Table of Contents

1. Scientific Theories
2. Axiom Systems
3. Explanation
4. Postulates
   1. Examples
   2. Postulates of Newton’s Theory of Gravitation
5. Predictions
   1. A single refuted prediction disproves a theory
   2. Confirmed predictions support a theory
   3. Famous Scientific Predictions
6. Big Picture Graphics of Scientific Theories
   1. Newton’s Theory of Gravitation
   2. Mendel’s Theory of Heredity
   3. Einstein’s Special Relativity
7. Addenda
   1. Measurement
   2. Principle of Simplicity
   3. Mathematical and Scientific Theories
   4. Scientific Laws and Theories
   5. Hypotheses, Conjectures, and Theories
   6. Unscientific Hypotheses
   7. How Scientific Theories Developed
   8. Dictionary Definitions of ‘Theory’
   9. Dictionary Definitions of ‘Fact’
   10. Two Sides of Science

Scientific Theories

• A scientific theory is an axiom system
  • designed to explain certain kinds of phenomena
  • defined by its postulates
  • supported or disproved by its predictions
• A scientific theory is not an amorphous collection of ideas.

Axiom Systems

View Axiom System

Explanation

Scientific Theories are designed to explain certain kinds of phenomena

View Explanation

Postulates

Scientific theories are defined by their postulates.

Examples

Postulates of Newton’s Theory of Gravitation

• Law of Universal Gravitation
  • Informal: Things with mass attract each other
  • Precise: For any pair of physical bodies there’s a force on each, toward the other, with magnitude Gm₁m₂/r²
    • where G = 6.67384 10^-11 (the gravitational constant), m₁ and m₂ are the masses of the bodies, and r is the distance between them
  • Formula: F = Gm₁m₂/r²
• Equation of Motion
  • Informal: Forces makes things move
  • Precise: The acceleration A a physical body equals the net force F on the body divided by its mass M
    • Acceleration is the rate of change of velocity
    • Formula: F = MA
• Action and Reaction
  • Forces come in pairs, equal in magnitude and opposite in direction

Predictions

A scientific theory is supported or disproved by its predictions

A single refuted prediction disproves a theory

• A theory is disproved by the valid deductive argument-form modus tollens
  • If T then P
  • It’s false that P
  • Therefore, it’s false that T
• Example of modus tollens
  • If 2019 is a leap year, it’s (evenly) divisible by 4.
  • 2019 is not divisible by 4.
  • Therefore, 2019 is not a leap year.
• Disproof of Newton’s Theory of Gravitation
  • If Newton’s theory is true, Mercury’s orbit precesses 5557 seconds of arc per century
  • Mercury’s orbit does not precess 5557 seconds of arc per century.
  • Therefore, Newton’s theory is false.

Confirmed predictions support a theory

• A scientific theory is supported by its confirmed predictions.
• The more unexpected the truth of prediction (apart from the theory), the greater the support.
  • Unexpected predictions:
    • Electromagnetism’s prediction of Radio Waves
    • Special Relativity’s prediction of Time Dilation
    • Relativistic Quantum Mechanics’ prediction of Antimatter
    • Big Bang Theory’s prediction of CMB Radiation
    • General Relativity’s prediction of Gravitational Waves
• The greater the variety of verified predictions, the greater the support.
  • General Relativity’s prediction of
    • Orbits of Heavenly Bodies, Motion of Falling Bodies, Time Dilation, Gravitational Redshift, Gravity Waves, Deflection of Light

