The Dual Null Hypothesis and Pascal’s Wager
by Joseph Mark Haykov
May 28, 2024
Abstract
Inspired by Bertrand Russell, who acknowledged the possibility of being wrong, and Mark Twain, who warned against the dangers of certainty in false knowledge, I began contemplating Pascal’s Wager by asking: How does one convince oneself of the existence of God? My background is in applied mathematics and probability theory, which I have utilized for the past 30 years in statistical arbitrage trading on Wall Street, never experiencing a losing year. This success stems from the principle that, in mathematical arbitrage, we only bet on certainties; anything else would inevitably lead to losses. Statistical arbitrage is akin to playing blackjack with superior knowledge of probable outcomes compared to your trading counterpart, ensuring consistent profitability.
Although this is a mathematical treatise, it may lead you to agree with Bertrand Russell that Karl Marx, who described religion as the opium of the people, might have been wrong, and that Jesus Christ was a mathematical genius who intuitively understood probability in a way that took me 30 years of hard work to master, surpassing even Euclid and Aristotle and far exceeding the oversimplifications of Euler and Gauss. Blaise Pascal's prudent advice of believing in God to be on the safe side forms the basis of this work, to which it is dedicated with eternal gratitude for his ingenious wager. Rather than dismissing Pascal's Wager, we approach it directly using hypothesis testing and uncover an intriguing result: a dual null hypothesis. This paper fully explains this concept and demonstrates how to model it formally.
Introduction
This essay is inspired by Blaise Pascal, to whom it is dedicated, as we embrace the challenge posed by Pascal's famous wager regarding the existence of God. In this article, the concept of "God" serves not only as an axiom—our null hypothesis—but also as a rational bet, as Pascal ingeniously proposed. Avoiding religious debate, this article focuses instead on applying rigorous hypothesis testing to Pascal's Wager. Our analysis introduces a novel concept: the double null hypothesis, which enables us to present surprising evidence in support of our properly chosen null hypothesis. The only way to describe our findings is through Arthur Conan Doyle's words:
"When you have eliminated the impossible, whatever remains, however improbable, must be the truth."
— Arthur Conan Doyle, The Sign of the Four
What Exactly is Pascal's Wager
Blaise Pascal (1623-1662) was a French mathematician, philosopher, scientist, and inventor known for his significant contributions to probability theory, Pascal's Triangle, and various practical achievements. Pascal is credited with inventing an early digital calculator, a syringe, a hydraulic press, and the roulette wheel. However, this discussion will focus on his famous philosophical argument known as Pascal's Wager.
Pascal's Wager
Pascal's Wager addresses the rationality of belief in God, framing the decision to believe or not believe in God as a bet. The wager can be summarized as follows:
If God exists and you believe in God, you gain infinite happiness (heaven).
If God exists and you do not believe in God, you suffer infinite loss (hell).
If God does not exist and you believe in God, you lose very little (a finite amount of time, resources, etc.).
If God does not exist and you do not believe in God, you gain very little (a finite amount of time, resources, etc.).
Given these outcomes, it is rational to believe in God because the potential gain (infinite happiness) outweighs the potential loss (finite resources). Even if the probability of God's existence is low, the infinite value of the reward justifies belief (Pascal, 1670).
Clarifying the Concept of Belief: Statistical Hypothesis Testing vs. Religious Faith
In discussing concepts such as God and religion within this paper, it is crucial to preempt any misunderstandings by stating unequivocally that our focus is purely mathematical, concerning probability and hypothesis testing under uncertainty. The practices discussed have been successfully applied by the author on Wall Street for over three decades, distinctly separate from conventional religious beliefs.
In mathematical terms, 'belief' operates dualistically. It involves choosing between competing hypotheses based on which one appears more likely to be true given the evidence. This is fundamentally different from the religious use of 'belief,' which often involves conviction without the need for comparative alternatives.
In this analysis, we engage with Pascal’s Wager not as a theological endorsement but as a statistical hypothesis. We start with the assumption that God exists as our initial hypothesis (H0) and consider alternative hypotheses accordingly. For instance, in a statistical framework like regression analysis, H0 might assert no correlation between cigarette smoking and cancer mortality. If data reveals a high correlation, we then shift our belief to H1, the alternative hypothesis, suggesting a significant relationship exists.
