Unbounded operator

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

In mathematics, more specifically functional analysis and operator theory, the notion of unbounded operator provides an abstract framework for dealing with differential operators, unbounded observables in quantum mechanics, and other cases.

The term "unbounded operator" can be misleading, since

• "unbounded" should sometimes be understood as "not necessarily bounded";
• "operator" should be understood as "linear operator" (as in the case of "bounded operator");
• the domain of the operator is a linear subspace, not necessarily the whole space;
• this linear subspace is not necessarily closed; often (but not always) it is assumed to be dense;
• in the special case of a bounded operator, still, the domain is usually assumed to be the whole space.

In contrast to bounded operators, unbounded operators on a given space do not form an algebra, nor even a linear space, because each one is defined on its own domain.

The term "operator" often means "bounded linear operator", but in the context of this article it means "unbounded operator", with the reservations made above. The given space is assumed to be a Hilbert space. Some generalizations to Banach spaces and more general topological vector spaces are possible.

Short history

The theory of unbounded operators developed in the late 1920s and early 1930s as part of developing a rigorous mathematical framework for quantum mechanics. The theory's development is due to John von Neumann and Marshall Stone. Von Neumann introduced using graphs to analyze unbounded operators in 1932.

Definitions and basic properties

Let X, Y be Banach spaces. An unbounded operator (or simply operator) T : D(T) → Y is a linear map T from a linear subspace D(T) ⊆ X—the domain of T—to the space Y. Contrary to the usual convention, T may not be defined on the whole space X.

An operator T is said to be closed if its graph Γ(T) is a closed set. (Here, the graph Γ(T) is a linear subspace of the direct sum X ⊕ Y, defined as the set of all pairs (x, Tx), where x runs over the domain of T .) Explicitly, this means that for every sequence {x_n} of points from the domain of T such that x_n → x and Tx_n → y, it holds that x belongs to the domain of T and Tx = y. The closedness can also be formulated in terms of the graph norm: an operator T is closed if and only if its domain D(T) is a complete space with respect to the norm:

${\displaystyle \Vert x{\Vert }_T=\sqrt{\Vert x{\Vert }^2+\Vert Tx{\Vert }^2}.}$

An operator T is said to be densely defined if its domain is dense in X. This also includes operators defined on the entire space X, since the whole space is dense in itself. The denseness of the domain is necessary and sufficient for the existence of the adjoint (if X and Y are Hilbert spaces) and the transpose; see the sections below.

If T : X → Y is closed, densely defined and continuous on its domain, then its domain is all of X.

A densely defined symmetric operator T on a Hilbert space H is called bounded from below if T + a is a positive operator for some real number a. That is, ⟨Tx|x⟩ ≥ −a ||x||² for all x in the domain of T (or alternatively ⟨Tx|x⟩ ≥ a ||x||² since a is arbitrary). If both T and −T are bounded from below then T is bounded.

Example

Let C([0, 1]) denote the space of continuous functions on the unit interval, and let C¹([0, 1]) denote the space of continuously differentiable functions. We equip ${\displaystyle C([0,1])}$ with the supremum norm, ${\displaystyle \Vert \cdot {\Vert }_{\mathrm{\infty }}}$, making it a Banach space. Define the classical differentiation operator d/dx : C¹([0, 1]) → C([0, 1]) by the usual formula:

${\displaystyle \left(\frac{d}{dx}f\right)(x)=\underset{h\to 0}{lim}\frac{f(x+h)-f(x)}{h},\mathrm{\forall }x\in [0,1].}$

Every differentiable function is continuous, so C¹([0, 1]) ⊆ C([0, 1]). We claim that d/dx : C([0, 1]) → C([0, 1]) is a well-defined unbounded operator, with domain C¹([0, 1]). For this, we need to show that ${\displaystyle \frac{d}{dx}}$ is linear and then, for example, exhibit some ${\displaystyle \{f_n{\}}_n\subset C^1([0,1])}$ such that ${\displaystyle \Vert f_n{\Vert }_{\mathrm{\infty }}=1}$ and ${\displaystyle \underset{n}{sup}\Vert \frac{d}{dx}{f}_n{\Vert }_{\mathrm{\infty }}=+\mathrm{\infty }}$.

This is a linear operator, since a linear combination a f  + bg of two continuously differentiable functions  f , g is also continuously differentiable, and

${\displaystyle \left(\frac{d}{dx}\right)(af+bg)=a\left(\frac{d}{dx}f\right)+b\left(\frac{d}{dx}g\right).}$

The operator is not bounded. For example,

${\displaystyle \{\begin{array}{l}{f}_n:[0,1]\to [-1,1]\\ f_n(x)=\sin (2\pi nx)\end{array}}$

satisfy

${\displaystyle {\Vert f_n\Vert }_{\mathrm{\infty }}=1,}$

but

${\displaystyle {\Vert \left(\frac{d}{dx}{f}_n\right)\Vert }_{\mathrm{\infty }}=2\pi n\to \mathrm{\infty }}$

as ${\displaystyle n\to \mathrm{\infty }}$.

The operator is densely defined, and closed.

The same operator can be treated as an operator Z → Z for many choices of Banach space Z and not be bounded between any of them. At the same time, it can be bounded as an operator X → Y for other pairs of Banach spaces X, Y, and also as operator Z → Z for some topological vector spaces Z. As an example let I ⊂ R be an open interval and consider

${\displaystyle \frac{d}{dx}:\left(C^1(I),\Vert \cdot {\Vert }_{C^1}\right)\to \left(C(I),\Vert \cdot {\Vert }_{\mathrm{\infty }}\right),}$

where:

${\displaystyle \Vert f{\Vert }_{C^1}=\Vert f{\Vert }_{\mathrm{\infty }}+\Vert f^{\prime }{\Vert }_{\mathrm{\infty }}.}$

Adjoint

The adjoint of an unbounded operator can be defined in two equivalent ways. Let ${\displaystyle T:D(T)\subseteq H_1\to H_2}$ be an unbounded operator between Hilbert spaces.

