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Volume 2, Issue 1

On the Logic of Being and Wigner’s Astonishment Regarding the Applicability of Mathematics 31

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
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 the 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.



32 Letters and Notes

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 "
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.


