Introduction to Mathematical Philosophy

by Bertrand Russell

2017 Reprint of 1919 Edition. Full facsimile of the original edition, not reproduced with Optical Recognition software. Introduction to Mathematical Philosophy has been a seminal work for more than nine decades. It gives the general background necessary for any serious discussion on the foundational crisis of mathematics in the beginning of the twentieth century. Requiring neither prior knowledge of mathematics nor aptitude for mathematical symbolism, the book serves as essential reading for anyone interested in the intersection of mathematics and logic and in the development of analytic philosophy in the twentieth century. Russell offers to his readers a penetrating discussion on certain issues of mathematical logic that embodies the dawn of modern analytic philosophy.

Genres: Philosophy, Mathematics, Science, Nonfiction, Logic, Classics, History

220 pages, Paperback

First published January 1, 1918

About the author

Bertrand Arthur William Russell, 3rd Earl Russell, OM, FRS, was a Welsh philosopher, historian, logician, mathematician, advocate for social reform, pacifist, and prominent rationalist. Although he was usually regarded as English, as he spent the majority of his life in England, he was born in Wales, where he also died.

He was awarded the Nobel Prize in Literature in 1950 "in recognition of his varied and significant writings in which he champions humanitarian ideals and freedom of thought."

Reviews

[Ben Labe · Rating 5 out of 5]

Ordered for its didactic utility rather than for the linearity of its assertions, this book makes a terrific supplement to a study of mathematical analysis. Logic is Russell's forte; here we find him at the top of his game. Starting with cardinal numbers, Russell begins by analyzing the mathematical concepts that people take most for granted, providing their logical foundations, clarifying their meanings, and identifying common pitfalls in our language about them. Once we have recognized the component parts of our concept of numbers, we discover that we must further identify the component parts of the first level of component parts. This naturally introduces an infinite regress, which is a problem that Russell acknowledges. Thus, he devotes entire chapters to the set theoretic axioms of infinity, reducibility, and multiplication, explaining where they are necessary and under what circumstances they might be true. Interspersed are discussions of where previous mathematical philosophers--most notably Leibniz--have gone wrong. Russell even provides a brief and (as much as is possible under such demands of precision) intuitive description of the logical theory of types and its evolution. The final chapter might be the best merely because it holds the rest of the chapters together. For the lay reader, it is also the most useful for its definitions of mathematics and logic and its explanation of why they represent essentially the same sort of pursuit, i.e. the pursuit of tautological truth, or as Leibniz would say, truth in all possible worlds. Lastly, it states how such a pursuit seems to be possible.

[hayatem · Rating 5 out of 5]

على مر التاريخ كانت الرياضيات والمنطق مبحثين متمايزين تماماً . ‏كانت الرياضيات مرتبطة بالعلوم، والمنطق مرتبطاً بالإغريق . لكن الاثنين تطورا في العصور الحديثة : أصبح المنطق أكثر رياضية، وأصبحت الرياضيات أكثرت تمنطقاً. ‏نتج عن ذلك أنه قد بات من المستحيل تماماً ‏الآن رسم خطٍّ بين الاثنين؛ في الحقيقة الاثنان هما واحد. يختلفان كما يختلف الرجل والصبي : المنطق هو شباب الرياضيات، ‏والرياضيات هي مبلغ رجولة المنطق ‏. امتعض المنطقيون من هذا المنظور،‏أولئك الذين أفنوا أعمارهم في دراسة النصوص الكلاسيكية باتوا غير قادرين على تتبع جزئية في التفكير المنطقي‏ الرمزي، ‏وكذلك امتعض الرياضيون، أولئك الذين تعلموا التقنية دون أن يزعجهم التساؤل عن معناها أو مبررها.لحسن الحظ كلا النزعتين تصبح ‏أندر فأندر. ص(333)

