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

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.
What is in the title “Certainty, Mystery, and the Classroom” other than a catchy phrase?
Certainty, Mystery, and the Classroom

3. The allure of mathematics
Certain Knowledge
Proof
Transcendence
Beauty
Utility
It sells
Common conceptions
(1.) Transcendence: Mathematics is the same across time, language, and culture.
(2.) There is a romantic beauty in mathematics that we see in numbers such as π and φ (the Golden Ratio).
(3.) Abstract mathematics perfectly describes the natural world.
Certainty, Mystery, and the Classroom

4. 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.
We say axioms are self-evident – but are they?
We say 1+1=2 the world over – but is it?
We say that proof grants certainty – but do proofs change?
Certainty, Mystery, and the Classroom

5. Mystery in mathematics
Why do we see numbers like pi, the golden ratio, e, and infinity appearing?
Certainty, Mystery, and the Classroom

6. Certainty, Mystery, and 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.
I picture the common conception of math education this way: Suppose we piled algebra books one on top of another. The oldest at the bottom and the most recent at the top. How much would they really change?
Certainty, Mystery, and the Classroom

7. Certainty, Mystery, and the Classroom
The Question
What is math and where does it come from?
Have you ever wondered that with the emphasis in mathematics on precise definitions, we never provide a definition for mathematics itself?
Here are just a few books whose titles capture these questions.
Certainty, Mystery, and the Classroom

8. Certainty, Mystery, and the Classroom
The Stakes
Some may see the philosophy of mathematics as an esoteric topic that doesn’t impact the “real world.”
Ideas have consequences:
Science: The current definitions have led to a schism between mathematical and scientific research.
Economics: Our economic meltdown perhaps demonstrates the risk of letting mathematical formulas dictate policy w/o common sense
Philosophy: Philosophy is about the love of knowledge. Mathematics is the archetype of abstract (and perhaps apriori) knowledge.
Education: American mathematical education is in a troubled state – you can’t build on a sandy foundation for long
Certainty, Mystery, and the Classroom

9. Four Views on Mathematics
The Naturalist
The Platonist
The Formalist
The Humanist
The naturalist (clarifies the mystery)
The Platonist (the dominant view)
The Formalist (the established view)
The Humanist (making in roads)
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10. Certainty, Mystery, and the Classroom
The Naturalist
I’ve always wondered where math comes from?
Is it part of our physical world?
Something like anybody could dig out their garden?
Maybe clean off a little
Sell to college students?
Mmm?
Certainty, Mystery, and the Classroom

11. 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.
Abstract: Having no physical substance.
Certainty, Mystery, and the Classroom

12. Certainty, Mystery, and the Classroom
Discard naturalism
Because mathematics is clearly abstract, I think we can safely discard a material/natural view of mathematics.
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13. Certainty, Mystery, and the Classroom
Three viable Answers
And thus the mystery … mathematics exists and yet where does it live and come from?
The Platonist
The Formalist
The Humanist
The Platonist (it is out there)
The Formalist (it exists necessarily)
The Humanist (it is a part of us)
Next: Are there any Trekkies out there? If so, then perhaps you will connect with mathematical Platonism.
Certainty, Mystery, and the Classroom

14. Platonism Mathematics is “out there”
Now I am not trying to dismiss either Trekkies or Platonism. I did my undergraduate studies at a school famous for offering classes on Star Trek and I personally find aspects of Platonism quite compelling.
Rather, I want to communicate that the Platonist views mathematics as existing in another world … a non-physical Platonic realm. This is the kind of place that would keep Captain Kirk busy for a number of episodes.
Certainty, Mystery, and the Classroom

