Remarks on the foundations of mathematics

by Ludwig Wittgenstein

This analyzes in depth such topics logical compulsion & mathematical conviction; calculation as experiment; mathematical surprise, discovery, & invention; Russell's logic, Godel's theorem, cantor's diagonal procedure, Dedekind's cuts; the nature of proof & contradiction; & the role of mathematical propositions in the forming of concepts.
Translator's Note
Editors' Preface
The Text
Index

Genres: Philosophy, Mathematics, Science, Nonfiction, Logic

204 pages, Paperback

First published January 1, 1956

About the author

Ludwig Josef Johann Wittgenstein (Ph.D., Trinity College, Cambridge University, 1929) was an Austrian-British philosopher who worked primarily in logic, the philosophy of mathematics, the philosophy of mind, and the philosophy of language.

Described by Bertrand Russell as "the most perfect example I have ever known of genius as traditionally conceived, passionate, profound, intense, and dominating", he helped inspire two of the twentieth century's principal philosophical movements: the Vienna Circle and Oxford ordinary language philosophy. According to an end of the century poll, professional philosophers in Canada and the U.S. rank both his Tractatus Logico-Philosophicus and Philosophical Investigations among the top five most important books in twentieth-century philosophy, the latter standing out as "...the one crossover masterpiece in twentieth-century philosophy, appealing across diverse specializations and philosophical orientations". Wittgenstein's influence has been felt in nearly every field of the humanities and social sciences, yet there are widely diverging interpretations of his thought.

Reviews

[Jimmy · Rating 4 out of 5]

Here are some lines I found interesting:

15. It is important that in our language--our natural language--'all' is a fundamental concept and 'all but one' less fundamental; i.e. there is not a single word for it, nor yet a characteristic gesture.

154. Would it be possible that people should go through one of our calculations to-day and be satisfied with the conclusions, but to-morrow want to draw quite different conclusions, and other ones again on another day?

167. The mathematician is an inventor, not a discoverer.

Appendix 1: 8. . . . (What is called "losing" in chess may constitute winning in another game.)

Appendix 1: 17. . . . (The superstitious fear and awe of mathematicians in face of contradiction.)

Appendix 2: 6. Why should we say: The irrational numbers cannot be ordered?--We have a method of upsetting any order. . . .

Part II: 1. 'A mathematical proof must be perspicuous.' . . .

5. . . . In philosophy it is always good to put a question instead of an answer to a question. For an answer to the philosophical question may easily be unfair; disposing of it by means of another question is not. . . .

71. It could be said: a proof subserves mutual understanding. An experiment presupposes it.

Or even: a mathematical proof moulds our language.

But it surely remains the case that we can use a mathematical proof to make scientific predictions about the proving done by other people.--

If someone asks me: "What colour is this book?" and I reply: "It's green"--might I as well have given the answer: "The generality of English-speaking people call that 'green'"?

Might he not ask: "And what do you call it?" For he wanted to get my reaction.

'The limits of empiricism.'

Part III: 7. A mathematical proposition stands on four feet, not on three; it is over-determined.

29. . . . So much is clear: when someone says: "If you follow the rule, it must be like this", he has not any clear concept of what experience would correspond to the opposite.

Or again: he has not any clear concept of what it would be like for it to be otherwise. And this is very important.

30. What compels us so to form the concept of identity as to say, e.g., "If you really do the same thing both times, then the result must be the same too"?--What compels us to procee according to a rule, to conceive something a a rule? What compels us to talk to ourselves in the forms of the languages we have learnt?

For the word "must" surely expresses our inability to depart from this concept. (Or ought I to say "refusal"?)

And even if I have made the transition from one concept-formation to another, the old concept is still there in the background.

33. . . . Imagine that a proof was a work of fiction, a stage play. Cannot watching a play lead me to something?

I did not know how it would go,--but I saw a picture and became convinced that it would go as it does in the picture.

The picture helped me to make a prediction. Not as an experiment--it was only midwife to the prediction.

For, whatever my experience is or has been, I surely still have to make the prediction. (Experience does not make it for me.)

No great wonder, then, that proof helps us to predict. Without this picture, I should not have been able to say how it will be, but when I see it I seize on it with a view to prediction.

59. . . . The proposition that contradicts itself would stand like a monument (with a Janus head) over the propositions of logic.

