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Volume 2, Issue 1

How to Explain Behavior: Author Precis
Sam S. Rakover

DOI: 10.33014/issn.2640-5652.2.1.rakover.1

Abstract

This is an author precis of the book How to Explain Be-
havior: A Critical Review and a New Approach by Sam
S. Rakover (2018, Lexington). The precis has two sections.
The first, basic methodological origin-points, treats the fun-
damental ideas and premises concerning explanation and
understanding. The second section outlines the book’s ar-
rangement, and summarizes the content of each chapter.

1 Introduction

This section describes the infrastructure on which I built
the entire book. It covers matters from the philosophy of
explanation in general to explanations in psychology more
particularly.

I start the discussion by emphasizing the following two gen-
eral cornerstone ideas in scientific research methodology (ci-
tations are taken from Einstein & Infeld, 1938/1950. See
also Popper, 1959/1995):

(A) The scientific attempt is to explain and understand re-
ality: "The purpose of any physical theory is to explain
as wide a range of phenomena as possible. It is justi-
fied in so far as it does make events understandable."
(P. 43)

"All the essential ideas in science were born in dra-
matic conflict between reality and our attempts at un-
derstanding." (P. 280)

(B) Scientific knowledge is provisional: "There are no eter-
nal theories in science. It always happens that some
of the facts predicted by a theory are disproved by
experiment. Every theory has its period of gradual de-
velopment and triumph, after which it may experience
a rapid decline." (P. 77)

Now I move on to discuss several methodological ideas that
are more specific to the subject of the book:

(1) To o!er an explanation for the studied phenomenon, a
suitable scientific procedure has to be used.

To propose explanations one has to know how, that is, sci-
entists need a justified procedure for giving an explana-
tion that is appropriate for the given phenomenon under
scrutiny, just as they need a procedure suitable for making
an observation or performing an experiment.

In a way similar to the justifications for the use of the pro-
cedure of an experiment that are based on several logical,
epistemological, ontological considerations, so the proce-
dure for giving scientific explanations is justified by various
considerations—logical, epistemological, ontological, and so
on up to common-sense considerations of daily life. A model
of scientific explanation is not like a scientific theory, but is
like a set of rules, norms, on how to behave. From this per-
spective it follows that an explanation model, procedure,
may not be tested empirically like a theory (which is sub-
ject to an empirical test) but is evaluated by theoretical
considerations and the degree of match of the procedure to
the field under study.

(2) A theory (or a hypothesis, model, law, mechanism) in
itself is not able to propose an explanation.

To explain a phenomenon, suitable explanatory informa-
tion is needed. In science this information is provided by a
theory, hypothesis, model and so on. However, the theory
in itself is not able to explain the occurrence of the studied
phenomenon. For the theory to do this, it has to function in
the framework of a procedure created to provide scientific
explanations.

(3) The functioning of an explanation-procedure is based
on rational relations.

Rational relations are of di!erent kinds: from logical, math-
ematical, statistical and causal to practical. To propose a
scientific explanation based on non-rational relations is im-
possible because what one will get is fortuity and chaos.

(4) A procedure of explanation is an integral part of the



18 How to Explain Behavior: Author Precis

methodology of science and is closely connected espe-
cially to a procedure of empirical testing.

The methodology of science is a collection of methods/pro-
cedures for research whose main purpose is to bring about
understanding of the world. Explanation-procedures are in-
dissolubly bound to a procedure for empirical testing, such
as the Hypothetico-Deductive (H-D) method (see Hempel,
1965, 1966). Without an empirical test we will never be
able to know if the given theory is false or true. However,
it should be stressed (as mentioned above) that scientific re-
search never ends, and anyone who believes that a certain
theory is wholly correct in fact drops out of the empirical
science game (see Popper, 1972, 1959/1995).

(5) The scientific explanation is an attempt to approximate
the hidden process involved in the occurrence of the
studied phenomenon. This can be expressed by the fol-
lowing three basic assumptions:

(a) There exists in nature some kind of Unknown
Real-Process (URP) that is responsible for
the occurrence of the studied phenomenon
under certain conditions. (Note that except
for the assumption regarding reality, the rest
is unknown.);

(b) Science proposes a theory (hypothesis, model,
mechanism) which tries to approximate the
URP and its function, thereby giving a
reasonable explanation for the studied phe-
nomenon;

(c) The proposed theory is based on the scientific
knowledge collected in the area under discus-
sion. The degree of proximity of this theory
to the URP is estimated by the degree of suc-
cess/failure of this theory.

The approach here (and of course in the entire book) is
realistic in the broad sense. That is, I perceive science as
a system that attempts to answer the question how nature
(inanimate, plant and animal) functions by suggesting a
theory that attempts to decipher nature’s code: how it is
built, how it works, and according to what.

(6) To date no one has been able to develop a theory that
will describe and explain the connection between neu-
rophysiological activity in the brain and consciousness,
i.e., no one has succeeded in developing a mind/body
theory. This carries the following implications:

(a) Behavior cannot be explained solely by mech-
anistic explanations (which are accepted in
the sciences, in behaviorism, in cognitive psy-
chology, and in physiological psychology), be-
cause it neglects conscious behavior;

(b) To propose a better account of behavior one
must consider the conscious aspect explana-
torily.
This may be done as follows:

(1) A procedure must be developed for
proposing mentalistic explanations
(e.g., explanations based on one’s
will and belief);

(2) It has to be shown that a procedure
for mentalistic explanation meets
the accepted methodological require-
ments of science;

(3) A theoretical framework has to be
developed that allows coherent use
of these two kinds of explanation
(mechanistic, mentalistic). This
account will be better than a purely
mechanistic explanation.

Idea (6) is one of the most important ideas sustaining
the present book. Chapter 5 summarizes the empirical-
theoretical arguments showing that the mind/body theory
has not yet been developed, and chapter 6 describes in de-
tail (a) Methodological Dualism (MD), which shows that
like the mechanistic explanation, the mentalistic explana-
tion meets the methodological requirements of science, and
(b) the Multi-Explanation Framework (MEF), which allows
one to construct a coherent theory in any psychological do-
main based on these two kinds of explanation: the mecha-
nistic and the mentalistic.

The book does not aim to o!er another solution for the
mind/body or consciousness/brain problem. Quite the re-
verse: the goal is to circumvent this problem, thereby to
achieve another goal, but no less important, namely to im-
prove the explanation of behavior. This purposeful e!ort
takes place within the framework of science by constructing
an argument showing that a mentalistic explanation can be
encompassed by the accepted methodological framework of
science.

2 Chapter Summaries

The book’s seven chapters plot a course of study from chap-
ter 1, on the relation between explanation and understand-



Volume 2, Issue 1

2. CHAPTER SUMMARIES 19

ing, to chapter 7, on a general procedure for giving expla-
nations for the behavior of the individual.

Chapter 1 discusses the relation between explanation and
understanding. The prevalent view from 1948 (the year
Hempel and Oppenheim published their influential paper)
was that scientific understanding was based on scientific ex-
planation. About fifty years later the literature began to
suggest that scientific understanding was not built entirely
on explanation, as it had additional sources such as the
ability of the scientist herself. The article by Lipton (2009)
even proposed the idea of understanding without explana-
tion. In the spirit of that article I proposed a relatively
new approach, namely the Dual Theory of Understanding,
which posits a certain non-dependence between explanation
and understanding.

First, scientific understanding is based on consciousness of
the scientific content associated with the phenomenon un-
der study. Hence a sophisticated robot may be imagined
that will supply an appropriate explanation for a given
phenomenon without understanding the significance of the
explanation. Accordingly, while giving an explanation is a
mechanical rational process, understanding the explanation
requires a human’s consciousness.

Secondly, even before 1948 (e.g., in antiquity), the human
being tried to understand the world and himself by means
of certain procedures that di!ered from the explanation-
procedures (models) known today. The chapter sets forth
several illustrations of procedures for an understanding of
that kind: a match between the phenomenon under study
and some scheme that exists in the individual’s cognitive
system; understanding person A when person B gets into
person A’s shoes; the passage of a square through a round
hole when the diagonal of the square is smaller than the
diameter of the circle; etc. These procedures are based
on the underlying assumption that there is some Unknown
Real-Process (URP) that is responsible for the appearance
of the phenomenon in question (in ancient times and even
today God represents URP).

