Real number

From Wikipedia, the free encyclopedia

https://en.wikipedia.org/wiki/Real_number

 

A symbol for the set of real numbers

In mathematics, a real number is a value of a continuous quantity that can represent a distance along a line (or alternatively, a quantity that can be represented as an infinite decimal expansion). The adjective real in this context was introduced in the 17th century by René Descartes, who distinguished between real and imaginary roots of polynomials. The real numbers include all the rational numbers, such as the integer −5 and the fraction 4/3, and all the irrational numbers, such as ${\displaystyle {\sqrt {2}}}$ (1.41421356..., the square root of 2, an irrational algebraic number). Included within the irrationals are the real transcendental numbers, such as π (3.14159265...). In addition to measuring distance, real numbers can be used to measure quantities such as time, mass, energy, velocity, and many more. The set of real numbers is denoted using the symbol R or ${\displaystyle \mathbb {R} }$ and is sometimes called "the reals".

Real numbers can be thought of as points on an infinitely long line called the number line or real line, where the points corresponding to integers are equally spaced. Any real number can be determined by a possibly infinite decimal representation, such as that of 8.632, where each consecutive digit is measured in units one-tenth the size of the previous one. The real line can be thought of as a part of the complex plane, and the real numbers can be thought of as a part of the complex numbers.

Real numbers can be thought of as points on an infinitely long number line

These descriptions of the real numbers are not sufficiently rigorous by the modern standards of pure mathematics. The discovery of a suitably rigorous definition of the real numbers—indeed, the realization that a better definition was needed—was one of the most important developments of 19th-century mathematics. The current standard axiomatic definition is that real numbers form the unique Dedekind-complete ordered field (${\displaystyle \mathbb {R} }$ ; + ; · ; <), up to an isomorphism, whereas popular constructive definitions of real numbers include declaring them as equivalence classes of Cauchy sequences (of rational numbers), Dedekind cuts, or infinite decimal representations, together with precise interpretations for the arithmetic operations and the order relation. All these definitions satisfy the axiomatic definition and are thus equivalent.

The set of all real numbers is uncountable, in the sense that while both the set of all natural numbers and the set of all real numbers are infinite sets, there can be no one-to-one function from the real numbers to the natural numbers. In fact, the cardinality of the set of all real numbers, denoted by ${\displaystyle {\mathfrak {c}}}$ and called the cardinality of the continuum, is strictly greater than the cardinality of the set of all natural numbers (denoted ${\displaystyle \aleph _{0}}$, 'aleph-naught').

The statement that there is no subset of the reals with cardinality strictly greater than ${\displaystyle \aleph _{0}}$ and strictly smaller than ${\displaystyle {\mathfrak {c}}}$ is known as the continuum hypothesis (CH). It is neither provable nor refutable using the axioms of Zermelo–Fraenkel set theory including the axiom of choice (ZFC)—the standard foundation of modern mathematics. In fact, some models of ZFC satisfy CH, while others violate it.

History

Real numbers ${\displaystyle (\mathbb {R} )}$ include the rational numbers ${\displaystyle (\mathbb {Q} )}$, which include the integers ${\displaystyle (\mathbb {Z} )}$, which in turn include the natural numbers ${\displaystyle (\mathbb {N} )}$

Simple fractions were used by the Egyptians around 1000 BC; the Vedic "Shulba Sutras" ("The rules of chords") in c. 600 BC include what may be the first "use" of irrational numbers. The concept of irrationality was implicitly accepted by early Indian mathematicians such as Manava (c. 750–690 BC), who were aware that the square roots of certain numbers, such as 2 and 61, could not be exactly determined. Around 500 BC, the Greek mathematicians led by Pythagoras realized the need for irrational numbers, in particular the irrationality of the square root of 2.

The Middle Ages brought about the acceptance of zero, negative numbers, integers, and fractional numbers, first by Indian and Chinese mathematicians, and then by Arabic mathematicians, who were also the first to treat irrational numbers as algebraic objects (the latter being made possible by the development of algebra). Arabic mathematicians merged the concepts of "number" and "magnitude" into a more general idea of real numbers. The Egyptian mathematician Abū Kāmil Shujā ibn Aslam (c. 850–930) was the first to accept irrational numbers as solutions to quadratic equations, or as coefficients in an equation (often in the form of square roots, cube roots and fourth roots).

In the 16th century, Simon Stevin created the basis for modern decimal notation, and insisted that there is no difference between rational and irrational numbers in this regard.

In the 17th century, Descartes introduced the term "real" to describe roots of a polynomial, distinguishing them from "imaginary" ones.

In the 18th and 19th centuries, there was much work on irrational and transcendental numbers. Johann Heinrich Lambert (1761) gave the first flawed proof that π cannot be rational; Adrien-Marie Legendre (1794) completed the proof, and showed that π is not the square root of a rational number. Paolo Ruffini (1799) and Niels Henrik Abel (1842) both constructed proofs of the Abel–Ruffini theorem: that the general quintic or higher equations cannot be solved by a general formula involving only arithmetical operations and roots.

Évariste Galois (1832) developed techniques for determining whether a given equation could be solved by radicals, which gave rise to the field of Galois theory. Joseph Liouville (1840) showed that neither e nor e² can be a root of an integer quadratic equation, and then established the existence of transcendental numbers; Georg Cantor (1873) extended and greatly simplified this proof. Charles Hermite (1873) first proved that e is transcendental, and Ferdinand von Lindemann (1882), showed that π is transcendental. Lindemann's proof was much simplified by Weierstrass (1885), still further by David Hilbert (1893), and has finally been made elementary by Adolf Hurwitz and Paul Gordan.

The development of calculus in the 18th century used the entire set of real numbers without having defined them rigorously. The first rigorous definition was published by Georg Cantor in 1871. In 1874, he showed that the set of all real numbers is uncountably infinite, but the set of all algebraic numbers is countably infinite. Contrary to widely held beliefs, his first method was not his famous diagonal argument, which he published in 1891. For more, see Cantor's first uncountability proof.

Definition

Main article: Construction of the real numbers

The real number system ${\displaystyle (\mathbb {R} ;{}+{};{}\cdot {};{}<{})}$ can be defined axiomatically up to an isomorphism, which is described hereafter. There are also many ways to construct "the" real number system, and a popular approach involves starting from natural numbers, then defining rational numbers algebraically, and finally defining real numbers as equivalence classes of their Cauchy sequences or as Dedekind cuts, which are certain subsets of rational numbers. Another approach is to start from some rigorous axiomatization of Euclidean geometry (say of Hilbert or of Tarski), and then define the real number system geometrically. All these constructions of the real numbers have been shown to be equivalent, in the sense that the resulting number systems are isomorphic.

Axiomatic approach

Let ${\displaystyle \mathbb {R} }$ denote the set of all real numbers, then:

• The set ${\displaystyle \mathbb {R} }$ is a field, meaning that addition and multiplication are defined and have the usual properties.
• The field ${\displaystyle \mathbb {R} }$ is ordered, meaning that there is a total order ≥ such that for all real numbers x, y and z:
  • if x ≥ y, then x + z ≥ y + z;
  • if x ≥ 0 and y ≥ 0, then xy ≥ 0.
• The order is Dedekind-complete, meaning that every non-empty subset S of ${\displaystyle \mathbb {R} }$ with an upper bound in ${\displaystyle \mathbb {R} }$ has a least upper bound (a.k.a., supremum) in ${\displaystyle \mathbb {R} }$.

The last property is what differentiates the real numbers from the rational numbers (and from other more exotic ordered fields). For example, ${\displaystyle \{x\in \mathbb {Q} :x^{2}<2\}}$ has a rational upper bound (e.g., 1.42), but no least rational upper bound, because ${\displaystyle {\sqrt {2}}}$ is not rational.

These properties imply the Archimedean property (which is not implied by other definitions of completeness), which states that the set of integers has no upper bound in the reals. In fact, if this were false, then the integers would have a least upper bound N; then, N – 1 would not be an upper bound, and there would be an integer n such that n > N – 1, and thus n + 1 > N, which is a contradiction with the upper-bound property of N.

