Jurado: Mario Gómez-Torrente (presidente), Miguel Ángel Fernández Vargas (vocal), Santiago Echeverri Saldarriaga (secretario). [Graduado con Mención Honorífica.].

Juliette Kennedy’s new book brims with intriguing ideas. I don’t understand all of them, and I’m not convinced that the ones I do understand all fit together, b.

Heinz Von Forester characterizes the objects “known” by an autopoietic system as eigen-solutions, that is, as discrete, separable, stable and composable states of the interaction of the system with its environment. Previous articles have presented the FBST, Full Bayesian Significance Test, as a mathematical formalism specifically designed to access the support for sharp statistical hypotheses, and have shown that these hypotheses correspond, from a constructivist perspective, to systemic eigen-solutions in the practice of science. In this article several issues related to (...) the role played by language in the emergence of eigen-solutions are analyzed. The last sections also explore possible connections with the semiotic theory of Charles Sanders Peirce. (shrink)

This paper puts forward a new account of rigorous mathematical proof and its epistemology. One novel feature is a focus on how the skill of reading and writing valid proofs is learnt, as a way of understanding what validity itself amounts to. The account is used to address two current questions in the literature: that of how mathematicians are so good at resolving disputes about validity, and that of whether rigorous proofs are necessarily formalizable.

According to the simple modal account of essence, an object has a property essentially just in case it has it in every world in which it exists. As many have observed, the simple modal account is implausible for a number of reasons. This has led to various proposals for strengthening the account, for example, by adding a restriction to the intrinsic or sparse properties. I argue, however, that these amendments to the simple modal account themselves fail. Drawing on lessons from (...) these failures, I propose a new version of a modal account, inspired by Ruth Barcan Marcus's defense of the coherence of quantified modal logic, according to which an object has a property essentially just in case (i) it has it in every world in which it exists, (ii) the property is discriminating (or non-trivial), and (iii) the property is qualitative. The resulting account of essence does not face any of the standard objections other accounts face, and I defend it from other potential objections. (shrink)

Penelope Maddy has recently addressed the set-theoretic multiverse, and expressed reservations on its status and merits ([Maddy, 2017]). The purpose of the paper is to examine her concerns, by using the interpretative framework of set-theoretic naturalism. I first distinguish three main forms of 'multiversism', and then I proceed to analyse Maddy's concerns. Among other things, I take into account salient aspects of multiverse-related mathematics , in particular, research programmes in set theory for which the use of the multiverse seems to (...) be crucial, and show how one may provide responses to Maddy's concerns based on a careful analysis of 'multiverse practice'. (shrink)

Approaching Infinity addresses seventeen paradoxes of the infinite, most of which have no generally accepted solutions. The book addresses these paradoxes using a new theory of infinity, which entails that an infinite series is uncompletable when it requires something to possess an infinite intensive magnitude. Along the way, the author addresses the nature of numbers, sets, geometric points, and related matters. The book addresses the need for a theory of infinity, and reviews both old and new theories of infinity. It (...) discussing the purposes of studying infinity and the troubles with traditional approaches to the problem, and concludes by offering a solution to some existing paradoxes. (shrink)

I present a general metaphysical framework for any formal system that works with truth-values. To establish such a framework, I start with the notion of absolute nothingness, from which I construct a nothingness which is akin to the notion of an empty set in mathematics. Then I provide a formal system that its ability to produce symbols is an integral property and an inseparable part of its metaphysics.

ABSTRACTDo truth tables—the ordinary sort that we use in teaching and explaining basic propositional logic—require an assumption of consistency for their construction? In this essay we show that truth tables can be built in a consistency-independent paraconsistent setting, without any appeal to classical logic. This is evidence for a more general claim—that when we write down the orthodox semantic clauses for a logic, whatever logic we presuppose in the background will be the logic that appears in the foreground. Rather than (...) any one logic being privileged, then, on this count partisans across the logical spectrum are in relatively similar dialectical positions. (shrink)