60. The pernicious thing is not, to produce a contradiction in the region in which neither the consistent nor the contradictory proposition has any kind of work to accomplish; no, what is pernicious is: not to know how one reached the place where contradiction no longer does any harm.

Part IV: 2. Does a calculating machine calculate?

Imagine that a calculating machine had come into existence by accident; now someone accidentally presses its knobs (or an animal walks over it) and it calculates the product 25 x 20. . . .

3. . . . A human calculating machine might be trained so that when the rules of inference were shewn it and perhaps exemplified, it read through the proofs of a mathematical system (say that of Russell), and nodded its head after every correctly drawn conclusion, but shook its head at a mistake and stopped calculating. One could imagine this creature as otherwise perfectly imbecile. . . .

4. . . . Imagine that calculating machines occurred in nature, but that people could not pierce their cases. And now suppose that these people use these appliances, say as we use calculation, though of that they know nothing. . . .

These people lack concepts which we have; but what takes their place? . . .

How far does one need to have a concept of 'proposition', in order to understand Russellian mathematical logic?

7. Imagine set theory's having been invented by a satirist as a kind of parody on mathematics.--Later a reasonable meaning was seen in it and it was incorporated into mathematics. (For if one person can see it as a paradise of mathematicians, why should not another see it as a joke?)

The question is: even as a joke isn't it evidently mathematics?--

9. . . . What if someone were to reply to a question: 'So far there is no such thing as an answer to this question'?

So, e.g., the poet might reply when asked whether the hero of his poem has a sister or not--when, that is, he has not yet decided anything about it.

14. Suppose children are taught that the earth is an infinite flat surface; or that God created an infinite number of stars; or that a star keeps on moving uniformly in a straight line, without ever stopping.

Queer: when one takes something of this sort as a matter of course, as it were in one's stride, it loses its whole paradoxical aspect. It is as if I were to be told: Don't worry, this series, or movement, goes on without ever stopping. We are as it were excused the labour of thinking of an end.

'We won't bother about an end.'

It might also be said: 'for us the series is infinite'.

'We won't worry about an end to this series; for us it is always beyond our ken.'

48. 'Mathematical logic' has completely deformed the thinking of mathematicians and of philosophers, by setting up a superficial interpretation of the forms of our everyday language as an analysis of the structures of facts. Of course in this it has only continued to build on the Aristotelian logic.

50. If you look into this mouse's jaw you will see two long incisor teeth.--How do you know?--I know that all mice have them, so this one will too. . . .

53. The philosopher is the man who has to cure himself of many sicknesses of the understanding before he can arrive at the notions of the sound human understanding.

If in the midst of life we are in death, so in sanity we are surrounded by madness.

Part V: 16. . . . It is my task, not to attack Russell's logic from within, but from without.

That is to say: not to attack it mathematically--otherwise I should be doing mathematics--but its position, its office.

My task is, not to talk about (e.g.) Godel's proof, but to pass it by.

18. . . . Godel's proposition, which asserts something about itself, does not mention itself.

26. But in that case isn't it incorrect to say: the essential thing about mathematics is that it forms concepts?--For mathematics is after all an anthropological phenomenon. . . .

29. What sort of proposition is: "The class of lions is not a lion, but the class of classes is a class"? How is it verified? How could it be used?--So far as I can see, only as a grammatical proposition. . . .

[Alec Julien · Rating 5 out of 5]

I gave this five stars even though I'm pretty sure I don't understand it. (I'm reasonably sure that nobody understand Wittgenstein, but that's another story.) Nonetheless, the book provides a wealth of brain food for thinking about issues in the philosophy of math and logic, and gives obscure but invaluable insights into Wittgenstein's takes on such matters.

[Zedder · Rating 3 out of 5]

The wood sellers!!!

[Djordjevicc · Rating 3 out of 5]

still in progress..

[Illiterate · Rating 5 out of 5]

Wittgenstein tackles maths using the approach of PI. He distinguishes its various uses and contexts, eg. pattern, paradigm, experiment. He presents its statements as ’grammatical’, not referential. He suggests even particular proofs are inventions, not discoveries. It’s all part of our ‘natural history’. To follow the details requires a little maths but mainly familiarity with contemporary logic, especially Frege and Russell.