View Hypothetico-Deductive Support

Famous Scientific Predictions

• John Couch Adams in 1843 and Urbain Le Verrier in 1845 independently predicted the existence of Neptune, based on Newton’s theory of gravitation and the wayward orbit of Uranus.  Neptune was observed in 1846.
• James Clerk Maxwell in 1873 predicted the existence of electromagnetic waves, based on his theory of electromagnetism.   Heinrich Hertz used Maxwell’s equations to generate radio waves in 1887.
• Albert Einstein in 1915 predicted gravitational time dilation, based on his theory of General Relativity.  Hafele and Keating verified the prediction in 1971 using atomic clocks on board airplanes.
• Wolfgang Pauli in 1930 predicted the existence of the neutrino, based on the law of the conservation of energy and the process of beta decay.  The neutrino was detected in 1956.
• Paul Dirac in 1931 predicted the existence of antimatter, based on his theory of relativistic quantum mechanics.  Carl Anderson discovered the positron in 1932.
• Ralph Alpher and Robert Herman in 1948 predicted the existence of CMB Radiation, based on the Big Bang Theory.  Arno Penzias and Robert Wilson detected the CMB radiation in 1964. 
• In 1964 Peter Higgs and others predicted the Higgs Boson based on the Standard Model of Particle Physics. The particle was detected at the Large Hadron Collider at CERN in 2012

Big Picture Graphics of Scientific Theories

Newton’s Theory of Gravitation

View Newton’s Theory of Gravitation

Mendel’s Theory of Heredity

View Mendel’s Theory of Heredity

Einstein’s Special Relativity

View Special Relativity

Addenda

1. Dictionary Definitions of ‘Fact’
2. Dictionary Definitions of ‘Theory’
3. How Scientific Theories Developed
4. Hypotheses, Conjectures, and Theories
5. Mathematical and Scientific Theories
6. Measurement
7. Principle of Simplicity
8. Scientific Laws and Theories
9. Two Sides of Science
10. Unscientific Hypotheses

Measurement

• Measurement associates numbers with physical quantities: length, mass, volume, time, electric charge, velocity, location, distance, etc.
• Physics uses the International System of Units (SI), where:
  • mass is measured in kilograms, length in meters, time in seconds, energy in joules, temperature in kelvins, electric current in amperes, electric charge in coulombs, and force in newtons.
  • View International System of Units (SI), Fundamentally Revised
• Measurements are made relative to a reference frame, a system of abstract space and time coordinates imposed on a latticework of measuring rods and synchronized clocks.

Principle of Simplicity

• Principle of Simplicity
  • The simpler theory is more likely to be true, other things being equal.
• Occam’s Razor (Ockham’s Razor)
  • “Entities should not be multiplied beyond necessity.”
• Newton’s Law of Parsimony
  • “We are to admit no more causes of natural things than such as are both true and sufficient to explain their appearances.”
• Einstein
  • “The grand aim of science is to cover the greatest number of empirical facts by logical deduction from the smallest number of hypotheses or axioms.”
    • That is, the objective of a scientific theory is to predict the greatest range of empirical facts by deriving them from the smallest, simplest set of postulates
    • A scientific theory should predict the mostest from the leastist.

Mathematical and Scientific Theories

• Mathematical and scientific theories are both axiom systems.
• They differ in their subject matter: abstract entities vs physical phenomena.
• Theory of Arithmetic
  • Axioms
    • Zero is a natural number.
    • Every natural number has a successor in the natural numbers.
    • Zero is not the successor of any natural number.
    • If the successor of two natural numbers is the same, then the two original numbers are the same.
    • If a set contains zero and the successor of every number is in the set, then the set contains the natural numbers.
  • Theorems (logically derived from axioms)
    • For any natural numbers x, y, and z, x · (y + z) = (x · y) + (x · z)
• Newton’s Theory of Gravitation
  • Postulates
    • Law of Universal Gravitation
    • Equation of Motion
    • Action and Reaction
  • Predictions and Laws (logically derived from postulates)
    • Kepler’s Laws of Planetary Motion

Scientific Laws and Theories

• A scientific law (or law of nature) is an empirical generalization that holds universally and necessarily, for example:
  • Conservation of Energy
  • Kepler’s Laws of Planetary Motion
  • Heisenberg Uncertainty Principle
  • Law of Universal Gravitation
• By contrast, accidental generalizations just happen to be true, like all gold spheres are less than one mile in diameter. 

