The Benefits of Belief in God in Mathematics
by Joseph Mark Haykov – version 1.00
On the day of our Lord and Savior, Jesus Christ, May 20, 2024
(Just to be on the safe side, guys, you never know, he could be right – not my idea – that’s according to Bertrand Russell.)
Abstract
In this paper, we take Blaise Pascal up on his bet and seriously attempt to believe in God. Unlike the intellectuals who have previously tackled this challenge, our background is different. We are formally trained in mathematical economics, finance, and computer science, and have traded statistical arbitrage on Wall Street for 30 years, applying mathematics—specifically probability theory—in reality. We do not fantasize about math; we actually use applied mathematics and probability theory to make money.
The only way to describe our findings is:
"When you have eliminated the impossible, whatever remains, however improbable, must be the truth."
— Arthur Conan Doyle, The Sign of the Four
Preface
In this paper, we would like to report an accidental finding related to our research into improving cryptocurrency payment processing—an accidental finding that relates to theoretical physics and perhaps mathematics, though we are not sure. Below, we present an extract from our paper about the use value of money, available at “tnt.money”. However, we add an introduction about God to put these findings into perspective. If you wish to skip the God part and go directly to the theoretical physics and money part of this paper , please proceed to the section entitled “Our Approach”.
Introduction
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).
Dually Defined Null Hypothesis
An interesting aspect of this wager, if we are to take it seriously using probability theory, is the construction of null and alternative hypotheses. Pascal posits as an axiom (an educated guess), known as H0 in hypothesis testing, that God, as well as hell and heaven, are all real. In applied mathematics, we would try to disprove H0 by proving the opposite hypothesis—generally referred to as the alternative hypothesis, or H1, positing as an axiom that there is no God. However, this approach is insufficient.
In the context of the wager, the number of gods is a natural number: 0, 1, 2, 3, and so on, as defined by Peano’s fifth axiom. Due to this, there is an inherent duality in the hypothesis H0 because there are two ways in which it could turn out to be true. Therefore, we must consider multiple hypotheses.
H0: There are no gods at all, except for Yahweh, the God specifically referenced in the wager. Pascal was a devout believer in Christ, meaning the God he referred to specifically was Yahweh, also known as "the Father" (or The Godfather, to be more accurate—akin to a good version of Marlon Brando in the movie of the same name).
H1: There are multiple gods, but Yahweh is the only one we should worship above all others. This is akin to the advice to diversify to spread risk in finance, by buying the S&P 500 instead of trying to pick stocks (as per John Bogle and many others), but exactly in reverse—the opposite of diversification.
H2: There are multiple gods, but Yahweh is not the one we should worship above all others, aligning with the advise to diversify.
H3: There are no gods at all, period, and we are all alone ;(
Mathematically, let N denote the number of gods. The prior estimate of N, given our lack of knowledge, could be infinitely many—not just 0 or 1. This is an important consideration when discussing probability, aligning with Nassim Taleb's observation that just because we have never seen a black swan, it does not mean one does not exist.
Which one do we pick as our null hypothesis, H0 or H1? Having two different hypotheses is a problem because, in applied mathematics, we don't just throw darts at the board—we only ever bet on sure things, having never lost money trading stat-arb on Wall Street. Absolute certainty in the objective reality we all live in and share is strictly limited to things that can be independently verified, which means we can only ever be absolutely certain about empirical facts and deductive reasoning. Logical deduction guarantees with absolute certainty that as long as our axioms are true, the theorems will also be true. The accuracy of deductive logic in mathematics—showing that if A (axioms) is true, then B (theorems) must logically follow—can be independently verified (e.g., proving the Pythagorean Theorem in middle school).
However, axioms are nothing but educated guesses, accepted as true without proof and based on being 'self-evidently' true to those who first propose them—in this case, ourselves. This prompts us to consider which of the alternative hypotheses, H0 or H1, should be utilized. We can avoid guessing by heeding the wise advice of Bertrand Russell to future generations1 and consulting the original sources that Pascal referenced, rather than dogma. According to the Torah, Yahweh, the deity Pascal discusses, commands specifically, and we quote from the original: "You shall have no other gods before me" (Exodus 20:3, NIV). This directive indicates that we should utilize H1 as our null hypothesis, which posits Yahweh as the primary deity, deserving of exclusive worship ahead of others. This acknowledgment of Yahweh as the foremost deity aligns with the concept of multiple gods found in the Bhagavad Gita within Indian culture, where a hierarchy of divine beings can coexist.