[Anthony · Rating 4 out of 5]

What are we measuring when we put two yardsticks together? Are a fortune teller's predictions about numbers mathematical propositions? What does the knowledge that an infinity of different proofs could prove the same proposition do to our understanding of any particular proof and what exactly it proves? Wittgenstein dares to ask sublimely inane questions about basic mathematical concepts like, um, counting--the results are wonderful. My favorite crazy little question comes in section V:
"The class of cats is not a cat." --How do you know?

[Volbet · Rating 3 out of 5]

Have you ever seen a moth bash its head against a window? That observation is very similar to reading Remarks on the Foundation of Mathematics, witnessing Wittgenmoth repeatedly run into his own methodological and wondering why it's begun cracking.

The main thing I took away from the book is that this is Ludwig Wittgenstein's respons and negation of the methodology of his positivist predecessors. Especially the logic of Gottlob Frege and Bertrand Russell is under fire, with Wittgenstein doing what he does best to demonstrate the paradoxical and self-contraditory nature of the logical systems that both Frege and Russell build.
And it's all done through the lens of mathematics, with Wittgenstein trying to demonstrate how math, and by extension, logic, can be considered true. As was the case with both Frege and Russel, they considered logic to be true via logic adhering to some kind law of logic that's independent of human thought. Whether or not humans are around or not, logic is still true.
Now, Wittgenstein doesn't necessarily refute that point, but rather moves the goal of when something can be considered "true." The analysis of the inherent truth of logical propositions can't be considered in the tautological nature of logic itself, but rather has to be considered in the abstract construct of logic itself.
Logic isn't true in it of itself, but is true in that it adheres to the rules that we have collectively agreed to.

My only problem with Remarks on the Foundation of Mathematics is the tangents and weird examples that Wittgenstein comes up with. I usually enjoy the borderline psychotic ways Wittgenstein can circumvent his arguments (be they his own or from others), but here they're just strange. The snarky remarks and games that Wittgenstein is so famous for takes on an almost sophistic character here. They're still funny in a strange way, but they only serve to completely derail the discourse that Wittgenstein is going for.

[Steven H · Rating 5 out of 5]

THE PHILOSOPHER’S THOUGHTS ON LOGIC AND MATHEMATICAL THEORY

Ludwig Josef Johann Wittgenstein (1889-1951) was an Austrian-British philosopher whose books such as Tractatus Logico-Philosophicus and Philosophical Investigations are among the acknowledged “classics” of 20th century philosophy. Born into a wealthy family, he gave all of his inheritance away, served in the Austrian Army during World War I, taught schoolchildren in remote Austrian villages, but ultimately taught at Cambridge for many years.

The Tractatus was the only book he published during his lifetime, but his papers have been posthumously edited, and notes of lectures taken by his students have been transcribed, and have resulted in many published books, such as 'Lectures & Conversations on Aesthetics, Psychology, & Religious Belief,' 'Philosophical Grammar,' 'Philosophical Remarks,' 'The Blue and Brown Books,' 'Remarks on the Philosophy of Psychology,' 'Remarks on Colour,' 'Zettel,' etc.

The Editor’s Preface states, “The remarks on the philosophy of mathematics and logic, which are published here, were written in the years 1937-1944. After that time Wittgenstein did not again return to this topic. He had written a great deal on this subject in the period 1929 to roughly 1932, part of which we hope to publish later… This earlier work belongs to a stage in Wittgenstein’s development which is still fairly close to the Tractatus Logico-Philosophicus. The remarks presented in THIS volume are of a piece with the thought of Philosophical Investigations.”

He says, “There is a transition from one proposition to another VIA other propositions, that is, a chain of inferences… There is nothing occult about this; it is s derivation of one sentence from another according to a rule… We call it a ‘conclusion’ when the inferred proposition CAN in fact be derived from the premise… Now what does it mean to say that one proposition CAN be derived from another by means of a rule? Can’t anything be derived from something by means of SOME rule---or even according to any rule, with a suitable interpretation? What does it mean for me to say e.g.: this number can bed got by multiplying these two numbers?” (I, §6 & 7)

He observes, “I might also say as a result of the proof: ‘From now on an H and a P are called ‘the same in number.’ Or: this proof doesn’t EXPLORE the essence of the two figures, but it does express what I am going to count as belonging to the essence of the figures from now on. I deposit what belongs to the essence among the paradigms of language. The mathematician creates ESSENCE.” (§32)

