Certainty, Mystery and the Classroom Dusty Wilson Highline Community College.

Certainty, Mystery and the Classroom

Dusty WilsonHighline Community College

Certainty, Mystery, and the Classroom 2

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.


The allure of mathematics

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


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.


Mystery in mathematics


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.


The Question

• What is math and where does it come from?


The Stakes


Four Views on Mathematics

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


The Naturalist


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.


Discard naturalism

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


Three viable Answers

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


Platonism

Mathematics is

“out there”


How do we know what is real?


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


The Platonic Mathematician

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


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.


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.


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.


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


∞ Infinity ∞

• What is infinity?• Where does it come

from?• Does it obey the laws of

the finite?• Why does it lead to

paradox?


∞ 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)


Superhero or Myth

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

• The Platonic realm is special.


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.


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).


Examples of Axioms

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


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.


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?


What is one?


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.


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.


Principia Mathematica (1910 -1913)

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

• An unreadable masterpiece.


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.


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.


Incompleteness in Logicomix


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.


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.


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?


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.


… 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.


Mathematical Humanism

• The hypercube – does it exist?

• The Four Color Theorem– proved by a computer.


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.


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.


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.


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.


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.


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.


Does it matter?

– Philosophy.– Education.

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

• Ideas have consequences.– Science.– Economics


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.


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.


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)


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.


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.


The Question

• One of my students asked me the following:

What was the most interesting thing you

learned while onyour sabbatical?


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?


Questions


References

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

Certainty, Mystery and the Classroom Dusty Wilson Highline Community College.

Documents

Transcript of Certainty, Mystery and the Classroom Dusty Wilson Highline Community College.