15. How do we know what is real?
Allegory of the Cave (Plato’s Republic, Book VII)
Plato imagines a group of people who have lived chained in a cave all of their lives, facing a blank wall. The people watch shadows projected on the wall by things passing in front of a fire behind them, and begin to ascribe forms to these shadows. According to Plato, the shadows are as close as the prisoners get to seeing reality. He then explains how the philosopher is like a prisoner who is freed from the cave and comes to understand that the shadows on the wall are not constitutive of reality at all, as he can perceive the true form of reality rather than the mere shadows seen by the prisoners.
Certainty, Mystery, and the Classroom

16. Certainty, Mystery, and the Classroom
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
Certainty, Mystery, and the Classroom

17. The Platonic Mathematician
The mathematician is a discoverer searching the Platonic realm for the eternal truths of mathematics.
Platonism is a powerful and pervasive view of knowledge. Alfred North Whitehead put it famously that philosophy is but a series of footnotes on Plato. Most mathematicians are Platonists (believing math is eternally “out there” for us to discover it). Platonism can be mixed with other philosophies: Russell and Gödel were Platonists. Hersh writes, “The working mathematician is a Platonist on weekdays, a formalist on weekends.” (39)
Certainty, Mystery, and the Classroom

18. 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.
Next I would like to talk about non-Euclidean geometry … but that only makes sense if we first have a basic understanding of Euclidean geometry.
Certainty, Mystery, and the Classroom

19. 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.
The phrase “self-evident” should cause you to stop and take stock. In the case of Euclid, were the postulates truly self-evident?
Certainty, Mystery, and the Classroom

20. 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.
Phrased another way, parallel lines never cross in Euclidean geometry.
Certainty, Mystery, and the Classroom

21. Certainty, Mystery, and the Classroom
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
Question: Must the interior angles in a triangle sum to 180 degrees?
Elliptic geometry: no parallel lines (all lines cross)
Hyperbolic geometry: infinite parallel lines (many lines don’t cross)
Kline: “The most significant fact about non-Euclidean geometry is that it can be used to describe the properties of physical space as accurately as Euclidean geometry does.” (84)
Another challenge to the eternal existence of a Platonic mathematical realm came thru the search for infinity.
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22. Certainty, Mystery, and the Classroom
∞ Infinity ∞
What is infinity?
Where does it come from?
Does it obey the laws of the finite?
Why does it lead to paradox?
How can infinity be just?
The guy that is credited with first comprehending that not all infinities are created equally is Georg Cantor.
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23. Certainty, Mystery, and the Classroom
∞ Infinity & Beyond ∞
On Transinfinities
A grave disease
Ridden through and through with the pernicious idioms of set theory
Utter nonsense
On Cantor
Corrupting the youth
A scientific charlatan
No one shall expel us from the Paradise that Cantor has created
Georg Cantor (1845 – 1918)
These are comments made regarding the Russian mathematician Georg Cantor ( ) and his study of transinfinities.
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24. Certainty, Mystery, and the Classroom
Superhero or Myth
The Platonic mathematician took a drink from a magical potion.
The Platonic realm is special.
The Platonic mathematician took a drink from a magical potion:
And can communicate with the Platonic realm.
The Platonic realm is special:
Its abstract members are immensely practical to our physical existence. Why should this be (like the ultimate in PC compatibility)?
What is out there? Is it numbers, shapes, theorems, proofs, and other mathematical worlds?
Having discussed the basic tenets and some of the objections to Platonism, we now move on to Formalism.
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25. Certainty, Mystery, and the Classroom
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.
Technically “Formalism” is one of the major philosophies of the “Foundations Movement.” The others, logicism and intuitionism, are quite distinct … however as this is just an introduction to the topic, I will group them together under the single label of “Formalism.”
Certainty, Mystery, and the Classroom

26. Certainty, Mystery, and the Classroom
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).
Logic is built upon self-evident axioms.
From these axioms, theorems/conclusions necessarily follow.
Formalism equates formal logic and mathematics.
Formal logic is built upon the concept of sets (or collections).
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27. Certainty, Mystery, and the Classroom
Examples of Axioms
Axiom of the empty set:
Axiom of extensionality:
Axiom of the empty set:
The empty set exists.
Axiom of extensionality:
Two sets are equal (are the same set) if they have the same elements.
Certainty, Mystery, and the Classroom

