Summary

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Summary
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What weve seen

• Evolution of ideas over time
• Resistance to certain concepts
• No idea persists in its original form
• Inspiration for new mathematics
• Tendency towards generalization
• Eventual applications
• What a math historian does

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Evolution of ideas over time

• Deductive mathematics born in Greece
• Number theory to explain the natural world
• Crisis of the irrational lead to geometry
• Geometry good practice for rhetoric
• Algebra a natural extension of practical math
• Trigonometry required for astronomy
• Coordinate geometry needed symbolic algebra
• Calculus used coordinate geometry to understand
  motion

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Resistance to certain concepts

• Positional number system
• Zero
• Negative numbers
• Irrational numbers
• Symbolic algebra
• Infinity

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No idea persists in its original form

• Algebraic equations
• Axioms of Euclidean geometry
• p
• Conic sections
• Sine and cosine
• x-y coordinates
• Infinitesimals

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Inspiration for new mathematics

• Practical problems of daily life
• Understanding the natural world
• Mathematics itself
• Foundations of mathematics
• Making connections through generalization

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Tendency towards generalization

• Numbers
• Vectors
• Matrices
• Polynomials
• Permutations
• Symmetries
• We can study all of these objects at once by
  abstracting their common properties

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Eventual applications

• Pythagorean theorem
• Conic sections
• Method of exhaustion
• Platonic solids
• Prime numbers
• Cubic equations
• Non-Euclidean geometry
• iPod math

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What a math historian does

• Original sources
• Early translations
• Commentaries
• What people in different times and places were
  interested in and why, what they could do, and
  how they did it

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What I hope youll take away

• This is mathematics.
• People did math because it interested them.
• When you teach math, please dont separate the
  result and its application from its motivation,
  discovery, and justification.
• Math has a history, a present, and a future.
• Math is one of the greatest human inventions ever.

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God created the integers all the rest is the
work of man.
Leopold Kronecker