Value, value, and value

Friday, 16th September, 2016

(in Chapter 1 of Marx’s Capital)

** Three types of value

In Chapter 1 of Capital, Marx describes three types of value: use-value, value, and exchange-value.

*** use-value

Every useful thing has use-value. The use-value of a thing is just the ways people might use it. That use-value might be historically conditioned (e.g. uranium had little use-value until recently), and it might be shall we say abstract (e.g., looks nice on my mantelpiece).

Use-value is a property of useful things in general, and is not limited to commodities. Use-value is not a property of commodities per se. For something to be a commodity that thing must have use-value, but things that are not commodities can have use-value; and things that are not products of human labour can have use-value.

A thing can be a use value, without having value. This is the case whenever its utility to man is not due to labour. Such are air, virgin soil, natural meadows, &c. A thing can be useful, and the product of human labour, without being a commodity. Whoever directly satisfies his wants with the produce of his own labour, creates, indeed, use values, but not commodities. In order to produce the latter, he must not only produce use values, but use values for others, social use values. Lastly nothing can have value, without being an object of utility. If the thing is useless, so is the labour contained in it; the labour does not count as labour, and therefore creates no value.
(Last paragraph of Section 1.)

*** value

Value is congealed homogenised socially-necessary human labour time. “The value of a commodity represents human labour in the abstract, the expenditure of human labour in general” (from section 2). Value is entirely abstract.

Value is a property of useful products of human labour, but still not necessarily of commodities. Value is just human labour, in the abstract.

A couple of points on what I interpret as Marx’s humanism in Capital, in the way that he conceives of labour:

First, for Marx the very definition of labour includes the production of use-values: if the product is useless, “the labour does not count as labour”. So, usefulness, and all that entails about co-operation and togetherness and community, is in there right at the beginning.

Second, an interesting footnote about Adam Smith in section 2. This note is an extended comment on Smith’s labour theory of value. Deep in the middle, Marx says, “[Smith] has a presentiment that labour, in so far as it manifests itself in the value of commodities, counts only as expenditure of labour power, but he treats this expenditure as the mere sacrifice of rest, freedom and happiness, not as at the same time the normal activity of living beings. But, then, he has the modern wage-labourer in his eye.” Marx does not (only) have the modern wage-labourer in his eye; and he sees labour, and the expenditure of labour power, as more fundamental to humanity.

*** exchange-value

Exchange-value is the form value takes in commodity exchanging societies. Section 3 looks at this form in some detail (“20 yards of linen = 1 coat” etc.) before arriving at the universal equivalent, money.

Exchange-value is a property of commodities, and the exchange-value of a commodity has a concrete form (or forms) — two chickens, 50 quid, twenty packets of fags — depending on concrete circumstances.

Putting aside the money form for the moment, two commodities which are equated are (implicitly) judged as having the same value, i.e. as being embodiments of the same amount of abstract socially-necessary labour time.

Exchange value could be thought of as a kind of replacement value.

** The value, value and value of human labour

In the kinds of societies Marx is talking about, human labour becomes a commodity just like any other: it has a use-value, a value and an exchange-value. At the same time (in fact, just because of that reason), it becomes a special commodity unlike any other.

The use-value of human labour is to create value. Value is congealed abstract human labour (measured in units of socially-necessary time). The use or expenditure of labour, congeals that labour into its product.

As with any other commodity, the value of human labour is the (abstract, socially-necessary, etc.) human labour that has been congealed into it. Depending on the amount of labour you want to value, that congealed labour might be the reproductive labour that has reproduced yesterday’s worker, or it might be the extended training that has produced a heart surgeon or a web developer.

The exchange-value of a unit of human labour is just this value given concrete form, e.g., in money: a daily wage or an annual salary.


In order to be able to extract value from the consumption of a commodity, our friend, Moneybags, must be so lucky as to find, within the sphere of circulation, in the market, a commodity, whose use-value possesses the peculiar property of being a source of value, whose actual consumption, therefore, is itself an embodiment of labour, and, consequently, a creation of value.
(Cheating slightly as that is from Chapter 6.)

The exciting thing for humanity in general, and for Mr. Moneybags in particular, is that — at a certain point in history, when productive forces rise up to a certain level — the expenditure of a unit of human labour creates more value than the value embodied in (and hence the exchange-value of) that unit of human labour. A unit of labour can then be bought and used at the correct — fair, just — price, and the value returned is greater than the value laid out. The difference between the value laid out and the value returned is a fourth type of value: surplus-value.