The chapter also suggests that compared with understand-
ing, explanation is interwoven with scientific knowledge
based on the development of a rational scientific method-
ology, which emphasizes for instance the importance of the
development of a scientific model. As an example, the chap-
ter considers the model developed by Rakover & Cahlon
(1989) called the "Catch model." It is intended to reproduce
from a witness’s memory the picture of the accused’s face.
The phenomenon whose understanding is being sought is
presented as the "reality-model," and what explains it (the
reality-model) is a "theoretical-model" (hence the Catch
model is based on two sub-models: the reality-model and

the theoretical-model).

Chapter 2 reviews critically nine models of explanation
discussed in the philosophical literature, which (a) exerted
great influence on the professional literature; and (b) are
likely to have a great e!ect on providing explanations in
psychology.

The chapter opens with a review and discussion of the three
models proposed by Hempel: the Deductive-Nomological
(D-N) model, the Deductive-Statistical (D-S) model and
the Inductive-Statistical (I-S) model. Together these are
known as the "covering-law theory" because the common
idea is that a natural law, a statistical law or a statistical
generalization cover explanatorily the studied phenomenon.
The three models gave rise to strong criticism, and as a
result other researchers proposed alternative models of ex-
planation.

The first alternative model is the Statistical-Relevance (S-
R) model proposed by Salmon (1971, 1990). The important
ideas behind this model are that an explanation need not
necessarily rest first on a logical argument, and secondly
on high probabilities close to 1, as Hempel’s I-S model re-
quires. This model experienced sharp criticism, and even-
tually Salmon himself abandoned it for a new explanatory
model based on the concept of causality.

The Causal-Mechanical (C-M) model proposed by Salmon
(1984) is based on the idea that the world is a net-
work of interactive causal processes that spread through
space and time. The explanation places the studied phe-
nomenon within this network. Another causal model is
the Manipulationist-Interventionist (M-I) model suggested
by Woodward (2003). Its basic idea is the causal connec-
tion between X and Y through a certain manipulation that
changes X, hence Y changes accordingly and systematically;
thus the causal explanation allows seeing what is likely to
happen as a result of the counterfactual operation of a cer-
tain manipulation. An additional explanatory model is the
Kairetic model proposed by Strevens (2008). Its basic idea
is the application of a method of elimination to explanation:
di!erent factors are removed from the list of causes poten-
tially responsible for the studied phenomenon; the goal is to
see the elimination of which factor causes the phenomenon
to disappear (or be likely to disappear). The elimination is
repeated until only the factors (or factor) are (is) left with-
out which the given phenomenon would not have occurred.

The chapter concludes with a discussion of two models. One
is the Unificationist model developed by Kitcher (1989), the
other is the Pragmatic model proposed by van Fraassen
(1980). The former posits that understanding the world
depends on the ability to derive from the same patterns



20 How to Explain Behavior: Author Precis

of inference many phenomena that are observed in scien-
tific research (e.g., the Newtonian theory). van Fraassen’s
Pragmatic model draws attention to the practical aspects,
the background factors, of explanation as giving an answer
to the questions: the subject of the question, the contrast-
class (the possible alternatives for giving an answer), and
the relevance-relation, a kind of relation between giving the
answer and the question (e.g., “Compared with the alterna-
tives of the answer, the phenomenon occurred because...”).

An important conclusion that arises from this review is that
there is no single concept for scientific explanation, which
is enormously complex and multi-dimensional.

Chapter 3 examines three additional models of explana-
tion: rule-explanation; functional, teleological models of
explanation; and the new-mechanistic model of explana-
tion. Although these models are intended to handle hu-
mans’ and animals’ behavior, they are unable to handle
the most important component of this behavior, namely
conscious-experience. My proposal how to deal with this
appears in the last three chapters of the book.

It is not possible to apply any of the models reviewed in
the foregoing chapter to human behavior, for example, to
the following cases: Dan stopped his car when the tra"c
light changed to red; or Dan stood on a chair in order to
take a book down from the shelf. This is because for these
kinds of behavior no law, theory or appropriate empirical
generalization exists. For example, stopping at a red light
is not explained by an appeal to a law or empirical general-
ization that covers behavior of this kind, but by looking up
a tra"c rule created by a human being and which a driver
is obliged to obey. Seven important di!erences between be-
havioral rules and laws are discussed: for example, laws are
universal and rules are local; laws are subject to an empir-
ical test and rules are not (rules are evaluated according to
the purpose for which they were devised). Although not
all rules are present in consciousness (e.g., basic rules for
generating language are innate), they di!er from laws of
nature in that they concern information.

Explanation by rules is also di!erent from teleological ex-
planation. For example, people are not always aware of the
aim of the rules and they obey rules automatically. Two
approaches attempt to characterize functional, teleological,
explanations as causal. The approach developed by Nagel
(1961) tries to answer the question why a certain component
appears in a given system; Cummins’ (1975, 1983) approach
tries to answer the question how a certain component suc-
ceeds in contributing causally to the e"cient function of a
given system.

The chapter ends with a discussion of the new-mechanistic

explanatory model. According to this model, an expla-
nation for the given phenomenon is proposed by an ap-
peal to a mechanism. It can be decomposed into di!erent
components with di!erent functions, unique in their array
and interaction, which produce the studied phenomenon.
Bechtel (2008a, b) draws a distinction between a new-
mechanistic explanatory model, which suits the sciences,
and one that suits psychology, that is, a mentalistic mecha-
nism, which processes information—the Information Pro-
cessing (IP) mechanism. There is an important di!er-
ence between Bechtel’s (2008b, c) approach to a mentalis-
tic mechanism and Rakover’s approach. While according to
Bechtel processing information occurs on the non-conscious
cognitive level, according to Rakover mentalistic activation
takes place on the conscious level.

Chapter 4 discusses whether an explanation model in it-
self can be subjected to empirical test. The chapter raises
arguments against the idea that the correctness of the ex-
planation model can be tested empirically.

First, the prediction that emerges from a certain theory
under certain conditions can be confirmed or refuted. In
both cases the empirical result carries no implications for
the explanatory model itself, but only for the theory in-
serted in the explanatory model. The reason is that the
explanation model is just a procedure that guides the re-
searcher on how to propose scientific explanations. An ex-
planation model is not like a hypothesis, theory or scientific
model, which may pass/fail an empirical test. A procedure
is assessed by its rational course of action (e.g., it does not
create situations that contradict each other) and its appro-
priateness for a given field. In this sense, a procedure for
giving explanations is akin to an administrative procedure,
such as tra"c rules that advise people how to behave in
certain situations. A rule of the road is assessed as suit-
able or unsuitable for a certain field of behavior but it is
not judged right or wrong. The chapter goes on to discuss
this distinction as against the accepted distinction between
descriptive, normative, theories and prescriptive ones.

Secondly, we may look at the argument that slight use of an
explanation model Mx attests to its incorrectness (i.e., Mx
is refuted). Here the question that arises is how we may ex-
plain this finding of infrequency of Mx use. Clearly, it can-
not be explained by Mx itself (because it has already been
refuted), so we must seek another model—My. However,
it is reasonable to suppose (according to those who assume
that an explanation model can be tested empirically) that
this model too fails empirically. Why? Because there is
very high probability that very often theories that used My
proposed predictions that were disconfirmed. Hence, not
only is My not suitable for use as a procedure for giving
explanations, no explanation model exists that can explain



Volume 2, Issue 1

2. CHAPTER SUMMARIES 21

the finding of slight use of Mx —because all explanation
models at some time have been associated with refutation
of empirical results of di!erent theories. Clearly then, em-
pirical success/failure has no implications for the degree of
correctness of a procedure for explanation.

Based on a review of the explanation models found in the
literature, chapter 4 o!ers a list of several constraints, char-
acteristics and properties for constructing an explanation
model, which have been described above. These make up
the infrastructure for building explanation models for psy-
chology (see especially chapters 6 and 7).

Chapter 5 discusses the following question: is it possi-
ble to comprehend mentalistic concepts and explanations
(based on the individual’s inner world) with the aid of
mechanistic concepts and explanations (based on and ac-
cepted in the sciences)? The answer is negative, and it
forges the way to the development of a new methodological
approach to proposing explanations, namely the Method-
ological Dualism (MD) and Multi-Explanation Framework
(MEF), whose principal purpose is to propose a procedure
to explain behavior that is saturated with consciousness.
This methodology is based on the fact that to date no one
has succeeded in developing a mind/body theory. Were it
possible to propose such a theory, it would be possible to
forgo the DM & MEF approach, because everything would
obtain a mechanistic explanation. But as chapter 6 shows,
this is not the situation so far.