The real numbers are uniquely specified by the above properties. More precisely, given any two Dedekind-complete ordered fields ${\displaystyle \mathbb {R} _{1}}$ and ${\displaystyle \mathbb {R} _{2}}$, there exists a unique field isomorphism from ${\displaystyle \mathbb {R} _{1}}$ to ${\displaystyle \mathbb {R_{2}} }$. This uniqueness allows us to think of them as essentially the same mathematical object.

For another axiomatization of ${\displaystyle \mathbb {R} }$, see Tarski's axiomatization of the reals.

Construction from the rational numbers

The real numbers can be constructed as a completion of the rational numbers, in such a way that a sequence defined by a decimal or binary expansion like (3; 3.1; 3.14; 3.141; 3.1415; ...) converges to a unique real number—in this case π. For details and other constructions of real numbers, see construction of the real numbers.

Properties

Basic properties

• Any non-zero real number is either negative or positive.
• The sum and the product of two non-negative real numbers is again a non-negative real number, i.e., they are closed under these operations, and form a positive cone, thereby giving rise to a linear order of the real numbers along a number line.
• The real numbers make up an infinite set of numbers that cannot be injectively mapped to the infinite set of natural numbers, i.e., there are uncountably infinitely many real numbers, whereas the natural numbers are called countably infinite. This establishes that in some sense, there are more real numbers than there are elements in any countable set.
• There is a hierarchy of countably infinite subsets of the real numbers, e.g., the integers, the rational numbers, the algebraic numbers and the computable numbers, each set being a proper subset of the next in the sequence. The complements of all these sets (irrational, transcendental, and non-computable real numbers) in the reals are all uncountably infinite sets.
• Real numbers can be used to express measurements of continuous quantities. They may be expressed by decimal representations, most of them having an infinite sequence of digits to the right of the decimal point; these are often represented like 324.823122147..., where the ellipsis (three dots) indicates that there would still be more digits to come. This hints at the fact that we can precisely denote only a few, selected real numbers with finitely many symbols.

More formally, the real numbers have the two basic properties of being an ordered field, and having the least upper bound property. The first says that real numbers comprise a field, with addition and multiplication as well as division by non-zero numbers, which can be totally ordered on a number line in a way compatible with addition and multiplication. The second says that, if a non-empty set of real numbers has an upper bound, then it has a real least upper bound. The second condition distinguishes the real numbers from the rational numbers: for example, the set of rational numbers whose square is less than 2 is a set with an upper bound (e.g. 1.5) but no (rational) least upper bound: hence the rational numbers do not satisfy the least upper bound property.

Completeness

Main article: Completeness of the real numbers

A main reason for using real numbers is so that many sequences have limits. More formally, the reals are complete (in the sense of metric spaces or uniform spaces, which is a different sense than the Dedekind completeness of the order in the previous section):

A sequence (x_n) of real numbers is called a Cauchy sequence if for any ε > 0 there exists an integer N (possibly depending on ε) such that the distance |x_n − x_m| is less than ε for all n and m that are both greater than N. This definition, originally provided by Cauchy, formalizes the fact that the x_n eventually come and remain arbitrarily close to each other.

A sequence (x_n) converges to the limit x if its elements eventually come and remain arbitrarily close to x, that is, if for any ε > 0 there exists an integer N (possibly depending on ε) such that the distance |x_n − x| is less than ε for n greater than N.

Every convergent sequence is a Cauchy sequence, and the converse is true for real numbers, and this means that the topological space of the real numbers is complete.

The set of rational numbers is not complete. For example, the sequence (1; 1.4; 1.41; 1.414; 1.4142; 1.41421; ...), where each term adds a digit of the decimal expansion of the positive square root of 2, is Cauchy but it does not converge to a rational number (in the real numbers, in contrast, it converges to the positive square root of 2).

The completeness property of the reals is the basis on which calculus, and, more generally mathematical analysis are built. In particular, the test that a sequence is a Cauchy sequence allows proving that a sequence has a limit, without computing it, and even without knowing it.

For example, the standard series of the exponential function

${\displaystyle e^{x}=\sum _{n=0}^{\infty }{\frac {x^{n}}{n!}}}$

converges to a real number for every x, because the sums

${\displaystyle \sum _{n=N}^{M}{\frac {x^{n}}{n!}}}$

can be made arbitrarily small (independently of M) by choosing N sufficiently large. This proves that the sequence is Cauchy, and thus converges, showing that ${\displaystyle e^{x}}$ is well defined for every x.

"The complete ordered field"

The real numbers are often described as "the complete ordered field", a phrase that can be interpreted in several ways.

First, an order can be lattice-complete. It is easy to see that no ordered field can be lattice-complete, because it can have no largest element (given any element z, z + 1 is larger).

Additionally, an order can be Dedekind-complete, see § Axiomatic approach. The uniqueness result at the end of that section justifies using the word "the" in the phrase "complete ordered field" when this is the sense of "complete" that is meant. This sense of completeness is most closely related to the construction of the reals from Dedekind cuts, since that construction starts from an ordered field (the rationals) and then forms the Dedekind-completion of it in a standard way.

These two notions of completeness ignore the field structure. However, an ordered group (in this case, the additive group of the field) defines a uniform structure, and uniform structures have a notion of completeness; the description in § Completeness is a special case. (We refer to the notion of completeness in uniform spaces rather than the related and better known notion for metric spaces, since the definition of metric space relies on already having a characterization of the real numbers.) It is not true that ${\displaystyle \mathbb {R} }$ is the only uniformly complete ordered field, but it is the only uniformly complete Archimedean field, and indeed one often hears the phrase "complete Archimedean field" instead of "complete ordered field". Every uniformly complete Archimedean field must also be Dedekind-complete (and vice versa), justifying using "the" in the phrase "the complete Archimedean field". This sense of completeness is most closely related to the construction of the reals from Cauchy sequences (the construction carried out in full in this article), since it starts with an Archimedean field (the rationals) and forms the uniform completion of it in a standard way.

But the original use of the phrase "complete Archimedean field" was by David Hilbert, who meant still something else by it. He meant that the real numbers form the largest Archimedean field in the sense that every other Archimedean field is a subfield of ${\displaystyle \mathbb {R} }$. Thus ${\displaystyle \mathbb {R} }$ is "complete" in the sense that nothing further can be added to it without making it no longer an Archimedean field. This sense of completeness is most closely related to the construction of the reals from surreal numbers, since that construction starts with a proper class that contains every ordered field (the surreals) and then selects from it the largest Archimedean subfield.

Advanced properties

See also: Real line

The reals are uncountable; that is, there are strictly more real numbers than natural numbers, even though both sets are infinite. In fact, the cardinality of the reals equals that of the set of subsets (i.e. the power set) of the natural numbers, and Cantor's diagonal argument states that the latter set's cardinality is strictly greater than the cardinality of ${\displaystyle \mathbb {N} }$. Since the set of algebraic numbers is countable, almost all real numbers are transcendental. The non-existence of a subset of the reals with cardinality strictly between that of the integers and the reals is known as the continuum hypothesis. The continuum hypothesis can neither be proved nor be disproved; it is independent from the axioms of set theory.

As a topological space, the real numbers are separable. This is because the set of rationals, which is countable, is dense in the real numbers. The irrational numbers are also dense in the real numbers, however they are uncountable and have the same cardinality as the reals.

The real numbers form a metric space: the distance between x and y is defined as the absolute value |x − y|. By virtue of being a totally ordered set, they also carry an order topology; the topology arising from the metric and the one arising from the order are identical, but yield different presentations for the topology—in the order topology as ordered intervals, in the metric topology as epsilon-balls. The Dedekind cuts construction uses the order topology presentation, while the Cauchy sequences construction uses the metric topology presentation. The reals form a contractible (hence connected and simply connected), separable and complete metric space of Hausdorff dimension 1. The real numbers are locally compact but not compact. There are various properties that uniquely specify them; for instance, all unbounded, connected, and separable order topologies are necessarily homeomorphic to the reals.