Thoroughgoing relativists typically dismiss the realist conviction that competing theories describe just one definite and mind-independent world-structure on the grounds that such theories fail to be relatively translatable even though they are equally correct. This line of argument allegedly brings relativism into direct conflict with the metaphysics of realism. I argue that this relativist line of reasoning is shaky by deriving a theorem about relativistic inquiry in formal epistemology—more specifically, in the approach Kevin Kelly has dubbed “logic of reliable inquiry”. (...) According to the theorem, two scientists, who share some background knowledge but follow different appropriately reliable methods, will converge to relatively formally translatable competing theories, even if meaning, truth, logic and evidence are allowed to vary in time depending on each scientist’s conjectures, actions, or conceptual choices. Some final remarks on the relevance of the theorem to the incommensurability thesis that has vexed twentieth century philosophy of science are adduced. (shrink)

Axiomatization is uncommon outside mathematics, partly for being often viewed as embalming, partly because the best-known axiomatizations have serious shortcomings, and partly because it has had only one eminent champion, namely David Hilbert. The aims of this paper are to describe what will be called dual axiomatics, for it concerns not just the formalism, but also the meaning of the key concepts; and to suggest that every instance of dual axiomatics presupposes some philosophical view or other. To illustrate these points, (...) a theory of solidarity will be crafted and axiomatized, and certain controversies in both classical and quantum physics, as well as in the philosophy of mind, will be briefly discussed. The upshot of this paper is that dual axiomatics, unlike the purely formal axiomatics favored by the structuralists school, is not a luxury but a tool helping resolve some scientific controversies. (shrink)

In this work we discuss logical structures related to indistinguishable particles. Most of the framework used to develop these structures was presented in [17, 28] and in [20, 14, 15, 16]. We use these structures and constructions to discuss possible ontologies for identical particles. In other words, we use these structures in order to characterize the logical structure of quantum systems for the case of indistinguishable particles, and draw possible philosophical implications. We also review some proposals available in the literature (...) which may be considered within the framework of the quantum logical tradition regarding the problem of indistinguishability. Besides these discussions and constructions, we advance novel technical results, namely, a lattice theoretical structure for identical particles for the finite dimensional case. This kind of approach was not present in the scarcely literature of quantum logic and indistinguishable particles. (shrink)

It is usually stated that quantum mechanics presents problems with the identity of particles, the most radical position—supported by E. Schrödinger—asserting that elementary particles are not individuals. But the subject goes deeper, and it is even possible to obtain states with an undefined particle number. In this work we present a set theoretical framework for the description of undefined particle number states in quantum mechanics which provides a precise logical meaning for this notion. This construction goes in the line of (...) solving a problem posed by Y. Manin, namely, to incorporate quantum mechanical notions at the foundations of mathematics. We also show that our system is capable of representing quantum superpositions. (shrink)

A multifactorial and growing crisis of health care systems in the developed world has affected medicine. In order to provide rational responses, some central concepts of the past, such as the definitions of health and disease, need to be updated. For this purpose physicians should initiate a new debate. As a point of departure the following definitions are proposed: Health is a dynamic state of wellbeing characterized by a physical, mental and social potential, which satisfies the demands of a life (...) commensurate with age, culture, and personal responsibility. If the potential is insufficient to satisfy these demands the state is disease. This term includes sickness, illness, ill health, and malady. The described potential is divided into a biologically given and a personally acquired partial potential. Their proportions vary throughout the life cycle. The proposed definitions render it empirically possible to diagnose persons as healthy or diseased and to apportion some of the responsibility for their state of health to individuals themselves. Treatment strategies should always consider three therapeutic routes: improvements of the biologically given and of the personally acquired partial potentials and adaptations of the demands of life. These consequences favourably contrast with those resulting from the WHO-definition of health. (shrink)

Quasi-set theory has been proposed as a means of handling collections of indiscernible objects. Although the most direct application of the theory is quantum physics, it can be seen per se as a non-classical logic (a non-reflexive logic). In this paper we revise and correct some aspects of quasi-set theory as presented in [12], so as to avoid some misunderstandings and possible misinterpretations about the results achieved by the theory. Some further ideas with regard to quantum field theory are also (...) advanced in this paper. (shrink)

Today, in any mature information society, we live neither online nor offline but on life, that is, we increasingly live in that special space that is both analogue and digital, both online and offline. Imagine someone asking whether the water is fresh or salty in the estuary where the river meets the sea. That someone has not understood the special nature of the place. Our information society is that place. And our technologies are perfectly evolved to take advantage of it, (...) like mangroves growing in brackish water. In the mangrove society, machine-readable data, new forms of agency and interactions are constantly evolving. The consequences of such a radical transformation are many, but one is particularly significant and rich in consequences: how should we design the mangrove society, when it comes to the relationship between the natural and the digital? In this text, I argue that the digital blue is a great ally of the environmental green. The two work best in tandem. And if the world will enjoy a sustainable development for the environment that is also socially preferable for humanity, it will be because the green and the blue will have created a virtuous circle of nature and technology. It can be done. There is just no time to waste. (shrink)