Chapter 5 breaks down the above fundamental question
into the following four questions:

(1) Can human behavior be explained mechanistically
alone, without recourse to the conscious-experience
concept?

(2) Has a theory been successfully developed that explains
the connection between neurophysiological processes in
the brain and conscious-experience?

(3) Can mentalistic explanations anchored to the concept
of conscious-experience be reduced to mechanistic ex-
planations (e.g., to a neurophysiological explanation)?

(4) Has a very complex and sophisticated computer proven
able, like us, to behave with conscious-experience?

An a"rmative answer to any of these four questions means
that there is no need to develop MD and MEF because the
standard methodology developed in the sciences can han-
dle research in psychology. Only a negative answer to all
these questions will oblige one to think about the develop-
ment of a new explanatory approach suited to the field of

psychological research. The answers to these questions are
negative.

Here are three popular arguments that support the negative
answers. First, it seems that mentalistic theories cannot
be reduced to mechanistic theories. One of the important
factors that prevents this kind of reduction is multiple re-
alizations. If a mental state or process is defined by its
function, it may be realized in various ways and with var-
ious materials. Hence, if reduction of a mentalistic theory
to a mechanistic depends on the possibility of identifying a
concept in one theory with another, this possibility is de-
stroyed because according to multiple realizations there are
many ways to realize and identify mentalistic concepts.

Secondly, a computer, as sophisticated as it may be, is inca-
pable of reaching a condition similar to humans’ conscious-
experience. One of the most powerful and famous argu-
ments in support of the negative answer is Searle’s (1980)
"Chinese Room" thought experiment. According to this,
activation of physical signs by means of syntactic rules
(which are operated by the computer) cannot impart se-
mantic meaning to these signs.

Finally, a very important obstacle blocking the development
of a mind/body theory is "consciousness epiphenomenal-
ism," which states that mental concepts and explanations
are inessential. The reason is that the entire explanation
of behavior falls on the shoulders of the neurophysiological
process.

Chapter 6 describes and justifies the approach for
Methodological Dualism (MD) and Multi-Explanation
Framework (MEF), which is based on chapter 5. Chap-
ter 6 is divided into several sections. The first proposes an
argument justifying why MD & MEF should be developed.
Namely, if we attempt to o!er an improved explanation for
behavior saturated with consciousness there is no way other
than adding to a mechanistic explanation a mentalistic ex-
planation.

The second part of the chapter develops Methodological
Dualism (MD). The chapter focuses particularly on the tele-
ological explanation that combines the individual’s will, be-
lief and behavior. It presents a new proposition to conceive
the following as a mentalistic explanation-model – a proce-
dure for creating specific will/belief explanations:

[Will/Belief]: If X wants G and believes that behavior B
will realize his/her will, then X will perform B.

This proposition immediately raises the following ques-
tion: does this mentalisitic explanation-procedure satisfy
the methodological requirements for explanation accepted
in science? The answer is yes.



22 How to Explain Behavior: Author Precis

The third section develops a theoretical framework, the
Multi-Explanation Framework (MEF), by means of which
it will be possible to construct a specific theory (a specific
"MEF-theory") for a specific research project based on two
kinds of explanation: mechanistic and mentalistic. Because
the theory rests on multiple explanation models, a major
di!erence arises between (a) the way a theory supplies an
explanation in science and (b) the way an MEF theory sup-
plies an explanation in psychology.

An MEF-theory is based on a match between explanation-
procedures (mechanistic or mentalistic) and behavior (it is
called the "explanation/behavior match" for short). While
there are behaviors that can be accounted for satisfacto-
rily by an appeal to mechanistic explanations only, most
behaviors and their decomposed components must be ap-
proached by both mechanistic and mentalistic explanations.
The coherence of a specific MEF-theory is achieved by
matching appropriately the explanation-procedure to the
behavior and its components by means of several guidelines
based on theoretical-empirical knowledge and the princi-
ple of explanation-matching. According to the latter, the
components of a mechanistic behavior (a whole behavior
that has been accounted for mechanistically) cannot be ex-
plained by mentalist explanations, while the components
of a mentalistic behavior (a whole behavior that has been
accounted for mentalistically) can be explained mentalis-
tically as well as mechanistically. The appropriate expla-
nation/behavior match solves several methodological prob-
lems raised by an MEF-theory.

Finally, the last section summarizes the main ideas of the
present approach and compares it to other relevant ap-
proaches.

Chapter 7 presents A General Explanation Procedure
(GEP) for understanding an individual’s behavior. As
can be seen from the previous chapters, psychology has
used a fairly large number of explanation models or pro-
cedures: di!erent variations of Hempel’s models, di!erent
kinds of causal processes, mechanisms based on the com-
puter metaphor, and di!erent kinds of procedures based on
neurophysiological processes. This employment of multi-
plicity of explanation models raises the following question:
is it possible to propose for psychology a general procedure
for giving an explanation that is likely to spread its wings
over other models too? The answer is a"rmative: the GEP.
It is based on three major theoretical ideas:

(a) It is assumed that in the individual a certain real-process
exists that is involved in the generation of the behavior un-
der study; although the nature of this process is unknown,
it is further assumed that it presents the real explanation
for the behavior in question [this process is called the "Un-

known Real-Process" (URP)];

(b) A researcher may approximate the URP by construct-
ing a theory (or a hypothesis, model, mechanism, etc.) that
portrays the properties and operation of the URP in gen-
erating the studied behavior. If the proximity between the
theory and URP is high, it may be proposed that the theory
provides us with a good account of the behavior in question;

(c) The degree of proximity of the theory to the URP is
estimated by the development of a "proximity index.” It
is based on the theory’s successful predictions. Clearly, a
confirmed theory is closer than a refuted theory to the URP
and to the real explanation.

The generality of GEP is expressed in two respects. First,
the URP can be approximated by diverse theories or hy-
potheses, models and mechanisms;

Secondly, the GEP may handle various explanation-
procedures (models) that researchers have been using in
their studies. These models may be viewed as various routes
for approaching the real explanation.

To illustrate the GEP, the chapter analyzes several explana-
tions given for learning in animals, perception and recogni-
tion of faces, and everyday purposive behavior (will/belief
accounts). Finally, the chapter discusses the basic philo-
sophical concepts and assumptions that underlie the GEP.

References

Bechtel, W. (2008a). Explanation: Mechanism, modular-
ity, and situated cognition. In P. Robbins & M. Ayd-
ede (Eds.), Cambridge handbook of situated cognition.
Cambridge, UK: Cambridge University Press.

Bechtel, W. (2008b). Mechanisms in cognitive psychology:
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9994.

Bechtel, W. (2008c). Mental mechanisms: Philosophical
perspectives on cognitive neuroscience. London: Rout-
ledge.

Cummins, R. (1975). Functional analysis. The Journal of
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Cummins, R. (1983). The nature of psychological explana-
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Einstein, A. & Infeld, L. (1938/1950). The evolution of
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Hempel, C. G. (1965). Aspects of scientific explanation and
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Hempel, C. G. (1966). Philosophy of natural science. En-



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glewood Cli!s, NJ: Prentice-Hall.
Hempel, C. G. & Oppenheim, P. (1948). Studies in the logic

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Lipton, P. (2009). Understanding without explanation. In
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Volume 2, Issue 1

Two Methods of Calculating Axiom Size
Jonathan Bartlett

DOI: 10.33014/issn.2640-5652.2.1.bartlett.1

Abstract

1 Introduction

Modeling insight has been a long-standing problem for cog-
nitive psychology. Bartlett (2012) noted that humans tend
to be able to infer axioms about a system in order to solve
problems. However, the framework proposed there su!ers
from a few theoretical problems. The first is that axioms
can overlap. Therefore, identifying distinct axioms is di"-
cult. Second, the framework relies on an ordering of axioms
from “small” to “large” (i.e., easy to di"cult), but fails to
provide a mechanism to measure the size of axioms.

Here we identify two possible mechanisms for calculating
the size of an axiom, and how it may help future progress
of cognitive research.

2 Dependent Axioms

Bartlett (2012) di!erentiated between two kinds of
axioms—independent axioms and dependent axioms. An
independent axiom is one which defines the parameters
of a formal axiomatic system. However, as Gödel points
out, any formal axiomatic system has truths which are not
provable within the formal axiomatic system (Raatikainen,
2018). These truths cannot be theorems within the ax-
iomatic system, because then they would be provable within
the system. However, these truths are dependent on the
structure of the axiomatic system.

In other words, given a particular axiomatic system A, some
truth B is set for certain, but cannot be proved within A.
Therefore, B is not a theorem within A, but its truth value
is set by the parameters of A. Because of this, we identify
such a truth as a dependent axiom.