Book details

Dialectical Logic: essays on its history and theory
Evald V. Ilyenkov
1974 (tr. English 1977)


This is a short book of two halves (both good). Here is the table of contents (I’ve added names of philosophers covered to the essays in Part One):

  • Introduction

Part One: From the History of Dialectics

  • Essay 1: The Problem of the Subject Matter and Sources of Logic [Descartes, Locke, Leibnitz]
  • Essay 2: Thought as an Attribute of Substance [Spinoza]
  • Essay 3: Logic and Dialectics [Kant]
  • Essay 4: The Structural Principle of Logic. Dualism or Monism [Fichte, Schelling]
  • Essay 5: Dialectics as Logic [Hegel]
  • Essay 6: Once More about the Principle of Constructing a Logic. Idealism or Materialism? [Feuerbach]

Part Two: Certain Problems of the Marxist-Leninist Theory of Dialectics

  • Essay 7: A Contribution to the Problem of a Dialectical Materialist Critique of Objective Idealism
  • Essay 8: The Materialist Conception of Thought as the Subject Matter of Logic
  • Essay 9: On the Coincidence of Logic with Dialectics and the Theory of Knowledge of Materialism
  • Essay 10: Contradiction as a Category of Dialectical Logic
  • Essay 11: The Problem of the General in Dialectics
  • Conclusion

So part one traces a development of dialectics and/or logic, while part two looks as particular aspects of dialectical logic from a Marxist perspective.


*** “thought” and “thinking”

In the short introduction, Ilyenkov defines Logic as the science of thought. The text always qualifies the word with the word “thinking”, e.g.:

Our ‘object’ or ‘subject matter’ in general, and on the whole, is thought, thinking; … (p. 7)

we understand thought (thinking) as the ideal component of the real activity of social people transforming both external nature and themselves by their labour. (p. 8)

It is so invariable I thought I’d check the Russian, and indeed, the Russian always uses “мышление” (thinking) and not “мысль” (thought). Perhaps a tiny echo of the influence on Ilyenkov of Vygotsky’s “Мышление и Речь” (Thinking and Speech).

Essay Two’s title has a much stronger Vygotskian and Spinozan and radical ring to it when translated properly as “Thinking as an Attribute of Substance”.

*** five essays not translated from the russian (after Essay 9 in the english)

While checking the Russian (here), I noticed the Russian original has five essays not translated into the English edition. Three of these are in Part Two, after Essay 9:

  • Essay 10. Логика “Капитала” — The logic of “Capital”
  • Essay 11. Абстрактное и конкретное в диалектической логике — The abstract and the concrete in dialectical logic
  • Essay 12. Диалектическая взаимосвязь логического и исторического — The dialectical inter-relationship between the logical and the historical

The last two are in a third part entitled “Dialectics and the Present” (“Диалектика и Современность”) (Современность is contemporaneity or “the now”):

  • Essay 15. Материализм воинствующий – значит диалектический — Militant materialism means dialectical materialism
  • Essay 16. Диалектика и мировоззрение — Dialectics and “Weltanschauung”

*** Part One

The Russian title for Part One is “Как Возникла и в чем Состоит Проблема”, which means something like “how the problem sprang up and in what it consists”. It’s a wordy but better title: these six essays plot the development of dialectics and/or logic in the modern era. Each essay takes a pivotal figure and describes how they took the field forward.

The essays in the part are a delight. Each is purely positive — Why was Kant important? How did his work enrich our understanding of logic? Criticism is made part of the historical process, so limitations of Kant’s approach are seen through the eyes of Fichte, Schelling and Hegel.

It’s a nice literary trick — and the untranslated Essay 12, on the relationship between the logical and the historical, might show the substance behind the rhetoric.

Given Ilyenkov’s enthusiasm for Plato, which surprised me, I’m slightly disappointed there’s no essay here on logic and dialectics in the Classical world. However, it’s easy to forget the hostile environment under which Ilyenkov was writing.

*** Part Two

Each essay in Part Two explores a particular aspect of logic from a Marxist perspective. In the Introduction Ilyenkov describes these essays (indeed the whole book) as preliminary sketches, statements of the problem: Lenin has bequeathed to us the task of creating a Logic (with a capital “L”) … these essays attempt to concretise some points of departure.

A Contribution to the Problem of a Dialectical Materialist Critique of Objective Idealism

How did Hegel’s *Idealism* constrain his development of dialectics? Why did this Idealist dialectics necessarily tend toward an apologetics of the current order? This essay relates Marx’s Capital (especially it’s first chapter) to Hegel’s Science of Logic, and there’s a nice (though not over-stretched) analogy between Marw’s M-C-M’ and Word-Act-Word.

The Materialist Conception of Thought as the Subject Matter of Logic

Again this is “thinking” — and “conception” is “понимание”, “understanding”.

This essay takes as its starting point Marx’s criticism of Feuerbach’s materialism: “that the thing, reality, sensuousness, is conceived only in the form of the object or of contemplation, but not as sensuous human activity, practice, not subjectively” (“Theses on Feuerbach” #1). Ilyenkov develops a “materialist conception of thought” as a conception of this sensuous — practical, social — human activity.

Spinoza gets a look-in here, in particular Spinoza’s rule (“On the Improvement of the Understanding” #96.1) that a definition of a thing should include its proximate cause:

(1) If the thing in question be created, the definition must (as we have said) comprehend the proximate cause.
(2) For instance, a circle should, according to this rule, be defined as follows: the figure described by any line whereof one end is fixed and the other free.

