Infinite Dilemma 3: No More Universes
The End of the Infinite Multiverse
Infinity: Where Heaven and This World Intersect
In the history of human thought, the concept of infinity has played a prominent role in philosophy, religion, and mathematics. Ancient civilizations used the idea of infinite deities to explain the mysteries of the world around them. Philosophers and mathematicians then began to grapple with the nature of infinity, developing concepts such as Zeno’s paradox.
One of the most famous examples of Zeno’s paradox concerns the race between Achilles and a tortoise. In this paradox, Zeno argued that Achilles could never catch up to the tortoise because the tortoise would have moved a little further by the time he reached the spot where the tortoise had been. And by the time Achilles reached that new spot, the tortoise would have moved a little further again. This process could continue infinitely, meaning that Achilles could never actually catch the tortoise.
This paradox seems to contradict our intuition about motion and change and has puzzled thinkers for centuries. However, mathematicians have developed ways to resolve the paradox by using the concepts of infinite series and limits. In modern calculus, for example, we can describe the motion of Achilles and the tortoise using a mathematical function, and we can prove that Achilles will eventually overtake the tortoise.
It’s important to note that infinity does not necessarily exist in nature. Rather, it is a useful concept that we have developed to explain the natural phenomena we observe. This idea intersects with the subtle philosophical debate between Platonic idealism and Aristotelian this-worldliness, in which Platonic idealism suggests that mathematical concepts exist independently of the physical world, while Aristotelian this-worldliness argues that they are derived from the observation of the physical world.
The Troubles with Infinity
Infinity, although not present in the natural world, has been historically associated with the supernatural creator or designer in religious contexts. However, as mathematics developed to explain the natural world, infinity became a concept to describe reality. Ironically, the chance atheism movement, which seeks to reject supernaturalism by supplanting causality with chance, encounters a paradoxical situation where they are required to treat infinity as a concrete, physical entity instead of a mere abstract concept.
The first two installments of the “Infinite Dilemma” series explored the challenges that arise from the paradoxical nature of infinity. In “Not Enough Time” (CP Lu, 2021a), I demonstrated that the notion of the random creation of complex structures, such as a watch or life, is flawed. This idea is often based on a misinterpretation of the implications of infinite time and resources, which, in reality, cannot be achieved due to the finite nature of the universe. In “Lost in Chance” (CP Lu, 2021b), I challenged the belief that natural phenomena are entirely random and lack causality. This view arises from a misunderstanding of probability and random processes, as shown.
Since the finiteness and causality of our universe are insufficient for the chance occurrence of our creative biosphere, the idea of an infinite number of universes seems necessary. Our universe in the infinite multiverse is then seen as one that hits the jackpot.
Make Sense of Our Lucky Star
The universe we inhabit seems to have incredibly precise values of physical constants and distribution of matter that are just right for life to exist (Barrow & Tipler, 1986). This apparent fine-tuning has led some to argue that there must be a creator or designer behind it (Collins, 2003). However, naturalists and modern atheism reject this explanation.
Instead, the idea of an infinite multiverse has gained popularity (Aguirre & Tegmark, 2010). This theory suggests that there are an infinite number of universes, each with different physical constants and laws of nature. According to this theory, our universe is fine-tuned for life simply because we exist to observe it.
While the infinite multiverse theory offers a naturalistic explanation for the fine-tuning of our universe, it is not without its challenges. One significant problem is the lack of direct evidence for other universes (Lineweaver, 2018). The theory also raises questions about the nature of reality and the existence of objective truth (Sorensen, 2016). If there are infinite universes with infinite variations of physical laws, what does that mean for the concept of reality itself?
Ironically, although the infinite multiverse theory aims to explain the probability of our universe, it encounters significant difficulties when dealing with probabilities, especially concerning the infinitesimally small. As Guth (2000) pointed out, this theory fails to address the issue of the probability of a particular universe with the precise physical constants and matter distribution required for life, which is effectively zero. This is due to the intricate interplay between infinity and continuity.
