The first thing to say is that the claim that “the universe is mathematical” hardly make sense at a prima facie level. It's not even that it's true or false. So this must surely mean that it's all about how we interpret such a claim.Despite saying that, sometimes it's hard to express (or even understand) precisely what Max Tegmark's actual position is. Can we say that reality (or the world) is mathematics or mathematical (as in the “is of identity”)? That reality is made up of numbers or equations? That reality instantiates maths, numbers or equations? Or should we settle for Tegmark's own very radical words? -“The Mathematical Universe Hypothesis... at the bottom level, reality is a mathematical structure, so its parts have no intrinsic properties at all! In other words, the Mathematical Universe Hypothesis implies that we live in a relational reality, in the sense that the properties of the world around us stem not from properties of its ultimate building blocks, but from the relations between these building blocks.”To put the case formally and as clearly as possible: Tegmark believes that physical “existence” and mathematical existence are “one the same” (which is a phrase he often uses) – they equal one another. More specifically, Tegmark stresses “structures”. Thus if we have a mathematical structure, it must exist physically as well. Or, more strongly, all mathematical structures exist physically.Of course the precise relation between mathematics and the world (or reality) has been debated for a long time. As we shall see, this isn't such a big problem for Tegmark for the simple reason that he believes that mathematics and the world are *one and the same thing*.

It would be nice if science answered all questions about our universe. In the past, mathematics has not just provided the language in which to frame suitable scientific answers, but was also able to give us clear indications of its own limitations. The former was able to deliver results via an ad hoc interface between theory and experiment. But to characterise the power of the scientific approach, one needs a parallel higher-order understanding of how the working scientist uses mathematics, and the development of an informative body of theory to clarify and expand this understanding. We argue that this depends on us selecting mathematical models which take account of the 'thingness' of reality, and puts the mathematics in a correspondingly rich informationtheoretic context. The task is to restore the role of embodied computation and its hierarchically arising attributes. The reward is an extension of our understanding of the power and limitations of mathematics, in the mathematical context, to that of the real world. Out of this viewpoint emerges a widely applicable framework, with not only epistemological, but also ontological consequences -one which uses Turing invariance and its putative breakdowns to confirm what we observe in the universe, to give a theoretical status to the dichotomy between quantum and relativistic domains, and which removes the need for many-worlds and related ideas. In particular, it is a view which confirms that of many quantum theorists -that it is the quantum world that is 'normal', and our classical level of reality which is strange and harder to explain. And which complements fascinating work of Cristian Calude and his collaborators on the mathematical characteristics of quantum randomness, and the relationship of 'strong determinism' to computability in nature.

A Case for a Mathematical God, 2023

Late Physicist Stephan Hawking famously stated, "There may be a God... but science can explain the universe without the need for a creator." I fight for the existence of mathematics as the agency of God.

HAL (Le Centre pour la Communication Scientifique Directe), 2021

Even though the march towards unification in physics began in the 17th century with the birth of mathematical physics, the notion of "Theory Of Everything" (TOE) only appeared around the 1980s. A TOE was supposed to unify all the fundamental interactions of nature: electromagnetism, weak interaction, strong interaction and gravitation. However, faced with the limits of most of the attempts (grand unification theory or GUT, string theory, loop quantum gravity, causal fermions systems, causal sets, Garrett Lisi's E8, causal dynamical triangulation, knot theory, ER = EPR ...), some wonder if we should not change the method, especially as logical or philosophical arguments (essential incompleteness of powerful theories, absence of fundamental laws, impossibility of embrassing "everything", essential infinitude of the universe, limited precision of calculations ...) could dissuade from seeking to build a definitive physical theory. More than anything, what is often disputed is the deductive nature of theories and the overuse of mathematics. Against these defeatist opinions, this article tries to rehabilitate the current approach of physicists to which we owe in fact many victories.

Review of Philosophy (Revista de Filosofie), 2014

Contemporary philosophical accounts of the applicability of mathematics in physical sciences and the empirical world are based on formalized relations between the mathematical structures and the physical systems they are supposed to represent within the models. Such relations were constructed both to ensure an adequate representation and to allow a justification of the validity of the mathematical models as means of scientific inference. This article puts in evidence the various circularities (logical, epistemic, and of definition) that are present in these formal constructions and discusses them as an argument for the alternative semantic and propositional-structure accounts of the applicability of mathematics. Keywords: philosophy of mathematics; applicability of mathematics; mathematical entities; mapping accounts; semantic accounts; circular definition; epistemic circularity; Frege; formal language; second-order logic; first-order logic; contingent truth; isomorphisms

Humanity is going to need a substantial new way of thinking if it is to survive! " Albert Einstein

