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We describe in dialogue form a possible way of discovering and investigating 10-adic numbers starting from the naive question about a `largest natural number'. Among the topics we pursue are possibilities of extensions to transfinite…
Some relations between physics and finitary and infinitary mathematics are explored in the context of a many-minds interpretation of quantum theory. The analogy between mathematical ``existence'' and physical ``existence'' is considered…
Is speculative mathematics dangerous? Recent interactions between physics and mathematics pose the question with some force: traditional mathematical norms discourage speculation, but it is the fabric of theoretical physics. In practice…
Throughout this book, we discuss some open problems in various branches of science, including mathematics, theoretical physics, astro-physics, geophysics etc. It is of our hope that some of the problems discussed in this book will find…
More than hundred years ago the 'classic physics' was it in its full power, with just a few unexplained phenomena; which however led to a revolution and the development of the 'modern physics'. Today the computing is in a similar position:…
The concept of informal mathematical proof considered in intuitionism is apparently vulnerable to a version of the liar paradox. However, a careful reevaluation of this concept reveals a subtle error whose correction blocks the…
I argue that scientific determinism is not supported by facts, but results from the elegance of the mathematical language physicists use, in particular from the so-called real numbers and their infinite series of digits. Classical physics…
This essay considers the special character of mathematical reasoning, and draws on observations from interactive theorem proving and the history of mathematics to clarify the nature of formal and informal mathematical language. It proposes…
Physics is formulated in terms of timeless classical mathematics. A formulation on the basis of intuitionist mathematics, built on time-evolving processes, would offer a perspective that is closer to our experience of physical reality.
Are symmetries discovered or rather invented by humans ? The stand you may take firmly here reveals a lot of your epistemological position. Conversely, the arguments you may forge for answering to this question, or to one of its numerous…
The chapter advances a reformulation of the classical problem of the nature of mathematical objects (if any), here called "Plato's problem," in line with the program of a philosophy of mathematical practice. It then provides a sketch of a…
Math is widely considered as a powerful tool and its strong appeal depends on the high level of abstraction it allows in modelling a huge number of heterogeneous phenomena and problems, spanning from the static of buildings to the flight of…
Any act of problem-solving combines prior knowledge, local search, and a third element that is less often discussed: the extraction of information from search to update understanding. I propose a model of mathematical problem-solving as a…
Since the early twentieth century, it has been understood that mathematical definitions and proofs can be represented in formal systems systems with precise grammars and rules of use. Building on such foundations, computational proof…
This is an essay that considering the knowledge structure and language of a different nature, attempts to build on an explanation of the object of study and characteristics of the mathematical science. We end up with a learning cycle of…
Present day mathematics is a human construct, where computers are used more and more but do not play a creative role. This situation may change however: computers may become creative, and since they function very differently from the human…
Equations are about more than computing physical quantities or constructing formal models; they are also about understanding. The conceptual systems physicists use to think about nature are made from many different resources, formal and…
Questions concerning origin of mathematical knowledge and roles of language and intuition (imagery) in mathematical thoughts are long standing and widely debated. By introspection, mathematicians usually have some beliefs regarding these…
In the past century many fundamental results on unpredictability, undecidability and uncertainty have compelled scientists to grapple with the idea that some questions may never be resolved within our current theories. While this…
There are good reasons to believe that we are classical algorithms run on (effectively) classical machines. However, the fact that a physical state of a system in a universe described by a classical deterministic model doesn't contain any…