Philosophy of Mathematics

1
Philosophy of Mathematics

• Where do you stand?

2
What is a number?
3
What is a number?

• A symbol
• A character designed to represent a certain type
  of mathematical idea
• One symbol for each type of number
• Has properties, but no existence outside the
  human reason

• A real entity
• An object that exists somewhere in reality
• Every number is different
• Properties unique to each number

4
Take a Stand

• Where could a number exist?
• Based on interaction?
• What do you think?

5
Where does math come from?
6
Where does math come from?

• Math is a product of the human mind
• Made-up
• Used to give both reason and order to the world
  around us

• Math exists and is only discovered
• All mathematical ideas are already in existence
• We find the properties of each new discovery

7
Take a Stand

• Why is math everywhere if it is made-up?
• If all concepts exist, where do they exist?
• Does math explain experience?

8
Can Math Change?
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Can Math Change?

• Yes
• Relative truths
• Assignments of rules to reason
• Must be true, so doesnt lose meaning
• Interpretation

• No
• Discoveries can differ
• Leads only to different mathematics
• Can exist without opposition

10
Take a Stand

• Euclidean Geometrys 5th Postulate

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Where do you end up?

• Mostly this column?
• Mathematical Realism

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Where do you end up?

• Mostly this column?
• Formalism

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Mathematical Realism

• Platonism
• From Pythagoreans-All is number.
• Gödel

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Formalism

• Deductivism
• Hilbert
• Compromise Mathematical Logicians

15
Gödel vs. Hilbert
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Quasi-Empiricism

• Math exists without objective truths
• Medieval English Mathmatics

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Action

• Math is obvious in the doing
• Algebra of seeing

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Unification

• Math is sererate from philosophy
• Above human perception

19
Language

• Nothing but symbols used to communicate an idea
• Mathematical grammar

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Conclusion

• Know where you stand.
• Use your ideas about math to reflect on students.
• Be willing to listen and evolve as a
  mathematician

21
Works Cited

• Philosophy of Mathematics." Wikipedia
  Encyclopedia. 27 Nov. 2005. 29Nov. 2005
  lthttp//en.wikipedia.org/wiki/Philosophy_of_mathem
  aticsgt.