Abstracts.

Slides:

Set theory fully vindicates the concept of actual infinite, as, through the very simple and intuitive notion of set, it is possible to provide a fully satisfactory theory of infinities of different sizes. After Cantor's creation of the Transfinitum, and his early naive formulation of the notion of 'set' (Menge), the axiomatisation resulting in the theory known as ZFC (due to Zermelo, Fraenkel, Skolem and von Neumann) secured the internal consistency of the early infinitary set-theoretic intuitions and methodologies.

Zermelo's 1930 seminal paper on isomorphic models was based on second-order categorical axioms, and the increasing prominence of first-order logic thereafter raised the issue of whether there really is an intended model of set theory.

Shortly afterwards, Gödel's and Cohen's model-theoretic manipulations (inner models and forcing) broke ground for the full emergence of the set-theoretic multiverse. In recent times, the notion has been introduced and discussed by some authors (see [6], [5] and [4], partly based on [1]), but we argue that all these accounts are, in some respects, insufficient.

In our talk, we wish to present the main features of the aforementioned conceptions and state our own theory, that is, the hyperuniverse theory. The hyperuniverse is a mathematical construct which has been introduced by S. Friedman and T. Arrigoni (see, in particular, [3], [2]) in order to deal with the notion of set-theoretic multiverse. The work done by Friedman and Arrigoni has led to the Hyperuniverse Programme, which is currently being pursued by the four authors at the Kurt Gödel Research Centre at the University of Vienna.

Our main theoretical goal is to provide a philosophically acceptable theory of the multiverse which is primarily concerned with the search for new set-theoretic axioms. We would like to identify philosophically justifiable principles and turn them into mathematical criteria, whereby it is possible to select preferred universes across the hyperuniverse. All the truths holding in such universes would, then, be thought of as 'truths in V', although we do not commit ourselves to any form of realism about set-theoretic truth. Finally, some of these truths may be viewed as new set-theoretic axioms.

• [1] M. Balaguer. A Platonist Epistemology. Synthèse, 103:303-25, 1995.
• [2] S. Friedman and T. Arrigoni. Foundational Implications of the Inner Model Hypothesis. Annals of Pure and Applied Logic, 163:1360-66, 2012
• [3] S. Friedman and T. Arrigoni. The Hyperuniverse Program. Bulletin of Symbolic Logic, 19(1):77-96, 2013.
• [4] J. D. Hamkins. The Set-Theoretic Multiverse. Review of Symbolic Logic, 5(3):416-449, 2012
• [5] S. Shelah. Logical Dreams. Bulletin of the American Mathematical Society, 40(2):203-228, 2003.
• [6] W. H. Woodin. Infinity. New Research Frontiers, chapter IV: The Realm of the Infinite, pages 89-118. Cambridge University Press, Cambridge, 2011.

The Treatise of Configurations, written by Nicole Oresme in the beginning second part of the XIVth century, is a singular work because it brings a broadly integrating and "critical" approach to the question: where does infinite fit within the structure of reality.

I will focus on this specific work of Oresme, which is already very rich, and not discuss the author's other works.

I propose to show that the appearance of mathematical infinite in this work has a musical undercurrent. This occurs in three ways.

First, a new style of polyphony, that musicologists describe as ars nova, is based on speed differences. This fits together with the musical analysis of the second part the Treatise and with its mathematical introduction, where we find what could be considered to be a sketch of a derivative process.

It should be mentioned that the physical relations are understood by Oresme according to a musical pattern of harmony.

Secondly, sound's components and structure are analysed in a new way which requires a microscopic, or more precisely, a micro-acoustic analysis. The study of geometrical figures is transformed in the same manner. An example of this microscopic geometry is the attempt to theorise curvature.

Thirdly, as musical composition (in the work of Guillaume de Machot, for instance), becomes more geometrical, the combination of possibilities, geometrically analysed, is a remarkable feature of Oresmian study of intensive phenomena. Imagination produces and is helped by geometrical signs. This appears in a clever summation of a series (result which would not be true if the summation was finite) that we find at the end of his Treatise. Thus, imagination leads one to understand possibilities that explain the real world but are never completely adequate.

I will conclude by raising a methodological question: should this analysis of the infinite be considered original or significant?

Slides:

In the history of thought the comparison between the infinitely small and the infinitely big has been made by several authors. These conceptions are thought by Pascal with a different meaning, implying the notion of "infinite distance" between man and the extremities of nature, and also between each one of the three orders exposed in one of the fragments of the Pensées (the orders of body, of mind and of heart). In Pascal's thought the notions of infinity and of disproportion can't be separated, at least in the anthropological realm (even if in his mathematical practice Pascal proposed some sort of calculus of the infinite).

Cantor attributed to Pascal one of the first attempts to make explicit the concept of actual infinity, and he also saw marks of the actual infinity in the works of Saint Augustine, who was one of the most important influences to Pascal and to the jansenists of Port-Royal.

Could the Pascalian view of infinity as an actual infinity, and as a "negative" concept in the sense that it is an attribute to a distance, could have a double origin in Saint Augustine? Has Pascal developed a conception of infinity and a theological accent in the fall of man that were both in Saint Augustine's works but with a different association?

Not only Pascal's dialogue with philosophy and with theology had a role in the formation of his concept of infinity, but also his mathematical practice was important as Pascal saw that he could make infinite sums and that a difference between two incommensurable magnitudes could be made as small as we want.

Finally, we shall ask: in what sense Pascal's three orders of reality can be formalized in terms of Cantor's transfinite numbers? The relation between Pascal's infinity and transfinite numbers was already analyzed (by Gardies 1984, Descotes 1993 and Merker 2001, for instance). We shall resume these proposals and present a new formalization for Pascal's three orders, identifying the order of the body with the finite, the order of the mind with \(\aleph_1\) and the order of the heart with Cantor's conception of the "absolute infinity".

Slides:

This paper focuses on the concepts of infinity Kant used in the solution of cosmological dilemmas in the Critique of the Pure Reason. In this paper, it is showed that the mathematical concepts of infinity were crucial to reconcile Kant's epistemology with his cosmology.

Kant's arguments for the indefinite extension of the universe in space and time have been investigated from an historical and philosophical perspective that stressed the crucial role a number of mathematicians, such as Leibniz and Lambert, played in informing Kant's notion of infinity. Although historians emphasised that the notion of approximation is crucial for the understanding of Kant's epistemology, they have not adequately stressed the importance of the conception of series ad infinitum, in infinitum and in indefinitum in Kant's solution of the Antinomy of the Pure Reason in the light of his cosmology. Kant, indeed, coupled mathematical applications of the concepts of infinity with the principle of his general logic in order to obtain progresses of cosmology, by limiting metaphysics.

Kantian studies widely discussed whether Kant's early cosmology, Universal Natural History and Theory of the Heavens (1755), and his later achievements in the Critique of the Pure Reason (1781; 1787) are contradictory. Indeed, in 1755 Kant formulated a model of an expanding universe, whereas he later seemed to assert the impossibility of knowing the universe as a whole and its intimate nature. This paper shows that there is no contradiction between Kant's Critical achievements concerning the dilemma on the finite or infinite nature of the cosmos, solved via a regulative cosmological principle in the Antinomy of the Pure Reason, and his 1755 cosmology. Yet, this paper expounds the concepts of mathematical infinity implied in Kant's 1755 cosmology and shows that the development of the notion of synthesis applied to the mathematical concept of series was the key that disclosed the possibility for constructing the Antinomy's resolution in 1781 in order to accommodate Kant's epistemology with his cosmological hypothesis.

As shown in this paper, Kant successfully used the results of the antinomies in his late cosmology (1791 onwards), though the result of their resolution was also implicit in his 1755 cosmology. Kant assumed the universe as expanding (and the paper will show whether for Kant this expansion goes ad infinitum or in infinitum) and its boundaries extend in indefinitum; according to him, we must suppose the beginning of motion of matter, and we can determine it, but we cannot know whether and when chaotic matter was created. The most intriguing consequences of Kant's reasoning concerns the question of the origin of the universe in time. In the conclusion, the paper deals with this question (which is still an intriguing one for us) and assesses the relevance of the mathematical concept of regressum in indefinitum for its formulation in the light of Zermelo's observations on Kant's antinomies (Zermelo 1930).

