A prominent version of emotional cognitivism is the view that emotions are preceded by awareness of value. In a recent paper, Jonathan Mitchell (2019) has attacked this view (which he calls the content-priority view). According to him, extant suggestions for the relevant type of pre-emotional evaluative awareness are all problematic. Unless these problems can be overcome, he argues, the view does not represent a plausible competitor to rivaling cognitivist views. As Mitchell supposes, the view is not mandatory since its core (...) motivations can be accommodated by competing views, too. I argue that Mitchell's case against the content-priority view is unconvincing as it misconceives the principal motivation for the view. As I show, properly reconstructed, this motivation provides a strong case for the indispensability of the view to any adequate cognitivist treatment of emotion. Moreover, Mitchell's survey of candidates for pre-emotional value awareness can be seen to rest on problematic phenomenological assumptions. (shrink)

I examine Bayle's infamous statement that Christian mysteries are not only "above" human reason, but are "against" it. I put it back in the context of 16th-17th century Reformed thought. I then discuss the relation between reason and faith according to Bayle.

I first explain the scholastic (Scotist) thesis on the independence of essences Bayle alludes to in the passage of the Continuation des Pensée Diverses where he presents the Stratonicians' and the Chinese philosophers' retorsion. Then, I show that this retorsion applies to the argument of the existence of God based on "aseity", but not to the occasionalist argument based on the "quod nescis" principle. I conclude that materialism (the "Stratonician hypothesis") cannot be, for Bayle, a satisfying system.

Ludwig Feuerbach is famous for his critical hermeneutics of religion. At the heart of it lie arguments of philosophical anthropology that directly anticipate contemporary developments in the theory of recognition. He counts amongst the great philosophers who, immediately following Kant, emphasised the constitutive importance for human beings of interpersonal and social relations. Indeed, his theory of intersubjectivity contains features that are highly original, notably the link between individual and community, and between recognition and recollection.

This summary is not exactly the way I would have done it myself but I must admit that my writing is sometimes a challenge to read. So I asked an AI to do this summary expecting that it will give an easily understandable although not totally accurate view on Ontology of Knowledge and from this general understanding help the reader to read the original papers. Jean-Louis Boucon’s works, "Introduction to the Ontology of Knowledge" and "Time, Space, and World as (...) Knowledge," delve into the intricate nature of reality, meaning, and the process of knowing. Boucon challenges traditional dichotomies in philosophy and science, presenting a cohesive framework where the knowing subject and the world are intertwined aspects of a single dynamic reality. His ideas provide a robust foundation for exploring ontological and epistemological questions, including the interpretation of quantum mechanics. (shrink)

Est-il possible de parler de ce qui n’est pas ou d’y penser sans présupposer une forme d’être pour cela même que nous pensons ne pas exister? La vieille énigme parménidienne, qui hante toujours la philosophie contemporaine, est au cœur non seulement de la philosophie médiévale mais aussi des études médiévales, comme en témoigne le récent ouvrage d’Alain de Libera sur la référence vide. L’objet de cette étude est en comparaison beaucoup plus...

There is a wide range of things we do out of emotion. For example, we smile with pleasure, our voices drop when we are sad, we recoil in shock or jump for joy, we apologize to others out of remorse. It is uncontroversial that some of these behaviors are actions. Clearly, apologizing is an action if anything is. Things seem less clear in the case of other emotional behaviors. Intuitively, the drop in a sad person’s voice is something that happens (...) to her, rather than something she actively performs. Perhaps more interestingly, even jumping for joy can seem a problematic case: although its execution involves the active performance of certain movements, it has been argued to contrast, e.g., with an act of apology, in that it is not performed in order to achieve some end, such as repairing a relationship. This can make this behavior seem considerably different from paradigm actions. Our central concern in this paper is with which emotional behaviors should be classed as actions and why... (shrink)

