Mathecon
introduction, should be on all the web sites, somewhere
A Comprehensive Introduction to Mathematical Economics (Mathecon)
by Joseph Mark Haykov
December 12, 2023
Abstract
Mathematical Economics (or Mathecon), formally introduced here for the first time, represents an interdisciplinary approach that amalgamates behavioral economics with mathematical game theory, all supported by a structured axiomatic framework. This framework mirrors those utilized in algebra for theorem proving, yet it is distinctively tailored to the economic context. Mathecon harnesses formal axiomatic rules of discourse to establish a rigorous, mathematically grounded foundation for analyzing economic behaviors and strategies. The integration of behavioral insights and mathematical precision provides a unique perspective for comprehending the complexities and subtleties of economic interactions and decision-making processes.
Introduction
Hilbert's program, inaugurated by the esteemed German mathematician David Hilbert in the early 20th century, was a visionary endeavor aiming to lay a solid foundation for the entirety of mathematics. In Mathecon, while we do not claim to have completely resolved Hilbert's program, we do engage with some of its critical aspects, albeit not all.
Hilbert's vision included the formulation of mathematics on a finite, complete set of axioms. These axioms were intended to be straightforward and self-evident, forming the bedrock of all mathematical fields. A crucial part of his program was ensuring the consistency of these axiomatic systems. Consistency here meant that the systems should be contradiction-free, making it impossible to prove a statement and its negation simultaneously.
Additionally, Hilbert strived for completeness in these systems, where for every statement within the system, either the statement or its negation could be proven based on the axioms. Essentially, there should be no true statement that remains unprovable within the system. The final goal was decidability, which would involve a clear method or algorithm to determine the truth or falsity of any mathematical statement within the system.
However, Hilbert's ambitious program encountered significant challenges. In 1931, Kurt Gödel's Incompleteness Theorems revealed that any sufficiently powerful and consistent axiomatic system could not be complete; there will always be true statements in the system beyond the reach of the axioms. Gödel's second theorem further stated that such a system cannot demonstrate its own consistency. Around the same period, Alan Turing introduced the Halting Problem, underscoring the limits of decidability. He showed the impossibility of a general algorithm to definitively determine whether a program will halt or run indefinitely for every possible input.
These findings by Gödel and Turing reshaped our understanding of the foundations of mathematics. While Hilbert's program sought a complete, consistent, and decidable foundation, these revelations showed that such a foundation is beyond reach. Mathematics can indeed be formulated on consistent and robust axiomatic systems, but these systems inherently cannot be complete, and their consistency cannot be internally proven.
The Reality of Mathematics: Consistency over Completeness
In practical terms, mathematics operates on a foundation of consistency while accepting its inherent incompleteness. This equilibrium is struck by mathematicians through the application of dual definitions in mathematical concepts, ensuring consistency across both theoretical constructs, like algebra or geometry, and their real-world applications. For instance, in the realm of algebra, we see this duality clearly:
The number 0 is used to signify the absence of an object, while 1 indicates its existence, together creating a primary-dual pair that encapsulates the concept of absence and existence.
Similarly, the operation of addition (+) is interpreted as acquiring an object, while subtraction (−) represents the act of giving one away. This interplay of addition and subtraction, or the give-and-take actions, and the corresponding gain and loss of objects, lay the groundwork for the foundational dualities in algebra.
While algebra and other fields of mathematics may relinquish completeness — recognizing that not every truth can be proven within their systems — they uphold consistency. In this framework, any claim that stands in contradiction to an established axiom is deemed false, and its converse is thus established as true through the method of contradiction. This approach not only fortifies consistency within the mathematical framework of induction, deduction, and non-contradiction but also echoes one of the earliest methods of proof, famously utilized by Euclid in his demonstration of the infinitude of prime numbers.
Axiomatic Rules of Discourse in Algebra and Applied Mathematics
To illuminate the formal axiomatic framework that underpins applied mathematics, we introduce a novel concept from abstract algebra known as a 'logical claim ring,' or simply a 'claim ring.' Although this concept might seem abstract initially, it is intuitively accessible and effectively encapsulates the rules governing theorem proofs in mathematics. Furthermore, the concept of a claim ring aligns with our inherent understanding of reality, drawing parallels between the methodologies of mathematics and common sense in proficiently describing the world. This parallel arises because both mathematical constructs and real-world phenomena are logically grounded in fundamental axioms. These axioms are based on empirical, real-world facts, observable in our daily experiences, where everything operates according to the law of causality, positing that every action leads to an equal and opposite reaction, and every event has an underlying cause.