Megethology is the second-order theory of the part-whole relation developed by David Lewis, and it is obtained by combining plural quantification with classical extensional mereology. It can express some hypotheses about the size of the domain such as that there are inaccessibly many atoms. This will prove enough to get the orthodox set theory. Then, megethology is a possible foundation for mathematics. This paper is an introduction to megethology.

A dilemma has stymied interpretations of the Stranger’s method of dividing kinds into subkinds in Plato’sSophistandStatesman. The dilemma assumes that the kinds are either extensions or intensions. Now kinds divide like extensions, not intensions. But extensions cannot explain the distinct identities of kinds that possess the very same members. We propose understanding a kind as like an animal body—the Stranger’s simile for division—possessing both an extension and an intension. We find textual support in the Stranger’s paradigmatic four steps for collecting (...) a subkind. (shrink)

I present 3 arguments that the laws of physics, by themselves, cannot explain the origin of phenomenal consciousness. First, physics investigates the objective, while consciousness is subjective. Second, the laws of physics are syntactical, while consciousness is semantic. Third, the concept of consciousness cannot even be defined in terms of physics.

Points of view are a central phenomenon in human cognition. Although the concept of point of view is ambiguous, there exist common elements in different notions. A point of view is a certain way to look at things around us. In conceptual points of view, things are looked at or interpreted through conceptual lenses. Conceptual points of view are important for epistemology, cognitive science, and philosophy of science. In this article, a new method to formalize conceptual points of view is (...) introduced. It is based on the conceptual space approach, where concepts are regions of multi-dimensional quality spaces. Points of view, as defined in this article, consist of a selection of relevant dimensions, referred to here as determinables, and of a certain supposition, referred to here as a theory, about the subject content. After considering some early efforts to formalize points of view, the notion of conceptual space is defined and explored. One concept of point of view is defined and developed in the framework of conceptual space, and a new logic for points of view is also outlined. The problem of mind–body correlation is discussed as an example of the application of the notion of points of view. To conclude, the meaning and applications of the new concepts and tools developed in the article are discussed. (shrink)

The state of affairs of some things falling under a predicate is supposedly a single entity that collects these things as its constituents. But whether we think of a state of affairs as a fact, a proposition or a possibility, problems will arise if we adopt a plural logic. For plural logic says that any plurality include themselves, so whenever there are some things, the state of affairs of their plural self-inclusion should be a single thing that collects them all. (...) This leads to paradoxes analogous to those that afflict naïve set theory. Here I suggest that they are the very same paradoxes, because sets can be reduced to states of affairs. However, to obtain a consistent theoretical reduction we must restrict the usual axiom scheme of Comprehension for plural logic to ‘stratified’ formulas, to avoid viciously circular definitions. I prove that with this modification to the background plural logic, the theory of states of affairs is consistent; moreover, it yields the axioms of the familiar set theory NFU. (shrink)

The initial motivating question for this thesis is what the standard of rigour in modern mathematics amounts to: what makes a proof rigorous, or fail to be rigorous? How is this judged? A new account of rigour is put forward, aiming to go some way to answering these questions. Some benefits of the norm of rigour on this account are discussed. The account is contrasted with other remarks that have been made about mathematical proof and its workings, and is tested (...) and illustrated by considering a case study discussed in the literature. On the view put forward here one can obtain a manner of informal, rigorous mathematics founded on any of a variety of proof systems. The latter part of the thesis is concerned with the question of how we should decide which of these competing proof systems we should base our mathematics on: i.e., the question of which proof system we should take as a foundation for our mathematics. A novel answer to this question is proposed, in which the key property we should require of a proof system is that for as many different kinds of structures as possible, when the proof system allows a generalization about that kind of structure to be proved, the generalization actually holds of all real examples of that kind of structure which exist. This is the requirement of soundness of the proof system. It is argued that the best way to establish the soundness of a proof system may be by giving an interpretation of its axioms on which they are established as true. As preparation for this discussion, the thesis first investigates the logical and conceptual basis of various arithmetic concepts, with the results obtained used in the final discussion of soundness. (shrink)