يشرح رسل ويفند في الكتاب العديد من الفرضيات الرياضية لكل من: جيوسيبي بيانو ، كانتور، لودفيج فتغينشتاين، تسيرميلو، بولزانو، وديدكايند؛ ويضع يده على مكمن الخلل، و يتناول أهم اسهاماتهم في الرياضيات، عبر التحليل المنطقي الرياضي -الفلسفي. وهو يؤمن بالأساليب التحليلية المستمدة من عمله المنطقي. تأثر بشكل كبير بأسلوب بيانو في المنطق الرياضي. " كان عمل بيانو مهمًّا للغاية؛ لأنه حفز من رغبة راسل لاشتقاق الرياضيات من المنطق، وقدَّم له الوسيلة اللازمة لتنفيذ ذلك. وكرس راسل السنوات ما بين عام 1900 و1910 لهذه المهمة أساسًا، ونتج عن ذلك قدرٌ كبير من الإنجاز الفلسفي القيِّم."
اشتغل في مادة الكتاب على الأسس المنطقية للرياضيات. وعلم دلالة المنطق، مع اهتمامه بالنسق المنطقي الذي تندرج فيه. إضافة إلى إبراز أهمية استخدام البرهان المنطقي الرياضي أو القياس المنطقي في الرصد والتجربة. ليميز بين القضايا والحجج الجيدة من السيئة. أو إعطاء مؤشرات عن قضية ما أنها صحيحة أو خاطئة. ( من خلال دراسة النظم الشكلية للاستدلال inference، أو من خلال دراسة الحجج في اللغات الطبيعية.) و كذلك (الاستقصاء عن طريقة حصولنا على المعرفة واختبارنا لها). كما نلمس في المادة وجه المنعطف اللغوي في الفلسفة المعاصرة عند راسل. (انطلاقاً من كون فلسفة اللغة عنده تقوم على التواصل والترابط بين البحث المنطقي والرياضي والفلسفي.)

ما هي القضايا المنطقية؟

القضايا المنطقية هي تلك التي من الممكن معرفتها بشكل مسبق دون دراسة العالم الفعلي. نعرف فقط من دراسة الحقائق الإمبريقية أن سقراط هو رجل، لكننا نعرف ‏صحة القياس المنطقي في صورته المجردة ( أي عند النص عليه في ضوء المتغيرات ) دون الحاجة إلى أي توسل بالخبرة. ‏هذه سمة مميزة للطريقة التي نعرف بها القضايا المنطقية وليست كامنة فيها هي نفسها. ‏مع ذلك فلها حمولتها على التساؤل عما قد تكون طبيعتها، حيث إن هناك بعض أنواع القضايا التي سوف يكون من الصعب جداً افتراض أننا نقدر على معرفتها دون الخبرة. (ص347)


كتاب جامع في مادته بين العقل الرياضي المنطقي والعقل اللغوي، و بلغة جامعة بين لسان« التركيب المنطقي للغة » Logical Syntax of Language، و«الفلسفة والتركيب المنطقي» Philosophy and Logical Syntax . و المتصل بشكل أو بآخر في المناهج الاستقرائية.

[Myat Thura Aung · Rating 5 out of 5]

I doubt most of us will ever have time to read an intellectually demanding, voluminous masterpiece like Principia Mathematica but luckily for us, this book exists.

Wittgenstein once allegedly said, “Russell's books should be bound in two colours, those dealing with mathematical logic in red – and all students of philosophy should read them; those dealing with ethics and politics in blue – and no one should be allowed to read them.”

Well, this is the sort of red book that one should definitely love to read. It really made me happy and finishing it was a way of ending my 2019 on a high note.

[Ahmed · Rating 4 out of 5]

One of the hardest books I have read.....didnt understand most of it but I was determined to finish it....I somehow was able to grasp the essence of Russels teaching and I think my persistence paid off in the end....So I really glad that I kept the reading progress with this book intact...mind-blowing literally and educationally....very much motivated to read more about symbolic logic and mathematical philosophy.