He argues, “How is it established which pattern is the multiplication of 13 X 13? Isn’t it DEFINED by the rules of multiplication? But what if, using these rules, you get different results today from what all the arithmetic books say? Isn’t that possible?---‘Not if you apply the rules as THEY do.’ Of course not! But that is a mere pleonasm… Well, it never in fact happens that somebody who has learnt to calculate goes on obstinately getting different results… But if it should happen, then we should declare him abnormal, and take no further account of his calculation.” (§112)

He comments, “The laws of logic are indeed the expression of ‘thinking habits’ but also of the habit of THINKING. That is to say that they can be said to shew: how human beings think, and also WHAT human beings call ‘thinking’… The propositions of logic are ‘laws of thought,’ ‘because they bring out the essence of human thinking’---to put it more correctly: because they bring out, or shew, the essence, the technique, of thinking. They shew what thinking is and also shew kinds of thinking.” (§131, 133)

He notes, “In philosophy it is always good to put a QUESTION instead of an answer to a question. For an answer to a philosophical question may easily be unfair; disposing of it by means of another question it is not. Then should I put a question here, for example, instead of the answer that the arithmetical proposition cannot be proved by Russell’s method?” (II, §5)

He argues, “I want to say: with the logic of Principia Mathematica it would be possible to justify an arithmetic in which 1000 + 1 = 1000; and all that would be necessary for this purpose would be to doubt the sensible correctness of calculations. But if we do not doubt it, then it is not our conviction of the truth of logic that is responsible. When we say in a proof” ‘This MUST come out’---then this is not for reasons that we do not SEE. What convinces us---THAT is the proof; a configuration that does not convince us is not the proof, even when it can be shewn to exemplify the proved proposition. That means: it must not be necessary to make a physical investigation of the proof-configuration in order to shew us what has been proved.” (§39)

He goes on: “We incline to the belief that LOGICAL proof has a peculiar, absolute cogency, deriving from the unconditional certainty in logic of the fundamental laws and the laws of inference. Whereas propositions proved in this way can after all not be more certain than is the correctness of the way those laws of inference are APPLIED. The logical certainty of proofs… does not extend beyond their geometrical certainty.” (§43)

He states, “EXPERIENCE teaches us that we all find this calculation correct. We start ourselves off and get the result of the calculation. But now… we aren’t interested in having---under such and such conditions say---actually produced this result, but in the pattern of our working; it interests us as a convincing, harmonious pattern---not, however, as the result of an experiment, but as a PATH.” (§69)

He notes, “We went sleepwalking along the road between abysses. But even if we now say: ‘Now we are awake’---can we be certain that we shall not wake up one day? (And then say: so we were asleep again.) Can we be certain that there are no abysses now that we do not see? But suppose I were to say: The abysses in a calculus are not there if I don’t see them! Is no demon deceiving us at present? Well, if he is, it doesn’t matter. What the eye doesn’t see the heart doesn’t grieve over.” (§78)

He comments: “‘Only the proof of consistency shews me that I can rely on the calculus.’ What sort of proposition is it, that only THEN can you rely on the calculus? But what if you do rely on it WITHOUT that proof! What sort of mistake have you made?” (§84)

He says, “Consider also the rule which forbids one digit in certain places, but otherwise leaves the choice open. Isn’t it like this? The concepts of infinite decimals in mathematical propositions are not concepts of series, but of the unlimited technique of expansion of series.” (IV, §19)

He asserts, “Everything I say really amounts to this, that one can know a proof thoroughly and follow it step by step, and yet at the same time not UNDERSTAND what it was that was proved. And this in turn is connected with the fact that one can form a mathematical proposition in a grammatically correct way without understanding its meaning. Now when does one understand it? I believe: when one can apply it.” (§25)

He contends, “The curse of the invasion of mathematics by mathematical logic is that now any proposition can be represented in a mathematical symbol, and this makes us feel obligated to understand it. Although of course this method of writing is nothing but the translation of vague ordinary prose.” (§46)

He concludes, “The philosopher is the man who has to cure himself of many sicknesses of the understanding before he can arrive at the notions of the sound human understanding. If in the midst of life we are in death, so in sanity we are surrounded by madness.”(§53)

As always, Wittgenstein’s ideas are provocative, stimulating, and often profound. This book will be of great value to anyone studying his thought (particularly in its less-“linguistic” manifestations).