First, it can be defined in a way analogous to how one defines the adjoint of a bounded operator. Namely, the adjoint ${\displaystyle T^{\ast }:D\left(T^{\ast }\right)\subseteq H_2\to H_1}$ of T is defined as an operator with the property:

$${\displaystyle ⟨Tx\mid y{⟩}_2={⟨x\mid T^{\ast }y⟩}_1,x\in D(T).}$$

More precisely, ${\displaystyle T^{\ast }y}$ is defined in the following way. If ${\displaystyle y\in H_2}$ is such that ${\displaystyle x\mapsto ⟨Tx\mid y⟩}$ is a continuous linear functional on the domain of T, then ${\displaystyle y}$ is declared to be an element of ${\displaystyle D\left(T^{\ast }\right),}$ and after extending the linear functional to the whole space via the Hahn–Banach theorem, it is possible to find some ${\displaystyle z}$ in ${\displaystyle H_1}$ such that

$${\displaystyle ⟨Tx\mid y{⟩}_2=⟨x\mid z{⟩}_1,x\in D(T),}$$

since Riesz representation theorem allows the continuous dual of the Hilbert space ${\displaystyle H_1}$ to be identified with the set of linear functionals given by the inner product. This vector ${\displaystyle z}$ is uniquely determined by ${\displaystyle y}$ if and only if the linear functional ${\displaystyle x\mapsto ⟨Tx\mid y⟩}$ is densely defined; or equivalently, if T is densely defined. Finally, letting ${\displaystyle T^{\ast }y=z}$ completes the construction of ${\displaystyle T^{\ast },}$ which is necessarily a linear map. The adjoint ${\displaystyle T^{\ast }y}$ exists if and only if T is densely defined.

By definition, the domain of ${\displaystyle T^{\ast }}$ consists of elements ${\displaystyle y}$ in ${\displaystyle H_2}$ such that ${\displaystyle x\mapsto ⟨Tx\mid y⟩}$ is continuous on the domain of T. Consequently, the domain of ${\displaystyle T^{\ast }}$ could be anything; it could be trivial (that is, contains only zero). It may happen that the domain of ${\displaystyle T^{\ast }}$ is a closed hyperplane and ${\displaystyle T^{\ast }}$ vanishes everywhere on the domain. Thus, boundedness of ${\displaystyle T^{\ast }}$ on its domain does not imply boundedness of T. On the other hand, if ${\displaystyle T^{\ast }}$ is defined on the whole space then T is bounded on its domain and therefore can be extended by continuity to a bounded operator on the whole space. If the domain of ${\displaystyle T^{\ast }}$ is dense, then it has its adjoint ${\displaystyle T^{\ast \ast }.}$ A closed densely defined operator T is bounded if and only if ${\displaystyle T^{\ast }}$ is bounded.

The other equivalent definition of the adjoint can be obtained by noticing a general fact. Define a linear operator ${\displaystyle J}$ as follows:

$${\displaystyle \{\begin{array}{l}J:H_1\oplus H_2\to H_2\oplus H_1\\ J(x\oplus y)=-y\oplus x\end{array}}$$

Since ${\displaystyle J}$ is an isometric surjection, it is unitary. Hence: ${\displaystyle J(\mathrm{\Gamma }(T){)}^{\mathrm{\perp }}}$ is the graph of some operator ${\displaystyle S}$ if and only if T is densely defined. A simple calculation shows that this "some" ${\displaystyle S}$ satisfies:

$${\displaystyle ⟨Tx\mid y{⟩}_2=⟨x\mid Sy{⟩}_1,}$$

for every x in the domain of T. Thus ${\displaystyle S}$ is the adjoint of T.

It follows immediately from the above definition that the adjoint ${\displaystyle T^{\ast }}$ is closed. In particular, a self-adjoint operator (meaning ${\displaystyle T=T^{\ast }}$) is closed. An operator T is closed and densely defined if and only if ${\displaystyle T^{\ast \ast }=T.}$

Some well-known properties for bounded operators generalize to closed densely defined operators. The kernel of a closed operator is closed. Moreover, the kernel of a closed densely defined operator ${\displaystyle T:H_1\to H_2}$ coincides with the orthogonal complement of the range of the adjoint. That is,

$${\displaystyle \ker (T)=\mathrm{ran}(T^{\ast }{)}^{\mathrm{\perp }}.}$$

von Neumann's theorem states that ${\displaystyle T^{\ast }T}$ and ${\displaystyle T{T}^{\ast }}$ are self-adjoint, and that ${\displaystyle I+T^{\ast }T}$ and ${\displaystyle I+T{T}^{\ast }}$ both have bounded inverses. If ${\displaystyle T^{\ast }}$ has trivial kernel, T has dense range (by the above identity.) Moreover:

T is surjective if and only if there is a ${\displaystyle K>0}$ such that ${\displaystyle \Vert f{\Vert }_2\le K{\Vert T^{\ast }f\Vert }_1}$ for all ${\displaystyle f}$ in ${\displaystyle D\left(T^{\ast }\right).}$ (This is essentially a variant of the so-called closed range theorem.) In particular, T has closed range if and only if ${\displaystyle T^{\ast }}$ has closed range.

In contrast to the bounded case, it is not necessary that ${\displaystyle (TS{)}^{\ast }=S^{\ast }{T}^{\ast },}$ since, for example, it is even possible that ${\displaystyle (TS{)}^{\ast }}$ does not exist. This is, however, the case if, for example, T is bounded.

A densely defined, closed operator T is called normal if it satisfies the following equivalent conditions:

• ${\displaystyle T^{\ast }T=T{T}^{\ast }}$;
• the domain of T is equal to the domain of ${\displaystyle T^{\ast },}$ and ${\displaystyle \Vert Tx\Vert =\Vert T^{\ast }x\Vert }$ for every x in this domain;
• there exist self-adjoint operators ${\displaystyle A,B}$ such that ${\displaystyle T=A+iB,}$${\displaystyle T^{\ast }=A-iB,}$ and ${\displaystyle \Vert Tx{\Vert }^2=\Vert Ax{\Vert }^2+\Vert Bx{\Vert }^2}$ for every x in the domain of T.

Every self-adjoint operator is normal.