The following remarks, in the first section, draw together ideas which are directly addressed in Hacking's article 'Leibniz and Descartes: Proof and Eternal Truths' with ideas from the philosophy of intuitionist mathematics concerning notions of infinity, and an exploration of the types of cogito that we can discern in moving from Leibniz to Descartes to Wittgenstein.

It is argued that Hacking is incorrect in concluding that the problems traditionally posed within a Leibnizian-Cartesian framework are left untransformed by Wittgenstein. Rather, if one takes seriously the fact that Descartes was an epistemologist–and that Leibniz was not–, Descartes' eternal truths align with the Kantian transcendental project of knowledge through intuition, both projects which Wittgenstein rejects. Proof and proof-theoretic reasoning is explored, with a highlighting of modal considerations and a resultant modern conception of agency. A critical translation of passages in the sequences numbered under 1 and 2 in Tractatus Logico-Philosophicus constructs the logical space for a post-Enlightenment agency.

The second section examines Wittgenstein's unrelenting criticism of Cantor, Euler, and Gödel, which falls within a larger strategy to disarm a philosophy of mathematics which relies on completed infinite sets. Because transfinite numbers are seen to resolve the Zeno paradoxes, creating "a paradise from which we shall not be expelled", according to Hilbert , Wittgenstein's mathematics began to be seen as backward looking, particularly in the period of Turing's work on the Entscheidungproblem and computable numbers. It is argued that Wittgenstein offered consistent criticism and alternative approaches to paradoxes of the infinitely large and small, placing him within an Einsteinian position of general relativity and a post- Leibnizian position of infinite plenitude.

The final section explores Wittgenstein's sophisticated critique, which was developed within a type of constructivist mathematics of potential infinity, through a reappraisal of the more overtly mathematical key concepts of what I have elsewhere called the Cosmic Fragment passages (and the Fragment's precursor mathematical notebooks MSS 149 and 152). This provides support for the Einsteinian/post-Leibnizian thesis in section two, and, obliquely, for the cultural conclusion of the first section ,allowing the bridging concepts of Härte des logischen Zwangs, Unendliche Reihe, Maschine als Symbol ihrer Funktion and the recurring question "Aber sind denn die Übergange also durch die algebraische Formel bestimmt?" to function pivotally and consistently in a developing philosophy of mathematics, cosmology and culture by Wittgenstein.

In addition to Cantor's well-known systems of infinite cardinals and ordinals, there were a variety of other systems of actual infinite numbers that emerged in the decades bracketing the turn of the twentieth-century. Two grew out of the work of Paul du Bois-Reymond [1870-71, 1875, 1877, 1882], Otto Stolz [1883], Felix Hausdorff [1909] and G. H. Hardy [1910] on the rates of growth of real functions, and the others emerged from the pioneering investigations of non-Archimedean ordered algebraic and geometric systems by Giuseppe Veronese [1892], Tullio Levi-Civita [1892, 1898], David Hilbert [1899] and Hans Hahn [1907]. Unlike Cantor's systems, which solely embrace finite numbers alongside his familiar infinite numbers, the other just-said non-Cantorian number systems, like the more recent hyperreal number systems associated with Abraham Robinson's nonstandard approach to analysis, embody finite, infinite and infinitesimal numbers.

In [Ehrlich 2012], we show how the above-mentioned Cantorian and non-Cantorian number systems admit a striking unification in the author's [Ehrlich 2001] algebraico-tree-theoretic approach to J. H. Conway's system of surreal numbers. Building on the above, in this paper we will provide introductions to the aforementioned non-Cantorian theories of the finite, infinite and infinitesimal that emerged in the decades bracketing the turn of the twentieth-century, explain the motivation for their introduction, outline the roles these and related theories play in contemporary mathematics and discuss the relations between these theories and the better-known theories of Cantor and Robinson that emerge from the just-said unification. It is the author's hope that by drawing attention to the spectrum of theories of the infinite and the infinitesimal that have emerged from non-Archimedean mathematics since the latter decades of the 19th century, it will become clear that the standard 20th-century histories and philosophies of the actual infinite and the infinitesimal that are motivated largely by Cantor's theory of the infinite and by non-standard analysis are not only limited in scope but are inspired by an account of late 19th- and early 20th-century mathematics that is as mathematically myopic as it is historically flawed.

• Philip Ehrlich, Number Systems with Simplicity Hierarchies: A Generalization of Conway's Theory of Surreal Numbers, The Journal of Symbolic Logic 66 (2001), pp. 1231-1258.
• Philip Ehrlich, The Absolute Arithmetic Continuum and the Unification of All Numbers Great and Small, The Bulletin of Symbolic Logic 18 (2012), pp. 1-45.

Slides:

A natural notion of size for sets ("numerosity") should abide by the Euclidean common notions, with sum corresponding to disjoint union and ordering to inclusion. Denoting by \(n(A)\) the numerosity of \(A\), the 2nd and 3rd Euclidean notions might be combined into the "Aristotle's Principle"

• AP: \(n(A) = n(B)\) if an only if \(n(A\setminus B) = n(B\setminus A)\),

• EP: If \(A\) is a proper superset of \(B\), then \(n(A) > n(B)\).

Classically, one should also postulate the "Zermelian Principle"

• ZP: Numerosities are always comparable.

If the term "epharmozonta" (exactly overlapping) in the 4th Euclidean notion is interpreted as "equipotent", one gets the "Cantorian Principle"

• CP: If \(A\) and \(B\) are in 1-1 correspondence, then \(n(A) = n(B)\).

• CP[\(T\)]: \(n(A) = n(B)\) if \(t[A] = B\) for some map \(t \in T\).

The corresponding numerosities satisfy all the Euclidean common notions, and constitute the non-negative part of a discretely ordered group, which becomes a ring if the natural product \(n(A)\cdot n(B) = n(A\times B)\) is suitably chosen, so as to avoid the clashes with EP that could arise, depending on different set-theoretic codings of the Cartesian products.

Several theories that specify and strengthen the Euclidean principles AP,EP,and CP[\(T\)] in a "Cantorian-Zermelian direction" have been developed by various authors in the last decade (see the references in [1]), aiming to meet several desiderata, which up to now have been obtained separately, but not alltogether:

• find classes \(T\) suitable for getting a biconditional in CP[\(T\)], or, at least, obtain the converse of CP;
• let the natural ordering provided by EP, namely \(n(A) > n(B)\) if and only if \(A\) is equinumerous to a proper superset of \(B\), be total, so as to ensure also a subtraction of numerosities;
• let numerosity be continuous w.r.t. "normal approximations" (see [1]);
• for Lebesgue measurable sets of reals, let numerosity be proportional to measure, up to infinitesimals.

In general, such numerosities are hypernatural numbers obtained by means of various special (old and new) classes of ultrafilters, whose existence is often independent of ZFC. So they might suggest interesting axiomatic extensions of ZFC.

• [1] M. Di Nasso, M. Forti - Numerosities of point sets over the real line, Trans. Amer. Math. Soc. 362 (2010), 5355-5371.

A generally accepted idea is that learning to deal with infinity and integrating it into normal mathematics constituted a definite progress. History of mathematical infinity thus has a whiff of Whig history, one barely checked by Gödel's disenchantment. For all the fame and the attraction they exert, opponents (whether of the Kronecker or the Brouwer type, for instance) are mostly understood as obstructionists, obsessed by an overly extreme rigor, and their positions are seen as disavowed by the development of mathematics itself. However, alternative approaches to set theory or to the geometrical use of elements at infinity might be accompanied, at the end of the nineteenth century and later on, by an appeal to concreteness, naturalness and above all, mathematical fruitfulness. While Cantor's famous sentence, that "integers, both separately and in their total actual infinity, exist as eternal ideas in God's mind, at the highest level of reality" has been abundantly studied, the point of view of the mathematician to whom this sentence was addressed—Charles Hermite—is, for instance, almost never discussed. My talk will focus on his alternative, and extremely influential, perspective on infinity, on the opposition between discrete and continuous, as well as some of their variants, and will explore how taking these into account might change the questions we ask of mathematical infinity.

