Non-power positional number representation systems, bijective numeration, and the Mesoamerican discovery of zero

Rojo-Garibaldi, Berenice; Rangoni, Costanza; González, Diego; Cartwright, Julyan H. E.

doi:10.1016/j.heliyon.2021.e06580

Cited by 3 publications

(2 citation statements)

References 25 publications

(22 reference statements)

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“…where k can equal 0 [ 0  ] and k = 1 represents the first of N ordered geometric elements in the metric space of cycle fractions. Rt unit natural-number counts are consistent with Babylonian sexagesimal place-value notation, a 60-base system using finite sequences of sexagesimal digits 00, 01, …, 58, and 59, where both the first and last digits are non-null [52] [53], and the Mesoamerican vigesimal base-20 counting system (00, 01, …, 18, 19) [54]. In comparison to the Rt system, 00 corresponds to both 0 and 60 in a cycle, 1 0/60 = 60/60, or 1 0/20 = 20/20.…”

Section: Dimensional Quantities and Rt Unitsmentioning

confidence: 94%

Separating Space and Time for Dimensional Analysis and Euclidean Relational Modeling

Moore

2023

JAMP

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The theory of relativity links space and time to account for observed events in four-dimensional space. In this article we describe an alternative static state causal discrete time modeling system using an omniscient viewpoint of dynamical systems that can express object relations in the moment(s) they are observed. To do this, three key components are required, including the introduction of independent object-relative dimensional metrics, a zerodimensional frame of reference, and application of Euclidean geometry for modeling. Procedures separate planes of matter, extensions of space (relational distance) and time (duration) using object-oriented dimensional quantities. Quantities are converted into base units using symmetry for space (Dihedral 360 ), time (Dihedral 12 ), rotation (Dihedral 24 ), and scale (Dihedral 10 ). Geometric elements construct static state outputs in discrete time models rather than continuous time using calculus, thereby using dimensional and positional natural number numerals that can visually encode complex data instead of using abstraction and irrationals. Static state Euclidean geometric models of object relations are both measured and expressed in the state they are observed in zero-time as defined by a signal. The frame can include multiple observer frames of reference where each origin, point, is the location of a distinct privileged point of reference. Two broad and diverse applications are presented: a one-dimensional spatiotemporal orbital model, and a thought experiment related to a physical theory beyond Planck limits. We suggest that expanding methodologies and continued formalization, novel tools for physics can be considered along with applications for computational discrete geometric modeling.

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Section: Dimensional Quantities and Rt Unitsmentioning

confidence: 94%

Separating Space and Time for Dimensional Analysis and Euclidean Relational Modeling

Moore

2023

JAMP

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“…Also consistent with the Mesoamerican vigesimal base-20 counting system (0, 01, •••, 18, 19). [33] [34] An axiom in Rt relationalism relevant to this article is that physical timelines can be divided and subdivided to intervals that reach quantum limits, where an interval in time cannot be reduced any further. Therefore, there is zero of time in physical dimensional reality as defined by signals with no dimensional duration between them.…”

Section: Units; Dimension Symmetry Order and Zeromentioning

confidence: 99%

Relational and Euclidean Temporal Space

Moore

2024

JAMP

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In mathematics, space encompasses various structured sets such as Euclidean, metric, or vector space. This article introduces temporal space-a novel concept independent of traditional spatial dimensions and frames of reference, accommodating multiple object-oriented durations in a dynamical system. The novelty of building temporal space using finite geometry is rooted in recent advancements in the theory of relationalism which utilizes Euclidean geometry, set theory, dimensional analysis, and a causal signal system. Multiple independent and co-existing cyclic durations are measurable as a network of finite one-dimensional timelines. The work aligns with Leibniz's comments on relational measures of duration with the addition of using discrete cyclic relational events that define these finite temporal spaces, applicable to quantum and classical physics. Ancient formulas have symmetry along with divisional and subdivisional orders of operations that create discrete and ordered temporal geometric elements. Elements have cyclically conserved symmetry but unique cyclic dimensional quantities applicable for anchoring temporal equivalence relations in linear time. We present both fixed equivalences and expanded periods of temporal space offering a non-Greek calendar methodology consistent with ancient global timekeeping descriptions. Novel applications of Euclid's division algorithm and Cantor's pairing function introduce a novel paired function equation. The mathematical description of finite temporal space within relationalism theory offers an alternative discrete geometric methodology for examining ancient timekeeping with new hypotheses for Egyptian calendars.

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How to frame innovation in mathematics

Fisseni,

Sarikaya,

Schröder

2023

Synthese

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We discuss conceptual change and progress within mathematics, in particular how tools, structural concepts and representations are transferred between fields that appear to be unconnected or remote from each other. The theoretical background is provided by the frame concept, which is used in linguistics, cognitive science and artificial intelligence to model how explicitly given information is combined with expectations deriving from background knowledge. In mathematical proofs, we distinguish two kinds of frames, namely structural frames and ontological frames. The interaction between both kinds of frames can drive mathematical interpretation. We first discuss two examples where structural frames (formulaic notation) drive ontological development (the discovery or exploration of mathematical objects). The development of Boole’s Boolean algebra may at first appear as a metaphorical treatment of the (then) new area of logic. In the analysis, we discuss how different (aspects of) certain algebraic frames change in the transfer, how arising difficulties are solved and overall argue that Boole uses the numerical algebra frame as a research template for the discovery of a system for calculations in logic. Following Ifrah, we analyse the discovery of zero as an extension to the number ontology as driven by the development of notation. Both structural and ontological frames are extended and simplified as notation progresses. Finally, we discuss two examples from infinite combinatorics, viz. topological graph theory, and one foundational issue. In both examples, the two simultaneous frames about one object are maintained independently. They motivate different research questions, but may also fruitfully interact: shifting between multiple synchronously maintained perspectives acts as a motor of innovation. The analysis shows how a frame-based approach allows to model how different perspectives drive mathematical innovation because they highlight different aspects, questions and heuristics.

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Non-power positional number representation systems, bijective numeration, and the Mesoamerican discovery of zero

Cited by 3 publications

References 25 publications

Separating Space and Time for Dimensional Analysis and Euclidean Relational Modeling

Separating Space and Time for Dimensional Analysis and Euclidean Relational Modeling

Relational and Euclidean Temporal Space

How to frame innovation in mathematics

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