Ilyenkov: “This definition provided the /mode of constructing the thing/ in real space.”

On the Coincidence of Logic with Dialectics and the Theory of Knowledge of Materialism

Kant, and especially Hegel, were already pushing the territory of Logic beyond its traditional (Scholastic) boundaries. This essay takes as its kernel a note of Lenin’s from his Philosophical Notebooks:

In Capital Marx applied to a single science logic, dialectics, and the theory of knowledge of materialism (three words are not needed; it is one and the same thing).

It’s a side issue to the main drift of the essay, but there’s a very strange passage which seems to touch very gently on the “dialectics of nature” debate (roughly: is dialectics only about human thought, or does it also cover natural processes. Engels seemed gung-ho on the latter. Lenin’s position is unclear (to me at least), leaning towards dialectics of nature in his “Empirio-criticism”, not so much in his Philosophical Notebooks). Ilyenkov seems perhaps to hint that Lenin might have misinterpreted a point of Hegel’s (leading Lenin to make a pro-DN statement) — but Ilyenkov changes the subject as soon as possible and it’s gone. I’ll have a look at the Russian for these paragraphs perhaps.

The passage is four paragraphs starting with the paragraph that starts, “In appearing as a practical act thought included things outside consciousness in its movement, …” Here’s a precis (italics by Ilyenkov, bold by me):

Logic consequently proved to be precisely a theory of knowledge of things also, and not solely a theory of the self-knowledge of the spirit.

Lenin wrote: ‘Logic is the science not of external forms of thought, but of the laws of development “of all material, natural and spiritual things”, i.e., of the development of the entire concrete content of the world and of its cognition, i.e., the sum-total, the conclusion of the History of knowledge of the world.’

There is no such a formulation, and furthermore no such a conception of the subject matter of logic in Hegel himself. In this passage Lenin did not simply translate Hegel’s thought ‘into his own words’, but reworked it materialistically. Hegel’s own text, in which Lenin discovered the ‘rational kernel’ of his conception of logic, does not sound at all like that.

[Ilyenkov quotes the passage from Hegel, including the phrase quoted by Lenin:]

… But these thoughts of all natural and spiritual things [Only these words are found in Lenin’s formulation – EVI] even the substantial content, are yet such as to possess manifold determinations and to contain the distinction between Soul and Body, between a concept and its respective reality; …

The difference between Hegel’s formulation and Lenin’s is one of principle, because there is nothing in Hegel about the development of natural things, and could not even be. It would therefore be a gross error to think that the definition of logic as the science of the laws of development of all material and spiritual things is only Hegel’s idea transmitted by Lenin, or even simply cited by him. It is nothing of the sort; it is Lenin’s own idea, formulated, by him in the course of a critical reading of Hegel’s words.

I find this whole passage very strange and strong. It would be very easy to pas over it as praising Lenin’s materialist re-working of Hegel’s idealist dialectics, but I can’t help reading it as Ilyenkov pointing to a mis-reading by Lenin, an over-reach. The rest of the book — the rest of Ilyenkov — argues firmly against a dialectics of nature position and for a position that the subject matter of logic/dialectics is thought (granted, thought understood as practical social activity).

n.b.: I don’t think Ilyenkov is negative about Lenin generally. He puts Lenin’s Philosophical Notebooks to good use as a kind of commentary to read alongside Hegel. The title of the untranslated Essay 15 “Militant materialism means dialectical materialism” is presumably a reference to Lenin’s essay “On the Significance of Militant Materialism” (which actually I haven’t read …).

Contradiction as a Category of Dialectical Logic

Contradiction is a central category in Hegel’s Logic. This essay looks at Marx’s use of the category in Capital, especially Marx’s assessment and use of the logical contradictions in Ricardo’s economics.

The Problem of the General in Dialectics

A companion to the previous essay, this one takes the category of the general or universal ( a complementary category perhaps to that of contradiction) — as a “standard” category in Hegel, and how Marx “materialises” the category in Capital.

Next steps

Well, those untranslated essay are calling, and I’ll have a look at the Russian for that odd passage in Essay 9.

Otherwise the obvious next step is to re-read Chapter 1 of Capital, this time noticing the dynamics and the dialectics behind the politics and economics.

Notes on “A History of Mathematics”

Sunday, 24th April, 2016

Book details

A History of Mathematics (3rd ed.)
Uta C. Merzbach & Carl B. Boyer
1968, 1989, 1991, 2011


There is a lot of maths in this book. If you don’t like maths, you probably won’t like this book. This book is definitely aimed at the lay reader however (or perhaps the beginning maths student), so concepts are explained, worked examples are given, some are even fun to try out for yourself (e.g. in arithmetic and geometry). Having said that (negation upon negation), the bulk of this book was written in days when “popular exposition” and “accessibility” were taken far more earnestly than they are today: the reader is not pandered to, and there are no silly jokes or narcissistic digressions.

