Theory and Practice of Science
Science C1001-1002, Columbia College
Syllabus

"Theory and Practice of Science" was developed at Columbia College, by Professors Herbert Goldstein, Jonathan L. Gross, and Robert E. Pollack, and first taught in 1981. The two-semester course introduced sufficient mathematics so that original science papers could be used as the basis for study, and presented case studies of scientific practice in the context of general education.

The following is a syllabus of the course as it was taught by Goldstein, Gross, Pollack, Blumberg and others, in the late 1980s. This may also be considered a table of contents for the textbook for the course, written by Goldstein, Gross, Pollack and Blumberg, entitled The Scientific Experience. The book currently exists in xeroxed form, and the chapter entitled Measurement is available at MendelWeb. For more information about the course see the WWW94 Conference paper on MendelWeb. For the current status of the course at Columbia, see the Columbia College Bulletin (listed under "Colloquia", under "Program and Degree Requirements"). For information about the textbook, write to

Semester 1

Mathematics I: The Foundations of Quantification

1. Numbers and Their Representations
1.1 Integers (positional and non-positional notations)
1.2 Rational fractions
1.3 Closing the analytic gaps (sequences, differences, and convergence)
1.4 Algebraic completion of the number system


2. Computation

2.1 Algorithmic languages and computer programs
2.2 Computational paradigms (iteration and recursion)
2.3 Efficient information processing (search algorithms)
2.4 The Kolmogorov definition of randomness


3. Measurement

3.1 Nominal, ordinal, and interval scales
3.2 What observations can you trust? (reliability, accuracy, precision, validity)
3.3 From observation to prediction: the role of models
3.4 Obstructions to measurement


4. Mathematical Proof

4.1 Preliminary remarks about arguments
4.2 The role of symbolic logic
4.3 The induction principle
4.4 Different sizes of infinity
4.5 The existence of random sequences

Mathematics II: Deterministic Patterns and Models

5. Finding Patterns in Perfect Data
5.1 The problem of extending a sequence
5.2 Iterating the difference operation (closed formulas and recursion formulas)
5.3 Polynomial functions eventually yield constant rows
5.4 Reconstructing a polynomial function from a sequence
5.5 Difference tables for exponential functions (iterated rows or row multiples)


6. Recovering Patterns from Flawed Data

6.1 Linear interpolation
6.2 Difference tables for approximate sequences (building chains
of hypothesis and confirming them)
6.3 Explosion of random error under difference operations
6.4 Disguised patterns


7. Discrete Systems

7.1 Sets and orderings
7.2 Graphs and networks
7.3 Permutations and combinations


8. Continuous Systems

8.1 Differential calculus and slopes of curves (the analogy with the difference operation)
8.2 Derivative formulas for polynomials and exponentials
8.3 Integral calculus and areas under curves
8.4 Integrals as antiderivatives
8.5 Differential equations

Topics in Modern Physics

9. The Construction of the Physical Sciences
9.1 Models of the practice of science
9.2 Black Boxes, models, and theories
9.3 More on basic theories: Maxwell's theory as an example
9.4 Derivation of the wavelength-frequency formula


10. Discovery of the Electron: Faraday to Thomson

10.1 Faraday and electrolysis
10.2 The atomic theory, and Faraday's model of electrolysis
10.3 The reception given the atomic model of the elements and of electricity
10.4 Enter cathode rays
10.5 The measurement of e/m
Paper: Researches in Electricity, by Michael Faraday [1833-34] (excerpts)
Paper: "Cathode Rays" by J.J. Thomson (1897)


11. Electrons Everywhere

11.1 Electrons in atoms: spectroscopy
11.2 Electrons from the photoelectric effect
11.3 Electrons and the thermionic effect
11.4 Other sources and properties of electrons
Paper: "On the Influence of Magnetism on the Nature of the Light emitted by a Substance," by Dr. P. Zeeman (1897)


12. Discovery of Radioactive Elements

12.1 Fin-de-Siecle: X-rays and radioactivity
12.2 The Curies and new radioactive elements
12.3 The Curies technique in hunting for new elements
12.4 The Curies' paper announcing the discovery of radium
Paper: "On a New, Strongly Radio-active Substance Contained in Pitchblende," by P. Curie, Mme. Curie, and G. Bemont (1898)


13. Radioactive Transformations

13.1 The multiplication of radioactive species
13.2 The transformation model of radioactivity
13.3 The natural radioactivity decay chains
13.4 Transformation theory of decay chains
13.5 Exploration of other properties of radioactivity
Paper: "A Radioactive Substance emitted from Thorium Compounds," by E. Rutherford (1900).


