Introduction to Mathematical Philosophy by Bertrand Russell

Introduction to Mathematical Philosophy Reviews

• 
  hayatem

  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.
• 
  Ben Labe

  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.
• 
  Myat Thura Aung

  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

  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

  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.
• 
  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!
• 
  Robert

  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.
  
• 
  Utsob Roy

  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।
• 
  Russell

  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.
• 
  Hmd Book

  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!
• 
  Adam Pasztory

  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

  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

  
  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.
• 
  нєνєℓ ¢ανα

  Excellent!
• 
  Mark Moon

  Quite interesting as a historical artifact, but not otherwise.
• 
  Bernard

  'Mathematics, rightly viewed, possesses not only truth, but supreme beauty—a beauty cold and austere...'
  
  An accurate summation of my time with big Bertie in this text. It got me to consider simple aspects in mathematics that I take for granted under a new light, and made me more appreciative of mathematics overall. Sadly I did have to blast through the majority of Introduction since it was a library loan but I'd love to revisit it and digest it more slowly alongside a copy of Principles of Mathematics. The chief points that make this book weaker are when Russell's outdated chauvinism rears its ugly head in some of the introductory chapters that I could have done without. It's also a point to make that whilst the style is consistently lucid and to the point, I'd have appreciated more use of symbols to help break down wordier passages that became a bit of a struggle to work through in the repetitiveness of the prose. Of particular note are the chapters on infinititude in this regard. Otherwise, stellar effort, would recommend anyone interested in the subject.
• 
  Carlo

  Read it for the third time, I found out the now I have many things to learn and understand
  Great book that invites you to begin the mathematical journey
• 
  Zachary G. Augustine

  "For the moment, I do not know how to define "tautology"....It would be easy to offer a definition which might seem satisfactory for a while; but I know of none that I feel to be satisfactory, in spite of feeling thoroughly familiar with the characteristic of which a definition is wanted. At this point, therefore, for the moment, we reach the frontier of knowledge on our backward journey into the logical foundations of mathematics." -Bertrand Russell (p.204)
  
  So it is with modern math's struggle for ever-increasing rigor. I started reading this book in high school and quickly realized that I didn't have the math background to make sense of it.
  
  Now after I've taken logic and math classes at the University of Chicago, I revisited the book. Published in 1919, Russell offers what is, by any account, an excellent attempt at building a working system of mathematics from the bare bones of logical deduction. This alone is fascinating, as I've heard that it was possible but didn't know what it would look like (there's some things they don't teach you, even at college).
  
  Russell is an endless well of information on the fields of logic, math, and the gray areas between the two. He never fails to let you know where the current questions or problems in set theory or definitions or propositions are. A perfect example is in the statement and discussion surrounding two hypotheses of (I think) purely deductive set theory: multiplication and infinity. Both of them are not logically necessary, and yet these processes appear to function as if they are true. Intuitively, they should be true, yet we just don't know if they are!. This is the problem of proving things that have absolutely no grounding in the world as we know it, and Russell makes you aware of how pure mathematicians deal with these problems.
  
  In 1919, Russell skirts around the edges of a key aspect of logic--one that Godel later proved in his incompleteness theorems in 1931--that is:
  
  1) In a logically deduced system that is complex enough to perform math as we know it,
  2) There are certain hypotheses that are true and can never be proved or disproved.
  
  This is astounding, and Russell is right on the edge of predicting what I'm assuming is a long-conjectured, impossibly-strange aspect of logic. This particular concept blows my mind in a way a symbolic logic class never can. Russell covers these difficulties perfectly; he exemplifies intellectual honesty and objectivity.
  
  This book is essentially an extended essay that condenses Russell's own Principles of Mathematics--no short work at 576-pages--and the gigantic, 3-volume, nearly 2,000 page Principia Mathematica co-authored with Whitehead. He does this in plain english, which is perhaps the book's greatest merit. One can see how reading this slim book in order to better understand Russell's philosophy is a huge savings of time and difficulty.
  