28. 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.
Infinity: Have you ever seen an infinite set or number?
Choice: Imagine an infinite number of sets. Powers of 2, 3, 5, …. Now imagine choosing exactly one element from each of these sets …
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29. Certainty, Mystery, and the Classroom
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?
Frege doesn’t exist in a vacuum and he built upon the work of others such as Boole and Peano. However, he dedicated his life to understanding numbers and arithmetic … and Russell broke his heart
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30. Certainty, Mystery, and the Classroom
What is one?
One
One is anything that can be placed into a one to one correspondence with the set containing the empty set.
Hmmm
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31. Certainty, Mystery, and the Classroom
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.
In his famous 1900 talk at the International Congress of Mathematicians, in Paris, Hilbert attempted to give a bird’s-eye view of the mathematics of the twentieth century, by way of twenty-three great open questions. Of these:
11 solved
7 partly solved
His second problem demanded a proof of the consistency of arithmetic. This was the question that later prompted Russell to coauthor Principia (more later) and in the end undercut his own formalist school of thought thru Gödel (even later).
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32. Certainty, Mystery, and the Classroom
Bertrand Russell (1872 – 1970)
One of the greatest logicians of all time.
Coauthored (with Alfred North Whitehead) Principia Mathematica ( ) in an effort to set mathematics on a solid foundation.
Gödel addressed the decidability of propositions of Principia.
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.
The quote says that math reduces to logic the thus math can be analyzed thru logic itself.
Certainty, Mystery, and the Classroom

33. Principia Mathematica (1910 -1913)
23rd most influential non-fiction work of the 20th century.
An unreadable masterpiece.
The Modern Library. See this most famous of quotations on page 362 where it is shown that 1+1 = 2.
Certainty, Mystery, and the Classroom

34. 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.
*****be concise******
The intuitionists question the axiom of choice.
Lay users question the loss of transcendence.
*****be long winded*****
Kurt Gödel's Incompleteness Theorems
The unreasonable effectiveness of mathematics
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35. Certainty, Mystery, and the Classroom
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.
The story told by Oskar Morgenstern of Gödel's studies leading to citizenship. He found an inner contradiction in the Constitution that would make it perfectly legal for someone to become a dictator and set up a Fascist regime.
Examiner: Now, Mr. Gödel, where do you come from?
Gödel: Where I come from? Austria.
Examiner: What kind of government did you have in Austria?
Gödel: It was a republic, but the constitution was such that it finally was changed into a dictatorship.
Examiner: Oh! This is very bad. This could not happen in this country.
Gödel: Oh yes, I can prove it.
Certainty, Mystery, and the Classroom

36. Incompleteness in Logicomix
Certainty, Mystery, and the Classroom

37. 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.
Gödel proved mathematically that no attempt to establish mathematics upon axioms such as those in Principia could succeed. This answered Hilbert’s second question.
Consistency: In logic, a consistent theory is one that does not contain a contradiction. A system is consistent if a proof never exists for both P and not P.
Completeness: A formal system is "semantically complete" when all tautologies are theorems whereas a formal system is "sound" when all theorems are tautologies.
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38. Certainty, Mystery, and the Classroom
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.
Wigner wrote an article titled: The Unreasonable Effectiveness of Mathematics in the Natural Sciences (1960)
This is a famous article in the philosophy of mathematics and is a must read for all mathematicians.
What did Wigner say?
Certainty, Mystery, and the Classroom

