The scientific articles presented here are the result of my thinking about the foundations of physics, mathematics, logic, and knowledge in general. To understand the articles and follow the mathematical details, the mathematics set out in the Understanding and Doing Math books series is quite sufficient. While I find all the articles cited here valuable, I warn the reader that articles that have not yet been published have no confirmed scientific value by the scientific community. Although this confirmation sometimes seems an overly rigid mechanism, it is a very important mechanism of the scientific community that does not allow any claims to be presented as scientific.
Set theory is today positioned as a basic mathematical theory. It is the basis on which various mathematical ideas are modeled and interconnected. However, the notion of a set, unlike the notion of a natural number, is not a clear enough notion for such a foundational theory. The basic intuition is simple – we collect objects into sets. But sets are objects that we can also collect into sets. When we try to look at the whole world of sets, we come across paradoxes – there appear to be collections of sets that we cannot consistently combine into sets and it is not clear to us why this is so. We call them paradoxical collections. The most elementary example is Russell’s collection of all sets that are not their own elements. A set belongs to Russell’s collection just when it doesn’t belong to itself. If Russell’s collection were a set, according to the previous criterion, it would belong to itself just when it would not belong to itself – we got a contradiction. So Russell’s collection is not a set but a paradoxical collection. The article Logic of paradoxes in classical set theories explains the logical mechanism of paradoxes, called the principle of productivity. No matter how you imagine the notion of a set in classical language, this logical principle puts limits on collecting sets into sets – it tells us when we can do it and when we can’t. Contrary to the basic intuition about sets, the principle says that we cannot have all imaginable sets and all imaginable operations with sets at the same time. Then we could imagine an operation that, when applied to every set, gives a new element and we could also imagine a set that is closed under the operation. This is a contradiction in itself in the same way as the classical puzzle of the omnipotence of God. The basic religious intuition is that God is omnipotent, but then he could make a stone that he could not move. And this is a contradiction. The same conflict of basic intuition and logical bounds appears in the idea of a set.
The previous result suggests that the basic intuition about the notion of a set can be retained if we discard the classical language in which we describe sets. In my search for a suitable non-classical language, I was guided by the basic similarity between the paradoxes of the notion of a set and the paradox of the notion of truth. The simplest such paradox is the liar paradox. Consider a sentence that says about itself that it is not true. If it were true, then it would not be true, if it were not true, then it would be true. So that sentence is neither true nor false. Analyzing the paradoxes of truth, I came to a non-classical language in which some sentences are neither true nor false, and through it to a classical language (in which every sentence is true or false) in which the paradoxes of truth are simply resolved. This analysis of the paradoxes of truth is done in detail in the article The concept of truth, while in the article How to Conquer the Liar – an informal exposition it is applied in an informal way in the analysis of various paradoxes. Since the paradoxes of truth and the paradoxes of a set have the same basis, I now try to transfer these results to the notion of a set in order to obtain a set theory that is more meaningful than today’s set theory.
In modern times we should have before our eyes the words of Hannah Arendt: “The ideal subject of totalitarian rule is not the convinced Nazi or the convinced Communist, but people for whom the distinction between fact and fiction (ie, the reality of experience) and the distinction between true and false (ie, the standards of thought) no longer exist. ” Article The Synthetic Concept of Truth and its Descendants analyzes the notion of truth in the broader context of its meaning for science, mathematics, and logic.
I set out my thoughts on the role of mathematics in human cognition in the article Mathematics – an imagined tool for rational cognition. In short, as the title of the article says, I consider mathematics primarily our imagined tool by which, with certain assumptions about its applicability, we explore nature and synthesize our rational cognition of it. Such a view of mathematics also has methodological advantages when thought to students – it is not the inviolable truth about the world, but the world of ideas that we create ourselves. And anyone can actively participate in its creation. Of course, this approach to mathematics is also present in the Understanding and Doing Math books series, as well as in my other books and professional articles.
The questions of foundation naturally lead to the question of the foundation of all human knowledge. I summarized my thoughts on this in the article The Language Essence of Rational Cognition with some Philosophical Consequences. As the title itself says, I believe that language is the basic mechanism of our rational cognition, that our rational cognition is ultimately the creation and use of language. Going back to mathematics, the great importance of language for mathematics is indisputable. Language should also be given special attention in learning mathematics. My personal experience confirms, that almost all of those who think they do not understand mathematics, in fact, do not understand mathematical language.
Until the early 19th century, Euclidean geometry was considered an a priori mathematical theory formed by obvious truths about space. The modern view of Euclidean geometry is that it is a physical theory that describes well the space in which most of our activities take place. Such a view is at odds with the essential importance of Euclidean geometry in modern mathematics. Article Apriority of Euclidean Geometry argues that Euclidean geometry is a priori in the same way that numbers are a priori, the result of modeling, not the world, but our activities in the world. This philosophical view of geometry is based on a mathematical result published in the article An Elementary System of Axioms for Euclidean Geometry Based on Symmetry Principles, in which Euclidean geometry is based on certain principles of symmetry and to which we can give a natural a priori interpretation.
Lately, I have been turning more and more to the issues of founding physics. The first results of this thinking are the article A Simple Interpretation of Quantity Calculus which deals with the nature of physical quantities and calculations with them, and the article An analysis of the concept of inertial frame in classical physics and special theory of relativity which deals with inertial frames and their relationship to various forms of symmetry.
Helping my granddaughter in her mathematical development, I realized how limiting educational mathematics standards for the youngest are. Instead of developing their mathematical abilities with joy, in the current education system, children often develop an aversion to mathematics. In my opinion, this is unacceptable violence against children. First, I wrote a somewhat passionate article What is mathematics for the Youngest? and then a more measured and better-argued article Early Years Mathematics Education – the Missing Link in which I presented my views on this subject which I believe are in line with what many teachers and educators do or would like to do if the education system would allow them to do so. I hope the article will contribute to the critical mass needed for the necessary changes.