This pragmatic application of belief—to choose the more supported hypothesis—contrasts sharply with religious faith, where belief is typically singular and not subject to alternative propositions. It is important to emphasize that our discussion is confined to the realms of applied mathematics and statistical reasoning. Readers seeking spiritual guidance on faith matters are advised to consult resources beyond the scope of this paper.
Dually Defined Null Hypothesis
The foundational brilliance of Pascal's Wager lies in its requirement to formulate hypotheses, which enables serious engagement with the proposition. Pascal posits an axiom akin to the null hypothesis (H0) in statistical hypothesis testing, suggesting the existence of God, heaven, and hell as real entities. Typically, in applied mathematics, the approach is to invalidate H0 by substantiating an alternative hypothesis (H1), which, in this context, would assert the non-existence of God. However, this binary framework is insufficient for the nuanced considerations of Pascal's Wager. Pascal’s wager implicitly introduces a variable, 𝑁, the number of gods, which can be any natural number, including infinity as per Peano’s fifth axiom. This expands the potential outcomes from the point of probability, illustrating both how the hypothesis could be false and how it could be true.
Hypotheses Framework:
H0: The only existing god is Yahweh, as Pascal, a staunch Christian, specifically referenced.
H1: Multiple gods exist, but Yahweh is preeminent and should be worshiped above all, analogous to investing heavily in a single robust stock.
H2: Multiple gods exist, and Yahweh is not the one to be prioritized, akin to diversifying investments across various entities.
H3: No gods exist at all; humanity is entirely alone.
Given our foundational understanding of probability and the infinite potential number of gods, we must move beyond the standard binary model of hypothesis testing. This complexity echoes Nassim Taleb’s concept that the absence of evidence (e.g., of a black swan) is not evidence of absence.
Selecting a null hypothesis can be challenging: should we adopt H0 or H1 as our starting point? This decision is crucial because, unlike in traditional statistical applications where probabilities are assigned to known outcomes, the realm of deities involves profound uncertainties that defy simple probabilistic assignments. In applied mathematics, if our hypotheses, referred to as axioms, are true, then any theorems proven true via deductive logic are guaranteed to hold universally. This is because the two sets of claims—axioms and theorems—are proven to be logically equivalent.
For example, the Pythagorean Theorem’s truth can be empirically verified, demonstrating the strength of deductive reasoning. However, if the axioms turn out to be false, then the theorems are also false. This is exemplified by the fact that the shortest distance between two points in our reality is not a straight line; if it were, your GPS would not work. GPS systems use Riemannian geometry to triangulate your iPhone’s position on Earth, accounting for the time dilation effects by having satellite clocks operate at a different rate than those on Earth—as per Einstein, who used Riemannian geometry to model space-time as it exists.
The first thing we learn on Wall Street is to never assume or trust anyone, especially existing axioms. Axioms are merely educated guesses, posited because they are ‘self-evidently true’ to whoever is doing the initial guessing, and therefore represent a centuries-old perspective-based judgment. Starting with zero knowledge, nothing is ‘self-evident.’ The surest way to lose money is to invest in an asset you do not understand. We avoid guessing by consulting the original sources that Pascal referenced, rather than dogma. The Torah, a primary source for Pascal, dictates in Exodus 20:3, "You shall have no other gods before me." This commandment supports selecting H1 as our null hypothesis, asserting Yahweh's primacy and aligning with the hierarchical divine structure seen in texts like the Bhagavad Gita.
By adopting H1 as our null hypothesis, we acknowledge the potential plurality of deities but prioritize Yahweh, respecting both the mathematical structure and a broader understanding of religious belief systems.
Addressing Common Objections
What is truly amazing is that as we spend a bit of time going through the common objections as to why Pascal’s Wager was never taken seriously, we can see that all of them are inherently rooted in implicitly selecting H0, not H1, as the foundational axiom. The root cause of the problem is that all these arguments against taking the wager seriously are based on assuming there are only two hypotheses, rather than the four possible hypotheses that actually exist. This misunderstanding is why the wager was never fully considered. This fully aligns with Daniel Kahneman’s definition of theory-induced blindness, which is induced by incorrect implicit underlying assumptions. Such blindness is, in fact, assumption-dependent and axiom-induced, as we can see. But that’s a whole other story.
