Philosophy of Mathematics

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

• 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)

• 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)

• As Coffa notes,• [t]he 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)

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.

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.

‘philosophy-last’

• holds that mathematics is an autonomous science that doesn’t 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.

What Place Does Philosophy Have in

Teaching Mathematics?

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

What is Philosophy of Mathematics?

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?

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?

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.

What is Philosophy of Mathematics?

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

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.

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.

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”.

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.

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.

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?

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.

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?

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.

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.

The EndThank you

Questions?Comments?Discussion?