Philosophy%20of%20Mathematics

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Philosophy of Mathematics

• Efrah Ismael
• Philosophy of Mathematics
• 5400

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References

• Benacerraf, P., "Mathematical Truth", in The
  Philosophy of Mathematics, Hart, W.D., (Ed.),
  (1996), Oxford University Press, New York, p.
  14-30.
• Coffa, J.A., The Semantic Tradition from Kant to
  Carnap To the Vienna Station, (1991), Cambridge
  University Press, New York.
• Davis, P.J., Hersh, R., The Mathematical
  Experience, (1990), Penguin Books, Toronto.
• Hart, W.D., (Ed.), The Philosophy of Mathematics,
  (1996), Oxford University Press, New York.
• Shapiro, Stewart, Philosophy of Mathematics
  Structure and Ontology, (2000), Oxford University
  Press

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• Dieudonné characterizes the mathematician as
  follows
• we believe in the reality of mathematics, but
  of course when philosophers attack us with their
  paradoxes we rush to hide behind formalism and
  say "mathematics is just a combination of
  meaningless symbols" . . . Finally we are left in
  peace to go back to our mathematics, with the
  feeling each mathematician has that he is working
  on something real. (Dieudonné in Davis and Hersh,
  1981, p. 321)

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• Davis and Hersh, also, describe mathematician as
  
• the typical mathematician is a realist on
  weekdays and a formalist on Sundays. That is,
  when he is doing mathematics he is convinced that
  he is dealing with an objective reality . . . But
  then, when challenged to give a philosophical
  account of this reality, he finds it easiest to
  pretend that he does not believe in it after all.
  (Davis and Hersh, 1981, p.321)

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• As Coffa notes,
• the semantic tradition consisted of those who
  believed in the a priori but not in the
  constitutive powers of the mind . . . the root of
  all idealist confusion lay in misunderstandings
  concerning matters of meaning. Semanticists are
  easily detected They devote an uncommon amount
  of attention to concepts, propositions, and
  senses . . . (Coffa, 1991, p.1)

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All these authors were questioning

• about what objects mentioned in mathematical
  statements exist,
• about what mathematical statements we can know,
• about what mathematical statements are true or
  false.

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mathematical practice and philosophical
theorizing

• Stewart responds to the concern
• philosophy-first
• the principles of mathematics receive their
  authority, if any, from philosophy. Because we
  need a philosophical account of what mathematics
  is about only then can we determine what
  qualifies as correct mathematical reasoning.

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philosophy-last

• holds that mathematics is an autonomous science
  that doesnt need to borrow its authority from
  other disciplines.
• On this view, philosophers have no right to
  legislate mathematical practice but must always
  accept mathematicians own judgment.

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What Place Does Philosophy Have in Teaching
Mathematics?
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What is Mathematics? B. Russell

• Mathematics may be defined as the subject in
  which we never know what we are talking about,
  nor whether what we are saying is true.
• Bertrand Russell, Mysticism and Logic (1917) ch.
  4

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What is Philosophy of Mathematics?

• Ontology for Mathematics Being
• Ontology studies the nature of the objects of
  mathematics.
• It is the claim that mathematical objects exist
  independently of their linguistic expression.
• What we are talking about.
• What is a number?
• What is a point? line?
• What is a set?
• In what sense do these objects exist?

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What is Philosophy of Mathematics?

• Epistemology for Mathematics Knowing
• Epistemology studies the acquisition of knowledge
  of the truth of a mathematical statement.
  whether what we are saying is true.
• Does knowledge come from experience and evidence?
• Does knowledge come from argument and proof?
• Is knowledge relative or absolute?

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Platonism

• Platonism is one of the main theories in the
  philosophy of mathematics, and is one the major
  explanations of what mathematics really is. 
• The question it attempts to answer is whether
  mathematical truth has an independent existence
  do
• mathematicians discover mathematical truths that
  are, in some sense, out there to be found, or do
  they invent or create them?
• The answer to this question will determine the
  very way in which mathematician will look at his
  or her subject, and it is also part of the
  question of whether mathematics is a science or
  an art, or even possibly a game.
• The Platonist position, based (as the name
  implies) on ideas in the works of Plato, is that
  mathematical truth is discovered.  The idea is
  that mathematics consists of absolute truths,
  which were a prior (needing no other foundation,
  but being inescapable consequences of logical
  deduction).
• This position was reiterated by Kant in the 18th
  century, particularly with respect to Euclidean
  geometry.