Slides:

Stewart Shapiro has argued that mathematical concepts, as well as empirical concepts, can display open-texture. Shapiro's focus is the concept of computability in the early days of computability theory. In my talk, I will look at the relation between what Shapiro says and the distinction between concepts and conceptions. I will then examine whether Shapiro's diagnosis can be applied to the concept of finitism.

In this paper I argue that Aristotle with his introduction of the distinction between the potential and actual infinite does not aim to offer a general account of infinity, applying to infinity of every kind and in every domain in the same way (for example as he is commonly interpreted, as holding that there is no infinity of any kind, but at most a series which always is finite, but can be extended indefinitely, or, as the distinction came to be interpreted later in history, as holding that there are infinite series without them being actual wholes, e.g., infinite series of sets without these series themselves being sets). Rather, Aristotle tries to solve a problem with one type of infinity, namely that of the infinite divisibility of magnitude: he wants to show how a magnitude may be divisible to infinity without this entailing that this it can be divided to infinity at the same time. Normally for Aristotle the potentiality for something entails the possibility of its actuality, but since Aristotle believes that an infinite division is impossible because of Zeno's paradoxes, he cannot accept this entailment in this case. To solve this problem, he introduces a distinction between two types of existence: the way independent objects (substances) actually exist (existing by being there all at once) and the way things spread out over time (events) actually exist (existing/going on by only one part of them being there, as the day is there by being the present hour being there). The infinite, Aristotle claims, exists in the second way: the potentiality for an infinite division is there if there are infinitely many possibilities of division, and this potentiality is already actualised by merely one (or a limited number) of these possibilities being realised. As it happens, all other types of infinity, like the infinity of numbers, of geometrical magnitudes or of the generations of mankind, also adhere to this conception of the potentiality and actuality of the infinite, but that is a matter of course: unlike divisions, in these cases 'things do not remain' and can therefore never all be actual at the same time. For when the present generation of mankind is there, the previous one has gone again, and when the mind thinks of the one number or of the one geometrical magnitude, and does not think of the other – Aristotle holds that mathematical entities have only mind-dependent existence.

(Not in the paper yet:) This distinction between things which 'remain' and things which 'do not remain', which is crucial for Aristotle's conception of the potential infinite, was ignored by the ancient commentators, first by Simplicius and then most famously by Philoponus, who argued that for Aristotle the infinity of the past constitutes a completed and thus actual infinity – which should be impossible for Aristotle, so that he should have held that the universe came into being. Thus they started the history of reinterpretation of Aristotle's potential infinite as a general theory of infinite.

Slides:

Cardinality is a notoriously counterintuitive topic for students to come to terms with. In this presentation I report a study which investigated the strategies adopted by successful and unsuccessful mathematics students when asked to develop an understanding of cardinal arithmetic. Participants were asked to read a textbook introduction to the topic, while the location of their attention was recorded using a remote eye tracker. In the talk I will compare the reading strategies adopted by the two groups, as well as their reflections on the conceptual difficulties they encountered when coming to terms with the new material.

A decisive step in the history and philosophy of the concept of infinity was made in the early 20th century by two prominent mathematicians: Poincaré and Brouwer. Indeed, each one independently attacked most of Cantor's work on infinite sets. Their common goal was to have mathematical theories in general, and set theory in particular, fall under the direction of "mathematical intuition". I will explain their arguments and why they concluded that only two kinds of infinity should be admissible : the denumerable and the continuum.

First, they stressed the fundamental role of the "recurrence principle", which would allow us to deal with denumerable infinity. According to them, this principle is not a logical rule, such as modus ponens, because it contains an infinity of syllogisms. Hence, justifying it requires an appeal to what they called a "pure intuition". For Poincaré (1902), this was the "pure intuition" of the number, which would pre-exist in our mind before any experience and would justify the validity of all arithmetical laws. For Brouwer (1907), in a more Kantian way, the "intuition of time", in which any moment can be separated into two moments, would yield arithmetic. Finally, even if their approaches differed, they agreed on the key role of the recurrence principle and the perfect admissibility of denumerable sets.

However, they were sceptical about the other aleph numbers. As Poincaré famously stated, in response to Zermelo's axioms for set theory (1909), "I am not sure that aleph-one exists". For Poincaré, "mathematical existence" means "free from contradiction" and does not refer to any ontological commitment. He argued that Zermelo's axioms can be proved consistent for finite sets, but could lead to contradiction for infinite sets. Brouwer had a different view, since he carefully distinguished between "logical consistency" and "mathematical truth". Here, "true" means given in intuition. For him, no matter if Zermelo's axioms lead to contradiction or not, they would be in any case wrong when applied to the continuum.

Indeed for Brouwer, intuition shows that the continuum cannot be a set of points: it is given as a whole. So all the classical ways of constructing the real numbers would be "false", even if consistent. This is why he proposed a new theory of the continuum, based on "free choices", that would contradict many accepted results about real function theory. Poincaré had more mixed feelings: he also believed that the intuitive continuum should not be made of points, but on the other hand he recognized as a matter of fact that most of Cantor's work yields true results about it.

Interestingly, Brouwer's approach was criticized not only by proponents of Cantor, but also by philosophers such as Wittgenstein who agreed partly with Brouwer's views. I will finish by showing how Wittgenstein's arguments shed new light on Brouwer's position and how they could be compatible with Poincaré's views.

Cantorean set theory as axiomatized by the Zermelo-Fraenkel axioms ZFC including the axiom of choice constitutes the established mathematical theory of infinity. There are infinite sets which are not countable, and indeed there is a proper class of different infinite cardinalities. Higher cardinals trivialize all smaller ones, and these differences in size are the basis for strong arguments in infinitary combinatorics. Conversely it was shown that some concrete results like the determinacy of Borel games require the existence of many uncountable cardinals. Indeed by postulating large cardinals, i.e., the existence of size differences which are very large under certain combinatorial criteria, even stronger concrete results like the determinacy of all projective games may be established.

Uncountable, and even more so large cardinals pose ontological problems: Do large cardinals exist in the mathematical universe? Is the mathematical universe a model of the standard axioms of set theory? Which large cardinals axiom should be true?

I consider an approach whose ontological assumptions appear milder, but which still allows for large cardinal argumentations and constructions. We consider a canonical theory HC for the structure of hereditarily countable sets, which together with all their elements and iterated elements are countable. A model \(R\) of HC contains natural numbers, countable ordinals, and real numbers as objects, but, by Cantor's theorem, not the collection of all real numbers. Nevertheless there may be submodels of \(R\) which are models of ZFC or even large cardinal axioms. The existence of such models can be ensured by regularity assumptions within the descriptive set theory of the model \(R\). If the family of such submodels is non-trivial then it forms a rich multiverse of models of set theory which is, e.g., closed under forcing extensions, which can be interpreted as a kind of forcing absoluteness. Now a property like Borel determinacy can be proved in \(R\) by moving to an appropriate model in the multiverse in which the desired strategy for the Borel game can be constructed in the familiar way.

In my talk I shall present some mathematical details of the theory HC, some of which were obtained jointly with Michael Möllerfeld, compare the family of submodels to multiverse theories, and discuss reasons for adopting or rejecting HC.

This paper is about the conception of infinity in metaphysics, mathematics and natural philosophy of the well-known English materialist Thomas Hobbes. It begins with some general reflections that show that Hobbes's adopts the standard Aristotelian conception of potential infinity. The second section concentrates on the more specific discussion that we find in Anti-White (ca. 1643), a critique of Thomas White's De Mundo dialogi tres (1642); a work not much discussed outside the circle of Hobbes specialists. The problem of the second chapter of Anti-White is whether the world is finite or infinite. Hobbes's who defends the latter position gives three arguments:

• A. The theological argument says that an omnipotent God could create an infinite world and therefore it is possible that the world is infinite;
• B. The consistency argument says that from the concept of infinity (if understood correctly, i.e., as indefinite) nothing inconsistent follows and therefore it is possible that the world is infinite, and
• C. The intelligibility argument says that we have the conception of infinity and therefore it is not impossible that the world is infinite.

This analysis, which is in accordance with Hobbes's materialism, empiricism and nominalism, is rather traditional and the real novelty is that in addition to the conventional metaphysical and mathematical conceptions of the infinite, Hobbes's analysis introduces a concrete aspect of infinity likely inspired by the technical development of optical devices. The last section of the paper concentrates on this conception. The paper, then, aims to show that despite Hobbes's reputation as a poor mathematician he is, first, interesting philosopher of mathematics, and, second, that his empirically inspired analysis of infinity is a small chapter in the history of the concept that is worth remembering.

