concepts in mathematics

concepts in mathematics

Geocentric

Based on Aristotelian logic, Dominated pre-modern cosmology; challenged by heliocentrism, Sun-centered model – Proposed by Copernicus, confirmed by Galileo and Kepler – Reoriented the cosmos, dethroned Earth, and sparked the Scientific Revolution.marked the fall of philosophy over evidence.

Heliocentric

Sun-centered model, Copernicus, confirmed by Galileo and Kepler, Reoriented the cosmos, dethroned Earth, and demanded new physics/tools to find “truth”

Kepler’s Laws

Math-based laws of planetary motion, elliptical orbit (not circular) – Marked the shift from philosophy to empirical truth.

Galileo

Father of experimental science, quantitative measurement – telescope disproved geocentrism – observation > doctrine; laid groundwork for physics (relativity

Copernicus

Father of heliocentricm, dethroned humanity, empirical knowledge > doctrine/philosophy, massive shift in universal understanding

Kepler

calculated planetary orbit to be elliptical, empirical knowledge > philosophical/cultural narrative/doctrine

Aristotelian logic

logic system based in syllogism and deductive reasoning, FIXED TRUTHS, abstraction of the physical/given knowledge

Galilean relativity

all laws of motion are applicable in inertial (uniform) frames of reference, early basis for relativity

renaissance perspective

mathematicized optics, illusion of depth using vanishing points, visualized space as predictable, rational (measurable)

Leonardo/leonardo’s designs

empirical, rational, measure-based drawings, blending art+science/math

Binocular vision HELP

2-eye depth perception, 2 eyes to create one unified vision, renaissance obsession with understanding how we visualize optical space;

Newtonian logic

predictable, rational, certain (fails at quantum, relativistic scales)

Classical physics

based on newtonian logic; predictable, rational, certain (fails at quantum, relativistic scales) moves us into Einstein relativity

Laws of motion

Newtonian laws, unified celestial/terrestrial mechanics to predict motion and forces in the world

Law of inertia

The natural state of motion is continuous. either at rest or in the straightest line possible unless a force is acted upon it

f=ma

second law of motion by Newton; an object’s mass and acceleration determine it’s force, allows us to predict motion given a force,

Equal and opposite force

explains how forces interact with each other, forces work in pairs, for every action; equal opposite reactionewtoN

Newton’s 1st law of motion

Law of inertia, The natural state of motion is continuous. either at rest or in the straightest line possible unless a force is acted upon it

Newton;s 2nd law of motion

f=ma, allows us to predict motion given a force,

Newton’s 3rd law of motion

for every action, there’s an equal opposite reaction, forces work in pairs. establishes how forces interact with each other constantly

Universal gravitation

constant invisible attraction between masses, inversely proportional to the square of the distance between them, unifies terrestrial and celestial mechanics

Relativity

no absolute truths in space and time; everything is relative to the observer's motion, revolutionized our understanding of space, time, and the universe

Speed of light

Maxwell says is constant ( c), central to both special and general relativity, fundamental limit in physics, defied classical newtonian physics, led to special relativity

Maxwell’s equations

unified electricity and magnetism (like Newton unified celestial and terrestrial mechanics) Electromagnetic fields act as waves, Light is an electromagnetic wave, not just a particle or a force, SPEED OF LIGHT IS CONSTANT

Ether Wind Hypothesis

if the speed of light is always constant, there must be a medium for light to propagate through (ether), disproven by Michelson Morley Experiment

Michelson Morley Experiment

Detect the ether wind, Earth’s motion through space SHOULD cause change interference patterns
No ether wind was detected, confirmed constant speed of light, regardless of Earth’s motion/ time of year, led Einstein to special relativity

Special relativity

Einstein, space and time are RELATIVE, speed of light IS CONSTANT; time dilation, length contraction, relative/loss of simultaneity

Time dilation

Time passes more slowly at higher acceleration relative to observer; twin trip

Length contraction

objects at higher speed appears contracted along the line of motion (more noticeable at the speed of light

Loss of/relativity of simultaneity

Two events that seem simultaneous for one observer may not be simultaneous for another

Simultaneity

the state of two or more events happening at the same time, distorted in special relativity

Lorentz factor (gamma)

symbol that quantifies relativistic effects like time dilation and length contraction in Einstein’s relativity equations

Stanford linear accelerator

used to study high-energy physics and relativity near the speed of light

Minkowski spacetime

space and time are one entity, operates like a fluid, 4-dimensional fabric.

Principle of equivalence

Acceleration + gravity are indistinguishable. in an elevator, upward acceleration feels like gravitational pull down, in fee fall, gravity is gone —> is gravity even a force if it can just disappear?

Tidal forces

manifestation of gravity as explained in einstiens general relativity

General relativity

Einstein proposed gravity isnt a real force, but a manifestation of the curvature of spacetime CAUSED BY mass and energy. (extends special relativity to non-inertial frames of reference)

Gravitational time dilation

closer to a gravity well (massive bodies—stronger gravitational fields, black holes), time slows down infinitely,

Twin paradox/trip

two identical twins; one stays on Earth while the other travels in space at near-light speed OR closer to a gravitational well (string gravitational field). time slows for the traveling twin, they are younger due to time dilation experienced during their journey.

