On Infinite (0-1599)

"Things [...] are some of them continuous [...] which are properly and peculiarly called 'magnitudes'; others are discontinuous, in a side-by-side arrangement, and, as it were, in heaps, which are called 'multitudes,' a flock, for instance, a people, a heap, a chorus, and the like. Wisdom, then, must be considered to be the knowledge of these two forms. Since, however, all multitude and magnitude are by their own nature of necessity infinite - for multitude starts from a definite root and never ceases increasing; and magnitude, when division beginning with a limited whole is carried on, cannot bring the dividing process to an end [...] and since sciences are always sciences of limited things, and never of infinites, it is accordingly evident that a science dealing with magnitude [...] or with multitude [...] could never be formulated. […] A science, however, would arise to deal with something separated from each of them, with quantity, set off from multitude, and size, set off from magnitude." (Nicomachus, cca. 100 AD) 

"A quantity divided by zero becomes a fraction the denominator of which is zero. This fraction is termed an infinite quantity. In this quantity consisting of that which has zero for its divisor, there is no alteration, though many may be inserted or extracted; as no change takes place in the infinite and immutable God when worlds are created or destroyed, though numerous orders of beings are absorbed or put forth." (Bhaskara II, "Bijaganita", 12th century)

"Now, it is possible that an infinite collection of number stands to an infinite collection in

every numerical ratio and even in every non-numerical ratio. And some infinites are more than others and others less than others; thus, the collection of all numbers, both odd and even, is infinite and it is greater than the collection of all the even numbers, even though this too is infinite, for it exceeds it by the collection of all the even numbers, even though this too is infinite, for it exceeds it by the collection of all the odd numbers. The collection of all numbers proceeding without break from unity by doubling is also infinite, as too is the collection of all the [numbers] corresponding to those doubles as their halves, and yet the collection of the halves must be half of the collection of their doubles. In the same way, the collection of all numbers proceeding without break from unity by tripling is triple the collection of all the thirds corresponding to those triples. The same clearly holds for numerical ratios of every kind, since one infinite can stand to another infinite in any one of these ratios." (Robert Grosseteste, "De Luce", cca. 1220)

"This world is finite, the other infinite,
reality is blocked by form and image." (Jalaluddin Rumi, "Masnavi-ye Ma ‘navi" Vol. I ["Spritual Verses"], 1262-1264)

"The existence of an actual infinite multitude is impossible. For any set of things one considers must be a specific set. And sets of things are specified by the number of things in them. Now no number is infinite, for number results from counting through a set of units. So no set of things can actually be inherently unlimited, nor can it happen to be unlimited." (St. Thomas Aquinas, "Summa Theologica", cca. 1266-1273)

"Having gotten, with God’s help, to the very desired place, i.e. the mother of all cases called by the people 'the rule of the thing' or the 'Greater Art', i.e. speculative practice; otherwise called Algebra and Almucabala in the Arab language or Chaldean according to some, which in our [language] amounts to saying 'restaurationis et oppositionis', Algebra id est Restau­ ratio. Almucabala id est Oppositio vel contemptio et Solutio, because by this path one solves infinite questions. And one picks out those which cannot yet be solved." (Luca Pacioli, "Summa de arithmetica geometria proportioni et proportionalita", 1494)

"The infinity of All ever bringing forth anew, and even as infinite space is around us, so is infinite potentiality, capacity, reception, malleability, matter." (Giordano Bruno, "De immenso", 1591)