The Probability Of One Is Greater Than The Probability Of Infinity
The Probability of 1 is Greater Than the Probability of Infinity

A Treatise On Benford’s Law (Part I)

i.e. Exploring the Quantification Effect

Abstract (TL;DR):

Occam’s razor posits the problem-solving heuristic that “entities should not be multiplied without necessity,” or similarly, “of two [or more] competing theories, the simpler explanation of an entity is to be preferred.” That is to say, the simplest explanation is usually the correct one. The polyonymous Benford’s Law (a.k.a. Newcomb-Benford Law, the Law of Anomalous Numbers, The First-Digit Phenomenon, etc.) is no exception and is, in fact, a rather exemplary instance of Occam’s razor.

Benford’s Law has been observed, defined (both naturally and formally in mathematical language and notation), investigated, recognized, AND applied in fields ranging from accounting to forensics and beyond. Although the phenomenon’s existence has been established for nearly 150 years, as of the writing of this treatise, no generally accepted proof nor any other alternative explanation for its raison d’etre in mathematics, science, and nature has been deemed as complete and satisfactory.

This treatise aims to demonstrate both via formal numerical analysis and through analogical reasoning (thought experiments) that Benford’s law is a natural and predictable pattern formed by the unilinear mechanism (serial method of operation) inherent in quantitative analysis i.e. quantification.

Quantification is defined as:

“the act of counting and measuring that maps human sense observations and experiences into quantitiese.g. quantitative analysis or calculation of distances, amounts, weights, etc.

Within this treatise the following aspects of Benford’s Law are explored:

  1. Its Abstract Nature: The law’s fairly recent discovery in the course of human history is in no small part due to its abstruse nature, being difficult to detect and even more difficult to comprehend. Countless attempts have been made to derive a proof of Benford’s Law, but none have gained broad acceptance. This result (or rather lack thereof) is not unexpected, given that Benford’s Law is a byproduct of quantification. In other words, Benford’s Law is the evidence (or side effect) of the mechanism (algorithms) and materials (symbols) used in quantification; however, it is neither the mechanism (adding/multiplying/etc.) nor the material (numbers). It is analogous to the concept of heat, which can be felt, but is neither the chemical reaction (combustion), nor the material (fuel) used in the process of combustion; rather heat is a byproduct of combustion. Likewise, Benford’s Law is the byproduct of quantitative analysis.
  2. Its Logarithmic Nature: As the orders of magnitude increase, so does the “strength” of Benford’s Law. In other words, the larger the quantitative range (say intergalactic distances) the easier it is to percieve the presence of Benford’s Law. For this reason, there is a common misconception that multiple magnitudes of order are required for Benford’s Law to manifest. In actuality, Benford’s Law is present at all orders of magnitude including the single digit range 1–9, which will be shown in due course.
  3. Its Biased Nature: Numbers are “biased” towards one. In other words, the probability of any calculated result is weighted toward the number one. A result is more likely to be 1 than 2 and more likely to be 2 than 3, and more likely to be 11 than 23, ad infinitum. This concept requires us to contend with the notion that in any quantitative analysis, the probability of 1 is greater than the probability of infinity and any number in between for that matter.
  4. Its Universal Nature: An infinite variation of Benford’s Law can be predicted and observed for all positional numeral systems aside from the unary & binary (this has already been proven, confirming that the law is not an artifact of the Decimal System). Additionally, Benford’s Law holds true for all real numbers, including negative numbers.
  5. Its Substratal Nature: Benford’s Law isn’t just built into the fabric of existence, it IS the fabric of existence. Extrapolating on it’s universal nature, Benford’s Law is the foundation of the Central Limit Theorem, commonly known as the classic Bell Curve that permeates all of nature. This explanation of the law’s deeper meaning is diametrically opposite to modern approaches attempting to rationalize the existence of Benford’s Law by way of the Central Limit Theorem. Current approaches focus attention on the fact that the Central Limit Theorem is natural and ubiquitous, therefore since Benford’s Law appears to be connected with the Central Limit Theorem, the Central Limit Theorem must somehow be the cause of Benford’s Law. In actuality, the Central Limit Theorem describes a natural pattern, however it does not explain the reason nature tends towards a central limit (bell curve). The architecture of Benford’s Law justifies the Central Limit Theorem.

In principle this treatise is a treatment of Benford’s Law, in practice, however, it is much more far reaching in its application, as for all intents and purposes, it is an introduction to understanding the fundamental building blocks of quantitative analysis upon which rests the very foundations of our existence. It is an analysis of the mathematical logic that defines the framework intrinsic to quantitative analysis — and thereby intrinsic to nature itself. Upon recognizing & appreciating the full implications of Benford’s Law, one is required to re-calibrate their understanding of the core principles of the physical world, and the way in which numbers are taught. It is oft said that nature is not random — understanding Benford’s Law is critical to the explanation of this phenomenon; from the number of trees in forests, to the widths of the mouths of volcanoes, to the measurement of galactic distances, anything, in fact, that applies discrete mathematics and quantitative analysis-in other words, quantification.

Keywords:

Benford’s Law, Newcomb-Benford Law, Newcomb-Benford Effect, The Law Of Anomalous Numbers, First Digit Law, First-Digit Phenomenon, Significant Digit Law, Leading Digit Phenomenon, Pure Mathematics, Abstract Mathetmatics, Applied Mathematics, Discrete Mathematics, Real Numbers, Algorithms, Scale Variance, Scale Invariance, Measure Theory, Counting Theory, Set Theory, Quantification Theory, Quantification Effect, Quantitative Analysis

Overview:

This treatise opens by addressing prior knowledge which could skew and hinder grasping central ideas discussed herein. Following that, I present a standard description of Benford’s Law and briefly touch upon the history of its discovery as it will be referenced in later sections. Continuing on, I establish the key elements or “ingredients” of the numerical system we use in mathematics. Following that, extrapolations of the law are proposed and explored. In closing, thought experiments are conducted to further understanding of Benford’s Law as it correlates to real world applications, before concluding with a demonstration of the claim that even the simplest single digit range 1–9 follows Benford’s Law.

Outline:

~ Exordium

~ The Newcomb-Benford Law (Definition)

~ Ab Initio - Discovering The Law (History)

~ The Elements of Benford’s Law (Ingredients)

  • First Principles Of Benford’s Law
  • Understanding Algorithms
  • The Hindu-Arabic Numeral System Algorithm
  • The Base 10 (Decimal System) Hindu-Arabic Number System Algorithm

~ Numerical Analysis (Extrapolations) :

  • Base 1
  • Bases 2–15/20
  • An Infinite Pattern
  • Natural & Negative Numbers

~ Analogical Reasoning (Thought Experiments):

  1. The Brick Wall
  2. The Swimming Pool
  3. The Motel
  4. Shapes
  5. Symbols
  6. Colours
  7. Breaking Benford’s Law

~ Conclusion

~ Epilogue

~ References

(End of Part I)

Coming Next — Exordium, etc.

If you’ve enjoyed Part I of this series, please subscribe, clap & share.

Part II will be coming as soon as time permits.

Human Being, b. circa 1990 ~ planet Terra, Via Lactea Galaxia

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