14. Models of the Atom

14.1 How many electrons in the atom?
14.2 Thomson's plum-pudding model of the atom
14.3 The Rutherford model of the atom
14.4 Bohr extends the Rutherford model
Paper: "The Structure of the Atom," by Sir Ernest Rutherford (1914)


15. The Nuclear Atom and the Systematics of the Elements

15.1 Moseley and the numbering of the elements
15.2 The periodic table revisited
15.3 The radioactive elements are put in their places
15.4 Tidying up the details of the Bohr-Rutherford picture
15.5 Reflections in the Rutherford models of the atom and the nucleus
Paper: "Intra-atomic Charge," by Frederick Soddy (1913)


16. Nuclear Physics is Born and Promptly Languishes

16.1 Rutherford gets inside the nucleus
16.2 Nuclear physics languishes
16.3 Some advances in the rest of physics
16.4 Meanwhile, what of nuclear physics?
Paper: "Collision of α Particles with Light Atoms. IV. An Anomalous Effect in Nitrogen," by Professor Sir E. Rutherford (1919)


17. Anni Mirabiles

17.1 New tools
17.2 New discoveries
17.3 Discovery of the neutron, 1932
17.4 Artificial radioactivity
Paper: "A New Type of Radioactivity," by Irene Curie and Frederic Joliot (1934)
17.5 Model building
Paper: ""Possible Existence of a Neutron," by J. Chadwick (1932)


18. Enter Fermi and His Gang

18.1 Fermi and his gang start to bombard with neutrons
18.2 The bombardment of uranium and thorium with neutrons
18.3 Reactions to the discovery of the transuranics
Paper: "Possible Production of Elements of Atomic Number Higher than 92," by E. Fermi (1934)
Paper: "Artificial Radioactivity produced by Neutron Bombardment," by E. Fermi, E. Amaldi, O. D'Agostino, F. Rasetti, and E. Segré (1934)


19. The Scene Shifts to Germany and France

19.1 Chains of isomers
19.2 The experiments of Irene Curie and Paul Savitch
19.3 The final paper of Hahn and Strassman
Paper: "Concerning the Existence of Alkaline Earth Metals Resulting from Neutron Irradiation of Uranium," by O. Hahn and F. Strassman (1938)


20. Nuclear Models, Old and New

20.1 The old picture: the independent single particle model
20.2 The new picture: the liquid drop model
Paper: Radiations from Radioactive Substances (excerpt), by Ernest Rutherford, James Chadwick, and C.D. Ellis (1930)


21. Nuclear Fission Explained and Verified

21.1 The Meitner-Frisch letter to Nature
21.2 The Frisch experiment
21.3 The properties of neutron induced nuclear fission
21.3.1 The conditions for nuclear fission
21.3.2 Modes of fission
21.3.3 Radioactive decay of fission products
21.3.4 The transuranics
Paper: "Disintegration of Uranium by Neutrons: a New Type of Nuclear Reaction," by Lise Meitner and O.R. Frisch (1939)
Paper: Physical Evidence for the Division of Heavy Nuclei under Neutron Bombardment," by O.R. Frisch (1939)


22. Aftermath

22.1 How they brought the news from Copenhagen
22.2 Paris and nuclear energy
22.3 The locust effect
22.4 The veil of secrecy descends
22.5 Final assessment
Paper: "The Mechanism of Nuclear Fission" (excerpt), by Niels Bohr and John Archibald Wheeler (1939)


Semester 2

Mathematics I: Probabilistic Models

1. Probability Measures
1.1 Finite and infinite outcome spaces
1.2 Uniform and non-uniform likelihood
1.3 Counting uniformly likely cases
1.4 Iterated independent trials: binomial distributions


2. Inference and Decisions

2.1 Conditional probability
2.2 More conditional probability
2.3 Random variables and their expected values
2.4 Sum of independent random variables


3. Game Theory: a case study

3.1 How to play matrix games: simple and mixed strategies
3.2 Optimization by graphical representation of strategies
3.3 Eliminating dominated strategies


4. Evaluating Information: a case study

4.1 Shannon measure of information by bits
4.2 Application to a priori non-uniform distributions
4.3 Non-additivity of information value


Mathematics II: Statistical Analysis

5. Sampling
5.1 Hypothesis testing: two sorts of errors
5.2 Biased and unbiased samples


6. Variance and Plausibility

6.1 Standard deviation
6.2 Chebyshev's inequality
6.3 The benefit of increasing the number of trials


7. The Normal Distribution

7.1 How to use a normal distribution table
7.2 Normal approximation of the binomial distribution
7.3 What events are unlikely?