Every nonnegative real number has a square root in ${\displaystyle \mathbb {R} }$, although no negative number does. This shows that the order on ${\displaystyle \mathbb {R} }$ is determined by its algebraic structure. Also, every polynomial of odd degree admits at least one real root: these two properties make ${\displaystyle \mathbb {R} }$ the premier example of a real closed field. Proving this is the first half of one proof of the fundamental theorem of algebra.

The reals carry a canonical measure, the Lebesgue measure, which is the Haar measure on their structure as a topological group normalized such that the unit interval [0;1] has measure 1. There exist sets of real numbers that are not Lebesgue measurable, e.g. Vitali sets.

The supremum axiom of the reals refers to subsets of the reals and is therefore a second-order logical statement. It is not possible to characterize the reals with first-order logic alone: the Löwenheim–Skolem theorem implies that there exists a countable dense subset of the real numbers satisfying exactly the same sentences in first-order logic as the real numbers themselves. The set of hyperreal numbers satisfies the same first order sentences as ${\displaystyle \mathbb {R} }$. Ordered fields that satisfy the same first-order sentences as ${\displaystyle \mathbb {R} }$ are called nonstandard models of ${\displaystyle \mathbb {R} }$. This is what makes nonstandard analysis work; by proving a first-order statement in some nonstandard model (which may be easier than proving it in ${\displaystyle \mathbb {R} }$), we know that the same statement must also be true of ${\displaystyle \mathbb {R} }$.

The field ${\displaystyle \mathbb {R} }$ of real numbers is an extension field of the field ${\displaystyle \mathbb {Q} }$ of rational numbers, and ${\displaystyle \mathbb {R} }$ can therefore be seen as a vector space over ${\displaystyle \mathbb {Q} }$. Zermelo–Fraenkel set theory with the axiom of choice guarantees the existence of a basis of this vector space: there exists a set B of real numbers such that every real number can be written uniquely as a finite linear combination of elements of this set, using rational coefficients only, and such that no element of B is a rational linear combination of the others. However, this existence theorem is purely theoretical, as such a base has never been explicitly described.

The well-ordering theorem implies that the real numbers can be well-ordered if the axiom of choice is assumed: there exists a total order on ${\displaystyle \mathbb {R} }$ with the property that every non-empty subset of ${\displaystyle \mathbb {R} }$ has a least element in this ordering. (The standard ordering ≤ of the real numbers is not a well-ordering since e.g. an open interval does not contain a least element in this ordering.) Again, the existence of such a well-ordering is purely theoretical, as it has not been explicitly described. If V=L is assumed in addition to the axioms of ZF, a well ordering of the real numbers can be shown to be explicitly definable by a formula.^[17]

A real number may be either computable or uncomputable; either algorithmically random or not; and either arithmetically random or not.

Applications and connections to other areas

Real numbers and logic

The real numbers are most often formalized using the Zermelo–Fraenkel axiomatization of set theory, but some mathematicians study the real numbers with other logical foundations of mathematics. In particular, the real numbers are also studied in reverse mathematics and in constructive mathematics.

The hyperreal numbers as developed by Edwin Hewitt, Abraham Robinson and others extend the set of the real numbers by introducing infinitesimal and infinite numbers, allowing for building infinitesimal calculus in a way closer to the original intuitions of Leibniz, Euler, Cauchy and others.

Edward Nelson's internal set theory enriches the Zermelo–Fraenkel set theory syntactically by introducing a unary predicate "standard". In this approach, infinitesimals are (non-"standard") elements of the set of the real numbers (rather than being elements of an extension thereof, as in Robinson's theory).

The continuum hypothesis posits that the cardinality of the set of the real numbers is ${\displaystyle \aleph _{1}}$; i.e. the smallest infinite cardinal number after ${\displaystyle \aleph _{0}}$, the cardinality of the integers. Paul Cohen proved in 1963 that it is an axiom independent of the other axioms of set theory; that is: one may choose either the continuum hypothesis or its negation as an axiom of set theory, without contradiction.

In physics

In the physical sciences, most physical constants such as the universal gravitational constant, and physical variables, such as position, mass, speed, and electric charge, are modeled using real numbers. In fact, the fundamental physical theories such as classical mechanics, electromagnetism, quantum mechanics, general relativity and the standard model are described using mathematical structures, typically smooth manifolds or Hilbert spaces, that are based on the real numbers, although actual measurements of physical quantities are of finite accuracy and precision.

Physicists have occasionally suggested that a more fundamental theory would replace the real numbers with quantities that do not form a continuum, but such proposals remain speculative.

In computation

With some exceptions, most calculators do not operate on real numbers. Instead, they work with finite-precision approximations called floating-point numbers. In fact, most scientific computation uses floating-point arithmetic. Real numbers satisfy the usual rules of arithmetic, but floating-point numbers do not.

Computers cannot directly store arbitrary real numbers with infinitely many digits. The achievable precision is limited by the number of bits allocated to store a number, whether as floating-point numbers or arbitrary-precision numbers. However, computer algebra systems can operate on irrational quantities exactly by manipulating formulas for them (such as ${\displaystyle {\sqrt {2}},}$ ${\displaystyle \arcsin(2/23),}$ or ${\displaystyle \textstyle \int _{0}^{1}x^{x}\,dx}$) rather than their rational or decimal approximation. It is not in general possible to determine whether two such expressions are equal (the constant problem).

A real number is called computable if there exists an algorithm that yields its digits. Because there are only countably many algorithms, but an uncountable number of reals, almost all real numbers fail to be computable. Moreover, the equality of two computable numbers is an undecidable problem. Some constructivists accept the existence of only those reals that are computable. The set of definable numbers is broader, but still only countable.

"Reals" in set theory

In set theory, specifically descriptive set theory, the Baire space is used as a surrogate for the real numbers since the latter have some topological properties (connectedness) that are a technical inconvenience. Elements of Baire space are referred to as "reals".

Vocabulary and notation

Mathematicians use mainly the symbol R to represent the set of all real numbers. Alternatively, it may be used ${\displaystyle \mathbb {R} }$, the letter "R" in blackboard bold, which may be encoded in Unicode (and HTML) as U+211D ℝ (&reals;, &Ropf;). As this set is naturally endowed with the structure of a field, the expression field of real numbers is frequently used when its algebraic properties are under consideration.

The sets of positive real numbers and negative real numbers are often noted ${\displaystyle \mathbb {R} ^{+}}$ and ${\displaystyle \mathbb {R} ^{-}}$, respectively; ${\displaystyle \mathbb {R} _{+}}$ and ${\displaystyle \mathbb {R} _{-}}$ are also used. The non-negative real numbers can be noted ${\displaystyle \mathbb {R} _{\geq 0}}$ but one often sees this set noted ${\displaystyle \mathbb {R} ^{+}\cup \{0\}.}$ In French mathematics, the positive real numbers and negative real numbers commonly include zero, and these sets are noted respectively ${\displaystyle \mathbb {R_{+}} }$ and ${\displaystyle \mathbb {R_{-}} .}$ In this understanding, the respective sets without zero are called strictly positive real numbers and strictly negative real numbers, and are noted ${\displaystyle \mathbb {R} _{+}^{*}}$ and ${\displaystyle \mathbb {R} _{-}^{*}.}$

The notation ${\displaystyle \mathbb {R} ^{n}}$ refers to the set of the n-tuples of elements of ${\displaystyle \mathbb {R} }$ (real coordinate space), which can be identified to the Cartesian product of n copies of ${\displaystyle \mathbb {R} .}$ It is an n-dimensional vector space over the field of the real numbers, often called the coordinate space of dimension n; this space may be identified to the n-dimensional Euclidean space as soon as a Cartesian coordinate system has been chosen in the latter. In this identification, a point of the Euclidean space is identified with the tuple of its Cartesian coordinates.

In mathematics, real is used as an adjective, meaning that the underlying field is the field of the real numbers (or the real field). For example, real matrix, real polynomial and real Lie algebra. The word is also used as a noun, meaning a real number (as in "the set of all reals").

