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The paradox of mathematics
Using Kurt Godël's Incompleteness Theorem and Kenneth Arrow's Impossibility Theorem, I will explore the possibility of finding a 'perfect' distribution model for a commodity within a society
Using Mathematics to determine an optimal method of vaccine distribution
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A year after the beginning of the pandemic, a wide rollout of various vaccines took place. This gave way to disputes on how to most effectively distribute the vaccine. This concept of an ideal social welfare is what I will be exploring through the lens of theoretical mathematics.
There are two mathematical theorems that are crucial to this study: Godel's Incompleteness Theorem and Arrow's Impossibility Theorem. The former describes the self-destructive nature of mathematics and how the truth of a mathematical system can never be proved within that same system. The latter discusses the incompatibility between certain conditions of voting in a democratic society. The combination of these two theorems as well as some discussion of the ethical implications of this distribution serve as the tools to solving one of the most pressing problems in our world at the moment.
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Godel's Incompleteness Theorem
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“This statement is false”
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“This statement is false”
Godel’s impossibility theorem revolves around the concept embedded into this phrase: self-referentialism. Because the statement refers to itself, if the statement is true, then by its implied meaning it is false. But if it is false, then its implied meaning is true. We are in a paradox.
Any theorem must abide by certain properties:
Consistency: there exists no formula F in which both F and its negation (not F) can be proven
Efficiency: set of axioms are recursively enumerable
You could write a finite amount of computer code that, if ran, would give an output of the theory’s axioms
Completeness:
A system is complete “with respect to a particular property” if all formulas with that property can be derived in the system
For our purposes, we will be using notation derived from Peano arithmetic where:
0 = zero
S = successor operation
+, for plus and multiplication
^ = conjunction
v = disjunction
∀ = universal quantifier (“for all…”)
∃ = existential quantifier (“there exists…”)
= equal
ㄱ = negation
< = less than
( ) parenthesis
x = variable symbol
* = distinguishing symbol
You can make multiple forms of x → x*, x**, x***....
A theorem is composed of a well-formed sequence of formulae that can be constructed using these symbols. The proof is centered around Godel Numbering: each symbol in the list above is represented by a natural number and the formulas are now represented by a finite list of these numbers that are called “Godel Numbers”, created specifically for the proof of this theorem. For example, the number 3 can be represented as SSS0 = 4. In Godel Numbering, we can write: 27 * 37 *57 * 76 = …. You can write many of the arithmetic laws used today and their proofs using solely Godel Numbers:
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Using the image where proof “B” is a proof of statement “A”, we can construct a couple functions:
Proof(p,s) where p is the Godel number of a proof for a statement with a godel number s
Pr(s) = ∃p Proof(p,s) represents all the Godel numbers of the proofs of provable statements within the system
Pr(s) = P(A) = P(g(A)) P is a predicate, meaning it expresses a relationship between the godel numbers of the proof and the godel numbers of the statements
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We can consider a function ㄱP(Fn(n)) and assume that it must be present in the table below as it’s a function with one free variable, “x”. We can also name this function Fj(n) such that:
∀n Fj(n) ↔ㄱP(Fn(n))
This can be read as: For all n, there is a function Fj(n) if and only if not P(Fn(n)). In blue highlight, we can note the function Fj(n) for a value j and by diagonlizaton, we get Fj(j). We can rewrite Fj(j) = X, and therefore:
X = ㄱP(X)
In logical mathematics, we can write:
∀n Fj(n) ↔ㄱP(Fn(n)) (1)
∃n Fj(n) : Fj(j) (2)
If Fj(j) = X (3)
By combining (1) and (3),
X = ㄱP(X)
In this statement, X is not only being contradicted by the negation of itself but is stating its Godel number before the proof for the Godel number. This self-referential statement states its own contradiction, showing that although this statement is true, it cannot be proven within the system it is created. In a way, even if the perfect vaccine distribution method is able to be represented in Peano arithmetic, it would not be able to be proven.
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Proposition of a Different Model
The most effective method to translate this problem into a mathematical model would be to approach it using concepts in engineering of dynamic systems. By taking into account the amount of healthy, infected, and deceased people as well as the rates of infection and death, I will construct a series of first order linear differential equations that model the effects of vaccination rates over time. Because it is a time-dependent function, it could become an anomaly in Godel’s Theorem.
In an engineering course I took over the summer, I realized I could create and analyze the system of the United States using a core formula:
Input + production = output + accumulation (1)
It’s a general formula that can be used to evaluate any dynamic system. For our purposes, we will be using humans as the currency in this formula as we can find, for example, the input of vaccinated people, output of deceased, and accumulation of infected people. We will have three systems to analyze, the systems of the healthy, infected, and deceased.