In the realm of applied mathematics, the terms 'logically emanate' or ‘logically follow’ denote the necessity for all true claims, exemplified by theorems like Fermat’s Last Theorem, and all valid assertions about reality, to originate from the dual processes of induction and deduction. For example, induction in reality could involve observing the daily pattern of sunrise and sunset and inferring that this pattern will continue indefinitely. Based on this, certain relationships between the sun and the moon are deduced. This mirrors how mathematical proofs function, using formal, mathematically precise definitions of recursive operations of induction and deduction. These processes, similar to the sine-cosine relationship in trigonometry, are constrained by the principle of non-contradiction, which dictates that initially established truths take precedence over and invalidate any subsequent conflicting claims. Therefore, any claims that contradict these initial truths are considered false under the established rules of mathematical proof, thereby affirming the accuracy of the initial claims.
The structure of a 'claim ring' reflects this logical reasoning, drawing on the foundational mathematical constructs of rings and fields. In this framework, a field is a specialized type of ring, and a ring, in turn, serves as a logical claim ring. This approach melds abstract mathematical structures with the logical development of claims and truths in the realm of applied mathematics.
Logical Claim Rings in Mathematics
The 'logical claim ring' represents a theoretical framework in mathematics that reinterprets traditional ring structures with a focus on logic and mathematical proof. Its defining characteristics are:
Initial Axioms as Seed: The foundation of a logical claim ring is a set of foundational axioms. These axioms are accepted as intrinsic truths without requiring proof, serving as the cornerstone for developing further claims. They are more than just starting points, forming the bedrock of the ring's framework.
Non-Contradiction Axiom: A core principle in any claim ring is the axiom of non-contradiction. It mandates that any claim derived through induction or deduction must not contradict existing true claims that logically follow from the initial axioms. This ensures the internal consistency and logical integrity of the claim ring as it expands.
Dual-Definition Axiom: The concept of dual-definition is central to claim rings and applied mathematics. It applies to both objects (such as 0 and 1, points and lines, truth and falsehood) and operations (like and/or, addition/subtraction, multiplication/division). This duality enables recursive reasoning and the progression of logical thought through induction and deduction, in line with the principle of non-contradiction.
Growth of the Claim Ring: As new theorems are deduced or induced within a logical claim ring, the structure of the ring expands, reflecting the evolving nature of mathematical theory. Deduction is akin to the operation of multiplication, while induction resembles the addition operation in traditional rings. This growth, rooted in the initial axioms about dually-defined object-action pairs, enhances the complexity and depth of the ring, illustrating the adaptive character of mathematical theory.
Construction of Infinite Claim Rings: Logical claim rings can extend to infinite sizes. An example is the application of Peano's fifth axiom, the principle of induction, which introduces an endless series of claims about natural numbers using the duality of 0 (absence) and the '+1' successor function (addition to existence), applied recursively.
The logical claim ring framework provides a unique lens through which to view the development and structure of mathematical theories. It showcases how mathematical concepts are constructed from fundamental axioms, expand through logical reasoning, and are bounded by the need for internal consistency. This model mirrors the evolution of mathematical thought, where each new theorem or concept contributes to the overall structure's growing complexity and depth.
Application of Claim Rings to Behavioral and Mathematical Economics
The concept of claim rings, with its roots in mathematical logic, offers an intuitive and effective framework for the field of Mathematical Economics (Mathecon). This innovative application reshapes the claim ring concept to fit the dual nature of theoretical and empirical analysis in economics, guided by a foundational set of axioms known as the 'one-truth' postulate in applied mathematics.
Ensuring Non-Contradiction in Theoretical Postulates:
In Mathecon, every theory or proposition put forward must be consistent and non-contradictory. This non-contradiction axiom is vital for maintaining the internal consistency of economic theories, ensuring that they do not conflict with themselves or any other established truths.
Dual-Definition of Object-Action Pairs in Mathecon:
A pivotal aspect of Mathecon is the dual-definition approach to object-action pairs. This approach involves analyzing each element of an economic model from two complementary perspectives, enhancing the depth and breadth of economic analysis.
Subjective-Objective Duality in Costs and Benefits:
In economic models, the evaluation of costs and benefits must incorporate both subjective and objective viewpoints. Subjectively, this involves assessing costs and benefits from the perspective of individual economic actors, considering their personal preferences, perceptions, and circumstances.
Objectively, costs and benefits are evaluated in terms of what is optimal for the collective, taking into account societal welfare. This objective analysis often aligns with game theory scenarios, such as the prisoner’s dilemma.
Expanding Dual-Definition to Applied Mathematics:
This principle of dual-definition extends beyond economic models to encompass all objects and operations in applied mathematics. For instance, the concepts of '0' for absence and '1' for presence, as well as 'addition' for increase and 'subtraction' for decrease, are fundamental. This dual approach is essential in economics, where the interplay of concepts like supply-demand and cost-benefit drive analysis.
Implications for Economic Modeling:
By integrating these dual definitions, economic models can more accurately reflect the complex nature of real-world economic behavior.
Economists are thus equipped to weigh individual motivations against broader societal implications, fostering the development of robust and realistic models.
This methodology is particularly significant in policy analysis, where the implications of decisions can have varied impacts on individuals and society as a whole.