  Russell is an excellent writer, especially when it comes to making impossibly dense subjects appear clear. That being said, this book is considerably dry at points. It is also way over my head but I think it's interesting and I learned a lot about the philosophy of higher mathematics.
• 
  Arya Tabaie

  Update: Many of the ideas in the book have made it into the mainstream, and are no longer considered cutting edge or "philosophical". I consider any introductory text into Analysis (say, Baby Rudin) a better source for learning them.
  
  My old review:
  1) Bible for the obsessive:
  Mathematics is known to be the most rigorous and precise of all the branches of learning. If you feel even that is not good enough, this is the read for you!
  
  2) The struggle that I faced reading this book -which is intended for the "general public"- shows how us engineers sometimes overestimate our math abilities, especially considering that my field has been Electrical Engineering and Computer Science, which are more profoundly connected to theoretical mathematics than other engineering fields.
  
  3) Gonna be honest, comparing Russell's writing to that of Einstein, I can't help but to feel a bit of disappointment. This was an area I had greater passion for and stronger background in, yet he still managed to bore me from time to time. But again, maybe that was only because he went into greater depth, and what greatness can ever be achieved without indulging in some difficult and dismal technical detail?
• 
  Maurizio Codogno

  I fondamenti della matematica visti da uno che li ha creati
  
  Cosa può fare uno mentre è in carcere a causa delle sue idee pacifiste? Per Bertrand Russell la risposta è stata ovvia: scrivere un libro "divulgativo" per spiegare quanto aveva formalizzato nei Principia Mathematica. Attenzione: stiamo parlando di un testo pubblicato più di un secolo fa. Nonostante il titolo, quello che troviamo è una trattazione dei fondamenti della matematica. Dobbiamo ricordarci che Russell non è un matematico ma un logico. Questo significa - lo si vede anche nei testi di Gabriele Lolli - che i matematici sono considerati scienziati di serie B che hanno delle idee ma non sanno metterle bene in pratica. Significativa la frase con cui liquida la scelta di Dedekind di postulare che dai suoi tagli si ottengano gli irrazionali: qui poi entra anche in gioco l'ego di Russell che non è certo infimo. La traduzione di Luca Pavolini è corretta (salvo un paio di svarioni verso il fondo dove si deve essere annodato il cervello leggendo il testo, e non posso dargli torto), anche se si sente il mezzo secolo abbondante passato da quando l'ha scritta. Non posso infine dire nulla sulla prefazione di Odifreddi, perché ho letto la versione originale italiana del 1962.
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  Whoof

  Can be hard to understand in certain chapters (my own fault I'm sure) but the fundamentals come through.
  Among other things, came away with solid understanding of Cantor's power-set proof as well as a deeper understanding of Russell's paradox and pet "theory of types". Axiom of Choice is still confusing. He has a delightful writing style and sometimes his short tangents into areas just past the border of "pure logic" are the most interesting parts of a section. Loved the emphasis on the "incompatibility" operator--there's a reason NAND gates are useful.
  
  I like when he uses unicorns in his examples.
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  Mike

  I finally finished this short book. You really have to love this stuff in order to commit to reading it and philosophical meanderings of this sort are not my thing. This book seems more of an esoteric digression than a practical assistance to those trying to understand and employ math. The Goodreads description of this book reports that it is a seminal work, but not for me. Perhaps I am too limited in my scope of appreciation because elements of the book remind me of "database" theory, e.g. one to one, one to many, many to many and so forth.
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  Nick Black

  Worth reading for the sensible, thoughtful introduction to and deduction of "number as cardinality", which is one of the single most crucial realizations for grokking any of modern mathematics (look to Landau's
  Foundations of Analysis for a rigorous treatment of this topic from an Analytic standpoint). I gave this away to a mathematically-minded freshman when I graduated high school, with my blessings.
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  Roberto Rigolin F Lopes

  Sitting on his logic expertise, Russell tries to educate us on fundamental concepts of mathematics. The whole thing sounds like a talk trying to reach layperson using several analogies (he insults philosophers making logical mistakes as well) or to motivate young students. The highest point is a construction stating that is nonsense discuss the existence of god. Warning: You may fall in love with Mathematics!