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The three systems are outlined where dNHdt , dNIdt , and dNDdt represent the accumulations of healthy people, infected people, and deceased people respectively. The arrows moving from one system to another represent the inputs and outputs where Ri, Rv, and Rd are the rates of infection, vaccination, and death respectively. For our purposes, because we are focusing our system within the topic of an epidemic, we will take the production of humans to be 0.
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Using equation (1), we can create 3 first order linear differential equations:
dNHdt = Rv - Ri
dNIdt = Ri - Rv - Rd
dNDdt = Rd
The values for Ri, Rv, and Rd can be created using proportions where
Ri = α*I*H
Rv = Ɣ*I*D
Rd = ꞵ*I
Where I is the number of infected, H is the number of healthy people, and D is the number of deceased respectively. Alpha, beta, and gamma are all constants that can take an infinite amount of values based on the location, who the state decides to vaccinate first, and voting. The functions found in the Table (put number here) are derived from the resulting change in the constants that make up the transfers of the system.
Using the computer program Matlab, the set of differential equations can be solved and plotted.
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To evaluate the plots, we can propose certain characteristics of an ideal vaccination method:
Asymptomatic at high values of remaining healthy population, indicating low amount of deaths
Asymptomatic at low values of deceased population, indicating low and stable amount of deaths
Thinner or short-lasting spike in infections, indicating less time taken to control infections
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The constants and initial conditions can be modified to fit the conditions of the United States. Gathering data from the New York Times Covid Tracker and the CDC Covid Database after the vaccine rollout that began December 11 2020, I have created an initial series of plots of an idealized societal vaccination method. Although the data is from a reliable source, the selection of data could be slightly unreliable as many assumptions had to be made to create the initial conditions and constants from limited information.
*All plots are shown below in order of United States of America, California, and North Dakota*
The United States of America:
Initial Conditions:
Beta = 0.000006;
Gamma = 0.0005;
Ho = 359.5 (intial number of healthy people)
Io = 50.89 (initial number of infected)
In the United States graphs, the spike of infection occcurs around 800 days, approximately 2.2 years after December 11 2020 and begins to decrease and steady around 3000 days after that date. These plots can serve as control for the subsequent plots.
California:
Initial Conditions:
Beta = 0.0001
Gamma = 0.5
Ho = 39.25
Io = 0.95
In the California graphs, the spike of infection occurs around 300 days (~ 1 year after December 11 2020) and begins to stabilize around 900 days (~3 years after December 2020). This plot exhibits significantly better results than those of the United States, suggesting that maybe the solution to this problem lies in states managing COVID-19 independently.
North Dakota:
Initial Conditions:
Beta = 0.0001
Gamma = 0.005
Ho = 0.756
Io = 0.3
In the North Dakota graphs, the spike of infection lies around 900 days, almost triple the time of California yet are only marginally worse than the plots for the United States. For the trade-off between a significantly better result for California with a slightly worse for North Dakota and the long recovery time for the United States as a whole, the plots suggest that the vaccination and safety protocols should lie in the hands of the states themselves.
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Returning to Godel’s logic, the series of first order linear differential equations can be truncated into a function Fn(H, I, D) meaning that since it can be expressed in Peano Arithmetic, it has a godel number associated with this condensed function. Following the Godel proof, we conclude with a self-referential equation that states its own impossibility and improvability:
Proof(f, s) is a function where f is the godel number of the proof of the function Fn(H, I, D) that has a godel number s .
Even in a transient solution, there will be a function Fn(H, I, D) at a time t that will prove it’s own unprovability through Godel’s theorem.
While Godel’s Incompleteness theorem disproves the possibility of a “perfect” vaccination method, the graphs and transient solutions give an alternative solution. The results per state are overall better than a general solution for the United States. If each state could create their own plan for vaccination rates catered to their priorities, the outcome would benefit those states.
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Arrow's Impossibility Theorem
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Arrow’s Impossibility Theorem is mainly centered around voting systems, which could present itself as a plausible way of solving this problem. The definition of his theorem goes as follows: “Suppose that there are at elast 3 candidates and finitely many voters. Any social welfare function that satisfies universal domain, independence of irrelevant alternatives and unanimity is a dictatorship.” In a democratic society such as the United States, voting seems like a possible route. Arrow’s Theorem applies to ranked voting systems in which there are more than 2 options to be voted on. This process of collective-decision making will concern the distribution of a certain commodity: vaccines. To introduce the system of voting, we can imagine that there are 3 different methods of vaccine distribution (there are many more in reality) that we can name A, B, and C. Voters a, b, and c rank their preferences from 1st to 3rd choice and their preferences can be modeled in the table below:
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Voters a and b prefer option B to C and voters b and c prefer C to A, creating a social ordering that goes as follows B>C>A. Yet voters a and c prefer A to B (A>B) which contradicts the current social ordering. We find ourselves in a voting paradox.