According to a well-known view in the philosophy of mind, intentional attitudes by their very nature satisfy requirements of rationality (e.g. Davidson 1980; Dennett 1987; Millar 2004). This view (which I shall call Constitutivism) features prominently as the ‘principle of minimal rationality’ in de Sousa’s monograph The Rationality of Emotion (1987). By explicating this principle in terms of the notion of the formal object of an attitude, de Sousa articulates an interesting and original version of Constitutivism, which differs in important (...) respects from other extant versions. In this paper, I explore this version of Constitutivism against the background of recent developments in the theory of rationality and make explicit its ramifications for the long-standing dispute over whether the mind is essentially normative. My focus will be on how to conceive of the form of the rationality requirements that attitudes as such must satisfy according to this principle. I argue that, although de Sousa seems officially to endorse a structuralist conception of rationality, according to which these requirements are requirements of coherence, his considerations on formal objects suggest that they are more aptly conceived in terms of a reasons-responsive conception of rationality. I further argue that which of these two readings we choose makes a significant difference to the prospect of vindicating the essential normativity of mind by invoking the principle of minimal rationality. (shrink)

Physical reality is all the reality we have, and so physical theory in the standard sense is all the ontology we need. This, at least, was an assumption taken almost universally for granted by the advocates of exact philosophy for much of the present century. Every event, it was held, is a physical event, and all structure in reality is physical structure. The grip of this assumption has perhaps been gradually weakened in recent years as far as the sciences of (...) mind are concerned. When it comes to the sciences of external reality, however, it continues to hold sway, so that contemporary philosophers B even while devoting vast amounts of attention to the language we use in describing the world of everyday experience B still refuse to see this world as being itself a proper object of theoretical concern. Here, however, we shall argue that the usual conception of physical reality as constituting a unique bedrock of objectivity reflects a rather archaic view as to the nature of physics itself and is in fact incompatible with the development of the discipline since Newton. More specifically, we shall seek to show that the world of qualitative structures, for example of colour and sound, or the commonsense world of coloured and sounding things, can be treated scientifically (ontologically) on its own terms, and that such a treatment can help us better to understand the structures both of physical reality and of cognition. (shrink)

The text “The Ontology of Knowledge, Logic, Arithmetic, Set Theory, and Geometry” by Jean-Louis Boucon explores a deeply philosophical interpretation of knowledge, its logical structure, and the foundational elements of mathematical and scientific reasoning. -/- Here’s an overview condensed by an AI of the key themes and ideas, summarized into a quite general conceptual structure. These two pages are instructive on their own, but their main purpose is to facilitate the reading of the entire article, allowing the reader to (...) situate in the coherence of the whole both the meaning and the usefulness of each proposed idea. (shrink)

Contrary to what many interpretations claim, according to Bayle faith does not completely eliminate reason. It intervenes to reveal factual truths that can only be known through revelation (for example, that God allowed Adam and Eve to sin). To these factual truths can be applied a rational principle (an axiomatic and evident one, according to Bayle, which he calls a "common notion"), namely, that "what God does is well done." God allowed sin, so we must think it was justified, even (...) if we don't understand why. This approach is what I call, cum grano salis, Pierre Bayle's theodicy. (shrink)

The paper challenges the assumption, common amongst philosophers, that the reality described in the fundamental theories of microphysics is all the reality we have. It will be argued that this assumption is in fact incompatible with the nature of such theories. It will be shown further that the macro-world of three-dimensional bodies and of such qualitative structures as colour and sound can be treated scientifically on its own terms, which is to say not only from the perspective of psychology but (...) also ontologically. A new sort of emergentist position will be defended, one which yields the basis of a method for describing the perceptually salient macroscopic world in mathematical terms. Broadly, it will be argued that the macroscopic world exists in virtue of certain specific sorts of boundary-patterns in the field of what is captured by the theories of microphysics. (shrink)

This paper shows that the NMA eluding the knowing subject would be nothing more than a profession of faith. Then it re-introduces the subject with the help of Ontology of Knowledge Using the example of the theory of Relativity it shows the difficulties resulting from the elision of the subject and propose a solution to include the subject into the NMA.

