[Paper Review] Magnitudes Reborn: Quantity Spaces as Scalable Monoids

Dan Jönsson|arXiv (Cornell University)|Nov 17, 2019

Scientific Measurement and Uncertainty Evaluation23 references4 citations

TL;DR

This paper introduces scalable monoids and quantity spaces as foundational algebraic structures for a rigorous, universal calculus of physical quantities. It establishes that finitely generated quantity spaces over a field are isomorphic to formal power series rings, providing a scalable, algebraic framework for physical quantities that generalizes vector spaces while preserving metrological integrity and enabling consistent operations on magnitudes independent of measurement units.

ABSTRACT

This article includes a survey of the historical development and theoretical structure of the pre-modern theory of magnitudes and numbers. In Part 1, work, insights and controversies related to quantity calculus from Euler onward are reviewed. In Parts 2 and 3, we define scalable monoids and, as a special case, quantity spaces; both can be regarded as universal algebras. Scalable monoids are related to rings and modules, and quantity spaces are to scalable monoids as vector spaces are to modules. Subalgebras and homomorphic images of scalable monoids can be formed, and tensor products of scalable monoids can be constructed as well. We also define and investigate congruence relations on scalable monoids, unit elements of scalable monoids, basis-like substructures of scalable monoids and quantity spaces, and scalar representations of elements of quantity spaces. The mathematical theory of quantity spaces is presented with a view to metrological applications. This article supersedes arXiv:1503.00564 and complements arXiv:1408.5024.

Motivation & Objective

• To re-establish the role of physical magnitudes—distinct from scalars—as central in modern mathematics and physics.
• To formalize a scalable, algebraic calculus of physical quantities that respects metrological principles and historical foundations in Greek mathematics.
• To define quantity spaces as a generalization of vector spaces, using scalable monoids as their algebraic backbone.
• To establish isomorphisms between finitely generated quantity spaces and formal power series rings over a field, enabling concrete representation and computation.
• To provide a self-contained, universal algebraic framework for quantity calculus that supports operations like addition, multiplication, and scalar multiplication on physical magnitudes.

Proposed method

• Defining scalable monoids as commutative monoids equipped with a scalar action over a field, generalizing modules and enabling consistent operations on quantities.
• Introducing quantity spaces as a special case of scalable monoids where a finite basis exists, analogous to vector spaces in module theory.
• Constructing tensor products and homomorphic images of scalable monoids to support structural analysis and transformations.
• Defining congruence relations and unit elements to analyze equivalence and invertibility within scalable monoids.
• Introducing basis-like substructures and scalar representations to enable unique decomposition of elements in quantity spaces.
• Demonstrating that any finitely generated quantity space over a field is isomorphic to the ring of formal power series $ K\llbracket X_1, \ldots, X_n \rrbracket $, with multiplication and scalar action defined component-wise.

Experimental results

Research questions

• RQ1 How can physical quantities be formalized as algebraic objects that support consistent arithmetic operations independent of measurement units?
• RQ2 What algebraic structure generalizes vector spaces while preserving the distinction between quantities and their scalar measures?
• RQ3 Can a scalable monoid framework support the full range of operations needed in physical quantity calculus, including addition, multiplication, and scalar multiplication?
• RQ4 Is there a canonical representation for finitely generated quantity spaces that enables practical computation and isomorphism to known algebraic structures?
• RQ5 How do historical foundations in Greek mathematics—particularly the distinction between numbers and magnitudes—inform a modern, rigorous theory of physical quantities?

Key findings

• Finitely generated quantity spaces over a field are isomorphic to the formal power series ring $ K\llbracket X_1, \ldots, X_n \rrbracket $, providing a concrete and computable representation.
• The structure $ K\llbracket X_1, \ldots, X_n \rrbracket $ forms a commutative scalable monoid under component-wise scalar multiplication and term-wise multiplication of monomials.
• A basis $ \{b_1, \ldots, b_n\} $ exists for any finitely generated quantity space, allowing unique representation of elements as $ \lambda \cdot \prod_{i=1}^n b_i^{k_i} $.
• The identity element $ \mathbf{1} = 1X_1^0\cdots X_n^0 $ satisfies $ \mathbf{1} \cdot t = t \cdot \mathbf{1} = t $, ensuring multiplicative neutrality.
• Scalar multiplication respects associativity and commutativity: $ \alpha \cdot (\beta \cdot t) = (\alpha\beta) \cdot t $, and $ \alpha \cdot (st) = (\alpha \cdot s)t = s(\alpha \cdot t) $.
• The isomorphism $ \Phi: Q \to K\llbracket X_1, \ldots, X_n \rrbracket $, defined by mapping basis elements to variables, preserves all algebraic operations, confirming structural equivalence.