Do Mathematicians Really Mean What They Say

1
Do Mathematicians Really Mean What They Say?

• questions from
• the philosophy of mathematics
• Jason Douma
• University of Sioux Falls
• April 9, 2005
• presented for
• Mathematics on the Northern Plains
• at South Dakota State University

2
What distinguishes mathematics from the usual
natural sciences?

• Mathematics is not fundamentally empirical it
  does not rely on sensory observation or
  instrumental measurement to determine what is
  true.
• Indeed, mathematical objects themselves cannot be
  observed at all!

3

• Does this mean that mathematical objects are not
  real?
• Does this mean that mathematical knowledge is
  arbitrary?
• Good questions!
• These are the things that keep mathematical
  ontologists and epistemologists awake at night.

4

• Do all of our heroic mathematical accomplishments
  really mean anything at all!
• Scary question!
• Obsessing over this could lead you quickly along
  the path of Cantor. However, a bit of thoughtful
  musing over this might help us understand the
  proper place of mathematics in the greater
  context of human thought.

5
The Philosophy of Mathematicsan unreasonably
concise history

• Through most of the 17th Century, an
  understanding that mathematics was in some way
  part of natural philosophy was widely accepted.
• Beginning in the 17th Century, the philosophical
  status of mathematics began to take on a more
  subtle (and perhaps less mystical) character,
  through the epistemological methods of Spinoza
  the empiricism of Locke, Hume, and Mill and
  especially through the synthetic a priori
  status assigned to mathematics by Immanuel Kant.

6
The Philosophy of Mathematicsan unreasonably
concise history

• In the 19th Century, several developments
  (non-Euclidean geometry, Cantors set theory,
  anda little laterRussells paradox, to name a
  few) triggered a foundational crisis.
• The final decades of the 19th Century and first
  half of the 20th Century were marked by a heroic
  effort to make the body of mathematics
  axiomatically rigorous. During this time,
  competing foundational philosophies emerged, each
  with their own champions.

7
The Philosophy of Mathematicsan unreasonably
concise history

• After lying relatively dormant for half a
  century, these philosophical matters are now
  receiving renewed attention, as reflected by the
  Philosophy of Mathematics SIGMAA unveiled in
  January, 2003.
• Based on the furious rate of postings to the
  newly launched Philosophy of Mathematics
  listserv, interest in these issues is high,
  indeed.

8

• In the modern mathematical community, there is
  very little controversy over what it takes to
  show that something is truethis is what
  mathematical proof is all about.
• Most disagreements over this matter are questions
  of degree, not kind.
• (Exceptions proofs by machine, probabilistic
  proof, and arguments from a few extreme
  fallibilists)
• However, when discussion turns to the meaning of
  such truths (that is, the nature of
  mathematical knowledge), genuine and substantial
  distinctions emerge.

9
Arithmetic of Irrational Numbers

• What, exactly, do we mean by ?
• The most obvious answer to this question (its
  what we get when we multiply by itself) is
  perhaps among the least legitimate.

...lets see..carry the 1and
10
Gabriels Horn

• Gabriels Horn can be gener-ated by rotating the
  curve
• over 1,8) around the x-axis.
• As a solid of revolution, it has finite volume.
• As a surface of revolution, it has infinite area.

11
The Peano-Hilbert Curve

• (from analysis)
• There exists a closed curve that completely fills
  a two-dimensional region.
• Image produced by Axel-Tobias Schreiner, Image
  produced by John Salmon
• Rochester Institute of Technology, and Michael
  Warren, Caltech
• Programming Language Concepts, Parallel,
  Out-of-core methods for
• http//www.cs.rit.edu/ats/plc-2002-2/html/skript.
  html N-body Simulation,
• http//www.cacr.caltech.edu/johns/pubs/siam
  97/html/online.html

12
The Banach-Tarski Theorem

• (from topology)
• An orange can be sliced into five pieces in such
  a way that the five pieces can be reassembled
  into two identical oranges, each the same size as
  the original.
• Better yet, a golf ball can be taken apart (in a
  similarly kookie way) and reassembled into a
  sphere the size of the sun!

13
A Theorem of J.P. Serre

• (from homotopy theory)
• If n is even, then is a finitely generated
  abelian group of rank 1.