The Sincerity Objection
Believing in God simply to avoid hell may seem insincere, potentially resulting in that very outcome. However, under the properly selected H1 hypothesis, even attempting to believe in the right God, specifically Yahweh, results in a relative risk reduction of going to hell. Thus, this objection does not hold in a rational argument about God.
The Infinite Utility Problem
This objection highlights issues with using infinite rewards (heaven) and punishments (hell) in rational decision-making, arguing that infinite values make all finite outcomes seem irrelevant.
Response: This argument is based on a misunderstanding of what a null hypothesis, in this case H1, means in probability theory. Pascal's argument relies on the belief in the infinite nature of the rewards and punishments as an axiom that needs to be accepted. Questioning the infinite nature of these outcomes undermines the fundamental premise of the wager itself. Therefore, this objection misunderstands the framework of Pascal's argument, which requires accepting the infinite stakes as a starting point (Pascal, 1670).
The Moral Objection
Believing in God purely out of self-interest is morally questionable, suggesting that such belief reduces faith to a selfish gamble.
Response: Even if initial belief is insincere, it is still better than non-belief in terms of potential consequences. Pascal's Wager posits that pragmatic belief can lead to genuine faith and moral growth over time (Pascal, 1670). This again relates to the relative risk reduction under our H1 null hypothesis.
The Probability Objection
This objection questions the assumption that even a small probability of God's existence justifies belief due to the infinite reward, arguing that assigning probabilities to metaphysical claims is problematic.
Response: This objection reflects a misunderstanding of probability. Just because the probability is unknowable does not mean it is zero. With zero knowledge about the true probability of God's existence, the initial estimate should be 50%, aligning with the principle of indifference. Thus, the probability of God's existence is not inherently low (see Roger Penrose on this), and the potential infinite reward still justifies belief (Pascal, 1670).
The Cost Objection
This objection highlights that Pascal's Wager underestimates the potential costs of belief, such as sacrifices in time, resources, and personal freedoms.
Response: One does not need to spend excessive resources to hold a belief in God. Simple and moderate religious practices can be integrated into one's life without significant sacrifices, minimizing potential costs while still gaining the potential infinite reward (Pascal, 1670).
The Agnosticism Objection
This objection points out that Pascal's Wager presents belief as a binary choice without addressing the rational stance of agnosticism.
Response: This objection is rooted in a fundamental misunderstanding of how probability is defined and works in applied mathematics, and the difference between reality and hypothesis. The wager is based on the fact that the objective reality is binary: God, Yahweh, either exists or does not—either God is real or God is not real, and the two are mutually exclusive logical claims; they cannot both be true. Agnosticism does not change this binary, real-world fact. Pascal's Wager encourages a proactive decision in the face of this objective reality, arguing that the potential infinite reward of belief outweighs the finite costs (Pascal, 1670).
The Many Gods Objection
This objection contends that due to the multitude of belief systems, devotion to the "wrong" god could still lead to damnation.
Response: While numerous belief systems feature a pantheon of gods, Pascal specifically advocated for the belief in Yahweh, the deity referred to in the Ten Commandments: "You shall have no other gods before me" (Exodus 20:3, NIV). Yahweh is also recognized as "The Father" in the New Testament and "Allah" in the Quran, underscoring a monotheistic continuity across these traditions. Pascal's Wager essentially argues for a commitment to this particular divine entity.
Reflecting on cognitive biases in decision-making, Mark Twain’s observation is pertinent: “It’s not what you don’t know that gets you into trouble. It’s what you know for sure that just ain’t so.” This highlights the pitfalls of dogmatic adherence without empirical scrutiny, a caution echoed by Bertrand Russell who in 19591 emphasized the importance of examining facts over accepting dogma. This principle is especially relevant in financial contexts like Wall Street, where reliance on unverified beliefs can lead to financial losses.
Furthermore, the classification of the worship of non-Yahweh entities as idolatry aligns with Moses’s directives in the Torah and supports the H1 hypothesis as a valid framework. Idol worship, by definition, involves the veneration of false gods. However, the Torah also mentions various God-like entities such as angels, cherubim, seraphim, nephilim, and giants. While some of these beings are obedient to God, others are not. According to our H1 null hypothesis, these entities are categorized as "false gods." They should not be worshipped but are acknowledged as self-aware, conscious entities distinct from humans.
