get real the challenges of mathematical epistemology jason douma university of sioux falls november...

Get RealGet Realthe challenges of mathematical the challenges of mathematical

epistemologyepistemology

Jason DoumaJason DoumaUniversity of Sioux FallsUniversity of Sioux Falls

November 18, 2003November 18, 2003

presented to thepresented to theSDSU Senior Seminar in MathematicsSDSU Senior Seminar in Mathematics

What distinguishes What distinguishes mathematics from the usual mathematics from the usual

“natural sciences?”“natural sciences?”Mathematics is not fundamentally Mathematics is not fundamentally

empirical empirical —it does not rely on sensory —it does not rely on sensory observation or instrumental observation or instrumental measurement to determine what is measurement to determine what is true.true.

Indeed, mathematical objects Indeed, mathematical objects themselves cannot be observed at all!themselves cannot be observed at all!

Does this mean that mathematical Does this mean that mathematical objects are not real?objects are not real?

Does this mean that mathematical Does this mean that mathematical knowledge is arbitrary?knowledge is arbitrary?

Good questionsGood questions!!These are the things that keep mathematical These are the things that keep mathematical

epistemologists awake at night.epistemologists awake at night.

The Question of Epistemology:The Question of Epistemology:an unreasonably concise an unreasonably concise

historyhistoryThrough the 18Through the 18thth Century, an understanding Century, an understanding

that mathematics was in some way part of that mathematics was in some way part of “natural philosophy” was widely accepted.“natural philosophy” was widely accepted.

In the 19In the 19thth Century, several developments Century, several developments (non-Euclidean geometry, Cantor’s set (non-Euclidean geometry, Cantor’s set theory, and—a little later—Russell’s theory, and—a little later—Russell’s paradox, to name a few) triggered a paradox, to name a few) triggered a foundational crisis.foundational crisis.

The Question of Epistemology:The Question of Epistemology:an unreasonably concise an unreasonably concise

historyhistoryThe final decades of the 19The final decades of the 19thth Century and first Century and first

half of the 20half of the 20thth Century were marked by a Century were marked by a heroic effort to make the body of heroic effort to make the body of mathematics axiomatically rigorous. During mathematics axiomatically rigorous. During this time, competing epistemologies this time, competing epistemologies emerged, each with their own champions.emerged, each with their own champions.

After lying relatively dormant for half a After lying relatively dormant for half a century, these philosophical matters are now century, these philosophical matters are now receiving renewed, as reflected by the receiving renewed, as reflected by the Philosophy of Mathematics SIGMAA unveiled Philosophy of Mathematics SIGMAA unveiled in January, 2003.in January, 2003.

In the modern mathematical community, there In the modern mathematical community, there is very little controversy over what it takes to is very little controversy over what it takes to showshow that something is “true”…this is what that something is “true”…this is what mathematical proof is all about.mathematical proof is all about.

Most disagreements over this matter are Most disagreements over this matter are questions of degree, not kind. questions of degree, not kind.

(Exceptions: proofs by machine, probabilistic (Exceptions: proofs by machine, probabilistic proof, and arguments from a few extreme proof, and arguments from a few extreme fallibilists)fallibilists)

However, when discussion turns to the However, when discussion turns to the meaning of such “truths” (that is, the nature meaning of such “truths” (that is, the nature of mathematical knowledge), genuine and of mathematical knowledge), genuine and substantial distinctions emerge.substantial distinctions emerge.

Picture and equations generaPicture and equations generated by Mathematica.ted by Mathematica.

Gabriel’s HornGabriel’s HornGabriel’s Horn can be Gabriel’s Horn can be

gener-ated by rotating gener-ated by rotating the curvethe curve

over [1,∞) around over [1,∞) around the x-axis. the x-axis.

As a solid of revolution, it As a solid of revolution, it has finite volume.has finite volume.

As a surface of revolution, it As a surface of revolution, it has infinite area.has infinite area.

3/2xy

The Peano-Hilbert CurveThe Peano-Hilbert Curve(from analysis)(from analysis)

There exists a closed curve that completely There exists a closed curve that completely fills a two-dimensional region.fills a two-dimensional region.

Image produced by Axel-Tobias Schreiner,Image produced by Axel-Tobias Schreiner, Image produced by John SalmonImage produced by John SalmonRochester Institute of Technology,Rochester Institute of Technology, and Michael Warren, Caltechand Michael Warren, Caltech““Programming Language Concepts,”Programming Language Concepts,” “Parallel, Out-of-core methods for“Parallel, Out-of-core methods forhttp://www.cs.rit.edu/~ats/plc-2002-2/html/skript.htmlhttp://www.cs.rit.edu/~ats/plc-2002-2/html/skript.html N-body Simulation,”N-body Simulation,”

http://www.cacr.caltech.edu/~johns/pubs/siam97/http://www.cacr.caltech.edu/~johns/pubs/siam97/html/online.htmlhtml/online.html

A Theorem of J.P. SerreA Theorem of J.P. Serre(from homotopy theory)(from homotopy theory)

If If nn is even, then is even, then is a finitely is a finitely generated abelian group of rank 1.generated abelian group of rank 1.