Slides:

Broadly speaking, there are two different conceptions of infinity in foundations of mathematics and physics. One is set-theoretical or Cantorian, and regards an infinity (especially, continuum) as an enormous amount of discrete points or elements, where "discrete" means that those points exist independently of each other, and there is no cohesiveness among them. The other is geometric or Brouwerian, and considers an infinity like a continuum to be a cohesive totality, or rather a finitary law to generate it (in infinite time), which gives rise to intrinsic continuity as seen in Brouwer's theory of choice sequences. Category-theoretical foundations of mathematics, in particular topos theory and more recent "homotopy type theory", support the geometric view on infinity. Indeed, topos theory gives categorical models of Brouwer's intuitionistic mathematics, in particular his continuity principle, and homotopy type theory yields fibrational models of Martin-Loef's intuitionistic type theory with its identity type intensional rather than extensional. The distinction between the Cantorian and Brouwerian conceptions of infinity would be more or less parallel to that between Aristotle's ideas of actual and potential infinity.

In this talk, I aim at examining and articulating conceptual underpinnings of the dichotomy between Cantorian extensional discrete infinity and Brouwerian intentional continuous infinity, by placing it in a wider context of (both analytic and continental) philosophy. Wittgenstein says, "What makes it apparent that space is not a collection of points, but the realization of a law?", where space basically means a space continuum. This is highly analogous to Brouwer's concept of a spread as a law; note that Brouwer also explicitly uses the term "law" when defining his notion of a spread. At the same time, Wittgenstein contrasts the arithmetical and geometrical conceptions of a space continuum, which I argue is parallel (up to a certain point) to the distinction between discrete and continuous infinity. Looking at continental philosophy, I would emphasise Cassirer's dichotomy between the concept of function and that of substance, arguing that the functional conception of space leads us to the Brouwerian perspective on space, whilst the substance-based conception is suitable for the understanding of infinity as discrete. This sort of dichotomy may be observed in quantum physics as well, in its idea of state-observable duality or micro-macro duality. Whitehead's process philosophy also comes into the picture when we focus on the observable-based view. In this talk I shall attempt to elucidate both analogies and disanalogies between such ideas on the nature of infinity in mathematics and physics.

Inspired by Caleb Gattegno's remark that "something is mathematical only when it is shot through with infinity", I propose a phenomenological state of attention between experience of Aristotle's two kinds of infinity: potential infinity (as an uncompleted on going process) and completed infinity: the state of awareness of a generality in which the as-yet-unspecified is perceived by "seeing the general through the particular". It is an infinity which is at least potential yet not unfolding, and which can shift into a sense of completedness. I conjecture that students might find accepting and working with completed infinity less of an epistemological obstacle if they were familiar and confident with expressing generality.

Slides:

ZFCU is Zermelo-Fraenkel set theory with Choice, modified to allow for the existence of urelements, or (for purposes here) "objects". An important component if David Lewis's modal realism is that anything can coexist with anything, expressed initially in his unqualified principle of Recombination:

• (R) For any objects (in any worlds), there is a world that contains any number of duplicates of those objects.

Consider the following consequence of R:

• (RC) For any cardinal number \(\kappa\), it is possible that there are at least \(\kappa\) objects.

In the context of modal realism, RC entails:

• (A*) For any cardinal number \(\kappa\), there are at least \(\kappa\) objects.

But this is problematic for Lewis. Consider:

• (SoA) There is a set of all objects.

In the context of ZFCU, it follows that (A*) is inconsistent with (SoA). Faced with such inconsistencies, Lewis opted to abandon (RC) and, hence, Recombination. For, as properties are sets of objects and propositions are sets of worlds for Lewis, many important properties and propositions will not exist if there is no set \(O\) of all objects and, hence, many of the applications of modal realism will fail.

But there is another option here. In a forthcoming paper I develop a modification ZFCU* of ZFCU that accommodates the existence of "wide" sets, i.e., sets that, like \(O\) (assuming (A*) and (SoA)) are too big to have a definite cardinality but which have a definite rank. The key modifications are to Replacement and Powerset. Say that a set is (mathematically) determinable if it is equipotent to some pure set. We then our modified Replacement F* applies to sets \(S\) that are either determinable or for which we have a replacement mapping that is "bounded above" by rank on \(A\).

On the modified version PS* of Powerset, only the determinable subsets of a given set constitute a further set. The motivation for this modification stems from Cantor's conception of the "absolute infinite", i.e., the "size" that characterizes non-determinable collections. For Cantor, this "size" is an "absolute quantitative maximum" that is subject to no mathematically definite increase. Accordingly, given PS*, Cantor's theorem fails for wide sets. A final axiom further enforces the Cantorian intuition: Only determinable sets are smaller than some other set. ZFCU* thus yields a rather different, "cylindrical" picture of the cumulative hierarchy in which there is no mathematically definite increase in its "girth". (ZFCU* is consistent relative to ZFCU + "There exists an inaccessible cardinal".)

Assuming the set \(O\) of objects is wide, ZFCU* permits the construction of, even if not the full intuitive power set of O, infinitely many complex sets of arbitrarily high rank over \(O\). The central focus of my presentation will be to investigate the extent to which Lewis's program–with full Recombination–can be restored if the modal realist adopts ZFCU*. Secondarily, I will address other implications of ZFCU* in regard to the nature of the infinite, the structure of the comulative hierarchy, and the possibility of absolutely general quantification.

During the mid-fifteenth century, Nicholas of Cusa, in his major work Of Learned Ignorance [1], transformed the Aristotelian "potential" and "actual" infinity into a "negative" and "positive" infinity. In order to explain these new features of the infinite, Cusa developed the method of the coïncidentia oppositorum. In the context of what he coined as our "learned ignorance," he established a proportion between what we know and what we do not know. First of all, Cusa's method does not treat the infinite as a logical statement in the way the philosophers of the Middle Ages did. Their logical approach was based on speculations upon the sophisms and paradoxes of the infinite to overcome the gap between language and reality. For example, one of the most influential philosophers of that period, Gregory of Remini, came up with a new logical expression of the infinite as "categorematical" or "syncategorematical" [2]. Secondly, Cusa's perspective on the infinite was not quantitative as was Leibniz' calculus by the end of the seventeenth century [3,4]. Instead of conducting a logical or quantitative analysis, Cusa identified various orders of knowledge to define the infinite. According to his method of the coincidentia oppositorum, the maximum infinite can be identified with the minimum infinite. Cusa thought that geometry was the best science to use in order to get an understanding of the infinite by progressing through the different orders of knowledge. He studied geometrical figures such as lines, triangles and circles, and demonstrated how these figures can also be said to correspond to an infinite line, triangle, or circle. Cusa then applied this model to the world, concluding that the relationship between the finite and infinite figures is analogous to the relationship of the maximum infinite "to all things" [1].

In this presentation, I will examine how Cusa's method to reach the infinite through geometrical analysis provides a unique standpoint for research on mathematical and philosophical infinity and the pursuit of solutions to philosophical, mathematical and theological issues. In showing how Cusa's reasoning in his methodology of the infinite differs from a logical analysis (Gregory of Remini) or calculation (Leibniz) of the infinite, I will be able to underline a new way to manage the infinite. Finally, I will explain how the different fields of geometry, theology, and philosophy interplay with each other to define the infinite.

References.

• [1] Nicolaus Cusanus, Of learned Ignorance, Hyperion Press Inc : Wesport, (1954).
• [2] Françoise Monnoyeur, Infini des mathematiciens, infini des philosophes, Editions Belin: Paris, 1st Ed (1992).
• [3] Françoise Monnoyeur, Infini des philosophes, infini des astronomes, Editions Belin: Paris, 1st Ed (1995).
• [4] Françoise Monnoyeur, Journal of the History of Philosophy 48 (4):527-528 (2010).