First recorded in the 14th century, the analogy of spit-roast meat argues that expecting the Sun to rotate around a strictly immobile Earth would be just as ludicrous as trying to move the fire around the roasting meat. On the contrary, it should be the Earth that spins upon itself in order to glean, from all possible angles, all the benefits of the Sun, just as it is the meat’s responsibility to turn on the spit before the motionless fire for (...) it to be perfectly cooked on all sides. Aimed at demonstrating, in geocentricism, the plausibility of the rotation of the Earth and, in heliocentrism, at supporting the physical reality of this terrestrial rotation, nowadays this analogy barely elicits more than an amused, or even condescending, smile. However, its continued long-term existence and its frequency of use — from the 14th to the 19th century, we found it mentioned by no less than 45 different authors — incited us to finally pay attention to it. We then noticed that beyond its explicitly cosmological scope, it raises, in its own way, the question of the purpose of the natural world: had the latter been conceived based upon humankind, then it is normal, despite what this analogy advocates, that the Sun is at the service of something more important than itself by rotating, to the profit of humankind, around the Earth; conversely, since such an infringement of the rule of common sense, as illustrated by this analogy, can neither be justified nor tolerated, it is up to the Earth to move around the Sun. Tracing the vicissitudes of this analogy, over 6 centuries, is thus not only about reconstructing the largely forgotten history of a presently obsolete analogy, but also about discovering the deeper meaning, henceforth incomprehensible, behind it. In describing how this meaning became progressively lost, one can finally pro­vide an understanding of why, in the present day, we are no longer capable of more than an amused smile when happening upon it! (shrink)

Le réalisme scientifique occupe une place centrale dans le système philosophique de Mario Bunge. Au cœur de cette thèse, on trouve l’affirmation selon laquelle nous pouvons connaître le monde partiellement. Il s’ensuit que les théories scientifiques ne sont pas totalement vraies ou totalement fausses, mais plutôt partiellement vraies et partiellement fausses. Ces énoncés sur la connaissance scientifique, à première vue plausible pour quiconque est familier avec la pratique scientifique, demandent néanmoins à être clarifiés, précisés et, ultimement, à être inclus dans (...) un cadre théorique plus large et rigoureux. Depuis ses toutes premières publications sur ces questions et jusqu’à récemment, Mario Bunge n’a cessé d’interpeller les philosophes afin qu’ils développent une théorie, au sens propre du terme, de la vérité partielle afin de clarifier les enjeux épistémologiques liés au réalisme scientifique. Bunge a lui-même proposé plusieurs parties de cette théorie au fil des années, mais aucune de ces propositions ne l’a satisfait pleinement et la construction de cette théorie demeure un problème entier. Dans ce texte, nous passerons rapidement en revue certaines des approches proposées par Bunge dans ses publications et nous esquisserons certaines pistes qui devraient servir à tout le moins de desiderata pour la construction d’une théorie de la vérité partielle. (shrink)

Structuralism has recently moved center stage in philosophy of mathematics. One of the issues discussed is the underlying logic of mathematical structuralism. In this paper, I want to look at the dual question, namely the underlying structures of logic. Indeed, from a mathematical structuralist standpoint, it makes perfect sense to try to identify the abstract structures underlying logic. We claim that one answer to this question is provided by categorical logic. In fact, we claim that the latter can be seen—and (...) probably should be seen—as being a structuralist approach to logic and it is from this angle that categorical logic is best understood. (shrink)

This thesis provides an account of acquaintance with abstract objects. The notion of acquaintance is integral to theorising on reference and singular thought, since it is generally taken to be the relation that must exist between a subject and an object, in order for the subject to refer to, and entertain singular thoughts about the object. The most common way of understanding acquaintance is as a form of causal connection. However, this implies a problem. We seem to be able to (...) refer and have singular thoughts about abstract objects. But given that abstract objects are causally inert, this would mean that we are unable to become acquainted with them. This problem shall be the focus of this thesis. I first argue that these traditional causal interpretations of acquaintance are lacking. Instead, I show that acquaintance is dependent to some degree on factors internal to the subject, namely the skills that they possess. From doctors to sommeliers to mathematicians (and possibly even philosophers!) – these subjects seem to succeed in becoming acquainted with certain objects precisely in virtue of their respective skills. Thus, building off from Evans’ The Varieties of Reference, I present a novel account of acquaintance, which I term as Skill-based Acquaintance (SBA). On SBA, a subject is said to be acquainted with an object when they possess discriminating knowledge of that object, gained through the use of their capacities and skills. The SBA account is applied to virtual (Ch.3), fictional (Ch.4), and mathematical objects (Ch.5), as well as God (Ch.6). SBA is successful in explaining how subjects can indeed become acquainted with these problematic categories of objects - some of which are abstract – thus being able to refer and entertain singular thoughts about them. Overall, then, SBA is shown to have greater explanatory power than competing accounts and should thus be preferred. (shrink)

