Fighting With Infinity: A Proposal for The Addition of New Terminology (original) (raw)
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Educational Studies in Mathematics, 2005
This paper applies APOS Theory to suggest a new explanation of how people might think about the concept of infinity. We propose cognitive explanations, and in some cases resolutions, of various dichotomies, paradoxes, and mathematical problems involving the concept of infinity. These explanations are expressed in terms of the mental mechanisms of interiorization and encapsulation. Our purpose for providing a cognitive perspective is that issues involving the infinite have been and continue to be a source of interest, of controversy, and of student difficulty. We provide a cognitive analysis of these issues as a contribution to the discussion. In this paper, Part 1, we focus on dichotomies and paradoxes and, in Part 2, we will discuss the notion of an infinite process and certain mathematical issues related to the concept of infinity. KEY WORDS: actual and potential infinity, APOS Theory, classical paradoxes of the infinite, encapsulation, history of mathematics, human conceptions of the infinite, large finite sets
Some historical issues and paradoxes regarding the concept of infinity: An APOS analysis: Part 2
Educational Studies in …, 2005
This is Part 2 of a two-part study of how APOS theory may be used to provide cognitive explanations of how students and mathematicians might think about the concept of infinity. We discuss infinite processes, describe how the mental mechanisms of interiorization and encapsulation can be used to conceive of an infinite process as a completed totality, explain the relationship between infinite processes and the objects that may result from them, and apply our analyses to certain mathematical issues related to infinity.
The Philosophical Implications of Set Theory in Infinity
What does the term " Infinity " mean? There are mathematical, physical and metaphysical definitions of the concept of limitlessness. This study will focus on the scription of the three philosophical foundations of mathematics – formalism, intuitionism and logicism – in set theory. Examples will also be provided of the concept of infinity for these three schools of thought. However, none of them cannot prove whether there is an infinite set or the existence of infinity. It forms the foundational crisis of mathematics. Further elaboration on these schools of philosophy leads to the ideas of actual, potential and absolute boundlessness. These correspond to three basic aforementioned definitions of infinity. Indeed for example, by using Basic Metaphor Infinity, cognitive mechanisms such as conceptual metaphors and aspects, one can appreciate the transfinite cardinals' beauty fully (Nũńez, 2005). This implies the portraiture for endless is anthropomorphic. In other words, because there is a connection between art and mathematics through infinity, one can enjoy the elegance of boundlessness (Maor, 1986). Actually, in essence this is what mathematics is: the science of researching the limitless.
The Resilience of Formal Infinitism Through Puzzles & Paradoxes
2022
Is infinity possible, or inescapably problematic? The advancement of commerce, agriculture, engineering, physics, and the life sciences all rely on classical continuous mathematics that incontrovertibly assumes the possibility of the infinite, and yet infinity leads to paradoxes. The finitist position attempts to resolve such problems by denying the possibility of infinity altogether, and therefore must embrace a research program aimed at rebuilding the foundations of mathematics to be discrete and finite, rather than continuous and infinite. The infinitist position grants the possibility of infinity, and therefore must embrace a research program aimed at resolving, by one means or another, the paradoxes that result. The present work reflects an effort to contribute to the infinitist research program. The paradoxes that present challenges to infinitism are the direct subjects of active research in their own right, and also underlie other broader controversies in the philosophy of mathematics, metaphysics, and the philosophy of religion. Plenty of good work has been done to reconstruct historically significant paradoxes, trace their development over time, and exposit standard resolutions where such exist. Other efforts develop and defend each author's unique approach to a narrow research question. Although such work can be elucidatory, much of it neglects critical developments in set theory or else entirely rests on axiomatic foundations that ultimately beg the salient philosophical questions. This work attempts to defend infinitism while avoiding both trends.