)(12n

n S

Pictures courtesy of the MacPictures courtesy of the MacTutor History of MathematicsTutor History of Mathematics Archive, http://www-gap.dcs. Archive, http://www-gap.dcs.st-and.ac.uk/~history/st-and.ac.uk/~history/

The Platonist ViewThe Platonist View

Mathematical objects are Mathematical objects are real (albeit intangible) real (albeit intangible) and independent of the and independent of the mind that perceives mind that perceives them.them.

Mathematical truth is Mathematical truth is timeless, waiting to be timeless, waiting to be “discovered.”“discovered.”

Picture courtesy of the MacTPicture courtesy of the MacTutor History of Mathematics utor History of Mathematics Archive, http://www-gap.dcs.Archive, http://www-gap.dcs.st-and.ac.uk/~history/st-and.ac.uk/~history/

The Formalist ViewThe Formalist ViewMathematical objects have no Mathematical objects have no

external meaning; they are external meaning; they are structures that are formally structures that are formally postulated or formally postulated or formally defined within an axiomatic defined within an axiomatic system.system.

Mathematical truth refers only Mathematical truth refers only to consistency within the to consistency within the axiomatic system.axiomatic system.


The Intuitionist/Constructivist The Intuitionist/Constructivist ViewView

Mathematical objects finitely Mathematical objects finitely derived from the integers derived from the integers have real meaning; the rest have real meaning; the rest is mathematical fantasy.is mathematical fantasy.

Appeal to the law of the Appeal to the law of the excluded middle is not a excluded middle is not a valid step in a valid step in a mathematical proof.mathematical proof.

Picture courtesy of the HarvaPicture courtesy of the Harvard University Department of rd University Department of Philosophy, http://www.fas.haPhilosophy, http://www.fas.harvard.edu/~phildept/html/emrvard.edu/~phildept/html/emereti.htmlereti.html

The Empiricist and Pragmatist The Empiricist and Pragmatist ViewsViews

Mathematical objects have a Mathematical objects have a necessary existence and necessary existence and meaning inasmuch as they meaning inasmuch as they are the underpinnings of are the underpinnings of the empirical sciences.the empirical sciences.

The nature of a mathematical The nature of a mathematical object is constrained by object is constrained by what we are able to what we are able to observe (or comprehend).observe (or comprehend).


The Logicist ViewThe Logicist ViewMathematical objects are Mathematical objects are

values taken on by logical values taken on by logical variables.variables.

Mathematical truth is logical Mathematical truth is logical tautology.tautology.


The Humanist ViewThe Humanist ViewMathematical objects are mental Mathematical objects are mental

objects with reproducible objects with reproducible properties. properties.

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

Name that Epistemology:Name that Epistemology:

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

Eugene WignerEugene Wigner


““Certain things we want to say in science may Certain things we want to say in science may compel us to admit into the range of values compel us to admit into the range of values of the variables of quantification not only of the variables of quantification not only physical objects but also classes and physical objects but also classes and relations of them; also numbers, functions, relations of them; also numbers, functions, and other objects of pure mathematics.” and other objects of pure mathematics.”

““To be is to be the value of a variable.”To be is to be the value of a variable.”

W.V. QuineW.V. Quine


““Mathematical knowledge isn’t infallible. Like Mathematical knowledge isn’t infallible. Like science, mathematics can advance by science, mathematics can advance by making mistakes, correcting and making mistakes, correcting and recorrecting them.recorrecting them.

A proof is a conclusive argument that a A proof is a conclusive argument that a proposed result follows from accepted proposed result follows from accepted theory. ‘Follows’ means the argument theory. ‘Follows’ means the argument convinces qualified, skeptical convinces qualified, skeptical mathematicians.”mathematicians.”

Reuben HershReuben Hersh


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

Mary SomervilleMary Somerville


““Despite their remoteness from sense Despite their remoteness from sense experience, we do have something like a experience, we do have something like a perception also of the objects of set theory, perception also of the objects of set theory, as is seen from the fact that the axioms as is seen from the fact that the axioms force themselves upon us as being true.”force themselves upon us as being true.”

Kurt GődelKurt Gődel

Every Rose has its Thorn:Every Rose has its Thorn:a perfect epistemology is hard to a perfect epistemology is hard to

findfind

A Critique of Platonism:A Critique of Platonism:

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


findfind

A Critique of Formalism:A Critique of Formalism:

Three words: Gődel’s Incompleteness Theorem.Three words: Gődel’s Incompleteness Theorem.