Generalizations and extensions

The real numbers can be generalized and extended in several different directions:

• The complex numbers contain solutions to all polynomial equations and hence are an algebraically closed field unlike the real numbers. However, the complex numbers are not an ordered field.
• The affinely extended real number system adds two elements +∞ and −∞. It is a compact space. It is no longer a field, or even an additive group, but it still has a total order; moreover, it is a complete lattice.
• The real projective line adds only one value ∞. It is also a compact space. Again, it is no longer a field, or even an additive group. However, it allows division of a non-zero element by zero. It has cyclic order described by a separation relation.
• The long real line pastes together ℵ₁* + ℵ₁ copies of the real line plus a single point (here ℵ₁* denotes the reversed ordering of ℵ₁) to create an ordered set that is "locally" identical to the real numbers, but somehow longer; for instance, there is an order-preserving embedding of ℵ₁ in the long real line but not in the real numbers. The long real line is the largest ordered set that is complete and locally Archimedean. As with the previous two examples, this set is no longer a field or additive group.
• Ordered fields extending the reals are the hyperreal numbers and the surreal numbers; both of them contain infinitesimal and infinitely large numbers and are therefore non-Archimedean ordered fields.
• Self-adjoint operators on a Hilbert space (for example, self-adjoint square complex matrices) generalize the reals in many respects: they can be ordered (though not totally ordered), they are complete, all their eigenvalues are real and they form a real associative algebra. Positive-definite operators correspond to the positive reals and normal operators correspond to the complex numbers.

Rights of nature

From Wikipedia, the free encyclopedia

https://en.wikipedia.org/wiki/Rights_of_nature 

Rights of nature or Earth rights is a legal and jurisprudential theory that describes inherent rights as associated with ecosystems and species, similar to the concept of fundamental human rights. The rights of nature concept challenges twentieth-century laws as generally grounded in a flawed frame of nature as "resource" to be owned, used, and degraded. Proponents argue that laws grounded in rights of nature direct humanity to act appropriately and in a way consistent with modern, system-based science, which demonstrates that humans and the natural world are fundamentally interconnected.

This school of thought is underpinned by two basic lines of reasoning. First, since the recognition of human rights is based in part on the philosophical belief that those rights emanate from humanity's own existence, logically, so too do inherent rights of the natural world arise from the natural world's own existence. A second and more pragmatic argument asserts that the survival of humans depends on healthy ecosystems, and so protection of nature's rights in turn, advances human rights and well-being.

From a rights of nature perspective, most environmental laws of the twentieth century are based on an outmoded framework that considers nature to be composed of separate and independent parts, rather than components of a larger whole. A more significant criticism is that those laws tend to be subordinate to economic interests, and aim at reacting to and just partially mitigating economics-driven degradation, rather than placing nature's right to thrive as the primary goal of those laws. This critique of existing environmental laws is an important component of tactics such as climate change litigation that seeks to force societal action to mitigate climate change.

As of June 2021, rights of nature laws exist at the local to national levels in at least 39 countries, including dozens of cities and counties throughout the United States. They take the form of constitutional provisions, treaty agreements, statutes, local ordinances, and court decisions. A state constitutional provision is being sought in Florida.

Basic tenets

Proponents of rights of nature argue that, just as human rights have been recognized increasingly in law, so should nature's rights be recognized and incorporated into human ethics and laws. This claim is underpinned by two lines of reasoning: that the same ethics that justify human rights, also justify nature's rights, and, that humans' own survival depend on healthy ecosystems.

Thomas Berry - a U.S. cultural historian who introduced the legal concept of Earth Jurisprudence who proposed that society's laws should derive from the laws of nature, explaining that "the universe is a communion of subjects, not a collection of objects"

First, it is argued that if inherent human rights arise from human existence, so too logically do inherent rights of the natural world arise from the natural world's own existence. Human rights, and associated duties to protect those rights, have expanded over time. Most notably, the 1948 adoption by the United Nations, of the Universal Declaration of Human Rights (UDHR) that formalized recognition of broad categories of inalienable human rights. Drafters of the UDHR stated their belief that the concept of fundamental human rights arose not from "the decision of a worldly power, but rather in the fact of existing."

Some scholars have contended thereafter that, given that basic human rights emanate from humans' own existence, nature's rights similarly arise from the similar existence of nature, and so humans' legal systems should continue to expand to recognize the rights of nature.

Some notable proponents of this approach include U.S. cultural historian Thomas Berry, South African attorney Cormac Cullinan, Indian physicist and eco-social advocate Vandana Shiva, and Canadian law professor and U.N. Special Rapporteur for Human Rights and the Environment David R. Boyd.

Vandana Shiva - an Indian scholar and activist who has written extensively on Earth Jurisprudence and Earth Democracy that she describes as based on "local communities – organized on principles of inclusion, diversity, and ecological and social responsibility"

Thomas Berry introduced a philosophy and ethics of law concept called Earth jurisprudence that identifies the earth's laws as primary and reasons that everything by the fact of its existence, therefore, has an intrinsic right to be and evolve. Earth Jurisprudence has been increasingly recognized and promoted worldwide by legal scholars, the United Nations, lawmakers, philosophers, ecological economists, and other experts as a foundation for Earth-centered governance, including laws and economic systems that protect the fundamental rights of nature.

Second, support for rights of nature also is supported through the utilitarian argument that humanity can only thrive in the long term by accepting integrated co-existence of humans with the natural world. Berry noted that the concept of human well-being derived from natural systems with no fundamental right to exist is inherently illogical, and that by protecting nature's rights, humans advance their own self-interest.

The legal and philosophical concept of rights of nature offers a shift from a frame of nature as property or resource, to nature as an interconnected Earth community partner. This school of thought aims at following the same path that human rights movements have followed, where at first recognition of rights in the rightless appeared "unthinkable", but later matured into a broadly-espoused worldview.

Christopher Stone, a law professor at the University of Southern California, wrote extensively on this topic in his seminal essay, "Should Trees Have Standing", cited by a U.S. Supreme Court dissent in Sierra Club v. Morton for the position that "environmental issues should be tendered by [nature] itself." As described by Stone and others, human rights have increasingly been "found" over time and declared "self-evident", as in the U.S. Declaration of Independence, even where essentially non-existent in the law. The successes of past and current human rights movements provide lessons for the current movement to widen the circle of Earth community to include natural systems and species populations as rights-bearing entities.

Underpinnings and development

Critique of anthropocentric legal systems

Proponents of a shift to a more environmentally protective system of law contend that current legal and economic systems fail because they consider nature fundamentally as property, which can be degraded for profit and human desire. They point out that the perspective of nature as primarily an economic resource already has degraded some ecosystems and species so significantly that now, prominent policy experts are examining "endangered species triage" strategies to decide which species will be let go, rather than re-examine the economics driving the degradation. While twentieth and twenty-first century environmental laws do afford some level of protection to ecosystems and species, it is argued that such protections fail to stop, let alone reverse, overall environmental decline, because nature is by definition subordinated to anthropogenic and economic interests, rather than biocentric well-being.

Rights of nature proponents contend that re-envisioning current environmental laws from a nature's rights frame demonstrates the limitations of current legal systems. For example, the U.S. Endangered Species Act prioritizes protection of existing economic interests by activating only when species populations are headed toward extinction. By contrast, a "Healthy Species Act" would prioritize achievement of thriving species populations and facilitate economic systems that drive conservation of species.

As another example, the European Union's Water Framework Directive of 2000, "widely accepted as the most substantial and ambitious piece of European environmental legislation to date", relies on a target of "good status" of all EU waters, which includes consideration of needed "ecological flows". However, decades after the Directive's adoption, despite scientific advances in identifying flow-ecology relationships, there remains no EU definition of "ecological flow", nor a common understanding of how it should be calculated. A nature's rights frame would recognize not only the existing human right to water for basic needs, but would also recognize the rights of waterways to adequate, timely, clean water flows, and would define such basic ecological flow needs accordingly.

