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The mathematical manuscripts ofKarl Marx
Mario Natiello
Centre for Mathematical Sciences
Lund University
Sweden
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Outline
This talk will shortly consider:
• Publishing details of the manuscripts
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Outline
This talk will shortly consider:
• Publishing details of the manuscripts
• A brief history of differential calculus
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Outline
This talk will shortly consider:
• Publishing details of the manuscripts
• A brief history of differential calculus
• The mathematical content of the manuscripts
The mathematical manuscripts of Karl Marx – p.2/14
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Outline
This talk will shortly consider:
• Publishing details of the manuscripts
• A brief history of differential calculus
• The mathematical content of the manuscripts
• Other usages of mathematics by Marx
The mathematical manuscripts of Karl Marx – p.2/14
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Outline
This talk will shortly consider:
• Publishing details of the manuscripts
• A brief history of differential calculus
• The mathematical content of the manuscripts
• Other usages of mathematics by Marx
• Philosophy of mathematics
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Outline
This talk will shortly consider:
• Publishing details of the manuscripts
• A brief history of differential calculus
• The mathematical content of the manuscripts
• Other usages of mathematics by Marx
• Philosophy of mathematics
• Final remarks
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Outline
This talk will shortly consider:
• Publishing details of the manuscripts
• A brief history of differential calculus
• The mathematical content of the manuscripts
• Other usages of mathematics by Marx
• Philosophy of mathematics
• Final remarks
END
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Publishing details
• About 1000 manuscript pages
• Partial edition in german/russian from 1933.
• Extended (sovietic) edition from 1968commented by mathematicians.
• Translation to some european languages in the70’s
• Present english version (1983): Translation of1968 edition.
• Lately: Marxist writers comment the text.
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Differential calculus• Newton & Leibnitz introduce infinitesimal
variations.
• D’Alembert introduces the finite difference
• Lagrange treats “analytic” functions
• Cauchy attempts the first formalization of theconcept of limit
• Bolzano, Weierstrass: “modern” (current) versionof the concept of limit and continuity.
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Differential calculus II• Uniform convergence of function series
• Fourier
• Lacroix, Cauchy, Moigno
• Abel
• Seidel
Robinson and non-standard analysis.
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Differential calculus III
The problem:
1600-1900: The scientific conception of the worldstruggles to impose its views to that of preexistentalternative conceptions.
D’Alembert (1743): ..Up to the present...more concernhas been given to enlarging the building than toilluminate the entrance, to raising it higher than togiving proper strength to the foundations...
Hegel (1812-6): ...For the infinitesimal calculus permitsand requires modes of procedure which mathematicsmust wholly reject when operating with finitequantities...
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The manuscripts
Index
• On the concept of the derived function
• On the differential (including three drafts and a“final version” of this manuscript).
• On the history of differential calculus (includingloose pages).
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On the concept of the derived function• Marx works with examples.
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On the concept of the derived function• Marx works with examples.
• For any polynomial, p(x) − p(x0) is a polynomial,that can be divided by (x − x0).
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On the concept of the derived function• Marx works with examples.
• For any polynomial, p(x) − p(x0) is a polynomial,that can be divided by (x − x0).
• Consider the polynomial g(x) =p(x)−p(x0)
x−x0.
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On the concept of the derived function• Marx works with examples.
• For any polynomial, p(x) − p(x0) is a polynomial,that can be divided by (x − x0).
• Consider the polynomial g(x) =p(x)−p(x0)
x−x0.
• g(x0) is the derivative of p(x) at the point x0.
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On the concept of the derived function• Marx works with examples.
• For any polynomial, p(x) − p(x0) is a polynomial,that can be divided by (x − x0).
• Consider the polynomial g(x) =p(x)−p(x0)
x−x0.
• g(x0) is the derivative of p(x) at the point x0.
• A similar reasoning can be done for otherelementary functions (exponentials, roots, Marxmentions also log and trig).
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On the concept of the derived function• Marx works with examples.
• For any polynomial, p(x) − p(x0) is a polynomial,that can be divided by (x − x0).
• Consider the polynomial g(x) =p(x)−p(x0)
x−x0.
• g(x0) is the derivative of p(x) at the point x0.
• A similar reasoning can be done for otherelementary functions (exponentials, roots, Marxmentions also log and trig).
• Without the idea of limit, a recipe is needed toproduce g(x0) starting from p(x) without falling in“0/0”.
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On the differential
Considerations on the relationship among:
Df , df, Dx, dx
and their quotes.Includes reasoning around the idea of derivative of aproduct.
Back to list
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On the history of differential calculus• Newton & Leibnitz: Mystical calculus. du dv is set
equal to zero.
