Mysteries: Understanding Infinities of Space and Time

Hint: You’re Never Going to Understand It

Note: This piece has been prepared using ChatGPT 4.0 for research and ghostwriting.

Author's Preface:

Young people, curious ones, ponder the infinite. It makes no sense to them. They think about it. Eventually, they decide it's too mysterious to worry about and has no practical relevance. They grow away from it, or they become mathematicians. Now, that's my story anyway and I’m stickin’ to it.

So, space—does it go on without limit? Is it infinitely divisible? Time—does it extend towards eternity without end? Is there a shortest time? And space and time together—speed. Can we have the infinitely fast, the infinitely slow?

So, mathematicians have their view. Physicists, particle physicists, cosmologists have their view. The common man probably doesn't think much about it since it doesn't affect his day-to-day living as far as he knows, as far as I know. Mathematicians play with it and come up with notions of multiple types of infinity, which make sense mathematically, but make no sense in terms of our understanding of the concepts beyond the mathematical abstractions. So, I'm returning to this topic. Never a hot topic, I suppose, among the populace at large, but if you're young enough or young at heart, you may still be interested, or you could be a mathematician, physicist, astronomer or cosmologist.

Introduction

Infinity captivates and confounds, sparking questions that stretch beyond everyday understanding. Many ponder it briefly—whether space or time goes on forever—before leaving it behind as an unsolvable mystery. Yet some pursue these questions further, reaching into science and mathematics to explore infinity’s many forms.

Scientists like cosmologists and physicists investigate infinity through the cosmos’s vastness and the minuscule scales of matter, forming theories that often seem at odds with common sense. Mathematicians, by contrast, study infinity as an abstract framework, discovering multiple infinities and creating intricate structures that reveal patterns in the endless. For both, infinity is a doorway to deeper insights, one that challenges our understanding of knowledge’s limits.

Discussion

Infinity Through the Eyes of the Curious:

A Common Person’s Encounter with Boundaries, Divisibility, and the Limits of Time and Speed

Curious young minds often find themselves drawn to the concept of infinity, not something they have to deal with in throughout everyday life, finding it impossible to fully grasp. They may look at space, for instance, and imagine its extension. Based on common experience, most people conclude that everything has a boundary; every room, every plot of land, even the sky seems confined by horizons. However, this notion of boundaries doesn’t map neatly onto space itself. If space has a boundary, what lies beyond it? And if it has no boundary, is it infinite in extent? Both options seem nonsensical. So, many eventually stop contemplating the idea, deeming it one of those intractable mysteries, or they turn their curiosity to another domain—like the smallest things.

The very small, the infinitesimal, presents its own puzzles. Take a simple object, like an orange. One can slice it in half, quarters, eighths, and so on. How far can this subdivision go? Some might assume that there must be an eventual limit, perhaps at the level of atoms, as science once posited. But atoms, they learn, are themselves composed of smaller particles, which in turn may contain further substructures. So, if there’s no smallest part, why can’t this process of division continue indefinitely? This pursuit of an endpoint, of a smallest possible unit, ultimately yields no clear answer, leaving the curious with another unsatisfying mystery.

Time evokes similar questions. The Abrahamic religions speaks of a beginning, and cosmologists propose a beginning in the form of the Big Bang. But what existed before the beginning? Everyday experience teaches that all events are preceded by something else. The notion of an uncaused beginning, or an origin without a precursor, feels like a paradox. Equally confounding is the concept of an end to time. Physicists predict a final state—a heat death of the universe—where all motion and energy dissipate. Yet even this “end” leads to another question: what happens after that? The notion of a time that simply stops seems as absurd as the idea of a beginning.

Speed, too, raises similar conundrums. In theory, one might assume there’s always a faster speed to reach by adding another increment. But modern physics, following Einstein’s insights, tells us that nothing can surpass the speed of light. For many, this limitation is difficult to reconcile with the idea of an endless scale. On the other end of the spectrum, what does it mean to be infinitely slow? We observe slow processes, like the growth of plants or the erosion of rocks, yet these can still be measured in human terms. There are even technologies, like femtosecond photography, that capture events at near-light speed, further stretching our concept of time. The existence of these extremes—of incredibly rapid processes and others so slow they seem to defy perception—leaves one grasping at vague concepts without ever reaching true understanding.

These questions, which arise naturally in a curious mind, seem to dissolve into mystery with each attempt to answer them. To the common person, infinity remains an enigma—a set of abstract ideas that don’t align easily with everyday experience, or with what they can readily observe.

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Infinity Examined:

The Scientist’s Perspective—Cosmologists, Theoretical Physicists, Particle Physicists, and Other Reprobates

Infinity, as explored by scientists, is approached through an array of theories and hypotheses, many of which remain hotly debated, speculative, and always counterintuitive. Within cosmology, particle physics, and theoretical physics, scholars have proposed models to explain the vastness of the universe and the smallest scales of matter, but these models often depend on unverified assumptions and involve significant theoretical disputes (Smolin, 2001; Krauss, 2012).

Cosmologists, for example, propose the Big Bang as the beginning of the universe, suggesting that all matter originated from an infinitely dense point before expanding outward. While this theory is widely accepted, it is not without detractors; some cosmologists argue that the Big Bang model fails to account for certain astronomical observations, and others question the assumption of an origin from "nothing" (Hawking & Penrose, 1996). Even among supporters of the Big Bang theory, there is contention over the universe’s fate—whether it will expand indefinitely or collapse into a "Big Crunch." These debates, based on abstract physics and complicated data, often seem disconnected from any observable reality, casting doubt on their practical relevance and reliability (Krauss, 2012).

