This study examines the arguments developed by Ibn Sīnā (Avicenna, d. 1037) against the possibility of actual infinity in nature and the evaluations of these arguments by Fakhr alDīn al-Rāzī (d. 1210). The three main arguments under scrutiny are the collimation argument (burhān al-muwāzāt), the ladder argument (burhān al-sullamī), and the mapping argument (burhān al-taṭbīq). Before delving into the specifics of these arguments, the study first defines how infinity is conceptualized by Ibn Sīnā and identifies the types of infinity addressed by these arguments.

Ibn Sīnā’s works, including al-Najāt, Uyūn al-ḥikma, al-Shifā’: al-Tabīiyāt (Physics), and al-Ishārāt wa’l-Tanbīhāt, as well as his treatise Fī ḥaqīqat mā lā nihāyat lahu (On the Reality of the Infinite), reveal his in-depth treatment of the topic. According to Ibn Sīnā, infinity is defined as something that, regardless of what portion is taken from it, always leaves more behind. This definition applies primarily to quantities and inherently implies inexhaustibility. Ibn Sīnā’s nuanced view distinguishes between different types of infinities, such as quantitative, spatial, and temporal infinities, each of which requires separate consideration within his arguments. 

Ibn Sīnā’s arguments against actual infinity specifically target entities that are sequentially ordered by nature or position. He does not reject the existence of potential infinity, such as the divisibility of bodies, time, or motion. Moreover, he does not deny the possibility of infinite time or motion, as their parts do not coexist simultaneously but instead unfold successively. His rejection of actual infinity focuses on objects that are simultaneously present and sequentially ordered by their nature or position. Ibn Sīnā’s careful delineation between potential and actual infinity is critical, as it allows for infinite divisibility in theory while maintaining the impossibility of infinite magnitudes in practice. 

The collimation argument asserts the impossibility of infinite dimensions by illustrating the contradictions that arise when imagining a finite ray intersecting an infinite line. Ibn Sīnā argues that such an intersection would require the identification of a specific point on the infinite line, which is inherently contradictory. Fakhr al-Dīn al-Rāzī critiques this argument, suggesting that its premises fail to conclusively demonstrate the impossibility of infinity. He raises questions about the logical coherence of the assumptions underlying the argument and points to alternative interpretations of infinite dimensions.

Ibn Sīnā uses the ladder argument to show that dimensions cannot extend infinitely in space. He constructs a scenario where an infinite series of dimensions leads to contradictions regarding the measurement and addition of finite increments. Al-Rāzī identifies weaknesses in the premises of the argument, particularly questioning the necessity of equal increments in an infinite series. He concludes that the ladder argument is invalid unless its premises are supplemented with additional proofs. Al-Rāzī further notes that the argument’s reliance on hypothetical constructs undermines its applicability to real-world scenarios.

The mapping argument is the most comprehensive, addressing not only the impossibility of infinite physical bodies but also infinite numerical sequences. Ibn Sīnā employs this argument to demonstrate that infinite magnitudes lead to logical inconsistencies when compared or mapped. Al-Rāzī largely supports this argument, defending it against hypothetical objections and extending its application to both physical and conceptual infinities. He acknowledges the robustness of the mapping argument in addressing not only the impossibility of infinite bodies but also its implications for mathematical and metaphysical infinity.

While Al-Rāzī critiques the collimation and ladder arguments, he acknowledges the validity of the mapping argument. He also engages with objections raised by earlier philosophers and theologians, such as the differentiation between infinite voids and infinite bodies. Al-Rāzī argues that the mapping argument applies equally to voids and bodies, dismissing the distinction made by theologians as inconsistent. Al-Rāzī’s broader critique reflects his philosophical rigor in addressing the foundational assumptions underlying each argument. His engagement with theological perspectives further highlights the interdisciplinary implications of the debate on infinity.

As a result Ibn Sīnā’s three arguments collectively aim to demonstrate the impossibility of actual infinity in nature, with a specific focus on sequentially ordered entities. Al-Rāzī’s critiques reveal significant weaknesses in the collimation and ladder arguments while affirming the robustness of the mapping argument. Ultimately, the study underscores the philosophical depth of both Ibn Sīnā and Al-Rāzī in addressing one of the most challenging concepts in metaphysics and natural philosophy. By examining the strengths and limitations of each argument, this study contributes to a deeper understanding of the intellectual legacy of these two prominent thinkers and their enduring relevance in contemporary philosophical discourse.