If B is added to A as an axiom, A becomes A!. This new
system can prove B (because B is an axiom), and may also
be able to prove other new truths as well. However, there
are an infinite number of dependent axioms, so no finite set

of dependent axioms will cover all of the truths of A.

3 The Halting Problem

The most practical outgrowth of this line of reasoning is
in Turing’s Halting Problem. Turing showed that, given
a program p in a general-purpose programming language
(where p includes both the program and all its input), there
is no general, finite program h such that providing h with
p as a parameter will tell if p will ever complete (Turing,
1936).

More specifically, the problem comes in the fact that, while
we can tell if p does complete, we can never tell if it won’t
complete. The reason is simple—if we run p a step at a
time, at any given moment, p may complete. At that point,
p will be known to complete. However, we won’t know if
p doesn’t complete until we have waited an infinitely long
time, and it hasn’t happened. Therefore, while knowledge
of completion takes a finite amount of time, knowledge of
non-completion takes an infinite amount of time.

If p will not ever complete, knowledge of that fact can be
considered a dependent axiom. Knowledge of whether a
particular p or class of ps will not ever complete can be
added to h as dependent axioms. However, because there
are an infinite number of dependent axioms, we can never
add all of the axioms to h and have h remain a finite pro-
gram.

4 Example Axioms

Let us presume that there is a machine language such that
each instruction is one byte long, and the hexadecimal code
CC means “jump relative,” which looks at the next byte and
modifies the program counter based on that signed value.

Therefore, if the byte sequence CC-00 occurs, the machine
will go into an infinite loop, because it will jump relative
to the current instruction, but the amount of the jump will
be 0, because it doesn’t go anywhere.



26 Two Methods of Calculating Axiom Size

There are several ways of constructing an axiom which cov-
ers this fact. Non-exhaustively, these include:

1. If the entirety of the program contains exactly the code
CC-00 it will not complete.

2. If the program contains the code CC-00 within a line of
code that will provably execute (i.e., contains no prior
jumps or some similar criterion), then the program will
not complete.

3. If the program contains the code CC-XX, where X X is
a value less than or equal to zero which moves the
program to a place that provably returns back to the
same line of code, then the program will not complete.

Let us note several things about these di!erent dependent
axioms. First of all, note that all of the proposed axioms
include the extraordinarily simple program CC-00. Thus,
there is definitely overlap among the axioms. Also note
that while Axiom 1 is specifically for a single program, Ax-
iom 2 and Axiom 3 both match multiple programs. In fact,
Axiom 3 matches a much larger number of programs than
Axiom 2.

Also of interest, in these examples, the longer the descrip-
tion of the item, the more potential programs are matched.
This is not necessarily true (i.e., there could be a shorter
program that captures more potential programs), but it is
true that more length allows for more coverage. That is,
if we had a shorter program with more coverage, we could
get yet even more coverage by adding bits to our program.

5 Measuring the Axioms

I have identified two potential ways of measuring the “size”
of each of these axioms, though there are probably yet oth-
ers.

5.1 Measuring Occurrence Probabilities

The first way is to measure the probability of occurrence
of the axiom. That is, what is the likelihood of a given
arbitrary program that contains the axiom occurring? This
probability could be converted into bits in order to give a
size of an axiom.

To see how this would work, look at Axiom 1. Imagine that

programs were encoded as prefix-free codes.1 As a simplifi-
cation, the prefix-free code will be achieved by saying first
N bits are a tally of the number of bytes in the code, ter-
minated by a single 0 bit. Therefore, the prefix-free ver-
sion of the program identified by Axiom 1 would be 0b110
CC-00.2 The probability of this program would be 1

524288 ,
or 19 bits. Thus, the size of Axiom 1 using this metric is
19 bits. The size of the other two axioms would require
significantly more investigation (and specification as to the
programming system) to determine. However, for an axiom
that matches a finite number of programs, the size of the
axiom using this method is relatively easy to determine.

5.2 Measuring the Size of the Axiom
Identifier

Another method for size measurement is to identify the
shortest possible program which is able to identify the given
axiom in code. Using this method, Axiom 1 could be iden-
tified by the pseudo-code, “if the size of the program is
two bytes, and the first byte is CC and the second byte is
00, then return true, otherwise return false.“ Obviously, we
would need to spell out the rest of the language in order to
determine the size of this pseudo-code, but the procedure
is straightforward.

6 Implications

The most important result here is that dependent axioms
are in fact measurable. Whether these are the most impor-
tant measures of axioms is only partially relevant—the idea
that they can be measured has been validated.

Nonetheless, it seems that these measurements do have
some importance. Measuring the occurrence probability
measures the likelihood that the axiom will come into ef-
fect in any given program, thus whether or not someone
will likely be thinking about the axiom.

Measuring the size of the detector has even more interesting
applications. It was noted in Bartlett (2012) that some
axioms require other axioms to be known before discovery.
Measuring detector size fits in well with this idea because,
given an existing detector A, and a goal detector B, there
may exist some mutual information between A and B which

1This is a requirement so that the individual programs can be
represented as probabilities (Kraft, 1949).

2Note that the first part of the program is not necessarily fall on
an even byte mark, so it is just given as the list of bits, while the rest
of the code is given in hexadecimal.



Volume 2, Issue 1

REFERENCES 27

makes the size of B smaller given a detector for A.

The proposed equation for generating axioms using insight
in Bartlett (2012) was

A = I (Q, p, i, B) (1)

where I is the “insight” function of cognition, Q is the deci-
sion problem you are trying to solve (i.e., the halting prob-
lem), p is the program with i as the input, and B is a set
of axioms. The point of the function is to say that, given
a su"ciently inclusive B, the remaining axiom(s) A needed
to decide Q for p and i can be found. Using the detector
size metric, we might posit that the critical feature is the
size of A given B. That is, insight allows an amount of
information of a detector to be built. Given a specifically
inclusive B, the amount of mutual information between B
and A can reduce the number of bits required to build the
detector. This provides some amount of directionality for
insight problems, while still allowing for them to be found
in various ways.

References

Bartlett, J (2012). “Using Turing Oracles in Cognitive Mod-
els of Problem-Solving”. In: Engineering and the Ulti-
mate: An Interdisciplinary Investigation of Order and
Design in Nature and Craft. Ed. by J Bartlett, D
Halsmer, and M R Hall. Broken Arrow, OK: Blyth In-
stitute Press, pp. 99–122.

Kraft, L G (1949). “A Device for Quantizing, Grouping,
and Coding Amplitude-modulated Pulses”. PhD thesis.
Massachusetts Institute of Technology. url: http://
hdl.handle.net/1721.1/12390.

Raatikainen, P (2018). “Gödel’s Incompleteness Theo-
rems”. In: The Stanford Encyclopedia of Philoso-
phy. Ed. by E N Zalta. url: https : / / plato .
stanford . edu / archives / fall2018 / entries /
goedel-incompleteness/.

Turing, A M (1936). “On Computable Numbers, with Ap-
plication to the Entscheidungsproblem”. In: Proceedings
of the London Mathematical Society 42, pp. 230–265.





Volume 2, Issue 1

Letters and Notes

Divergent series and its assigned
value in a hyperreal context
Bas van der Reijden
DOI: 10.33014/issn.2640-5652.2.1.reijden.1

Abstract

This letter discusses the deep connection between the infi-
nite sum of natural numbers and the value " 1

12 . Aside of
more widely known facts, we consider a nontrivial way in
which we show the veracity of this connection; more pre-
cisely this concerns the BGN method (Bartlett, Gaastra,
and Nemati, 2020) applied on the so-called damped oscil-
lated Abel summed variant of the series. Moreover, we have
found a generalization of this method which ‘correctly’ as-
signs finite values to other divergent series. We conclude
with some questions concerning whether and how we can
analytically relate our hyperreal terms to frame the method
in a more justifiable and applicable context.

It is obvious that the sum of natural numbers 1+2+3+ · · ·
tends to infinity and can thus not be equal to " 1

12 . There
does however exist some connection between this series
and value and it is highly probable that this connection
is implicitely used (i.e. ‘under the hood’) in e.g. physics
(which often turns out to be perfectly justifiable, as can be
shown by various experiments).