Naive Infinitism: The Case for an Inconsistency Approach to Infinite Collections
Notre Dame Journal of Formal Logic, 2015
This paper expands upon a way in which we might rationally doubt that there are multiple sizes of infinity. The argument draws its inspiration from recent work in the philosophy of truth and philosophy of set theory. More specifically, elements of contextualist theories of truth and multiverse accounts of set theory are brought together in an effort make sense of Cantor's troubling theorem. The resultant theory provides an alternative philosophical perspective on the transfinite, but has limited impact on everyday mathematical practice. This relativity of cardinalities is very striking evidence of how far abstract formalistic set theory is removed from all that is intuitive. One can indeed construct systems that faithfully represent set theory down to the last detail. But as soon as one applies the finer instruments of investigation all this fades away to nothing. Of all the cardinalities only the finite ones and the denumerable one remain. Only these have real meaning; every thing else is formalistic fiction. (von Neumann, 1967) The aim of this paper is to take seriously the idea that we have, in some sense, misunderstood the message of Cantor's theorem; or at the least, that in hindsight we have driven headlong into the transfinite when we could have paused a moment longer to consider an alternative. My goal is to demonstrate that Cantor's theorem can be understood more like the liar paradox, as a kind of fork in the road. The crucial idea is that, in admitting there are multiple sizes of infinity, we have done irreparable damage to our naïve conception of the infinite. My goal in this paper is to demonstrate that we may coherently reject the multiplicity of infinite cardinalities and to illustrate the value of this perspective. 1
Infinite as an ontological problem
Proceedings NUMTA conference 2019
Since the birth of philosophy in ancient Greece, the concept of infinite has been closely linked with that of contradiction and, more precisely, with the intellectual effort to overcome contradictions present in an account of Totality as fully grounded. The present work illustrates the ontological and epistemological nature of the paradoxes of the infinite, focusing on the theoretical framework of Aristotle, Kant and Hegel. Interestingly, Aristotle solves the dilemmas of the infinite by hypothesizing a finite universe, by denying the existence of "actual" infinite and by supposing a non-dimensional entity which is at the origin of any movement and alteration of matter and which is beyond empirical verification: such a move cannot be embraced by modern philosophy and by modern mathematics, for different reasons. The work also attempts to realize an ontological interpretation of the necessity of mathematical instruments such as limits and infinitesimals and of their coexistence with paradoxes of the infinites and infinitesimals. Recent conceptual mathematical solutions to these paradoxes, such as Sergeyev's notion of gross-one, are then compared with Hegel's notion of true infinite in their pragmatic significance.
I consider two longstanding paradoxes and how their resolution is central to contemporary reasoning about infinity, and conclude that the foundational ideas supporting contemporary reasoning about infinity cannot be consistently employed to talk about infinities of differing sizes, in the sense of Cantorian cardinalities.
Is There an Ontology of Infinity?
Foundations of Science, 2020
In this article I try to articulate a defensible argumentation against the idea of an ontology of infinity. My position is phenomenologically motivated and in this virtue strongly influenced by the Husserlian reduction of the ontological being to a process of subjective constitution within the immanence of consciousness. However taking into account the historical charge and the depth of the question of infinity over the centuries I also include a brief review of the platonic and aristotelian views and also those of Locke and Hume on the concept to the extent that they are relevant to my own discussion of infinity both in a purely philosophical and epistemological context. Concerning the latter context, I argue against Kanamori's position, in The Infinite as Method in Set Theory and Mathematics, that the mathematical infinite can be accounted for solely in terms of epistemological articulation, that is, in the way it is approached, assimilated, and applied in the course of the construction of mathematical hierarchies. Instead I point to a subjectively constituted immanent 'infinity' in virtue of the a priori as well as factual characteristics of subjective constitution, underlying and conditioning any talk of infinity in an epistemological sense. From this viewpoint I also address some other positions on the question of a possible ontology of the mathematical infinite. My whole approach to the question of the infinite in an epistemological sense hinges on the assumption that the mathematical infinite subsumes the infinite of physical theories to the extent that physics and science in general deal with the infinite in terms of the corresponding mathematical language and the specific techniques involved.
Mathematical discourse does not appear to be syntactically or semantically different in kind from discourse about other subject-matters. Just as we use singular terms like -Evelyn‖ and -that cat‖ in [1] and [3] [1] Evelyn is prim.
The Infinite as Method in Set Theory and Mathematics
Resumen. El infinito como método en la teoría de conjuntos y la matemática Este artículo da cuenta de la aparición histórica de lo infinito en la teoría de conjuntos, y de cómo lo tratamos dentro y fuera de las matemáticas. La primera sección analiza el surgimiento de lo infinito como una cuestión de método en la teoría de conjuntos. La segunda sección analiza el infinito dentro y fuera de las matemáticas, y cómo deben adoptarse.