MATHEMATICS

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Bellwork Can you imagine a world in which 2+2=5? For example, what if ever time you brought two pairs of objects close to one another, a fifth one popped into existence?

Mathematics and certainty Analytical: True by definition Synthetic: Not Analytic A Priori: True Independent Experience A Posteriori: Not a true independent experience

Box 1: All definitions in this box Box 2: Empty Distinction matrix Box 1: All definitions in this box Box 2: Empty Box 3: Empirical Knowledge Box 4: Unkown Analytic Synthetic A priori Yes ? A posteriori No

Empirical Mathematics “Mathematical Truths are empirical generalizations based on a vast number of experiences” Common knowledge being 2+2=4, but not necessarily that metal expands when heated.

Discussion Question 1 on Page 199

Analytical Mathematics Truth is already present in mathematics, the “wrapper” simply has to be taken off Good short-term memory is important to mathematics Goldbach’s Conjecture almost contradicts analytical mathematics

MATHEMATICS AS SYNTHETIC A PRIORI It could also be argued that mathematics fits neither the empirical or analytical boxes, but in synthetic a prior knowledge. Reason must be used to arrive at natural truths “To speak freely, I am convinced that it “To speak freely, I am convinced that it [mathematics] is a more powerful instrument of knowledge than any other that has been bequeathed to us by human agency, as being the source of all others” - Rene Descartes

Mathematical truths are empirical generalizations The three views of math Empiricism Box 3 Mathematical truths are empirical generalizations Formalism Box 1 Mathematical truths are true by definition Platonism Box 4 Mathematical truths give us a priori insight into the structure of reality

Discovered or invented? Platonists believe that mathematical entities are discovered Formalists argue that they are invented and only exist in the mind

Group Discussion Discuss how you would explain mathematics as being both invented and discovered

Mathematics is more certain than perception Plato’s argument Mathematics is more certain than perception Mathematics is timelessly true Two objections to Plato If there are an infinite number of math entities, it is hard to prove or argue the observation of infinity If the entities have some ideal existence naturally, how do humans learn them?

Differed from Euclid’s Two points may determine more than one line Riemann’s Axioms Differed from Euclid’s Two points may determine more than one line All lines are finite in length but endless - i.e. circles There are no parallel lines Theorems Derived From Above Axioms All perpendiculars to a straight line meet at one point Two straight lines enclose an area The sum of the angles of any triangle is greater than 180 degrees

Problem of consistency Riemann believed that because Euclid’s axioms ran into contradictions, his would have contradictions as well However, intuition alone cannot determine contradiction Gödel’s Incompleteness Theorem Didn’t discover any contradictions Stated that math isn’t certain for we may one day discover a contradiction

Math only serves a purpose when it is applicable to the real world Applied Mathematics Math only serves a purpose when it is applicable to the real world An example is Appalonius studying ellipses in Greek times. Ellipses were useless to them at the time, yet it was still a mathematical discovery Einstein says that mathematics is invented, but it is a matter of discovery which of the various systems apply to reality. Buffon’s Needle Problem