Aristotle's antipathy to actual infinites (e.g., Physics 206b12-14) and his gnomic remark that nature flees the infinite (Generation of Animals 715b15-17) are well known. Aristotle is often taken to rule out infinite sequences of various sorts on this basis and he does indeed argue for an unmoved mover by rejecting an infinite causal sequence (Physics 256a4ff); further, he does claim that certain causal sequences without first members are unacceptable (Metaphysics 994a2-19). However, pace the common scholarly view, Aristotle does in fact admit certain infinite causal sequences: for instance, he takes all humans to have a human parent (e.g., Physics 206a26-9). This requires explanation. I will first clarify how the distinction between actual and potential infinites applies to infinite (diachronic) causal sequences and will then show how Aristotle's conception of principles (archai) explains both: (i) his ruling out of various infinite sequences (e.g., Metaphysics 994a1ff); and (ii) why he should allow some others. Aristotle takes all causal chains to have principles and he takes a principle (arche) to be: a member of a causal chain; something which has no members prior to it in that causal chain; and something upon which all subsequent members in the causal chain depend upon in some way for their own causal efficacy. However, such a conception does in fact allow certain infinite causal sequences. I will show how medieval philosophers (e.g., Aquinas ST I q.46 a.2) expanded upon Aristotle's brief remarks (e.g., Physics 256b5ff) about the distinction between a per se sequence of causes and a per accidens sequence of causes in a manner congenial to Aristotle's needs. In a per accidens sequence, when x has generated y, and y has generated z, y's coming into existence depends upon x, and z's coming into existence depends upon y; however, y's exercising of causal power does not depend upon x and z's exercising of causal power does not depend upon y (or x). By contrast, in a per se sequence, when x has generated y, and y has generated z, both y and z depend upon x for their causal power. This distinction, which turns upon how causal power is transmitted and exercised will be elaborated and I will show how it allows Aristotle to recognise the admissibility of certain infinite causal sequences: while a per accidens causal sequence may be infinite, a per se causal sequence may not. This has intrinsic historical interest: showing the manner in which Aristotle does in fact accept certain actual infinites (and may do so in a consistent manner). It also provides a metaphysical parallel to contemporary discussions concerning warrant transfer and emergence.

One class of questions which one might ask about the nature and structure of mathematical infinity includes those such as "Which infinities exist?" and various others concerning the relationships obtaining between such infinities (such as the question "Is the Continuum Hypothesis (CH) true?"). A different class of questions asks whether each meaningful question in the first class admits of a determinate answer (alternatively, whether each meaningful mathematical statement about the infinite realm admits of a determinate truth value), and, if not, why not. Although it might, at first, appear that the advocate of Platonism about mathematical reality can give an unequivocally affirmative answer to the second sort of question, recent debates around the multiverse and universe views of set theory have highlighted the fact that this ultimately depends on the sort of Platonist view which one endorses. In this context, the present paper will be concerned with the following conditional question: On the assumption that some version of Platonism about the domain of set theory is correct, which version is it? Specifically, under the aforementioned assumption, I aim to provide an examination, and ultimately a rejection, of the multiverse view of set-theoretic reality (MV) in favour of the universe view (U). This aim will be achieved in three stages. In the first, I characterise the multiverse view as a type of, what Mark Balaguer calls, Plenitudinous Platonism (PP): In particular, it is a position which is both more specific than (PP) (being concerned only with set-theoretic reality) and stronger than (PP) (embodying the claim (SP) that our concept of set gives rise to a plurality of distinct, instantiated set-theoretic universes). On this basis, I suggest that (MV) is best construed as consisting of two claims: The first being (PP); the second (SP). In the second stage, I consider (and find wanting) three prominent arguments in support of (MV) which have emerged in the literature in recent years. Namely, arguments to the effect that:

• (i) (PP) is the only version of Platonism which can survive Benacerraf's, so-called, epistemological argument against this generic view.
• (ii) (MV) is the version of Platonism which fits most naturally with the best approaches to: (a) Benacerraf's, so-called, non-uniqueness argument against Platonism; (b) problems which are independent of Zermelo-Fraenkel set theory, such as that of CH.
• (iii) Adopting (MV) provides the more fruitful framework for conducting certain important, contemporary mathematical investigations in set theory.

In the third and final stage, I briefly sketch an argument for (U), in the light of Joel David Hamkins' recent defence of (MV) against distinct arguments of Donald A. Martin and Daniel Isaacson.

Slides:

This talk focuses on the psychological process of creating the concept of infinity and it is part of a larger project of naturalizing the epistemology of arithmetic based on progress in the empirical study of numerical cognition in the past decades. I want to pursue the theory that we originally get the concept of natural number from the primitive way of processing observations in terms of discrete quantities. We share this ability with many nonhuman animals and it shows up, e.g., in subitizing, the ability to grasp small quantities without counting. Once the inductive step is introduced, the knowledge about small numbers (usually from one to five) can be extended indefinitely.

A further step is needed, however, in order to turn all this into a mathematical concept of infinity. For that purpose, I will argue, one must have two things: an informal understanding of something being limitless, plus a basic conception of axiomatization or implicit definition to be able to formulate this understanding mathematically.

In this paper I argue that nothing else is needed in order to reach the infinity of natural numbers. In particular, there is no need to evoke either actual infinities, understanding of an infinite set, or any such higher abstract reasoning. Philosophically, such an account can accommodate various approaches. A plausible account of how the concept of denumerable infinity was acquired and defined can be reached from purely empirical grounds. This does not by itself contradict ontologically heavier positions such as Platonism: it is still a possibility that infinity was in some sense discovered rather than invented. However, I will try to show that no such ontological demands are required from a perfectly satisfactory account of the infinity of natural numbers.

Slides:

Galileo, Leibniz, and Bolzano were all aware of the principle, now associated with Hume and Cantor, that one-to-one correspondence implies equal number. But they were also aware that, for infinite collections, this contradicts the Euclidean principle that the whole is always greater than the part, a principle which so gripped their intuitions that they could not accept the Hume-Cantor principle in the infinite case.

Today Cantor's theory of cardinal number is widely regarded as the only correct one. Gödel argued explicitly for this view in the opening paragraphs of "What is Cantor's Continuum Problem?", seemingly as an uncontroversial paradigm case of a uniquely correct mathematical concept. Yet recent work has developed alternative theories of set size that eschew the Cantorian bijection principle in favor of the Euclidean part-whole principle. Such theories are consistent with the standard axioms of set theory and have some intuitive appeal as well as algebraic virtues. Thus it at least appears that Cantor's theory is not forced on us, and Gödel's argument demands a closer look.

We lay out the argument's structure and premises, explicit and tacit. It centers on a thought experiment in which the physical objects in one set are transformed to resemble those of another. An important tacit premise is that if the elements of one set are indistinguishable in their properties and mutual relations from those in another, then the sets are equal in size. But this should not be taken for granted, as an ethical thought experiment illustrates.

More generally, Gödel relies on intuitions about number that are indeed persuasive but nonetheless debatable (and in some cases false), and which are contradicted by other intuitions–in particular the intuition that the whole is always greater than the part. Thus he overlooks the possibility that cardinality is over-determined by our intuitions.

However, if we set intuitions aside, we can give an argument related to Gödel's for a more pragmatic conclusion: Non-Cantorian theories of infinite size are limited. If qualitatively identical sets have different sizes, such sizes are not very informative, and this limits their usefulness.

Handout:

In the early twentieth century, major thinkers on the foundations of mathematics argued that accepted logic, while reliable in reasoning about finite domains, could be questioned in application to the infinite. This view was worked out in detail by Brouwer, presented in some striking remarks by Weyl, and at least partly embraced by Hilbert, although only the first rejected classical logic in his actual mathematical practice. The paper will examine some of their arguments and then inquire whether there are arguments for a similarly critical view of logic that have force today, when the focus is more likely to be on the higher infinite.

The distinction between actual and potential infinity goes back to Aristotle. Does it have a role in contemporary philosophy of mathematics?

Today, Bernhard Riemann is best known in mathematics and philosophy of mathematics for his Göttingen inaugural lecture "On the hypothesis which lie at the foundations of geometry" (1854), at the end of which the validity of the hypothesis of geometry is linked to the question whether physical reality is structured in a discrete or continuous manner.

Hermann Weyl took Riemann's reflections as evidence for (1) the thesis that the leading idea behind Bernhard Riemann's geometry and philosophical achievements was to "understand the world from the infinitely small". More specifically, Weyl and others (like Max Jammer, for example) also linked Riemann's physical geometry (2) directly to Einstein's understanding of space in his theory of general relativity, presented about 70 years later. Riemann was made a kind of 'forerunner' of Einstein.

