A Slightly Economics Related Post
Recently this blog has been considered with some of the more practical considerations of growth and proper representatives models, along with the theory of distribution. I intend to continue with this, but for now, I believe it is worth considering a meta-theoretical note on formal mathematical theories in economics.
The philosophy of mathematics is quite an extensive subject with varying interpretations, just as economics remains. Economics has concerned itself with the much more formalistic aspects of modeling for a great deal of time, particularly with the proper methodology of proofs for the existence of a mathematical phenomena and counter-examples to conjectures, e.g., the marginal productivity theory of remuneration of factors of production and the counter-example reswitching posited itself as. The reason for the incorporation of such topics into our economic understanding is due to the fact that economics has blurred the lines with pure mathematics as time has gone on, with the counter-revolutions starting in the 1960s in terms of the “Rational Expectations Hypothesis” and, e.g., the minor revolution led by Piero Sraffa at Cambridge in the 1960s. All of these have established a necessity for a meta-theory of economical proofs. The importance of a meta-theory for mathematics, and by the transitive property, economics, should not be understated in any attempt to further discourse on the formal aspects of economics.
Mathematics itself is riddled with peculiarities, since the older Greco-Roman branch of mathematics established itself as an authority on the nature of mathematical objects as a real, but abstract, object. The existence of such fundamental objects in mathematics, e.g., the constant of π (Pi), are treated as real, but abstract objects that directly impact the functionality of the world. This position is generally called “Platonism” and is held as one of the oldest positions on the philosophy of mathematics. I do not find this view to be particularly useful with regards to the ability to justify economic proofs, if we treat mathematical objects as real bodies then the very obvious question arises of how we are to acquire knowledge of said objects. As mentioned above, these objects are held as abstracts, and hence the self evidence of these objects is not certain, which leads to dissent over the philosophy of certain mathematical objects. The question arises, however, of how individuals are able to recognize these objects for themselves. On a theoretical note, how do we acquire knowledge of these abstract objects? All justifications of such an abstract argument imply an element of mathematical anti-realism, the use of constructive intuition or by knowledge of empirical facts. The problem with such a position is, however, when introducing such individual dependent concepts into mathematics, the ability to verify the “independence” of Platonic objects becomes dubious, there is no premise to then state that mathematical objects exist independent of the observer.
On the opposite end of such a spectrum is the concept of “formalism”; or that mathematics is fundamentally a game that evolves with societally given rules, and is nothing but the manipulation of alpha-numerical strings according to such a rule. This approach, pioneered by David Hilbert, essentially treated mathematics as a concept completely dependent on society and its own evolution. Assume, for the sake of argument, that as time evolves we forget everything we knew about mathematics, due to some catastrophe. Further, assume that we change a great deal in terms of our biological structure, so that our ability to manipulate said alpha-numerical strings changes. In such a situation, a formalist would hold that mathematics itself remains without any similarities to its older version prior to such a destructive, Schumpeter styled, evolution. Hilbert sought to reconstruct mathematics using his conception of mathematical objects as game-styled manipulation, using an axiomatic method. The problem with such an approach, however, was that its validity was dependent upon the validity of the all-encompassing nature of mathematics, that the axiomatic method was truly complete. One of the most notorious discoveries in the philosophy of mathematics was Gödel’s incompleteness theorems, stating that any system that was true was mathematically unprovable. I will refrain from discussing this, leaving it to a future post on the implications for economics, but in essence Gödel’s incompleteness theorem held that a true mathematical proposition, using the axiomatic method, cannot be proved within its own system, implying a fundamental element of uncertainty within our mathematical computational abilities.
