Get Real
the challenges of mathematical
epistemology
Jason Douma
University of Sioux Falls
November 18, 2003
presented to the
SDSU Senior Seminar in Mathematics
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!
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
epistemologists awake at night.
The Question of Epistemology:
an unreasonably concise history
Through the 18th Century, an understanding that
mathematics was in some way part of “natural
philosophy” was widely accepted.
In the 19th Century, several developments (nonEuclidean geometry, Cantor’s set theory, and—a
little later—Russell’s paradox, to name a few)
triggered a foundational crisis.
The Question of Epistemology:
an unreasonably concise history
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 epistemologies emerged, each with
their own champions.
After lying relatively dormant for half a century,
these philosophical matters are now receiving
renewed, as reflected by the Philosophy of
Mathematics SIGMAA unveiled in January, 2003.
In the modern mathematical community, there is
very little controversy over what it takes to show
that something is “true”…this 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.
Gabriel’s Horn
Gabriel’s Horn can be generated by rotating the curve
y  x 2 / 3 over [1,∞)
around the x-axis.
As a solid of revolution, it
has finite volume.
As a surface of revolution, it
has infinite area.
Picture and equations generated by
Mathematica.
The Peano-Hilbert Curve
(from analysis)
There exists a closed curve that completely fills a
two-dimensional region.
Image produced by Axel-Tobias Schreiner,
Rochester Institute of Technology,
“Programming Language Concepts,”
http://www.cs.rit.edu/~ats/plc-2002-2/html/skript.html
Image produced by John Salmon
and Michael Warren, Caltech
“Parallel, Out-of-core methods for
N-body Simulation,”
http://www.cacr.caltech.edu/~johns/pubs/siam97/html/online.html
A Theorem of J.P. Serre
(from homotopy theory)
n

(
S
If n is even, then 2 n1 ) is a finitely
generated abelian group of rank 1.
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.”
Pictures courtesy of the MacTutor
History of Mathematics Archive,
http://www-gap.dcs.stand.ac.uk/~history/
The Formalist View
Mathematical objects have 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.
Picture courtesy of the MacTutor
History of Mathematics Archive,
http://www-gap.dcs.stand.ac.uk/~history/
The Intuitionist/Constructivist View
Mathematical objects finitely
derived from the integers have
real meaning; the rest is
mathematical fantasy.
Appeal to the law of the excluded
middle is not a valid step in a
mathematical proof.
Picture courtesy of the MacTutor
History of Mathematics Archive,
http://www-gap.dcs.stand.ac.uk/~history/
The Empiricist and Pragmatist Views
Mathematical objects have a
necessary existence and
meaning inasmuch as they are
the underpinnings of the
empirical sciences.
The nature of a mathematical
object is constrained by what
we are able to observe (or
comprehend).
Picture courtesy of the Harvard
University Department of Philosophy,
http://www.fas.harvard.edu/~phildept
/html/emereti.html
The Logicist View
Mathematical objects are values
taken on by logical variables.
Mathematical truth is logical
tautology.
Picture courtesy of the MacTutor
History of Mathematics Archive,
http://www-gap.dcs.stand.ac.uk/~history/
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.
Picture courtesy of the MacTutor
History of Mathematics Archive,
http://www-gap.dcs.stand.ac.uk/~history/
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
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
Name that Epistemology:
“Mathematical knowledge isn’t 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
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 man…all of which must have
existed in that sublimely omniscient Mind from
eternity.”
Mary Somerville
Name that Epistemology:
“Despite their remoteness from sense experience,
we do have something like a perception also of
the objects of set theory, as is seen from the fact
that the axioms force themselves upon us as
being true.”
Kurt Gődel
Every Rose has its Thorn:
a perfect epistemology is hard to find
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.
Every Rose has its Thorn:
a perfect epistemology is hard to find
A Critique of Formalism:
Three words: Gődel’s 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.
Every Rose has its Thorn:
a perfect epistemology is hard to find
A Critique of Intuitionism/Constructivism:
Some notion of the continuum—such as our real
number line—seems both plausible and almost
universal, even among those not educated in
modern mathematics.
What’s more, the mathematics of the real numbers
works in practical application.
Every Rose has its Thorn:
a perfect epistemology is hard to find
A Critique of Empiricism/Pragmatism:
This doctrine tends to lead inexorably to the
conclusion that “inconceivable implies impossible.”
Yet history is filled with examples that were for
centuries inconceivable but are now common
knowledge. What’s more, mathematics provides
us with objects that yet seem inconceivable, but
are established to be mathematically possible.
Every Rose has its Thorn:
a perfect epistemology is hard to find
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.
Every Rose has its Thorn:
a perfect epistemology is hard to find
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?
When assessing metaphysical or
epistemological paradigms, it’s often helpful
to compare the various paradigms against
the “sticky wickets” to see which view is
best able to make sense out of the puzzling
case at hand.
Let’s give it a whirl…
Gabriel’s Horn
Gabriel’s Horn can be generated by rotating the curve
y  x 2 / 3 over [1,∞)
around the x-axis.
As a solid of revolution, it
has finite volume.
As a surface of revolution, it
has infinite area.
Picture and equations generated by
Mathematica.
The Peano-Hilbert Curve
(from analysis)
There exists a closed curve that completely fills a
two-dimensional region.
Image produced by Axel-Tobias Schreiner,
Rochester Institute of Technology,
“Programming Language Concepts,”
http://www.cs.rit.edu/~ats/plc-2002-2/html/skript.html
Image produced by John Salmon
and Michael Warren, Caltech
“Parallel, Out-of-core methods for
N-body Simulation,”
http://www.cacr.caltech.edu/~johns/pubs/siam97/html/online.html
A Theorem of J.P. Serre
(from homotopy theory)
n

(
S
) is a finitely
If n is even, then 2 n1
generated abelian group of rank 1.
Whaddaya think?
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.
Edward Rothstein, Emblems of Mind, Avon Books, 1995.