Certainty, Mystery and the Classroom

Dusty WilsonHighline Community College

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The Philosophy of Mathematics

• This talk is an introduction to the philosophy of mathematics.

• It outlines:– Questions in the philosophy of math.– Four Three philosophical camps.– The implications for us.

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The allure of mathematics

• Certain Knowledge• Proof• Transcendence• Beauty• Utility• It sells

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Certainty in mathematics

• Common conceptions– Mathematics is natural

and its axioms self evident.

– No matter where you go in the universe, you will always find that 1+1 = 2.

– Mathematics offers proof where the rest of science rests on theory.

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Mystery in mathematics

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The Classroom

• Conceptions– Mathematics is static

and unchanging.– There is only one answer

in mathematics.– Mathematics is a useful

tool but packaged as a necessary evil.

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The Question

• What is math and where does it come from?

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The Stakes

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Four Views on Mathematics

• The Naturalist• The Platonist• The Formalist• The Humanist

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The Naturalist

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Just your garden variety math

• Because of its relevance, there is a tendency to see mathematics as a part of the universe.– For example, π is a part of the circle.

• But where is it? Mathematics is separate from the figures we draw and the symbols we write.

• Mathematics is abstract.

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Discard naturalism

• Because mathematics is clearly abstract, I think we can safely discard a material/natural view of mathematics.

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Three viable Answers

• And thus the mystery … mathematics exists and yet where does it live and come from?– The Platonist– The Formalist– The Humanist

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Platonism

Mathematics is

“out there”

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How do we know what is real?

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Just Shadows

• Have you ever seen a true triangle or circle?• What is 3? • What characteristic is shared by:– Three blind mice– Three musketeers– Three branches of government

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The Platonic Mathematician

• The mathematician is a discoverer searching the Platonic realm for the eternal truths of mathematics.

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Contradictory Eternal Truths

• Through the early 19th century, most mathematicians believed in the objective existence of mathematical reality.

• But discoveries were made that seemed to imply contradictory eternal truths:– non-Euclidean geometry.– Cantor’s search to understand infinity.

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Euclid’s Elements (circa 300 BC)

• Euclid’s Elements begins with five postulates.

• The first is that we can draw a straight line between any two points.

• These postulates of Euclid had always been considered self-evident.

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Geometry sparked the search

• Euclid’s fifth (or parallel) postulate caused a great deal of consternation.

• It is most commonly expressed as: Given a line and a point not on the line, it is possible to draw exactly one line parallel to the given line through that point.

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Self-evident?

• But the discovery of non-Euclidean geometries (around 1830) began a mathematical revolution.

• Key players included Janos Bolyai, Nikolai Lobachevsky, Carl Gauss, and Bernhard Riemann.

Elliptic Geometry Hyperbolic Geometry

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∞ Infinity ∞

• What is infinity?• Where does it come

from?• Does it obey the laws of

the finite?• Why does it lead to

paradox?

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∞ Infinity & Beyond ∞

On Transinfinities• A grave disease• Ridden through and

through with the pernicious idioms of set theory

• Utter nonsenseOn Cantor• Corrupting the youth• A scientific charlatan

• No one shall expel us from the Paradise that Cantor has created

Georg Cantor (1845 – 1918)

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Superhero or Myth

• The Platonic mathematician took a drink from a magical potion.

• The Platonic realm is special.

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Formalism

• Mathematics rests upon the foundation of logic which exists necessarily.

• Mathematics is a game played according to certain simple rules with meaningless marks on paper.

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Enter Logic

• If the foundations of mathematics are not self-evident, upon what are they based?

• Logic: The science of the most general laws of truth (Frege).

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Examples of Axioms

• Axiom of the empty set:• Axiom of extensionality:

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Some Axioms are less Self-Evident

• Axiom of infinity:– There exists a set having infinitely many members.

• Axiom of choice– Given any set of pair-wise disjoint non-empty sets,

call it X, there exists at least one other set that contains exactly one element in common with each of the sets in X.

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Gottlob Frege (1848 – 1948)

• The first to dedicate himself to building the foundation of arithmetic upon logic.

• What are numbers? What is the nature of arithmetical truth?

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What is one?

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David Hilbert (1862 – 1943)

• Hilbert is the founder of mathematical formalism.

• Hilbert’s problems.• Mathematics is a game

played according to certain simple rules with meaningless marks on paper.

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Bertrand Russell (1872 – 1970)

• The fact that all mathematics is symbolic logic is one of the greatest discoveries of our age; and when this fact has been established, the remainder of the principles of mathematics consists in the analysis of symbolic logic itself.

• One of the greatest logicians of all time. • Coauthored (with Alfred North Whitehead) Principia

Mathematica (1910-1913) in an effort to set mathematics on a solid foundation.

• Gödel addressed the decidability of propositions of Principia.

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Principia Mathematica (1910 -1913)

• 23rd most influential non-fiction work of the 20th century.

• An unreadable masterpiece.

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Objections to Formalism

• While formalism remains the party line in mathematics, it has suffered at least four major objections:

• Of these, we will discuss the latter two.– Kurt Gödel's Incompleteness Theorems.– The unreasonable effectiveness of mathematics.