8. Pattern Recognition and Verification: a case study

8.1 Periodic sequences
8.2 How a sequence becomes distorted
8.3 Frequency analysis and decimations
8.4 Digraphic and trigraphic analysis
8.5 Redundancy of information and data recovery
8.6 Competing models


Topics in Modern Biology

9. The Use of Numbers in Science
9.1 Significant figures
9.2 The central limit theorem
9.3 The [[radical]]n rule
9.4 A map of living things
9.5 Complexity and size


10. Charles Darwin, and The Origin of Species (1859)

10.1 The diversity of living things
10.2 Darwin's axioms
10.3 Darwin's hypothesis
10.4 The origin of species
10.5 Speciation and selection
10.6 Comments on Origin of Species
10.7 What Darwin didn't say


11. Gregor Mendel (1865)

11.1 The garden pea
11.2 The crosses
11.3 The types of dominance and the question of ratios
11.4 The model
11.5 Testing the prediction of independent assortment
11.6 Phenotype and genotype
11.7 The last crosses confirm the model
11.8 A look ahead
Paper: "Experiments in Plant Hybridization" by Gregor Mendel (1865)


12. Cells, Chromosomes and Genes

12.1 A chemical for inheritance: a new idea in science
12.2 Chromosomes and evolution
12.3 Microscopes and the universality of cells
12.4 The predictable patterns of inheritance
12.5 Mitosis and meiosis
12.6 Reading the chromosome: a look ahead


13. Thomas Hunt Morgan (1910)

13.1 A new system of breeding: the fruit fly
13.2 Morgan's assumptions and results
13.3 The backcross
13.4 The model
Paper: "Sex Limited Inheritance in Drosophila", by T. H. Morgan (1910).


14. Alfred Henry Sturtevant (1913)

14.1 A second trait on the X chromosome
14.2 A background to Sturtevant's paper
14.3 Mapping the chromosome
14.4 The model of inheritance after Morgan and Sturtevant
Paper: "The Linear Arrangement of Six Sex-linked Factors in Drosophila, as Shown by their Mode of Association", by A. H. Sturtevant (1913)


15. Large Molecules

15.1 Building molecules: an introduction to chemical bonds
15.2 Information in molecules
15.3 Seeing molecules?
15.4 Macromolecules
15.5 Crystals and three-dimensional visualization
15.6 The cell in 1945-50


16. Avery, MacLeod, and McCarty (1944)

16.1 A bioassay for the genetic material
16.2 Virulence as a bacterial allele
16.3 Transformation in a tube
16.4 Antibody selection: a bioassay for transformation
16.5 Properties of the transforming principle: conclusions of the paper
Paper: "Studies on the Chemical Nature of the Substance Inducing Transformation of Pneumococcal Types", by O.T. Avery, C. M. MacLeod, and Maclyn McCarty (1944)


17. Erwin Schrödinger's What is Life? (1944)

17.1 Drosophila salivary chromosomes: a missed connection
17.2 What is life? -- An aperiodic crystal
17.3 X-raying the gene
17.4 Schrödinger's epilogue


18. Joshua and Esther Lederberg (1952)

18.1 Selective systems
18.2 The life of a bacterium
18.3 Pre-adaptation versus directed mutation
18.4 Replica plating and the Lederbergs' results
18.5 Streptomycin resistance
Paper: "Replica Plating and Indirect Selection of Bacterial Mutants" by Joshua Lederberg and Esther M. Lederberg (1952)

19. James Watson and Francis Crick (1953)

19.1 The bases of DNA
19.2 X-ray crystallography and what the biochemists knew
19.3 The first paper by Watson and Crick
19.4 The second paper
Paper: "Molecular Structure of Nucleic Acids", by J. D. Watson and F. H. C. Crick (1953)
Paper: "The Structure of DNA" by J. D. Watson and F. H. C. Crick (1953)


20. James Watson's The Double Helix (1958)

20.1 Some comments on Watson's memoir
20.2 More comments: pages 100-114
20.3 Storing information


21. Crick, Barnett, Brenner, and Watts-Tobin (1961)

21.1 DNA and protein" each needs the other
21.2 Decoding DNA into protein
21.3 The T4 bacterial virus
21.4 The genetic code
21.5 DNA evolving
Paper: "General Nature of the Genetic Code for Protein", by Dr. F. H. C. Crick, F. R. S. Leslie Barnett, Dr. S. Brenner and Dr. R. J. Watts-Tobin (1961)


22. Epilogue: Molecular Biology Today

22.1 The state of molecular biology
22.2 The future of science