14
A Few Additional Notes

• what we know (or what we think we know) about
  mathematical knowledge
• From cognitive science
• Abstract ideas and relationships are understood
  through conceptual metaphors.
• From the mathematics education research
• Students who succeed in their mathematical
  studies tend to view mathematical knowledge more
  as a coherent system of ideas and
  relationshipsand less as the product of a
  procedure or validationwhen compared with
  students who have been less successful in
  mathematics.

15
The Platonist View

• Mathematical objects are real (albeit intangible)
  and independent of the mind that perceives them.
• Mathematical truth is timeless, waiting to be
  discovered.

16
The Logicist View

• Mathematical knowledge is analytic a priori,
  logically derived from indubitable truths.
• Definitions (or linguistics, in general) are
  essentially all that distinguishes specific
  mathematical content from generic logical
  propositions.

17
The Formalist View

• Mathematical objects are formulas with no
  external meaning they are structures that are
  formally postulated or formally defined within an
  axiomatic system.
• Mathematical truth refers only to consistency
  within the axiomatic system.
• Curry the essence of mathematics is the process
  of formalization.

18
The Intuitionist/Constructivist View

• Mathematical knowledge is produced through human
  mental activity.
• Appeal to the law of the excluded middle (and the
  axiom of choice) is not a valid step in a
  mathematical proof.

19
The Empiricist and Pragmatist Views

• Mathematical objects have a necessary existence
  and meaning inasmuch as they are the
  underpinnings of the empirical sciences.
  (Indispensability)
• The nature of a mathematical object is
  constrained by what we are able to observe (or
  comprehend).

20
The Humanist View

• Mathematical objects are mental objects with
  reproducible properties.
• These objects and their properties (truths) are
  confirmed and understood through intuition, which
  itself is cultivated and normed by the
  practitioners of mathematics.

21
The Structuralist View

• Mathematical knowledge is knowledge of
  relationships (structures), not objects.
• For example, to say we understand is to say we
  understand the relationship between the
  circumference and diameter of a circle.
• Knowledge of a theorem is knowledge of a
  necessary relationship.

22
Name that Epistemology

• I would say that mathematics is the science of
  skillful operations with concepts and rules
  invented for just this purpose. The principal
  emphasis is on the invention of concepts. ... The
  great mathematician fully, almost ruthlessly,
  exploits the domain of permissible reasoning and
  skirts the impermissible.
• Eugene Wigner

23
Name that Epistemology

• Certain things we want to say in science may
  compel us to admit into the range of values of
  the variables of quantification not only physical
  objects but also classes and relations of them
  also numbers, functions, and other objects of
  pure mathematics.
• To be is to be the value of a variable.
• W.V. Quine

24
Name that Epistemology

• Mathematical knowledge isnt infallible. Like
  science, mathematics can advance by making
  mistakes, correcting and recorrecting them.
• A proof is a conclusive argument that a proposed
  result follows from accepted theory. Follows
  means the argument convinces qualified, skeptical
  mathematicians.
• Reuben Hersh

25
Name that Epistemology

• Nothing has afforded me so convincing a proof of
  the unity of the Deity as these purely mental
  conceptions of numerical and mathematical
  science, which have been by slow degrees
  vouchsafed to manall of which must have existed
  in that sublimely omniscient Mind from eternity.
• Mary Somerville

26
Name that Epistemology

• The essence of a natural number is its relations
  to other numbers.
• The subject matter of arithmeticis the pattern
  common to any infinite collection of objects that
  has a successor relation, a unique initial
  object, and satisfies the induction principle.
• Stewart Shapiro

27
Every Rose has its Thornor, mathematical truth
is one slippery fish

• A Critique of Platonism
• The Platonistic appeal to a separate realm of
  pure ideas sounds a lot like good ol Cartesian
  dualism, and is apt to pay the same price for
  being unable to account for the integration of
  the two realms.

28
Every Rose has its Thornor, mathematical truth
is one slippery fish

• A Critique of Logicism
• Attempts to reduce modern mathematics to logical
  tautologies have failed miserably in practice and
  may have been doomed from the start in principle.
  Common notion, local convention, and intuitive
  allusion all appear to obscure actual mathematics
  from strictly logical deduction.

29
Every Rose has its Thornor, mathematical truth
is one slippery fish

• A Critique of Formalism
• Three words Godels Incompleteness Theorem.
• In any system rich enough to support the axioms
  of arithmetic, there will exist statements that
  bear a truth value, but can never be proved or
  disproved. Mathematics cannot prove its own
  consistency.