A more analytical reading of Riemann's "Hypothesis", which also takes into account his earlier fragments and notes on "Mathematical philosophy of nature", on epistemology and on philosophy of science from his Nachlass supports Weyl's general claim (1), while it has to reject his more specific claim (2). One aim of my talk would be to qualify claim (1) and falsify claim (2) by an analysis of the "Hypothesis" in the broader context of Riemann's philosophical and scientific thinking. Riemann's scientific realism' and the role of pure mathematics – which for him is basically a mathematics of infinity – are crucial in this respect.

A second aim would be to show that - if we take the concept of infinity as a key concept of modern analytical mechanics and theoretical physics seriously - the case of Riemann can shed light on the difficult foundational relation between geometry and 'mathematical philosophy of nature' that reflects the epistemic premises and basic laws of mathematical physics. My claim here is that - contrary to the received wisdom – a new understanding of how to 'apply' mathematics to nature did not start with the foundational debate in geometry, but in the realm of mathematical physics. 'Infinity' in Riemann is a central element of this new interpretation. Norman Sieroka. Anaximanders notion of the "apeiron" – A forerunner of the modern concept of infinity? Abstract: It is often claimed that the origin of the concept of infinity goes back to the presocratic notion of the "apeiron". The paper critically evaluates this claim. It shows that it is not infinity in a modern sense (nor boundlessness, for which the Greek used a different term) which is at stake here. Rather, the term "apeiron" denotes the horizon or fictive bounds of what can be directly experienced and what (for all practical purposes) exceeds human capacities; especially the capacities involved in actual processes of counting and surveying.

The term "apeiros" was originally used to refer to things experienced to be in(de)finite in the sense of being inexhaustible, countless or untraversable. Assuming that Anaximander's notion of the "apeiron" was highly affected by this usage, I will supplement recent work on Anaximander in the history of both, philosophy and science. In particular, the role of land and water being "apeiros" is related to the depiction of the boundary regions on Anaximander's world map (one, if not the, first occidental world map).

By the same token, an evaluation and qualification is provided of claims concerning the "apeiron" being the first philosophical and scientific concept in occidental history; a claim recently made prominent by the eminent physicist Carlo Rovelli in his book on Anaximander (title: The First Scientist). I will argue that, rather than being the first philosophical and scientific concept itself, the "apeiron" is an important forerunner for the subsequent introduction of theoretical entities. The notion of the "apeiron" paved the way, as it were, for introducing concepts and entities which traverse the bounds of what is accessible in a direct and exhaustible fashion. Thus, the "apeiron" is surely an important forerunner also of the modern concept of infinity, but only in this wider sense.

Slides:

In his 1922 article, "Neubegründung der Mathematik", David Hilbert characterized Frege's and Dedekind's earlier projects in the foundations of mathematics as attempts to "explain the finite in terms of the infinite". He found these attempts daring and dazzling, but, because of the set-theoretic antinomies, judged them to be failures in the end. In this talk, I will reconsider the case of Dedekind. However, I will not look at Dedekind's contributions from a usual foundational perspective (at least not primarily). Instead, I will focus on the ways in which he, more than Frege and even, arguably, more than Cantor, brought the infinite into standard mathematical practice. Doing so will involve making connections between Dedekind's writings on the natural and real numbers, on the one hand, and his other mathematical work, especially his work on algebraic number theory and Galois theory, on the other hand. More specifically, I will build on a recent remark by Akihiro Kanamori that it was in the latter work where the infinite first entered mainstream mathematical in a serious way. In addition, I will take the word 'explanation' in Hilbert's remark about Frege and Dedekind seriously, by clarifying what might be meant by it in this context and by emphasizing its philosophical significance. Indeed, I will argue that, if it is understood in a relevant sense, Dedekind's explanatory project was a success after all, despite the antinomies. Overall, the discussion is meant to illustrate that, besides the usual foundational (set-theoretic and proof-theoretic) and metaphysical (realist or nominalist) aspects, there is another dimensions to the use of the infinite in mathematics that deserves attention, especially in the context of a "philosophy of mathematical practice".

It has been often noted that the notion of infinity that we find in pre-19th century sources is vastly different from the notion that we inherit from Cantor and that is tacit in the axioms of ZFC set theory. For instance, this pre-Cantorian notion seems to validate the principle that a proper subset of the natural numbers should be smaller than the natural numbers themselves, a principle which the Cantorian theory of cardinality happily violates. If one had to isolate a distinctive feature of the Cantorian conception of the infinite, it might well be the feature that we can apply induction to ordinal and cardinal numbers. However, induction was not the only feature of the natural numbers which was transferred to the infinite: for, in Cantor's works and practice, we find that the notion of recursion was simultaneously transferred to the infinite.

While much in the extant literature on the philosophy of set theory has attended to the idea of a well-order inherent in the Cantorian conception of the infinite, less attention has been paid to the notion of recursion, and this talk (and associated paper) seeks to fill this gap in the literature. To be sure, part of this lack of attention may be traceable to the fact that the iterative conception of set is sometimes thought not to justify reflection and replacement, both of which are used crucially in the standard proof of transfinite recursion. Our guiding questions in this talk are thus: what notions of recursion and computation are available for Cantor's infinite collections, and what philosophical principles about the notion of set can we avail ourselves of to validate these principles?

Part of the difficulty in answering these questions resides in the comparative opacity of the conception of computation even in the ordinary setting of the natural numbers. There we have a confluence of different models (Turing programs, general recursive functions, etc.), but this confluence does not seemingly deliver an axiomatic conception of computation in the same way that the Peano axioms deliver an axiomatic conception of number. Despite this, I suggest that some light can be shed on these conceptual questions by asking after the pre-theoretic ideas implicit in recent work by Hamkins, Koepke, and Welch on infinite time and tape Turing machines. So, in this talk and the associated paper, I proceed by contrasting these pre-theoretic ideas to those behind Kreisel and Sacks' work on higher recursion theory and the ideas arguably behind Jensen's notion of a rudimentary function.

In the Physics, Aristotle draws a distinction between actual and potential infinity. This distinction is introduced as a sort of "yes-and-no" answer to the question "Does the infinite exist?", and there is a hint in Aristotle's discussion that he regards this answer as perhaps a little misleading. It is nonetheless a distinction that he routinely uses, and it seems to be at work in his own analysis of several super-task paradoxes. Concentrating on the discussion in the Physics, the first part of this paper considers the relevance of the distinction between actual and potential infinity to Aristotle's analysis of super-task paradoxes. A second part of the paper investigates the extent to which Aristotle's own analysis anticipates contemporary debates.

The main problems intuitionistic mathematics deals with, the conceptual problems from which it also originates, are essentially the problem of how to formally represent the continuum by means of finite resources without deceiving its nature, and the problem of finding out how to (mathematically) reason about infinite entities, namely entities whose construction cannot be completed and whose representation is necessarily finite.

The main tools L.E.J. Brouwer (the father of intuitionistic mathematics) devised for facing such problems are the notions of "spread", the notion of "choice sequence", and the principle of reasoning called "continuity principle".

Spreads are collections of choice sequences, choice sequences (elements of spreads) are infinite sequences of natural numbers, and the continuity principle states a way of reasoning about elements of a spread (namely about choice sequences) by considering only their initial (and therefore finite) segments.

Spreads and choice sequences formally represent by means of finite resources infinite entities (infinite sets, infinite sequences of natural numbers). The continuity principle is the tool which allows us to make mathematical reasonings about infinite entities by considering their finite representation.

In my proposed talk, after having introduced the basic notions and methods of Brouwerian set theory (the notions of spread and choice sequence, and the continuity principle), I will explore the problem of counting /enumerating elements of a spread under the rule of the contintuity principle. In particular I will show how the classical notion of "infinite countable / denumerable set" intuitionistically splits into several non-equivalent notions. After having offered some examples of how intuitionistic set theoretical operations (such as union) over Brouwerian infinite collections have different properties from the classical case, I will propose a uniform, simple definition of intuitionistic "infinite denumerable set".