The first nineteen chapters are geographical and chronological. Chapter 19 is about Gauss (1777-1855) and his immediate influence. After the Gauss chapter there are three topic surveys on geometry, algebra, and analysis, and a chapter on “Twentieth-Century Legacies”. Perhaps the authors as historians made an implicit judgement that Gauss separates past from present. However, these later chapters are still “historical” in the sense that they are about the development (and continuing development) of the discipline. There is a brief final chapter sampling some “Recent Trends”.

A book this size (600 pages before bibliographies and index) doesn’t need to present a single linear “path of progress” and full acknowledgement is given to dead-ends, decadence, re-appearances, simultaneous discoveries, etc. It’s a good “history”.

More than anything else I got an idea of the gradualness and lumpiness of the development of the discipline. For example, the piecemeal move from natural language to symbols for ideas like “the thing” (i.e., “x”), arithmetic operations, exponents, etc. Famously “zero” appeared long after all the other numbers, but using the same base for fractions as for integers was not settled for a long time. The equals sign first appeared in print in 1557.

There is a constant interaction, from the earliest times, between what we might call “applied” and “pure” approaches to (or even “conceptions of”) mathematics. From earliest traces (calculations for temple design, rituals, games, patterns), via figures like Archimedes and Euclid, through to figures like Newton and Gauss who were as involved in practical projects as they were in purely mathematical exploration. The institutionalisation of “pure” and “applied” mathematics in academe and schooling isn’t touched on, but it is clear that it is a very very recent development.

Pedagogy is a strong theme in the book. Mathematics is presented as a discipline in which the elite are centrally concerned with defining, presenting, and renewing their discipline. There’s Euclid’s Elements, and several similar projects, but the theme really takes off in the chapter on the French revolution (Chapter 18, “Pre- to Postrevolutionary France”). An interesting 20th century example is Nicolas Bourbaki, a “polycephalic” mathematician producing mathematics textbooks since the 1930s.

I am not a fan of revisionist histories, but it is miserable that only five women were mentioned in the whole book: Hypatia, Sofia Kovalevskaya, Sophie Germain, Mary Winston Newson, Emmy Noether (so that’s no women at all between 415 and 1850).

Next steps

On the mathematical side, I am reminded of my interest in Riemann’s non-Euclidian geometry (for Schrödinger‘s wave mechanics). I don’t think reading this history has been a mathematical “preparation” but Schrödinger (“Space-Time Structure”, 1950) or Weyl (“Space-Time-Matter”, 1922) might be a nice next read.

Philosophical or foundational issues were not touched on much, apart from as occasional asides, but there is plenty of historical grist for such considerations. Two introductions to the philosophy of maths I have my eye on are “Thinking About Mathematics” (Stewart Shapiro, 2001) and “Introducing Philosophy of Mathematics” (Michele Friend, 2007).

Book details

Externalism: putting mind and world together again
Mark Rowlands


I liked this book a lot. It describes the internalist position and gives a kind of chronological survey of the externalist response. After highlighting the limitations of the “standard” externalism (which Rowlands calls Content Externalism), Rowlands outlines a more radical (and much more interesting imho) variant which he calls Vehicle Externalism.

** The Internalist position: Cartesianism (Chapter 2) and Idealism (Chapter 3)

Rowlands roots internalism in Cartesian dualism: the Mind/Body split, and the individual pairing between an individual Mind (or Soul) and that Mind’s own Body. In “Cartesian materialism” the explicit dualism is dropped and instead of the Mind/Soul being linked to the Body, the mind is somehow located or implemented in the body. Internalism is then characterised as holding two claims (p. 13):

  • Location: “any mental phenomenon is spatially located inside the boundaries of the subject, S, that has or undergoes it.”
  • Possession: “the possession of any mental phenomenon by a subject S does not depend on any feature that is external to the boundaries of S.”

Although the dualism has been officially expunged, idealism still has its influence, and Chapter 3 describes things like the Sapir-Whorf hypothesis, structuralism, and so on, as examples of a post-Kantian idealism.

Chapter 2 effectively shows the modern mainstream internalist position as a more-or-less direct descendant of the Cartesian position. Chapter 3 otoh is perhaps the weakest of the book, but also the least central (at least for my own interest).

There is a brief nod to historical context in Chapter 2, pointing to the influence on Descartes of “the rise of mechanism associated with the scientific revolution” (p. 18). Marx was very critical of the “mechanical” materialism of his own day, saying the the dynamic idealism of e.g. Hegel was closer to the truth than the mechanical materialism of e.g. Feuerbach. Interesting if the mechanical approach was not a consequence of materialism.

Repeatedly Rowlands conflates the Ideal and the Mental (e.g., pp 4, 32, 60). On p. 4 Rowlands is characterising Kant, and on p. 60 Berkeley. However, on p. 32 Rowlands seems to be defining Idealism per se as “the view that reality is mental”. This might be a fair characterisation of Kant’s idealism, but I don’t know if it is a fair characterisation of Plato’s idealism. It is certainly not accurate of Hegel.