• D’Alembert: Rational calculus. The incrementquote Df /Dx.
• Lagrange: Pure algebraic caalculus. “...The wholeproblem is then resolved into finding (algebraic)methods of developing all kinds of functions ofx + h in integral ascending powers of h...”
• Some notes on Taylor and McLaurin’sformulæand their use in the theory of functionsby Lagrange.
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Other mathematics in Marx
• In some of the drafts the expressions “limit”,“limit value” are found.
• Moigno is named on a bibliographic list.
• The concept of operational rule.
• The use of the equal sign.
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Philosophy of Mathematics I• Hegel, Marx, Engels had different views about the roles of
mathematics and science.
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Philosophy of Mathematics I• Hegel, Marx, Engels had different views about the roles of
mathematics and science.
• Mathematics at the end of 1800: Foundations revisited.
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Philosophy of Mathematics I• Hegel, Marx, Engels had different views about the roles of
mathematics and science.
• Mathematics at the end of 1800: Foundations revisited.
• Logicism (Russell) To structure mathematics in terms of its
logical substrate.
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Philosophy of Mathematics I• Hegel, Marx, Engels had different views about the roles of
mathematics and science.
• Mathematics at the end of 1800: Foundations revisited.
• Logicism (Russell) To structure mathematics in terms of its
logical substrate.
• Intuitionism (Brouwer) To accept only finite, intuitively
evident methodology.
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Philosophy of Mathematics I• Hegel, Marx, Engels had different views about the roles of
mathematics and science.
• Mathematics at the end of 1800: Foundations revisited.
• Logicism (Russell) To structure mathematics in terms of its
logical substrate.
• Intuitionism (Brouwer) To accept only finite, intuitively
evident methodology.
• Formalism (Hilbert) Axiomatic foundation (axioms free of
content).
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Philosophy of Mathematics I• Hegel, Marx, Engels had different views about the roles of
mathematics and science.
• Mathematics at the end of 1800: Foundations revisited.
• Logicism (Russell) To structure mathematics in terms of its
logical substrate.
• Intuitionism (Brouwer) To accept only finite, intuitively
evident methodology.
• Formalism (Hilbert) Axiomatic foundation (axioms free of
content).
• Gödel and the breakdown of the search for consistency.
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Philosophy of Mathematics I• Hegel, Marx, Engels had different views about the roles of
mathematics and science.
• Mathematics at the end of 1800: Foundations revisited.
• Logicism (Russell) To structure mathematics in terms of its
logical substrate.
• Intuitionism (Brouwer) To accept only finite, intuitively
evident methodology.
• Formalism (Hilbert) Axiomatic foundation (axioms free of
content).
• Gödel and the breakdown of the search for consistency.
• Appearance of different equivalent axiom systems
(relativism).
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Philosophy of Mathematics I• Hegel, Marx, Engels had different views about the roles of
mathematics and science.
• Mathematics at the end of 1800: Foundations revisited.
• Logicism (Russell) To structure mathematics in terms of its
logical substrate.
• Intuitionism (Brouwer) To accept only finite, intuitively
evident methodology.
• Formalism (Hilbert) Axiomatic foundation (axioms free of
content).
• Gödel and the breakdown of the search for consistency.
• Appearance of different equivalent axiom systems
(relativism).
• Mathematics as a calculational tool (Marx,
post modernism).The mathematical manuscripts of Karl Marx – p.12/14
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Philosophy of Mathematics IIImre Lakatos (Popper’s school).
• Mathematics is quasi-empirical (alike natural sciences).
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Philosophy of Mathematics IIImre Lakatos (Popper’s school).
• Mathematics is quasi-empirical (alike natural sciences).
• Euclidean theories have truth-value injections at the top.
Truth propagates downwards inundating the whole
system.
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Philosophy of Mathematics IIImre Lakatos (Popper’s school).
• Mathematics is quasi-empirical (alike natural sciences).
• Euclidean theories have truth-value injections at the top.
Truth propagates downwards inundating the whole
system.
• Quasi-empirical theories have truth-value injections
elswhere. Eventual falsity propagates upwards
individuating “guilty” assumptions.
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Philosophy of Mathematics IIImre Lakatos (Popper’s school).
• Mathematics is quasi-empirical (alike natural sciences).
• Euclidean theories have truth-value injections at the top.
Truth propagates downwards inundating the whole
system.
• Quasi-empirical theories have truth-value injections
elswhere. Eventual falsity propagates upwards
individuating “guilty” assumptions.
• Mathematics is conjectural (the question on final absolute
truth is improper).
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Philosophy of Mathematics IIImre Lakatos (Popper’s school).
• Mathematics is quasi-empirical (alike natural sciences).
• Euclidean theories have truth-value injections at the top.