Infinity and Continuity: Two Sides of the Same Coin
The scale of microscopic structures in our universe is mind-bogglingly small. For instance, a single DNA molecule has a diameter of only about 2 nanometers. However, when stretched out, a DNA molecule’s length can increase by up to 10 million-fold. To put this into perspective, if a pre-stretched DNA molecule were the size of a human hair, a post-stretched DNA molecule would be about 1.5 times the distance from the Earth to the Sun. This incredible increase in length is just one example of the mind-bending scales at which the universe operates, from the microscopic to the macroscopic and everything in between.
As we zoom in even further, we encounter a concept of infinity that defies our intuition. Consider comparing the infinitesimal points densely packed on the tip of a pin, leaving no gap, and the numbers you can count indefinitely. Which is bigger? Surprisingly, the former is infinitely larger. This type of infinity is known as uncountable infinity, which is often associated with “continuity,” while the latter is called countable infinity.
The difference between these infinities lies in their “cardinality” or the size of their sets. Countable infinite sets are those that can be put in a one-to-one correspondence with the set of natural numbers, meaning that we can count them. For instance, the set of even numbers or the set of fractions can be put in a one-to-one correspondence with the set of natural numbers, such as:
2 <-> 1
4 <-> 2
6 <-> 4
…
Even the set of rational numbers, which seems much denser, has a one-to-one correspondence with the set of natural numbers using a zigzag pattern shown in the following diagram:
In contrast, uncountable infinite sets are those that cannot be put in a one-to-one correspondence with the set of natural numbers. The set of real numbers, which includes all the points on a number line, is an example of an uncountable infinite set.
Georg Cantor was the German mathematician who introduced the concept of sets and revolutionized the understanding of infinity. One of his most notable contributions to mathematics is his proof that there are more real numbers than natural numbers. This proof is known as Cantor’s diagonalization argument, and it demonstrates that even though both sets are infinite, the set of real numbers is a larger infinity than the set of natural numbers. To understand why this is the case, consider listing out all the real numbers between 0 and 1. We run into problems because they cannot be listed in a sequence like natural numbers. To prove this by contradiction, he assumed there was a list of all the real numbers between 0 and 1 and then constructed a new number that was not on the list. This new number was constructed by looking at the first digit of the first number on the list, the second digit of the second number on the list, the third digit of the third number on the list, and so on, and changing each digit to a different value. For example, we can write this list as follows:
- 0.123456789…
- 0.987654321…
- 0.246801357…
- 0.135790864…
- …
Now let’s construct a new number by changing each digit on the diagonal to a different value: 0.23250… This new number is not on the list since it differs from the first number in the first digit, the second number in the second digit, and so on.
The concept of uncountable infinity can be challenging to grasp, but it has significant implications in mathematics and physics. For instance, the uncountable infinity of real numbers allows for the existence of a continuum or a smooth spectrum of values, which is essential in calculus and analysis. In physics, the ideas of continuity and uncountable infinity have been used to explain the behavior of particles in a continuous field.
In mathematics, a densely packed set or a continuum is a collection of points that are so closely spaced together that there are infinitely many points in any finite interval. For example, the set of real numbers between 0 and 1 is a continuum.
One interesting feature of such a densely packed set, or a continuum, is that the probability of hitting any particular real number within the set is zero. This is because the set is infinitely dense, meaning there are infinitely many numbers between any two adjacent numbers in the set, so the probability of hitting a single point is infinitesimally small.
However, the probability of hitting a range of values within the set is non-zero. For example, the probability of picking a real number between 0.5 and 0.6 from the set between 0 and 1 is non-zero because there are infinitely many numbers within that range.
From Infinite Dilemma to Infinite Fantasy
The concept of the multiverse, which proposes the existence of an uncountably infinite number of parallel universes, presents a conundrum regarding the probability of the existence of a universe according to specific cosmological parameters. Traditional probability theory assigns a probability of effectively zero to any single universe existing according to a specific set of cosmological parameters due to the infinite number of universes in the multiverse (Carroll & Chen, 2004). This is analogous to the aforementioned scenario of the probability of hitting a specific real number in the range of 0 to I being zero.