In science as in mathematics, it is popular to know little and resent much about category theory. Less well known is how common it is to know little and like much about set theory. The set theory of almost all scientists, and even the average mathematician, is fundamentally different from the formal set theory that is contrasted against category theory. The latter two are often opposed by saying one emphasizes Substance, the other Form. However, in all known systems of mathematics (...) throughout history, mathematicians have moved fluidly between ideas conceived of as thing-like, property-like, and process-like. On the other hand one way to advance science is to better distinguish between thing, property, and process. All this constitutes a distracting background for those interested in, or distressed by, the possible application of category theory to science, and to mathematics as well. (shrink)

An important perquisite for verification of the correctness of software is the ability to write mathematically precise documents that can be read by practitioners and advanced users. Without such documents, we won't know what properties we should verify. Tabular expressions, in which predicate expressions may appear, have been found useful for this purpose. We frequently use partial functions in our tabular documentation. Conventional interpretations of expressions that describe predicates are not appropriate for our application because they do not deal with (...) partial functions. Experience with this documentation has led us to choose a logic in which predicates are total but functions remain partial. We have found that this particular interpretation results in simpler expressions and is easily understood by practitioners. (shrink)

It is a commonplace of set theory to say that there is no set of all well-orderings nor a set of all sets. We are implored to accept this due to the threat of paradox and the ensuing descent into unintelligibility. In the absence of promising alternatives, we tend to take up a conservative stance and tow the line: there is no universe. In this paper, I am going to challenge this claim by taking seriously the idea that we can (...) talk about the collection of all the sets and many more collections beyond that. A method of articulating this idea is offered through an indefinitely extending hierarchy of set theories. It is argued that this approach provides a natural extension to ordinary set theory and leaves ordinary mathematical practice untouched. (shrink)

Appeal to standard set theory in minimalist syntax is shown to be in conflict with the goal of analyzing dependency formation, a.k.a. movement, as involving genuine constituent copies. The underlying tension is due to extensionality, which—other things being equal—favors a perspective on dependencies in terms of multidominance. The above argument is developed against the backdrop of a recent exposition of minimalist syntax :229–261, 2019), which can be seen as exemplary. The resulting critical assessment should be taken as removing obstacles on (...) the way toward proper minimalist foundations for syntactic theory. (shrink)

Wittgenstein's conception of infinity can be seen as continuing the tradition of the potential infinite that begins with Aristotle. Transfinite cardinals in set theory might seem to render the potential infinite defunct with the actual infinite now given mathematical legitimacy. But Wittgenstein's remarks on set theory argue that the philosophical notion of the actual infinite remains philosophical and is not given a mathematical status as a result of set theory. The philosophical notion of the actual infinite is not to be (...) found in the mathematics of set theory, only in a certain associated philosophy – what Wittgenstein calls a certain kind of “prose”. (shrink)

Can Quine’s criterion for ontological commitment be comparatively applied across different logics? If so, how? Cross-logical evaluations of discourses are central to contemporary philosophy of mathematics and metaphysics. The focus here is on the influential and important arguments of George Boolos and David Lewis that second-order logic and plural quantification don’t incur additional ontological commitments over and above those incurred by first-order quantifiers. These arguments are challenged by the exhibition of a technical tool—the truncation-model construction of notational equivalents—that compares the (...) ontological role and increased expressive strength of non-first-order ideology to first-order ideology. (shrink)

Wittgenstein was hostile towards set theory; see his remark in §22 of RFM II: ‘I believe and hope that a future generation will laugh at this hocus pocus’. At the same time, he says that what philosophy owes set theory is ‘tremendous’ and that this is something ‘deep’. I want to clarify these two statements and to reconcile them. The hocus‐pocus remark is mainly directed at the temptation to talk about an own world of sets, focussing on extensions and thereby (...) forgetting the actual mathematical manoeuvres leading to them. Wittgenstein does not shy away from using set‐theoretical symbols and expressions, but he regularly deprives them of their genuine set‐theoretical character. When speaking of the ‘depth’ we encounter with regard to set theory, what he means is the depth explained in PI §111: It arises ‘through a misinterpretation of our forms of language’, and this is particularly so in set theory. (shrink)