No-Arbitrage Constraint on 𝐸: The Transpose of Its Own Hadamard Inverse
In this paper, we start with an assumption—our null hypothesis, H1—that many gods exist and that God is omnipotent and benevolent, aligning not only with Pascal’s wager, but also with the traditional narratives about Yahweh found in historical texts like the Torah, which are themselves heavily influenced by works such as the Emerald Tablets of Thoth and encapsulated by the Hermetic principle "as above, so below." This principle offers a unique lens through which to examine the exchange rates between all goods and services within the economy.
Our discussion begins with an analysis of the Forex market. In the real-world FX market, approximately n=30 of the most actively traded currencies are exchanged, and their exchange rates can be mathematically represented as an exchange rate matrix, denoted as E. In this matrix, the value in row i and column j represents the exchange rate from currency i to currency j. This matrix serves as a model for understanding how exchange rates between not only currencies but also the exchange rates between all goods and services that can be either bought or sold in an arm's length commercial transaction in an economy are structured to prevent arbitrage.
Arbitrage, by its very nature, is not possible if a uniform price is maintained for an asset in different markets. Specifically, in the foreign exchange market, if the exchange rate of currency A to B is set, then it must be the reciprocal of the exchange rate of B to A. For example, if $1 buys £0.50, then £1 should buy $2. This reciprocal relationship is crucial to eliminating arbitrage opportunities arising from exchange rate discrepancies. Let the matrix E represent a set of exchange rates, such as those observed between the 30 or so currencies in the FX market, where n, the number of rows and columns, would be 30. The no-arbitrage condition dictates that E must be equal to its own transpose once the Hadamard inverse is applied. Mathematically, the exchange rate matrix is constrained as follows: E=Ei,j=1÷Ej,i=ET.
This 𝐸=𝐸𝑇 “no-arb” constraint is somewhat similar to the property of a matrix being involutory, where an involutory matrix is its own inverse, 𝐸=𝐸−1. However, while 𝐸⋅𝐸−1=𝐼 (the identity matrix), 𝐸⋅𝐸𝑇=𝑛⋅𝐸. As we can see, the resulting matrix is not the identity matrix but rather a scalar multiple of 𝐸 scaled by its row count, 𝑛. The reason for this is that the constrained matrix 𝐸=𝐸𝑇 has a single eigenvalue, which is also its trace, and is invariably equal to 𝑛, due to the fact that the exchange rate of a currency with itself is, by definition, always 1. For any no-arbitrage 𝐸=𝐸𝑇 matrix, we have:
𝐸⋅𝐸𝑇=𝐸𝑇⋅𝐸=𝐸⋅𝐸=𝐸𝑇⋅𝐸𝑇=𝐸2=𝑛⋅𝐸
This shows that the matrix 𝐸 has one Hadamard squared root. This aligns with the principle that multiplication is associative. However, just as division is not associative, the product of 𝐸 with its transpose (𝐸⋅𝐸𝑇) is not the same as the product of its transpose with itself (𝐸𝑇⋅𝐸), resulting in two, not one, non-Hadamard, regular transpose roots. However, regardless of whether 𝐸 is pre-multiplied or post-multiplied by its real transpose, the result, when squared, is always (𝐸𝑇)4=𝐸4. Conceptually, the reciprocal operation of raising 𝐸 to the fourth power represents taking the fourth root of 𝐸, which generally results in four such roots of the 𝐸4 (or 𝑛2⋅𝐸) matrix: 𝐸 pre- and post-multiplied by its transpose, and the reciprocal Hadamard transpose. Since the two Hadamard roots are the same, there are three distinct roots. Both of the non-Hadamard roots, when raised to the fourth power, result in the same 𝑛2⋅𝐸=E4 matrix, as illustrated below:
(E⋅ET)⋅(ET⋅E)=E⋅(ET⋅ET)⋅E=E4=n2⋅E
and of course, similarly
(ET⋅E)⋅(E⋅ET)=ET⋅(E⋅E)⋅ET=E4=n2⋅E
By imposing the E=ET condition, the matrix E simplifies, having only a single eigenvalue, n, and reducing to a vector-like structure. This simplification occurs because each row or column of E can define the entire matrix, dramatically reducing the dimensionality of the information required to quote exchange rates. For example, the entire matrix E is equal to the outer product of its first column and its first row, which also happens to be the reciprocal of the first column, producing the full matrix. Consequently, each row or column of E is proportional to the others, meaning that all rows or columns are scalar multiples of one another. This characteristic renders E a rank-1 matrix, indicating that all of its information can be captured by a single vector.
