Please enable JavaScript.
Coggle requires JavaScript to display documents.
Mathematics - Coggle Diagram
Mathematics
Big picture
-
Key Concepts
Axioms
Definition: Axioms are basic statements or propositions that are assumed to be true without proof. They serve as the foundational building blocks for any logical system or mathematical theory.
Example: In Euclidean Geometry, the statement "the angles of a triangle add up to 180 degrees" is an axiom. In Set Theory, the existence of an infinite set is considered an axiom, even though it cannot be derived from other axioms.
Bertrand Russell’s Critique: Bertrand Russell criticized the reliance on axioms in mathematics, arguing that if the foundational axioms are unproven or incorrect, then all subsequent knowledge based on them could be flawed.
Proofs
Definition: Proofs in mathematics are logical arguments that derive conclusions from axioms or previously established theorems. Deductive reasoning ensures that if the axioms are true, the conclusion must also be true.
Example: The proof of 1+1=2 is famously detailed in the Principia Mathematica by Bertrand Russell and Alfred North Whitehead, requiring 300 pages to rigorously establish this fundamental arithmetic truth from first principles.
-
Conjectures
Definition: A conjecture is a mathematical statement that is proposed as true based on empirical evidence or intuition but has not yet been proven. Conjectures often drive research and exploration in mathematics.
Example: Goldbach’s Conjecture is a famous unsolved problem in number theory, which states that every even integer greater than two is the sum of two prime numbers. Despite extensive numerical evidence, it remains unproven.
Certainty & Truth
Definition: Certainty in mathematics comes from the logical structure of deductive reasoning. If the axioms are true, then the conclusions drawn from them are certain.
Truth: While mathematical certainty is achievable within a given logical system, the truth of the axioms themselves, and thus the truth of the entire system, can be questioned.
Example: Albert Einstein's theory of General Relativity was mathematically certain when formulated, but it wasn't considered objectively true until it was confirmed by empirical evidence during the solar eclipse of 1919.
-
-