The Probability of One 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.

At its most rudimentary, Benford’s Law simply describes a fundamental property of real numbers — much akin to the description of homo sapiens having the fundamental property of bipedalism.

It has been recognized, defined (both naturally and formally in mathematical language and notation), investigated, applied, and identified and observed in fields ranging from accounting to forensics, to geology, and beyond.

The earliest known record of the phenomenon’s existence was first noted nearly 150 years ago, yet as of the writing of this treatise, there has been no broadly accepted proof nor explanation for its raison d’etre.

Benford’s Law is a naturally occurring and predictable pattern resulting from the unilinear (serial) mechanics of numerals inherent to quantitative analysis

This treatise aims to elucidate the first principles underlying Benford’s Law, both via formal numerical analysis and through analogical reasoning i.e. thought experiments.

Although the subject of what Benford’s Law is will certainly be addressed, it is not the primary focus of this treatise, in large part due to that particular topic having been covered extensively ad nauseam in countless prior essays attempting to explain and rationalize the existence of Benford’s Law.

The focus of this treatise is to define and describe the fundamental principles (and constraints) underlying cardinal (numerical) logic, thereby providing a foundation for comprehension of why Benford’s Law exists and why we see it in every discipline from mathematics, to science, to nature — an endeavor that up til now has never been thoroughly nor satisfactorily accomplished.

This treatise will show that Benford’s Law is a naturally occurring and predictable pattern resulting from the unilinear (serial) mechanics inherent to numerals used in quantitative analysis. Not only should we expect the existence of Benford’s Law, we should be surprised when it is not present in quantitative analysis. The fact that almost every description of Benford’s Law states that its presence is counter-intuitive, underscores just how misunderstood it really is.

Stated another way, Benford’s Law is the trace evidence of cardinal numeral behavior in quantitative analysis. The resulting pattern we know as Benford’s Law is produced from the process of quantification.

Quantification is defined as:

“the act of counting and measuring that maps human sense observations and experiences into quantities”

e.g. quantitative analysis; similarly understood as any form of calculation be it distances, amounts, weights, areas, etc.

Just as heat is generated from combustion, Benford’s Law is generated from quantification.

Within this treatise the following traits of Benford’s Law will be discussed:

1. It is Abstract: The law’s fairly recent discovery in the course of human history is in no small part due to its elusive nature, being difficult to detect, difficult to perceive, and even more difficult to conceptualize. Countless attempts have been made to derive a proof of Benford’s Law, however, 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 the process of quantification; however, it is neither the mechanism (addition/multiplication/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, just as heat is a naturally occurring and inevitable byproduct inherent to combustion, so Benford’s Law and the numeric pattern it generates, is a naturally occurring and inevitable byproduct inherent to numeral behavior in quantitative analysis.
2. It is Logarithmic: The greater the orders of magnitude, the greater the “strength” of Benford’s Law, making the effect (or numeric pattern) more and more pronounced. In other words, the larger the quantitative range (say intergalactic distances measured in meters as opposed to astronomical units) the easier it is to detect and perceive the pattern predicted by Benford’s Law. For this reason, a common misconception prevalent in most explanations and descriptions of Benford’s Law states that multiple orders of magnitude are required when conducting quantitative analysis for Benford’s Law “to work.” In actuality, Benford’s Law is present at all orders of magnitude including the single-digit range 1–9. This will be proven in due course.
3. It is Biased: In quantitative analysis, results “prefer” the number one. In other words, the probability of any calculated result is weighted toward the number one. Any result that has been quantified 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. Stated differently, in any quantitative analysis, the probability of 1 is greater than the probability of infinity and any number in between for that matter.
4. It is Ubiquitous: An infinite variation of Benford’s Law can be predicted and observed for all positional numeral systems aside from the unary & binary (this has previously been proven, demonstrating that the law is not an artifact of only the Decimal System). In addition (and what has never before been posited prior to this treatise), Benford’s Law applies to all real numbers, meaning that Benford’s Law can be observed not only in calculations of positive sets of numbers but also in calculations of negative sets of numbers. This too will be proven in due course.
5. It is Elemental: Benford’s Law isn’t simply a cool, useful pattern; it is much more than that. It is evidence of how the framework of existence works to produce the observable universe—generated by the fundamental constructs (as well as constraints) of numeral logic. Benford’s Law describes a pattern woven into the fabric of existence, by existence — it is self-referring. Without existence, the law wouldn’t exist, nor can existence exist without the law. Understanding the deeper meaning behind the statistical pattern known as Benford’s Law is the key to understanding why mathematical constructs such as the Normal Distribution (the Bell Curve) exist and permeate all of nature.

Benford’s Law describes a pattern woven into the fabric of existence, by existence…

This treatise is prima facie an exposé on Benford’s Law. On a more pragmatic level, however, it is an introduction to one of the simplest, most elegant, and most fundamental principles concerning the nature of numbers (and quite possibly one of the most misunderstood). Fully grasping Benford’s Law has far-reaching implications and ramifications for the entire mathematical and scientific community.

This treatise is much more useful in its application, as for all intents and purposes, it is an introduction to understanding the fundamental principles of quantitative analysis upon which rests our very existence. It is an analysis of the mathematical logic that defines the framework intrinsic to quantitative analysis — and thereby intrinsic to nature itself.

Upon being able to recognize & fully appreciate the far-reaching implications of Benford’s Law, one must re-evaluate and reconsider their perspective on the principles underlying the framework of the physical world; on how numbers work; and maybe even most importantly on how numbers are taught in schools.

It is oft said that nature is not random — comprehending the deeper meaning of the phenomenon known as Benford’s Law is vital to understanding why nature is not as random as might be expected; from the number of trees in forests to the widths of the mouths of volcanoes to galactic distances; anything at all 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 Mathematics, Applied Mathematics, Discrete Mathematics, Real Numbers, Algorithms, Scale Variance, Scale Invariance, Measure Theory, Counting Theory, Set Theory, Number Theory, Quantification Theory, Quantification Effect, Quantitative Analysis

Overview:

This treatise begins by addressing possible issues that could result from having prior knowledge of a deeply misunderstood topic. Preconceived notions could skew or hinder grasping the core ideas presented herein. Following that, I provide a basic explanation of Benford’s Law and briefly touch upon the history of its discovery as it will be pertinent and referenced in later sections. Continuing, I lay out the key elements or “ingredients” of the numerical systems developed and in use in mathematics today. Following that, extrapolations of the law based on the fundamental principles outlined are proposed and explored. In closing, thought experiments are presented in an attempt to connect the abstract ideas presented and correlate them to tangible, real-world applications.

Outline:

Part II:

~ Exordium

• On Naming Convention
• On Prior Knowledge a.k.a. Schemata
• On Abstract Concepts
• On Quantitative vs. Qualitative vs. Statistical Analysis

Part III:

~ The Newcomb-Benford Law (Definition)

~ Ab Initio — Discovering The Law (History)

~ The Elements of Benford’s Law (A Closer Look)

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

Part IV:

~ Numerical Analysis (Extrapolations) :

• Base 1
• Bases 2–15/20
• An Infinite Pattern
• Real Numbers

Part V:

~ Analogical Reasoning — Part I (Thought Experiments):

1. The Brick Wall
2. The Swimming Pool
3. The Motel

Part VI:

~ Analogical Reasoning — Part II (Thought Experiments):

1. Shapes
2. Symbols
3. Colours

Part VII:

~ Breaking Benford’s Law

~ Conclusion

~ Epilogue

~ References

(End of Part I)

Coming next….Part II — Exordium, etc.

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

Part II of this series is scheduled to be released October 2021.