by From Philip George What are vectors? . . . we defined a vector as a quantity having both magnitude and direction and represented it by an arrow. This makes sense in Euclidean 3-dimensional space. But in higher dimensions the idea of direction is not intuitive and we need a more formal definition that is consistent with the definition in three dimensions. In mathematics, an object is defined as a vector if it is an element in a vector space. This seems a circular definition but the additional requirements make it clear why it is defined in this way. Thus, when a vector is multiplied by a scalar (a real number, for our purpose) the result must be an element of the vector space, i.e., another vector. And a vector added to another vector must also be a vector in that vector space.
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from Philip George
What are vectors?
. . . we defined a vector as a quantity having both magnitude and direction and represented it by an arrow. This makes sense in Euclidean 3-dimensional space. But in higher dimensions the idea of direction is not intuitive and we need a more formal definition that is consistent with the definition in three dimensions. In mathematics, an object is defined as a vector if it is an element in a vector space. This seems a circular definition but the additional requirements make it clear why it is defined in this way. Thus, when a vector is multiplied by a scalar (a real number, for our purpose) the result must be an element of the vector space, i.e., another vector. And a vector added to another vector must also be a vector in that vector space.
Consider two 10-tuples of numbers, T1 = (t1, t2, t3, … , t10) and T1¢ = (t1¢, t2¢, t3¢,… , t10¢). Let these represent the temperatures at ten points along the lengths of two metal bars. Then it is obvious that these 10-tuples cannot be vectors because they fail the requirement of vector addition; it makes no sense to add t1 and t1¢ because, as shown in the section on temperature, adding temperatures is a meaningless operation.
Similarly, consider two 10-tuples of numbers P1 = (p1, p2, p3, … , p10) and P1¢ = (p1¢, p2¢, p3¢, … , p10¢). Let P1 be the prices of 10 goods which an individual consumes on Monday and P1¢ be the prices of the same goods which he consumes on Tuesday. Now it is meaningless to add the price of a good on Monday to the price of the good on Tuesday. Therefore, it is meaningless to add the elements of the two 10-tuples P1 and P1¢. Hence, the two 10-tuples cannot be vectors and, indeed, there cannot be such a thing as a price vector.
We are therefore forced to conclude that General Equilibrium Theory (GET), which in its modern version is nearly three quarters of a century old, is merely highfalutin nonsense.
A bit of history
Einstein described Gibbs as “the greatest mind in American history”. And later, when asked who were the most powerful thinkers he had known, Einstein said: “Lorenz”, adding, “I never met Willard Gibbs; perhaps had I done so, I might have placed him beside Lorenz.”
Gibbs’s influence on modern economics, especially on optimisation theory, is well known. That influence was through the effect on Samuelson, via Gibbs’s pupil, E.B. Wilson.
But Gibbs, along with Oliver Heaviside, was also the inventor of vectors, the subject matter of this paper. So, his influence runs through the other important strand of modern mathematical economics as well, mainly in proofs of general equilibrium, though the price vector has since ramified into other areas such as international trade theory.
Critics of mathematical economics would say that Gibbs was doubly unfortunate in his disciples, at one remove. But, of course, that was hardly his fault.
Samuelson must be blamed for erecting a huge mathematical edifice, without first ascertaining that the utility and profit functions are differentiable.
In GET, the error that Arrow and Debreu made was in blindly transferring mathematical ideas to the real world without first ascertaining that those ideas were transferable. It is significant that both of them were mathematicians who wandered into economics.
Conclusion
Debreu noted in his Nobel Prize lecture that the success of the mathematization of economic theory depended “on the fact that the commodity space has the structure of a real vector space”. We have shown that this is incorrect. The “price vector” is not a vector, and GET is therefore false. But we may go further and assert that not only was the proof incorrect, what was set out to be proved was not true in the first place. The real economy cannot be brought into equilibrium by adjusting prices. And indeed, the real economy is never in equilibrium. http://www.paecon.net/PAEReview/issue101/George101.pdf