To investigate the cosmos, astronomers rely on enormous telescopes, both ground-based and space-borne, including projects like the Hubble Space Telescope, each costing billions of dollars (Sagan, 1980). While these instruments have undeniably expanded our view of the universe, the interpretations drawn from them rely heavily on theoretical frameworks that remain unproven. For instance, the cosmological models used to analyze light from distant stars assume certain fundamental principles about the nature of space and time—principles that are still matters of conjecture rather than fact (Greene, 2004).

In particle physics, scientists examine infinity on the subatomic scale through experiments like those at the Large Hadron Collider (LHC), where particles are accelerated to near-light speeds and collided to observe the results. These collisions provide data used to develop hypotheses about the smallest units of matter and energy, including the so-called Planck scale, which suggests a theoretical limit to divisibility (Smolin, 2001). However, these results depend on indirect observation and require substantial interpretative leaps, leading some critics to question the robustness of the findings (Lederman & Hill, 2011). The Planck limit and other supposed "fundamental scales" of matter may, in the end, prove to be artifacts of the models themselves rather than reflections of actual physical boundaries.

Theoretical physics goes even further, delving into realms where empirical testing is practically or even theoretically impossible. String theory, for instance, proposes that the universe’s fundamental building blocks are not particles but vibrating strings in multiple dimensions (Greene, 2004). Similarly, multiverse theories hypothesize countless parallel universes, each with its own unique set of physical laws. These theories, however, lack empirical validation and remain speculative at best. Since they cannot be directly tested or falsified, some argue they lie closer to philosophy or metaphysics than to science (Susskind, 2006). The enthusiasm for these ideas among some theoretical physicists has sparked criticism for diverting attention from more grounded, testable inquiries (Smolin, 2001).

For the layperson, these scientific theories about infinity can appear as intricate but implausible constructs that lack clear connections to reality. The sheer complexity and speculative nature of these models raise questions about their validity and relevance. Ultimately, the models proposed by cosmologists, particle physicists, and theoretical physicists often reflect the limitations and assumptions of human thought more than any definitive understanding of infinity. In the end, these theories may reveal more about our intellectual ambitions and biases than about the true nature of the universe (Hawking & Penrose, 1996).

The Strange World of the Mathematician:

A different way of thinking

Not everyone can become a mathematician. It requires a unique blend of curiosity and a certain sort of intelligence to engage with the abstract realms of mathematics, where infinity plays a central role. To most people, infinity is a straightforward concept of endlessness—a notion of boundless space, time, or quantity. For mathematicians, however, infinity is a far more intricate and varied idea. In fact, where most see infinity as a singular idea, mathematics has revealed multiple types of infinities, each with unique properties and implications (Rucker, 1982; Stillwell, 2010).

Historically, infinity began as a somewhat mystical and singular notion, with philosophers and early mathematicians pondering its implications in both the finite and metaphysical realms (Aczel, 2000). But over time, mathematics evolved, building upon foundational ideas of counting and measurement, and branching into more elaborate concepts that only a select few could fully understand. The discipline requires significant dedication and, often, specialized intelligence to master its complexities, making it “not for everyone” (Dunham, 1991). Ironically, while mathematicians may excel in these abstractions, balancing a checkbook or managing practical arithmetic can sometimes prove just as challenging—or even harder—than pure theoretical math (Devlin, 2000).

A turning point in understanding infinity came with the work of Georg Cantor, who revealed that there is not just one form of infinity but many (Stillwell, 2010). Cantor’s discoveries, particularly in set theory, showed that infinities could be ordered by size and cardinality, meaning that some infinities are larger than others. For instance, the set of real numbers between 0 and 1 is “larger” than the set of all positive integers, as it encompasses infinitely many values that the integers cannot capture (Rucker, 1982). This distinction introduced the concept of multiple infinities, fundamentally reshaping the way mathematicians and philosophers think about the boundless.

To understand infinity, one might start with a quantity, any simple number, and consider that adding one to it infinitely never reaches an endpoint. Extending this, one could double it or add any other quantity indefinitely, approaching an unending series of values. Moving beyond integers, fractional values between two numbers constitute yet another type of infinity, one that continues indefinitely and includes irrational numbers like pi (π), whose decimal representation never repeats or terminates (Aczel, 2000). These fractional values, or the “real numbers,” illustrate the density of infinity even within a finite segment (Stillwell, 2010).

Mathematicians also explore infinity through various functions and operations. Doubling, tripling, or applying exponents yields ever-larger numbers, while inverse functions decrease values, approaching but never quite reaching zero. This process reveals the nuances of infinity: there are infinities that expand endlessly, and infinities that contract, but never fully vanish. In this bizarre mathematical world, each manipulation of infinity can be explored, tested, and classified, resulting in an intellectual pursuit that blends discovery with intricate play (Rucker, 1982).

Ultimately, infinity in mathematics is not an unreachable or abstract concept. Rather, it represents a multifaceted landscape where mathematicians work to uncover patterns, relationships, and structures. Infinity becomes more than a mystery—it is a system of ideas that offers both a challenge and a fascination, revealing new aspects of the boundless through rigorous study and continuous exploration.

Summary

Infinity represents both the mystery and potential of human understanding. Scientists examine infinity through the universe's size and matter’s smallest components, often reaching conclusions that defy intuition. Mathematicians, meanwhile, explore it in abstract terms, defining multiple types of infinity and manipulating them within theoretical structures.

While infinity challenges comprehension, it also fuels discovery. Whether as a scientific question or mathematical concept, infinity reveals the boundaries of what we know—and inspires a continued quest to explore what lies beyond.

References

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