To address this issue, some scientists propose using a Bayesian approach to probability (Hajek, 2012). In Bayesianism, probability is defined as a measure of the degree of belief rather than a strict mathematical ratio (Jaynes, 2003). Therefore, instead of considering a single point in the space of cosmological parameters, Bayesian probability allows us to consider a range of possible values and our degree of belief in the existence of a universe with those parameters (Hajek, 2012). This approach takes into account all available evidence and updates our beliefs as new evidence arises (Jaynes, 2003).
While the Bayesian approach offers a solution to the zero-probability problem in the infinite multiverse, it also introduces new challenges. By considering a range of cosmological parameters rather than a specific point, we can assign non-zero probabilities to the existence of a universe with those parameters. However, this approach raises the question of how to determine the range of parameters to consider.
Moreover, Bayesian probability introduces a degree of subjectivity into the calculation of probabilities. The degree of belief that an individual assigns to the existence of a universe with certain cosmological parameters is based on their prior beliefs, which can be influenced by personal biases or limited knowledge. This means that different individuals can arrive at different probabilities for the same universe, depending on their subjective beliefs.
Additionally, the utilization of Bayesian probability to solve the zero-probability issue in the infinite multiverse theory may seem to contradict the initial aim of providing a naturalistic explanation for the unlikelihood of our universe, and it raises concerns about subjectivity and belief. This approach assumes that a subject can have a mind with a priori beliefs across uncountably infinite numbers of universes. We are moving from infinitesimally dense universes to infinitesimally dense states of mind, venturing into a realm of infinite fantasy.
Worth the Trouble?
The proposition that “anything and everything is possible in the infinite multiverse” may seem too convenient to be a serious explanation for the existence of our universe. This approach lacks the ability to distinguish between different possibilities, making it a catch-all explanation that can be used to justify any argument without offering any testable predictions or insights. As such, it is not a productive path toward deeper understanding.
Furthermore, the infinite multiverse hypothesis undermines the concept of scientific progress. If anything is possible in an infinite multiverse, it is difficult to see how scientific discovery can lead to deeper understanding or insights. The infinite multiverse hypothesis implies that there is no ultimate truth or natural laws governing the universe, which is antithetical to the scientific enterprise.
Given the challenges associated with the infinite multiverse hypothesis, it may not be worthwhile to rely solely on this theory for a naturalistic explanation of the universe. Fixing the theoretical shortfalls would require significant effort and conceptual work without necessarily leading to a more productive avenue for scientific inquiry or a more meaningful path toward deeper understanding.
Conclusion: No Need for More Universes
With the development of Artificial General Intelligence (AGI) on the horizon, the rise of advanced language models such as ChatGPT and GPT-4 is posing a challenge to the idea of chance atheism. These models are capable of detecting patterns and regularities in data, revealing profound insights that were previously beyond our understanding. This suggests that events in the natural world do not occur without a cause and that the concept of chance is inadequate in explaining the complexity of the universe.
The implications of causality have significant consequences for our understanding of the universe. As we continue to push the boundaries of AGI, it is important to remember that our ability to reason and understand causality is a crucial tool in this endeavor grounded in a realistic and practical approach. Unlike the infinite multiverse hypothesis, which proposes that anything and everything is possible, the belief in causality allows us to trust that the universe follows a set of laws and principles that we can gradually uncover and comprehend.
It is reassuring to know that, despite our finite time and resources, we can eventually unravel the causes underlying the phenomena we encounter during our exploration of the natural world and ourselves. Reason will be our guide in this pursuit, sparing us from relying on mere chance or serendipity. The universe in which we live will continue to provide us with a vast wealth of knowledge and understanding to discover. We do not need infinite universes that distract us from unraveling its mysteries.
- CP Lu (2021a), Infinite Dilemma 1: Not Enough Time.
- CP Lu (2021b), Infinite Dilemma 2: Lost in Chance.
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