We believe it is worthwhile to highlight what might be an obvious connection. In the context of Einstein's equation E=mc2, if E is a bounded E=ET matrix, we see that E4=(ET)4=n2·ET=n2·E=m·c2. As we can see, mass is simply the fourth root of energy, but whereas E has 4 roots in theory, in reality, only 3 roots exist on account of the E=ET constraint imposed on E. Thus, m=E−4.
While this remains purely conjectural, it intriguingly aligns not only with the principle of supersymmetry in theoretical physics but also, surprisingly, with the ancient Hermetic axiom 'as above, so below.' This concept also echoes the geometry of the Egyptian pyramids and is reminiscent of the notion that '42' is the 'answer' to the ultimate question of life, the universe, and everything, as humorously proposed in The Hitchhiker's Guide to the Galaxy. Although this reference is not directly related to quantum physics, it touches on probability.
At this point, we must caution that our expertise in theoretical physics is limited to interactions with physicist colleagues during our tenure managing the stat-arb book at RBC on Wall Street. Therefore, please treat our comments about physics with considerable skepticism, particularly the ideas about quantum set theory outlined below, which are purely speculative. This may assist a physicist who is not on Wall Street, unlike those making real money at hedge funds such as Renaissance, founded by the late Jim Simons.
Condensation of Information
In a matrix that simplifies to a vector-like structure, the entirety of the matrix can be described by any of its rows or columns. Here’s what happens in such a scenario:
Reduced Dimensionality: Instead of needing to know all elements of a matrix (which in a full matrix would be 𝑛×𝑚 values), you only need to know the elements of a single vector (either 𝑛 or 𝑚 values, depending on whether it's a row or a column vector). This drastically reduces the dimensionality of the information required.
Data Compression: This vector represents a form of data compression, where instead of storing or processing multiple independent pieces of information, one vector informs the entire structure. This simplification could improve the efficiency of computations and analyses involving 𝐸.
Basis for a New Set Theory
Extending this idea to a theoretical framework, especially in contexts like quantum mechanics, can lead to intriguing possibilities:
Modeling Quantum States: In quantum mechanics, states can be superposed and entangled. A matrix that simplifies to a vector-like structure might analogously suggest a system where states are not independently variable but are intrinsically linked—a form of quantum entanglement at a mathematical level.
Set Theory and Quantum States: A new set theory that models such matrices could consider sets where elements are fundamentally interconnected. Traditional set theory deals with distinct, separate elements, but this new theory could focus on sets where elements are vector-like projections of one another.
Applications: Such a theory could be useful in fields like quantum computing or quantum information, where understanding entangled states in a compressed, simplified form could lead to more efficient algorithms and a better understanding of quantum systems.
By utilizing a matrix that reduces to a vector-like structure as a basic element, we could potentially model a system where traditional notions of independence between elements are replaced by a more interconnected, entangled state representation. This could open new avenues in both theoretical and applied physics, especially in handling complex systems where interdependencies are crucial.
We note in passing, as illustrated here in this video from MIT online lectures2, the axiom of pairing from ZF set theory is used to derive Bell’s Inequality. At approximately the 1 hour and 15 minute mark, the lecturer uses the axiom of pairing, for example, to split up the set 𝑁(𝑈,¬𝐵) into 𝑁(𝑈,¬𝐵,¬𝑀) and 𝑁(𝑈,¬𝐵,𝑀) In this particular case, when set elements are pairs of entangled particles, the axiom of pairing does not work, simply because such a set cannot be split up into two separate subsets. However, if we replace set elements with vectors that are all entangled on account of being constrained by 𝐸=𝐸𝑇, we may—with hard work that no one in their right mind would do for free—develop a better set theory that will more accurately model quantum entanglement, akin to the way Riemannian geometry was derived from a set of axioms that more accurately reflect the reality of how space-time operates.
If you want to find out more, please visit us at tnt.money, as we may have a way to properly motivate people to do such research using one-true money!
Computer Science Applications of 𝐸=𝐸𝑇 Vector/Matrix
As we mentioned at the outset, we are not theoretical physicists, and that is a fact. However, we are well-versed in computer science, with extensive experience in programming and algorithm development. While this section is equally speculative as the theoretical physics section, we feel we are making far more educated guesses here, having actively programmed in C++ and R for the last 30 years. In particular, the 𝐸=𝐸𝑇 matrix we are describing bears resemblance to an automated Prolog-like theorem prover.
