Lecture 7 - Exactness & Gene


Department
History and Philosophy of Science and Technology
Course Code
HPS211H1
Professor
Chen- Pang Yeang
Lecture
7

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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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