Humans and organizations have limitations of computational capacity and information management. Such constraints are synonymous with bounded rationality. Therefore, in order to extend the human and organizational boundaries to more advanced models of cognition, this research proposes concepts of cognitive machines in organizations. From a micro point of view, what makes this research distinct is that, beyond people, it includes in the list of participants of the organization the cognitive machines. From a macro point of view, this paper relies on (...) the premise that cognitive machines can improve the cognitive abilities of the organization. From such perspectives, it presents rationale and principles of a class of cognitive machines with capabilities to carry out complex cognitive tasks in organizations. It also introduces analyses of the cognitive machines in organizations through theories of bounded rationality, economic decision-making, and conflict resolution. The analyses indicate that these machines can solve or reduce intra-individual and group dysfunctional conflicts which arise from decision-making processes in the organization, and thus they can improve the degree of organizational cognition. From all these backgrounds, this research outlines implications of cognitive machines for organizations. (shrink)

In this paper, I generalise the logical concept of compatibility into a broader set-theoretical one. The basic idea is that two sets are incompatible if they produce at least one pair of opposite objects under some operation. I formalise opposition as an operation ′ ∶ E → E, where E is the set of opposable elements of our universe U, and I propose some models. From this, I define a relation ℘U × ℘U × ℘U^℘U, which has (mutual) logical compatibility (...) as its more natural interpretation. (shrink)

Beginning from some passages by Vācaspati Miśra and Bhāskararāya Makhin discussing the relationship between a crown and the gold of which it is made, this paper investigates the complex underlying connections among difference, non-difference, coreferentiality, and qualification qua relations. Methodologically, philological care is paired with formal logical analysis on the basis of ‘Navya-Nyāya Formal Language’ premises and an axiomatic set theory-based approach. This study is intended as the first step of a broader investigation dedicated to analysing causation and transformation in (...) non-difference. (shrink)

In science as in mathematics, it is popular to know little and resent much about category theory. Less well known is how common it is to know little and like much about set theory. The set theory of almost all scientists, and even the average mathematician, is fundamentally different from the formal set theory that is contrasted against category theory. The latter two are often opposed by saying one emphasizes Substance, the other Form. However, in all known systems of mathematics (...) throughout history, mathematicians have moved fluidly between ideas conceived of as thing-like, property-like, and process-like. On the other hand one way to advance science is to better distinguish between thing, property, and process. All this constitutes a distracting background for those interested in, or distressed by, the possible application of category theory to science, and to mathematics as well. (shrink)

In economically meaningful interactions negotiations are particularly important because they allow agents to improve their information about the environment and even to change accordingly their own characteristics. In each step of a negotiation an agent has to emit a message. This message conveys information about her preferences and endowments. Given that the information she uses to decide which message to emit comes from beliefs generated in previous stages of the negotiation, she has to cope with the uncertainty associated with them. (...) The assessment of the states of the world also evolves during the negotiation. In this paper we analyze the intertwined dynamics of beliefs and decision, in order to determine conditions on the agents that allow them to reach agreements. The framework for decision making we consider here is based on defeasible evaluation of possibilities: an argument for a choice defeats another one if it is based on a computation that better uses all the available information. (shrink)

I construe the question Are species sets? as a question about whether species can be conceived of as sets, as the term set is understood by contemporary logicians. The question is distinct from the question Are species classes?: The conception of classes invoked by Hull and others differs from the logician's conception of a set. I argue that species can be conceived of as sets, insofar as one could identify a set with any given species and that identification would satisfy (...) three desiderata: the set would be a set of organisms, the identification would be apposite, and the identification would permit the formulation of statements about species in set-theoretic terms. One cannot, however, identify a species with any given set. Understanding the claim that species are sets in this way enables one to understand better the dispute between some who accept the claim (e.g., Kitcher 1984, 1987) and some who apparently reject it (e.g., Mayr 1987). (shrink)

This paper examines the proposition that adopting a socially responsible, or philanthropic, management posture is not antithetic to the capitalist business model but rather can be seen as a sound approach to the development of long-term sustainability in business in a modern business environment, wherein a strand of corporate social responsibility is one core aspect of the composite utility function of the modern business. We suggest further that for many of the prominent/significant examples of the successful adoption of a policy (...) of informed benevolence in the management of employees that policy seems to be faith rooted – be that faith Christian, Buddhist, Jewish, Muslim or other. Indeed, even if one observed an organisation subject to an humanist or even atheistic based socially responsible management, one could see that as being rooted in that organisation’s ‘non-faith’ faith, or ethical bubble. Our argument also gives the lie to the proposition, in currency in some circles, that the term ‘business ethics’ is an oxymoron. To underpin the argument we cite examples (/set out mini-cases, or a mixture of these) from a range of faith based traditions including, Buddhism, the Quaker-Christian tradition, Judaism and Islam. (shrink)