39. 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?
“The laws of gravity which Newton reluctantly established and which he could verify with an accuracy of about 4% has proved to be accurate to less than a thousandth of a percent and became so closely associated with the idea of absolute accuracy that only recently did physicists become again bold enough to inquire into the limitations of its accuracy.”
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40. Certainty, Mystery, and the Classroom
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.
In 1956, Bertrand Russell wrote the following on the end of formalism.
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41. Certainty, Mystery, and the Classroom
… 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.
Portrait from Memory (1956).
If math isn’t “out there” in a Platonic realm or simply a meaningless game we play as in formalism, perhaps it is part of being human?
Certainty, Mystery, and the Classroom

42. Mathematical Humanism
The hypercube
does it exist?
The Four Color Theorem
proved by a computer.
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43. Certainty, Mystery, and the Classroom
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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44. Certainty, Mystery, and the Classroom
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.
Proofs and Refutations (1976) is a surprisingly readable dialogue. In it, Lakatos makes a compelling case that the standards and depth in rigor within proof changes as our knowledge grows and evolves. The book has been translated into more than 15 languages worldwide, including Chinese, Korean and Serbo-Croat, and went into its second Chinese edition in It is based upon his PhD thesis of 1961, but wasn’t published until after his death.
This quote is a rebuke of formalism which denies the status of mathematics to most of what has been commonly understood to be mathematics, and can say nothing about its growth (p2).
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45. Certainty, Mystery, and the Classroom
Reuben Hersh ( )
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.
In many ways, Hersh is a popularizer of the ideas of Lakatos.
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46. Certainty, Mystery, and the Classroom
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.
Lakoff: a professor of linguistics at Berkeley
Nunez: a professor of cognitive science at UC, San Diego
This book has been controversial and received mixed reviews within the mathematical community.
Certainty, Mystery, and the Classroom

47. 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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48. Certainty, Mystery, and the Classroom
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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49. Certainty, Mystery, and the Classroom
Does it matter?
Perhaps you believe that questions in the philosophy of mathematics are irrelevant …
Ideas have consequences.
Science.
Economics
Philosophy.
Education.
We will focus on the implications to mathematical education.
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50. Certainty, Mystery, and the Classroom
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.
Classic example, the “New Math” of the 1960’s was based upon an axiomatic formalist view of mathematics.
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51. Certainty, Mystery, and the Classroom
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.
Platonist (an emphasis on discovery)
Formalist (an emphasis on theory and proof)
Humanist (an emphasis on the invention and evolution of mathematical knowledge)
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52. Certainty, Mystery, and the Classroom
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)
John L. Synge – mathematical physicist
All mathematicians (including math teachers) have a philosophy of mathematics.
This philosophy is passed on to our students.
Most mathematicians are unaware of their philosophy.
We pass on a philosophy w/o being aware of the fact.
Teachers need to know themselves.
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53. Certainty, Mystery, and the Classroom
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.
Discover and invention is still possible.
The history provides beautiful segues to a variety of mathematical topics.
It ties math to the humanities.
Students need to think about why mathematics works.
Certainty, Mystery, and the Classroom

54. Certainty, Mystery, and the Classroom
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.
See Morris Kline’s Loss of Certainty for examples of the last point.
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55. Certainty, Mystery, and the Classroom
The Question
One of my students asked me the following:
What was the most
interesting thing you
learned while on
your sabbatical?
Mathematics gives the appearance of being valueless, amoral, and philosophically neutral. What surprised me more than anything else was that every single book I read made reference to the existence of God (most of them quite directly). Certainly some books were written by Christians and so their inclusion of God wasn’t surprising. But the atheists and humanists brought God up just as much – if only to ridicule and argue against His existence.
So, the most interesting thing I learned is that mathematics is not a completely neutral topic and that “God talk” is alive and well in the philosophy of mathematics.
Certainty, Mystery, and the Classroom

56. 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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57. Certainty, Mystery, and the Classroom
Questions
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58. Certainty, Mystery, and the Classroom
References
A list of references and works cited is available upon request.
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59. Certainty, Mystery, and the Classroom