Many thanks to @yanzhiao for sharing the poem, and for help with translation.

贺麟《斯宾诺莎像赞》

宁静淡泊,持躬卑谦。
道弥六合,气凌云汉。
神游太虚,心究自然。
辨析情意,如治点线。
精察性理,揭示本源。
知人而悯人,知天而爱天。
贯神人而一物我,超时空而齐后先。
与造物游,与天地参。
先生之学,亦诗亦哲;
先生之品,亦圣亦仙。
世衰道微,我生也晚;
高山仰止,忽后瞻前。

zh, py, en

贺麟《斯宾诺莎像赞》
Hè Lín “sī bīn nuò shā xiàng zàn”

He Lin “Inscriptions on a Portrait of Spinoza”

1. 宁静淡泊,持躬卑谦。
níng jìng dàn bó, chí gōng bēi qiān.

Living life simply, peacefully,
keeping yourself modestly.

2. 道弥六合,气凌云汉。
dào mí liù hé, qì líng yún hàn.

Dao fills the universe,
Spirit soars to the stars.

3. 神游太虚,心究自然。
shén yóu tài xū, xīn jiū zì rán.

Mind travelling the universe,
your heart investigated nature.

4. 辨析情意,如治点线。
biàn xī qíng yì, rú zhì diǎn xiàn.

Analysing affection and idea
by the rules of point and line.

5. 精察性理,揭示本源。
jīng chá xìng lǐ, jiē shì běn yuán.

A close examination of properties
brings to light the roots.

6. 知人而悯人,知天而爱天。
zhī rén ér mǐn rén, zhī tiān ér ài tiān.

Knowing humanity but pitying humanity,
Knowing nature but loving nature.

7. 贯神人而一物我,超时空而齐后先。
guàn shén rén ér yī wù wǒ, chāo shí kōng ér qí hòu xiān.

Connecting God and man, unifying object and self;
Beyond time and space, uniting before and after.

8. 与造物游,与天地参。
yǔ zào wù yóu, yǔ tiān dì cān.

both immersed in substance,
and engaged part in the world.

9. 先生之学,亦诗亦哲;
xiān shēng zhī xué, yì shī yì zhé;

Teacher of science, both poetic and wise.

10. 先生之品,亦圣亦仙。
xiān shēng zhī pǐn, yì shèng yì xiān.

Teacher of morals, both sainted and celestial.

11. 世衰道微,我生也晚;
shì shuāi dào wēi, wǒ shēng yě wǎn;

The world wanes and the way is dim,
I was born late.

12. 高山仰止,忽后瞻前。
gāo shān yǎng zhǐ, hū hòu zhān qián.

A high mountain, a summit to aim for,
Forget the past and face the future.

Notes

He Lin 贺麟 was a twentieth century philosopher, a representative of the Chinese new Confucianism, also with interests in Hegel and Spinoza.

The title is an allusion to 《历代古人像赞》 (“Lì dài gǔ rén xiàng zàn”, “Praise for the ancients”), a 15th century book of portraits of classic thinkers. Each portrait had a short poem inscription.

*** 12.1 Book of Songs

The first part of line 12 is an allusion to a poem in the Book of Songs (《诗经》, “Shī jīng”). Although the allusion is perhaps to the line’s Confucian reuse in the Shǐjì (史記):

《诗》有之:”高山仰止,景行行止”。虽不能至,然心向往之。

The poem says: “Look up at the high mountains and walk along the wide road”. Although I can not reach this level, my heart desires to be there.

(source)

*** 12.2 Confucius

The second part of line 12 is an allusion to analect 9.11 from Confucius’ Analects:

颜渊喟然叹曰:‘仰之弥高,钻之弥坚,瞻之在前,忽焉在后。夫子循循然善诱人,博我以文,约我以礼。欲罢不能, 既竭吾才,如有所立卓尔。遂欲从之,末由也已。’

Yan Hui, with a deep sigh, said, ‘The more I look up at it, the higher it soars; the more I penetrate into it, the harder it becomes. I am looking at it in front of me, and suddenly it is behind me. The Master is good at drawing me forward a step at a time; he broadens me with culture and disciplines my behavior through the observance of ritual propriety. Even if I wanted to quit, I could not. And when I have exhausted my abilities, it is as though something rises up right in front of me, and even though I want to follow it, there is no road to take.’

(tr. Roger T. Ames)
(source)

XIX Международная научная конференция «ИЛЬЕНКОВСКИЕ ЧТЕНИЯ» Э.В. Ильенков и проблема человека в революционную эпоху” (Москва, 20-21 апреля 2017 г.)

http://tovievich.ru/news/7831-xix-mezhdunarodnaya-nauchnaya-konferenciya-ilenkovskie-chteniya-ev-ilenkov-i-problema-cheloveka-v-revolyucionnuyu-epohu-moskva-20-21-aprelya-2017-g.html

http://www.caute.ru/ilyenkov/conf.html

Sex and gender

Saturday, 28th January, 2017

Working definitions and some corollaries.

Sex

Sex is a biological category, part of the system of reproduction. Some creatures (animals and plants) reproduce asexually: a single individual of a species can reproduce on its own. Some creatures reproduce sexually: two individuals of distinct kinds (sexes) must come together to reproduce. The sexes are generally called male and female. Each sex plays its own distinct role in reproduction. Categorisation of the sexes depends on the logically prior category of reproduction. A sex is defined by its role in reproduction.

We could imagine a species requiring three sexes for reproduction — e.g. providing sperm, egg and incubator. I don’t know if there are there any such species, animal or plant.

What makes a particular individual a male or a female of its species will be a set of biological features, each of which will be more or less important, more or less directly related to reproduction. In a given species, the biological feature sets of the different sexes might overlap — less so in spiders, more so in rabbits.

