Simplicity as a Surrogate

John D. Norton

Department of History and Philosophy of Science

Center for Philosophy of Science

University of Pittsburgh

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Center for Philosophy of ScienceUniversity of Pittsburgh, November 27, 2012.

The Claim of this Talk

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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.

How it works

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Bird Tracks

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What caused these tracks?

One bird walking?

Two coordinated one-legged birds hopping?

Many one-legged birds touching down just once?

“Nature is pleased with simplicity, and affects not the pomp of superfluous causes.”

Simplicity as an Epistemic Criterion

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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

…ONE bird.

Bird Tracks again

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What caused these tracks?

Rule II. Therefore to the same natural effects we must, as far as possible, assign the same causes.

…ONE bird?…MANY birds?

(in ONE flock).

One bird walking a lot?

Many birds birds each walking a little?

Background knowledge…

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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.

…is what really decides,

but we use simplicity talk

to avoid having to

explain lots of little details.

A Brief Farewell to the

Metaphysics of Simplicity

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Nature is Simple

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“…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.

Nature is NOT Simple.

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Nature is NOT NOT Simple, either.

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

“Einstein’s theory of gravity is simple; Newton’s is complex.”Misner, Thorne and Wheeler, 1973

"...the complications of the theory of relativity are altogether too much...I fear it will always remain beyond my grasp..."Hale, 1920

General relativity in 1973

Curve Fitting

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Hierarchy of Functions

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constant

linear

quadratic

quartic

cubic

Choose the simplest that works.Real least squares fit to the data.

Distinct projects

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Data Compression

Present experimental data in a compact usable form. Most engineering uses of curve fitting.

versus

The mark of truth

Simplicity strips away confounding error noise to reveal truth.

My concern here.

Simplicity is pragmatic,not an indicator of truth.

Search

More efficient to check the simpler hypotheses first, independently of whether the truth is simple or not.

Background Assumptions make simplicity is a mark of truth.

Fails in data compression in engineering applications.There may be no true curve.

I. Error model holdserror laden data

= true curve

+ error

Fails for density of primes

true data

=error laden curve

+ error

density of

primes in 0 to x

y =

Background Assumptions make simplicity a mark of truth.

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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.

II. The right parameterization is used.

1, x, x2, x3, x4, x5, x6, x7, …

rescale z = x3

1, z, z2, …

The right parametrization well-adapted to the true processes.

Reparametrize

Simplicity in curve fitting is a surrogate for these background assumptions.

II. and III. Combined.

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Data generated by true curve y=x

True curvey = sin z = z – (1/3!)z3 + (1/5!)z5 - …cannot be found in finite ascent of polynomial hierarchy.

Reparameterizesame data withz = sin-1x

Curve FittingIllustrated

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Fitting trajectories to planets, comets…

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Newton’s theory of gravity holds.Object deflected by sun.No other object exerts a perceptible deflecting force.

Fit ellipse, hyperbola, parabola.(Not straight line.)

Background assumptions

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

Harmonic analysis of tides: the toy theory

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Harmonic analysis of tides: the real theory

20Joe S. Depner, “Mathematical Description of Oceanic Tides,” 2012

Physical Basis of 37 Harmonic Constituents Fitted

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Model Selection

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Which Model?

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constant

linear

quadraticcubic

Less simple models eventually perform better by overfitting

= conforming to error noise.

Akaike Information Criterion

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Which model?

constant

linear

quadraticcubic

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

“Performance” = log likelihood of data

inflated by overfitting

(lack of) simplicity penalty

Akaike Information Criterion

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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.

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.

Simplicity description

is an imprecise surrogate

for

the precise procedure of bias correction.

Values, Virtues…

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Accuracy, Consistency, Scope, Simplicity, Fruitfulness…Explanatory Power

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Are they properly called…

Criteria? for theory choice that might lead us to the truth

Virtues, values?

Sought because they might lead us to the truth.

Prized as ends in themselves.

Whether they do this is a matter of further analysis.

Virtues, values are endpoints of analysis.

Whether they do is imposed on us by the external world.

Virtues, values are agreed upon by social convention.

of theories selected by the scientific community.

“Virtues, values” encodes a skepticism that the criteria are not guides to the truth.

“Criteria” is neutral.

Does the Difference Really Matter?

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virtues, values

criteria

“science and values”

“science,criteria for theory choice

and ethical values”

“Objectivity, Value Judgment, and

Theory Choice.”

“Objectivity, Criteria-Based Judgment and

Theory Choice.”

CRITERIA-BASED JUDGMENT

Conclusion

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The Claim of this Talk

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Simplicity is a surrogate for background facts or assumptions that warrant the relevant inductive inference.

Application of the material theory of induction to simplicity.

Elliot Sober has been defending this view of simplicity for decades.

In so far as it has any epistemic power…

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