In summary, the adoption of the Dual-Definition of Object-Action Pairs in Mathecon highlights the critical need for multi-perspective analysis in economics. This approach supports the creation of economic models and policies that are comprehensive, balanced, and effective. The dual perspective not only deepens theoretical understanding but also bolsters practical applications in policy-making and forecasting. Significantly, it allows for the modeling of the economy with a level of mathematical precision akin to that found in disciplines like geometry or algebra. This precise approach is in line with the Austrian school of economics, the principles of which we will explore in the following sections.
Alignment with Empirical Reality in Mathematical Economics
Empirical Verification as a Core Principle:
In the sphere of mathematical economics, it's imperative that theories and hypotheses consistently align with observable and verifiable empirical facts. This alignment goes beyond being a mere guideline; it's a fundamental requirement for the validity of any economic model.
Adhering to this principle ensures that economic theories are not just abstract constructs but concrete reflections of actual economic phenomena and behaviors.
Immediate Reevaluation of Contradicting Theories:
Economic theories that find themselves at odds with empirical data must be promptly reconsidered or invalidated. Holding onto a theory that fails to pass empirical muster, even in a single instance, is unacceptable.
Upholding this strict standard is vital for preserving the integrity and relevance of mathematical economics, affirming its role as a practical and reliable tool for interpreting economic realities.
Rejecting the 'Reasonable Explanation Fallacy':
Within mathematical economics, any discrepancy between theoretical predictions and real-world observations cannot be dismissed or excused under the guise of "reasonable explanations."
Should a theory deviate from empirical realities, the onus is on the theory to convincingly account for this divergence. Any such explanations must be subjected to stringent scrutiny and validation.
The plausibility of explanations offered as part of a theory is evaluated as a core component of the theory, not merely as a supplementary justification.
Importance of Plausibility in Theoretical Adjustments:
Modifications to theories, in light of empirical inconsistencies, must be grounded in plausible, economically sound reasoning and evidence.
Theories in mathematical economics need to adapt and evolve in response to empirical evidence, ensuring they remain pertinent and accurate in explaining and forecasting economic trends and patterns.
In summary, the principle of Alignment with Empirical Reality is a critical safeguard in mathematical economics, ensuring that theories are not only mathematically robust but also practically viable. This principle promotes a discipline where theories are continuously tested and refined based on real-world data, thus enhancing the reliability and applicability of economic models in both academic research and policy-making.
Mathecon: Axiomatic Foundations and Principles
In Mathecon, or Mathematical Economics, the development and meticulous testing of theories are governed by a robust set of axiomatic rules. These rules play a crucial role in accurately depicting and forecasting real-world economic behavior. For instance, an economic theory must be consistent with verifiable facts, like the comparative analysis of income tax rates in different regions such as Florida and California. Should a theory conflict with such empirical facts, the discrepancy is attributed to the theory, not to reality. This methodology highlights the significance of logical consistency, the dual-definition of concepts, and empirical validation through non-contradiction. It lays out a structured approach for formulating economic theories that are not only mathematically sound but also practically relevant, effectively capturing the complexities of real-world economic systems.
To define a theory in Mathecon – a well-constructed and axiomatically sound mathematical economics theory under the 'first, one-truth postulate' of applied mathematics – we consider a theory that adheres to two essential rules:
Dual-Definition: Every concept in the theory must have a dual definition, such as cost-benefit or objective-subjective. This dual-definition is imperative for a holistic analysis of economic phenomena from various perspectives.
Alignment with Reality and Consistency: Any truth proposed within the theory must not contradict observed reality, the theory itself, or any other established truth. This fundamental principle ensures the theory’s relevance and accuracy in mirroring real-world economics.
Given these foundational principles, it logically follows under the 'first, one-truth postulate' of Mathecon that any economic theory derived from these core postulates – via induction and deduction, while adhering to the principle of non-contradiction – has a high probability of accurately representing economic realities. This increased likelihood stems from the theory's limited reliance on assumptions about reality, which are susceptible to being disproven and could invalidate the entire theoretical structure. Essentially, a theory validated as true through induction and deduction, and in harmony with empirical facts, minimizes assumptions, thereby optimizing its alignment with empirical reality. This method ensures that the theory, as defined by its own criteria, aligns closely with observed economic phenomena.
In practical terms, consider two theories, A and B, both based on the same foundational assumptions. If theory B incorporates an additional assumption that is later disproven, B would be invalidated, while theory A would still stand valid. A clear example is the variance in how individual economic actors subjectively perceive the costs and benefits of decisions compared to a collective or objective societal perspective. Any theory that extends beyond this fundamental observation by adding extra assumptions is less likely to be accurate than an alternative explanation that does not necessitate such additional assumptions. Therefore, the fewer the assumptions a theory makes beyond what is already accepted as true, the more likely it is to remain valid and applicable. With this foundation laid, we now turn our attention to our next topic: exploring the concept of ‘surplus value’ as initially defined by Karl Marx.