Related papers: Physical Constraints of Numbers
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…
Physical laws are a set of rules in the relationship between observations made by the experimenter. All these observations are made through a mechanism that links the external world to the experimenter's awareness, a mechanism which is not…
If there is a "platonic world" M of mathematical facts, what does M contain precisely? I observe that if M is too large, it is uninteresting, because the value is in the selection, not in the totality; if it is smaller and interesting, it…
Both metamathematics and physics are posited to emerge from samplings by observers of the unique ruliad structure that corresponds to the entangled limit of all possible computations. The possibility of higher-level mathematics accessible…
Metaphysics is traditionally conceived as aiming at the truth -- indeed, the most fundamental truths about the most general features of reality. Philosophical naturalists, urging that philosophical claims be grounded on science, have often…
Can mathematics help us find our way through all the wonders and mysteries of the universe? When physicists describe the laws governing the physical world, mathematics is always involved. Is this due to the fact that the universe is, at…
We claim that human mathematics is only a limited part of the consequences of the chosen basic axioms. Properly human mathematics varies with time but appears to have universal features which we try to analyze. In particular the functioning…
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.
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…
Despite the success of modern physics in formulating mathematical theories that can predict the outcome of quantum-scale experiments, the physical interpretations of these theories remain controversial. In this manuscript, we propose a new…
Measures in the context of Category Theory lead to various relations, even differential relations, of categories that are independent of the mathematical structure forming objects of a category. Such relations, which are independent of…
Despite its apparent complexity, our world seems to be governed by simple laws of physics. This volume provides a philosophical introduction to such laws. I explain how they are connected to some of the central issues in philosophy, such as…
The human mind is endowed with innate primordial perceptions such as space, distance, motion, change, flow of time, matter. The field of cognitive science argues that the abstract concepts of mathematics are not Platonic, but are built in…
It is a widespread belief that results like G\"odel's incompleteness theorems or the intrinsic randomness of quantum mechanics represent fundamental limitations to humanity's strive for scientific knowledge. As the argument goes, there are…
Do we have two kinds of reality: physical and mathematical? What is the role of mathematics in physics? These fundamental questions have intrigued original and brilliant minds since ancient times. A recent article (Aharonov, Cohen and…
Mathematics cannot anymore be assimilated to a linguistic game, where formal proofs are strongly differentiated with conjectural thinking, without building any category of knowledge to understand the passage (Wittgenstein's gist). Nowadays,…
Every physical theory has (at least) two different forms of mathematical equations to represent its target systems: the dynamical (equations of motion) and the kinematical (kinematical constraints). Kinematical constraints are…
While philosophy of science is the study of problems of knowledge concerning science in general, there also exists - or should exist - a '' philosophy in science'' directed at finding out in what ways our actual scientific knowledge may…
We state the defining characteristic of mathematics as a type of symmetry where one can change the connotation of a mathematical statement in a certain way when the statement's truth value remains the same. This view of mathematics as…
The concept of number is fundamental to the formulation of any physical theory. We give a heuristic motivation for the reformulation of Quantum Mechanics in terms of non-standard real numbers called Quantum Real Numbers. The standard axioms…