Transpose

See also: Transpose of a linear map

Let ${\displaystyle T:B_1\to B_2}$ be an operator between Banach spaces. Then the transpose (or dual) ${\displaystyle {}^{t}T:{B_2}^{\ast }\to {B_1}^{\ast }}$ of ${\displaystyle T}$ is the linear operator satisfying:

$${\displaystyle ⟨Tx,y^{\prime }⟩=⟨x,\left({}^{t}T\right)y^{\prime }⟩}$$

for all ${\displaystyle x\in B_1}$ and ${\displaystyle y\in B_2^{\ast }.}$ Here, we used the notation: ${\displaystyle ⟨x,x^{\prime }⟩=x^{\prime }(x).}$

The necessary and sufficient condition for the transpose of ${\displaystyle T}$ to exist is that ${\displaystyle T}$ is densely defined (for essentially the same reason as to adjoints, as discussed above.)

For any Hilbert space ${\displaystyle H,}$ there is the anti-linear isomorphism:

$${\displaystyle J:H^{\ast }\to H}$$

given by ${\displaystyle Jf=y}$ where ${\displaystyle f(x)=⟨x\mid y{⟩}_H,(x\in H).}$ Through this isomorphism, the transpose ${\displaystyle {}^{t}T}$ relates to the adjoint ${\displaystyle T^{\ast }}$ in the following way:

$${\displaystyle T^{\ast }=J_1\left({}^{t}T\right)J_2^{-1},}$$

where ${\displaystyle J_j:H_j^{\ast }\to H_j}$. (For the finite-dimensional case, this corresponds to the fact that the adjoint of a matrix is its conjugate transpose.) Note that this gives the definition of adjoint in terms of a transpose.

Closed linear operators

Main article: Closed linear operator

Closed linear operators are a class of linear operators on Banach spaces. They are more general than bounded operators, and therefore not necessarily continuous, but they still retain nice enough properties that one can define the spectrum and (with certain assumptions) functional calculus for such operators. Many important linear operators which fail to be bounded turn out to be closed, such as the derivative and a large class of differential operators.

Let X, Y be two Banach spaces. A linear operator A : D(A) ⊆ X → Y is closed if for every sequence {x_n} in D(A) converging to x in X such that Ax_n → y ∈ Y as n → ∞ one has x ∈ D(A) and Ax = y. Equivalently, A is closed if its graph is closed in the direct sum X ⊕ Y.

Given a linear operator A, not necessarily closed, if the closure of its graph in X ⊕ Y happens to be the graph of some operator, that operator is called the closure of A, and we say that A is closable. Denote the closure of A by A. It follows that A is the restriction of A to D(A).

A core (or essential domain) of a closable operator is a subset C of D(A) such that the closure of the restriction of A to C is A.

Example

Consider the derivative operator A = d/dx where X = Y = C([a, b]) is the Banach space of all continuous functions on an interval [a, b]. If one takes its domain D(A) to be C¹([a, b]), then A is a closed operator which is not bounded. On the other hand if D(A) = C^∞([a, b]), then A will no longer be closed, but it will be closable, with the closure being its extension defined on C¹([a, b]).

Symmetric operators and self-adjoint operators

Main article: Self-adjoint operator

An operator T on a Hilbert space is symmetric if and only if for each x and y in the domain of T we have ${\displaystyle ⟨Tx\mid y⟩=⟨x\mid Ty⟩}$. A densely defined operator T is symmetric if and only if it agrees with its adjoint T^∗ restricted to the domain of T, in other words when T^∗ is an extension of T.

In general, if T is densely defined and symmetric, the domain of the adjoint T^∗ need not equal the domain of T. If T is symmetric and the domain of T and the domain of the adjoint coincide, then we say that T is self-adjoint. Note that, when T is self-adjoint, the existence of the adjoint implies that T is densely defined and since T^∗ is necessarily closed, T is closed.

A densely defined operator T is symmetric, if the subspace Γ(T) (defined in a previous section) is orthogonal to its image J(Γ(T)) under J (where J(x,y):=(y,-x)).

Equivalently, an operator T is self-adjoint if it is densely defined, closed, symmetric, and satisfies the fourth condition: both operators T – i, T + i are surjective, that is, map the domain of T onto the whole space H. In other words: for every x in H there exist y and z in the domain of T such that Ty – iy = x and Tz + iz = x.

An operator T is self-adjoint, if the two subspaces Γ(T), J(Γ(T)) are orthogonal and their sum is the whole space ${\displaystyle H\oplus H.}$

This approach does not cover non-densely defined closed operators. Non-densely defined symmetric operators can be defined directly or via graphs, but not via adjoint operators.

A symmetric operator is often studied via its Cayley transform.

An operator T on a complex Hilbert space is symmetric if and only if the number ${\displaystyle ⟨Tx\mid x⟩}$ is real for all x in the domain of T.

A densely defined closed symmetric operator T is self-adjoint if and only if T^∗ is symmetric. It may happen that it is not.

A densely defined operator T is called positive (or nonnegative) if its quadratic form is nonnegative, that is, ${\displaystyle ⟨Tx\mid x⟩\ge 0}$ for all x in the domain of T. Such operator is necessarily symmetric.

The operator T^∗T is self-adjoint and positive for every densely defined, closed T.

The spectral theorem applies to self-adjoint operators  and moreover, to normal operators, but not to densely defined, closed operators in general, since in this case the spectrum can be empty.

A symmetric operator defined everywhere is closed, therefore bounded, which is the Hellinger–Toeplitz theorem.

Extension-related

See also: Extensions of symmetric operators

By definition, an operator T is an extension of an operator S if Γ(S) ⊆ Γ(T). An equivalent direct definition: for every x in the domain of S, x belongs to the domain of T and Sx = Tx.

Note that an everywhere defined extension exists for every operator, which is a purely algebraic fact explained at Discontinuous linear map § General existence theorem and based on the axiom of choice. If the given operator is not bounded then the extension is a discontinuous linear map. It is of little use since it cannot preserve important properties of the given operator (see below), and usually is highly non-unique.

An operator T is called closable if it satisfies the following equivalent conditions:

• T has a closed extension;
• the closure of the graph of T is the graph of some operator;
• for every sequence (x_n) of points from the domain of T such that x_n → 0 and also Tx_n → y it holds that y = 0.

Not all operators are closable.