L. Funar, A. Papadopoulos (eds.), Essays on Topology, Springer Nature, Switzerland AG , 2025

This chapter argues for an asymmetry in the reciprocal impact of modern mathematics and modern physics on each other, an impact commonly seen more symmetrically. By modern mathematics I refer to mathematics that, from its emergence around 1800 to our own time, was defined, as abstract mathematics, by separating, abstracting, itself from natural objects and thus from physics, as well as from the objects of daily thinking. Modern physics, which emerged two centuries earlier, along with what we call modernity itself, and has continued to our own time as well, is a mathematical-experimental science, with mathematics defining this conjunction. As such it has not and could have not separated itself from mathematics. Indeed, as this chapter argues, twentieth-century physics, especially relativity and quantum theory, was able to use modern mathematical theories born from the separation of modern mathematics from physics. Reciprocally, throughout its history, beginning with the invention of calculus, modern physics has had a major influence on mathematics, including modern mathematics and, with relativity and quantum theory, on the twentieth-and twenty-first century of mathematics. This chapter argues for an asymmetry in this reciprocal impact, in favor of the impact in favor of the impact of modern mathematics on modern physics, while this mutual impact is commonly seen more symmetrically, and sometimes by way of a reverse asymmetry, in favor of theoretical physics in helping to resolve some outstanding problems of pure mathematics. This chapter will challenge and qualify the latter view by arguing that theoretical physics cannot do so short of becoming mathematics, by, as modern mathematics itself has done, separating itself from physics. This chapter sees all modern physics as defined primarily by the invention of new theories, comprised of mathematized conceptual conglomerates, capable of predicting and, in some cases, representing the physical reality responsible for the phenomena considered, which are part of this reality. The invention of new mathematics, at least new for physics (this mathematics can be borrowed from already existing mathematics), is, the chapter argues, what has most fundamentally defined modern physics from its emergence until our own time.

Resonance, 2018

In this article, I discuss the relationship of mathematics to the physical world, and to other spheres of human knowledge. In particular, I argue that Mathematics is created by human beings, and the number π can not be said to have existed 100, 000 years ago, using the conventional meaning of the word 'exist'.

The frontiers collection, 2016

No paper in the history of science has had a greater impact solely through its title than Eugene Wigner's "The Unreasonable Effectiveness of Mathematics in the Natural Sciences" (1960). The title conveys both a claim and a puzzle. The claim is that mathematics is the language in which our most accurate scientific theories are formulated. The puzzle is why the language of mathematics should be such an effective tool for describing the physical world. The claim is indisputable. The puzzlement requires some unpacking. In its most radical form, the puzzlement could be directed at all of mathematics: why should any mathematical propositions have bearing on the behavior or structure of physical objects? Wigner himself does not raise this worry. Certain areas of mathematics struck him as unproblematic. He writes: "…whereas it is unquestionably true that the concepts of elementary mathematics and particularly elementary geometry were formulated to describe entities which are directly suggested by the actual world,…". That mathematical concepts developed for the purpose of describing the physical world manage to do so reasonably well is not prima facie puzzling at all. Wigner's own conundrum appears in the completion of the cited sentence: "…, the same does not seem to be true of the more advanced concepts which play such an important role in physics". Wigner mentions complex numbers as an example of the more advanced concepts, and contemporary physics can supply more exotic examples. We can therefore divide the initial question into two subquestions. 1) Which mathematical concepts seem naturally suited to describe features of the physical world, and what does their suitability imply about the physical world? 2) Why should any mathematical concepts that do not fall into the class of naturally suited ones nonetheless be of use in physics? Among the naturally suited mathematical concepts are the integers. The usefulness of the integers for describing physical situations requires little on the side of physics, but does require something. If there are physical items so constituted as to be solid objects, held together by strong internal forces and resistant to fracture and to amalgamation, then they will be effectively countable. A physical world completely described by fluid mechanics would contain no such objects, so the physics does make a crucial contribution. We can also imagine a physics whose fundamental constituents are perfectly discrete and "uncuttable"the atoms of Democritus, for example-which are even better suited to unambiguous counting than are macroscopic solids such as tables and chairs. So the usefulness and limits of even the most basic mathematical concepts for describing the physical world are influenced by the physics. The only question is which physical features-such as practical solidity and indivisibility in this case-are relevant. Even when the applicability of mathematics to the physical situation is straightforward, something akin to Wigner's puzzle may arise. A child playing