Knowledge of the subject only ever speaks to the subject himself and of nothing more than the subject himself. And yet, when due to their mode of emergence, objects of this knowledge seem to occupy part of a space (whatever it may be) which would contain them, it is tempting for the subject to give this space the name of Reality. The conventional western philosophies (and science) gave in to this temptation. They came to distinguish the Subject {place-of-the-mind} from the (...) Reality {place-of-the-objects} whereas Reality is only the "objectified" or rather "objectivized" aspect of knowledge of the subject. From this artificial distinction followed a great confusion, a dispute that lasted for centuries, and still does, between different -isms with infinitesimal differences, from which the most influential parties emerge victorious each at his turn. In reaction to this confusion a "pivotal" re-orientation is suggested, considering the way objects of knowledge emerge and are organized, not in correlation with objects of Reality but with the subject for himself and knowledge for itself. For this re-orientation the OK proposes to transcend the subject's mind to reveal its logical principles. That's the specific subject I would like to delve into here : the transcendence of meaning. This is necessary for understanding the OK and could be beneficial to “mental health counseling”. (shrink)

Bringing together two aspects of Thomas Aquinas's thought that have been studied separately: his theory of God's concurrence and his theory of instrumental causality, I show how he uses the latter (which I discuss first) to clarify the Proclusian principle that the first cause has a greater influence on an effect than the proximate causes. Thanks to this theory, Aquinas accounts for the fact that it is God who confers existence to every new being that is produced by natural processes, (...) without however negating the agency of secondary, created causes. (shrink)

In this chapter, I explore my affective engagement with Charlotte Wood’s The Weekend (2019). Adopting definitions that reveal the nested hierarchies of feeling, affect, and emotion, I situate emotion as a semantic experience within the framework of thought, arguing that thought itself is an affectual process that carries meaning. Cognition, in other words, is an affective process. Thought’s affectual status is often overlooked, however, with the focus on its semantic content drawing attention from this; yet meaning affects us, and this (...) is the function of thought as affect: it organises experience in ways that are, in turn, affecting. My approach to Wood’s novel aims to emphasise this and find firmer ground on which to perceive emotion as a kind of thought, noting that reading stimulates thinking in terms of grammatically established points of view. (shrink)

In this paper, I examine the problem of Bayle's skepticism. I show that he is not a wholesale skeptic. Rather, he thinks that reason is plagued by internal conflicts. But its principles, which clash with each other, can be adopted separately from each other. It is often what we have to do when dealing with metaphysical problems. This also entails that reason is not to be rejected as a whole when it happens to be contradicted by faith; only some of (...) its principles have to be denied, and solely in that case. Further, Bayle believes that we possess absolute certainly in our knowledge of ethical principles. I show how this is compatible with his relativism in other domains by putting back his contention in the context of Reformed thought. (shrink)

Anne TIHON, Théorie et réalité : l’exemple de l’astronomie an­cienne (pp. 7-23) ; Isabelle DRAELANTS, Les encyclopédies com­me sommes des connaissances, d’Isidore de Séville au XIIIe siè­cle, avec les fondements antiques (pp. 25-50) ; Andrée COLINET, Alchimie antique et médiévale avant 1300 : mystères et réalités (pp. 51-70) ; Baudouin VAN DEN ABEELE, Quelques pas de grue à travers l’histoire naturelle médiévale : un regard diversifié sur le réel (pp. 71-98) ; Régine LEURQUIN, L’astrolabe plan (pp. 99- 117) ; Patricia (...) RADELET-DE GRAVE, Copernic, Stevin, Galilée et la réalité des orbites célestes (pp. 119-151) ; Brigitte VAN TIGGELEN, Étiqueter ou définir : le réalisme dans la nomen­cla­ture chimique aux XVIIe et XVIIIe siècles (pp. 153-183) ; Michel GHINS, L’existence de l’espace et du temps selon Leon­hard Euler (pp. 185-193) ; Chantal TILMANS-CABIAUX, Le rap­port de l’idée théorique à l’expérimentation chez Hahnemann : le système homéo­pathique naissant et le réalisme (pp. 195-214) ; Jean MAWHIN, La Terre tourne-t-elle ? À propos de la philosophie scientifique de Poincaré (pp. 215-252) ; Michel WILLEM, Paul Lévy et les fondements du calcul des probabilités (pp. 253-263). (shrink)