The first evidence of this connection is retrieved when one
considers the Riemann zeta function ! (s). It is known
that ! (s) is equal to " 1

12 when s = "1 and it is interesting
that one retrieves the sum of natural numbers when one
‘plugs in’ s = "1 at the defining series of the Riemann zeta
function

!#
k=1

1
k s . Plugging s = "1 in the above series is

unfortunately not justifiable (given that ! (s) is only equal
to this series when $(s) > 1) but it remains an interesting
thing to mention.

Another evidence of the connection can be revealed
when one considers the ‘smoothed version’ of the partial
sums

!N
k=1 k, see also Tao, 2010. It turns out that these

smoothed partial sums have the same behaviour as the

Figure 1: Smoothed partial sums
!N

k=1 k with a y-
intercept of " 1

12 .

regular partial sums (i.e. they have the same asymptotic
expansion) and thus tend to infinity when N % #.
However, one probably recognizes the constant value in its
asymptotic expansion, which is (according to Tao (2010))
given by CN2 " 1

12 +O( 1
N ) (with C some coe"cient of little

importance in this case). Moreover, it is given that " 1
12 is

attained when one looks at the intersection with the y-axis
in Figure 1.

The first two evidences we mentioned are quite widely
known but a more unknown fact can be observed when
one considers a damped oscillating variant of the series
1 + 2 + 3 + · · · , namely

#"

k=1

ke"k! cos(k" ). (1)

This variant was also discussed in a previous letter
(Bartlett and Khurshudyan, 2019). In this letter it was
also mentioned that, in the context of hyperreal numbers
by introducing # := # (i.e. by appying the BGN method
on it), (1) can be written in a closed-form expression (or at



30 Letters and Notes

least as an approximation of it). It remained however still
unclear which value/magnitude the infinitesemal quantity
" must be1 in order that the BGN method applied on (1)
equals the ‘appropriate’ value " 1

12 ; only numerical evidence
was given. In particular, it was shown that if " = 1

" , the
computing software “Wolfram Mathematica” will include
the constant " 1

12 in its BGN expression (which is similar
to the observation of the previous paragraph).

At the time that Bartlett and Khurshudyan (2019)
was written, it only seemed clear that there is a numer-
ical evidence that (1) equals " 1

12 when we take " in a
su"ciently small interval. Recently, we have found that
Sugiyama (2014) (Section 2.3) provides a more theoretical
derivation of this matter. Although the website and its
choice of words are somewhat vague and confusing, the
derivation seems correct. In this derivation there is being
made use of a so-called ‘damped oscillated Abel summation
method’, which is a kind of generalization of the more
common Abel summation method used to assign finite
values to divergent series. In this article, this method of
‘damped oscillated Abel summation’ is consequently used
on a larger class of divergent series as well; furthermore
it turns out that the ‘damping’ and ’vibrating’ constant
should not be necessarily equal to each other. We thus
in fact have that (see also Section 5.2 and Section 6.1 of
Sugiyama (2014), we here write " instead of x)

!#
k=1 k i can

be transformed to (letting i & 1 be an integer)

""

k=1

k ie"k! cot !
2i+2 cos(k" )

and
""

k=1

k ie"k
i+1
2 ! cos(k

i+1
2 " )

and consequently taking the limit " % 0 yields the ‘ap-
propriate’ assigned value; we also numerically verified this2.

It remains of course interesting how this damped os-
cillated Abel summation method can be stated in our more
‘detailed’ hyperreal context; i.e. in which we know the
exact values of " (possibly in terms of #) in order that the
BGN method assigns the ‘appropriate’ value to a divegent
series. Unless it is still untrivial which values " must have
in order that the mentioned method assigns this value, we
can however say from Equation (5.57) in Sugiyama (2014)
that in general 1

! must be a lot smaller than # (this was
also shown by numerical experiments: if we set " = 0.01, #
must be a lot larger than 100).

In conclusion, we can thus say that the connection
1in relation to "
2by again letting ! be in a su!ciently small interval

between the often assigned value of a divergent series is
hidden in its asymptotic expansion. Furthermore, some
slight variations (performed in the context of hyperreals) of
the terms in the divergent series will alterate its asymptotic
expansion in such a manner that that the BGN method
assigns the ‘desired value’ to it. As it is at this point
still untrivial when equality holds, and how in this case "
and # thus must be related, remains an interesting topic
for further research. To state this in a more general and
mathematically way: Consider a divergent series with
BGN expansion A(" (#))#2 +C +O(1/#) (here A is a value
dependent of " which is in turn dependent of # and C is
the ‘appropriate’ value we want to have), the question is
now which variations (in terms of " (#)) we have to make
in order to make A(" (#))#2 equal to zero.

Bartlett, J, L Gaastra, and D Nemati (2020). “Hyperreal
Numbers for Infinite Divergent Series”. In: Communi-
cations of the Blyth Institute 2.1, pp. 7–16.

Bartlett, J and A Khurshudyan (2019). “Numberphile’s
Proof for the Sum 1+2+3+...” In: Communications of
the Blyth Institute 1.1, pp. 54–55.

Sugiyama, K (2014). New proof that the sum of natural
numbers is -1/12 of the zeta function. url: https :
/ / xseek - qm . net / Regularization _ e . htm #
_Toc524947400 (visited on 11/16/2019).

Tao, T (2010). The Euler-Maclaurin formula, Bernoulli
numbers, the zeta function, and real-variable analytic
continuation. url: https://terrytao.wordpress.
com/2010/04/10/the-euler-maclaurin-formula-
bernoulli - numbers - the - zeta - function - and -
real-variable-analytic-continuation/ (visited on
01/07/2019).

Do Mutation Rates Match the Kelly
Criterion?
Jonathan Bartlett
DOI: 10.33014/issn.2640-5652.2.1.bartlett.2

The Kelly Criterion defines an optimal betting strategy for
games that have a defined risk and payo!. It was developed
by John Kelly, Jr. at Bell Labs (Kelly, 1956). Given a bet
with a probability of success P and a payout of B, the Kelly



Volume 2, Issue 1

On Logic of Being and Wigner’s astonishment regarding the applicability of Mathematics 31

criterion tells you the size of your bet compared to your
overall bankroll. The Kelly criterion is given as

PB + P " 1
B

(1)

This formula can be derived from a formula for an expected
total payo! of the bet given by the equation

T = A(1 + B f )NP (1 " f )N (1"P) (2)

where T is the total winnings, A is your starting amount,
N is the number of trials, and f is the bet size. Optimizing
for T yields Equation 1.

Many researchers have discussed the concept of mutations
in populations as “bet hedging.” (Philippi and Seger, 1989;
Bartlett, 2008; Simons, 2011; Grimbergen et al., 2015)
Since the Kelly criterion allows one to at least theoreti-
cally calculate the optimum bet size for each configuration,
it might be possible to calculate various optimum mutation
rates at di!erent sites and compare them to their optimal
size according to the Kelly criterion, or an adjusted version
of it.

Most analysis of bet hedging has merely checked to see if
the hedging strategy is empirically beneficial (Childs, Met-
calf, and Rees, 2010; Simons, 2011) or potentially evolvable
(King and Masel, 2007). Applying the Kelly criterion may
be able to help determine how optimal organisms’ various
bet hedging strategies are.

One possible experimental approach would be to provide
organisms with a long-term, continually-varying environ-
ment. After many generations, it would be interesting to
check if the mutation rates for adaptive switching between
environments had any relation to the theoretical considera-
tions of the Kelly criterion, or any other theoretical hedging
system.

Bartlett, J L (2008). “Statistical and Philosophical Notions
of Randomness in Creation Biology”. In: Creation Re-
search Society Quarterly 45, pp. 91–99.

Childs, D Z, C J E Metcalf, and M Rees (2010). “Evolution-
ary bet-hedging in the real world: empirical evidence
and challenges revealed by plants”. In: Proceedings fo
the Royal Society B: Biological Sciences 277, pp. 3055–
3064. doi: 10.1098/rspb.2010.0707.

Grimbergen, A J et al. (2015). “Microbial bet-hedging: the
power of being di!erent”. In: Current Opinion in Mi-
crobiology 25, pp. 67–72.

Kelly, J L (1956). “A New Interpretation of Informa-
tion Rate”. In: The Bell System Technical Journal
35, pp. 917–926. url: http://www.herrold.com/
brokerage/kelly.pdf.

King, O D and J Masel (2007). “The evolution of bet-
hedging adaptations to rare scenarios”. In: Theoretical
Population Biology 72, pp. 560–575. doi: 10.1016/j.
tpb.2007.08.006.

Philippi, T and J Seger (1989). “Hedging One’s Evolution-
ary Bets, Revisited”. In: Trends in Ecology and Evolu-
tion 4.2, pp. 41–44.

