With respect to Aristotle, Thomas Aquinas pointed out some important aspects of the problem of infinity that have in more recent times been confirmed by the mathematical approach to infinity as it was developed by Georg Cantor and which is today considered to be classical. Bearing in mind that within the context of actually infinite objects in an absolute sense there can exist only one Absolute or Subsistent Being—which are philosophical attributes Aquinas recognizes as proper only to God —the medieval philosopher proposes a distinction regarding three kinds Lat.

It would therefore be erroneous to maintain that the adjustment carried out by Cantor—using the methods of set theory to rigorously confront the different types of infinity that exist within mathematics—rejects the Scholastic philosophical doctrine that considered the notion of infinity in act contradictory, except when applied to divinity.

For example, Aquinas says that it is perfectly consistent to affirm that the totality of the natural numbers is infinite because there does not exist, nor can there exist, a maximum natural number of each infinite sequence of natural numbers. Regarding infinity considered as an attribute of God, one should keep in mind that St. Within the logic of this question, the infinity of God derives from the reality of God being Pure Act, that is, from the convergence between His fullness of Being and His unconditional capacity to be causa omnium formarum , without having within Himself any composition of matter.

The infinity that is attributed to God is not the infinity that refers to quantity but rather that which refers to Pure Act. This Act determines, and therefore transcends, every quantity in the material order. It follows that there cannot be absolute actual infinities in the dimensional order, i.

The first case addresses a notion that any human being is by nature predisposed to appreciate the meaning of, while the second case is rather a conclusion reached by a thinking believer in this case, Anselm who compares his knowledge of God with everything that is not God. On the other hand, it must be kept in mind that the term is predominantly philosophical and as such it is hardly present in Scripture.

Cavalieri maintained this regardless of the fact that he never affirmed explicitly that the continuum was composed of an infinity of indivisible infinitesimals, an assertion that is not rationally demonstrable cf. Therefore every continuum is composed of infinite undividable parts that are non-quanta cf. Lombardo-Radice, , p.

Employing such parallel lists, he discovered the law of falling bodies in the form of the square of the times.

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In fact, he realized that, assuming each time t was the geometric mean between the number 2 and the length of the pendulum, the two tables become correlated row for row. Properly varying the two magnitudes in question, he further verified, in short, that the relation was biunivocal only if this relationship was satisfied cf.

### The Review of Modern Logic

Drake, He attempted to lead the new experimental science back to the great Greek tradition, which did not consider unity a fundamental of discreet quantification a number, but rather a generator of reciprocally irreducible numerical sets, obtained through the sequence operation. On the other hand, the beginning of modern mathematics can be identified precisely by the renunciation of this reciprocal irreducibility between numerical sets founded on the non-numerical character of unity; modern mathematics considered as numbers those irrational limiting of ratios that the Greeks considered to be non-numbers.

This consideration, through the invention of infinitesimal calculus and the progressive definition of the concept of limit, brought us to the theory of real numbers as conceived by Dedekind, and therefore to the Cantorian theory of sets, which was the first attempt to give a coherent systematics to the whole problem. Leibniz , thus resolving the paradox of Zeno, implicit in the affirmation that the sum of the infinite quanta parts brings us always to an infinite result, thanks to the discovery of convergent series and their application to the physics created by I.

Newton This discovery, which was closely related to that algebraic formulation of geometry of R. Descartes , was a significant contribution. This led finally to the days of Georg Cantor Three Kinds of Infinity. Cantor considered three kinds of infinity cf.

Hallett, , pp. Also, for Cantor the latter infinity can be said only in reference to God, the Absolute Being. He made this distinction in order to oppose two kinds of ideologies typical of the Enlightenment regarding modern science and mathematics one of a Spinozian nature, the other Kantian and to refute their two fundamental principles.

The former is based on the Spinozian equivalence between God and Nature Deus sive Natura , which is a fundamental principle of modern theoretical pantheism. All of these antinomies are based on the conception of absolute actual infinity understood as the limit towards which the finite tends.

## David Deutsch Quotes (Author of The Beginning of Infinity)

Therefore he was led to believe that he had to systematically oppose the Scholastic philosophical doctrine with his own conception of actual infinity in mathematics. Thus, the said domain must not be itself something variable, otherwise every base founded for the study of mathematics would vanish.

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The three notions of infinity are strictly related to the three fundamental principles that give unity to all Cantorian thought: reductionism, finitism, and the instance or the postulation of an Absolute. At the same time, it was necessary that this absolute collection was not mathematically determinable, and in particular that it was not itself a set, in order to escape from the corresponding antinomies. Though he was close to various aspects of the thought of Von Neumann , Cantor distanced himself from him because the foundation Cantor gave for the absolute collections was not axiomatic but rather metaphysical.

It is something profoundly linked to the Platonic roots of his metaphysics regarding the ultimate non-predicability of the One that, at least in this context, is profoundly anti-Parmenidean. The second instance implies that existence depends only on logical non-inconsistency and therefore on belonging to the universal set.

The Notion of Transfinite. Paul Steinhardt, Princeton University, helped develop inflation, the theory of the origin of our universe.

## Abstracts.

So, we have not explained any feature of the universe by introducing inflation after all. We have just shifted the problem of the original big bang model how can we explain our simple universe when there is a nearly infinite variety of possibilities that could emerge from the big bang? Maybe the cosmos really is this way and we just have to accept it.

Certainly, there are many who are willing to defend the multiverse as a valid direction for thought. Comfortingly, if we do live in a multiverse, we can be assured that somewhere out there is an alternate version of you and me that have already figured all this out and won a Nobel prize for the effort. He will be delivering the Guardian masterclass on Is there life beyond Earth? If there is a multiverse, however, that quest could be doomed to failure. However, Steinhardt turned against his own theory. Topics Science Across the universe.

### Hegel on the dialectics of infinity

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