Underlying science and ethics

Modern environmental laws began to arise in the 1960s out of a foundational perspective of the environment as best managed in discrete pieces. For example, United States laws such as the Clean Water Act, Clean Air Act, Endangered Species Act, Marine Mammal Protection Act, and numerous others began to be adopted in the early 1970s to address various elements of the natural world, separately from other elements. Some laws, such as the U.S. National Environmental Policy Act, called for a more holistic analysis of proposed infrastructure projects and required the disclosure of expected negative environmental impacts. However, it did not require that actions be taken to address those impacts in order to ensure ecosystem and species health.

These laws reflected the science of the time, which was grounded in a reductionist analysis of the natural world; the modern, system-based understanding of the natural world, and the integrated place of humans with it, was still in development. The first major textbook on ecological science that described the natural world as a system rather than a collection of different parts, was not written until 1983. The Gaia Hypothesis, which offered a scientific vision of the world as a self-regulating, complex system, first arose in the 1970s. Systems dynamics similarly began to evolve from a business focus to include socioeconomic and natural systems starting in the 1970s. Since then, scientific disciplines have been converging and advancing on the concept that humans live in a dynamic, relationship-based world that "den[ies] the possibility of isolation".

While science in the late twentieth century shifted to a systems-based perspective, describing natural systems and human populations as fundamentally interconnected on a shared planet, environmental laws generally did not evolve with this shift. Reductionist U.S. environmental laws passed in the early 1970s remained largely unchanged, and other national and international environmental law regimes similarly stopped short of embracing the modern science of systems.

Nineteenth century linguist and scholar Edward Payson Evans, an early rights of nature theorist and author of "the first extensive American statement of (...) environmental ethics”, wrote that each human is “truly a part and product of Nature as any other animal, and [the] attempt to set him up on an isolated point outside of it is philosophically false and morally pernicious”.

Thomas Berry proposed that society's laws should derive from the laws of nature, explaining that "the universe is a communion of subjects, not a collection of objects". From the scientific perspective that all life arose from the context of the universe, Berry offered the ethical perspective that it is flawed to view humans as the universe's only subjects, with all other beings merely a collection of objects to be owned and used. Rather, consideration of life as a web of relationships extending back to a shared ancestry confers subject status to all, including the inherent rights associated with that status. Laws based on a recognition of the intrinsic moral value of the natural world, create a new societal moral compass that directs society's interactions with the natural world more effectively toward well-being for all.

Aldo Leopold - a scientist and forester who advocated to "see land as a community to which we belong" rather than as "a commodity belonging to us" (1946 photograph)

Scientists who similarly wrote in support of expanded human moral development and ethical obligation include naturalist John Muir and scientist and forester Aldo Leopold. Leopold expressed that "[w]hen we see land as a community to which we belong", rather than "a commodity belonging to us", we can "begin to use it with love and respect". Leopold offered implementation guidance for his position, stating that a "thing is right when it tends to preserve the integrity, stability, and beauty of the biotic community. It is wrong when it tends otherwise." Berry similarly observed that "whatever preserves and enhances this meadow in the natural cycles of its transformation is good; what is opposed to this meadow or negates it is not good." Physician and philosopher Albert Schweizer defined right actions as those that recognize a reverence for life and the "will to live".

The outgrowth of scientific and ethical advances around natural systems and species is a proposed new frame for legal and governance systems, one grounded in an ethic and a language that guide behavior away from ecological and social practices that ignore or minimize human-nature interconnections. Rather than a vision of merely "sustainable development", which reflects a frame of nature maintained as economic feedstock, scholars supporting rights of nature suggest that society is beginning to consider visions such as "thriving communities", where "communities" includes nature as a full subject, rather than simply an object to be used.

History

Common roots with Indigenous worldviews

The ethical and philosophical foundation of a nature's rights legal theory and movement is a worldview of respect for nature, as contrasted with the "nature domination" worldview that underlies the concept of nature as object and property. Indigenous law professor John Borrows observed that "[w]ithin indigenous legal traditions, creation stories... give guidance about how to live with the world", rather than live at odds with it. A 2012 international Declaration of Indigenous Peoples found that modern laws destroy the earth because they do not respect the "natural order of Creation". The Declaration observed that humans "have our place and our responsibilities within Creation's sacred order" and benefit from "sustaining joy as things occur in harmony with the Earth and with all life that it creates and sustains".

Indigenous worldviews align with and have accelerated the development of rights of nature law, including in Ecuador and Bolivia. Ecuador amended its constitution in 2008 to recognize the rights of nature in light of the perceived need to better protect and respect Pachamama, a term that embodies both the physical and the spiritual aspects of the natural world. Bolivia similarly amended its constitution and enacted nature's rights statutes to reflect traditional Indigenous respect for Pachamama, and a worldview of natural systems and humans as part of one family.

New Zealand law professor Catherine Iorns Magallanes observed that traditional Indigenous worldviews embody a connection with nature is so deep that nature is regarded as a living ancestor. From this worldview arises responsibilities to protect nature as one would a family member, and the need for a legal structure that reflects a primary frame of responsibilities to the natural world as kin.

Common roots with world religions

Many of the world's other religious and spiritual traditions offer insights consistent with a nature's rights worldview. Eastern religious and philosophical traditions embrace a holistic conception of spirituality that includes the Earth. Chinese Daoism and Neo-Confucianism, as well as Japanese Buddhism, teach that the world is a dynamic force field of energies known as bussho (Buddha nature or qi), the material force that flows through humans, nature, and universe. As the eleventh century pioneering Neo-Confucianist philosopher Zhang Zai explained, "that which extends throughout the universe I regard as my body and that which directs the universe I consider as my nature".

In both Hinduism and Buddhism, karma ("action" or "declaration" in Sanskrit) reflects the reality of humanity's networked interrelations with Earth and universe. Buddhist concepts of “co-dependent arising” similarly hold that all phenomena are intimately connected. Mahayana Buddhism's "Indra's Net" symbolizes a universe of infinitely repeated mutual relations, with no one thing dominating.

Western religious and philosophical traditions have recognized the context of Earth and universe in providing spiritual guidance as well. From the Neolithic through the Bronze ages, the societies of "Old Europe" revered numerous female deities as incarnations of Mother Earth. In early Greece, the earth goddess Gaia was worshipped as a supreme deity. In the Philebus and Timaeus, Plato asserted that the "world is indeed a living being endowed with a soul and intelligence (...) a single visible living entity containing all other living entities, which by their nature are all related". Medieval theologian St. Thomas Aquinas later wrote of the place of humans, not at the center of being, but as one part of an integrated whole with the universe as primary, stating that “The order of the universe is the ultimate and noblest perfection in things."

More recently, Pope Benedict XVI, head of the Catholic church, reflected that, "[t]he obedience to the voice of Earth is more important for our future happiness... than the desires of the moment. Our Earth is talking to us and we must listen to it and decipher its message if we want to survive." His successor, Pope Francis, has been particularly vocal on humanity's relationship with the Earth, describing how humans must change their current actions in light of the fact that "a true 'right of the environment' does exist". He warned against humanity's current path, stating that "the deepest roots of our present failures" lie in the direction and meaning of economic growth, and the overarching rule of a "deified market".

The Qur’an, Islam's primary authority in all matters of individual and communal life, reflects that "the whole creation praises God by its very being". Scholars describe the "ultimate purpose of the Shari'ah" as "the universal common good, the welfare of the entire creation," and note that "not a single creature, present or future, may be excluded from consideration in deciding a course of action."

Bringing together Western and Indigenous traditions, Archbishop Desmond Tutu spoke of "Ubuntu", an African ethical concept that translates roughly to "I am because you are", observing that: "Ubuntu speaks particularly about the fact that you can't exist as a human being in isolation. It speaks about our interconnectedness... We think of ourselves far too frequently as just individuals, separated from one another, whereas you are connected and what you do affects the whole world."

Common roots with human rights

Human rights have been developing over centuries, with the most notable outgrowth being the adoption of Universal Declaration of Human Rights (UDHR) by the United Nations in 1948. Key to the development of those rights are the concepts of natural rights, and rights of humans emanating from the existence of humanity.

