Department

History and Philosophy of Science and TechnologyCourse Code

HPS211H1Professor

Chen- Pang YeangLecture

7This

**preview**shows page 1. to view the full**5 pages of the document.**HPS211 Lecture 7 JUN05/2011

Exactness

- Natural philosophers had used numerical values/information in disciplines like astronomy and navigation

- Quality of measurement and calculations has longstanding issues especially in astronomy

- Concerned with the precision, accuracy and exactness of their calculation

- Issues became pressing in the 18th c.

- Context: sprit of quantification

I. Spirit of Quantification

- Newtonian mechanics and physical astronomy

Closer fit between data and prediction

Lunar table, orbits of Saturn and Jupiter, figure of earth

Navigation, map-making, and exploration

- Quantitative experiments in chemistry, optics, pneumatics, hydraulics, elasticity these experiments were

imposed by scientists in the 18th c.)

- Quantitative administration of state

Impose the cause of reason into government administration utilize statistical/numerical data

Demography, forestry, metric units, standardizing manufacturing

- Commerce and trade with more exact numbers

Accounting, actuaries, insurance business

- Questions with Numbers

Which numbers are more trustworthy?

How to improve the fit between data and theory?

How to make results of measurement or calculation consistent?

In particular: Observers, measurers, or experiments often obtained multiple sets of data under same

condition how to best represent the data?

II. The Single Best Measurement

- Dominant Approach: Single Best Measurement

Make multiple measurements, but only present the “best” data set

Epistemic and moral implications:

There was a “perfect” figure for a measurement, a “golden event” for experiment or

observation

The perfect figure was attainable by improving instruments or skills. No fundamental reason to

prevent the acquisition of the perfect figures

The failure to obtain the perfect figure was thus measurer’s “fault” symptom of laziness,

carelessness, incompetence

Artisan’s mentality

- Lavoisier represented an example of “best measurement” approach

Used weight balance to demonstrate that hydrogen and oxygen compose water (1780s)

Precise measurements of hydrogen, oxygen, and water (8 decimal figures)

- But Lavoisier’s “precision” did not convince phlogiston chemists

William Nicholson estimated that Lavoisier’s instruments were only sensitive within 3 decimal figures.

Extra digits were meaningless

In other words, Lavoisier lacked a concept of significant figures and a theory of errors

- Alternative Approach Averaging

Do not choose one data set among multiple sets, combine all sets

Dates back to Newton

Became more popular in the late 18th century

Breakthrough with the introduction of the least squares method

III. Least Squares Method

- Least square method was invented by French mathematician Adrien-Marie Legendre in 1805

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- Used to determine empirical formulas for orbits of comets from observational data

- Legendre’s Formulation of the Problem

Suppose a celestial body’s N measurable quantities are governed by an equation with

N+1 parameters

M (>N+1) measured data sets for :

Question: How to determine from

- Issues of Over-Determination

M equations

… … …

But only N+1 variables

- Legendre’s Least Squares Solution

Measured data ( ),( )… ( ) have errors; they do not always

follow the equation:

… … …

But the errors … should be small

Determine by minimizing sum squares of errors

Theorem in calculus: function is minimum when derivative is 0

N+1 linear equations, N+1 variables

Solved by standard algebraic procedure

IV. Probabilistic Theory of Errors

- Carl Friedrich Gauss (Göttingen) and Pierre Simon Laplace (Paris)

- Probabilistic Reframing of Least Squares Method (1809-27)

Employed probability theory

The errors …are random

The least squares estimate has maximum probability when the probability distribution of … follows

“normal” (Gaussian) distribution

- Theory of Errors in Measurements

First prevailed in astronomy and geodesy

Later applied to experimental science

Especially popular in German states: Gauss & Weber in Göttingen, Franz Neumann in Königsberg

(Kalinningrad)

Impossible to achieve “perfect figures”

Errors were not symptoms of incompetence, but unavoidable part of experiment/observation

What mattered was statistical distributions of errors

- Errors, Accuracy, Precision

Two types of errors in measurements

(1) Constant (systematic) error systematic bias

(2) Accidental (random) error random fluctuation

Accuracy: minimum system bias

Precision: minimum random fluctuation

Improve accuracy by eliminating biases in instruments and procedures

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