A closable operator T has the least closed extension ${\displaystyle \overline{T}}$ called the closure of T. The closure of the graph of T is equal to the graph of ${\displaystyle \overline{T}.}$ Other, non-minimal closed extensions may exist.

A densely defined operator T is closable if and only if T^∗ is densely defined. In this case ${\displaystyle \overline{T}=T^{\ast \ast }}$ and ${\displaystyle (\overline{T}{)}^{\ast }=T^{\ast }.}$

If S is densely defined and T is an extension of S then S^∗ is an extension of T^∗.

Every symmetric operator is closable.

A symmetric operator is called maximal symmetric if it has no symmetric extensions, except for itself. Every self-adjoint operator is maximal symmetric. The converse is wrong.

An operator is called essentially self-adjoint if its closure is self-adjoint. An operator is essentially self-adjoint if and only if it has one and only one self-adjoint extension.

A symmetric operator may have more than one self-adjoint extension, and even a continuum of them.

A densely defined, symmetric operator T is essentially self-adjoint if and only if both operators T – i, T + i have dense range.

Let T be a densely defined operator. Denoting the relation "T is an extension of S" by S ⊂ T (a conventional abbreviation for Γ(S) ⊆ Γ(T)) one has the following.

• If T is symmetric then T ⊂ T^∗∗ ⊂ T^∗.
• If T is closed and symmetric then T = T^∗∗ ⊂ T^∗.
• If T is self-adjoint then T = T^∗∗ = T^∗.
• If T is essentially self-adjoint then T ⊂ T^∗∗ = T^∗.

Importance of self-adjoint operators

The class of self-adjoint operators is especially important in mathematical physics. Every self-adjoint operator is densely defined, closed and symmetric. The converse holds for bounded operators but fails in general. Self-adjointness is substantially more restricting than these three properties. The famous spectral theorem holds for self-adjoint operators. In combination with Stone's theorem on one-parameter unitary groups it shows that self-adjoint operators are precisely the infinitesimal generators of strongly continuous one-parameter unitary groups, see Self-adjoint operator § Self-adjoint extensions in quantum mechanics. Such unitary groups are especially important for describing time evolution in classical and quantum mechanics.

Dialectical materialism

From Wikipedia, the free encyclopedia

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

Dialectical materialism is a materialist theory based upon the writings of Karl Marx and Friedrich Engels that has found widespread applications in a variety of philosophical disciplines ranging from philosophy of history to philosophy of science. As a materialist philosophy, Marxist dialectics emphasizes the importance of real-world conditions and the presence of functional contradictions within and among social relations, which derive from, but are not limited to, the contradictions that occur in social class, labour economics, and socioeconomic interactions. Within Marxism, a contradiction is a relationship in which two forces oppose each other, leading to mutual development.

In contrast with the idealist perspective of Hegelian dialectics, the materialist perspective of Marxist dialectics emphasizes that contradictions in material phenomena could be resolved with dialectical analysis, from which is synthesized the solution that resolves the contradiction, whilst retaining the essence of the phenomena. Marx proposed that the most effective solution to the problems caused by contradiction was to address the contradiction and then rearrange the systems of social organization that are the root of the problem.

Dialectical materialism recognises the evolution of the natural world, and thus the emergence of new qualities of being human and of human existence. Engels used the metaphysical insight that the higher level of human existence emerges from and is freerooted in the lower level of human existence. That the higher level of being is a new order with irreducible laws, and that evolution is governed by laws of development, which reflect the basic properties of matter in motion.

In the 1930s, in the Soviet Union, the book Dialectical and Historical Materialism (1938), by Joseph Stalin, set forth the Soviet formulation of dialectical materialism and of historical materialism, which were taught in the Soviet system of education. In the People's Republic of China, an analogous text was the essay On Contradiction (1937), by Mao Zedong, which was a foundational document of Maoism.

The term

The term dialectical materialism was coined in 1887 by Joseph Dietzgen, a socialist who corresponded with Marx, during and after the failed 1848 German Revolution. Casual mention of the term "dialectical materialism" is also found in the biography Frederick Engels, by philosopher Karl Kautsky, written in 1899. Marx himself had talked about the "materialist conception of history", which was later referred to as "historical materialism" by Engels. Engels further explained the "materialist dialectic" in his Dialectics of Nature in 1883. Georgi Plekhanov, the father of Russian Marxism, first used the term "dialectical materialism" in 1891 in his writings on Georg Wilhelm Friedrich Hegel and Marx. Stalin further delineated and defined dialectical and historical materialism as the world outlook of Marxism–Leninism, and as a method to study society and its history.

Historical background

Marx and Engels each began their adulthood as Young Hegelians, one of several groups of intellectuals inspired by the philosopher Hegel. Marx's doctoral thesis, The Difference Between the Democritean and Epicurean Philosophy of Nature, was concerned with the atomism of Epicurus and Democritus, which is considered the foundation of materialist philosophy. Marx was also familiar with Lucretius's theory of clinamen.

Marx and Engels both concluded that Hegelian philosophy, at least as interpreted by their former colleagues, was too abstract and was being misapplied in attempts to explain the social injustice in recently industrializing countries such as Germany, France, and the United Kingdom, which was a growing concern in the early 1840s, as exemplified by Dickensian inequity.

In contrast to the conventional Hegelian dialectic of the day, which emphasized the idealist observation that human experience is dependent on the mind's perceptions, Marx developed Marxist dialectics, which emphasized the materialist view that the world of the concrete shapes socioeconomic interactions and that those in turn determine sociopolitical reality.

Whereas some Hegelians blamed religious alienation (estrangement from the traditional comforts of religion) for societal ills, Marx and Engels concluded that alienation from economic and political autonomy, coupled with exploitation and poverty, was the real culprit.

In keeping with dialectical ideas, Marx and Engels thus created an alternative theory, not only of why the world is the way it is but also of which actions people should take to make it the way it ought to be. In Theses on Feuerbach (1845), Marx wrote a famous quote, "The philosophers have only interpreted the world, in various ways. The point, however, is to change it." Dialectical materialism is thus closely related to Marx's and Engels's historical materialism (and has sometimes been viewed as synonymous with it). Marx rejected Fichte's language of "thesis, antithesis, synthesis".

