Math
Mathematical Terms
Axioms: Considered self-evident truths within specific mathematical frameworks.
Conjectures: Propositions believed to be true but unproven within the system's logic.
Proofs: Deductive arguments confirming the truth of mathematical statements universally.
Perspectives
Formalist perspective
Real VS True
Scopes
Axioms and Deductive Logic
Pure vs. Applied Mathematics
Big picture
Mathematics in TOK explores "How do we know?" through axioms, which are foundational self-evident truths.
Mathematics is seen as a universal language with precise definitions, minimizing ambiguity.
TOK focuses on how mathematical knowledge is constructed and its cultural/social implications, not just content.
Justification in mathematics is based on internal consistency and logic, though real-world evidence has historically supported it.
Axioms vary by mathematical branch and consensus, with some, like the infinite set in set theory, widely accepted despite being conjectural.
Philosophical debates
Platonism: Views mathematical entities as discoverable, pre-existing realities independent of human thought.
Nominalism: Either mathematical objects, relations, and structures do not exist at all, or they do not exist as abstract objects
Fictionalism: Regards mathematical entities as useful fictions, facilitating understanding and problem-solving without claiming their independent existence.
Bertrand Russell's Perspective on Mathematics
Power and Criticism: Russell valued mathematics but criticized its reliance on unproven axioms.
"Principia Mathematica": Co-authored to establish a more proof-based foundation for mathematics.
Formalism: Considers mathematics a constructed language of thought, with its truths contingent upon human-defined axioms and systems.
Axioms: Self-evident truths used as the foundation of mathematics.
Conjectures: Unproven beliefs treated as true within mathematics.
Deductive Logic: Ensures consistency, but flawed axioms can lead to false conclusions.
Pure Mathematics: Abstract, focused on general patterns, not immediately practical.
Applied Mathematics: Used in real-world contexts, crucial for making precise predictions and decisions.
Plantonic perspective
Mathematics is inherent in the universe, much like natural laws, and is not created by humans.
The ability of mathematics to accurately predict natural phenomena supports its existence outside of human though
Mathematics is a human-constructed language, similar to other forms of symbolic communication.
Mathematical objects, like numbers, exist only in the human mind and do not have a physical existence.
While mathematics is useful for making predictions, it does not necessarily represent a 'real' existence.
Truth: Can be contextually and culturally constructed, changing over time.
Reality: Exists independently of human perception or cultural influence.
Believes mathematics exists independently of humans, discovered rather than invented.
Considers mathematics both real and true, existing outside human creation.
Considers mathematical concepts useful and valuable, but ultimately human inventions.