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TOK Mathematical Ideas - Coggle Diagram
TOK Mathematical Ideas
Scope
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Russell was critical of the foundations of mathematical knowledge, considering them to be on shaky ground.
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To establish a more solid foundation for mathematics, Russell co-authored the Principia Mathematica with Alfred North Whitehead.
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Even basic assumptions, like 1+1=2, required rigorous proof.
Proving 1+1=2 took them 300 pages, starting in volume one and concluding in volume two of the Principia Mathematica.
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Perspectives
Analogy: Like dinosaurs existed before their discovery, mathematics exists whether or not we are aware of it.
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Galileo Galilei Quote: Mathematics is seen as a language of the universe, necessary to understand its "book."
Mathematical Platonism: Also known as mathematical realism, posits that mathematics exists a priori, outside human experience.
Evidence for Platonism:
Experiential Certainty: Two objects plus two objects equals four, regardless of human consciousness.
Predictive Validity: Mathematics underpins scientific laws and theories with accurate predictions (e.g., theory of relativity, black holes, cosmological expansion).
Question: If mathematics did not exist, how could it have such predictive validity in the natural world?
Formalist Perspective: Also known as mathematical anti-realism or fictionalism, argues that mathematics is a human-invented method of description, similar to language.
Evidence for Formalism: Mathematical objects like numbers do not exist outside the human mind; they are abstract ideas invented by humanity.
Counterclaim to Platonism: Abstract ideas used for predictions do not necessarily imply that those ideas are real, just that they are useful.
Analogy: Mathematics is compared to a game like chess or basketball, where players can be creative within the set rules, leading to a valid body of knowledge according to those rules.
Rule Changes: If axioms (rules) change, the resulting knowledge would be different but still valid and effective within the new framework.
Impact on Knowledge: Both Platonist and formalist views require adherence to mathematical laws, axioms, and operations, so they do not greatly affect the knowledge produced.
Emotional Benefit: Platonism may offer emotional satisfaction; according to astrophysicist Mario Livio, "all working mathematicians are Platonists in their hearts."
Big Picture
Mathematics is often seen as straightforward because it is based on axioms, which are considered self-evident truths.
An example of an axiom-based truth in mathematics is that the angles in a triangle add up to 180 degrees in Euclidean geometry.
Despite being necessary for set theory, the axiom of an infinite set is technically a conjecture.
Mathematics: Universally agreed language, precise definitions, argument clarity
Axioms: Self-evident truths, foundational elements
Debate: Numbers, mathematical symbols
Existence: Objects of mathematical inquiry, numbers’ existence
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Mathematical Nominalists: Deny the existence of abstract mathematical objects, viewing them as names or symbols.
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