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Kurt Gödel (1906 – 1978)

• Perhaps the greatest logician of all time.

• Wrote, “On formally undecidable propositions of Principia Mathematica and related systems” in 1931.

• ...a consistency proof for [any] system ... can be carried out only by means of modes of inference that are not formalized in the system ... itself.

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Incompleteness in Logicomix

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The 2nd Incompleteness Theorem

• Theorem: For any self-consistent recursive axiomatic system powerful enough to describe the arithmetic of the natural numbers:

– If the system is consistent, it cannot be complete.– The consistency of the axioms cannot be proven within the

system.

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Eugene Wigner (1902 – 1995)

• Nobel prize in Physics, 1963• The miracle of the

appropriateness of the language of mathematics for the formulation of the laws of physics is a wonderful gift, which we neither understand nor deserve.

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The Unreasonable Effectiveness

• Mathematics is unreasonably effective in its descriptions and predictive explanations of the physical world.

• The enormous usefulness of mathematics in the natural sciences is something bordering on the mysterious.

• Not everyone agrees.– What is meant by “effective?”– What is “reasonable” effectiveness?

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Bertrand Russell on …

• I wanted certainty in the kind of way in which people want religious faith. I thought that cer-tainty is more likely to be found in mathematics than elsewhere. But I discovered that many math-ematical demonstrations, which my teachers ex-pected me to accept, were full of fallacies, and that, if certainty were indeed discoverable in mathematics, it would be in a new field of mathematics, with more solid foundations than those that had hitherto been thought secure.

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… the end of Formalism

• But as the work proceeded, I was continually reminded of the fable about the elephant and the tortoise. Having constructed an elephant upon which the mathematical world could rest, I found the elephant tottering, and proceeded to cons-truct a tortoise to keep the elephant from falling. But the tortoise was no more secure than the elephant, and after some twenty years of very arduous toil, I came to the conclusion that there was nothing more that I could do in the way of making mathematical knowledge indubitable.

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Mathematical Humanism

• The hypercube – does it exist?

• The Four Color Theorem– proved by a computer.

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Overview of humanism

• Mathematics describes the physical world because it was invented to describe the physical world.

• Mathematics is human and varies through time, culture, and society.

• Mathematics is fallible.• Mathematics is a language and

changes/adapts as do all languages.

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Imre Lakatos (1922 – 1974)

• Popularized subjectiveness in Proofs and Refutations.

• The history of mathematics, lacking the guidance of philosophy, [is] blind, while the philosophy of mathematics, turning its back on the most intriguing phenomena in the history of mathematics, is empty.

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Reuben Hersh (1927 - )

• A controversial author on the philosophy of math.

• Mathematical objects are created by humans.

• Mathematical knowledge isn’t infallible.

• Mathematical objects are a distinct social-historic object.

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Lakoff and Nunez

• Authors of Where Mathematics Comes From (2000)

• All the mathematical knowledge that we have or can have is knowledge within human mathematics.

• Where does mathematics come from? It comes from us! We create it ... through the embodiment of our minds.

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Objections to Humanism

• Some likely objections include:– Does it adequately explain the unreasonable

effectiveness of mathematics? – It seems to grant the mathematician the divine

power to create.– It denies the transcendence of math that seems so

self-evident.

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Time will tell …

• As the most recent of the mathematical philosophies, humanism hasn’t yet undergone the test of time.

• Much effort has gone into debunking Platonism and formalism, but humanism has yet to feel the weight of academic and mathematical critique.

• It may be early to hang your hat on a humanistic view of mathematics.

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Does it matter?

– Philosophy.– Education.

• Perhaps you believe that questions in the philosophy of mathematics are irrelevant …

• Ideas have consequences.– Science.– Economics

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Math Education

• Our philosophy of mathematics impacts education in a number of ways:– It impacts our

curriculum– It impacts our teachers– It impacts the

motivations of students– It impacts research.

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What to do: Curriculum

• In curriculum design– Authors write from a

philosophical perspective and a conception of mathematics.

– Our conception and definition of mathematics influences our receptivity to textbooks.

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What to do: Teaching

• In teaching (for teachers)– “… each young

mathematician who formulates his own philosophy – and all do – should make his decision in full possession of the facts.” (John Synge, 1944)

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What to do: Students

• In motivating students:– Some students are put

off by a fixed and static conception of mathematics.

– The story of the philosophy of mathematics can excite students

– It provokes interest in supplemental study.

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What to do: Research

• Philosophy impacts research:– Is mathematical research

a process of discovery or invention?

– The philosophy of math impacts the questions that are found interesting for research.

– Philosophy impacts the degree to which the researcher refers to outside disciplines.

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The Question

• One of my students asked me the following:

What was the most interesting thing you

learned while onyour sabbatical?

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Conclusion

• With the loss of certainty that comes through the philosophy of mathematics, we now have a side of mathematics so simple that a child can contribute and yet such an enigma that it can baffle a sage for a lifetime.

What is math and where does it come from?

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Questions

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References

• A list of references and works cited is available upon request.

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