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MATHEMATICAL THINKING A guest lecture by Mr. Chase Is mathematics invented or discovered? Aristotle Plato Is mathematics invented or discovered? Options: Poll! 1. 2. 3. 4. Invented Discovered Unresolvable I donβt know β« π π₯ ππ₯ π βNewton and Leibniz invented Calculus.β β conventions and symbols log 2 64 invented! π(π₯) our number system And if you think mathematics is discovered: if a mathematical theory goes undiscovered, does it truly exist? π+π long division β arbitrary notation Is 267 β 1 prime or composite? Are there an infinite number of βtwin primesβ? β« π π₯ ππ₯ discovered! math is like scienceβ itβs true, regardless of whether we discover it or not. π no contradictions π+π air-tight logic Correct answerβ¦ discovered! Is this always true? Arenβt you dying for a proof? Is 9π β 1 always divisible by 8? There exist two people in DC with the exact same number of hairs on their heads. Why? β Mathematics is a queen of science. Carl Friedrich Gauss β what mathematicians have to sayβ¦ Wherever there is number, there is beauty. Proclus It is impossible to be a mathematician without being a poet in soul. Sofia Kovalevskaya The mathematician does not study pure mathematics because it is useful; he studies it because he delights in it and he delights in it because it is beautiful. Jules Henri Poincaré what mathematics are we free to invent? the symbols and conventions we choose are arbitrary. FORMALISM Mathematics is a game played according to certain simple rules with meaningless marks on paper. David Hilbert the field axioms. Closure of π under addition and multiplication For all a, b in F, both π + π and π · π are in πΉ (or more formally, + and β are binary operations on πΉ). Associativity of addition and multiplication For all π, π, and π in πΉ, the following equalities hold: π + (π + π) = (π + π) + π and π · (π · π) = (π · π)π. Commutativity of addition and multiplication For all π and π in πΉ, the following equalities hold: π + π = π + π and π · π = π · π. Existence of additive and multiplicative identity elements There exists an element of πΉ, called the additive identity element and denoted by 0, such that for all π in πΉ, π + 0 = π. Likewise, there is an element, called the multiplicative identity element and denoted by 1, such that for all π in πΉ, π · 1 = π. To exclude the trivial ring, the additive identity and the multiplicative identity are required to be distinct. Existence of additive inverses and multiplicative inverses For every π in πΉ, there exists an element βπ in πΉ, such that π + (βπ) = 0. Similarly, for any π in πΉ other than 0, there exists an element πβ1 in πΉ, such that π · πβ1 = 1. (The elements π + (βπ) and π · π β1 are also denoted π β π and π/π, respectively.) In other words, subtraction and division operations exist. Distributivity of multiplication over addition For all π, π and π in πΉ, the following equality holds: π · (π + π) = (π · π) + (π · π). group ring domain Can we break or change the rules? YES. Abelian group skew field Epic math battles Prove the thing! I want to create a formal system in which we can prove all statements. David Hilbert Kurt Gödel You canβt prove the thing! In every formal system, there must be unprovable statements. Silly example Axioms: it is raining outside. if it is raining, I will take an umbrella. Statements: I will take an umbrella. Provably true. It is not raining outside. Provably false. I will take my pet hamster as well. Undecidable Math is useful Itβs like a gorgeous painting that also functions as a dishwasher! Ben Orlin Butβ¦WHY is it useful? Why study math? Liberal Education Glimpsing the mind of God In summaryβ¦ Math is different. It allows certain knowledge. Questions?