Dialectical materialism is an aspect of the broader subject of materialism, which asserts the primacy of the material world: in short, matter precedes thought. Materialism is a realist philosophy of science, which holds that the world is material; that all phenomena in the universe consist of "matter in motion," wherein all things are interdependent and interconnected and develop according to natural law; that the world exists outside consciousness and independently of people's perception of it; that thought is a reflection of the material world in the brain, and that the world is in principle knowable.

Marx criticized classical materialism as another idealist philosophy—idealist because of its transhistorical understanding of material contexts. The Young Hegelian Ludwig Feuerbach had rejected Hegel's idealistic philosophy and advocated materialism. Despite being strongly influenced by Feuerbach, Marx rejected Feuerbach's version of materialism (anthropological materialism) as inconsistent. The writings of Engels, especially Anti-Dühring (1878) and Dialectics of Nature (1875–82), were the source of the main doctrines of dialectical materialism.

Marx's dialectics

Main article: Marxist dialectic

The concept of dialectical materialism emerges from statements by Marx in the second edition postface to his magnum opus, Das Kapital. There Marx says he intends to use Hegelian dialectics but in revised form. He defends Hegel against those who view him as a "dead dog" and then says, "I openly avowed myself as the pupil of that mighty thinker Hegel". Marx credits Hegel with "being the first to present [dialectic's] form of working in a comprehensive and conscious manner". But he then criticizes Hegel for turning dialectics upside down: "With him it is standing on its head. It must be turned right side up again, if you would discover the rational kernel within the mystical shell.".

Marx's criticism of Hegel asserts that Hegel's dialectics go astray by dealing with ideas, with the human mind. Hegel's dialectic, Marx says, inappropriately concerns "the process of the human brain"; it focuses on ideas. Hegel's thought is in fact sometimes called dialectical idealism, and Hegel himself is counted among a number of other philosophers known as the German idealists. Marx, on the contrary, believed that dialectics should deal not with the mental world of ideas but with "the material world", the world of production and other economic activity. For Marx, a contradiction can be solved by a desperate struggle to change the social world. This was a very important transformation because it allowed him to move dialectics out of the contextual subject of philosophy and into the study of social relations based on the material world.

For Marx, human history cannot be fitted into any neat a priori schema. He explicitly rejects the idea of Hegel's followers that history can be understood as "a person apart, a metaphysical subject of which real human individuals are but the bearers". To interpret history as though previous social formations have somehow been aiming themselves toward the present state of affairs is "to misunderstand the historical movement by which the successive generations transformed the results acquired by the generations that preceded them". Marx's rejection of this sort of teleology was one reason for his enthusiastic (though not entirely uncritical) reception of Charles Darwin's theory of natural selection.

For Marx, dialectics is not a formula for generating predetermined outcomes but is a method for the empirical study of social processes in terms of interrelations, development, and transformation. In his introduction to the Penguin edition of Marx's Capital, Ernest Mandel writes, "When the dialectical method is applied to the study of economic problems, economic phenomena are not viewed separately from each other, by bits and pieces, but in their inner connection as an integrated totality, structured around, and by, a basic predominant mode of production."

Marx's own writings are almost exclusively concerned with understanding human history in terms of systemic processes, based on modes of production (broadly speaking, the ways in which societies are organized to employ their technological powers to interact with their material surroundings). This is called historical materialism. More narrowly, within the framework of this general theory of history, most of Marx's writing is devoted to an analysis of the specific structure and development of the capitalist economy.

For his part, Engels applies a "dialectical" approach to the natural world in general, arguing that contemporary science is increasingly recognizing the necessity of viewing natural processes in terms of interconnectedness, development, and transformation. Some scholars have doubted that Engels' "dialectics of nature" is a legitimate extension of Marx's approach to social processes. Other scholars have argued that despite Marx's insistence that humans are natural beings in an evolving, mutual relationship with the rest of nature, Marx's own writings pay inadequate attention to the ways in which human agency is constrained by such factors as biology, geography, and ecology.

Engels's dialectics

Engels postulated three laws of dialectics from his reading of Hegel's Science of Logic. Engels elucidated these laws as the materialist dialectic in his work Dialectics of Nature:

1. The law of the unity and conflict of opposites
2. The law of the passage of quantitative changes into qualitative changes
3. The law of the negation of the negation

The first law, which originates with the ancient Ionian philosopher Heraclitus, can be clarified through the following examples:

For example, in biological evolution the formation of new forms of life occurs precisely through the unity and struggle of opposites in heredity and variability. In physical processes the nature of light was explained precisely by means of the unity and struggle of opposites appearing, for example, as corpuscular and wave properties; this, moreover, cleared the path for a “drama of ideas” in physical science, whereby the opposition and synthesis of corpuscular and wave theories characterized scientific progress. The most basic expression of the unity and struggle of opposites in the world of commodity capitalism is that of use value and value; the most highly developed oppositions in capitalism are the working class and the bourgeoisie,

— The Great Soviet Encyclopedia (1979), Unity and Struggle of Opposites – Web page

The first law was seen by both Hegel and Vladimir Lenin as the central feature of a dialectical understanding:

It is in this dialectic as it is here understood, that is, in the grasping of oppositions in their unity, or of the positive in the negative, that speculative thought consists. It is the most important aspect of dialectic.

— Hegel, Science of Logic, § 69, (p. 56 in the Miller edition)

The splitting of a single whole and the cognition of its contradictory parts is the essence (one of the "essentials", one of the principal, if not the principal, characteristics or features) of dialectics. That is precisely how Hegel, too, puts the matter.

— Lenin's Collected Works: Volume 38, p. 359: On the question of dialectics.

The second law Hegel took from Ancient Greek philosophers, notably the paradox of the heap, and explanation by Aristotle, and it is equated with what scientists call phase transitions. It may be traced to the ancient Ionian philosophers, particularly Anaximenes from whom Aristotle, Hegel, and Engels inherited the concept. For all these authors, one of the main illustrations is the phase transitions of water. There has also been an effort to apply this mechanism to social phenomena, whereby population increases result in changes in social structure. The law of the passage of quantitative changes into qualitative changes can also be applied to the process of social change and class conflict.