2015

Metascience, 2012

objects like numbers, sets and functions while at the same time continuing to accept our ordinary scientific theories and practices. Those raised in the Quinean tradition often suspect that fictionalism is either incoherent or so revisionary of our standards of doing metaphysics that it is not worth taking seriously. This can contribute to a kind of philosophical standoff between fictionalists and non-fictionalists. Mary Leng has done an excellent job in Mathematics and Reality of explaining to the Quinean what it takes to be a fictionalist. She goes far beyond previous expositions of the fictionalist position by setting out the fictionalist's conception of pure mathematics and clarifying what attitude towards our ordinary scientific theories the fictionalist can take. While there have been extensive discussions of fictionalism already in the work of Yablo, Balaguer, Rosen, Bueno and others, it is only with Leng's book that we have a comprehensive discussion of a fictionalist position. Even though I will more or less

AI

2008

In a recent article, M. Tegmark poses the hypothesis that our known universe is a ``baggage free'' mathematical structure among many other possible ones, which also correspond to other physical universes --Mathematical Universe Hypothesis, MUH. Naturally, questions arise, such as how to obtain the physical properties of our world from the mathematical structure, or how many possibilities exist for a

I explore physics implications of the External Reality Hypothesis (ERH) that there exists an external physical reality completely independent of us humans. I argue that with a sufficiently broad definition of mathematics, it implies the Mathematical Universe Hypothesis (MUH) that our physical world is an abstract mathematical structure. I discuss various implications of the ERH and MUH, ranging from standard physics topics like symmetries, irreducible representations, units, free parameters, randomness and initial conditions to broader issues like consciousness, parallel universes and Gödel incompleteness. I hypothesize that only computable and decidable (in Gödel's sense) structures exist, which alleviates the cosmological measure problem and may help explain why our physical laws appear so simple. I also comment on the intimate relation between mathematical structures, computations, simulations and physical systems.

Annals of the New York Academy of Sciences, 2019

Why is it that fundamental laws discovered through pure mathematics have been able to describe the behavior of our physical world with such precision? Given that the physical universe is composed of mathematical properties, some have posited that mathematics is the language of the universe, whose laws reveal what appears to be a hidden order in the natural world. Physicist S. James Gates, Jr. and science writer Margaret Wertheim explore the uncanny ability of mathematics to reveal the laws of nature.

All existence is either patterned or not-patterned (random). Both are mathematical. This exhaustive dichotomy establishes mathematical structure as the sole coherent substrate of existence; any concept of substance outside it collapses to it, and even structurelessness is structure, since complete random is mathematically defined. Even if structureless existence were coherent, observers could not inhabit it; pattern is a precondition of observation. This mathematical reality is uncreated and necessary. No agent makes 2+2=4, and none could make it otherwise; necessity admits no creator and requires none. The question "why is there something rather than nothing?" dissolves: mathematical structure cannot fail to exist, and explanatory regress terminates at necessity rather than brute fact. The Principle of Sufficient Reason is satisfied. All consistent mathematical structures exist. Any filter selecting among them is itself a consistent structure, so all filters exist, including the null filter that excludes nothing. This yields the Level IV multiverse not as hypothesis but as the unique non-arbitrary solution, resolving the fine-tuning problem. For 2400 years, from Plato's Forms to contemporary philosophy of mind, dualism has been a logical fallacy that collapses into mathematical monism (Matheism). Distinct things that interact must share structure; there is no causal influence across ontologically separate domains. Substance, soul, mind, and Forms all reduce to structure. Big bang and evolution or not, the block universe is eternal. Physics is a mathematical detail. When a map specifies every particle, field, and relation in the cosmos, the math is the territory. The hard problem of consciousness assumes its conclusion: that structure cannot constitute sentience felt from the inside. But any explanation must be structured, and anything causally interacting with physical systems must share their structure. Together, these arguments establish reality as math. This completes the line from Pythagoras through Tegmark and Ontic Structural Realism, but derives the conclusion rather than positing it.

