fit ellipse, hyperbola, parabola. (Not straight line.) Newton’s theory of gravity holds. Object deflected by sun. No other object exerts a perceptible deflecting force. There must be another object deflecting. 1846: successful prediction of Neptune for perturbations in Uranus. 1915: anomalous motion of Mercury explained by general relativity. Background assumption fails. Fit ellipse whose elements change with time. Advancing perihelion

19. Harmonic analysis of tides: the toy theory

20. Harmonic analysis of tides: the real theory Joe S. Depner, “Mathematical Description of Oceanic Tides,” 2012

21. Physical Basis of 37 Harmonic Constituents Fitted

22. Model Selection

23. Which Model? cubic quadratic linear constant Less simple models eventually perform better by overfitting = conforming to error noise.

24. Akaike Information Criterion cubic quadratic Which model? linear constant Unbiased estimator of average performance of fitted curve, distribution over all data sets Performance of fitted curve, distribution on particular data set at hand Dimension of model containing fitted curve, distribution - = inflated by overfitting (lack of) simplicity penalty “Performance” = log likelihood of data

25. Akaike Information Criterion The analysis could proceed without any overt talk of simplicity.We introduce it since we find it a comfortable way to describe Akaike’s very simple formula. No posit of simplicity or principle of parsimony is assumed. The bias correction follows from ordinary statistical modeling. No general principle of parsimony is derived. Results hold only for those systems presumed. is an imprecise surrogate for the precise procedure of bias correction. Simplicity description

26. Values, Virtues…

27. Accuracy, Consistency, Scope, Simplicity, Fruitfulness…Explanatory Power Are they properly called… Criteria? Virtues, values? for theory choice that might lead us to the truth of theories selected by the scientific community. Sought because they might lead us to the truth. Prized as ends in themselves. Virtues, values are endpoints of analysis. Whether they do this is a matter of further analysis. Virtues, values are agreed upon by social convention. Whether they do is imposed on us by the external world. “Virtues, values” encodes a skepticism that the criteria are not guides to the truth. “Criteria” is neutral.

28. Does the Difference Really Matter? “science and values” CRITERIA-BASED JUDGMENT “science, criteria for theory choice and ethical values” virtues, values “Objectivity, Value Judgment, and Theory Choice.” “Objectivity, Criteria-Based Judgment and Theory Choice.” criteria

29. Conclusion

30. The Claim of this Talk Simplicity is a surrogate for background facts or assumptions that warrant the relevant inductive inference. In so far as it has any epistemic power… Application of the material theory of induction to simplicity. Elliot Sober has been defending this view of simplicity for decades.