Comments on: Debreu’s Theory of Value http://www.arsmathematica.net/archives/2007/06/28/debreus-theory-of-value/ Dedicated to the mathematical arts. Tue, 07 Oct 2008 07:34:57 +0000 http://wordpress.org/?v=2.5.1 By: Michael Greinecker http://www.arsmathematica.net/archives/2007/06/28/debreus-theory-of-value/#comment-53510 Michael Greinecker Sat, 07 Jul 2007 22:19:08 +0000 http://www.arsmathematica.net/archives/2007/06/28/debreus-theory-of-value/#comment-53510 Well, basically in the (informal) Bourbaki sense. In his "Valuation Equilibrium and Pareto Optimum", Debreu basically shows how the first welfare theorem is basically a simple argument about general vector spaces and the second welfare theorem a theorem on topological vector spaces. An important line is: "Its main interest, however,may be that by forcing one to a greater generality it brings out with greater clarity and simplity the basic concepts of the analysis and its logical structure. Not a single simplification of the proofs woul indeed be brought about by restriction to the finite dimensional case. " So it is really a methodological paper. Well, basically in the (informal) Bourbaki sense. In his “Valuation Equilibrium and Pareto Optimum”, Debreu basically shows how the first welfare theorem is basically a simple argument about general vector spaces and the second welfare theorem a theorem on topological vector spaces. An important line is:

“Its main interest, however,may be that by forcing one to a greater generality it brings out with greater clarity and simplity the basic concepts of the analysis and its logical structure. Not a single simplification of the proofs woul indeed be brought about by restriction to the finite dimensional case. ”

So it is really a methodological paper.

]]> By: Walt http://www.arsmathematica.net/archives/2007/06/28/debreus-theory-of-value/#comment-53287 Walt Wed, 04 Jul 2007 19:25:58 +0000 http://www.arsmathematica.net/archives/2007/06/28/debreus-theory-of-value/#comment-53287 Michael, I'm not familiar with the term "structural" as you're using it. Does it mean abstract? Michael, I’m not familiar with the term “structural” as you’re using it. Does it mean abstract?

]]> By: Jonathan Vos Post http://www.arsmathematica.net/archives/2007/06/28/debreus-theory-of-value/#comment-53056 Jonathan Vos Post Sun, 01 Jul 2007 17:58:06 +0000 http://www.arsmathematica.net/archives/2007/06/28/debreus-theory-of-value/#comment-53056 Mathematical Economics has evolved in curious directions, often towards what is capable of publishable proof, rather than what is observed about the real world with real humans in real markets. I speak as someone who has published refereed papers on Mathematical Economics, with a coauthor who hands out PhDs in the subject. I tell my students that, in its heart of hearts, mathematics is about 3 things: * Quantity * Structure * Change and various combinations of those 3. Quantity, they slightly know, with integers, and confused contact with fractions and decimals. Structure, they are amazed, is not only in Geometry. I teach even the weakest 9th graders about Triangular Numbers, and how young Karl Gauss added 1 + 2 + 3 + ... + 98 + 99 + 100 and I see the metaphorical light go on over their heads. Then I return to square numbers, and sum them to get square pyramidal numbers... Change, I tell them about what a genius/madman Newton was, and what Calculus is about. But then I tell them about Motion as change, and how we will study speed = distance/time, and cars, and busses, and trains, and rocketships. Then we talk about commuting. Soon we're on commutative laws... But they mostly HATE math. <a href="http://golem.ph.utexas.edu/category/2007/06/why_math_teachers_get_grumpy.html#comments" rel="nofollow">June 13, 2007 Why Math Teachers Get Grumpy Posted by John Baez</a> [truncated] Comments: [truncated] High Schoolers Hate Math; Re: Why Math Teachers Get Grumpy I asked 9th, 10th, 11th, and 12th grade summer school students at a Pasadena high school to write a paragraph on (their choice) “Why I Love Math” or “Why I Hate Math.” Their responses (names removed for privacy, no spelling or grammar corrections): I wouldn’t necessarily say that I hate math, I just dislike it a great deal... Mathematical Economics has evolved in curious directions, often towards what is capable of publishable proof, rather than what is observed about the real world with real humans in real markets. I speak as someone who has published refereed papers on Mathematical Economics, with a coauthor who hands out PhDs in the subject.

I tell my students that, in its heart of hearts, mathematics is about 3 things:

* Quantity

* Structure

* Change

and various combinations of those 3.

Quantity, they slightly know, with integers, and confused contact with fractions and decimals.

Structure, they are amazed, is not only in Geometry. I teach even the weakest 9th graders about Triangular Numbers, and how young Karl Gauss added 1 + 2 + 3 + … + 98 + 99 + 100 and I see the metaphorical light go on over their heads. Then I return to square numbers, and sum them to get square pyramidal numbers…

Change, I tell them about what a genius/madman Newton was, and what Calculus is about. But then I tell them about Motion as change, and how we will study speed = distance/time, and cars, and busses, and trains, and rocketships. Then we talk about commuting. Soon we’re on commutative laws…

But they mostly HATE math.

June 13, 2007
Why Math Teachers Get Grumpy
Posted by John Baez

[truncated]

Comments:

[truncated]

High Schoolers Hate Math; Re: Why Math Teachers Get Grumpy

I asked 9th, 10th, 11th, and 12th grade summer school students at a Pasadena high school to write a paragraph on (their choice) “Why I Love Math” or “Why I Hate Math.”

Their responses (names removed for privacy, no spelling or grammar corrections):

I wouldn’t necessarily say that I hate math, I just dislike it a great deal…

]]> By: Michael Greinecker http://www.arsmathematica.net/archives/2007/06/28/debreus-theory-of-value/#comment-52910 Michael Greinecker Fri, 29 Jun 2007 20:55:38 +0000 http://www.arsmathematica.net/archives/2007/06/28/debreus-theory-of-value/#comment-52910 Yes, it doesn't even have much structural flavor. For this, you can see his 1954 paper <a href="http://cowles.econ.yale.edu/P/cp/p00b/p0084.pdf" rel="nofollow">Valuation Equilibrium and Pareto Optimum</a>, which is essentially a propaganda piece for reducing things to structure. BTW: A badly looking version of the TOV can be found <a href="http://cowles.econ.yale.edu/P/cm/m17/index.htm" rel="nofollow">here</a>. Yes, it doesn’t even have much structural flavor. For this, you can see his 1954 paper Valuation Equilibrium and Pareto Optimum, which is essentially a propaganda piece for reducing things to structure.

BTW: A badly looking version of the TOV can be found here.

]]> By: John Armstrong http://www.arsmathematica.net/archives/2007/06/28/debreus-theory-of-value/#comment-52877 John Armstrong Fri, 29 Jun 2007 05:25:46 +0000 http://www.arsmathematica.net/archives/2007/06/28/debreus-theory-of-value/#comment-52877 Clearly Bourbaki would have written no such thing: economics is an application. Clearly Bourbaki would have written no such thing: economics is an application.

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