In computer science, there are two primary models of computation: the Turing machine and the self-defining recursive lambda function, as seen in languages like Scheme. Mathematically, both models are equivalent in that they are capable of computing anything that is theoretically computable.
Turing Machine Model:
The Turing machine is a theoretical construct that defines an abstract machine capable of manipulating symbols on a strip of tape according to a set of rules. It serves as a fundamental model of computation, underpinning the concept of algorithms and decidability.
In the context of 𝐸=𝐸𝑇, the matrix can be thought of as encoding a set of rules or operations that ensure consistency and non-contradiction, similar to how a Turing machine operates with a finite set of states and transitions.
Lambda Calculus Model:
Lambda calculus, particularly in its self-defining recursive form, is a mathematical framework for defining functions and their evaluations. It forms the basis of functional programming languages such as Scheme.
The 𝐸=𝐸𝑇 matrix can be seen as a self-referential structure where each element is defined in relation to others, ensuring that the overall system maintains logical consistency. This is akin to how recursive functions call themselves with modified parameters to achieve a desired result.
Both models, at their core level, utilize 0-1 bits, which are manipulated using basic logic operations such as AND, OR, and NOT—the same logic operations used in mathematical proofs. In the context of the 𝐸=𝐸𝑇 model, these bits become qubits, whose values are constrained by the 𝐸=𝐸𝑇 equality, akin to real-world quantum entanglement—guaranteeing consistency. This equivalence is why anything that can be stated or proven in mathematics can be equivalently computed. All of this is fundamentally rooted in the natural numbers, as defined by Peano’s fifth axiom.
If we represent individual bits as qubits, or elements of an 𝐸=𝐸𝑇 matrix, then no resulting logical claims are self-contradictory. This effectively imposes a non-contradiction constraint on mathematical logical deduction, thereby “solving” Gödel’s incompleteness theorem by finding all superimposed 𝐸 states where no logical claims contradict each other. Additionally, we note that all claims about natural numbers can be examined by looking only at the outer product of all prime numbers. This may have something to do with the Riemann hypothesis.
The fact that there are three, and not four, roots ensures non-contradiction in reality and drives evolution by bending probabilities in such a way as to make everything Pareto-efficient. This principle can be applied in various domains, including:
Statistical Arbitrage Trading:
In our work developing statistical arbitrage trading software, the principles of non-contradiction and logical consistency are crucial. The 𝐸=𝐸𝑇 matrix can model exchange rates and ensure that arbitrage opportunities are correctly identified and exploited.
The reciprocal relationships within 𝐸 can be used to validate trading strategies, ensuring that no arbitrage opportunities arise from inconsistencies in exchange rates.
Automated Theorem Proving:
The Prolog-like structure of 𝐸=𝐸𝑇 can be utilized to automate logical reasoning tasks. By defining a set of initial conditions and rules, the system can infer new knowledge and validate hypotheses.
This has applications in artificial intelligence, where automated theorem proving can enhance decision-making processes, optimize resource allocation, and solve complex problems.
Final Thought
In conclusion, let us finish with the following thought. David Hilbert sought a complete and consistent set of axioms for all of mathematics. Here, we propose such a framework. Given a set of initial logical claims, referred to as axioms, everything that logically follows by deduction from these axioms—regardless of whether the deductive logic is recursively applied a finite or infinite number of times (as exemplified by induction)—any claims that can thus be logically deduced, under the constraint of non-contradiction, are absolutely guaranteed to hold true in reality. This is due to the real-world nature of the 𝐸=𝐸𝑇 quantum entanglement constraint, as long as the axioms are true.
Thus, there is your complete and consistent set of axioms for all of mathematics, at least as they pertain to modeling real-world behavior in this universe. Oh yes, as for Albert Einstein’s question – God does indeed play dice with the universe, but in such a way that God is guaranteed to win and so is whoever God plays with. God plays with Pareto-efficiently loaded dice – loaded to drive us towards prosperity and love, the love of life. For if there is no randomness and all is perfect, we might as well all die, as we would all get bored to death. Even Gods need to have fun once in a while, eh?
Purpose of this Paper: SEC Approval
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