ROUDAUT (François) & STOFFEL (Jean-François), Avant-propos (pp. 371-372). BLAY (Michel), Soleil de Copernic et soleil de Galilée (pp. 373-389). MEHL (Édouard), Le Soleil au cinquième concile du Latran (1512-1516) : Copernic et la réforme du calendrier julien (pp. 391-408). STOFFEL (Jean-François), «Qui choisirait de poser ce flambeau dans un lieu autre ou meilleur que celui d’où il peut illuminer le tout simultanément ?»: examen de la pertinence d’un argument copernicien de convenance (pp. 409-458). ROUDAUT (François), Le Soleil chez (...) quelques lexicographes et vulgarisateurs (pp. 459-485). CÉARD (Jean), Le Soleil selon les physiciens de la Renaissance (pp. 487-498). MÉNAGER (Daniel), Le soleil dans les fêtes de cour sous les derniers Valois (pp. 499-510). SEIDENGART (Jean), La parenté du Soleil et des étoiles fixes dans la cosmologie de Giordano Bruno (pp. 511-529). ŠPELDA (Daniel), Les soleils et leurs observateurs au XVIIe siècle (pp. 531-577). MELLINGHOFF-BOURGERIE (Viviane), Connivences héliologiques entre théologie et astronomie chez François de Sales et Bérulle: le cas du barnabite Redento Baranzano (pp. 579-610). NEJESCHLEBA (Tomáš), De la métaphysique de la lumière à l’héliocentrisme: la vision du Soleil selon Valérien Magni (pp. 611-627). Les auteurs (pp. 629-630). (shrink)

Supra-Bayesianism is the Bayesian response to learning the opinions of others. Probability pooling constitutes an alternative response. One natural question is whether there are cases where probability pooling gives the supra-Bayesian result. This has been called the problem of Bayes-compatibility for pooling functions. It is known that in a common prior setting, under standard assumptions, linear pooling cannot be nontrivially Bayes-compatible. We show by contrast that geometric pooling can be nontrivially Bayes-compatible. Indeed, we show that, under certain assumptions, geometric and (...) Bayes-compatible pooling are equivalent. Granting supra-Bayesianism its usual normative status, one upshot of our study is thus that, in a certain class of epistemic contexts, geometric pooling enjoys a normative advantage over linear pooling as a social learning mechanism. We discuss the philosophical ramifications of this advantage, which we show to be robust to variations in our statement of the Bayes-compatibility problem. (shrink)

The paper reviewed financial resources which is a critical factor and component of Business Process Re-engineering in achieving academic performance of higher education institutions in the Democratic Republic of Congo. The objective of the study is to examine whether financial resources can contribute to improving and achieving academic performance of higher education institutions in general and students in particular. The study used a systematic literature review and content analysis was to establish the relationship between financial resources and academic performance. The (...) key findings are grouped in three dimensions: 1) financial resources; 2) budget and funding/financial model; 3) tuitions fees, scholarship and loans provision. The study found consistent evidence to show that there is a significant and positive relationship between financial resources and performance of higher education institutions. The study concluded that higher education sector in the Democratic Republic of Congo needs a radical change and fundamental rethinking in improving educational budget processes to meet the needs of the institutions, provision of scholarship and loans processes to students mainly those from poor background and designing innovative financial/funding model to sustain educational institutions. The study is based on academic capitalism theory. The recommendations were made based on the findings of the study. -/- . (shrink)

Successful biomedical data mining and information extraction require a complete picture of biological phenomena such as genes, biological processes, and diseases; as these exist on different levels of granularity. To realize this goal, several freely available heterogeneous databases as well as proprietary structured datasets have to be integrated into a single global customizable scheme. We will present a tool to integrate different biological data sources by mapping them to a proprietary biomedical ontology that has been developed for the purposes of (...) making computers understand medical natural language. (shrink)

In this paper, following the claims made by various mathematicians, I try to construct a theory of levels of abstraction. I first try to clarify the basic components of the abstract method as it developed in the first quarter of the 20th century. I then submit an explication of the notion of levels of abstraction. In the final section, I briefly explore some of main philosophical consequences of the theory.