The third law, "negation of the negation", originated with Hegel. Although Hegel coined the term "negation of the negation", it gained its fame from Marx's using it in Capital. There Marx wrote this: "The [death] knell of capitalist private property sounds. The expropriators are expropriated. The capitalist mode of appropriation, the result of the capitalist mode of production, produces capitalist private property. This is the first negation of individual private property ... But capitalist production begets, with the inexorability of a law of Nature, its own negation. It [this new negation] is the negation of negation."

Z. A. Jordan notes, "Engels made constant use of the metaphysical insight that the higher level of existence emerges from and has its roots in the lower; that the higher level constitutes a new order of being with its irreducible laws; and that this process of evolutionary advance is governed by laws of development which reflect basic properties of 'matter in motion as a whole'."

Lenin's contributions

After reading Hegel's Science of Logic in 1914, Lenin made some brief notes outlining three "elements" of logic. They are:

1. The determination of the concept out of itself [the thing itself must be considered in its relations and in its development];
2. The contradictory nature of the thing itself (the other of itself), the contradictory forces and tendencies in each phenomenon;
3. The union of analysis and synthesis.

Lenin develops these in a further series of notes, and appears to argue that "the transition of quantity into quality and vice versa" is an example of the unity and opposition of opposites expressed tentatively as "not only the unity of opposites but the transitions of every determination, quality, feature, side, property into every other [into its opposite?]."

In his essay "On the Question of Dialectics", Lenin stated, "Development is the 'struggle' of opposites." He stated, "The unity (coincidence, identity, equal action) of opposites is conditional, temporary, transitory, relative. The struggle of mutually exclusive opposites is absolute, just as development and motion are absolute."

In Materialism and Empiriocriticism (1908), Lenin explained dialectical materialism as three axes: (i) the materialist inversion of Hegelian dialectics, (ii) the historicity of ethical principles ordered to class struggle, and (iii) the convergence of "laws of evolution" in physics (Helmholtz), biology (Darwin), and in political economy (Marx). Hence, Lenin was philosophically positioned between historicist Marxism (Labriola) and determinist Marxism—a political position close to "social Darwinism" (Kautsky). Moreover, late-century discoveries in physics (x-rays, electrons), and the beginning of quantum mechanics, philosophically challenged previous conceptions of matter and materialism, thus matter seemed to be disappearing. Lenin disagreed:

'Matter disappears' means that the limit within which we have hitherto known matter disappears, and that our knowledge is penetrating deeper; properties of matter are disappearing that formerly seemed absolute, immutable, and primary, and which are now revealed to be relative and characteristic only of certain states of matter. For the sole 'property' of matter, with whose recognition philosophical materialism is bound up, is the property of being an objective reality, of existing outside of the mind.

Lenin was developing the work of Engels, who said that "with each epoch-making discovery, even in the sphere of natural science, materialism has to change its form". One of Lenin's challenges was distancing materialism, as a viable philosophical outlook, from the "vulgar materialism" expressed in the statement "the brain secretes thought in the same way as the liver secretes bile" (attributed to 18th-century physician Pierre Jean Georges Cabanis); "metaphysical materialism" (matter composed of immutable particles); and 19th-century "mechanical materialism" (matter as random molecules interacting per the laws of mechanics). The philosophic solution that Lenin (and Engels) proposed was "dialectical materialism", wherein matter is defined as objective reality, theoretically consistent with (new) developments occurring in the sciences.

Lenin reassessed Feuerbach's philosophy and concluded that it was in line with dialectical materialism.

Trotsky's contributions

In 1926, Trotsky said in a speech:

It is the task of science and technology to make matter subject to man, together with space and time, which are inseparable from matter. True, there are certain idealist books—not of a clerical character, but philosophical ones—wherein you can read that time and space are categories of our minds, that they result from the requirements of our thinking, and that nothing actually corresponds to them in reality. But it is difficult to agree with this view. If any idealist philosopher, instead of arriving in time to catch the 9 pm train, should turn up two minutes late, he would see the tail of the departing train and would be convinced by his own eyes that time and space are inseparable from material reality. The task is to diminish this space, to overcome it, to economise time, to prolong human life, to register past time, to raise life to a higher level and enrich it. This is the reason for the struggle with space and time, at the basis of which lies the struggle to subject matter to man—matter, which constitutes the foundation not only of everything that really exists, but also of all imagination ... Every science is an accumulation of knowledge, based on experience relating to matter, to its properties; an accumulation of generalised understanding of how to subject this matter to the interests and needs of man.

In his book, In Defence of Marxism, Leon Trotsky defended the dialectical method of scientific socialism during the factional schisms within the American Trotskyist movement in the period 1939–40. Trotsky viewed dialectics as an essential method of analysis to discern class nature of the Soviet Union. Specifically, he described scientific socialism as "the conscious expression of the unconscious historical process; namely, the instinctive and elemental drive of the proletariat to reconstruct society on communist beginnings".

Lukács's contributions

György Lukács, Minister of Culture in the brief Béla Kun government of the Hungarian Soviet Republic (1919), published History and Class Consciousness (1923), in which he defined dialectical materialism as the knowledge of society as a whole, knowledge which, in itself, was the class consciousness of the proletariat. In the first chapter "What is Orthodox Marxism?", Lukács defined orthodoxy as fidelity to the "Marxist method", not fidelity to "dogmas":

Orthodox Marxism, therefore, does not imply the uncritical acceptance of the results of Marx's investigations. It is not the "belief" in this or that thesis, nor the exegesis of a "sacred" book. On the contrary, orthodoxy refers exclusively to method. It is the scientific conviction that dialectical materialism is the road to truth and that its methods can be developed, expanded, and deepened, only along the lines laid down by its founders. (§1)

In his later works and actions, Lukács became a leader of Democratic Marxism. He modified many of his formulations of his 1923 works and went on to develop a Marxist ontology and played an active role in democratic movements in Hungary in 1956 and the 1960s. He and his associates became sharply critical of the formulation of dialectical materialism in the Soviet Union that was exported to those countries under its control. In the 1960s, his associates became known as the Budapest School.

