Number
What is philosophy and why it matters
Philosophy (big picture): understanding how things hang together in the broadest possible sense, i.e., the aim of understanding connections across domains (Sellars, 1963).
Two central images in philosophy:
Manifest image: ordinary, observable world (tables, chairs, trees, stones, people).
Scientific image: world as described by theoretical/unobservable entities (quarks, dark matter, black holes, genes, tectonic plates).
The philosophical task: reconcile the manifest image with the scientific image and relate scientific theory to human purposes.
Big History and the aim of big philosophy
Big history: tracing history from the Big Bang to the present.
Big = Philosophy + Big history: a composite view of the evolving universe.
Influences on big philosophy include: theoretical biology, metabolic theory of ecology, systems theory, complexity science, cybernetics, physics, theoretical neuroscience, information theory, statistics, machine learning, information philosophy, emergentism, and more.
The aim: develop a robust and coherent understanding of the world and our place in it through the lens of big philosophy.
The two images in Sellars’ framework (more detail)
Manifest image: common-sense world we observe.
Scientific image: world as described by science, including unobservables.
Task: relate the scientific world to our purposes and make it our world.
The Two Tables Paradox (Eddington)
The Nature of the Physical World (Eddington, 1928) introduces a paradox: a manifest table vs. a scientific table.
The question: How do we reconcile the manifest image of a table with the scientific image of the table, which might be mostly emptiness with charges moving at high speed?
Promise for LECTURE 2: More on the manifest image, the scientific image, and the two tables paradox.
Pythagoras and the mathematical worldview
Pythagoras: Everything is made of numbers; the world is ordered numerically.
Plato/Natural order: The world is made of numbers and the order is beautiful and symmetric.
Wilczek’s view: Numbers and symmetry underlie natural patterns and observable structures.
Pythagorean Theorem (classic result)
The area-based statement: The area of the square on the hypotenuse equals the sum of the areas of the squares on the other two sides for a right triangle.
In equation form: a^2 + b^2 = c^2 where a and b are the legs and c is the hypotenuse.
A standard geometric proof (Glynn, 2013) outline:
Start with a square of side length c containing four identical right triangles with legs a and b, and a central square with side length ??? (diagram-based reasoning).
Rearrange the four triangles to form a large square with area equal to the sum of the areas of the two smaller squares, yielding a^2 + b^2 = c^2 (QED).
Significance: Demonstrates the connection between geometry and algebra; foundational for understanding physical patterns and spatial relationships.
Pythagorean tuning and musical ratios
Pythagorean tuning relates musical intervals to simple frequency ratios:
Octave: length ratio \frac{1}{2} corresponds to frequency ratio 2:1 (two strings at same tension with half-length produce an octave).
Perfect fifth: ratio \frac{2}{3} in length corresponds to a frequency ratio 3:2.
Perfect fourth: ratio \frac{3}{4} in length corresponds to a frequency ratio 4:3.
Intuition: Both geometry (numbers) and music reveal same numerical relationships in physical systems.
Regular polygons and Platonic solids
Regular n-sided polygon: planar figure with all sides equal and all interior angles equal.
Examples: n = 3 (equilateral triangle), n = 4 (square), n = 5 (regular pentagon).
Interior angle of a regular n-gon:
\theta = \dfrac{(n-2) \cdot 180^{\circ}}{n}
For antipodal vertex arrangements in 3D solids (convex polyhedra), at each vertex the sum of face angles must be less than 360°; this constrains possible combinations of (n, m) where m is the number of faces meeting at a vertex.
Platonic solids: solids with identical regular faces and the same number of faces meeting at every vertex.
The five Platonic solids and their standard combinatorial data:
Tetrahedron: 4 faces (all equilateral triangles); exactly 3 faces meet at each vertex.
Cube (hexahedron): 6 faces (all squares); exactly 3 faces meet at each vertex.
Octahedron: 8 faces (all equilateral triangles); exactly 4 faces meet at each vertex.
Dodecahedron: 12 faces (all regular pentagons); exactly 3 faces meet at each vertex.
Icosahedron: 20 faces (all equilateral triangles); exactly 5 faces meet at each vertex.
Theorem: There are exactly five Platonic solids (Euclid, c. 300 BCE).
Sketch of the proof (logic):
For a regular n-gon face and m faces meeting at a vertex, interior angle θ = \frac{(n-2) \cdot 180}{n}.
At a convex vertex, m θ < 360°.
Solve for integer pairs (n, m) with n, m ≥ 3 that satisfy the inequality; the valid pairs are:
(n, m) ∈ { (3,3), (3,4), (3,5), (4,3), (5,3) }.
These correspond to the Tetrahedron, Octahedron, Icosahedron, Cube, and Dodecahedron respectively (QED).
Platonic solids in relation to classical elements (Timaeus):
In Plato's dialogue Timaeus, the philosopher associates each of the five Platonic solids with a classical element, providing a geometric explanation for the physical world's structure.
Fire → tetrahedron, Earth → cube, Air → octahedron, Water → icosahedron.
Cosmos (dodecahedron) associated with the cosmos in Plato’s scheme.
Kepler’s nesting model and celestial geometry
Kepler’s nesting idea (1596) combines Copernican heliocentrism with Platonic idealism.
Model: Six planets (Mercury, Venus, Earth, Mars, Jupiter, Saturn) orbit the Sun with nested spheres.
Nesting scheme: Each pair of spheres is separated by a Platonic solid, such that one solid fits between two spheres.
Kepler’s nesting order (from inner to outer): octahedron → icosahedron → dodecahedron → tetrahedron → cube.
Location mapping (examples from Kepler's Mysterium Cosmographicum): The octahedron fits between the spheres of Mercury and Venus; the icosahedron between Venus and Earth; the dodecahedron between Earth and Mars; the tetrahedron between Mars and Jupiter; and the cube between Jupiter and Saturn.