Incidentally, the error of conflating the Ideal and the Mental — and an argument that this was *not* Plato’s position — is one of Ilyenkov’s main points in his Dialectics of the Ideal.

** Externalist responses

The Externalist response is presented in three chronological episodes:

  • Husserl & Sartre (Chapter 4)
  • Wittgenstein (Chapter 5)
  • Putnam, Burge and others — “Content Externalism” (Chapters 6-8)

I don’t want to use the word “historical” because there is no development from one episode to the next, although Husserl’s ideas reappear in Vehicle Externalism.

Husserl & Sartre

Chapter 4 was the surprise highlight of the book for me.

An important part of Husserl’s phenomenology seems to have been a “principle of unity” of consciousness. Each phenomenon of consciousness does not exist alone, but only in a network of related phenomena. Consequently each experience brings with it anticipations or expectations or potentialities (all words used by Rowlands at different points) of other experiences.

It’s easy to psychologise this — in fact, it’s difficult for me not to psychologise it. It immediately brought to mind Mead about the horse:

A person who is familiar with a horse approaches it as one who is going to ride it. He moves toward the proper side and is ready to swing himself into the saddle. His approach determines the success of the whole process. But the horse is not simply something that must be ridden. It is an animal that must eat, that belongs to somebody. It has certain economic values. The individual is ready to do a whole series of things with reference to the horse, and that readiness is involved in any one of the many phases of the various acts. It is a horse that he is going to mount; it is a biological animal; it is an economic animal. Those characters are involved in the ideas of a horse. If we seek this ideal character of a horse in the central nervous system we would have to find it in all those different parts of the initiated acts. One would have to think of each as associated with the other processes in which he uses the horse, so that no matter what the specific act is, there is a readiness to act in these different ways with reference to the horse. We can find in that sense in the beginning of the act just those characters which we assign to “horse” as an idea, or if you like, as a concept.

(Mead, 1934, Mind, Self and Society, chapter 2, page 12)

Sartre’s contribution spells out the implications of Husserl’s phenomenology for the internalist/externalist debate. Specifically, if consciousness is nothing more than a network of expectations, there need be nothing at all “in” consciousness: consciouness is an entirely outward pointing attitude (so for Sartre the opposition is not between matter and mind, but between “Being and Nothingness”). Rowlands makes the connection with behaviourism (and connectionism for that matter) virutally explicit when he says (p. 65):

Not only are the intentional objects of consciousness not in consciousness, but also there are no /representations/ of these objects.

My prejudice of Husserl had always been of a kind of latter-day, vulgarised Hegel. That prejudice is not challenged here. However, I can see how Husserl’s work on consciousness (as presented by Rowlands) could be part of a kind of “greater behaviourism” (which would include such as Vygotky and Mead alongside Pavlov and Skinner).

Sartre comes over as a latter-day, vulgarised Marx. Sartre claims his interpretation of Husserl is a “radical reversal of idealism” (p. 64; Sartre’s Being and Nothingness p. 239). This obviously recalls Marx’ declaration of turning Hegel right-side-up in the introduction to Capital.


I can’t get excited about Wittgenstein. It’s not that I disagree with what he says. His work just seems very slight. This chapter covers things like the private language argument, and the notion of following a rule as conforming to a community practice.

Content Externalism

Chapters 6-8 cover the archetypal Externalism of Anglo-American “Analytical” philosophy, with Hilary Putnam and Tyles Burge and their twin-earth thought experiments as prime exemplars. I’m afraid I find this kind of philosophy absolutely tedious. I can say Rowlands seemed to be systematic and thorough.

Chapter 6 lays out the strength of the Content Externalist attack on internalism.

One interesting thing about Husserl and Wittgenstein, which disappears in Content Externalism, is that with H & W the subject of consciousness is an active agent: some form of “situatedness” seems to treated as fundamental to the subject, so each “consciouness event” is related to previously experienced or anticipated events. With the “Analytic” philosophers, the examples are isolated events: a moment of perception, the holding of a belief. The subject of consciousness is purely contemplative. This takes us back to Marx’s criticism of Feuerbach and mechanical materialism.

** Vehicle Externalism

Content Externalism addresses Internalism’s Possession claim and (according to Rowlands in chapter 7) is further limited to certain kinds of “mental contents”. Vehicle Externalism addresses the Location claim. Vehicle Externalism is the position that the thinking subject, or the system that thinks, is to some extent outside the boundaries of the human body.

Chapter 9 spends a lot of time in biology, e.g., how some animals (examples used are beavers and acanthocephalan worms) have evolved to outsource functions to their environment. James Gibson’s ecological approach to vision is used to show (a) how animals use objective structure in their physical environment and (b) how important is the subject’s subjectivity, activity, to understanding their cognition.