يالله كم ارهقني هذا الكتاب....قمة في الصعوبة اللغوية والعلمية...واصداره في عشرينيات القرن المنصرف مع اخذ بالاعتبار اسم الكاتب يعطيكم فكرة سريعة عن صعوبة اللغة الانكليزية المستخدمة....
لا اخادعكم القول اني كنت على وشك الاستسلام والتوقف عن قراءته بعد فصلين ولكن غيرت رايي بعد ذلك وصممت على انهائيه....
من اصعب الكتب التي قراتها في حياتي....واستيعابي كان اقل من ٣٠%....اكبر غلط وقعت به هو ان ابدا بعلم جديد مثل فلسفة الرياضيات والمنطق الرمزي بقراءة اصعب الكتب في هذا المجال...ولكن مع كل هذا استطعت ان اخرج بفكرة عامة ونضرة سوف تساعدني كثيرا لاحقا مع كتب اخرى في هذا المجال....
يحاول الكاتب برنارد روسل مناقشة وتحليل فلسفة الارقام الرياضية من منشاها الى انواعها وارتباطها بالمنطق السليم...يستخدم لغة المنطق الرمزي الصعب جدا....اسلوبه اللغوي التحليلي صعب جدا...ولاسيما لغتي الانكليزية المتواضعة صعبت هذا الشيئ اكثر...
سوف اعود لقراءة هذا الكتاب مرة اخرى في المستقبل بعد ان اقرا اكثر في هذا المجال....

[Ivan Vuković · Rating 5 out of 5]

At first, I wanted to give this book 4 stars, not 5, but that wouldn't be fair. The main reason I would often find myself struggling through the chapters was my lack of patience and overwhelming curiosity of what might I find on the next page.

The times in which I was patient and thorough enough to follow Russell's reasoning were a very rewarding experience and I think it really changed my approach to thinking about mathematical/logical precision.

To sum it up, I'd like to quote the man himself:
"If any student is led into a serious study of mathematical logic by this little book, it will have served the chief purpose for it which has been written."

And that is the most important reason why I want to give this book 5 stars after all. It's a masterpiece and it deserves much more time and thought than I was ready to give it and I definitely intend to make up for that some day soon.

[Robert · Rating 4 out of 5]

The first few chapters were mind-blowing: not necessarily difficult to understand, but not the kind of things one would have thought about (and even then, not to the same precision as Russell).

However, it started to get a bit more labyrinthine; the chapter concerning the multiplicative axiom and the axiom of infinity flew way over my head. Definitely not an easy read, and I think Russell could use clearer examples. That being said, the final few chapters were pretty clear and not difficult to follow.

[Xander]

Having read the first ten chapters, I will not go any further. It is definitely a very strong introduction to the logical foundations of mathematics and Russell's bright mind shines on all pages. Yet the subject matter cannot really sustain my attention, which has partly to do with reasons laying outside of this book. I'll definitely pick it up on a more suitable moment in the future!

[Neal · Rating 3 out of 5]

Honestly I didn't really finish this. Be ready for some hardcore logic shit that will rock you.

[T.R. Preston · Rating 5 out of 5]

I love the fact that Bertrand even took the time to write something like this. I am not one to pretend I understand something that I do not, so I'll get this out of the way and say I have no earthly idea what the hell Bertrand is on about in this book. The majority of these words and equations are gibberish to me. However, I was fascinated enough that I may start studying mathematics. Mathematical philosophy is the one category of philosophy that I have not dipped my toes into at all. I should probably change that.

Bertrand Russell remains on my current Mt. Rushmore of philosophy: Albert Camus, Friedrich Nietzsche, Bertrand Russell, and Marcus Tullius Cicero.

[T · Rating 3 out of 5]

My rating here is somewhat arbitrary because I did not have the patience to understand a significant portion of this book. Although it is an "introductory" text, it nevertheless requires a certain mathematical inclination to grasp. I've never been one to find math very interesting or stimulating, and Russell has reinforced this opinion for me. Owing to his reputation, however, I'm confident that the content itself of the book is impeccable, but I'm certainly not in a position to critique it.

