Related papers: Realizing Infinity
Ever since its foundations were laid nearly a century ago, quantum theory has provoked questions about the very nature of reality. We address these questions by considering the universe, and the multiverse, fundamentally as complex…
A major question in philosophy of science involves the unreasonable effectiveness of mathematics in physics. Why should mathematics, created or discovered, with nothing empirical in mind be so perfectly suited to describe the laws of the…
We present a coherent collection of finite mathematical theorems some of which can only be proved by going well beyond the usual axioms for mathematics. The proofs of these theorems illustrate in clear terms how one uses the well studied…
This text tries to give an elementary introduction to the mathematical properties of infinite sets. The aim is to keep the approach as simple as possible. Advanced knowledge of mathematics is not necessary for a proper understanding, and…
Models of computation operating over the real numbers and computing a larger class of functions compared to the class of general recursive functions invariably introduce a non-finite element of infinite information encoded in an arbitrary…
Any problem is concerned with the mind, but what do minds make a decision on? Here we show that there are three conditions for the mind to make a certain answer. We found that some difficulties in physics and mathematics are in fact…
One of the outstanding problems of philosophy of science and mathematics today is whether there is just "one" unique mathematics or the same can be bifurcated into "pure" and "applied" categories. A novel solution for this problem is…
There is a problem with the foundations of classical mathematics, and potentially even with the foundations of computer science, that mathematicians have by-and-large ignored. This essay is a call for practicing mathematicians who have been…
Traditional computers work with finite numbers. Situations where the usage of infinite or infinitesimal quantities is required are studied mainly theoretically. In this paper, a recently introduced computational methodology (that is not…
This paper offers a solution method that allows one to find exact values for a large class of convergent series of rational terms. Sums of this form arise often in problems dealing with Quantum Field Theory.
From 1873 to 1897, Georg Cantor worked on developing set theory, and despite a strong initial resistance, it rapidly became accepted as the foundation of mathematics. In this work, however, we'll demonstrate that Cantor's use of infinity is…
Present treatments of eternal inflation regulate infinities by imposing a geometric cutoff. We point out that some matter systems reach the cutoff in finite time. This implies a nonzero probability for a novel type of catastrophe. According…
We use fast-growing finite and infinite sequences of natural numbers and more complicated constructs to define models of hypercomputation and interpret non-arithmetic predicates, with the strongest extensions reaching full second order…
Since ancient times, mathematics has proven unreasonably effective in its description of physical phenomena. As humankind enters a period of advancement where the completion of the much coveted theory of quantum gravity is at hand, there is…
This contribution is a technical description of details for implementing infinite elements. The novelty is the closed, analytic form used for most quantities.
The natural logarithm can be represented by an infinite series that converges for all positive real values of the variable, and which makes concavity patently obvious. Concavity of the natural logarithm is known to imply, among other…
Mathematics and its relation to the physical universe have been the topic of speculation since the days of Pythagoras. Several different views of the nature of mathematics have been considered: Realism - mathematics exists and is…
Classical mathematics (involving such notions as infinitely small/large and continuity) is usually treated as fundamental while finite mathematics is treated as inferior which is used only in special applications. We first argue that the…
Mathematicians occasionally discover interesting truths even when they are playing with mathematical ideas with no thoughts about possible consequences of their actions. This paper describes two specific instances of this phenomenon. The…
The quantization of gravity offers a solution to the presence of singularities in cosmology. Infinities are removed because of the existence of finite quanta of spacetime. This is one of the most important prediction of Loop Quantum…