Do al-Ghazālī and Craig sufficiently distinguish between "actual infinite" and "potential infinite," or does this distinction dissolve in contemporary physics?

This question brings us to the heart of one of the most complex debates in the kalām argument: the distinction between actual infinite and potential infinite. Al-Ghazālī in the eleventh century and William Lane Craig in our era rely heavily on this distinction to prove that the universe has a beginning. But does this distinction hold up against contemporary mathematics and physics?

Inadequate responses to avoid

From some defenders of the kalām argument:

"The distinction between actual and potential infinite is intuitively clear and needs no defense." This is a misleading simplification. The distinction that seemed clear to Aristotle and al-Ghazālī became a subject of deep debate after Cantor and the development of set theory. Claiming that the distinction is intuitive ignores a century and a half of complex mathematical discussion.

"Contemporary physics has no relation to the metaphysical question about infinity." An artificial separation. Contemporary physics—especially in string theory and quantum physics—uses mathematical concepts of infinity in ways that al-Ghazālī or even Craig in his early work never imagined. Ignoring these developments weakens the kalām argument.

From some objectors:

"Cantor proved the existence of actual infinities, end of discussion." An unjustified leap. Cantor developed a consistent mathematical theory of transfinite numbers, but this doesn't mean they exist physically in the real world. Mathematical consistency doesn't equal physical existence.

"Modern physics routinely uses actual infinity." An exaggeration. Most physicists treat infinities in equations as mathematical tools, not as physical realities. The appearance of infinities in theories (singularities) is usually considered a sign of the theory's limits, not evidence of actual infinities.

Why these responses are inadequate

Responses from both sides fail to deal with the real complexity: the distinction between actual and potential infinite is clear in some contexts and ambiguous in others. The question isn't "Is the distinction correct?" but rather "In which contexts is the distinction meaningful, and do these contexts include the kalām argument?"

The distinction in al-Ghazālī

Al-Ghazālī in "Tahāfut al-Falāsifa" clearly distinguishes: potential infinite is a process that never ends (like counting: 1, 2, 3... you can always add one). Actual infinite is a completed collection of an infinite number of things existing together in reality.

Al-Ghazālī's basic argument: potential infinite is possible (you can count forever), but actual infinite is impossible in reality. Why? Because it leads to contradictions. For example: if the number of past days were actually infinite, then we added today, we would have "infinity + 1"—and this is a contradiction, because infinity cannot increase.

The distinction in Craig

Craig adopts al-Ghazālī's distinction but develops it with contemporary tools. He uses "Hilbert's Hotel" to show the contradictions of actual infinite: a hotel with infinite rooms, all occupied, but you can always add a new guest by moving each guest to the next room. This seems contradictory: the hotel is full and not full at the same time.

Craig adds: contemporary mathematics (Cantorian set theory) is logically consistent, but this doesn't mean that actual infinities can exist in physical reality. Logical consistency is one thing, metaphysical possibility is another.

The challenge from contemporary mathematics

Georg Cantor (1845-1918) revolutionized our understanding of infinity. He proved that there are different "sizes" of infinity: the infinity of natural numbers (aleph-null) is smaller than the infinity of real numbers (aleph-one). His theory is mathematically consistent and accepted today.

But—and this is important—Cantor himself distinguished between mathematical infinity and absolute infinity (God). He saw mathematical infinities as existing "in the mind of God," not necessarily in the physical world. This distinction is sometimes overlooked in contemporary discussions.

The challenge from contemporary physics

In contemporary physics, infinity appears in multiple contexts:

- Singularities: In general relativity, matter can compress to infinite density. But most physicists see this as a sign of the theory's breakdown, not as a description of reality.

- Hilbert space in quantum mechanics: Quantum states are represented in infinite-dimensional space. But this is a mathematical representation, and the question of its relation to physical reality is complex.

- Multiverse models: Some models assume an infinite number of universes. But these are speculative theoretical models, and even their supporters are divided about whether the infinity is actual or merely a mathematical tool.

Does the distinction hold up?

The answer is complex. The distinction between actual and potential infinite holds up in certain contexts:

Where it holds up: In constructive mathematics, the distinction is fundamental. In philosophy of mathematics, many philosophers (finitists and ultrafinitists) reject actual infinity. In applied physics, infinities are treated as approximations or signs of theoretical limits.

Where it weakens: In standard set theory, the distinction is unnecessary—actual infinities are part of the mathematical structure. In some interpretations of quantum mechanics (especially the many-worlds interpretation), there may be actual infinities. In some cosmological models, space may be actually infinite.

Implications for the kalām argument

If the distinction doesn't hold up decisively, this weakens but doesn't invalidate the kalām argument. Craig can retreat to a weaker position: even if actual infinities are mathematically possible, the burden of proof is on those who claim their physical existence. And cosmological evidence (Big Bang, BGV theorem) suggests that the universe has a beginning.

Contemporary positions

Defenders of the distinction: Robert Koons, Alexander Pruss (with reservations), David Oderberg. They see the distinction as metaphysically fundamental, regardless of mathematical developments.

Rejectors of the distinction: Quentin Smith, Graham Oppy, Wes Morriston. They see Cantorian mathematics as showing that actual infinities are consistent, and their metaphysical rejection as arbitrary.

Middle position: Adolf Grünbaum, Bas van Fraassen. The distinction is meaningful in certain contexts, but its application to the question of the universe's beginning is complex and doesn't lead to a decisive conclusion.

Where we stand today

The debate continues with no consensus. The distinction between actual and potential infinite hasn't disappeared, but it no longer has the clarity that al-Ghazālī envisioned. The kalām argument needs more precise formulations that take into account mathematical and physical developments. The reasonable position—consistent with the method of rational preponderance (rajḥān ʿaqlī)—is that the distinction provides partial support for the kalām argument, but doesn't settle it.

For advanced reading

- Advanced level: The distinction in constructive versus classical mathematics
- Advanced level: Infinity in modern cosmological models
- Page "Kalam Cosmological Argument and Infinity"
- Al-Ghazali, The Incoherence of the Philosophers, Discussion 1
- William Lane Craig & James Sinclair, "The Kalam Cosmological Argument" in The Blackwell Companion to Natural Theology (2009)
- Graham Oppy, "Infinity in the Kalam Cosmological Argument" in Philosophical Perspectives on Infinity (2006)