[Utsob Roy · Rating 5 out of 5]

ছোটবেলায় রবীন্দ্রনাথের শান্তিনিকেতনের জন্য লেখা টেক্সটবুক পড়ে জেনেছিলাম ১ সংখ্যাটা কোনো একটা জিনিসে নেই। রীতিমত তাজ্জব বিষয়। অ্যাবস্ট্রাকশনের সাথে পরিচয়টা ওখানেই।

তো সেইসব সংখ্যার যুক্তি ও ব্যাখ্যা বুঝতে অনেকটা সময় লেগেছে। তাতে খানিকটা কন্ট্রিবিউশন প্রোগ্রামিং-এরও। এমনকি রাসেল সাহেব লজিক ও ম্যাথেমেটিকসের যে অভিন্ন ধারা চিন্তা করেছেন, প্রোগ্রামিং ল্যাঙ্গুয়েজগুলো তাই-ই।

কেন জানি না, রাসেল-হোয়াইটহেড-উইটজেনস্টেইনের কাজ আরো প্রোফাউন্ড হলেও বাঙালি নীৎশে-কিয়ের্কেগার্ড আর আধা-মিস্টিক ফ্যালাসিমিশ্রিত মাম্বো-জাম্বোর চর্চার বেশি উদগ্রীব।

[Laura · Rating 4 out of 5]

Free download available at Project Gutenberg

I am converting this ebook to html5&svg images and Project Gutenberg will publish it pretty soon.

CONTENTS
CHAP.
PREFACE
EDITOR'S NOTE
1. THE SERIES OF NATURAL NUMBERS
2. DEFINITION OF NUMBER
3. FINITUDE AND MATHEMATICAL INDUCTION
4. THE DEFINITION OF ORDER
5. KINDS OF RELATIONS
6. SIMILARITY OF RELATIONS
7. RATIONAL, REAL, AND COMPLEX NUMBERS
8. INFINITE CARDINAL NUMBERS
9. INFINITE SERIES AND ORDINALS
10. LIMITS AND CONTINUITY
11. LIMITS AND CONTINUITY OF FUNCTIONS
12. SELECTIONS AND THE MULTIPLICATIVE AXIOM
13. THE AXIOM OF INFINITY AND LOGICAL TYPES
14. INCOMPATIBILITY AND THE THEORY OF DEDUCTION
15. PROPOSITIONAL FUNCTIONS
16. DESCRIPTIONS
17. CLASSES
18. MATHEMATICS AND LOGIC
INDEX

[Rahul Sandhu · Rating 5 out of 5]

id rather pull the trigger and paint my wall red then read this book again. Honestly, this book is as terrible as it looks. Horribly boring, basically just explains basic mathematical principals like set theory, real/imaginary numbers, addition, etc. Basically just explains the gray area of why and how math is done with logical principals. Chapter 10 is where things get semi-interesting but you need a basic understanding of calculus. His explanation of logical duction was pretty interesting actually. I don't know why i decided to read this for fun but most of it went over my head anyways- 5 STARS

[Russell · Rating 3 out of 5]

The first part is somewhat dry, Russell clearly prefers using precise language when describing numbers, cardinality, and relational sets over analogies and other less precise, but perhaps more accessible, language. I easily forgive him for that, after all he was one of the founders of analytic philosophy, but what knocked the book down in my eyes, surprisingly enough, were his logical flaws at the end in his intro to logic!

"Now if Fx is sometimes true, we may say there are x's for which it is true, or we may say 'arguments satisfying Fx exist'. This is the fundamental meaning of the word 'existence'."

Yeah, he defined existence as something that exists when it is an argument satisfying a function. Things exist because they exist. And I can't see anywhere on the interwebs anyone calling him on this. Or sure, there's the dance around 'functions' and whatnot, but the end result is the definition contains the word it is defining.

He then goes on to conflate material existence with all forms of existence, and declares that there is nothing else aside from the material existence as seen through our senses, "There is only one world, the 'real' world[.]"

Which was exceptional odd since he just spent a chapter talking about the unknown reality of the axiom of infinity.