Simons, A M (2011). “Modes of response to environmental
change and the elusive empirical evidence for bet hedg-
ing”. In: Proceedings of the Royal Society B: Biological
Sciences 278.1712, pp. 1601–1609. doi: 10.1098/rspb.
2011.0176. url: https://royalsocietypublishing.
org/doi/10.1098/rspb.2011.0176.

On Logic of Being and Wigner’s
astonishment regarding the
applicability of Mathematics
Gordon E Mullings
DOI: 10.33014/issn.2640-5652.2.1.mullings.1

The Nobel Prize winning Physicist, Eugene Wigner, fa-
mously posed a powerful challenge (1960) by asking why
is mathematics so e!ective, especially in the physical sci-
ences (Wigner, 1960). It is possible that the reason for the
e!ectiveness of mathematics is not because mathematics
is in any way causative, but instead because mathematics
studies the structure of logical possibility and constraint.
When plugged into a possible world, mathematics gives us
the tools to analyze the logically possible outcomes. There-
fore, when a possible world that is expressed mathemati-
cally su"ciently aligns with reality, mathematics becomes
e!ective at expressing relationships and outcomes.

For example, beings (as well as possible beings and things
impossible of being) can be understood in the context of
possible worlds. A “possible world” is a su"ciently complete
description of possible states of a!airs described through
chains of propositions. We may observe that things impos-
sible of being, such as a square circle, have in them mutu-
ally inconsistent required core attributes; they cannot be
realised in any possible world. Possible beings would exist
in at least one possible world were it actualised.

For instance, a contingent being B that depends on C might
exist in a world W and not in a closely neighbouring one W !



32 Letters and Notes

if C is present in W but not W !; C thus being an enabling,
necessary causal factor for B. By contrast, a necessary be-
ing F will exist in all possible worlds, showing itself to be
a framework element for such a world.

A key insight is that for any world W to be distinct from W !
it requires some factor A in W that is absent in W !. We may
then partition the factors of W as W = {A|¬A}. After parti-
tioning, we will have two distinct groups—the factor A and
all of the factors which are not A. The null set corresponds
to zero. Each particular set in the partition can be counted
as the number one, and the combination of both partitions
(even in a single world where A is an empty set) is two.
Thus, for any particular possible world W , the quantities
0, 1, 2 are necessarily present. Taking the von Neumann
construction, immediately we find N, thence (using addi-
tive inverses) Z, so also (taking ratios) Q and (summing
convergent power series) R; where Z provides unit-stepped
mileposts in R. That is, a structured core of quantities will
be present in any W , and we may regard mathematics as
the study of the logic of structure and quantity. Extensions
to the hyperreals R' follow by construction of some H that
has as reciprocal h = 1

H closer to 0 than 1
n for any n in N.

Therefore, relationships and linked operations across such
quantities will also be present, or may be constructed as
needed. Illustrating, after Abraham Robinson (Robinson,
1966), hyperreals allow calculus to be treated as extensions
of algebra in R'.

Thus, while bare distinct identity and coherence focused on
quantities will not cause things by the inherent potential or
action of such entities, they instead are logical constraints
on being and are tied to what can or must be or cannot
be or happens not to be. So, too, we may see that the
abstract logic model worlds that we may construct then
lead to key entities that if necessary are framework to any
possible world; thus applicable to our common world. By
contrast, if certain quantities and relationships are merely
part of the contingencies of some W !! that is close enough
to our own, they may provide adequate analogies for mod-
elling.

As a result, we have good reason to expect that mathe-
matical reasoning and core entities will in many cases be
highly relevant to and have powerful predictive power for
our common world.

Robinson, Abraham (1966). Non-Standard Analysis. Ams-
terdam: North-Holland Publishing Company.

Wigner, Eugene (1960). “The Unreasonable E!ectiveness of
Mathematics in the Natural Sciences”. In: Communica-

tions in Pure and Applied Mathematics 13.1, pp. 1–14.
doi: 10.1002/cpa.3160130102.

Independence Conservation and
Evolutionary Algorithms
Eric Holloway
DOI: 10.33014/issn.2640-5652.2.1.holloway.2

Levin’s Law of Independence Conservation

Leonid Levin’s 1984 article (Levin, 1984) is the first to this
author’s knowledge to prove a fully stochastic conservation
of information law. Levin titled his law ’independence con-
servation’ which he considered fairly obvious, describing it
as “Torturing an uninformed witness cannot give informa-
tion about the crime!”

Levin’s law is not well known, which is unfortunate since
the more commonly known conservation laws are focused
either only on the random or deterministic case. Levin’s
law is remarkable because it unifies both the random and
deterministic cases, showing that the combination also can-
not result in information increase.

The second remarkable thing about his law is how easy
it is to prove, given some preliminaries about algorithmic
information.

Algorithmic Information Theory
Background

First is required the notion of algorithmic information,
which is defined on bitstrings. Algorithmic information is
the length of the shortest program that generates a partic-
ular bitstring.

K (x) := min
y |U (y)=x

|y |. (1)

The shortest program is itself known as the elegant program
for that particular bitstring. Each bitstring has a unique
elegant program.



Volume 2, Issue 1

Independence Conservation and Evolutionary Algorithms 33

y' := arg min
y |U (y)=x

|y |. (2)

All programs either terminate after a fixed amount of time,
or never terminate. All the programs in question are known
as prefix free, which means that no terminating program
begins another terminating program.

Algorithmic mutual information is the length of this pro-
gram if we are also provided another bitstring as input,
subtracted from the length if we are not provided the extra
input bitstring.

I (x : y) := K (y) " K (y |x). (3)

Unfortunately, this basic definition of algorithmic mutual
information is only symmetric under a logarithmic error,
because we have to mark where one bitstring starts and
the other ends. This requires a number of bits logarithmic
on the size of the shortest bitstring, which is x in this case.

I (x : y) " I (y : x) = O(log(x)). (4)

We can improve the definition of algorithmic mutual infor-
mation to be completely symmetric under a constant that
is independent of the bitstrings we are looking at, which in
other words means we don’t have to worry about the con-
stant and the algorithmic mutual information is symmetric
as far as we are concerned. This improvement is to use
the elegant program of the input bitstring instead of the
bitstring itself.

I'(x : y) := K (y) " K (y |x'). (5)

Since the elegant program halts once it has generated the
input bistring, we know we can start on the next bitstring,
so we avoid having to encode the bitstring length. This
saves us from having to use the logarithimic term.

I'(x : y) " I'(y : x) = O(1). (6)

One final fascinating point on algorithmic information is
that we can also use it to create a universal distribution.

m(x) := 2"K (x) . (7)

“Universal” means is that we have a distribution that pro-
vides the highest probability for every bitstring possible,
within a multiplicative constant, under the assumption that
we are only dealing with computable generating sources for
the bitstrings.

m(x) & p(x) 'O(1). (8)

The computable distribution assumption is a reasonable as-
sumption for dealing with physical phenomena, since (as far
as we know) everything physical can be modeled to theo-
retically perfect accuracy with enough computational re-
sources.

Proving Levin’s Deterministic Law

Alright, so now onto proving Levin’s law.

We first start with a simple lemma, that providing more
information can only decrease conditional algorithmic in-
formation. In other words, the more we know about y, the
less information we need to describe y.

K (y |x) & K (y |x, z) +O(1). (9)

We now introduce another simple lemma that with a pro-
gram to generate x, namely z which we execute with Turing
machine U to generate x,

x = U (z), (10)

we can generate both x and z. Thus, the joint information
is the same between z and x, z.

K (z) = K (x, z) +O(1). (11)

This also means to generate the triple {y, x, z} we only need
y and z.

K (y, x, z) = K (y, z) +O(1). (12)



34 Letters and Notes

Since it is the case that

K (y |x, z) = K (y, x, z) " K (x, z). (13)

Then performing replacements to Equation 13 with Equa-
tions 11 and 12, we get

K (y |x, z) = K (y, z) " K (z) (14)
= K (y |z). (15)

Combining Equations 9 and 14 shows us that x can never
tell us more about y than z.

K (y |x) & K (y |x, z) (16)
= K (y |z). (17)

We can then use Equation 16 to show running a program
f on i does not increase mutual information with y. The
notation U ( f .i) to mean we’ve run program f with input i
using a universal Turing machine U.

First we decompose the mutual information.

I'( f (i) : y) = I'(U ( f .i) : y) (18)
= K (y) " K (y |U ( f .i)). (19)

(20)

Now, we set z = f .i and x = U (z) = U ( f .i), and then apply
Equation 16 to Equation 18.

