The ultimate tactics of self-referential systems

Mathematics is usually regarded as a kind of language. The essential behavior of physical phenomena can be expressed by mathematical laws, providing descriptions and predictions. In the present essay I argue that, although mathematics can be seen, in a first approach, as a language, it goes beyond this concept. I conjecture that mathematics presents two extreme features, denoted here by {\sl irreducibility} and {\sl insaturation}, representing delimiters for self-referentiality. These features are then related to physical laws by realizing that nature is a self-referential system obeying bounds similar to those respected by mathematics. Self-referential systems can only be autonomous entities by a kind of metabolism that provides and sustains such an autonomy. A rational mind, able of consciousness, is a manifestation of the self-referentiality of the Universe. Hence mathematics is here proposed to go beyond language by actually representing the most fundamental existence condition for self-referentiality. This idea is synthesized in the form of a principle, namely, that {\sl mathematics is the ultimate tactics of self-referential systems to mimic themselves}. That is, well beyond an effective language to express the physical world, mathematics uncovers a deep manifestation of the autonomous nature of the Universe, wherein the human brain is but an instance.Comment: 9 pages. This essay received the 4th. Prize in the 2015 FQXi essay contest: "Trick or Truth: the Mysterious Connection Between Physics and Mathematics

How Can Mathematical Objects Be Real but Mind-Dependent?

Taking mathematics as a language based on empirical experience, I argue for an account of mathematics in which its objects are abstracta that describe and communicate the structure of reality based on some of our ancestral interactions with their environment. I argue that mathematics as a language is mostly invented, and it is mind-dependent in a specific sense. However, the bases of mathematics will characterize it as a real, non-fictional science of structures

Mathematical Communication: What And How To Develop It In Mathematics Learning?

Mathematics is the language of symbols so that everyone who studied mathematics required having the ability to communicate using the language of these symbols. Mathematical communication skills will make a person could use mathematics for its own sake as well as others, so that will increase positive attitudes towards mathematics. Mathematical communication skills can support mathematical abilities, such as problem solving skills. With good communication skills then the problem will more quickly be represented correctly and this will support in solving problems. Students' mathematical communication skills can be developed in various ways, one with group discussions. Brenner (1998) found that the formation of small groups facilitate the development of mathematical communication skills. This paper describes the mathematical communication and how to develop the mathematical communication skills in learning mathematics. For further clarify the discussion, given also the example of learning that emphasizes the development of mathematical communication skills. Keywords: Mathematical Communication, Mathematics Learning

A reappraisal of online mathematics teaching using LaTeX

The mathematics language LaTeX is often seen outside of academic circles as a legacy technology that is awkward to use. MathML - a verbose language designed for data-exchange, and to be written and understood by machines - is sometimes by contrast seen as something that will aid online mathematics and lack of browser support for it bemoaned. However LaTeX can already do many of the things that MathML might promise. LaTeX is here proposed as a language from which small fragments, with concise syntax, can be used by people to easily create and share mathematical expressions online. The capability to embed fragments of LaTeX code in online discussions is described here and its impact on a group of educators and learners evaluated. Here LaTeX is posited as a useful tool for facilitating asynchronous, online, collaborative learning of mathematics

Kripkenstein from the mathematical point of view: a preliminary survey

This paper deals with the problem of the impact of Kripke’s skeptical paradox on the philosophy of mathematics. By perceiving mathematics as a huge rule-following discipline, one could argue that the Kripkean nonfactualist thesis should be adopted within the philosophy of mathematics en bloc to imply a refutation of objectivity and an enforcement of a particular view on the nature of mathematics. In this paper I will discuss this claim. According to Kripke’s skeptical solution we should reject the notion of fact and adopt the use theory of meaning that could be stated as follows: ’One understands the concepts embodied in a language to the extent that one knows how to use the language correctly.’ [Shapiro 1991, 211] [Kripke 1982]. Focusing on mathematical discourse, we should ask: what are the implications of the use theory of meaning for the philosophy of mathematics? Furthermore, is the answer to the skeptical paradox consistent with selected views in philosophy of mathematics? The supposed answer to the first question is that it demands the view that mathematics should be perceived as a strictly pragmatic discipline and the rules of mathematical discourse are mere conventions. But this is too simplistic a view and the matter at hand is far more complicated.This paper is a part of a research project financed by National Centre of Science (Poland) on the basis of the decision no. UMO-2016/20/T/HS5/00232