Lukács, in his philosophical criticism of Marxist revisionism, proposed an intellectual return to the Marxist method. So did Louis Althusser, who later defined Marxism and psychoanalysis as "conflictual sciences", stating that political factions and revisionism are inherent to Marxist theory and political praxis, because dialectical materialism is the philosophic product of class struggle:

For this reason, the task of orthodox Marxism, its victory over Revisionism and utopianism can never mean the defeat, once and for all, of false tendencies. It is an ever-renewed struggle against the insidious effects of bourgeois ideology on the thought of the proletariat. Marxist orthodoxy is no guardian of traditions, it is the eternally vigilant prophet proclaiming the relation between the tasks of the immediate present and the totality of the historical process. (§5)

...the premise of dialectical materialism is, we recall: 'It is not men's consciousness that determines their existence, but, on the contrary, their social existence that determines their consciousness'.... Only when the core of existence stands revealed as a social process can existence be seen as the product, albeit the hitherto unconscious product, of human activity. (§5)

Philosophically aligned with Marx is the criticism of the individualist, bourgeois philosophy of the subject, which is founded upon the voluntary and conscious subject. Against said ideology is the primacy of social relations. Existence—and thus the world—is the product of human activity, but this can be seen only by accepting the primacy of social process on individual consciousness. This type of consciousness is an effect of ideological mystification.

At the 5th Congress of the Communist International (July 1924), Grigory Zinoviev formally denounced Lukács's heterodox definition of Orthodox Marxism as exclusively derived from fidelity to the "Marxist method", and not to Communist party dogmas; and denounced the philosophical developments of the German Marxist theorist Karl Korsch.

Stalin's contributions

In the 1930s, Stalin and his associates formulated a version of dialectical and historical materialism that became the "official" Soviet interpretation of Marxism. It was codified in Stalin's work, Dialectical and Historical Materialism (1938), and popularized in textbooks used for compulsory education within the Soviet Union and throughout the Eastern Bloc.

Mao's contributions

In On Contradiction (1937), Mao Zedong outlined a version of dialectical materialism that subsumed two of Engels's three principal laws of dialectics, "the transformation of quantity into quality" and "the negation of the negation" as sub-laws (and not principal laws of their own) of the first law, "the unity and interpenetration of opposites".

Ho Chi Minh's contributions

In his 1947 article New Life, Ho Chi Minh described the dialectical relationship between the old and the new in building society, stating:

Not everything old must be abandoned. We do not have to reinvent everything. What is old but bad must be abandoned. What is old but troublesome must be corrected appropriately. What is old but good must be further developed. What is new but good must be done.

As a heuristic in science and elsewhere

Historian of science Loren Graham has detailed at length the role played by dialectical materialism in the Soviet Union in disciplines throughout the natural and social sciences. He has concluded that, despite the Lysenko period in genetics and constraints on free inquiry imposed by political authorities, dialectical materialism had a positive influence on the work of many Soviet scientists.

Some evolutionary biologists, such as Richard Lewontin and Stephen Jay Gould, have tried to employ dialectical materialism in their approach. They view dialectics as playing a precautionary heuristic role in their work. Lewontin's perspective offers the following idea:

Dialectical materialism is not, and never has been, a programmatic method for solving particular physical problems. Rather, a dialectical analysis provides an overview and a set of warning signs against particular forms of dogmatism and narrowness of thought. It tells us, "Remember that history may leave an important trace. Remember that being and becoming are dual aspects of nature. Remember that conditions change and that the conditions necessary to the initiation of some process may be destroyed by the process itself. Remember to pay attention to real objects in time and space and not lose them in utterly idealized abstractions. Remember that the qualitative effects of context and interaction may be lost when phenomena are isolated". And above all else, "Remember that all the other caveats are only reminders and warning signs whose application to different circumstances of the real world is contingent."

Gould shared similar views regarding a heuristic role for dialectical materialism. He wrote that:

...dialectical thinking should be taken more seriously by Western scholars, not discarded because some nations of the second world have constructed a cardboard version as an official political doctrine.

...when presented as guidelines for a philosophy of change, not as dogmatic precepts true by fiat, the three classical laws of dialectics embody a holistic vision that views change as interaction among components of complete systems and sees the components themselves not as a priori entities, but as both products and inputs to the system. Thus, the law of "interpenetrating opposites" records the inextricable interdependence of components: the "transformation of quantity to quality" defends a systems-based view of change that translates incremental inputs into alterations of state, and the "negation of negation" describes the direction given to history because complex systems cannot revert exactly to previous states.

This heuristic was also applied to the theory of punctuated equilibrium proposed by Gould and Niles Eldredge. They wrote that "history, as Hegel said, moves upward in a spiral of negations", and that "punctuated equilibria is a model for discontinuous tempos of change (in) the process of speciation and the deployment of species in geological time." They noted that "the law of transformation of quantity into quality... holds that a new quality emerges in a leap as the slow accumulation of quantitative changes, long resisted by a stable system, finally forces it rapidly from one state into another", a phenomenon described in some disciplines as a paradigm shift. Apart from the commonly cited example of water turning to steam with increased temperature, Gould and Eldredge noted another analogy in information theory, "with its jargon of equilibrium, steady state, and homeostasis maintained by negative feedback", and "extremely rapid transitions that occur with positive feedback".

Lewontin, Gould, and Eldredge were thus more interested in dialectical materialism as a heuristic than a dogmatic form of 'truth' or a statement of their politics. Nevertheless, they found a readiness for critics to "seize upon" key statements and portray punctuated equilibrium, and exercises associated with it, such as public exhibitions, as a "Marxist plot".

The Communist Party's official interpretation of Marxism, dialectical materialism, fit Alexander Oparin's studies on the origins of life as 'a flow, an exchange, a dialectical unity'. This notion was re-enforced by Oparin's association with Lysenko.

In 1972, the worst chaos of China's Cultural Revolution was over and scientific research resumed. Astrophysicist and cosmologist Fang Lizhi found an opportunity to read some recent astrophysics papers in western journals, and soon wrote his first paper on cosmology, "A Cosmological Solution in Scalar-tensor Theory with Mass and Blackbody Radiation", which was published on the journal Wu Li (Physics), Vol. 1, 163 (1972). This was the first modern cosmological research paper in mainland China. Fang assembled a group of young faculty members of USTC around him to conduct astrophysics research.