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What is Philosophy of Mathematics?

• According to Plato, knowledge is a subset of that
  which is both true and believed

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What is Philosophy of Mathematics?

• Formalism / Deductivism-
• is a school of thought that all work in
  mathematics should be reduced to manipulations of
  sentences of symbolic logic, using standard
  rules. 
• It was the logical outcome of the 19th-century
  search for greater rigor in mathematics. 
  Programmes were established to reduce the whole
  of known mathematics to set theory (which seemed
  to be among the most generally useful branches of
  the science). 
• First attempts to do this included those of
  Bertrand Russell and A.N. Whitehead in Principia
  Mathematica (1910), and the later Hilbert
  Programme.

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Semantic

• is a discipline concerned with inquiry into the
  meaning of symbols, and especially linguistic
  meaning.
• Semantics in this sense is often contrasted with
  syntax,
• which deals with structures, and pragmatics,
• which deals with the use of symbols in their
  relation to speakers, listeners and social
  context.

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Roles for Philosophy in Teaching and Learning

• For the Teacher/Mentor (T/M)
• Awareness of issues can alert the T/M to
  excessively authoritarian approaches.
• Alternative philosophical views can allow the T/M
  to use and/or develop alternatives to traditional
  approaches.
• Philosophical issues can illuminate the value of
  and need for developing a variety of mathematical
  tools for solving problems.

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Roles for Philosophy in Teaching and Learning

• For the Student/Learner (S/L)
• Helps the S/L understand the context, goals, and
  objectives of the mathematics being studied.
• Opens the S/L to considerations of the human
  values and assumptions made in developing and
  using mathematics.
• Alerts the S/L to the use of authority and the
  value of different approaches to mathematics.

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Exploring Examples

• Following are few examples of topics that can be
  used to introduce and explore some philosophical
  issues in mathematic subjects at a variety of
  levels.
• Consider how these examples can be expanded or
  transformed to other aspects of the philosophy of
  mathematics.
• Consider how these examples can be expanded or
  transformed to other mathematics topics and/or
  courses.

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The Square Root of Two

• Questions for Open Discussion
• Ontological
• Definition?
• Does it exist?
• What is the nature of this object?
• Epistemological
• How do we know it exists?
• How do we know it is between 1 and 2
• How do we know it is not a rational number?

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Squares, Diagonals, and Square Roots

• Learning Objectives
•   Students will
• Measure the sides and diagonals of squares.
• Make predictions about, and explore the
  relationship between, side lengths and diagonals.
  
• Formulate a rule for finding the length of a
  diagonal based on the side length.

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Teacher Reflection

• As the students were measuring the sides and
  diagonals of their squares, what did you observe
  about accuracy and precision? If students had
  trouble measuring their shapes, what can you do
  in the future to improve this skill?
• What alternative patterns or methods did students
  discover that you did not anticipate? If the
  students did not discover alternate patterns, do
  you think there are any? Could you have led the
  students in another direction?
• How did the students demonstrate understanding of
  the materials presented?
• Were concepts presented too abstractly? Too
  concretely? How will you change the lesson if you
  teach it in the future?

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2005 Curriculum Expectations

• Grade 6-8
• Measurement
• Select and apply techniques and tools to
  accurately find length, area, volume, and angle
  measures to appropriate levels of precision.
• Understand, select, and use units of appropriate
  size and type to measure angles, perimeter, area,
  surface area, and volume.

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2005 Curriculum Expectations

• Grade 6-8
• Geometry
• Understand relationships among the angles, side
  lengths, perimeters, areas, and volumes of
  similar objects.
• Use geometric models to represent and explain
  numerical and algebraic relationships.

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The EndThank you?

• Questions?
• Comments?
• Discussion?