Geodesics

natural line/path of motion is along the straightes/shortestt line possible (geodesics), general relativity explains planetary orbits and light in curved spacetime

Ultraviolet catastrophe

classical prediction that black body would emit infinite energy at short wavelengths; energy is not infinite, solved by photoelectric experiment and plancks constant

Photoelectric experiment

Einstein showed light can knock electrons off metal surfaces only above certain frequencies —> planks constant

Planck's constant

energy is quantized, emits in discrete chunks, not a continuous flow

Q particles

electrons, photons, follow quantum laws: discrete energy emission, uncertainty

Wave-particle duality

You see the behavior of light based on what you choose to measure, particle or a wave, breakdown of classical physics

Quantum statistics

describes how quantum particles distribute across states.
Replaces classical statistics in atomic and subatomic systems

Bosons

(symmetric) can occupy the same quantum state simultaneously (BOSE-EINSTIEN STATISTICS):

force carriers (photons) they mediate electromagnetism, gravity, and the strong and weak nuclear forces

Fermions

(asymmetric) cant occupy same state (Pauli Exclusion Principle): building blocks of matter (electrons, photons, neutrons) and atomic structure (more “tangible” than bosons)
Pauli exclusion principle

Pauli exclusion principle

No two identical fermions can occupy the same quantum state at the same time.
Explains the structure of atoms and why matter takes up space

Transcendentals

Numbers that are not solutions to any polynomial with rational coefficients (Denumerable π, e).
They show that not all real numbers are algebraic— the vastness of number systems.

Denumerable

A set that can be matched 1:1 with the natural numbers (like integers or rationals).
It’s infinite, but still countable—there’s an ordering even if it goes on forever.

Non-denumerable

A set that cannot be listed or matched with the naturals (real numbers).
This form of infinity is larger than countable infinity—proven through diagonalization.

Countability

Determines whether a set can be put into one-to-one correspondence with the natural numbers

cardinality

measure of a set’s size, including infinite sets (Aleph null ℵ₀ for naturals, 𝑐 for reals).
It allows comparison between different types and sizes of infinity

Cantor’s paradox

• For any infinite set, i can always make a larger infinite set (with the power set the set of all subsets of a set)

  • Not all infinities look the same

Transfinite numbers

numbers larger than any finite number, used to quantify infinities, aleph null. manages and structures infinity

Different by design proof

Real number diagonalization, prove that some infinities are uncountable—by constructing a number not in any list.
reals between 0 and 1 can’t be listed proves uncountability of the reals and logical undecidability.

Diagonalization of rational numbers

the rationals can be listed (denumerable), using a diagonal method to count them without missing any.
Shows that not all dense sets are uncountable.

Predicate logic

• extends syllogistic reasoning more abstractly, exploration of relationships and generalizations

boolean logic

• mathematics of true/false (binary)

  • computing and algorithm design, all possible outcomes for a set of logical statements, digital decision making (but has limits, does not fully account for complex relational dynamics)

“All Cretans are liars,” said by a Cretan.
Raises the issue of self-referential contradiction in logic.

Vicious circle argument

Circular definitions or logic that assume what they try to prove.
Russell and others used this idea to show flaws in naive set theory.

Russell’s paradox

The set of all sets that do not contain themselves—does it contain itself?
Reveals contradictions in naive set theory

Autology

Words that describe themselves (like “short” or “noun”).
Used in philosophy and logic to explore self-reference.

Heterology

Words that do not describe themselves (like “long” or “blue”).
Raises paradoxes when asking if “heterological” is heterological.

Frege

tried to build a perfect, complete logical system for all of math. Genius, but his system collapsed when Russell pointed out a contradiction (Russell’s paradox). (logicicist)

Logicist

• Modify rules for set membership, sets must include complete, consistent descriptions; model disproved them

Formalists

• Blame semantics (language), not the logical structure.

  • Use of symbols: ∀ (for all), ∃ (there exists), ∈ (is a member of)...

Godels incompleteness theorem

• some truths cannot be proven within the system

  • Any fix still leaves space for paradox

  • There's always a meta-level paradox

Fibonacci Sequence

• 1, 1, 2, 3, 5, 8, 13…

  • arrangement of petals in flowers, the spirals in pinecones

  • mathematical structures that appear in nature, mathematic rules govern natural forms

    
    

Golden ratio

• he ratio of two numbers where the ratio of the larger to the smaller is the same as the ratio of the sum to the larger number

• mathematical structures appear and shape aesthetics

Self-Organizing Shapes

• mathematical structures that appear in nature, mathematic rules govern natural forms

Torus

A donut-shaped surface that loops back on itself.
Represents continuous flow, recurrence, and spatial intuition in science and surface geometry for binocular vision

TYPOGRAPHY

•  visual logic system, often uses geometry in LANGUAGE, how we interpret meaning through form

Voyager Space Plaque

: Logic as universal language; legibility