At the time, conducting research on relativity theory and cosmology in China was very risky politically, because these theories were considered to be "idealistic" theories in contradiction with the dialectical materialism theory, which is the official philosophy of the Communist Party. According to the dialectical materialism philosophy, both time and space must be infinite, while the Big Bang theory allows the possibility of the finiteness of space and time. During the Cultural Revolution, campaigns were waged against Albert Einstein and the Theory of Relativity in Beijing and Shanghai. Once Fang published his theory, some of the critics of the Theory of Relativity, especially a group based in Shanghai, prepared to attack Fang politically. However, by this time the "leftist" line was declining in the Chinese academia. Professor Dai Wensai, the most well-known Chinese astronomer at the time and chair of the Astronomy Department of Nanjing University, also supported Fang. Many of the members of the "Theory of Relativity Criticism Group" changed to study the theory and conduct research in it. Subsequently, Fang was regarded as the father of cosmological research in China.

Criticism

Philosopher Allen Wood argued that, in its form as an official Soviet philosophy, dialectical materialism was doomed to be superficial because "creativity or critical thinking" was impossible in an authoritarian environment. Nevertheless, he considered the basic aims and principles of dialectical materialism to be in harmony with rational scientific thought.

Economist and philosopher Ludwig von Mises wrote a critique of Marxist materialism which he published as a part of his 1957 work Theory and History: An Interpretation of Social and Economic Evolution. H. B. Acton described Marxism as "a philosophical farrago". Max Eastman argued that dialectical materialism lacks a psychological basis.

Leszek Kołakowski criticized the laws of dialectics in Main Currents of Marxism, arguing that they consist partly of truisms with no specific Marxist content, partly of philosophical dogmas, partly of nonsense, and partly of statements that could be any of these things depending on how they are interpreted.

Of the term

Joseph Needham, an influential historian of science and a Christian who nonetheless was an adherent of dialectical materialism, suggested that a more appropriate term might be "dialectical organicism".

Marxist rejection

This section needs expansion. You can help by adding to it. (February 2023)

Anti-communist, formerly Marxist humanist Leszek Kołakowski argued that dialectical materialism was not truly Marxist.

Consumer culture

From Wikipedia, the free encyclopedia
https://en.wikipedia.org/wiki/Consumer_culture

Shopping malls have had a huge impact on consumer culture. Shown in the picture is the Mall of America, one of the largest malls in the US.

Consumer culture describes a lifestyle hyper-focused on spending money to buy material goods. It is often attributed to, but not limited to, the capitalist economy of the United States. During the 20th century, market goods came to dominate American life, and for the first time in history, consumerism had no practical limits. Consumer culture has provided affluent societies with alternatives to tribalism and class war.

Consumer culture began to increase rapidly during the extreme economic growth of the Roaring Twenties. The challenge for the future is to find ways to revive the valid portion of the culture of constraint and control the overpowering success of the twentieth century.

Types of culture

Social scientists Arthur Berger, Aaron Wildavsky, and Mary Douglas have suggested that there are four political and consumer cultures possible in a democratic society: hierarchical/elitist, individualist, egalitarian, and fatalist.

• Hierarchical/Elitist: Someone with the belief that a system or society should be ruled, dominated, or otherwise controlled by a group of individuals determined to be of higher standing than others.

• Individualist: Someone who puts the needs of the individual before the needs of others.
• Egalitarian: Someone that believes that peoples' needs and rights should be equal, fair, and cared for.
• Fatalist: Someone with the belief that future events are inevitable, and thus there is little to no point in attempting to alter them.

Mass market theory

Main article: Mass-market theory

Advertising and strategies

A cover for a collection of sheet music from 1899, showing a woman dressed in luxurious clothes spending money in multiple scenarios

To improve the effectiveness of advertisements, people of various age groups are employed by marketing companies to increase the understanding of the beliefs, attitudes, and values of the targeted consumers.

A quote by Shah states that "The sophistication of advertising methods and techniques has advanced, enticing and shaping and even creating consumerism and needs where there has been none before".

Richard Wilk has written an article about bottled water and the consumerist basis it has in society. The base point of this article is to point out how water is free, and it is abundant, but over time it has become a point of marketing. The debate has always existed if there is a real difference in taste between bottled water and tap water. When it comes down to it, during many blind tastes, people cannot even tell which was tap and which was bottled, and more often than not tap water won for better taste. Water has always been regarded as a pure substance and has connections in many religions. Throughout history, it has been shown that control over water is equivalent to control over untamed nature, and this has been shown in movies as well. To build on this idea, Wilk points out how having bottled water enforces the idea of this control over nature and the need that humans have for water. To further enforce this idea of natural and pure water, a 1999 report by the National Resources Defense Council found that many water companies use words like “pure” and “pristine” to aid in marketing. Wilk explains that it is more than just the marketing of it being pure, but that people want bottled water because they know the source. Public water comes from an anonymous source and Wilk concludes that the home is an extension of ourselves, so why would we want to bring an unknown specimen into our home? This is where the bottled water preference comes in, according to Wilk, because people can trace it back to where it came from. In addition to this idea, many brands and companies have started marketing water toward specific needs like special water for women, kids, athletes, and so on. This increases competition between brands and takes away from customers being able to choose what water they want. This is due to the larger companies being able to make more connections and pay the expensive fee to be sold on the shelves. Wilk concludes that since it is hard to trust either, it comes down to which is distrusted least.

Industrial Revolution

Main article: Industrial Revolution

Wage work

Pictured are both men and women working side-by-side in a factory

Before the Industrial Revolution, the home was a place where men and women produced, consumed, and worked. The men were highly valued workers, such as barbers, butchers, farmers, and lumbermen who brought income into the house. The wives of these men completed various tasks to save money which included, churning butter, fixing clothes, and tending the garden. This system created an equal value for all of the jobs and tasks in a community. Once the Industrial Revolution began, there was no such thing as equal and high valued work in a mass production industry. The only value these workers had were the wage they made. That meant the wives lost their value at home and had to start working for a living. This new system created the thought of everyone being replaceable.

Life of the worker

The life of a worker was a challenging one. Working 12 to 14-hour days, 6 days a week, and in a dangerous environment. The worst part was the infrequency of pay or not being paid at all. At times, employers paid their workers in script pay, non-U.S. currency, or even in-store credit.