I'( f (i) : y) = I'(x : y) (21)
= K (y) " K (y |x) (22)
( K (y) " K (y |x, z) (23)
= K (y) " K (y |z) (24)
= I'(z : y) (25)
= I'( f , i : y). (26)

Giving the final concise result,

I'( f (i) : y) ( I'( f , i : y). (27)

This Equation 27 states that executing function f on in-
put i does not produce any more information about y than
the function and input before they are executed. In other
words, running a program doesn’t produce any information.

Proving Levin’s Random Law

Now with the deterministic version out of the way, we can
move onto the random version.

The random version asks, what if we generate f randomly,
could that result in an information gain? This question is
based on the fact that generating f randomly will result
in an f with a lot of algorithmic information, since it will
be incompressible. So, even though running U ( f .i) doesn’t
give us anything new, the initial selection of f may start us
o! with a good amount of information about y.

Levin’s second step in proving the random law shows this
intuition is false. To prove the second law, we will rely
on the dominance property of the universal distribution in
Equation 8.

"

f

p( f )I'( f , i : y) =
"

f

p( f ) log2
m( f , i |y)
m( f , i)

(28)

(
"

f

p( f ) log2
m( f , i |y)
m( f )m(i)

(29)

( log2

"

f

p( f )m( f , i |y)
m( f )m(i)

(30)

( log2

!
f m( f , i |y)
m(i)

(31)

= log2
m(i |y)
m(i)

(32)

= K (i) " K (i |y) (33)
= I'(i : y) (34)

Which gives us the result that randomly generating an f is
not expected to provide any information about y.

"

f

p( f )I'( f , i : y) ( I'(i : y). (35)

To wrap up the independency conservation law, we apply
Equation 27 to Equation 35.

"

f

p( f )I'( f (i) : y) (
"

f

p( f )I'( f , i : y) ( I'(i : y). (36)



Volume 2, Issue 1

CrowdRank: A Simple Ranking Algorithm for Crowdsourced Rating Systems with Uneven Participation 35

Evolutionary Algorithms

What sort of impact, if any, does the law of independency
conservation in Equation 36 have on evolutionary algo-
rithms?

One of the simplest implications is that if there is a target
area independently designated by y, then it is not possi-
ble to randomly evolve population i towards y without any
fitness information.

However, there are also implications even if there is fitness
information provided for the evolution of i.

Let’s say that f represents one round of an evolutionary
algorithm applied to i, which consists of the following steps:

1. crossover population

2. vary population

3. select population

i is defined as a population of bitstrings selected at random.

We then define y as the set of bitstrings rated at a certain
level of fitness.

Part of f stays constant, and another part is varied ran-
domly, such as mutation and which bitstring sections are
crossed. We denote the random part as r.

This gives us a surprising result.

"

r

p(r)I'( f (i, r) : y) ( I'( f , i : y). (37)

Equation 37 states that each round of randomized evolution
provides no further information about the fitness region y
than already existed in the initial conditions of the original
population i and the evolutionary algorithm f . Thus, this
equation proves that evolutionary algorithms cannot gen-
erate algorithmic mutual information, even in regards to
regions specified according to fitness.

Levin, Leonid A (1984). “Randomness conservation inequal-
ities; information and independence in mathematical
theories”. In: Information and Control 61.1, pp. 15–37.

CrowdRank: A Simple Ranking
Algorithm for Crowdsourced Rating
Systems with Uneven Participation
Jonathan Bartlett
DOI: 10.33014/issn.2640-5652.2.1.bartlett.3

Introduction

Public rating systems are di"cult to score well. Voting
systems tend to simply favor what is already popular. Av-
eraging systems tend to have significant variance if there
are not enough people scoring.

For instance, let’s say that I run a songwriting contest and
have 100 entries. I then put it out to a public vote on the
Internet to see who wins. Most people are not going to lis-
ten to all 100 songs. If I do a simple “thumbs up” approach
and count how many votes a song has, then whichever song-
writer has the best existing following will simply tell their
fans to vote for them, and it will simply devolve into a
popularity contest.

Let’s say instead I do a rating system where you can rate
a song between 0 and 100. Now, songs by popular artists
will actually be negatively weighted because they will have
more visibility for negative ratings. It is not hard for a few
votes to be all 100s, but it is hard for a thousand votes to
be that way. Thus, those who have fewer ratings have an
advantage.

The goal, then, is to come up with a fair way of handling
public ratings which takes into account both the average
score that people assign and the relative certainty that we
have that the score is representative of the “true” score.

The Model

This problem actually becomes rather easy once an appro-
priate mental model is devised. Assuming a normal distri-
bution of actual scores that come in around a “true” value
for a particular score for an entry, what is the range of
possible score values based on the scores that have been
submitted so far?



36 Letters and Notes

Take a concrete example. Let’s say that Song A has 12
votes with an average score of 60. What is the range that
the “real” score should lie in? The main open question when
dealing with statistics is what confidence level we want to
deal with. For this example, let’s say that we want to main-
tain a 95% confidence interval. That means that we want
to know what the range is of two standard deviations from
the mean.

With only 12 samples, this leads to a fairly wide interval,
with the real score being between 32 and 88. However, as
we add more samples, this range narrows in to the aver-
age. If we have 24 samples and maintain the same average,
then our range is restricted to between 40 and 80. At 144
samples, the range narrows to 52–68.

So, with a few scores, the possible “real” score has a very
wide range. However, as more and more scores come in, the
range narrows further and further.

Now, even though these rankings get tighter variances with
more scores, the average value for the scores remain what
they were. So how do we convert this into a more legitimate
ranking system than we had before?

What we can do is simply rank the songs using their lowest
possible scores according to the chosen confidence interval.
That is, we have established statistically what the lower
bound for their score is. Therefore, we can definitively give
them that score because we know they have earned at least
that score. This minimal defensible score will be called the
CrowdRank score.

Let’s say that Song A has 144 rankings that average to 60,
and Song B has 25 rankings that average to 70. Which song
should be ranked higher? As we have already noted, Song
A’s “real” score has a potential range of 52–68. Song B,
because it has fewer score submissions, has a wider potential
range of 50–90. Since the lowest defensible score of Song A
is 52, and the lowest defensible score of Song B is 50, that
means that Song A will be ranked higher than Song B.

The actual ranking will be dependent on the confidence
level that is chosen for the rankings. The higher confidence
levels will take many more rankings for the scores to ap-
proach their averages.

The Calculation

The calculation of each entry’s score is fairly straightfor-
ward. It is basically the inverse of standard statistical
scores.

p The population size

n The number of samples (i.e., number of rankings on a
particular entry)

z The confidence level desired, expressed as a z-value (the
number of standard deviations that a given confidence
level uses—2.58 for 99% confidence, 1.96 for 95% con-
fidence, etc.)

e The margin of error for the confidence interval, expressed
as a decimal (i.e., 0.25 for ±25%)

s The average score of the samples expressed as a real num-
ber between 0 and 1. In the present example we would
divide all scores by 100.

m The expected value. Choosing 0.5 is a “most-safe” value.

Typically, the number of needed samples is determined from
the desired margin of error, using

n =
z2m(1 " m)

e2 . (1)

Rearranging to find the margin of error from the sample
size, we find

e =

#
z2m(1 " m)

n
. (2)

Since our results are distributed as a percentage anyway (a
score of zero to one), the crowdrank is just the score s " e.
Simplified using m = 0.5, the CrowdRank calculation for a
particular entry is

CrowdRank = s "
#

0.25 z2

n
. (3)

If the samples are taken from a restricted population of
size P (say, all the members of a club), you can get an even
better measurement from the following:

CrowdRank = s "
#

0.25 z2

n
P " n
P " 1

(4)

Di!culties

There are two primary di"culties with this system. The
first is that, if there are too few rankings for each entry, the
confidence level will fall o! to zero. This can be mitigated
by varying the desired confidence level based on the average
rankings per entry.



Volume 2, Issue 1

CrowdRank: A Simple Ranking Algorithm for Crowdsourced Rating Systems with Uneven Participation 37

The other di"culty is in communicating the results to end-
users. It is di"cult for them to understand why having 144
people all giving a ranking of 60 might translate to a Crow-
dRank of 52. Having scores whose origin is not transparent
can lead to a lack of confidence in the system. However, be-
cause the discount to the scores is fixed for the number of
entries, you can communicate this as the number of points
that are discounted for a given number of entries. For in-
stance, if you are using the 95% confidence interval, then
you can post that receiving 23–25 entries will result in a 20
percentage point discount.