To introduce the concept of a dictator within voting systems, we can first examine the p-voter dilemma. This problem is centered around a pivotal voter who can alter the social preferences in some way, thus becoming a dictator in the context of the voting system. In other words, this voter has the ability to move a preference A from the bottom of the social ranking to the top by switching his personal preference from A as his third preference to his first preference. This can be modeled in the tables below where p represents the pivotal voter who is assumed to be in the middle of the group of voters. In this system it can be assumed that all voters before voter p choose A as their first preference and all voters after voter p choose A as their last preference.
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In this voting system, since more than half of the voters put A as their first preference (including voter p), according to independence of irrelevant alternatives, society puts A at the top of social ordering. In short, choice A wins.
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In this voting system, since voter p changed his vote to C, more than half of the voters put C as their first preference. According to independence of irrelevant alternatives, society puts A at the top of the social ordering. Because this change in result is caused by one voter and that voter p’s personal preference will always be the same as the social preference, that voter is then characterized as a dictator within the system.
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Now that we have introduced the idea of preference, the possibility of paradox in voting, and the concept of a dictator in a voting system, we can now delve into the specific conditions, consequences, and implications of Arrow’s Theorem. Before we do so, it is essential that we lay out some of the foundations - axioms, definitions, and notation - that constitute the framework for the theorem.
Firstly, we can relate alternatives x and y by the notation R, meaning “prefered or indifferent to”. This is best explained by the first axiom:
(1) For all x and y, either xRy or yRx
meaning that x is preferred or indifferent to y OR y is preferred or indifferent to x. It’s easy to read it as “x has a relation to y” or vice versa. Both scenarios may occur, but at least one must occur. Since the voting system we will evaluate has more than two options, this axiom of preference extends to more than two options
(2) For all x, y, and z, xRy and yRz imply xRz
To expand our notation to encompass preference and indifference distinctly, we can state 2 definitions:
xPy is defined to mean not yRx
xIy means xRy and yRx
Definitions 1 and 2 can be read as x is preferred to y and x is indifferent to y respectively. P indicates preference and I indicates indifference.
(3)
Arrow’s Theorem consists of 5 main conditions built upon these axioms and definitions that I have summarized:
The “social welfare function is defined for every admissible pair of individual orderings, R1, R2”, meaning that with a sufficiently large pool of voters, the social welfare function will eventually produce a general social ordering
If an alternative x rises (or does not fall) in an individual ordering without any changes in those ordering, and if x was preferred to another alternative y before any change, then x is still preferred to y. In other words, the relationship between a previously preferred alternative A and a lower preferred alternative B will not change if A rises
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3. Independence of Irrelevant Alternatives. I find that the most effective way to communicate this condition is through an example. Suppose that there are 3 alternatives for vaccine distribution A, B, and C for which a finite group of voters will list their preferences. After the voting takes place, scientists discover that one of the methods, B, is extremely ineffective and decide to abandon it completely. Therefore the choice between the remaining alternatives A and C should be independent of the preferences when B was still an option. In other words, if a group of voters prefers A to C, then removing B as an alternative should not modify the results of the social ordering.
4. The social welfare function may not be imposed. If a social welfare function is “imposed”, I mean that if there is a pair of alternatives x and y, society cannot express a preference of y over x no matter how many individuals prefer y to x. The choice between a set of 3 or more alternatives must not be constrained prior to the social welfare function. Some preferences are taboo or ‘imposed’ on individuals. However, we’d like for our social welfare function to be able to accommodate the true individual preferences of the voters and respect definition 4. Note that if the system is imposed, definition 1 does not hold; the assertion that xRy holds for all sets of individual preferences is the same as stating that yRx does not hold for any individual preferences, thus contradicting definition 1.
5. The social welfare function is non-dictatorial, meaning that one voter cannot dictate the results of the vote (xP1y implies xPy regardless of individual preferences of voters other than voter 1 where P represents the true social ordering
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The final component of Arrow’s Theorem is the series of consequences that arise out of the conditions detailed above. There are 3 main consequences that can be proven using the conditions:
If xP1y and xP2y, then xPy
In other words, if voter 1 and voter 2 prefer x to y (assuming the vote takes place solely between these two voters), then society prefers x to y. We can prove this using condition 4 by suggesting that there are individual preferences R1 and R2 for voters 1 and 2 and that these preferences rank three alternatives x, y, z such that the resulting overall social preference is xPy. This can be modeled in the table below:
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Although the individual preferences may differ x is preferred to y in both cases and thus society prefers x to y. If there is a hypothetical change to the vote resulting in a new ranking (we will call these R’1 and R’2) where x is now higher in rank or goes to first place, xP’y remains and satisfies condition 2. Since condition 3 calls for the independence of irrelevant alternatives, the relation to be proven is only between x and y, thus is independent of the position of z in the individual preferences, thus xPy.