It’s a slight shame that the relevance of Husserl & Sartre’s arguments from chapter 4 is not spelt out explicitly in this part of the book. There are very strong echoes of Husserl’s anticipations/expectations in the potentialities & sensorimotor contingencies referred to here. Rowlands criticises a representational theory of mind, preferring what he calls an “extended” approach (as in extending into the environment) — with similarly clear echoes of Sartre’s outwardly pointing consciouness (and imho echoes of behaviourism).

That’s just a slight shame though. I enjoyed the book as a whole, and I thought Chapters 4 (on Husserl and Sartre) and 9 & 10 (on Vehicle Externalism) were very strong.

Next steps

What would a Marxist make of all this? Surely a Marxist would never even consider an Internalist position? But Ilyenkov’s polemic in Dialectics of the Ideal, against the “official” Soviet Marxist line, is precisely against an Internalist line where the Ideal is mental and the mental is neural. So these arguments still have to be had.

Rowlands equates the Ideal with the mental, which is a weakness, but the most important line in this book (Husserl/Sartre through to Vehicle Externalism) doesn’t use that equation at all. In fact I think this line is if anything more sympathetic to Ilyenkov’s position, that the Ideal is based on social practice.

To me Vehical Externalism is far more interesting than Content Externalism, which really does feel like a debate entirely internal to academic Analytical philosophy. Content Externalism still retains the absolute split between subject and object, while in Vehicle Externalism part of the subject is “out there” in the object (e.g., language, custom, features of the social environment).

So, two paths lead out of this:

  • The book was written over twelve years ago. Where is Vehicle Externalism now? How has the Externalist position in general developed?
  • After reading this book, Husserl suddenly seems interesting. I want to get a quick overview of his philosophy and a sense of the strengths and weaknesses of his approach. IIRC Lukacs gives Husserl short shrift (in the Destruction of Reason). Does Hegel (Marx, Vygotsky, and others) give us everything we need? What exactly does Husserl bring? Part of the light shining on this exploration will be the differences between Husserl and Brouwer (in which my preference is closer to Brouwer).

[update: Mark Rowlands has a more recent (2013) book out on externalism: The New Science of the Mind: From Extended Mind to Embodied Phenomenology.]

Book details

Dialectics of the Ideal: Evald Ilyenkov and Creative Soviet Marxism
Edited by Alex Levant and Vesa Oittinen


The focus of this book is Ilyenkov’s essay “Dialectics of the Ideal”, which he wrote in 1974 but which was not published until 2009 (long after Ilyenkov’s death in 1979). This is the essay’s first translation into English in full (an incomplete translation was published in 2012). As well as the essay itself, there are various articles providing context and commentary.

“Dialectics of the Ideal” itself is a good introduction to Ilyenkov and, perhaps, a good introduction to (a certain kind of) Marxism. Ilyenkov’s is a “Classical” Marxism, in the sense that his references are to Marx, Lenin, and then to earlier sources like Hegel. Ilyenkov was strongly influenced by Vygotsky, and the two share an influence in Spinoza.

The essay is primarily a polemic against a reductionist interpretation of the Ideal, then current in Soviet philosophy, which interpreted ideal phenomena (e.g. concepts) as mental states, and thenceforth reduced them to neural events. This position is perhaps comparable to eliminative materialism and similar positions clustering around neuroscience.

Ilyenkov’s position is that ideal phenomena are social — kind of representations of social practices — which confront the individual, and consequently that any mental or neural states are effects of this pre-existing Ideal. Ilyenkov’s position can perhaps usefully be compared with social externalism in the analytical tradition.

the other essays

The other essays are variable, but some are very good. In particular, the essays about Ilyenkov and his context in Soviet (and current Russian) Marxism:

  • Alex Levant, the translator, and the kind of Maitre D’ of the book provides opening and closing essays;
  • Andrei Maidansky, an academic philosopher at Belgorod State University, who has an Ilyenkov web site, with texts in Russian, English and some other languages, writes a very good commentary essay “Reality of the Ideal”;
  • There is an interview with Sergei Mareev, author of “Из истории советской философии. Лукач-Выготский-Ильенков” (2008, “From the history of Soviet philosophy: Lukacs – Vygotsky – Ilyenkov”). Very nice interview — and I could hardly believe my eyes when I saw the title of this book: my three favourite Marxists (after Marx and Lenin). I need to read it!

Next Steps

Read next/soon: Ilyenkov’s “Dialectical Logic” (1974).

Keep up with Alex Levant’s papers — which he publishes on his page.

Read Mareev’s 2008 “Из истории советской философии. Лукач-Выготский-Ильенков”. Ha ha ha! No, really. First I am working my way through Maidansky’s 2009 review “Диаграмма философской мысли”. Reading that is interesting in its own right, and it will tone up my Russian so I can (slightly more) sensibly embark on Mareev’s book.

