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Thanks Tom. This sounds like a good reading suggestions and good
observations. pat
=================================
In a message dated 3/12/01 7:57:44 PM Eastern Standard Time,
[log in to unmask] writes:
> I would recommend that the participants in this debate
> read Imre Lakatos's seminal book 'Proofs and
> Refutations: The Logic of Mathematical Discovery'.
>
> The debate here is not about mathematics it is about
> the philosophy of mathematics. Lakatos directly
> addresses what I see as the subject of the debate here
> in his book when he makes the distinction between
> formal and informal mathematics. Formal mathematics is
> contained in the proofs published in mathematical
> journals. Informal mathematics are the strategies that
> working mathemeticians use to make their work a useful
> exercise in mathematical discovery.
>
> I see the objection to the four color proof here. It
> was undoubtedly a valid formal proof. However it did
> nothing to advance the cause of mathematics beyond
> this.
>
> The reason that Lakatos equates proofs and refutation
> in his title is his contention that it is the
> refutations that are developed that show
> mathematicians the deficiencies in their theories. It
> is their attempts to deal with unwanted and unexpected
> refutations - to preserve a valuable theory in the
> face of imperfect axioms and proof methods - that
> teach mathemeticians the true depths of their
> conceptions and to point the way to new and deeper
> ones.
>
> Lakatos shows this by an account of the historical
> development of the concept of proof in mathematics and
> by showing in historical detail how certain valuable
> 'proofs' were preserved in the face of refutation. To
> this point Lakatos shows that the 'proofs' of the
> truth of Euler's number are no proofs at all. The
> great mathemetician Euler noticed that for any regular
> polyhedron the formula V-E+F=2 where V is the number
> of vertexes, E is the number of edges and F is the
> number of faces. Euler's and his successors proofs
> fall before any number of counterexamples. Does this
> prove that the theorem is 'incorrect?' Or does it mean
> as the mathemetician's actions show that they thought
> it meant was that their concept of what constituted a
> regular polyhedron was deficient. Lakatos shows how
> these conceptions were modified over a couple of
> hundred years as counterexample after counterexample
> was faced. These counterexamples all made mathematics
> stronger by deepening the conception of what polyhedra
> really are and by discovering new classes of them. In
> the end Euler's formula turned out not to have a proof
> but to be in effect a tautology. It is true for the
> regular polyhedra for which it is true by the
> definition of what constittues a polyhedron. It is
> true because human mathematicians in order to make
> progress need it to be true.
>
> The computer proof of the four color theorem was a
> triumph of formal mathematics. Its critics complained
> and. if interpreted according to what Lakatos wrote,
> they complained because it defeated the progress of
> informal mathematics.
>
> Mathematical proofs are useful tools. The tell us what
> we need to know. Formal mathematics is about finding
> them. Informal mathematics is about making them
> useful. Mathematics is not some Platonian ideal
> divorced from humanity, painting, poetry ... It is a
> human endeavor to meet human needs.
>
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<HTML><FONT FACE=arial,helvetica><FONT SIZE=3 FAMILY="SANSSERIF" FACE="Arial Narrow" LANG="0"><B>Thanks Tom. This sounds like a good reading suggestions and good
<BR>observations. pat
<BR>=================================
<BR>
<BR>In a message dated 3/12/01 7:57:44 PM Eastern Standard Time,
<BR>[log in to unmask] writes:
<BR>
<BR></FONT><FONT COLOR="#000000" SIZE=2 FAMILY="SANSSERIF" FACE="Arial" LANG="0"></B>
<BR><BLOCKQUOTE TYPE=CITE style="BORDER-LEFT: #0000ff 2px solid; MARGIN-LEFT: 5px; MARGIN-RIGHT: 0px; PADDING-LEFT: 5px">I would recommend that the participants in this debate
<BR>read Imre Lakatos's seminal book 'Proofs and
<BR>Refutations: The Logic of Mathematical Discovery'.
<BR>
<BR>The debate here is not about mathematics it is about
<BR>the philosophy of mathematics. Lakatos directly
<BR>addresses what I see as the subject of the debate here
<BR>in his book when he makes the distinction between
<BR>formal and informal mathematics. Formal mathematics is
<BR>contained in the proofs published in mathematical
<BR>journals. Informal mathematics are the strategies that
<BR>working mathemeticians use to make their work a useful
<BR>exercise in mathematical discovery.
<BR>
<BR>I see the objection to the four color proof here. It
<BR>was undoubtedly a valid formal proof. However it did
<BR>nothing to advance the cause of mathematics beyond
<BR>this.
<BR>
<BR>The reason that Lakatos equates proofs and refutation
<BR>in his title is his contention that it is the
<BR>refutations that are developed that show
<BR>mathematicians the deficiencies in their theories. It
<BR>is their attempts to deal with unwanted and unexpected
<BR>refutations - to preserve a valuable theory in the
<BR>face of imperfect axioms and proof methods - that
<BR>teach mathemeticians the true depths of their
<BR>conceptions and to point the way to new and deeper
<BR>ones.
<BR>
<BR>Lakatos shows this by an account of the historical
<BR>development of the concept of proof in mathematics and
<BR>by showing in historical detail how certain valuable
<BR>'proofs' were preserved in the face of refutation. To
<BR>this point Lakatos shows that the 'proofs' of the
<BR>truth of Euler's number are no proofs at all. The
<BR>great mathemetician Euler noticed that for any regular
<BR>polyhedron the formula V-E+F=2 where V is the number
<BR>of vertexes, E is the number of edges and F is the
<BR>number of faces. Euler's and his successors proofs
<BR>fall before any number of counterexamples. Does this
<BR>prove that the theorem is 'incorrect?' Or does it mean
<BR>as the mathemetician's actions show that they thought
<BR>it meant was that their concept of what constituted a
<BR>regular polyhedron was deficient. Lakatos shows how
<BR>these conceptions were modified over a couple of
<BR>hundred years as counterexample after counterexample
<BR>was faced. These counterexamples all made mathematics
<BR>stronger by deepening the conception of what polyhedra
<BR>really are and by discovering new classes of them. In
<BR>the end Euler's formula turned out not to have a proof
<BR>but to be in effect a tautology. It is true for the
<BR>regular polyhedra for which it is true by the
<BR>definition of what constittues a polyhedron. It is
<BR>true because human mathematicians in order to make
<BR>progress need it to be true.
<BR>
<BR>The computer proof of the four color theorem was a
<BR>triumph of formal mathematics. Its critics complained
<BR>and. if interpreted according to what Lakatos wrote,
<BR>they complained because it defeated the progress of
<BR>informal mathematics.
<BR>
<BR>Mathematical proofs are useful tools. The tell us what
<BR>we need to know. Formal mathematics is about finding
<BR>them. Informal mathematics is about making them
<BR>useful. Mathematics is not some Platonian ideal
<BR>divorced from humanity, painting, poetry ... It is a
<BR>human endeavor to meet human needs.
<BR></FONT><FONT COLOR="#000000" SIZE=3 FAMILY="SANSSERIF" FACE="Arial" LANG="0"></BLOCKQUOTE>
<BR></FONT><FONT COLOR="#000000" SIZE=3 FAMILY="SANSSERIF" FACE="Arial Narrow" LANG="0"><B>
<BR></B></FONT></HTML>
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