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Iqbal extends the anti-classical thesis from empirical method into pure mathematics and physics, arguing that where the Greek ideal was proportion and finite form, the Muslim ideal was infinity and dynamic relation. He reads Naṣīr al-Dīn al-Ṭūsī as a precursor of non-Euclidean geometry, al-Khwārizmī and al-Bīrūnī as having begun the transformation of number 'from being to becoming,' and refuses Einstein's spatialisation of time in favour of Whitehead's processual account.
Section 6 completed the evidentiary case for the Islamic origin of the experimental method, culminating in Iqbal's strongest formulation of the anti-classical thesis: the empirical spirit arose from 'intellectual warfare' with Greek thought, not compromise. Section 7 now shifts from the domain of empirical method to the domain of pure mathematics and theoretical physics, arguing that the Muslim engagement with the problems of space, time, and the infinite represents a second, parallel departure from the Greek intellectual orientation — and one that, in certain respects, anticipates the conceptual revolutions of modern mathematics and physics.
The section advances through three movements. First, Iqbal establishes the philosophical framework: knowledge must begin with the concrete but cannot rest there, because the finite, taken as self-sufficient, becomes 'an idol obstructing the movement of the mind.' The Qur'an directs the intellect beyond the finite toward an 'infinite cosmic life and spirituality.' Where the Greek ideal was proportion — the well-defined limits of the physically present finite — the Muslim ideal was infinity — the possession and enjoyment of the unbounded. In a culture animated by this ideal, 'the problem of space and time becomes a question of life and death.'
Second, Iqbal traces two Muslim responses to the problem of space. The Ash'arites rejected the absolute space that Democritean atomism presupposed, developing instead an atomism that anticipates certain features of modern physics (already treated in Lecture III). More remarkably, Naṣīr al-Dīn al-Ṭūsī, in his effort to demonstrate Euclid's parallel postulate, 'realized the necessity of abandoning perceptual space' — a move that Iqbal identifies as furnishing 'a basis, however slight, for the hyperspace movement of our time,' the tradition culminating in non-Euclidean geometry and Einstein's general relativity.
Third, Iqbal turns to al-Bīrūnī and al-Khwārizmī to argue that the mathematical idea of function — the representation of dynamic relationships between variables, which 'turns the fixed into the variable, and sees the universe not as being but as becoming' — was not, as Spengler claimed, a uniquely Western achievement but had its roots in Islamic mathematics. Al-Khwārizmī's movement from arithmetic to algebra began the transformation of number from 'pure magnitude to pure relation'; al-Bīrūnī, in generalising interpolation formulae to arbitrary functions, took 'a definite step forward' toward what Spengler called 'chronological number.' The section closes with a striking preference for Whitehead's theory of relativity over Einstein's, on the ground that Einstein's theory 'mysteriously translates time into utter space,' thereby losing the character of temporal passage that is central to the Islamic and Qur'anic understanding of reality.
This section is the most mathematically and scientifically dense in Lecture V. Its argumentative function is to extend the anti-classical thesis from empirical method (Sections 4–6) to pure mathematics and theoretical physics, showing that the Muslim departure from Greek thought was not merely practical but conceptual — a reorientation of the intellect's relationship to space, time, infinity, and the dynamic character of reality.