[Ege]

As a first approximation, for Wittgenstein arithmetical identities (such as ‘three times three is nine’) are not propositions, as their superficial grammar indicates – but rules. Importantly though, these rules are not arbitrary; in a sense (to be explicated later on), the rules in place are the only ones that could have been adopted or, as Steiner [2009, 12] put it, “the only rules available.” The typical conventionalist difficulty (that they might have an arbitrary character) is answered when it is added that the rules are grounded in objectively verifiable empirical regularities (Fogelin [1987]; Steiner [1996], [2000], [2009]); or, as Wittgenstein says, the empirical regularities are “hardened” into rules (RFM VI-22).

The diagnosis of PI §38 is that philosophical problems appear when ‘language goes on holiday’, or ‘idles.’ What does this mean? (The original reads “…wenn die Sprache feiert.” Stern [2004, 97] argues that Rhees’ initial translation of ‘feiert’ by ‘idle’ coveys better the idea that the language does no work than Anscombe’s standard translation ‘goes on holiday.’) Steps toward the clarification of this point can be made by observing that Wittgenstein has in mind here the situation in which a concept otherwise fully intelligible, oftentimes even unremarkable, is detached from its original contexts and uses (language-games). Once dragged into a new ‘territory’, the concept idles (just like a cogwheel separated from the gear), as it cannot engage other related concepts anymore. Anyone trying to employ the concept in this new ‘environment’ experiences mental cramps and disquietudes, precisely because, as it happens when one ventures into uncharted ‘terrain’, the familiar ‘paths’ and ‘crossroads’ no longer exist: one does not know her “way about” (PI §123), one does not see things clearly anymore. Hence, according to Wittgenstein, the whole point of philosophizing is not to defend ‘philosophical positions’ but to undo this perplexity, and achieve a liberating ‘perspicuous representation’, or over-view (übersichtliche Darstellung; PI §122).

Wrigley [1977, 50] sketches some of the reasons why he cannot be described as a finitist-constructivist, thus disagreeing with how Dummett [1959] and Bernays [1959] portray him.

Consider the multiplication 3 × 3 = 9. Obviously enough, one typical role of this equality is to license the replacement of one string of symbols (‘3’, ‘×’, ‘3’) with another symbol (‘9’) in extensional contexts. Wittgenstein does not disagree with this account, but has a more substantial story to tell. These identifications are not (cannot be) mechanical, disembodied actions – rather, they are grounded in ancestral, natural human practices, such as sorting and arranging objects.

Wittgenstein’s emphasis on conceiving mathematics as essentially presupposing human practices and human language should not be taken as an endorsement of a form of subjectivism of the sort right-is-what-I-(my-community)-take-to-be-right... What Wittgenstein opposes is not objectivity per se, but the ‘philosophical’ explanation of it... So, what Wittgenstein rejects is a certain “metaphysics of objectivity” (Gerrard [1996, 173]).

However, the wild variation described above does not exist. It is a contingent, brute natural fact (again!) that we do not live in such a world, but in one in which regularities prevail. Moreover, it is precisely the existence of such regularities – together with, as we will see in a moment, regularities of human behaviour – that makes possible the arithmetical practice in the first place. Wittgenstein, however, realized this rather late, as Steiner [2009] documents.

Yet, as stressed above, it is crucial to note that speaking in terms of behavioural agreement when it comes to understanding the mathematical enterprise should not lead one to believe that Wittgenstein is in the business of undermining the objectivity of mathematics... Wittgenstein does not regard the agreement among the members of the community’s opinions on mathematical propositions as establishing their truth-value. Convincing passages illustrating this point can be found virtually everywhere in his later works, and Gerrard [1996] collects several of them.

It [that we see regularities in nature like 2+3=5] is a contingent, brute natural fact (again!) that we do not live in such a world, but in one in which regularities prevail. Moreover, it is precisely the existence of such regularities – together with, as we will see in a moment, regularities of human behaviour – that makes possible the arithmetical practice in the first place.

So, it is simply not the case that the truth-value of a mathematical identity is established by convention. Yet behavioural agreement does play a fundamental role in Wittgenstein’s view. This is, however, not agreement in verbal, discursive behaviour, in the “opinions” of the members of the community. It is a different, deeper form of consensus – “of action” (LFM, p. 183; both italics in original), or agreement in “what [people] do” (LFM, p. 107).
The specific kind of behavioural agreement (in action) is a precondition of the existence of the mathematical practice. The agreement is constitutive of the practice; it must already be in place before we can speak of ‘mathematics.’ The regularities of behaviour (we subsequently ‘harden’) must already hold. So, we do not ‘go on’ in calculations (or make up rules) as we wish: it is the existent regularities of behaviour (to be ‘hardened’) that bind us.