• Differences between Laws and Theories
  • A law must be true, whereas a theory can be false. If what was thought to be a law is disproven, it was never a law.  A disproved theory stays a theory.
  • A theory may have more than one postulate but a law is a single proposition.
• Some laws are postulates of a theory
  • For example, the Law of Universal Gravitation is a postulate of Newton’s Theory of Gravitation
• Some laws are generalizations derived from the postulates of a theory
  • For example Kepler’s Laws of Planetary Motion are derived from Newton’s Theory.
• Principle is another name for Law
  • For example, the Heisenberg Uncertainty Principle is derived from the postulates of Quantum Mechanics.

Hypotheses, Conjectures, and Theories

• A hypothesis is a “a tentative assumption made in order to draw out and test its logical or empirical consequences” (Merriam-Webster)
• A conjecture is a “hypothesis that has been formed by speculating, usually with little hard evidence.” (Wordnet)
• A theory typically begins as a conjecture or hypothesis.  As evidence accumulates, it ceases being a hypothesis and becomes a well-confirmed theory.  It’s an established fact if the evidence becomes overwhelming.

Unscientific Hypotheses

• Unscientific hypotheses are untestable:
  • Everything that happens is God’s will
  • People have souls
  • God created life
  • A one-celled human zygote is a human being
  • Nothing is known with certainty
  • Doctrine of Transubstantiation

How Scientific Theories Developed

• In the 5th century BC the Greeks developed theories of nature, for example:
  • Empedocles (5th Century BC)
    • Things: Earth, Air, Fire, Water
    • Forces: Love, Strife
      • Strife makes an element withdraw from the others.
      • Love makes the elements mingle.
        • britannica.com/biography/Empedocles
  • The theories generated after-the-fact explanations but no testable predictions
• In 1623 Galileo wrote that:
  • “The book of nature is written in the language of mathematics.”
• In 1687 Newton set forth his theory of gravitation, which made precise, testable predictions (thanks to the magic of differential equations).

Dictionary Definitions of ‘Theory’

• Body of principles designed to explain and predict phenomena
  • merriam-webster.com/dictionary/theory
    • A plausible or scientifically acceptable general principle or body of principles offered to explain phenomena
  • ahdictionary.com/word/search.html?q=theory
    • A set of statements or principles devised to explain a group of facts or phenomena, especially one that has been repeatedly tested or is widely accepted and can be used to make predictions about natural phenomena.
  • britannica.com/science/scientific-theory
    • A scientific theory may be characterized as a postulational system (a set of premises) from which empirical laws are deducible as theorems.
• Conjecture
  • merriam-webster.com/dictionary/theory
    • an unproved assumption : CONJECTURE
  • ahdictionary.com/word/search.html?q=theory
    • An assumption based on limited information or knowledge; a conjecture.

Dictionary Definitions of ‘Fact’

• Established truth
  • ahdictionary.com/word/search.html?q=fact
    • 2a. Something demonstrated to exist or known to have existed
      • Genetic engineering is now a fact. 
      • That Chaucer was a real person is an undisputed fact.
  • merriam-webster.com/dictionary/fact
    • a piece of information presented as having objective reality
      • These are the hard facts of the case.
• Actual state-of-affairs
  • ahdictionary.com/word/search.html?q=fact
    • 2b. A real occurrence, an event
      • We don’t yet know all the facts.
  • merriam-webster.com/dictionary/fact
    • : something that has actual existence
      • space exploration is now a fact

Two Sides of Science

• Empirical Side
  • Observation
    • Passive: Investigator doesn’t control observational conditions, e.g. astronomy
  • Experiment
    • Active: Investigator has control of experimental conditions, e.g. physics
  • Use of Math
    • Counting objects, e.g. pea plants with yellow seeds
    • Measuring quantities, e.g. time, length, distance, velocity, mass, electric charge
• Theoretical Side
  • Theory
    • System of postulates from which predictions are derived.
  • Use of Math
    • Formulating the postulates of a theory
    • Deriving predictions from a theory