Russell goes on talking about how while "unicorn" has some meaning, but "a unicorn" does not, it describes nothing. And as far as I can tell, he means nothing material. Again, this confuses me since I have never seen a material existence of a cardinal number aleph. "To say that unicorns have an existence in heraldry, or in literature, or in imagination, is a most pitiful and paltry evasion." To say cardinal numbers have an existence in math books, or lectures, or in imagination, is a most pitiful and paltry evasion. The very arguments he marshals against non "real" objects can be applied to anything non material, including mathematical axioms and concepts.

Russell also lumps "unicorn" with "a round square". While the latter is violation of Euclid's axioms, the former isn't. Instead, a unicorn is something that might have existed in the past and it might exist in the future, but for the present we have no current material evidence. Conflating the two might have been due to limited scope of the book but since he had no qualms challenging Leibniz throughout the book it seems he'd could have easily clarified that point. He either didn't but gave no explanation or he failed to grasp the error; am I the only one that was amused by the irony of this category error?

In other words, while makes sense to dismiss the concept of a round square based on the clear violation of axioms, one cannot do the same with a unicorn. They are clearly in different categories, applying the same reasoning is a logical error. A unicorn wouldn't violate any Euclidean axioms, so it cannot be lumped with something that is in violation.

Despite this ideological flaw that colors his philosophy, it's an interesting and thought provoking work, well worth read for anyone that likes math, is interested in class theory, and enjoys reading what some consider to be a summary of Principia Mathematica.

[Rodrigo]

It's hard to understand, but I like it. The study of logic has a certain flavor to it that nothing else has.
I wonder how might this compare to the way information is arranged in our brain. Neumann wrote it was probabilistic, but I'm not sure I get what he meant my that.
So much to know, so much to discover!

[Yumeko (blushes) · Rating 3 out of 5]

I notice that the other philosophy math books I read basically echoed this. I suppose redundancy is understandable then. It was still enlightening in many aspects, but you get that usually when you learn something from more than one source.

[Eitan · Rating 4 out of 5]

Me, has a degree in Aerospace Engineering: "..."

Bertrand Russell from his prison cell: "You are like a little baby, I will teach you how to count"

[Amine Fidah · Rating 5 out of 5]

A stellar read, yet the verbiage leaves much to be desired. While it could be attributed to a predilection towards formal symbolism on the reader's end, the concepts presented, though fascinating as historical parcels and simple, are regrettably veiled with awkward protrusions. This could potentially render them obscure to an uninitiated reader through an over-reliance on ordinary grammar and syntax. Russell himself acknowledges this issue in the closing chapter. Although the inadequacy may be related to the reader's preferences, it is, all in all, a great read that I would recommend for someone seeking an entertaining read and a refresher on the concepts rather than as an introductory text, despite its many merits as that.

[Mateo Jaramillo · Rating 5 out of 5]

This short intro book had me falling in love with the subject all over again. Nobody in real life has the time to read Principia Mathematica so this scratches that itch. Granted I already had a background in the subjects of a few of the chapters but I just loved the attention to detail in defining the most fundamental building blocks of mathematics. I will never get bored of reading explanations of infinite cardinal numbers and transfinite ordinals, just blows my mind every time. This book helped re-light a flame in my brain that has been embering for the past year

[نورة · Rating 4 out of 5]

