What is Neologicism?

Linsky, Bernard; Zalta, Edward N.

doi:10.2178/bsl/1140640944

Cited by 23 publications

(8 citation statements)

References 37 publications

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“…One of the important questions in this context is the following: "What can be considered a rigorous mathematical proof?" There are several approaches to the foundations of mathematics (see, for example, [19][20][21]).…”

Section: Foundations Of Mathematics and Recognition-explicit And Implicit Definitionsmentioning

confidence: 99%

The Brain and the New Foundations of Mathematics

Melkikh

2021

Symmetry

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Many concepts in mathematics are not fully defined, and their properties are implicit, which leads to paradoxes. New foundations of mathematics were formulated based on the concept of innate programs of behavior and thinking. The basic axiom of mathematics is proposed, according to which any mathematical object has a physical carrier. This carrier can store and process only a finite amount of information. As a result of the D-procedure (encoding of any mathematical objects and operations on them in the form of qubits), a mathematical object is digitized. As a consequence, the basis of mathematics is the interaction of brain qubits, which can only implement arithmetic operations on numbers. A proof in mathematics is an algorithm for finding the correct statement from a list of already-existing statements. Some mathematical paradoxes (e.g., Banach–Tarski and Russell) and Smale’s 18th problem are solved by means of the D-procedure. The axiom of choice is a consequence of the equivalence of physical states, the choice among which can be made randomly. The proposed mathematics is constructive in the sense that any mathematical object exists if it is physically realized. The consistency of mathematics is due to directed evolution, which results in effective structures. Computing with qubits is based on the nontrivial quantum effects of biologically important molecules in neurons and the brain.

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Section: Foundations Of Mathematics and Recognition-explicit And Implicit Definitionsmentioning

confidence: 99%

The Brain and the New Foundations of Mathematics

Melkikh

2021

Symmetry

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“…One example is plenitudinous platonism; see Balauger (1998) and Linsky & Zalta (1995, 2006 for two different versions. The central idea is that any mathematical objects which can exist, do exist.…”

Section: Platonism Under the Lensmentioning

confidence: 99%

What the Applicability of Mathematics Says About Its Philosophy

Wilson

2018

Philosophy of Engineering and Technology

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But we have to talk about something, so in what follows I present some of the main ideas from the long history of this vaguely-defined area of philosophy. This is not an exhaustive study of all of the schools of the philosophy of mathematics, neither will we see all of the main areas of study. Those in the know might find it shocking that I do not mention Descartes, Locke, Berkeley, or Wittgenstein, and spend scant time on Kant and Hume. Their ideas fill these pages through their influence on their contemporaries and those who came after them and on whose ideas I focus. And while I try to present some historical development, this can only ever be cursory in a single chapter covering over 2,500 years from Pythagoras to the present. I am painfully aware of the Western bias in my presentation, with no mention of the great Indian, Chinese, and Arabic traditions. I hope that you are intrigued enough to follow the references. If you are eager to start right now, then Bostock (2009) gives a highly readable and comprehensive introduction, Benacerraf & Putnam (1983) contains selected key papers and readings, Horsten (2016) is an excellent starting point for an educational internet journey, and Mancosu (2008) is a survey of the modern perspective. But I hope you will read this chapter first. The chapter divides the philosophy of mathematics into four schools, each of which has its own section. This division is broadly accepted and historically relevant, but not without controversy. I have also tried to present the arguments of smaller subschools of the philosophy of mathematics. Sometimes this has required discussing a subschool when a theme arises, even if historically it does not belong in that section. I hope that historians of the philosophy of mathematics, and the philosophers themselves, will forgive me. Mostly I have tried to avoid jargon, but there are some important concepts that I have tried to develop as they arise. However, there are two words needed from the start: ontology and epistemology. Ontology concerns the nature of being. In terms of mathematics: what do we mean when we say that a mathematical object exists? Are mathematical objects pure and outside of space and time, as the platonist insists, or are they purely mental, as the intuitionist would argue, or the fairy tales of the fictionalist? Epistemology concerns the nature of knowledge, how we can come to have it, and what justifies our belief in it. Speaking loosely, we can say that if ontology is concerns the nature of what we know, then epistemology concerns how we know it.

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“…It may be worthwhile to compare our logicism with neologicism (for an introduction to neologicism see (13)). We just sketch here the most relevant for our purposes.…”

Section: The Logicist Flavor Of the Theory Of Logical Conceptsmentioning

confidence: 99%

A Notion of Logical Concept Based on Plural Reference

Carrara

Martino

2017

Acta Anal

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In To be is to be the object of a possible act of choice (6) the authors defended Boolos' thesis that plural quantification is part of logic. To this purpose, plural quantification was explained in terms of plural reference, and a semantics of plural acts of choice, performed by an ideal team of agents, was introduced. In this paper, following that approach, we develop a theory of concepts that -in a sense to be explained -can be labelled as a theory of logical concepts. Within this theory we propose a new logicist approach to natural numbers. Then, we compare our logicism with Frege's traditional logicism.

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What is Neologicism?

Cited by 23 publications

References 37 publications

The Brain and the New Foundations of Mathematics

The Brain and the New Foundations of Mathematics

What the Applicability of Mathematics Says About Its Philosophy

A Notion of Logical Concept Based on Plural Reference

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