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In order to answer the question, Prof. Barrow starts with Aristotle's philosophical interpretation of the concept of "infinity" - thereby implying that he was the first.
However, and according to Wikipedia, though Aristotle came up with the philosophical basis in 350 BC, "The Jain mathematical text Surya Prajnapti (c. 4th–3rd century BCE) classifies all numbers into three sets", of which "infinity" was one.
Like most other things in history, the philosophical basis of a concept came first and those that were proven scientifically got accepted later. Seemingly, "infinity" is such a concept.
Most children growing up in India are introduced to the Sanskrit word "Ananta" which PRECISELY means "infinity".
If the "mathematical basis" came in c. 4th–3rd century BCE, chances are that the "philosophical" basis of "infinity" - "Ananta" - must have come even earlier!!
Perhaps Prof Barrow should remove his colored Euro-centric glasses and address the philosophical basis of "infinity" more accurately and thereby more profoundly!!