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Unformatted text preview: Do Mathematicians Really 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 What distinguishes 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 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. Do all of our heroic mathematical 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. The Philosophy of Mathematics:
The Philosophy of Mathematics:
an 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. The Philosophy of Mathematics:
The Philosophy of Mathematics:
an unreasonably concise history
In the 19th Century, several developments (non
Euclidean geometry, Cantor’s set theory, and—a little later—Russell’s 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. The Philosophy of Mathematics:
The Philosophy of Mathematics:
an 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. In the modern mathematical community, there is 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. Arithmetic of Irrational Numbers
Arithmetic of Irrational Numbers
What, exactly, do we mean by ? π
2 The most obvious answer to this question (“it’s π
what we get when we multiply by itself”) is perhaps among the least legitimate.
...let’s see..carry the 1…and… Gabriel’s Horn Gabriel’s Horn can be gener
ated by rotating the curve
y = x −2 / 3over [1,∞) around the xaxis. As a solid of revolution, it has finite volume.
As a surface of revolution, it has infinite area. Pic ande uatio g ne db Mathe atic
ture
q
ns e rate y
m a. The PeanoHilbert Curve
The PeanoHilbert Curve
(from analysis) There exists a closed curve that completely fills a twodimensional region. Image produced by AxelTobias Schreiner,
Rochester Institute of Technology,
“Programming Language Concepts,”
http://www.cs.rit.edu/~ats/plc20022/html/skript.html Image produced by John Salmon
and Michael Warren, Caltech
“Parallel, Outofcore methods for
Nbody Simulation,”
http://www.cacr.caltech.edu/~johns/pubs/siam97/html/online.html The BanachTarski Theorem
The BanachTarski 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! A Theorem of J.P. Serre
A Theorem of J.P. Serre
(from homotopy theory) π 2 n−1is a finitely generated (S n )
If n is even, then abelian group of rank 1. A Few Additional Notes
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 relationships”—and less as the product of a procedure or validation—when compared with students who have been less successful in mathematics. 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.” Pic sc urte o theMac r His ryo Mathe atic Arc , http:/
ture o sy f
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m s hive 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. Pic c urte o theMac r Histo o Mathe atic Arc , http://
ture o sy f
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m s hive 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. Pic c urte o theMac r Histo o Mathe atic Arc , http://
ture o sy f
Tuto
ry f
m s hive 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. Pic c urte o theMac r Histo o Mathe atic Arc , http://
ture o sy f
Tuto
ry f
m s hive 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). Pic c urte o theHarvardUnive ity De
ture o sy f
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so 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. Pic c urte o theMac r Histo o Mathe atic Arc , http://
ture o sy f
Tuto
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m s hive The Structuralist View
Mathematical knowledge is knowledge of relationships (structures), not objects. π
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. Pic c urte o St. Andre Unive
ture o sy f
ws
rsityhttp://www.standre .ac
ws .uk Name that Epistemology:
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:
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:
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:
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:
Name that Epistemology:
“The essence of a natural number is its relations to other numbers.
The subject matter of arithmetic…is 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 Every Rose has its Thorn
Every Rose has its Thorn or, 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. Every Rose has its Thorn
Every Rose has its Thorn or, 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. Every Rose has its Thorn
Every Rose has its Thorn or, mathematical truth is one slippery fish 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
Every Rose has itsThorn or, mathematical truth is one slippery fish 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
Every Rose has its Thorn or, 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. Every Rose has its Thorn
Every Rose has its Thorn or, 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? Every Rose has its Thorn
Every Rose has its Thorn or, mathematical truth is one slippery fish
A Critique of Structuralism:
I couldn’t help but notice that the bestknown exponents of structuralism are philosophers, not mathematicians.
In practice, mathematicians still refer to ‘objects,’ certainly in their language and likely in their ontology. When assessing metaphysical or When assessing metaphysical or philosophical paradigms, it’s 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. Let’s give it a whirl… Arithmetic of Irrational Numbers
Arithmetic of Irrational Numbers
What, exactly, do we mean by ? π
2 The most obvious answer to this question (“it’s π
what we get when we multiply by itself”) is perhaps among the least legitimate.
...let’s see..carry the 1…and… Gabriel’s Horn Gabriel’s Horn can be gener
ated by rotating the curve
y = x −2 / 3over [1,∞) around the xaxis. As a solid of revolution, it has finite volume.
As a surface of revolution, it has infinite area. Pic ande uatio g ne db Mathe atic
ture
q
ns e rate y
m a. The PeanoHilbert Curve
The PeanoHilbert Curve
(from analysis) There exists a closed curve that completely fills a twodimensional region. Image produced by AxelTobias Schreiner,
Rochester Institute of Technology,
“Programming Language Concepts,”
http://www.cs.rit.edu/~ats/plc20022/html/skript.html Image produced by John Salmon
and Michael Warren, Caltech
“Parallel, Outofcore methods for
Nbody Simulation,”
http://www.cacr.caltech.edu/~johns/pubs/siam97/html/online.html The BanachTarski Theorem
The BanachTarski 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! A Theorem of J.P. Serre
A Theorem of J.P. Serre
(from homotopy theory) π 2 n−1is a finitely generated (S n )
If n is even, then abelian group of rank 1. …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 relationships”—and less as the product of a procedure or validation—when compared with students who have been less successful in mathematics. Whaddaya think?
Whaddaya think? A Brief Bibliography A Brief Bibliography for the (amateur) Philosopher of Mathematics
Paul Benacerraf and Hilary Putnam, Philosophy of Mathematics, PrenticeHall, 1964.
Philip Davis and Reuben Hersh, The Mathematical Experience, Houghton Mifflin, 1981.
Judith Grabiner, “Is Mathematical Truth TimeDependent?”, American Mathematical Monthly 81: 354365, 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. ...
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