Tradução do livro: Sociologias da alimentação: os comedores e o espaço social alimentar. Florianópolis, Editora da UFSC, 2004, 311 páginas. Obra traduzida juntamente com Rossana Pacheco da Costa Proença e Carmen Silvia Rial. ISBN 85.328.0290-7. [Título original: Sociologies de l'alimentation: Les mangeurs et l'espace social alimentair. Obra publicada originalmente pela PUF, Paris, 2002. Edição original 288 páginas. ISBN: 213050643. Janeiro 2003, 2e édition. Etnologia-Antropologia].

It is popular to hold that emotions are evaluative. On the standard account, the evaluative character of emotion is understood in epistemic terms: emotions apprehend or make us aware of value properties. As this account is commonly elaborated, emotions are experiences with evaluative intentional content. In this paper, I am concerned with a recent alternative proposal on how emotions afford awareness of value. This proposal does not ascribe evaluative content to emotions, but instead conceives of them as evaluative at the (...) level of intentional mode or attitude. I first argue that this proposal fails to make emotions intelligible as value apprehensions. There are reasons to suppose that emotions do not apprehend value to begin with, but are related to values in a different, non-epistemic sense. I then go on to show that the notion of an evaluative intentional mode can still help elucidate the evaluative character of emotion. I argue that there is a plausible non-epistemic understanding of the view that emotions are evaluative modes. On this account, emotions are not ways of apprehending values, but ways of acknowledging values. (shrink)

On entend généralement par « théorie des modèles » autant la métamathématique (ou sémantique formelle) que la sémantique des modèles des sciences non formelles. Cet article a pour objet la théorie des modèles scientifiques que Mario Bunge a développée dans Method, Models and Matter (1973). J’y analyse l’intégration théorique qu’opère Bunge des sciences formelles et des sciences expérimentales ou observationnelles, laquelle prend appui sur sa philosophie des sciences. Je la compare sommairement à la théorie des modèles de Gilles-Gaston Granger dans (...) le but évident d’en dégager les ressemblances et les dissimilitudes, mais aussi leur commun point d’achoppement : l’une comme l’autre usent en effet d’un concept non analysé dont la fonction épistémologique est pourtant capitale et produit les mêmes effets. Au centre de la théorie des modèles de Bunge se trouve le concept de simulation que je comparerai à celui qui est en usage dans les sciences de l’ordinateur et qui est de nos jours largement appliqué à diverses sciences, tant sociales que naturelles. Je conclurai sur les conséquences méthodologiques et métaphysiques de la théorie bungéenne des modèles. (shrink)

Worldwide populations are aging with economic development as a result of public health initiatives and advances in therapeutic discoveries. Since 1850, life expectancy has advanced by 1 year for every four. Accompanying this change is the rapid development of anti‐aging science. There are three schools of thought in the field of aging science. One perspective is the life course approach, which considers that aging is a good and natural process to be embraced as a necessary and positive aspect of life, (...) where the aim is to improve the quality of existing lifespan and “compress” morbidity. Another view is that aging is undesirable, and that rejuvenation and indeed immortality are possible since the biological basis of aging is understood, and therefore, strategies are possible for engineering negligible senescence. Finally, a hybrid approach is that life span can be extended by anti‐aging medicines but with uncertain effects on health. While these advances offer much promise, the ethical perspectives are seldom discussed in cross‐disciplinary settings. This article discusses some of the key ethical issues arising from recent advances in biogerontology. (shrink)

In this paper, we argue that, contrary to the view held by most philosophers of mathematics, Bourbaki’s technical conception of mathematical structuralism is relevant to philosophy of mathematics. In fact, we believe that Bourbaki has captured the core of any mathematical structuralism.

One of the key features of modern mathematics is the adoption of the abstract method. Our goal in this paper is to propose an explication of that method that is rooted in the history of the subject.

Categorical foundations and set-theoretical foundations are sometimes presented as alternative foundational schemes. So far, the literature has mostly focused on the weaknesses of the categorical foundations. We want here to concentrate on what we take to be one of its strengths: the explicit identification of so-called canonical maps and their role in mathematics. Canonical maps play a central role in contemporary mathematics and although some are easily defined by set-theoretical tools, they all appear systematically in a categorical framework. The key (...) element here is the systematic nature of these maps in a categorical framework and I suggest that, from that point of view, one can see an architectonic of mathematics emerging clearly. Moreover, they force us to reconsider the nature of mathematical knowledge itself. Thus, to understand certain fundamental aspects of mathematics, category theory is necessary (at least, in the present state of mathematics). (shrink)