The Probability Of One Is Greater Than The Probability Of Infinity
The Probability Of One Is Greater Than The Probability Of Infinity
The Probability Of One Is Greater Than The Probability Of Infinity

A Treatise On The Newcomb-Benford Effect (Part I)

i.e. Exploring The Newcomb-Benford Law

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Narrated by Jim Wigdahl

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Abstract (TL;DR):

Occam’s razor posits the problem-solving concept which states 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 Newcomb-Benford Law (a.k.a. Benford’s Law, the Law of Anomalous Numbers, The First-Digit Phenomenon, etc.) is no exception to the rule and is in fact a rather exemplary case of Occam’s razor.

Benford’s Law has been observed, recognised, defined (both naturally & formally in mathematical language & notation), investigated, AND applied in fields ranging from accounting to forensics & beyond. Yet a viable derivation or proof for the law’s existence in the fields of mathematics, science, & nature has eluded both the academic and amateur communities since it was first noted nearly one and a half centuries ago.

This treatise presents empirical evidence demonstrating that the law’s enigmatic nature is not the result of any mathematical principle that has yet to be discovered, but rather is the byproduct of a mathematical process. Namely, it is an inherent artifact of the process of quantification, e.g. calculating numerical data such as distance, amount, weight, etc.

As the obfuscated nature of the Newcomb-Benford Effect is herein elucidated, six long-standing mysteries are addressed:

  • First — the effect’s relatively recent discovery is in no small part due to its abstract nature, which clarifies why it has evaded detection until fairly recently in the course of human history.
  • Second — the infinite and predictable variation of NBE can be observed for all positional numeral systems aside from the unary & binary.
  • Third — the Newcomb-Benford Effect holds true for all real numbers, including negative numbers.
  • Fourth — as the order of magnitude increases, so does the “strength” of the effect. In other words, the greater the range of calculations (say intergalactic distances for instance) the easier it is to observe the Newcomb-Benford Effect. This explains for the common misconception that multiple magnitudes of order are required for the Newcomb-Benford Effect to hold. In actuality, NBE is present at all orders of magnitude including single digits.
  • Fifth — numbers are biased towards one. In other words the probability of any outcome 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 presents us with the rather difficult task of grasping the notion that in quantification, the probability of 1 is greater than the probability of infinity or any other number for that matter.
  • Sixth — and of primary import, although attempts have been made to find an acceptable proof of NBE, none have prevailed in deriving one. This result (or rather lack thereof) is to be expected given that NBE is an indirect & ancillary outcome of an altogether separate process and cannot exist autonomously. In other words, the Newcomb-Benford Effect is the evidence (or side effect) of the mechanism applied to quantitation but is not a proof of itself, neither is NBE the mechanism nor the resulting quantity.

This treatise lays the groundwork for understanding the foundational principles that are required for the Newcomb-Benford Effect to occur. Following that, extrapolations of the effect, based on the principles previously laid out, are proposed. Finally, thought experiments are presented in order to further explore the phenomenon and offer “hooks” or references to parallel concepts in order to provide as comprehensive a picture as possible, before illustrating that one only requires the single digit range 0-9 in order to demonstrate NBE.

Keywords:

Newcomb-Benford Effect (NBE), Newcomb-Benford Law, Benford’s Law, The Law Of Anomalous Numbers, First Digit Law, First-Digit Phenomenon, Significant Digit Law, Leading Digit Phenomenon, Pure Math, Abstract Math, Applied Math, Real Numbers, Algorithm, Scale Invariance, Measure Theory, Counting Theory, Set Theory, Quantification Theory

Outline:

-Exordium

-The Newcomb-Benford Law (An Overview of the Effect)

-Ab Initio: A Brief History (Discovery of the Effect)

-Required Concepts for the Effect to Occur

  • First Principles (A Closer Look at the Effect)
  • Algorithms
  • The Algorithm of the Hindu-Arabic Numeral System
  • The Base 10 Algorithm (a.k.a. The Decimal System)

-Extrapolations :

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

-Thought Experiments Applying The Newcomb-Benford Effect:

  1. The Motel
  2. The Swimming Pool
  3. The Brick Wall
  4. Shapes
  5. Symbols
  6. Colours
  7. Breaking The Newcomb-Benford Effect

-Conclusion:

-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 shortly.

There’s who I THINK I am. There’s who I WANT to be. And there’s who I ACTUALLY am. I’ll be damned if I can figure out which one is which.

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