Re-familiarising myself with social externalism might be a worthwhile thing to do, as it might broaden out again my understanding of this area.

book details

L. E. J. Brouwer — Topologist, Intuitionist, Philosopher: How Mathematics Is Rooted in Life
Dirk van Dalen

the biography

This is a very good biography. Exasperatingly thorough and comprehensive (830 pages not including bibliography and index). It’s a very humane book, and a good example of what a humanistic project a biography is, or should be.

the maths

During Brouwer’s work on his PhD thesis — perhaps especially in correspondence with his PhD supervisor Korteweg, and in parts of the thesis that were later rejected — we see already many of the foundation stones of Brouwer’s intuitionism (p. 86-7): e.g., that “the points of departure of the theory” should be determined by “how mathematics roots in life” (p. 86), that “the primeval phenomenon is simply the intuition of time”, the “shift from goals to means”.

The “life” in which maths roots is not a social life, and the rooting is not an instrumentalism. It seems to be a kind of subjective idealist solipsism: “my mathematical thinking is non-sensory internal architecture”, “consciousness gains access to free creation — which is my mathematics — as soon as it knows itself autonomous and immortal, ignoring objective knowledge and common sense.” (p. 190-1, interview with Weissing ca 1913).

Brouwer was aware of and did tackle preceding work on mathematical foundations, including Kant, and especially the constructivist approaches of Kronecker, Poincare and Borel (p. 232f). Brouwer’s position differed from these earlier constructivists in important ways.

The constructivists’ main concern seems to have been that mathematical objects should be constructible in finitely many steps from the natural numbers — e.g., irrational numbers were admissable mathematical objects only if derivable from natural numbers by an algorithm with finitely many steps. Brouwer held to this rather weakly even in the beginning, for example including “free selection” as an algorithmic step. He does seem to have kept the “finitely many steps” idea, but in a less technical form.

Brouwer also differed from the earlier constructivists in his rejection of the principle of the excluded middle (p. 104f, 196). See his 1908 paper. In 1918 he went further and demonstrated that the law of double negation cannot be proved (p. 307). n.b. these constraints on logic are for infinite domains like mathematics.

However, Brouwer’s focus seems not to have been on the objects of mathematics, but on the subject of mathematics — the mathematician. Van Dalen refers to Brouwer’s “subjectivist approach to mathematics” (p. 306). The creating subject at the centre of Brouwer’s maths is an idealised mathematician (p. 738-9).

next steps

The two points most interesting to me are (a) the nature of the mathematical subject and (b) the constraints on logic.

Unlike Husserl’s “transcendental” subject, Brouwer’s subject exists in time. The “intuition” of time is a foundational experience for this subject, but what is meant by an “intuition” and why the subject’s appreciation of time can’t be ordinary learned experience is not explored here. Of course I am tempted to reach for some kind of psychologism, or maybe a kind of “abstract” psychologism a la Hegel.

Reading about Brouwer’s constraints on logic I kept thinking about Hegel’s logic (all those old slogans: “the unity of opposites”, “the negation of the negation”).


  • read more on the nature of the subject and the constraints on logic in Brouwer’s maths (see references below)
  • swot up on Hegel’s “Science of Logic”
  • it might be useful to read an introduction to the philosophy of maths, as it is a completely new field to me. Two likely-looking candidates are “Thinking About Mathematics” (Shapiro, 2001) and “Introducing Philosophy of Mathematics” (Friend, 2007).

annotated references

Brouwer, L. E. J. (1908). De onbetrouwbaarheid der logische principes (The unreliability of the logical principles). Tijdschr. Wijsb. 2, 152-58. (There are two English translations available: one in volume 1 of Brouwer’s Collected Works (1975); and a new translation by Mark van Atten & Göran Sundholm

Brouwer’s “revolutionary rejection of the general validity of the principle of the excluded third” (p. 104f, 106).

Brouwer, L. E. J. (1913). Intuitionism and formalism. Bull. Am. Math. Soc. 20, 81-96. (English translation of “Wiskunde, Waarheid, Werkelijkheid”).

This was Brouwer’s inaugural lecture, in which he differentiates his own intuitionism from the contructivism of Kroneker, Poincare, et al. (p. 218-20, 233f).

Brouwer, L. E. J. (1929). Mathematik, Wissenschaft und Sprache. Monatschefte Math. Phys. 36, 153-164. English translation in Mancosu (1998).

Presents “Brouwer’s views on the genesis of the basic entities of the subject’s inner and outer world”, motivated and developed more clearly here than in Brouwer’s PhD thesis (according to van Dalen. Plausible).

Brouwer, L. E. J. (1949). Consciousness, Philosophy and Mathematics. Proceedings of the 10th International Congress of Philosophy, Amsterdam, 1948, pp. 1235-1249.

This paper sounds like a summing up and a restatement of B’s fundamental intuitionist positions (p. 755-6): the phases of consciousness, the “move of time”, the “cunning act” (aka the jump from ends to means), the creating subject, the solipsism.

