**Table of Contents**

- Scientific Theories
- Axiom Systems
- Explanation
- Postulates
- Predictions
- Big Picture Graphics of Scientific Theories
- Addenda

#### 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

- An
*axiom system*is a set of*axioms*(or*postulates*) from which*theorems*(or predictions) are logically derived. - The classic example is the system of Plane Geometry in Euclid’s Elements, consisting of:
- Axioms / Postulates
- Theorems
- Proofs of Theorems.

View Axiom System

#### Explanation

Scientific Theories are designed to explain certain kinds of phenomena

**Newton’s Theory of Gravitation**- Orbits of heavenly bodies
- Motion of falling bodies

**Mendel’s Theory of Heredity**- How certain characteristics of pea plants are transmitted to their offspring

**Quantum Mechanics**- Interactions of atoms and subatomic particles with one another and with electromagnetic radiation.

**Einstein’s Special Relativity**- Why the observed speed of light is the same, regardless of the relative motion sender and receiver.

View Explanation

#### Postulates

Scientific theories are defined by their postulates.

###### Examples

**Newton’s Theory of Gravitation**- Law of Universal Gravitation
- Equation of Motion
- Law of Action and Reaction

**Mendel’s Theory of Heredity**- Determination Postulate
- Inheritance Postulate

**Darwin’s Theory of Evolution**- Common Descent with Modification

**Einstein’s General Relativity**- Field Equation
- Geodesic Postulate

**Theory of Electromagnetism**- Maxwell’s Equations
- Lorentz Force Law
- Equation of Motion

**Quantum Mechanics**- State/Vector Representation
- Observable/Operator Representation
- Time Evolution (Schrodinger Equation)
- Prediction
- Collapse

**Einstein’s Special Relativity**- Principle of Relativity
- Constancy of Speed of Light

**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
**Gm**_{1}**m**_{2}**/r**^{2}- where
**G**= 6.67384 10^{-11}(the gravitational constant),**m**and_{1}**m**are the masses of the bodies, and_{2}**r**is the distance between them

- where
- Formula:
**F = Gm**_{1}**m**_{2}**/r**^{2}

**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

- Unexpected predictions:
- 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

- General Relativity’s prediction of

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* - 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**

**Einstein’s Special Relativity**

View Special Relativity

#### Addenda

- Dictionary Definitions of ‘Fact’
- Dictionary Definitions of ‘Theory’
- How Scientific Theories Developed
- Hypotheses, Conjectures, and Theories
- Mathematical and Scientific Theories
- Measurement
- Principle of Simplicity
- Scientific Laws and Theories
- Two Sides of Science
- 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

- mass is measured in
- 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.

- “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.”

##### 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)

- Axioms
- 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

- Postulates

##### 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

- For example, the
- 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.

- For example Kepler’s
*Principle*is another name for*Law*- For example, the
*Heisenberg Uncertainty Principle*is derived from the postulates of Quantum Mechanics.

- For example, the

##### 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.

- The theories generated after-the-fact explanations but no testable predictions

- Empedocles (5th Century BC)
- 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.

- merriam-webster.com/dictionary/theory
*Conjecture*- merriam-webster.com/dictionary/theory
- an unproved assumption
**:**CONJECTURE

- an unproved assumption
- ahdictionary.com/word/search.html?q=theory
- An assumption based on limited information or knowledge; a conjecture.

- merriam-webster.com/dictionary/theory

##### 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.*

- 2a. Something demonstrated to exist or known to have existed

- merriam-webster.com/dictionary/fact
- a piece of information presented as having objective reality
*These are the hard facts of the case.*

- a piece of information presented as having objective reality

- ahdictionary.com/word/search.html?q=fact
*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.*

- 2b. A real occurrence, an event
- merriam-webster.com/dictionary/fact
**:**something that has actual existence*space exploration is now a fact*

- ahdictionary.com/word/search.html?q=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

- Observation
*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

- Theory