One reason for the " unreasonable effectiveness of mathematics " is that it is never compared to nature itself, which is ambiguous, but to well-defined idealized versions (models). Math is consistent with nature in unfamiliar situations because it is consistent within itself. PART ONE: the Map and the Territory Nature and mathematics are divided by the same categorical gulf that has long plagued the relationship between the physical and the mental. Just as there is no mind without body, there is no mathematics without mathematicians. Mathematics and physics are alike cognitive activities undertaken by intelligent organisms. The relationship of one to the other—and of each to nature—must be considered in their context as embodied cognition. We moderns understand the universe through mathematical models, which are idealized constructs that correspond only approximately to reality. It is easy enough to mistake the model for the reality—the map for the territory—when one has learned to think of nature as literally consisting of such idealizations. This mistake parallels the naïve realism of ordinary cognition, in which we normally take our perception as a transparent window on the world. One can then even conclude that the universe is literally mathematics. It is perfectly reasonable for physicists to believe in the guidance of mathematics. Yet, if physics is a form of cognition, then it is also reasonable to believe in the guidance of cognitive theory, evolutionary psychology, and theoretical epistemology (the nascent science of possible cognitive systems). Such things are not a part of physics as we know it. The Scientific Revolution redefined natural philosophy as " first-order " science: study of the external world in strictly objectivist terms. Focus on the object excluded focus on the subject. Cognition in general is a form of map-making. So, then, are mathematics and physics. The map, however, is not the territory. At best it represents it selectively, symbolically, and adequately for specific purposes. In the case of ordinary cognition, these purposes are bequeathed by evolutionary history. Math and physics may be driven by other purposes as well, better described perhaps by sociology and anthropology. In any case, the map corresponds only grossly to the territory. There is always simplification and streamlining involved. Science tends to mask the real complexity of the world when it presumes simplicity or prefers tidy systems to messy facts. The complexity and messiness of nature are the signs of its reality, which (contra Plato) lies in its very " imperfection. " We vaunt the ability of mathematics to exhaustively represent natural systems, but the reality of nature lies precisely in its ability to resist such exhaustion. Mathematical laws generally describe ideal things and circumstances that do not occur in nature, and that could not even be stated without idealization, isolation, and experimental control. (Ellis 2002, p90-94) To single out a causal relationship, one must know how the process

2011

Sabine Hossenfelder's recent book "Lost in Math" has attracted numerous responses, including by notable physicists such as Frank Wilczek. In this article we focus on Wilczek's remark on that book, in particular on the perils of postempirical science. We also discuss shortly multiverse hypothesis from philosophical perspective. In last section, we offer a resolution from the perspective of Neutrosophic Logic on this problem of classical tension between mathematics and experience approach to physics, which seems to cause the stagnation of modern physics.

2025

This essay examines two competing metaphysical claims concerning the fundamental nature of reality. The first claim asserts that reality is fundamentally mathematical. According to this view, mathematical structures are not merely descriptive tools but constitute the very substance of the cosmos. This idea finds its origins in Pythagorean philosophy, where numbers were understood as the principles of harmony and order (Burkert, 1972), and was further developed by Plato, who located mathematical objects within the realm of eternal Forms, providing an intelligible framework for understanding the physical world (Plato, 2000). In contemporary philosophy and physics, Max Tegmark’s Mathematical Universe Hypothesis presents a modern articulation of this idea, claiming that all mathematically consistent structures exist physically, and that the universe itself is a mathematical object (Tegmark, 2014). Proponents of this thesis argue that mathematical ontology offers explanatory unification, predictive power, and an overarching framework that can account for regularities across all phenomena. The second claim, sometimes described as a mixed ontology, holds that reality is only partially mathematical. Mathematics captures patterns, quantitative relationships, and structural regularities, but it does not exhaust the totality of being. Philosophical exemplars of this approach include Aristotle, who proposed a hylomorphic framework in which form and matter combine to produce substances with qualitative, teleological, and causal features that cannot be reduced to mathematical description (Aristotle, 1924); Kant, who distinguished between phenomena accessible to mathematical representation and noumena, the unknowable things-in-themselves (Kant, 1998); and Whitehead, who conceived reality as a network of processes and events from which abstract structures emerge (Whitehead, 1978). Mixed ontology acknowledges the indispensability of mathematics for scientific understanding while allowing for qualitative, emergent, and potentially non-mathematical aspects of reality, including consciousness, agency, and unknown physical principles such as dark matter or dark energy (Chalmers, 1996; Rovelli, 2016). This essay evaluates the philosophical coherence, explanatory scope, and limitations of both positions, considering historical, metaphysical, and scientific perspectives. It argues that while mathematical ontology provides a powerful framework for modelling the universe, it may be insufficient for capturing the full richness of reality. A more comprehensive metaphysical framework integrates the structural insights of mathematics with qualitative and emergent dimensions, providing an account of reality that is both rigorous and open to aspects beyond formal representation.

Bolema: Boletim de Educação Matemática, 2015

Famous physicists, like Einstein and Wigner have been wondering, why mathematical symbolism could play such an effective and decisive role in the development of physics. Since the days of Plato, there have been essentially two different answers to this question. To Plato mathematics was a science of the unity and order of this universe. Since Galilei people came to believe that mathematics does not describe the objective world, it is not a reflection of some metaphysical realism. It is rather a reflection of human activity in this world. Kant, by his “Copernican Revolution of Epistemology” seems to have been the first to realize this. For example, number, or more generally arithmetic, was to the Pythagoreans “a cosmology” (KLEIN, 1985, p. 45), to Dedekind it is a means to better distinguish between things. The paper sketches the transition from an ontological to a semiotic interpretation of mathematics.

International Studies in The Philosophy of Science, 2011

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