• L.E.J. Brouwer, "Begründung der Mengenlehre unabhängig vom logischen Satz vom ausgeschlossenen Dritten." Erster Teil: "Allgemeine Mengenlehre." Koninklijke Nederlandse Akademie van Wetenschappen. Verhandeling, 1e sectie 12, no. 5, 43 p., 1918.
• A. Heyting, De Telbaarheidspraedicaten van Prof. Brouwer, Voordracht, gehouden voor het Wiskundig Genootschap, 1929.
• G. Ronzitti, "On the cardinality of a spread", Ph.D thesis, University of Genoa, 2002.
• W. Veldman, "Analytic, co-analytic and projective sets from Brouwer's intuitionistic perspective", Radboud University Nijmegen, Preprint.

Slides:

Following the observation that the informal infinitesimal calculus used historically in mathematics and to date in physics, produces constructive results, the question arises how can infinitary objects like infinitesimals produce constructive or concrete results? We identify a property of the aforementioned infinitesimal calculus, called \(\Omega\)-invariance, and show that it exactly corresponds to Turing computability. Intuitively speaking, an object is \(\Omega\)-invariant if it is independent of the choice of infinitesimal used in its definition. In other words, as long as the latter property is respected, any (in general infinitary and ideal) construction involving infinitesimals corresponds to a computable one. Thus, \(\Omega\)-invariance constitutes a 'computable' link between the infinite and the finite.

This paper focuses on philosophical work on infinity in the early modern period. In Two New Sciences, Galileo presents an argument whose conclusion is that the natural numbers are both equinumerous with and more than the even natural numbers. This came to be known as "Galileo's Paradox". Galileo himself proposed a solution which rejected Archimedes' Axiom that the whole is greater than its part in the case of infinite collections. Later, Leibniz proposed an alternative solution to the paradox that preserved Archimedes' Axiom and instead rejected the coherence or intelligibility of the notion of an infinite collection or an infinite whole. I summarize this dialectic and then discuss how a Cartesian treatment of infinity, grounded in Descartes' distinction between the infinite (which applies to God alone, "a real thing, which is incomparably greater than all those which are in some way limited") and the indefinite (which applies to, e.g., the divisibility of matter of the size of possible bodies), illuminates the core idea behind Leibniz's solution. The result, I suggest, is an important division between two very different approaches to paradoxes of infinity. Whereas Galileo's solution, and the Cantor-Dedekind approach it anticipates, adopts a logico-mathematical perspective on infinity, focusing on formal properties of collections (or "sets"), the Cartesian-Leibnizian solution turns on a distinctly metaphysical perspective on infinity. Bringing this division into greater focus might enable us to come to terms with assumptions underwriting the former, logico-mathematical perspective that subsequent work on infinity has largely taken for granted.

John Wallis's Arithmetica Infinitorum(1665) is one of the landmarks in the history of mathematics. He was one of the first to work on the generalization of exponents to include negative and fractional as well as positive and integral numbers. His legacy contains discussion of conics as curves of second-degree polynomials rather than as sections of a cone. Wallis was also one of the first to accept negative numbers and to recognize them as roots of equations. His Arithmetic of Infinitesimals certainly influenced Newton's work on calculus and the binomial theorem. In this paper I analyze the connection between Wallis's work on infinite series in Arithmetica Infinitorum and Newton's method of first and last ratios, with an eye towards the plethora of infinitesimal methods that circulated at the same time. If Newton extended Wallis's work, where can we identify this influence and where was Newton's specific contribution materialized? More importantly, is Newton's method of first and last ratios simply an extension of Wallis's method that includes infinite series? In this paper I argue that Newton did not simply extended Wallis's method. In particular, I will show that Wallis seems to have been unaware that his method of induction as it is used in infinite summation is prone to some problems and it makes it inapplicable to the cases of infinite products. This becomes clearer if we have a look at Newton's careful formulations and proofs. I have two specific examples of infinite series in mind: the case of infinite summation and the case of infinite product (the latter being used for calculating \(\pi\)). I argue that while the former can be approached by a certain method of induction similar to Wallis's, the latter use of infinite product is quite an bold move on Wallis's part. In Principia and in his 1676 letters to Oldenburgh and Leibniz, Newton seems aware of the difficulties raised by such examples. It becomes clear that Newton departs from the Archimedean understanding of an infinite series (as a shorthand for the sequence of finite summations) and approaches the notion of convergence, both in the geometrical uses and in Newton's Binomial series.

• Galluzi, Massimo. 2010. Newton's attempt to construct a unitary view of mathematics. Historia Mathematica, 2010, Vol.37(3), pp.535-562
• Mancosu, P. (ed) 2008. Philosophy of Mathematical Practice, Oxford University Press, UK.
• Mancosu, P. 1996. Philosophy of mathematics and mathematical practice in seventeenth century, New York: Oxford University Press.
• Newton, I. The Principia. Mathematical Principles of Natural Philosophy. Translation by I.B. Cohen and Anne Whitman, preceded by A guide to Newton's Principia by I.B. Cohen, 1999, University of California Press.
• Pourciau, B. 1998. The Preliminary Mathematical Lemmas of Newton's Principia. Archive for History of Exact Sciences, 1998, Vol.52(3), pp.279-295
• Pourciau, B. 2001. Newton and the Notion of Limit. Historia Mathematica, 2001, Vol.28(1), pp.18-30
• Scott, J.F. The mathematical work of John Wallis. Chapter IV–Arithmetica Infinitorum, pp. 26-65.
• Selles, Manuel. 2006. Infinitesimals in the foundations of Newton's Mechanics Historia Mathematica, 2006, Vol.33(2), pp.210-223
• Stedall, Jacqueline. The Arithmetic of Infinitesimals, John Wallis 1656
• Whiteside (ed.), D.T. The Mathematical Papers of Isaac Newton, Volumes I–VIII, Cambridge (England): Cambridge University Press, 1967–1981.

In 1748, Euler initiated a new approach to the "analysis of the infinite" studying functions through their representations by analytic means. These representations included infinite series and products, thereby shifting the focus of analysis from the study of curves to the study of "functions" defined through these means. Euler's approach proved very successful for the eighteenth century, but beginning in the 1820s doubts and objections began to emerge in the form of Cauchy's ban of arguments by "the generality of algebra", and of divergent series in particular. This was emphasized by Abel's observation, that certain behavior familiar for finite numbers of operations ceased to obtain when infinite series were considered. Thus, the intuition so skillfully mastered by Euler was seriously questioned. This development can be said to have come to a head when Weierstrass in 1872 presented his example of an every-where continuous, no-where differentiable function which thus defied basic intuitions about the connections between curves and functions. Importantly, Weierstrass' example was defined through an infinite series.

In this paper, I present key aspects of the development of intuitions about functions and curves between Euler and Weierstrass. Special emphasis is given to the reflections about the permissibility of drawing inferences from the finite domain into infinite operations such as series and products. In so doing, I provide both an overview of the standard narrative of rigorization of analysis as well as a historical framework for critically discussing recent cognitive-historical analyses of Abel's exception to an important theorem by Cauchy and of Weierstrass' monster function. This will lead to a brief discussion about the relative merits of cognitive-historical analyses in the sense of Núñez et al. as compared to more traditional history of mathematics.

Infinity plays a crucial role in several parts of Aquinas' metaphysics and theology. God is called "infinite", created things can be infinite "secundum quid", and some causal chains can "proceed to the infinite", while others cannot. In this talk, I will investigate the conceptual network in which Aquinas embeds his "infinity"-talk. Can one make sense of his talk of "essential infinity" of God? How does God's "essential infinity" relate to the quantitative concept of infinity? Is there any relation or is "infinity", in the end, merely a label for several distinct concepts?