In this work, formulas are inclusions \ and non-inclusions \ between Boolean terms \ and \. We present a set of rules through which one can transform a term t in a diagram \ and, consequently, each inclusion \ ) in an inclusion \ ) between diagrams. Also, by applying the rules just to the diagrams we are able to solve the problem of verifying if a formula \ is consequence of a, possibly empty, set \ of formulas taken as (...) hypotheses. Our system has a diagrammatic language based on Venn diagrams that are read as sets, and not as statements about sets, as usual. We present syntax and semantics of the diagrammatic language, define a set of rules for proving consequence, and prove that our set of rules is strongly sound and complete in the following sense: given a set \ of formulas, \ is a consequence of \ iff there is a proof of this fact that is based only on the rules of the system and involves only diagrams associated to \ and to the members of \. (shrink)

The method of approximate reasoning using a fuzzy logic introduced by Baldwin (1978 a,b,c), is used to model human reasoning in the resolution of two well known paradoxes. It is shown how classical propositional logic fails to resolve the paradoxes, how multiple valued logic partially succeeds and that a satisfactory resolution is obtained with fuzzy logic. The problem of precise representation of vague concepts is considered in the light of the results obtained.

This thesis comprises three main chapters—each comprising one relatively standalone paper. The unifying theme is fragmentalism about truth, which is the view that the predicate “true” either expresses distinct concepts or expresses distinct properties. -/- In Chapter 1, I provide a formal development of alethic pluralism. Pluralism is the view that there are distinct truth properties associated with distinct domains of subject matter, where a truth property satisfies certain truth-characterizing principles. On behalf of pluralists, I propose an account of logic (...) and semantics that shows how they can answer central conceptual and logical challenges for their view. -/- In Chapter 2, I motivate and develop a modal account of propositions on the basis of an iterative conception of propositions, where the modality is logico-mathematical. The modal account of propositions takes the conception to motivate an inherently potential hierarchy of propositions. I show that the account helps provide satisfying solutions to the intensional paradoxes of Russell-Myhill, Kaplan, and Prior. -/- In Chapter 3, I propose that “true” is polysemous. I suggest that “true” is initially polysemous between correspondence truth and disquotational truth, and further polysemous between the meanings corresponding to the subconcepts of the concept truth generated by the indefinite extensibility of that concept. I show that the proposal provides satisfying solutions to the semantic paradoxes. (shrink)

definitions and explanations frequently come together and permeate almost all fields of knowledge. This does not exclude mathematics, even when these definitions hold clear links and close connections with our physical world. Here we propose a rather different perspective. Making operational physical assumptions, we show how it is possible to rigorously reconstruct some features of both geometry and topology. Broadly speaking, assuming this operational and more concrete philosophy we not only are capable of defining primitive concepts like points, straight lines, (...) planes, angles and spaces, but we also go all the way up to more complex constructions. We show that within our method it is also possible to come up with what we call a compass-based topology as well as a normed and a metric space. We hope this operationalist point-of-view can assist other researchers and students to have a clearer understanding of these quite abstract concepts. (shrink)

This is the first, out of two papers, devoted to Andrzej Grzegorczyk’s point-free system of topology from Grzegorczyk :228–235, 1960. https://doi.org/10.1007/BF00485101). His system was one of the very first fully fledged axiomatizations of topology based on the notions of region, parthood and separation. Its peculiar and interesting feature is the definition of point, whose intention is to grasp our geometrical intuitions of points as systems of shrinking regions of space. In this part we analyze separation structures and Grzegorczyk structures, and (...) establish their properties which will be useful in the sequel. We prove that in the class of Urysohn spaces with countable chain condition, to every topologically interpreted representative of a point in the sense of Grzegorczyk’s corresponds exactly one point of a space. We also demonstrate that Tychonoff first-countable spaces give rise to complete Grzegorczyk structures. The results established below will be used in the second part devoted to points and topological spaces. (shrink)