أستغرق مني وقت بالقراءة أكثر مما توقعت الحقيقة، لأنني كل ما قرأت مسلمّة او أسم فيلسوف/عالم رياضياتي لا أعرفه هممت بالبحث عنه وأغرق ببحثي لشدة إعجابي بكمية المساهمات والنظريات في علم الرياضيات وربطه بالمنطق. كتاب دسم جدًا رغم بساطة المعلومات التي يحتويه، وهذا يمثل طبيعة الرياضيات.
—
من قبل لا اقرأ الكتاب وانا دائمًا لدي نظرة بخصوص الشكوك والتناقضات التي يمرّ فيها أي علم من العلوم، ودائمًا أرى ان هذه الشكوك تؤول إلى اكتشافات وتطبيقات بديعة في أغلب الاحيان، والكتاب أثبت نظرتي هذه.
نتيجة لهذه التناقضات وأزمة أسس الرياضيات بدأ التساؤل عن صحة واحدة من المسلمات الخمس لأقليدس ومن حسن الحظ أن هذه الشكوك حول مسلمات أقليدس ساهمت في اكتشاف الهندسة الإقليدية وهندسة ريمان (نسبةً للعالم جورج ريمان)
وباتت الشكوك حول البديهيات/المسلمات شيء مثير للاهتمام نظرًا لعواقب هذه الشكوك، ومن أهم نتيجة هذه الشكوك أن للهندسة الإقليدية التي تم اكتشافها بعد الشك حول مسلمّة أقليدس أدت إلى تطبيقات رائعة ومن أهمها هي نظرية النسبية العامة المشهورة لأنيشتاين.
—
لفتني بالكتاب التحدث ولو بشكل بسيط عن اسهامات الجبر لفروع متعددة للرياضيات أولها اقامة حساب معقول ومنطقي لغته الرموز ويُقام على معادلات وقوانين غير كمية وهذا ما يسمى بحساب التفاضل والتكامل/calculus.
—
الكتاب تحدث عن أشياء كثيرة كثيرة جدًا ولا أستطيع ان اذكرها في تقييم الكتاب بدون ما أذكر كل سطر بالكتاب. تحدّث راسل عن الاستنباطات وعلاقة المنطق بالرياضيات, وكيف أن الرياضيات تحديدًا يهتم بدراسة العلاقة بين المبرهنات والاستنباطات ولا يهتم كثيرًا بمطابقتها مع الواقع وكما ذَكر راسل بكتابه أن الرياضياتي لا يحتاج أن يشغل نفسه بالكيانات أو الطبيعة الجوهرية لنقاطه وخطوطه ومستوياته حتى عندما يراها كرياضياتي "تطبيقي"، ومن هنا وَجد راسل مدخلًا للتحدث عن عدم التوافق ونظرية الاستنباط والوصل وقيمة الصدق وعدم التوافق والاستلزام…إلى آخره. التي لها تطبيقات في علم الإحصاء بشكل أكبر، من وجهة نظري
—
كتاب تمنيت أن قرأته باللغة الانقليزية لأن تعليمي الأكاديمي/الذاتيّ للرياضيات والإحصاء كان باللغة الانقليزية ولم يسبق لي قراءة أي مصدر للتعلم مترجم أو باللغة العربي وكان كافي بالنسبة لي، أوصي بالكتاب بشكل خاص لأي شخص مهتم بالرياضيات كأصول وأسس وتاريخ.

[Hmd Book · Rating 3 out of 5]

کتاب را از کتابخانه دانشگاه گرفتم- ترجمه ابوالقاسم لاله منتشر شده در سال ۱۳۷۶.
راسل آدمی بود که از اعداد تا خدا را بدون منطق نمی‌پذیرفت- جایی خوانده بودم راسل در خانه تحصیل کرد و وقتی برادرش اصول هندسه اقلیدسی را به عنوان اصولِ بدون اثبات بیان می‌کرد، راسل اصرار بر بیان اثباتی بر آنها داشت!
با این مقدمه در مورد راسل، این کتاب تلاش راسل است برای فهم اعداد در قالب نظریه مجموعه‌ها. باید ذهنی بیش از حد شکّاک داشته باشید تا با کتاب جلو روید. خیلی وقت‌ها احساس می‌شود مسائل پیش رو بدیهی‌اند و نویسنده بی‌جهت کار را سخت می‌کند و لقمه را از آن سمت به دهان می‌گذارد- اما این طور نیست، اگر خط استدلال را از ابتدا دنبال کرده باشید که اعتراف می‌کنم به هیچ وجه کار آسانی نیست. در کنار مطالب گاه خسته کننده مثل فصل رابطه‌ها، نکات جذابی در کتاب وجود دارد مثل فصلی که به بی‌نهایت بودن تعداد اعداد اشاره می‌کند و همانند سازی آن با بی‌نهایت بودن تعداد اعداد حقیقی بین هر دو عدد.
همین طور اگر از خود پرسیده‌اید کتاب پرینکیپیا ماتماتیکا راسل و وایتهد با انصد صفحه-که ظاهرا به دلایل باز منطقی به اهدافش نرسید- در مورد چیست، بخشی از این کتاب خلاصه‌ای از آن کتاب عظیم را در بر دارد.
در پایان پارادوکس راسل که یکی از مشارکت‌های مشهور راسل در ریاضیات است بحث می‌شود، اما از نظر من مشخص نیست ترجمه بد یا خود اصل نوشته آن قدر خشک است که مطالب به سختی -حتی به خواننده‌ی به شدت متمرکز- هم منتقل نمی‌شود: و به همین دلیل این کتاب/ترجمه این کتاب بیش از سه ستاره نمی‌گیرد!