There also seems to be an innovation, in the “… phenomenon of play, occurring when conative activity or causal thinking or acting is performed playfully, i.e. without inducement of either desire or apprehension …”

Johan Huizinga is only mentioned once in the biography, when Brouwer mourns Huizinga’s death in a 1945 letter (p. 702), but I couldn’t help thinking of Homo Ludens when I read this passage. Huizinga published Homo Ludens in 1938.

Brouwer, L. E. J. (1955). The effect of intuitionism on classical algebra of logic. Proceedings of the Royal Irish Academy. Section A: Mathematical and Physical Sciences Vol. 57 (1954 – 1956), pp. 113-116.

Brouwer’s last paper, stresses “the basic differences between classical and intuitionist logic”, “worth reading … because of its reflections on the nature of logic” (p. 798-800).

Heyting, A. (1956). Intuitionism, an introduction. North-Holland.

“Its role should not be underestimated, … its readability has always been praised.” (p. 800)

Mancosu, P. (1998). From Brouwer to Hilbert: The Debate on the Foundations of Mathematics in the 1920s. Oxford UP.

Translated papers of Brouwer, Weyl and others. Contains Brouwer (1929) and Weyl (1921).

van Dalen, D. (2004). Kolmogorov and Brouwer on constructive implication and the Ex Falso rule. Russ. Math. Surv. 59, 247-257.

This paper traces” the construction-meaning of implication in Brouwer’s famous ‘jump from ends to means'” (p. 612).

Weyl, H. (1921). Uber die neue Grundlagenkrise der Mathematik. Math. Z. 10, 39-79. English translation in Mancosu (1998).

This sounds like quite an exciting read, “beautiful provocative paper” (p. 311).

Paper details

M. van Atten, D. van Dalen, and R. Tieszen
Brouwer and Weyl: the phenomenology and mathematics of the intuitive continuum
Philosophia mathematica 10(3), 203-226


I thought this was a nice accessible paper (hardly any maths!). As well as “doing what it says on the tin”, it provides an introductory overview and a discussion of choice sequences.

Next stop for me is van Dalen’s biography.


The “intuition” in Brouwer’s intuitionism seems to be to do with our appreciation of time. Weyl’s definitions seem to be explicitly Husserlian, but Brouwer doesn’t explicitly use Husserl’s intuitionism. ADT quote Brouwer describing (what they repeatedly call) “the primordial intuition of mathematics” as “the substratum … of any perception of change, a unity of continuity and discreteness”. Related “intuitions” are the role of memory, and the asymmetry between past and future.

My questions here would be:

  • What does Brouwer mean by (an) intuition? Does it follow a Kantian distinction between learned knowledge and direct appreciation?
  • Why choose these particular things to be the “primordial” intuitions of mathematics? Time is a good candidate I concede. An even more “primordial” foundation might be eye movement: saccadic movement for the discrete counting; smooth pursuit movement for the continuum.
  • (Why) is it important that these things be unlearned intuitions? From my reading in developmental psychology, I would say these things — appreciation of time, memory, appreciation of an asymmetry between past and future — are learned by the human infant.

Related is the question of how Brouwer’s intuitionism differs from other constructivist approaches to mathematics. How much of this talk of “intuition” is really necessary?

Brouwer vs Weyl; Brouwer vs Husserl

Weyl seems to be much closer to Husserl than Brouwer does. It wasn’t clear to me from reading this paper whether Brouwer was just unfamiliar with Husserl or actively disagreed with Husserl’s position. Where differences are outlined in this paper, Brouwer’s position seems preferable to me.

Bits I like

I like Brouwer’s emphasis on the “creative subject” (or “transcendental ego”). This subject seems to exist in time: e.g., it can carry out only “finitely many acts of intuition”.

In what sense is this subject “transcendental”? What does it “transcend”? (and, again, why?)

Why am I interested in all this anyway?

My interest in Brouwer — and more broadly in the Constructivist approach to mathematics — is a digression from my main interests. However, it’s a digression from three of my main interests at the same time, which makes it hard to ignore.

  • Work: I am a computer programmer. The importance and role of type systems is a hot topic atm, and consequently type theories and proof systems. The constructivist approach to maths is relevant to computer programming because, like Brouwer’s creative subject, the computer program exists in time.
  • Schrödinger: I want to understand Schrödinger’s wave mechanics! That is the maths I am really interested in. I think I first heard about Brouwer — especially the Brouwer-Weyl connection — when I was reading Schrödinger’s biography late last year. Weyl was Schrodinger’s best friend in Zurich, and helped S with some of the maths for his breakthrough wave mechanics paper (1926).
  • Marx & Hegel: As well as the reference to Hegel on nLab, lots of the Brouwer in this paper is very reminiscent to me of Hegel: Brouwer’s “unity of continuity and discreteness” recalls Hegel’s “unity of unity and disunity”; the “two-ity” described early in ADT recalls the fundamental place Hegel gave to contradiction.