[Remarks on the Foundations of Mathematics] EBOOK/EPUB
The Magistrates' Court: An Introduction (Fifth Edition) rThe wood sellers I can t give aating to a book in which I don t understand most of the content There s definitely food for thought I ll have to come back to it when I can better understand the topics and engage to a es Bemerkungen zu den Grundlagen der MathematikWittgensteins Gedanken zur Mathematik sind teils philosophisch teils aber auch unstrukturiert vor sich hingedacht Man findet Gedanken zur Logik und Beweisf hrung zu Axiomen und Gleichungen zu unendlichen Kardinalzahlen von Mengen zur Geometrie und zur Arithmetik zu Folgen und Reihen und Br chen 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 Wenn z B aus mathematischen Sprachspiele werden oder Mathematik mit Alchemie verglichen wird oder I gave this five stars even though I m pretty sure I don t understand it I m easonably sure that nobody understand Wittgenstein but that s another story Nonetheless the book provides a wealth of brain food for thinking 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 uestions about basic mathematical concepts like um counting the esults are wonderful My favorite crazy little uestion comes section VThe class cats is not a cat How do you know 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 ule for example I would only The Crimson Thread: A Retelling of Rumpelstiltskin recommendeading 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 The Dog Cancer Survival Guide: Full Spectrum Treatments to Optimize Your Dog's Life Quality and Longevity read it without understanding Wittgenstein s broader aims The collection ofemarks was never formulated into a fully cohesive book and much of the comments were just Wittgenstein s comments to himself so some parts were No Beast So Fierce: The Terrifying True Story of the Champawat Tiger, the Deadliest Animal in History repetitive and other parts without development That said there are plenty of interesting ideas For example Wittgenstein that basic arithmetical statements such as 32 5 are used asules 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 and are not empirical statements or statements giving knowledge Wittgenstein is directly against Russell in that he did not mathematics Close to the Land: The Way We Lived in North Carolina, 1820-1870 reuired aigorous foundation and takes aim at the idea that the Berlayar di Pamor Badik real proof of an arithmetical statement is the one found in a system such as Russell s PM One of theeasons for this is that PM or another foundational calculus cannot be considered the ground of 224 as one of the criteria someone would look for in a potential foundation is that it would have to prove statements like 224 Russell s PM would have been ejected if it had proved statements like 225 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 the interpretation they e 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 eg Fermat s theorem before the proof and to what a proof is There are also interesting emarks around nonconstructive existence proofs and ho. This analyzes in depth such topics logical compulsion mathematical conviction; calculation as experiment; mathematical surp. .
Ludwig Wittgenstein Ä 2 Characters.
Arm 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 productpresses its knobs or an animal walks over it and it calculates the product x 20 3 A human calculating machine might be trained so that when the ules of inference were shewn it and perhaps exemplified it ead 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 easonable 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 uestion is even as a joke isn t it not another see it as a joke The uestion is even as a joke isn t it mathematics 9 What if someone were to The Pink Pearl reply to a uestion So far there is no such thing as an answer to this uestion So eg the poet mighteply 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 ueer 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 Manyhimself of many 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 eg 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 still in progress 35 Migrant Resistance in Contemporary Europe reeading material great for public communication each point a different The Violinist and The Son of Redhead: Two Plays by Leonard Melfi riddle this is where Nassim Taleb took Wittgenstein suler from points 94 and 93. Contradiction; the ole of mathematical propositions in the forming of conceptsTranslator's NoteEditors' PrefaceThe TextInd. W starkly less clear they are in their meaning than constructive ones Wittgenstein considers as an example uestions 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 Here are some lines I found interesting 15 It
important that in our our natural language all is a fundamental concept and all but one less fundamental ie there isIs Important That In Our
Not A Single Word Fora single word for 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 uite 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 uestion instead of an answer to a uestion For an it is always good to put a uestion instead of an answer to a uestion For an to the philosophical uestion may easily be unfair disposing of it by means of another uestion 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 emains 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 Galaxies in the Universe: An Introduction 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 myeaction 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 Dying to Be Ill: True Stories of Medical Deception 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 eg If youeally do the same thing both times then the Octavio's Journey result must be the same too What compels us to procee according to aule to conceive something a a Philosophy in Social Work 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 sayefusal 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 Americana: The Kinks, the Riff, the Road: The Story 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 oneeached the place where contradiction no longer does any Rise discovery invention; Russell's logic Godel's theorem cantor's diagonal procedure Dedekind's cuts; the nature of proof. ,