# Remarks on the Foundations of Mathematics [E–pub New]

The wood sellers I can t give a rating to a book in which I don t understand most of the content There s definitely food for thought I l have to come back to it * WHEN I CAN BETTER UNDERSTAND THE TOPICS AND ENGAGE *I can better understand the topics and engage a respectable evel 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 Write It When I'm Gone: Remarkable Off-The-Record Conversations with Gerald R. Ford 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 dieogische Notation pauschal bem ngelt wird muss man sich schon fragen Wo ist da die Bedeutung seiner Gedanken wo ist da der Erkenntniswert Konstruktiv kritische Bemerkungen sind Ready for Summer leider Mangelware Eine sehrockere Philosophie der Mathematik 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 ogic and gives obscure but invaluable insights into Wittgenstein s takes on such matters 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 ike um counting the results are wonderful My favorite crazy ittle uestion comes in section VThe class of cats is not a cat How do you "know This book contains comments written over a decade of work of Wittgenstein A arge part "This book contains comments written over a decade of work of Wittgenstein A Write Your Novel!: Tips from a Bestseller large part the text was originally supposed to be the second half of the Philosophical Investigations and there areots 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 Dogs Behaving Badly 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 32 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 reuired 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 224 as one of the criteria someone wouldook for in a potential foundation is that it would have to prove statements First Year Teacher: Wit and Wisdom from Teachers Who've Been There like 224 Russell s PM would have been rejected if it had proved statementsike 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 re given and the significance they hold and the unusual statements that people make in connection with them. This analyzes in depth such topics ogical compulsion mathematical conviction; calculation as experiment; mathematical surp. ,

Kind of work to accomplish no what is pernicious is not to know how one reached the place where contradiction no onger 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 "accident now someone accidentally presses its 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 April 16th: Virginia Tech Remembers 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 mathematicalogic 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 uestion is even as a joke isn t it evidently mathematics 9 What if someone were to reply to a uestion So far there is no such thing as an answer to this uestion So eg 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 ine without ever stopping ueer when one takes something of this sort as a matter of course as it were in one s stride it oses 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 "**the abour of thinking an end We won t bother about an end It might also be said for

*the series is infinite We t worry about an end to this series for us it is always beyond our ken 48 Mathematical ogic has completely deformed the thinking of mathematicians and of philosophers by setting up a superficial interpretation of the forms of our everyday anguage as an analysis of the structures of facts Of course in this it has only continued to build on the Aristotelian ogic 50 If you The Habitat Guide to Birding look into this mouse s jaw you will see twoong 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 Space Kid life we are in death so in sanity we are surrounded by madness Part V 16 It is my task not to attack Russell sogic 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 Circumstantial Evidence: Death, Life, and Justice in a Southern Town lions is not aion 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 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. Contradiction; the role of mathematical propositions in the forming of conceptsTranslator's NoteEditors' PrefaceThe TextInd. There is some interesting discussion on whether or*

US THE SERIES IS INFINITE WE

you understand mathematical propositions without knowing a proof eg Fermat s theorem before the proof and to a proof is There are also interesting remarks around nonconstructive existence proofs and how starkly ess clear they are in their meaning than constructive ones Wittgenstein considers as an uestions about whether or not the string 777 occurs in particular irrational numbers and what "It Means To Say 777 Does Not Occur In "means to say that 777 does not occur in infinite decimal expansion of an irrational number Here are some ines I found interesting 15 It is important that in our Every Drop for Sale: Our Desperate Battle Over Water language our naturalanguage all is a fundamental concept and all but one Down We Go: Living Into the Wild Ways of Jesus less fundamental ie 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 uite different conclusions and other ones again on another day 167 The mathematician is an inventor not a discoverer Appendix 1 8 What is calledosing 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 answer 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 Reach for the Rainbow: Advanced Healing for Survivors of Sexual Abuse 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 Theimits 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 Troubled Waters 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 beike 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 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 The Allergy-Fighting Garden: Stop Asthma and Allergies with Smart Landscaping languages we haveearnt 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 ead 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 ike a monument with a Janus head over the propositions of ogic 60 The pernicious thing is not to produce a contradiction in the region in which neither the consistent nor the contradictory proposition has any. Rise discovery invention; Russell's ogic Godel's theorem cantor's diagonal procedure Dedekind's cuts; the nature of proof.not you understand mathematical propositions without knowing a proof eg Fermat s theorem before the proof and to