Conclusion

This note introduced a system of averaging crowdsourced
rankings that appropriately discounts ranking averages
based on the number of submissions. This can be used in
any place where a variable number of crowdsourced rank-
ings might be received. It removes the “popularity contest”
problem of simple voting, as well as the problem of having
too few rankings available in a generic averaging system.





Volume 2, Issue 1

News

Eric Holloway Publishes Paper on
Specified Complexity

David Nemati and Blyth Institute fellow Eric Holloway
recently published a new paper in the journal BIO-
Complexity (Nemati and Holloway, 2019). The paper, titled
“Expected Algorithmic Specified Complexity,” explores the
expected Algorithmic Specified Complexity (ASC) of a ran-
dom variable, concluding that the expected ASC is always
less than zero. This is true both of random variables and of
“processed” random variables—variables which have under-
gone some sort of transformation. This indicates that the
existence of positive ASC always counts as “surprise,” and
therefore always requires explanation.

New Thinking on Human Origins

This past year witnessed a plethora of new thinking on hu-
man origins. First up, new research by Ola Hössjer and
Ann Gauger recently showed a unique model for looking at
biodiversity (Hössjer and Gauger, 2019). Essentially, they
looked at the ways in which minor allele frequency distri-
butions can arise, and how long di!erent distributions take
to arise. New alleles essentially start out as one-o! events,
arising as a mutation in a single individual and later spread-
ing through the population or dying out (or somewhere in-
between). The frequency of these minor alleles can pro-
vide some amount of information about the history of the
species in question. It takes time for an allele to spread
through a population. Therefore, initially, from an initially
homozygous pair, the low-frequency side of the minor allele
frequency spectrum will contain all of the new mutations.
The mutations will take time to spread throughout the pop-
ulation. What Hössjer and Gauger discovered, though, is
that a heterozygous initial pair creates an allele frequency
spectrum that looks much older than it is. This is because
an allele can actually start as either 25%, 50%, 75%, or
100% of the population before any mutations even occur.
Thus, the frequency spectrum will fill very quickly from the
initial pair, and the initial heterozygosity will look equiva-
lent to ancient mutations.

Using standard assumptions, Hössjer and Gauger calcu-
lated that the current allele frequency spectrum could be
attained from an initial starting pair in 100,000–500,000
years. Using other alternative assumptions about the na-
ture of the starting pair could result in attaining the current
allele frequency spectrum in an even shorter period of time.

Another interesting paper was done by Nathaniel Jeanson
and Ashley Holland, which analyzed the human Y chromo-
some (Jeanson and Holland, 2019). Analyzing the human
genome as a whole leads to a number of model-specific is-
sues. If we imagine a starting pair for humanity, were the
original chromosomes identical or did they house diversity?
Additionally, might the first human female have eggs that
had additional diversity in their chromosomes?

However, nearly all considerations of single-couple human
origins have exactly one starting Y chromosome, leading to
fewer model-specific considerations. Jeanson and Holland
aimed to improve the data available about Y chromosome
mutation rates by examining pedigree-based studies which
used high-coverage sequencing. According to Jeanson and
Holland, (a) a Y chromosome molecular clock exists, and
(b) it suggests a paternal history of the human race of about
4,500 years.

Finally, the year ended with the publication of Joshua
Swamidass’s new book titled The Genealogical Adam and
Eve (Swamidass, 2019). This book aims to show that the
practical di!erence between popular and scientific concep-
tions of “Adam and Eve” are not too far o!. His viewpoint
is that, although, according to the consensus view, there
could be no single-couple origin of humans, there could be
a single couple to whom all modern humans could trace
their genealogies.

Communicating Science Through
New Venues

New media becomes old media very quickly in the modern
age. Email was once the best technological way to com-
municate interpersonally. This is now often replaced with
various social media platforms such as Facebook, Twitter,
and others. Likewise, YouTube was once at the center of



40 News

video-based media. While YouTube is still the de facto des-
tination for video content, it is useful to look at some of the
newer players.

First of all, some video is being distributed through apps on
standard platforms. Developing an app for Roku, Amazon
Fire, or Apple TV is a new way to get content to users.
Additionally, streaming services such as Twitch, and now
Mixer, are becoming the dominant social video platforms
for the youth. Science communicators should explore ways
of reaching the public through these outlets. One possibil-
ity would be to play video games with scientists, discussing
what they do while shooting zombies, or discussing the lat-
est theories of the universe while racing Nascar.

Growth opportunities are always with the latest technology.
Asserting a strong position while they are still unproven is
the best way to establish leadership for the future.

Austrian Society Zentrum für
BioKomplexität & NaturTeleologie
Opens with Special Symposium

A new Austria-focused scientific society, Zentrum für
BioKomplexität & NaturTeleologie, recently formed and
held their first symposium. The symposium featured many
European scientists and mathematicians, as well as a few
from the United States as well. The organizers of the con-
ference were Günter Bechly, a prominent German paleon-
tologist who presently works as a senior scientist for the
Biologic Institute, and Siegfried Scherer, a professor of mi-
crobiology and chair of microbial ecology at the Technical
University of Munich.

While the society is based in Austria, the sympo-
sium itself was largely in English. Many of the
talks from the three day symposium are available on
YouTube, at https://www.youtube.com/playlist?list=
PLkaKqUjdyg2JHNqeWQHnVTXZ-37h0tJMb.

Breaking the Weismann Barrier and
Closing the Loop for Lamarckian
Evolution in Multicellular Organisms

A recent review paper in the Royal Society shows that the
Weismann barrier is crumbling. In “The active role of sper-
matozoa in transgenerational inheritance,” Sciamanna et al

review the mounting evidence that there is a Lamarckian
feedback loop in DNA inheritance (Sciamanna et al., 2019).
The paper reviews evidence that mammalian somatic tis-
sues release RNA-containing vesicles, and that these vesi-
cles are then passed to epididymal spermatozoa. Addition-
ally, epididymal spermatozoa are known to be able to inter-
nalize foreign nucleic acids into their nuclei. This completes
the communication channel between somatic cells and germ
cells required for Lamarckian inheritance.

This communication channel was original proposed in the
1990’s. Steele et al’s Lamarck’s Signature suggested that so-
matic mutations might be passed to germ line cells through
an RNA channel (Steele, Lindley, and Blanden, 1999). Af-
ter decades of work across multiple groups, evidence is
mounting that Steele’s hypothesis was largely correct.

Overcoming Entrenched Dogma
About Pseudogenes

Biologists have long known that not all non-coding DNA is
junk DNA. However, pseudogenes have long been consid-
ered the standard bearer for the junk DNA concept. Since
pseudogenes look like defective, non-coding copies of ordi-
nary genes, it has been often assumed that these are evolu-
tionary leftovers—genes which once coded for something, or
a copy of a gene that once coded for something, but which
some accident of mutation incapacitated its activity.

While the evidence for the activity of pseudogenes has long
been known, the idea that pseudogenes indicate junk DNA
has been ingrained in biologists. A group of researchers re-
cently published a paper pointing out that the prejudices
that biologists have about the status of pseudogenes as
junk DNA is impeding the progress of understanding the
way that the genome functions (Cheetham, Faulkner, and
Dinger, 2019).

References

Cheetham, S W, G J Faulkner, and M E Dinger (2019).
“Overcoming challenges and dogmas to understand the
functions of pseudogenes”. In: Nature Reviews Genetics.
doi: 10.1038/s41576-019-0196-1.



Volume 2, Issue 1

REFERENCES 41

Hössjer, O and A Gauger (2019). “A Single-Couple Human
Origin is Possible”. In: 2019 (1), pp. 1–20.

Jeanson, N T and A D Holland (2019). “Evidence for a
Human Y Chromosome Molecular Clock”. In: Answers
Research Journal 12, pp. 393–404.

Nemati, D and E Holloway (2019). “Expected Algorithmic
Specified Complexity”. In: BIO-Complexity 2019 (2),
pp. 1–10.

Sciamanna, I et al. (2019). “The active role of spermatozoa
in transgenerational inheritance”. In: Proceedings of the
Royal Society B: Biological Sciences 286.20191263.

Steele, E J, R A Lindley, and R V Blanden (1999).
Lamarck’s Signature: How Retrogenes are Changing
Darwin’s Natural Selection Paradigm. Basic Books.

Swamidass, S J (2019). The Genealogical Adam and Eve:
The Surprising Science of Universal Ancestry. IVP
Academic.