In any system rich enough to support the In any system rich enough to support the axioms of arithmetic, there will exist axioms of arithmetic, there will exist statements that bear a truth value, but can statements that bear a truth value, but can never be proved or disproved. Mathematics never be proved or disproved. Mathematics cannot prove its own consistency.cannot prove its own consistency.


findfind

A Critique of Intuitionism/Constructivism:A Critique of Intuitionism/Constructivism:

Some notion of the continuum—such as our real Some notion of the continuum—such as our real number line—seems both plausible and number line—seems both plausible and almost universal, even among those not almost universal, even among those not educated in modern mathematics. educated in modern mathematics.

What’s more, the mathematics of the real What’s more, the mathematics of the real numbers numbers worksworks in practical application. in practical application.


findfind

A Critique of Empiricism/Pragmatism:A Critique of Empiricism/Pragmatism:

This doctrine tends to lead inexorably to the This doctrine tends to lead inexorably to the conclusion that “inconceivable implies conclusion that “inconceivable implies impossible.” Yet history is filled with examples impossible.” Yet history is filled with examples that were for centuries inconceivable but are now that were for centuries inconceivable but are now common knowledge. What’s more, mathematics common knowledge. What’s more, mathematics provides us with objects that yet seem provides us with objects that yet seem inconceivable, but are established to be inconceivable, but are established to be mathematically possible.mathematically possible.


findfind

A Critique of Logicism:A Critique of Logicism:

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


findfind

A Critique of Humanism:A Critique of Humanism:

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

When assessing metaphysical or When assessing metaphysical or epistemological paradigms, it’s often epistemological paradigms, it’s often helpful to compare the various helpful to compare the various paradigms against the “sticky paradigms against the “sticky wickets” to see which view is best wickets” to see which view is best able to make sense out of the puzzling able to make sense out of the puzzling case at hand.case at hand.

Let’s give it a whirl…Let’s give it a whirl…

Picture and equations generaPicture and equations generated by Mathematica.ted by Mathematica.

Gabriel’s HornGabriel’s HornGabriel’s Horn can be Gabriel’s Horn can be

gener-ated by rotating gener-ated by rotating the curvethe curve

over [1,∞) around over [1,∞) around the x-axis. the x-axis.

As a solid of revolution, it As a solid of revolution, it has finite volume.has finite volume.

As a surface of revolution, it As a surface of revolution, it has infinite area.has infinite area.

3/2xy

The Peano-Hilbert CurveThe Peano-Hilbert Curve(from analysis)(from analysis)

There exists a closed curve that completely There exists a closed curve that completely fills a two-dimensional region.fills a two-dimensional region.

Image produced by Axel-Tobias Schreiner,Image produced by Axel-Tobias Schreiner, Image produced by John SalmonImage produced by John SalmonRochester Institute of Technology,Rochester Institute of Technology, and Michael Warren, Caltechand Michael Warren, Caltech““Programming Language Concepts,”Programming Language Concepts,” “Parallel, Out-of-core methods for“Parallel, Out-of-core methods forhttp://www.cs.rit.edu/~ats/plc-2002-2/html/skript.htmlhttp://www.cs.rit.edu/~ats/plc-2002-2/html/skript.html N-body Simulation,”N-body Simulation,”

http://www.cacr.caltech.edu/~johns/pubs/siam97/http://www.cacr.caltech.edu/~johns/pubs/siam97/html/online.htmlhtml/online.html

A Theorem of J.P. SerreA Theorem of J.P. Serre(from homotopy theory)(from homotopy theory)

If If nn is even, then is even, then is a finitely is a finitely generated abelian group of rank 1.generated abelian group of rank 1.

)(12n

n S

Whaddaya think?Whaddaya think?

A Brief Bibliography A Brief Bibliography for the (amateur) Philosopher for the (amateur) Philosopher

of Mathematicsof MathematicsPaul Benacerraf and Hilary Putnam, Paul Benacerraf and Hilary Putnam,

Philosophy of MathematicsPhilosophy of Mathematics, Prentice-Hall, 1964., Prentice-Hall, 1964.Philip Davis and Reuben Hersh, Philip Davis and Reuben Hersh, The Mathematical The Mathematical

Experience, Experience, Houghton Mifflin, 1981.Houghton Mifflin, 1981.Judith Grabiner, “Is Mathematical Truth Time-Judith Grabiner, “Is Mathematical Truth Time-

Dependent?”, Dependent?”, American Mathematical MonthlyAmerican Mathematical Monthly 81: 81: 354-365, 1974.354-365, 1974.

Reuben Hersh, Reuben Hersh, What is Mathematics, Really?What is Mathematics, Really?, Oxford , Oxford Press, 1997.Press, 1997.

George Lakoff and Rafael Nuñez, George Lakoff and Rafael Nuñez, Where Mathematics Where Mathematics Comes FromComes From, Basic Books, 2000., Basic Books, 2000.

Edward Rothstein, Edward Rothstein, Emblems of MindEmblems of Mind, Avon Books, 1995., Avon Books, 1995.

get real the challenges of mathematical epistemology jason douma university of sioux falls november...

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