[Nemo · Rating 5 out of 5]

In Bertrand Russell's Introduction to Mathematical Philosophy, we are presented with a work of philosophical inquiry that explores the fundamental nature of mathematics and its relationship to the broader realm of human knowledge. At the heart of Russell's inquiry is the concept of the transfinite, which he defines as "the infinite which is reached by successive additions." This notion, which owes a great debt to the pioneering work of the German mathematician Georg Cantor, is explored in great depth throughout the book. For Russell, the study of the transfinite is not simply a matter of mathematical curiosity, but rather represents a fundamental challenge to our understanding of the nature of reality itself. As he notes early on in the book, "the infinite is in some sense the key to the whole of mathematics."
Throughout the book, Russell engages with a wide range of philosophical and mathematical thinkers, including Cantor himself, as well as a host of other luminaries such as Leibniz, Frege, and Dedekind. While he is deeply respectful of their contributions to the field, he is also unafraid to criticize their ideas when he feels that they have gone astray. Russell is realist, he believes that "The number is not a mere abstraction, it is something concrete and objective, existing in the same sense as any other physical object."

[Adam Pasztory · Rating 5 out of 5]

For a deep, mathematical text, this is quite brief and readable!
Some of these ideas outdated a few years later by Gödel's incompleteness theorems, but still it's fascinating stuff. Russell a clever, clear, writer, and most assertions are buttressed concrete, sensible examples. Even if some of it is over my head, I love how some of the simplest notions -- like trying to define What Is A Number -- so quickly spiral into complexity.
Finally, I recommend the Partially Examined Life podcast episode 38, which provides some helpful context for this book and explains Russell's place in the history of philosophy. http://www.partiallyexaminedlife.com/...

[Elia Mantovani · Rating 2 out of 5]

E' complicato recensire questo trattatello. Da un lato, prima facie, non posso nascondere il fatto che non mi sia piaciuto, dall'altro provo grande soggezione verso tutti gli illustri lettori che in quest'opera hanno visto il vero "gioiello" della prolifica produzione di Russell. Sinceramente, penso sia sopravvalutata la portata essoterica del libro: non mi è sembrato così divulgativo come è stato fatto passare e francamente mi risulta difficile credere che possa essere facilmente accessibile ad ogni lettore interessato. Certo, intriganti i tentativi logicisti di definire per esteso i concetti intuitivi quali numero, funzi0ne, ordine e infinito, ma troppo confusi e poco cristallini.

[Eleclyah · Rating 2 out of 5]

Abbandonato dopo qualche capitolo.
Non tanto perché fosse brutto o poco interessante, quanto perché non aggiungeva nulla a quelle che già sono le mie conoscenze sull'argomento.
Per chi ha una buona base matematica, più che un approfondimento, questo saggio di Russell può essere considerato... una sorta di ripasso.
Per chi non ce l'ha... beh, armatevi di pazienza. Potreste scoprire un mondo.

[Marco E. · Rating 4 out of 5]

...ma alla fine arriva Godel!

[Emadacuz · Rating 4 out of 5]

...ma alla fine arriva Godel!

[нєνєℓ ¢ανα · Rating 5 out of 5]

Excellent!

[Mark Moon · Rating 3 out of 5]

Quite interesting as a historical artifact, but not otherwise.

[Luke · Rating 5 out of 5]

If you thought you knew addition, think again.