Truth propagates downwards inundating the whole
system.
• Quasi-empirical theories have truth-value injections
elswhere. Eventual falsity propagates upwards
individuating “guilty” assumptions.
• Mathematics is conjectural (the question on final absolute
truth is improper).
• Mathematics evolves by proofs and refutations.
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Philosophy of Mathematics IIImre Lakatos (Popper’s school).
• Mathematics is quasi-empirical (alike natural sciences).
• Euclidean theories have truth-value injections at the top.
Truth propagates downwards inundating the whole
system.
• Quasi-empirical theories have truth-value injections
elswhere. Eventual falsity propagates upwards
individuating “guilty” assumptions.
• Mathematics is conjectural (the question on final absolute
truth is improper).
• Mathematics evolves by proofs and refutations.
• Demarcationism: Appraisal of knowledge is possible in
relatively objective (slightly platonic) terms.
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Philosophy of Mathematics IIImre Lakatos (Popper’s school).
• Mathematics is quasi-empirical (alike natural sciences).
• Euclidean theories have truth-value injections at the top.
Truth propagates downwards inundating the whole
system.
• Quasi-empirical theories have truth-value injections
elswhere. Eventual falsity propagates upwards
individuating “guilty” assumptions.
• Mathematics is conjectural (the question on final absolute
truth is improper).
• Mathematics evolves by proofs and refutations.
• Demarcationism: Appraisal of knowledge is possible in
relatively objective (slightly platonic) terms.
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Final remarks
• Marx did not discover the problems ofdifferential calculus (even Hegel knew aboutthem).
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Final remarks
• Marx did not discover the problems ofdifferential calculus (even Hegel knew aboutthem).
• He did not produce new mathematics. He doesnot prove theorems and considers only particularcases.
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Final remarks
• Marx did not discover the problems ofdifferential calculus (even Hegel knew aboutthem).
• He did not produce new mathematics. He doesnot prove theorems and considers only particularcases.
• Marx makes clear the evolutionary process:mystic → rational → algebraic (→ precise).
The mathematical manuscripts of Karl Marx – p.14/14
![Page 41: The mathematical manuscripts of Karl Marxctr.maths.lu.se/matematiklth/personal/mario/Static-typo3/talks/madri… · The mathematical manuscripts of Karl Marx – p.13/14. Philosophy](https://reader033.vdocuments.us/reader033/viewer/2022042420/5f3715d305878b10af5cb004/html5/thumbnails/41.jpg)
Final remarks
• Marx did not discover the problems ofdifferential calculus (even Hegel knew aboutthem).
• He did not produce new mathematics. He doesnot prove theorems and considers only particularcases.
• Marx makes clear the evolutionary process:mystic → rational → algebraic (→ precise).
• Marx does not hide his pleasure on making thisdiscovery (he finds an invention of “his own” inan independent and totally unexpected context).
The mathematical manuscripts of Karl Marx – p.14/14
![Page 42: The mathematical manuscripts of Karl Marxctr.maths.lu.se/matematiklth/personal/mario/Static-typo3/talks/madri… · The mathematical manuscripts of Karl Marx – p.13/14. Philosophy](https://reader033.vdocuments.us/reader033/viewer/2022042420/5f3715d305878b10af5cb004/html5/thumbnails/42.jpg)
Final remarks
• Marx did not discover the problems ofdifferential calculus (even Hegel knew aboutthem).
• He did not produce new mathematics. He doesnot prove theorems and considers only particularcases.
• Marx makes clear the evolutionary process:mystic → rational → algebraic (→ precise).
• Marx does not hide his pleasure on making thisdiscovery (he finds an invention of “his own” inan independent and totally unexpected context).
• Without knowing it, he enters the operationalschool of mathematics.
The mathematical manuscripts of Karl Marx – p.14/14
![Page 43: The mathematical manuscripts of Karl Marxctr.maths.lu.se/matematiklth/personal/mario/Static-typo3/talks/madri… · The mathematical manuscripts of Karl Marx – p.13/14. Philosophy](https://reader033.vdocuments.us/reader033/viewer/2022042420/5f3715d305878b10af5cb004/html5/thumbnails/43.jpg)
Final remarks
• Marx did not discover the problems ofdifferential calculus (even Hegel knew aboutthem).
• He did not produce new mathematics. He doesnot prove theorems and considers only particularcases.
• Marx makes clear the evolutionary process:mystic → rational → algebraic (→ precise).
• Marx does not hide his pleasure on making thisdiscovery (he finds an invention of “his own” inan independent and totally unexpected context).
• Without knowing it, he enters the operationalschool of mathematics.
ENDThe mathematical manuscripts of Karl Marx – p.14/14