30
Every Rose has its Thornor, mathematical truth
is one slippery fish

• A Critique of Intuitionism/Constructivism
• Some notion of the continuumsuch as our real
  number lineseems both plausible and almost
  universal, even among those not educated in
  modern mathematics.
• Whats more, the mathematics of the real numbers
  works in practical application.

31
Every Rose has its Thornor, mathematical truth
is one slippery fish

• A Critique of Empiricism/Pragmatism
• This doctrine inexorably leads to the conclusion
  that inconceivable implies impossible. Yet
  history is filled with examples that were for
  centuries inconceivable but are now common
  knowledge. Indeed, mathematics provides us with
  objects that yet seem inconceivable, but are
  established to be mathematically possible.

32
Every Rose has its Thornor, mathematical truth
is one slippery fish

• A Critique of Humanism
• This view is pressed to explain the universality
  of mathematics. What about individuals, such as
  Ramanujan, who produced sophisticated results
  that were consistent with the systems used
  elsewhere, yet did not have the opportunity to
  norm their intuition against teachers or
  colleagues?

33
Every Rose has its Thornor, mathematical truth
is one slippery fish

• A Critique of Structuralism
• I couldnt help but notice that the best-known
  exponents of structuralism are philosophers, not
  mathematicians.
• In practice, mathematicians still refer to
  objects, certainly in their language and likely
  in their ontology.

34

• When assessing metaphysical or philosophical
  paradigms, its often helpful to compare the
  various alternatives against the sticky wickets
  to see which view is best able to make sense out
  of our most puzzling cases.
• Lets give it a whirl

35
Arithmetic of Irrational Numbers

• What, exactly, do we mean by ?
• The most obvious answer to this question (its
  what we get when we multiply by itself) is
  perhaps among the least legitimate.

...lets see..carry the 1and
36
Gabriels Horn

• Gabriels Horn can be gener-ated by rotating the
  curve
• over 1,8) around the x-axis.
• As a solid of revolution, it has finite volume.
• As a surface of revolution, it has infinite area.

37
The Peano-Hilbert Curve

• (from analysis)
• There exists a closed curve that completely fills
  a two-dimensional region.
• Image produced by Axel-Tobias Schreiner, Image
  produced by John Salmon
• Rochester Institute of Technology, and Michael
  Warren, Caltech
• Programming Language Concepts, Parallel,
  Out-of-core methods for
• http//www.cs.rit.edu/ats/plc-2002-2/html/skript.
  html N-body Simulation,
• http//www.cacr.caltech.edu/johns/pubs/siam
  97/html/online.html

38
The Banach-Tarski Theorem

• (from topology)
• An orange can be sliced into five pieces in such
  a way that the five pieces can be reassembled
  into two identical oranges, each the same size as
  the original.
• Better yet, a golf ball can be taken apart (in a
  similarly kookie way) and reassembled into a
  sphere the size of the sun!

39
A Theorem of J.P. Serre

• (from homotopy theory)
• If n is even, then is a finitely generated
  abelian group of rank 1.

40
and Those Other Notes

• what we know (or what we think we know) about
  mathematical knowledge
• From cognitive science
• Abstract ideas and relationships are understood
  through conceptual metaphors.
• From the mathematics education research
• Students who succeed in their mathematical
  studies tend to view mathematical knowledge more
  as a coherent system of ideas and
  relationshipsand less as the product of a
  procedure or validationwhen compared with
  students who have been less successful in
  mathematics.

41

• Whaddaya think?

42
A Brief Bibliography for the (amateur)
Philosopher of Mathematics

• Paul Benacerraf and Hilary Putnam,
• Philosophy of Mathematics, Prentice-Hall, 1964.
• Philip Davis and Reuben Hersh, The Mathematical
  Experience, Houghton Mifflin, 1981.
• Judith Grabiner, Is Mathematical Truth
  Time-Dependent?, American Mathematical Monthly
  81 354-365, 1974.
• Reuben Hersh, What is Mathematics, Really?,
  Oxford Press, 1997.
• George Lakoff and Rafael Nuñez, Where Mathematics
  Comes From, Basic Books, 2000.
• Stewart Shapiro, Thinking About Mathematics,
  Oxford Press, 2000.