Roderick Fraser Nash, professor of history and environmental studies at the University of California, Santa Barbara, traced the history of rights for species and the natural world back to the thirteenth century Magna Carta's launch of the concept of "natural rights" that underlies modern rights discourse.

Peter Burdon, professor at the University of Adelaide Law School and an Earth Jurisprudence scholar, has expanded upon Nash's analysis, offering that seventeenth century English philosopher and physician John Locke's transformative natural rights thesis led to the American Revolution, through the concept that the British monarchy was denying colonists their natural rights. Building on that concept, U.S. President, attorney, and philosopher Thomas Jefferson argued that the "laws of nature and of nature's God" reveal "self-evident" truths that "all Men are created equal" in their possession of "certain unalienable rights", particularly "life, liberty, and the pursuit of happiness". The 1789 French Declaration of the Rights of Man and of the Citizen later recognized as well the "natural, inalienable and sacred rights of Man", adding that the "final end of every political institution is the preservation of the natural and imprescriptible rights of Man."

The expansion of rights continued out to animals, with eighteenth-nineteenth century English philosopher and legal theorist Jeremy Bentham claiming that the “day may come when the rest of the animal creation may acquire those rights which never could have been withholden from them but by the hand of tyranny”. Nineteenth century linguist and scholar Edward Payson Evans observed that:

"[i]n tracing the history of the evolution of ethics we find the recognition of mutual rights and duties confined at first to members of the same horde or tribe, then extended to worshippers of the same gods, and gradually enlarged so as to include every civilized nation, until at length all races of men are at least theoretically conceived as being united in a common bond of brotherhood and benevolent sympathy, which is now slowly expanding so as to comprise not only the higher species of animals, but also every sensitive embodiment of organic life."

The 1948 adoption of the Universal Declaration of Human Rights (UDHR) by the United Nations was another milestone, underpinned by the belief that fundamental human rights arise from "the fact of existing". The movement for rights of nature built on this belief, arguing that if "existence" is the defining condition for fundamental rights, this defining condition could not be limited to the rights of only one form of existence, and that all forms of existence should enjoy fundamental rights. For example, Aldo Leopold's land ethic explicitly recognized nature's "right to continued existence" and sought to "change the role of Homo sapiens from conqueror of the land-community to plain member and citizen of it".

Proponents of the rights of nature also contend that from the abolition of slavery, to the granting of the right to vote to women, to the civil rights movement, and the recognition of other fundamental rights, societies have continued to expand rights in parallel with a growing acceptance of the inherent moral worth of the potential new rights holders. And, that this expansion of the circle of community ought to continue to grow to encompass the natural world, a position that has seen growing acceptance in the late twentieth century and early twenty-first.

Proponents suggest that rights derived from existence in nature do not confer human rights to all beings, but rather confer unique rights to different kinds of beings. Thomas Berry put forth the theory that rights "are species specific and limited"; that is, "rivers have river rights", "birds have bird rights", and "humans have human rights". In his view, the difference is "qualitative, not quantitative".

Extending this point, the common ethical and moral grounding of human rights and the rights of nature gives rise to the concept of "co-violations" of rights, defined as a "situation in which governments, industries, or others violate both the rights of nature and human rights, including indigenous rights, with the same action". For example, in the Ecuadorian Amazon, pollution from Texaco's (now Chevron) oil drilling operations from 1967 to 1992 resulted in an epidemic of birth defects, miscarriages, and an estimated 1,400 cancer deaths, that were particularly devastating to indigenous communities. These operations further caused more than one million acres of deforestation and polluted local waterways with 18 billion gallons of toxic wastewater and contaminants, severely damaging a formerly pristine rainforest of extraordinary biodiversity. Asserting that the same human actions that created such impacts violated the fundamental rights of both people and natural systems, it is argued that ethical and legal theories that recognize both sets of rights will better guide human behavior to recognize and respect humans' interconnected relationships with each other and the natural world.

As with the recognition of human rights, legal scholars find that recognition of the rights of nature alters the framework of human laws and practices. Harvard Law professor Laurence Tribe theorized further that "choosing to accord nature a fraternal rather than an exploited role... might well make us different persons from the manipulators and subjugators we are in danger of becoming".

20th and 21st century developments

The adoption of the UDHR in 1948 formalized recognition of broad categories of inalienable human rights globally. These include recognition that "[a]ll human beings are born free and equal in dignity and rights", that "[e]veryone has the right to life, liberty and security of person", and that "[e]veryone has the right to an effective remedy by the competent national tribunals for acts violating the fundamental rights granted". Recognition of fundamental rights in "soft law" instruments such as the UDHR provided guidance to nations around the world, who have since developed constitutional provisions, statutes, court decisions, regulations, and other bodies of law based on the UDHR and the human rights it champions.

Decades later, USC law professor Christopher Stone called for recognition of the legal standing and associated rights of the natural world as well, consistent with the "successive extension of rights" throughout legal history. As was done in the UDHR, Stone outlined the necessary elements of nature's participation in human legal systems, describing such a legal system as necessarily including: recognition of injuries as subject to redress by public body, standing to institute legal actions (with guardians acting on behalf of the natural entity), redress calculated for natural entity's own damages, and relief running to the benefit of the injured natural entity.

In addition to Stone's legal work, other late twentieth and early twenty-first century drivers of the rights of nature movement include indigenous perspectives and the work of the indigenous rights movement; the writings of Arne Naess and the Deep Ecology movement; Thomas Berry's 2001 jurisprudential call for recognizing the laws of nature as the "primary text"; the publication of Cormac Cullinan's Wild Law book in 2003, followed by the creation of an eponymous legal association in the UK; growing concern about corporate power through the expansion of legal personhood for corporations; adoption by U.S. communities of local laws addressing rights of nature; the creation of the Global Alliance of the Rights of Nature in 2010 (GARN; a nonprofit advancing rights on nature worldwide); and mounting global concerns with species losses, ecosystem destruction, and the existential threat of climate change.

These and other factors supported the development of the 2010 Universal Declaration of the Rights of Mother Earth (UDRME). The UDRME was adopted by representatives of 130 nations at the World People's Conference on Climate Change and the Rights of Mother Earth, convened in Bolivia following the concerns of many regarding the disappointing results of the 2009 Copenhagen climate negotiations. Just as the U.N. recognized human rights as arising from existence, so did the UDRME find that the "inherent rights of Mother Earth are inalienable in that they arise from the same source as existence". Like the UDHR, the UDRME defends the rights-bearing entity (nature and her elements) from the excesses of governing authorities. These rights include, among others, the recognition that "Mother Earth and all beings of which she is composed have... the right to life and to exist" as well as the "right to integral health". The UDRME adds that "[e]ach being has the right to a place and to play its role in Mother Earth for her harmonious functioning".

Just as the rights protected by the UDHR are enforceable by the "right to an effective remedy by the competent national tribunals", so too does the UDRME specifically require humans and their institutions to "recognize and promote the full implementation and enforcement of the rights and obligations recognized in this Declaration". The UDRME addresses enforcement by requiring "damages caused by human violations of the inherent rights" to be "rectified", with those responsible "held accountable".[88] Moreover, it calls on states to "empower human beings and institutions to defend the rights of Mother Earth and of all beings".

Bolivian President Evo Morales urged then-U.N. Secretary-General Ban Ki-Moon to make U.N. adoption of the UDRME a priority. While that recommendation remains to be addressed, since then the UDRME has served to inform other international and national efforts, such as a 2012 Resolution by the International Union for Conservation of Nature (IUCN) proposing a Universal Declaration of the Rights of Nature. The Incorporation of the Rights of Nature was adopted at the IUCN World Conservation Congress in Hawaii (2016).

As of 2021 rights of nature has been reflected in treaties, constitutions, court decisions, and statutory and administrative law at all levels of government. Craig Kauffman, political science professor at the University of Oregon, and scholar of nature's rights and global governance, contends that evolving rights of nature initiatives and networks represent an "important new global movement" arising from "an informal global governance system... being constructed by citizens disillusioned by the failure of governments to take stronger actions to address the dual crises of climate change and biodiversity loss".