In a given individual, the extent to which it carries the features of one or other sex will presumably vary as well. However, given natural selection, we’d expect most functional individuals of a species that reproduces sexually to be functionally one sex or the other.

Update:
As I failed to imagine or know, but as @peter4logo points out, there are species that reproduce sexually in which individuals can change sex, e.g. snails, (aka Sequential hermaphroditism), and species in which an individual can function as both sexes (e.g. giant clams). So, the above paragraphs should be generalised. Note that above kinds of species retain the separate sex categories of male and female.

So, we have the biological system of reproduction and, within that system, we have the sexes male and female, defined in relation to each other, and in relation to the system of reproduction.

Gender

Gender is a human, social category, an artefact of the oppression of women. That is, oppression on grounds of sex.

By artefact I mean both “effect” and “means”.

A gender is a set of virtues (update: i.e. of required or acceptable behaviours) historically associated with a particular (human) sex. For example, the archetypal feminine virtues in Elizabethan England were obedience, silence and chastity.

A logical/historical development of gender might follow these lines:

  1. oppression of women
  2. specification of virtues (behaviours & other traits) required of women
  3. ideological elevation of this specification: femininity
  4. naturalisation of the specification: gender, masculinity

Whereas with sex we have the system of reproduction directly entailing both male and female, with gender we have the system of oppression entailing only constraints on women’s behaviour. “Femininity” is an ideological elevation of this set of constraints. “Gender”, as a naturalisation of femininity, comes later (logically, and I would hypothesise, historically). “Maculinity” is a corollary of gender.

So “feminine” and “masculine” genders don’t make a direct ‘pair’ in the way that “male” and “female” sexes do.

Some corollaries

  • Ideologies which take gender as given are reactionary, and will to some extent be complicit in the oppression of women.
  • Feminism is opposition to the oppression of women — i.e., opposition to oppression on grounds of sex.
  • So-called “trans women” are not women.
  • The question of whether trans people are oppressed, and what to do about it, are legitimate political questions, but they have nothing to do with feminism.

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.

Now,

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.

On Lukács’ alleged dualism

Tuesday, 31st May, 2016

It is on my books to write a piece on why I think McNally 2015 does not demonstrate (as it claims to) that “the social relations of race, gender and sexuality, among others, [are] internally constitutive of class”. However, that is a story for another day.

One of his footnotes recalled a snippet of Gramsci I came across once, so for now I’d just like to discuss that.

Quotations

Among these shortcomings [of Lukács] are a dualistic confinement of dialectics to society rather than to nature, and a related failure to theorise the mediating role of labour in the dialectic of humans and nature.

McNally, 2015, n.5.

It would appear that Lukács maintains that one can speak of the dialectic only for the history of men and not for nature. He might be right and he might be wrong. If his assertion presupposes a dualism between nature and man he is wrong because he is falling into a conception of nature proper to religion and to Graeco-Christian philosophy and also to idealism which does not in reality succeed in unifying and relating man and nature to each other except verbally. But if human history should be conceived also as the history of nature (also by means of the history of science) how can the dialectic be separated from nature? Perhaps Lukács, in reaction to the baroque theories of the Popular Manual, has fallen into the opposite error, into a form of idealism.

Gramsci, 1971.

[The Popular Manual Gramsci is referring to is Nikolai Bukharin’s Historical Materialism.]

Comments

These two texts point to the same shortcoming of Lukács. Lukács conceives of dialectics as applying to social/historical phenomena only, and not to nature. Ilyenkov shares this shortcoming. Ilyenkov is very clear in Dialectical Logic that dialectics/logic is the science of thinking, with thinking understood as a social practical activity. Ilyenkov includes as thinking not just speech, but “the whole objective body of civilisation, … tools and statues, workshops and temples, factories and chancelleries, political organisations and systems of legislation.”

I don’t share Gramsci & McNally’s perspective so I can only guess where the accusation of dualism might come from. It sounds like there is this thing called “dialectics”, and Lukács has decided to apply it to one set of things (society, history), but not another set of things (nature). Splitting reality into two like this — instead of using this given “dialectics” to understand all of reality — is dualist. Lukács has decided that reality is made up of two types of thing.

In his essays on the history of dialectics, Ilyenkov presents things the other way around. Logic and dialectics developed, over the centuries, through a long study (an empirical study if you like) of thought. Ilyenkov’s (and I should think Lukács’) position on what dialectics rightly applies to is based on understanding of how dialectics came to be.

The Gramsci & McNally conception of dialectics seems to be of something handed down to us, something pre-existing.

I am not very familiar with this “Dialectics of Nature” position (I haven’t read much Engels at all apart from his Socialism: Utopian and Scientific pamphlet). Dialectics would certainly have something to say about chemistry or biology as sciences (i.e. as thinking), but I can’t see how dialectics would be applied to chemical or biological matter.

If anyone can recommend recent applications of dialectics to nature, I would like to read and try to understand.

References

Gramsci, Antonio. (19??, tr. 1971). “The Concept of ‘Science'” section of “Problems of Marxism,” in Selections from the Prison Notebooks, ed. and trans. by Quintin Hoare and Geoffrey Nowell Smith (New York: International Publishers, 1971), page 448.
Gramsci on Lukács and the dialectic in human vs natural history

McNally, David. (2015). “The dialectics of unity and difference in the constitution of wage-labour: On internal relations and working-class formation”. Capital & Class, February 2015; vol. 39, 1: pp. 131-146.
http://cnc.sagepub.com/content/39/1/131.full.pdf+html

Book details

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

Contents

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.

Comments

*** “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
Wiley

Summary

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).