Addressing Common Objections
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, our Godfather, by definition 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 again 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 yet again 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 argues that given the multitude of belief systems, believing in the "wrong" God might still result in damnation.
Response: While there are many belief systems with many gods, Pascal advocated for belief in Yahweh, the God referred to in the Ten Commandments: "You shall have no other gods before me" (Exodus 20:3, NIV). Yahweh, also known as "The Father" in the New Testament and "Allah" in the Koran, is the one God we should believe in according to Pascal's Wager. Here, we cannot help but quote Mark Twain, who said, “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.” On Wall Street, we prefer to reference the original source material rather than listening to the opinions of mathematically illiterate intellectuals, as exemplified by Karl Marx, which have given us such diseases as socialism. Also, for God’s sake, check your source material carefully people, please!
Our Approach
In this paper, we posit, as an axiomatic assumption, that many gods exist, and also posit as an axiomatic assumption that God is all-powerful and all-loving, in line with the original teachings about Yahweh, which can be traced back to original source material, such as the Emerald Tablets of Thoth and the Hermetic principle "as above, so below." This principle is particularly interesting for understanding the exchange rates between all goods and services in the economy. We begin this discussion by closely examining such a matrix, as exemplified by the matrix of exchange rates in the Forex market, where roughly 30 of the most actively traded currencies are exchanged. In this case, the full exchange rate matrix E is a 30-by-30 symmetrical matrix.
Given the mechanics of arbitrage, it becomes evident that arbitrage is untenable if a uniform price exists for any given asset across different markets. In the foreign exchange (FX) market, this principle implies that for any two currencies, A and B, the exchange rate from A to B should be the reciprocal of the exchange rate from B to A. In other words, if the exchange rate of the US dollar to the British pound is $2 to £1, then the exchange rate of the pound to the dollar must be 50 pence. This relationship ensures that no arbitrage opportunities arise purely from differences in these exchange rates.
No-arbitrage constraint on E: The transpose of its own Hadamard inverse
In matrix form, if E is the exchange rate matrix, then the no-arbitrage condition imposes a constraint on E. The Hadamard inverse of E, denoted as , is defined by simply replacing each individual element of the matrix with its reciprocal. We use the notation ET rather than to refer to the transpose of the Hadamard inverse of a matrix: . Using this notation, the no-arbitrage condition can be restated as E=ET, which states that E becomes the reciprocal of its own transpose, ensuring for all j and i.
This 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:
This aligns with the principle that multiplication is associative. However, just as division is not associative, so too:
However, as the two formulas below show, the two different products of E with its real transpose, as opposed to the product of E with its reciprocal transpose, when cross-multiplied, do form the same result, aligning with the principle of having dual square roots of a number. These are the formulas below:
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 we represent the terms E and m as a vector that is also an E=ET matrix, we see that E4=m4c2, where the energy and mass vectors are inverse cross-products of each other.
By taking the square root of both sides, we see not only that mass is the reciprocal of energy but also that both mass and energy have four square roots—computed by not only post-multiplying but also pre-multiplying the E matrix by not only its actual transpose, but also the transpose of the Hadamard inverse. Yet (E⋅ET)2 has just one root, namely E, because E=ET.
While this is all purely conjectural, it aligns not only with the supersymmetry principle in theoretical physics, but also—shockingly—with the ancient Hermetic principle of "as above, so below." Additionally, the idea somehow mirrors the Egyptian pyramids geometrically, though we can’t prove it. At this point, we should emphasize: because we are not theoretical physicists, you should take everything we say about physics with a big grain of salt. We only know a wee bit of theoretical physics, having interacted with physicist colleagues during our time on Wall Street. For this reason, and also because the concept of money does not yet have an equivalent in physics, we should return to the main topic of this paper. However, given that we would like to give back a little to the community, before we do, let us speculate about quantum set theory, just in case it helps a physicist still not on Wall Street.
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 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!
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Pascal, Blaise. Pensées. Translated by W.F. Trotter. Project Gutenberg, 1670.
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The Holy Bible, New International Version. Grand Rapids: Zondervan, 1984.
Twain, Mark. Following the Equator: A Journey Around the World. Hartford, CT: The American Publishing Company, 1897.