Rights of nature law

The early 2000s saw a significant expansion of rights of nature law, in the form of constitutional provisions, treaty agreements, national and subnational statutes, local laws, and court decisions. As of 2021, nature's rights laws exist in 17 countries, including in Canada, seven Tribal Nations in the U.S. and Canada, and dozens of cities and counties throughout the United States. The total number of countries with either existing or pending rights of nature legal provisions was 28 as of 2019.

New Zealand - in 2012 a treaty agreement between the government and the indigenous group Maori iwi established the Whanganui River (top image), and its tributaries as a legal entity with its own standing.Similarly, Mount Taranaki (bottom image) was recognized in 2014 as "a legal personality, in its own right".

Treaties

New Zealand

Legal standing for natural systems in New Zealand arose alongside new attention paid to long-ignored treaty agreements with the Indigenous Maori. In August 2012, a treaty agreement signed with the Maori iwi recognized the Whanganui River and tributaries as a legal entity, an "indivisible and living whole" with its own standing. The national Te Awa Tupua Act was enacted in March 2017 to further formalize this status.

In 2013, the Te Urewera Forest treaty agreement similarly recognized the legal personhood of the Forest, with the Te Urewera Act signed into law in 2014 to formalize this status. In 2017 a treaty settlement with the Maori was signed that recognized Mount Taranaki as "a legal personality, in its own right".

Each of these developments advanced the indigenous principle that the ecosystems are living, spiritual beings with intrinsic value, incapable of being owned in an absolute sense.

Constitutional law

Ecuador

See also: Rights of nature in Ecuador

 

Yasuní National Park, Ecuador

In 2008, the people of Ecuador amended their Constitution to recognize the inherent rights of nature, or Pachamama. The new text arose in large part as a result of cosmologies of the indigenous rights movement and actions to protect the Amazon, consistent with the concept of sumak kawsay ("buen vivir" in Spanish, "good living" in English), or encapsulating a life in harmony with nature with humans as part of the ecosystem. Among other provisions, Article 71 states that "Nature or Pachamama, where life is reproduced and exists, has the right to exist, persist, maintain itself and regenerate its own vital cycles, structure, functions and its evolutionary processes." The Article adds enforcement language as well, stating that "Any person... may demand the observance of the rights of the natural environment before public bodies", and echoing Christopher Stone, Article 72 adds that “Nature has the right to be completely restored... independent of the obligation... to compensate people”.

Ho-Chunk Nation of Wisconsin

In 2015 the Ho-Chunk Nation of Wisconsin passed a resolution amending their constitution to include the rights of nature. By 2020 a working group was determining how to integrate the resolution into their constitution, laws, regulations, and processes.

Judicial decisions

Turag River, near Dhaka, Bangladesh

Bangladesh

In 2019, the High Court of Bangladesh ruled on a case addressing pollution of and illegal development along the Turag River, an upper tributary of the Buriganga.

Among its findings, the high court recognized the river as a living entity with legal rights, and it further held that the same would apply to all rivers in Bangladesh. The court ordered the National River Protection Commission to serve as the guardian for the Turag and other rivers.

Colombia

Atrato River in Colombia - in a 2016 ruling by the Constitutional Court involving the river's pollution, the court stated that the river is a subject of rights, and that humans are "only one more event within a long evolutionary chain [and] in no way... owner of other species, biodiversity or natural resources, or the fate of the planet".

Colombia has not adopted statutes or constitutional provisions addressing nature's rights (as of 2019). However, this has not prevented Colombian courts from finding nature's rights as inherent. In a 2016 case, the Colombia Constitutional Court ordered cleanup of the polluted Atrato River, stating that nature is a "true subject of rights that must be recognized by states and exercised... for example, by the communities that inhabit it or have a special relationship with it”. The court added that humans are “only one more event within a long evolutionary chain [and] in no way... owner of other species, biodiversity or natural resources, or the fate of the planet".

In 2018, the Colombia Supreme Court took up a climate change case by a group of children and young adults that also raised fundamental rights issues. In addition to making legal findings related to human rights, the court found that the Colombian Amazon is a "'subject of rights', entitled to protection, conservation, maintenance and restoration". It recognized the special role of Amazon deforestation in creating greenhouse gas emissions in Colombia, and as a remedy ordered the nation and its administrative agencies to ensure a halt to all deforestation by 2020. The court further allocated enforcement power to the plaintiffs and affected communities, requiring the agencies to report to the communities and empowering them to inform the court if the agencies were not meeting their deforestation targets.

Ecuador

A significant body of case law has been expanding in Ecuador to implement the nation's constitutional provisions regarding the rights of nature. Examples include lawsuits in the areas of biodigestor pollution, impaired flow in the Vilcabamba River, and hydropower.

India

Gangotri Glacier, a source of the Ganga river

 

The river Yamuna at Yamunotri Glacier

As in Colombia, as of 2019 no statutes or constitutional provisions in India specifically identified rights of nature. Nevertheless, the India Supreme Court in 2012 set the stage for cases to come before it on rights of nature, finding that "Environmental justice could be achieved only if we drift away from the principle of anthropocentric to ecocentric... humans are part of nature and non-human has intrinsic value."

The Uttarakhand High Court applied the principle of ecocentric law in 2017, recognizing the legal personhood of the Ganga and Yamuna rivers and ecosystems, and calling them "living human entities" and juridical and moral persons. The court quickly followed with similar judgments for the glaciers associated with the rivers, including the Gangotri and Yamunotri, and other natural systems. While the India Supreme Court stayed the Ganga and Yamuna judgment at the request of local authorities, those authorities supported the proposed legal status in concept, but were seeking "implementation guidance".

National, sub-national, and local law

Bolivia

Following adoption of nature's rights language in its 2009 Constitution, in 2010 Bolivia's Legislature passed the Law of the Rights of Mother Earth, Act No. 071. Bolivia followed this broad outline of nature's rights with the 2012 Law of Mother Earth and Integral Development for Living Well, Act. No. 300, which provided some implementation details consistent with nature's rights. It states in part that the "violation of the rights of Mother Earth, as part of comprehensive development for Living Well, is a violation of public law and the collective and individual rights". While a step forward, this enforcement piece has not yet risen to the level of a specific enforcement mechanism.

Mexico

State, regional, and local laws and local constitutional provisions have been arising in Mexico, including adoption in the constitutions of the Mexican states of Colima and Guerrero, and that of Mexico City.

Ponca

In 2017, the Ponca Nation enacted a rights of nature law which is a resolution that gives the Ponca Tribal Court the power to punish crimes against nature with prison and fines.

Uganda

Part 1, Section 4 of Uganda's 2019 National Environment Act addresses the Rights of Nature, stating in part that "Nature has the right to exist, persist, maintain and regenerate its vital cycles, structure, functions and its processes in evolution." Advocates who had sought inclusion of such language observed that "Ugandans' right to a healthy environment cannot be realised unless the health of Nature herself is protected," and that the language adoption reflected "recent gains for the growing African movement for Earth Jurisprudence".

United States

At the local level dozens of ordinances with rights of nature provisions have been passed as of 2019 throughout the United States, and in tribal lands located within the U.S. boundaries. Most were passed in reaction to a specific threat to local well-being, such as threats posed by hydrofracking, groundwater extraction, gravel mining, and fossil fuel extraction. For example, Pittsburgh, Pennsylvania passed an anti-fracking law that included the following provision to buttress protections: "Natural communities and ecosystems... possess inalienable and fundamental rights to exist and flourish." The ordinance continues that "Residents... shall possess legal standing to enforce those rights."

