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).

Idealism and Idealism

Sunday, 10th April, 2016

When I think of Idealism I think of Plato and Hegel.

For Plato, Ideas (the Greek word he used – ἰδέα – is where everyone else gets the word from) floated about in their own little realm, somewhere other than spacetime, completely independent of humanity. Our human conceptions were more-or-less clumsy graspings after these Ideas.

For Hegel, there was only one thing which existed: the Idea (Hegel used the German Geist or spirit). Hegel’s philosophy was all about exploring the multi-faceted contradictions of this single thing (especially in the Phenomenology of Spirit and the Science of Logic). Humanity, or rather human-type consciousnesses, tumble out of the proceedings at some point.

Reading Marx, and Marxist philosphy, this is the way terms like Idealism, the Ideal, etc. are used.

Now I am reading more broadly, I have started coming across people for whom Idealism means Something Else. This Idealism is rooted in Descartes and Kant, and it essentially understands “Ideal” as “Mental”. From my Big Book of Philosophy (entry on idealism, philosophical):

[Idealism] maintains that what is real is in some way confined to or at least related to the contents of our own minds. … It has been argued by Myles Burnyeat that idealism proper could not appear before Descartes had argued for the epistemological priority of access to our own minds.

The entry goes so far as to claim that Plato was not an Idealist (which of course he certainly wasn’t, according to that definition):

Plato’s theory of forms is sometimes said to be a species of Idealism on the grounds that his Forms are also called Ideas. But those so-called Ideas were not simply contents of our minds; indeed Plato explicitly rejects that supposition in his Parmenides.”

(That is D. W. Hamlyn writing in 1995)

Sadly this latter conception of Idealism seems to be common. I can’t see how it is an improvement on even Plato’s Idealism as it pre-supposes the most important thing to explain, i.e., the thinking subject. I can see now why Rowlands felt the need to devote a chapter to chasing after various types of contemporary neo-Kantianism (e.g., Sapir-Whorf, Chomsky, post-structuralism).

On these pages, Idealism will generally refer to proper Idealism (Plato, Hegel). If I need to discuss the degenerate (post-)Kantian version I’ll do so explicitly.

Book details

Externalism: putting mind and world together again
Mark Rowlands
2003
Acumen

Overview

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.

Wittgenstein

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
2014
Brill

Notes

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 academia.edu 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.

A New Year wish from Yang Lan

Saturday, 2nd January, 2016

Yang Lan (杨澜) sends a New Year message from her Weibo account:

http://www.weibo.com/1198920804/DbdtTgUAT

** gloss

In the New Year, with love at the centre, extending life’s radius, your world flourishing brilliantly!

** original

新的一年里,以爱为圆心,延长人生的半径,你的世界繁盛精彩!

** new words

圆心 yuán xīn centre (circle heart)
延长 yán cháng extend
半径 bàn jìng radius (half path)
繁盛 fán shèng thriving, flourishing, prosperous
精彩 jīng cǎi brilliant, splendid

A New Year Wish from Sammi Cheng

Friday, 1st January, 2016

Sammi Cheng sends a New Year message from her Weibo account:

http://www.weibo.com/1689219395/Db3zExUjl

** gloss

At last we are in 2016. All hope this year is better. But what do we mean by “good”? More successful? Richer? More beautiful? More “the best”? These definitions might conform to the standards of the world, but really in our hearts we might have a better “good”. For example: More loving. More forgiving. More kind-hearted. More humble. Let’s go!

** original

终于处身2016,任谁都希望一年比一年好。但是,「好」的定义又是什么?更成功?更富有?更美丽?更顶尖?这些也许是世界提供的标准,但其实我们心中可能有比这些更好的「美好」。例如:更有爱。更包容。更善良。更谦卑。共勉之!

** new words

终于 zhōng yú finally, at (long) last; 于 (yú) = at (time)
处身 chǔ shēn dwell
定义 dìng yì definition
顶尖 dǐng jiān peak, centre, “best”
也许 yě xǔ perhaps
标准 biāo zhǔn standard, criterion
其实 qí shí in truth; 实 (shí) = solid
包容 bāo róng forgive
善良 shàn liáng kind-hearted
谦卑 qiān bēi humble

** constructions

[notes forthcoming]

任谁都 rèn shuí dōu
一年比一年好 yī nián bǐ yī nián hǎo
yòu
比这些更好的「美好」 bǐ zhè xiē gèng hǎo de “měi hǎo”
共勉之 gòng miǎn zhī

book details

L. E. J. Brouwer — Topologist, Intuitionist, Philosopher: How Mathematics Is Rooted in Life
Dirk van Dalen
2013
Springer

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”).

So:

  • 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 https://www.academia.edu/17642508).

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).

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