The two other positions on the philosophy of mathematics that I will briefly discuss is Logicism and Intuitionism/Constructivism. Logicism essentially purports itself as a theory that describes the fundamental nature of mathematics through logic. All mathematical statements are reducible to logical, true or false statements, and in that situation mathematics is a branch of logic. Such a theory sits well with most individuals, it explains to a satisfactory degree how we acquire knowledge of mathematics, and we find it almost completely obvious that logic is the fundamental piece of branches of mathematics such as geometry. Henri Poincaré (1902) among others, however, ruthlessly criticized logicism as a theory that adds little to our pure mathematical knowledge. Reducing everything to a few true or false statements essentially means math itself is tautologous, and that any acquisition of any further mathematical knowledge is simply equivalent to the knowledge known before. On a more theoretical point, Poincaré held that mathematics is the application of the pure intuition in a construction of the intuitive continuum, arithmetical and topological. Our knowledge of geometry comes from our pure intuition, our ability to construct mathematical objects using the pure intuition of time and space. The pure intuition of space manifests itself into geometry, we “construct” geometrical objects using our ability to form a topological continuum and analyze the intuitive relationships between lines, points, rays, and segments in Euclidean geometry1. The pure intuition of time acts as our ability to create a numerical continuum of real numbers, irrationals, and imaginary numbers. Poincaré took an essentially Kantian view on mathematics when criticizing the views of Dedekind, Russell, and Frege on the foundations of mathematics, utilizing a similar conception of the intuitive “synthetic a priori” and transcendental idealism. Our mathematical knowledge was all constructed through our use of intuition, and deductive proofs which may be reducible to true or false statements lack the ability to expand our knowledge of mathematics. Take the example of 1 + 1 = 2; Kant argued that such a statement was not true by its premise, it was not knowable a priori without intuition. The mind constructs a numerical continuum using the intuition and constructively finds the final result, the truth of 1+1 is not deductively reducible to a logical a priori statement due to the inability to extend one’s mind and intuitively construct and arithmetical continuum of numbers. The Brouwerian continuum hypothesis (Poincaré being a precursor to an intuitionistic-conventionalistic philosophy of mathematics) can formally be framed in terms of:
Assume two predicates, A(α, x), α ranging over choice sequences (The intuitive idea of a set, a constructible set in essence, see Troelstra (1983)) and x over naturals. We can define extensionality as:
Hence, the weak continuity principle is:
Where α and β range over choice sequences, m and x over naturals, and α’m being the initial segment of length m. This holds that we can assign a number to every choice sequence, using a tree-like conception of number-generators and free-choice sequences, given the initial segment of such a sequence, α. The importance of such given above implies that we can provide geometrical and arithmetical knowledge of objects using the continuum hypothesis, but recasting it under intuitionistic terms, as the non-constructive proof given by classical mathematics remains unacceptable. Providing a continuum allows for construction of mathematical objects using the mind, i.e., mathematical objects are mind dependent, and hence, constructions. This implies that the use of the intuition is necessary in mathematics, and hence, mathematics is not reducible to pure logico-mathematical statements, meaningless alpha-numerical strings, or abstract mind-independent objects.
The proper interpretation of a meta-theory of proofs, seems to suggest in the direction of intuitionistic constructivism (To be distinguished from non-intuitionistic constructivism). In this line of thought, to prove an object is not to utilize the principle of the excluded middle, ¬(p∧¬p), cannot hold as a theory of proofs. The use of algorithmic constructive axiomatizations is the only valid manner to prove mathematical-economic propositions, formally. One peculiarity is the parallels to be found with Sraffa’s constructive, algorithmic method, and the BHK (Brouwer–Heyting–Kolmogorov) interpretation of constructive proofs. The manner of which Sraffa’s standard system is constructed is through a constructivist methodology of repeated algorithm like methods, along with the proof of the uniqueness of the standard system (Miyao 1977). Such parallels, in terms of the constructive existence proof go quite deep, seemingly implying a conscious choice on the part of Sraffa’s proof style. This interpretation seems to be suggested by Velupillai (2008), and it is interesting to note that Sraffa’s time as an academic friend to Ludwig Wittgenstein in Cambridge, an (in)famous philosopher with a constructivist philosophy of mathematics (Rodych 2013) may provide itself as a link to Sraffa’s peculiar proof style.
The purpose of this brief blogpost was to explore on a more meta-theoretical note, the foundations of mathematical-economics and the proper methodology for proofs in the aforementioned, along with a note on the mathematical foundations for Sraffa’s Production of Commodities by the Means of Commodities. I plan on exploring this more in depth later, but due to an unexpected sickness, I seem to lack the energy to write a sufficiently long blog-post. Old disclaimer applies with regards to equations, I apologize to have to screenshot equations from Google-Docs; but Blogger does not input it correctly otherwise.
Footnote:
Some hold that the intuitionistic methodology is incompatible with Non-Euclidean theories of geometry, found present in Einstein's Riemann Geometry based approach. Poincaré criticized this view through a conventionalist lense of geometry and the sciences, for further discussion with regards to the intuition of space and time and Non-Euclidean geometry, read Arledge (2014).
References:
Rodych, Victor. "Mathematical Sense: Wittgenstein’s Syntactical Structuralism". Wittgenstein and the Philosophy of Information: Proceedings of the 30th International Ludwig Wittgenstein-Symposium in Kirchberg, 2007, edited by Alois Pichler and Herbert Hrachovec, Berlin, Boston: De Gruyter, 2013, pp. 81-104. https://doi.org/10.1515/9783110328462.81
Velupillai, Kumaraswamy. (2008). Sraffa’s Economics in Non-Classical Mathematical Modes. 10.1057/9780230375338_15.
Miyao, Takahiro. “A Generalization of Sraffa’s Standard Commodity and Its Complete Characterization.” International Economic Review, vol. 18, no. 1, [Economics Department of the University of Pennsylvania, Wiley, Institute of Social and Economic Research, Osaka University], 1977, pp. 151–62, https://doi.org/10.2307/2525774.
Troelstra, A. S. “Analysing Choice Sequences.” Journal of Philosophical Logic 12, no. 2 (1983): 197–260. http://www.jstor.org/stable/30226270.
Poincaré, Henri, 1854-1912. Science and hypothesis. London, New York, Scott, 1905 (OCoLC)622773044
Arledge, Chris. (2014). Kant, non-Euclidean Geometry and A Priori Knowledge: A Reassessment.
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