Can an infinite number of things actually exist, or is infinity merely a mathematical concept?

Infinity is one of the most perplexing concepts in human thought. We deal with it daily in mathematics—numbers never end, we can always add one—but the deeper question is: can an infinite number of things actually exist in reality? For instance, could the universe be eternal (an infinite number of moments in the past)? Or is infinity merely a useful tool in the mind that cannot be realized in reality? The question is not merely mathematical, but has profound philosophical and theological dimensions.

Inadequate Responses to Avoid

From some believers, hasty responses:

"Infinity is impossible except for God." This confuses different concepts. When we say God is "infinite," we usually mean He is unlimited in His attributes (power, knowledge, mercy). This is different from the question: can an infinite number of created things exist? Conflating these concepts wastes the precise philosophical discussion.

"The Qur'an says everything has a beginning, so infinity is impossible." A hasty interpretation. Religious texts speak about the creation of the heavens and earth, but they do not explicitly address the philosophical question of actual infinity. Using religious text to settle a technical philosophical matter without careful analysis weakens the argument.

From some agnostics, oversimplified responses:

"Science proves the universe is infinite." An imprecise claim. Modern science does not settle the question of the universe's infinity. Yes, the universe might be "unbounded" in the sense that it has no edge, but this doesn't mean it's infinite in size. Moreover, Big Bang theory points to a beginning of time, complicating the question of temporal infinity.

"Infinity exists in mathematics, therefore it exists in reality." A logical leap. The existence of a concept in mathematics doesn't guarantee its existence in physical reality. Imaginary numbers (√-1) are very useful in mathematics and physics, but this doesn't mean there are "imaginary things" in reality.

Why These Responses Are Inadequate

The common problem: failure to distinguish between types of infinity. There's a fundamental difference between "potential infinity"—the ability to continue without end—and "actual infinity"—the existence of an infinite number of things together in reality. Most quick responses confuse them or ignore the philosophical complexities of the matter.

Serious Positions in the Debate

First, the classical Aristotelian position. Aristotle clearly distinguished between potential and actual infinity. He accepted the former and rejected the latter. A line can always be divided into smaller parts (potential infinity), but there cannot exist a line composed of an infinite number of points existing together (actual infinity). This position was adopted by many Muslim philosophers like al-Ghazālī, who used it in their arguments to prove that the universe has a beginning.

Second, the position of modern mathematics. Georg Cantor in the nineteenth century revolutionized the field by proving that actual infinity is mathematically possible, and that there are different "sizes" of infinity. The set of natural numbers is infinite, and the set of real numbers is a larger infinity. This mathematical development reopened the philosophical debate: if actual infinity is mathematically possible, why shouldn't it be possible in reality?

Third, the position of contemporary Islamic kalām. William Lane Craig and others defend the Aristotelian position with new arguments. The most famous is "Hilbert's Hotel"—a hotel with an infinite number of rooms, all occupied, yet it can always accommodate new guests! These paradoxes show, according to this position, that actual infinity leads to logical contradictions and therefore cannot exist in reality.

Fourth, the contemporary naturalistic position. Other philosophers like Graham Oppy and Quentin Smith argue that actual infinity is possible in reality. The apparent paradoxes (like Hilbert's Hotel) are not genuine contradictions but counterintuitive consequences of infinity's nature. Moreover, some contemporary cosmological theories include the possibility of infinite universes or eternal time.

Where We Stand in This Debate Today

The debate over actual infinity remains open in contemporary philosophy. Developments in mathematics (set theory) and physics (cosmology) add new dimensions to the ancient debate. The prevailing academic position today is to acknowledge that the matter is complex and that arguments from both sides carry weight. This aligns with the approach of "rational preference" (rajḥān ʿaqlī)—recognizing that some deep philosophical questions may not be settled with absolute certainty.

For Advanced Reading

If you wish to delve deeper:
- Intermediate level: The difference between potential and actual infinity in Aristotle
- Advanced level: Cantorian set theory and paradoxes of infinity
- "Infinity and God" family page
- The kalām cosmological argument and temporal infinity