My concern in this paper is with the role played by infinities in the realm of science. As it is well known, infinity enters science in two ways: indirectly, by sneaking in with classical mathematics (scientists are not shy in using the infinite involving mathematical apparatus whenever they need it, as long as the end result is physically meaningful, of course), and more directly by appearing in some models of phenomena (e.g., infinitely deep liquids, infinite plane waves, semi-infinite bodies of elastic material, systems containing an infinite number of particles – in thermodynamics, for example –, infinite interactions – in quantum field theory, for example, we have such a thing between an electron and its own field –, in the case of singularities in the description of black holes; in economics we have models with a single, infinitely-lived agent (representative-agent models) and models with infinitely many periods / agents (overlapping-generations model), etc.) An interesting issue that draws our attention if we look closely at these examples is this: how can we use such infinity involving mathematical models in trying to get a better understanding of some physically finite systems? This problem can be taken as a good starting point in an argument for the mysteriousness of the applicability of mathematics in science. An interesting way of dealing with this problem comes from the finitists. From a strict finitist perspective, "the mysterious applicability of mathematics can be handled in a radically different way. In a slogan: finite world, finite mathematics, end of problem." (Van Bendegem 2012: 148) This answer might work well in the case of the first way infinity enters science, but it is far from dealing well with infinity involving scientific models. My aim in this paper is twofold. First of all I will argue that infinite models don't generate applicability problems any more than the use of usual ideal models. Second of all, linking all this with the discussion surrounding the role of models in science, I will argue that infinity involving models are not explanatory but they do nonetheless provide understanding of physical phenomena. If I am right, then the following finitist claim falls: "in trying to get a better understanding of the world, a finite, empirical framework practically suggests itself as the most suited approach" (Van Bendegem 1987: 10)

Research in mathematics education indicates that in the transition from given systems to wider ones learners tend to attribute the properties that hold for the former to the latter. In particular, it has been found that, in the context of Cantorian Set Theory, learners tend to attribute properties of finite sets to infinite ones — using methods which are acceptable for finite sets, to infinite ones.

In the conference we shall present various didactical attempts to deal with the specific "peculiarities" of infinite sets. We shall describe several activities addressing intuitive and formal aspects of infinite sets, and discuss their impact on the learners' ways of thinking about infinite sets.

Slides:

Those familiar with the history of logic and grammar will know the distinction of words into categorematic and syncategorematic words, that is, words which have signification or meaning in isolation from other words (such as nouns, pronouns, verbs) and those which have signification only when combined with other words (such as conjunctions, quantifiers, and articles). Some words, however, are both categorematic and syncategorematic, and in medieval Latin one such word is infinitum 'infinity'. Many attempts have been made to identify the categorematic and syncategorematic uses of infinitum with the Aristotelian actual and potential infinites, but this, in our opinion, rather misses the point. The distinction between syncategorematic and categorematic is not a distinction between types of infinity but rather one concerning the use of the word infinitum. Infinitum and its cognates can be used either categorematically or syncategorematically, and conflating the two uses gives rise to paralogisms and fallacies.

In this paper, we look at how western European logicians in 13th- and early 14th-century identified paralogisms arising from conflation of the two uses of infinitum, and the rules that the introduced to prevent such fallacies. We take our material from two types of texts: Treatises on syncategorematic ters, such as those by Peter of Spain, William of Sherwood, and Nicholas of Paris, and treatises on sophismata (sophisms), such as those by William Heytesbury and Richard Kilvington. The former provide a theoretical basis for propositions with syncategorematic words (Peter of Spain notes that "Truth or falsity is caused in a proposition by syncategorematic words" Tractatus Syncategorematum, trans. J.P. Mullally, p. 17), while the latter show how these theoretical principles are applied to concrete problems. We also see, in the latter treatises, how the analysis of syncategorematic and categorematic uses of infinitum relates to theories of so-called "exponible terms".

Slides:

Alexander Yessenin-Volpin is usually mentioned as the "founding father" of strict finitist thinking (or ultra-intuitionism in his words), sometimes in combination with the "later" Wittgenstein. All too often a critique and rejection of the former's approach is seen as a rejection of strict finitism altogether. From the historical point of view, I will try to show that rather David Van Dantzig merits the title (if such needs to be awarded). Since then, on and off, contributions have been made to proposals for a strict finitist mathematics, sometimes in relation to physics. Although this position is (still) quite marginal within the broad field of foundational studies of mathematics, I will defend that, at least from the philosophical point of view, strict finitism is an interesting topic as it invites us to reflect on what mathematics could be all about and especially what its boundaries can be.

Slides:

It is well-known that there are natural mathematical statements that cannot be settled by the standard axioms of set theory. The Zermelo-Fraenkel axioms, with the Axiom of Choice (ZFC), are incomplete. In light of the incompleteness phenomenon, this paper explores an alternative approach to set theory that couples a naive conception of set with a notion of mathematical modality.

The ZFC axioms were proposed as an alternative to the naive conception of set, which takes an unrestricted comprehension axiom as its sole set existence principle: for any condition \(P\), there exists a set of all and only those things that satisfy the condition \(P\). According to this principle, there exists a set of all and only red things, there exists a set of all and only round things, etc. It is well-known that the naive conception of set is inconsistent. In response to the inconsistency of the naive conception, and to the extensive incompleteness of the ZFC axioms, we argue in favor of a modal conception of set. According to this conception, sets are merely possible, or potential, with respect to their members. The basic set existence principle that we articulate is a modal version of the unrestricted comprehension axiom: for any condition \(P\), it is possible that there exists a set of all and only those things that satisfy the condition \(P\).

We extend the language of set theory with modal operators. When formalizing modal notions, it is most common to extend the object language with two modal operators, the familiar \(\square\) for necessity and \(\lozenge\) for possibility. The object language we use has four modal operators: \(\square\), \(\lozenge\), \(\blacksquare\), and \(\blacklozenge\). These four modal operators let us scan the universe of sets in different directions: the familiar \(\square\) and \(\lozenge\) look out to the higher reaches of the set-theoretic universe, while \(\blacksquare\) and \(\blacklozenge\) track backwards to previous stages in the set-formation process.

Given this modal language of set theory, we explore the following basic proof-theoretic question: What can be derived from a modal version of the unrestricted comprehension axiom?

Modal Unrestricted Comprehension: \(\lozenge \exists y \forall x [x \in y \leftrightarrow \blacklozenge P(x)].\)

We build on contemporary work in modal set theory, which, though fruitful, has not been explored to its full potential. Early investigations in modal set theory include Fine (1981) and Parsons (1983b). Current work on modal set theory is also being done by Linnebo (2010, 2011) and Studd (forthcoming). The current project differs from these in several ways. The most significant difference concerns the proof-theoretic strength of the modal logic involved. Linnebo, Parsons, and Studd hone in on the modal logic S4.2 as the most appropriate logic for modal set theory, while Fine deploys the modal logic S5. We argue that the modal logic of set theory need not be as strong as either of these. We develop a modal set theory using a weaker modal logic, relying on the strength of the modal unrestricted comprehension axiom to minimize incompleteness.

• Fine, K. (1981). First order modal theories i: Sets. Nous 15, 177–205.
• Linnebo, O. (2010). Pluralities and sets. The Journal of Philosophy 107, 144–164.
• Linnebo, O. (2011). The potential hierarchy of sets. unpublished manuscript.
• Parsons, C. (1983a). Mathematics in Philosophy: Selected Essays. Ithaca, NY: Cornell University Press.
• Parsons, C. (1983b). Sets and modality. In Parsons (1983a), pp. 298–341.
• Studd, J. P. (forthcoming). The iterative conception of sets: A (bi-)modal axiomatisation. Journal of Philosophical Logic.

According to EIp18 of Spinoza's Ethics, "God is the immanent, not the transitive, cause of all things." This not only grounds his ontology, but ensures his methodology and why human nature is able to form an adequate idea of the absolute. Nevertheless, in EIp28, we come across a puzzle that the infinite, insofar as it is eternal, cannot act upon that which is finite and temporal. So how can it be that God remains the immanent cause of finite things? Gilles Deleuze offers an interesting interpretation in his text on Spinoza, Expressionism in Philosophy where he follows Spinoza's famous "Letter on the Infinite." Spinoza, here, makes four distinctions of the infinite, as follows: 1) absolutely, 2) through itself, 3) in virtue of its cause and 4) what is infinitely divisible (indefinite). The first two refer to substance and attribute, respectively, while the latter two refer to finite modes, 3) in reference to finite essences and 4) to finite existent objects. These two elements are not allowed to interact upon one another in Spinoza's ontology. Thus, it is my intention to explore, specifically, how Deleuze interprets the attributes of substance as necessarily expressing themselves in a bifurcated movement toward finite essences and finite existent objects, both obedient to the same laws but the former as comprehended within the divine attribute and the latter as "outside" it, though only perceptually so.