While the behavioural agreement constitutes the background for the arithmetical practice, Wittgenstein takes great care to keep it separated from the content of this practice (Gerrard [1996, 191])... The very fact of the existence of this background is not amenable to philosophical analysis. The question ‘Why do we all act the same way when confronted with certain (mathematical) situations?’ is, for Wittgenstein, a request for an explanation, and it can only be answered by advancing a theory of empirical science (neurophysiology, perhaps, or evolutionary psychology).

Related to Platonism, ‘mentalism’ is another target of Wittgenstein, as Putnam [1996] notes.

See Diamond [1991] and Conant [1997] for discussions of Wittgenstein’s location within the broader realist-antirealist landscape).

Source: https://iep.utm.edu/wittmath/

[Evan Hurley O'Dwyer · Rating 4 out of 5]

This book contains comments written over a decade of work of Wittgenstein. A large part of the text was originally supposed to be the second half of the Philosophical Investigations, and there are lots of themes in common - what it means to follow a rule, for example. I would only recommend reading it if you are already familiar with the later Wittgenstein's philosophy in general, as parts of this book are difficult to interpret if you were to read it without understanding Wittgenstein's broader aims. The collection of remarks was never formulated into a fully cohesive book, and much of the comments were just Wittgenstein's comments to himself so some parts were repetitive and other parts without development. That said, there are plenty of interesting ideas. For example, Wittgenstein that basic arithmetical statements such as "3+2 = 5" are used as rules or criteria to determine whether someone has calculated correctly, and are not empirical statements or statements giving knowledge.

Wittgenstein is directly against Russell in that he did not believe mathematics required a "rigorous" foundation, and takes aim at the idea that the "real" proof of an arithmetical statement is the one found in a system such as Russell's PM. One of the reasons for this is that PM or another foundational calculus cannot be considered the ground of "2+2=4", as one of the criteria someone would look for in a potential foundation is that it would have to prove statements like "2+2=4". Russell's PM would have been rejected if it had proved statements like "2+2=5".

There are some interesting discussions about Godel, Cantor and Dedekind. Wittgenstein tends to be attacked for his comments on these mathematicians, although Wittgenstein isn't disputing the proofs themselves, it's more the interpretation they're given and the significance they hold, and the unusual statements that people make in connection with them.

There is some interesting discussion on whether or not you understand mathematical propositions without knowing a proof (e.g. Fermat's theorem before the proof), and to what a proof is. There are also interesting remarks around nonconstructive existence proofs and how starkly less clear they are in their meaning than more constructive ones. Wittgenstein considers, as an example, questions about whether or not the string "777" occurs in particular irrational numbers, and what it means to say that "777" does not occur in the infinite decimal expansion of an irrational number.

[Reinhard Gobrecht · Rating 3 out of 5]

Bemerkungen zu den Grundlagen der Mathematik
Wittgensteins Gedanken zur Mathematik sind teils philosophisch, teils aber auch unstrukturiert vor sich hingedacht. Man findet Gedanken zur Logik und Beweisführung, zu Axiomen und Gleichungen, zu unendlichen Kardinalzahlen von Mengen, zur Geometrie und zur Arithmetik, zu Folgen und Reihen und Brüchen. Wittgenstein macht auch Bemerkungen zu Schriften von Frege und Russell. Manche Bemerkungen sind nicht philosophisch und liefern daher auch keinen Erkenntnisgewinn. Manche Bemerkungen werten die Mathematik geradezu ab, insbesondere die Logik. Wenn z. B. aus mathematischen Formulierungen Sprachspiele werden oder Mathematik mit Alchemie verglichen wird oder die logische Notation pauschal bemängelt wird, muss man sich schon fragen: Wo ist da die Bedeutung seiner Gedanken, wo ist da der Erkenntniswert. Konstruktiv kritische Bemerkungen sind leider Mangelware. Eine sehr "lockere" Philosophie der Mathematik.

[42 · Rating 5 out of 5]

re-reading material; great for public communication: each point a different riddle.
(this is where Nassim Taleb took Wittgenstein's ruler from - points 94 and 93).

[Kaidi Pan · Rating 4 out of 5]

Still cracking the damn code.

[Dionysius the Areopagite · Rating 5 out of 5]

Review in time.