Santa Monica State Beach - in 2013 the city adopted a "Sustainability Rights Ordinance", recognizing the "fundamental and inalienable rights" of "natural communities and ecosystems"

Residents in Santa Monica, California proactively sought to recognize nature's rights in local law after the U.S. Supreme Court's expansion of corporate rights in Citizens United v. FEC. In 2013 the Santa Monica City Council adopted a "Sustainability Rights Ordinance", recognizing the "fundamental and inalienable rights" of "natural communities and ecosystems" in the city to "exist and flourish". The ordinance emphasized that "[c]orporate entities... do not enjoy special privileges or powers under the law that subordinate the community's rights to their private interests". It specifically defined "natural communities and ecosystems" to include "groundwater aquifers, atmospheric systems, marine waters, and native species". Santa Monica updated its Sustainable City Plan in 2014 to reinforce its codified commitment to nature's rights. In 2018, the city council adopted a Sustainable Groundwater Management Ordinance that specifically referenced the inherent rights of the local aquifer to flourish.

In November, 2020, voters in Orange County, Florida passed a charter amendment for the "right to clean water" by a margin of 89% that protects waterways in the county from pollution and enables citizens to bring lawsuits to defend against such pollution, becoming the largest community in the country to enact such a rights of nature initiative. It has prompted the Florida Right To Clean Water direct initiative to incorporate the principle into the state constitution, which is gathering petition signatures to have an amendment put onto the 2024 ballot for consideration by all Florida voters. In his January 2022 monthly newsletter, Jim Hightower identified the Florida initiative as, "the epicenter of today’s Rights of Nature political movement".

Toledo, Ohio passed the “Lake Erie Bill of Rights” (LEBOR). In 2019 it was struck down by the Supreme Court of Ohio in 2020. BP North America spent almost $300,000 fighting the bill through a front group.

International bodies and soft law

United Nations

Advancements during the early twenty-firstst century in international "soft law" (quasi-legal instruments generally without legally binding force) have precipitated broader discussions about the potential for integrating nature's rights into legal systems. The United Nations has held nine "Harmony with Nature" General Assembly Dialogues as of 2019 on Earth-centered governance systems and philosophies, including discussions of rights of nature specifically. The companion United Nations Harmony with Nature initiative compiles rights of nature laws globally and offers a U.N. "Knowledge Network" of Earth Jurisprudence practitioners across disciplines. These U.N. Dialogues and the Harmony with Nature initiative may provide a foundation for development of a United Nations-adopted Universal Declaration of the Rights of Nature which, like the U.N.'s Universal Declaration of Human Rights, could form the foundation for rights-based laws worldwide. A model could be the 2010 UDRME, an informal, but widely-supported nature's rights agreement based on the UDHR.

International Union for Conservation of Nature

In 2012, the International Union for Conservation of Nature (IUCN, the only international observer organization to the U.N. General Assembly with expertise in the environment) adopted a resolution specifically calling for a Universal Declaration of the Rights of Nature. The IUCN reaffirmed its commitment to nature's rights at its next meeting in 2016, where the body voted to build rights of nature implementation into the upcoming, four-year IUCN Workplan. The IUCN's subgroup of legal experts, the World Commission on Environmental Law, later issued an "IUCN World Declaration on the Environmental Rule of Law" recognizing that "Nature has the inherent right to exist, thrive, and evolve".

Related initiatives

The development of stronger and more active transnational rights of nature networks during the early 2000s, is a likely cause for the greater adoption of those championed principles into law. This has occurred in close integration with other, system-changing initiatives and movements for rights, including: development and implementation of new economic and finance models that seek to better reflect human rights and nature's rights; indigenous leadership to advance both the rights of indigenous peoples and nature's rights; international social movements such as the human right to water; advancement of practical solutions consistent with a nature's rights frame, such as rewilding; and rights of nature movement capacity building, including through development of nature's rights movement hubs globally.

To illustrate implementation of nature's rights laws, the Global Alliance for the Rights of Nature has established International Rights of Nature Tribunals. These tribunals are a civil society initiative and they issue non-binding recommendations. The tribunals bring together advocates of rights of nature, human rights, and rights of indigenous peoples into a process similar to the Permanent Peoples' Tribunals. The goal of the tribunals is to provide formal public recognition, visibility, and voice to the people and natural systems injured by alleged violations of fundamental rights and marginalized in current law, and to offer a model for redress for such injuries.

As awareness of rights of nature law and jurisprudence has spread, a new field of academic research is developing, where legal scholars and other scholars have begun to offer strategies and analysis to drive broader application of such laws, particularly in the face of early implementation successes and challenges.

In popular culture

The 2018 documentary Rights of Nature: A Global Movement, directed by Isaac Goeckeritz, Hal Crimmel and Valeria Berros explores the challenges of creating new legal structures in relation to Rights of Nature.

A documentary film entitled Invisible Hand about the rights of nature movement, directed by Joshua Boaz Pribanic and Melissa Troutman of Public Herald, was released in 2020, executive-produced and narrated by actor Mark Ruffalo. It won four Best Documentary Awards.

The Overstory, which won the 2019 Pulitzer Prize for Fiction and spent over a year on the New York Times bestseller list, examined relationships with and rights of trees.

The podcast Damages explores the concept of the rights of nature in different contexts.

The Daily Show covered the concept of the rights of nature in an episode.

Notable documents

• Republic of Ecuador (2008). "Constitution, Chapter 7". (First constitutional provisions recognizing nature's rights)
• Plurinational State of Bolivia (2010). "Law of the Rights of Mother Earth, Law No. 071 2010". (Early national law recognizing nature's rights)
• Constitutional Court of the Republic of Colombia (2016). "Judgment in Case T-622 of 2016, Proceeding T-5.016.242" (PDF). (Stating that the contaminated Atrato River is a “true subject of rights”)
• Supreme Court of the Republic of Colombia (2018). "Judgment in Case STC-4360-2018, Filing no. 11001-22-03-000-2018-00319-01". (Recognition of the inherent rights of the Colombian Amazon to a healthy climate)
• Supreme Court of India (2012). "Judgment: T.N. Godavarman Thirumulpad Vs. Union of India & Others". (Constitutional law case calling for a shift from anthropocentric to ecocentric principles of justice)
• High Court of Uttarakhand at Naintal (2017). "Judgment by High Court of Uttarakhand at Naintal regarding Writ Petition (PIL) No. 126 of 2014". (Court decision recognizing the legal personhood of the Ganga and Yamuna rivers in India)
• High Court of Uttarakhand at Naintal (2017). "Judgment by High Court of Uttarakhand at Naintal regarding Writ Petition (PIL) No. 140 of 2015" (PDF). (Court decision recognizing the legal personhood of glaciers and associated natural systems in India)
• New Zealand (2017). Te Awa Tupua (Whanganui River Claims Settlement) Act. (First national law recognizing a river as a legal person)
• New Zealand (2014). Te Urewera Act. (National law recognizing a former national park as a legal person)
• Uganda (2019). "National Environment Act" (PDF). (First national law in Africa recognizing nature's rights)
• "Constitution of Mexico City" (PDF). 2017. (Constitutional provision recognizing “ecosystems and species as a collective entity subject of rights” in one of the world's largest cities)
• "Municipal Code, Title 6, Art 1, Ch. 618, 'Marcellus Shale Natural Gas Drilling Ordinance'". City of Pittsburgh, PA, U.S. 2010. (Largest U.S. city to recognize nature's rights in law, as of 2019)
• "Ordinance of the City Council of the City of Santa Monica Establishing Sustainability Rights, Santa Monica Municipal Code, Art. 4, Ch. 4.75, (April 9, 2013)". City of Santa Monica, CA, U.S. 2013. (First local law on the U.S. West Coast recognizing nature's rights)
• "Universal Declaration of the Rights of Mother Earth". World People's Conference on Climate Change and the Rights of Mother Earth. 2010. (First worldwide declaration of nature's rights; modeled on Universal Declaration of Human Rights)
• International Union for Conservation of Nature (2012). "Resolution 100, 'Incorporation of the Rights of Nature as the organizational focal point in IUCN's decision making'". (Call for a Universal Declaration of the Rights of Nature by international experts organization that holds both observer and consultative status at the United Nations)
• International Union for Conservation of Nature, World Commission on Environmental Law (2017). "IUCN World Declaration on the Environmental Rule of Law" (PDF). (Statement by IUCN legal experts recognizing that “Nature has the inherent right to exist, thrive, and evolve”)
• Declaration of the Rights of the Moon (2021)