If we suppose that for two alternatives x and y (and voters 1 and 2) that if xP1y and yP2x results in xPy, then whenever xP1y, xPy
Another way to put this consequence is that if voter 1 prefers x to y and voter 2 prefers y to x but the result shows that society prefers x to y, then whenever voter 1 prefers x to y, then society does the same. This case will then hold true if voter 2 is indifferent or changes his/her preference to xP2y. Now suppose there are two orderings, R1 in which xP1y and R2 that can be any ordering. A hypothetical change to these votes is made, resulting in two new orderings, R’1 and R’2, where R’1 = R1 and R’2 is derived from R2 by moving alternative x to the bottom of the ranking. From this hypothetical change, xP’1y and yP’2x. Our hypothesis states that xP’y (society prefers x to y in this hypothetical change) and the only difference between the four orderings - R1, R2, R’1, and R’2 - is that R2 has x at a higher ranking than in R’2. From condition 2, it can be stated from xP’y that xPy, thus proving the statement that for R1 and R2, if xP1y, then xPy.
If xP1y and yP2x, then xIy
If voters 1 and 2 have completely opposing preferences, then society will be indifferent to the alternatives. The most effective way to prove this statement is to prove that it is false and that it will ultimately lead to a contradiction. This would imply that either xPy or yPx to be true, but not xIy. We can choose the assumption that xPy. By Condition 3, we must have:
(i) If xP1y and yP2x, then xPy
Let’s assume that voter 1 has a ranking that follows: x>y>z. Let’s suppose further that voter 2 has a ranking y>z>x. According to (i), xPy, and according to the two rankings, yPz which satisfies condition 1. From the two orderings, we get a social welfare function that goes as follows: xP1z, zP2x, and xPz.
(ii) If xP1z and zP2x, then xPz
Let’s continue by assuming another set of orderings where voter 1 has a ranking of y>x>z and voter 2 has a ranking of z>y>x. The exact same logic as above appears here: Condition 1 tells us yPx and (ii) tells us that xPz so the new social welfare function is yPz.
(iii) If yP1z and zP2y, then yPz
If you continue to use this logic, you can prove that whenever voter 1 prefers one alternative to another, society will end up preferring that alternative to any other. However, voter 1 would become a dictator by definition 5 and be violating condition 5. Therefore, (i) must be false and consequence 3 is proven.
The Fall of The Social Welfare Function:
Voter 1 has an ordering x>y>z and voter 2 has an ordering z>x>y. We notice that yP1z and zP2y, therefore yIz. We can confidently say that through consequence 1, xPy. From the two previous statements, we can construct a social welfare function xPz (society prefers x to z). However, in the two rankings that is not true: xP1z and zP2x. This leads to a contradiction between the individual preferences and the social welfare function. Essentially, if the social welfare function accords with conditions 1-3, either conditions 4 or 5 are violated. In an ideal social welfare function, all conditions would hold.
Similar to the proof of Godel’s Theorem, theories cannot be effective while being complete and consistent (and include elementary arithmetic). The different possibilities offered in the voting system are similar to the columns and rows in the diagonalization table in Godel’s Incompleteness Theorem. If certians conditions hold, others will inevitably be violated. No matter how many possibilities are presented to the public of the United States, certain conditions will hold while others will inevitably be violated.
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The first attempts at applying a strictly theoretical mathematical approach led to an initial dead-end as it was proven that any system that can be written using arithmetic is inherently unprovable. But in transitioning to changing systems over time, while the theorem still holds, results showed that while there is not a single perfect solution, if each state dealt with vaccination procedures their own way, the general outcome would be better. In that sense, there is no single solution, but many. By transitioning to one of the core characteristics of the United States, democracy, there was still hope in finding a perfect method of vaccination distribution through voting. Yet, Arrow’s impossibility theorem shows that not only are voting systems (having three or more options to vote on) in the U.S. unjust, but so are any voting systems in general. Although these theorems have disproved any formal distribution method, it shows that extremely biased opinions about a perfect vaccine distribution method are false. There can be educated guesses and estimations, but there is no sure distribution method that will eradicate the epidemic. However, these are the types of results that are achieved in the world of abstract and complex math. In this realm, nothing is 100% definitive and it becomes the process of logical thinking that allows one to evaluate what works and what does not. The results of this research are not without value and represent only the first step to understanding and analyzing problems similar to this one.
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Works Cited
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says:, Norman Fulton, et al. “Optimal Strategy for a COVID-19 Vaccine Roll-Out.” Science in the News, 8 Jan. 2021, sitn.hms.harvard.edu/flash/2021/optimal-strategy-for-a-covid-19-vaccine-roll-out/.
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