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 NoteEdiThis 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 NoteEditors' PrefaceThe TextIndex...
|Title||:||Remarks on the Foundations of Mathematics|
|Number of Pages||:||444 Pages|
|Status||:||Available For Download|
|Last checked||:||21 Minutes ago!|
Remarks on the Foundations of Mathematics Reviews
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. . . .
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.
The wood sellers!!!
still in progress..
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?
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.
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 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.
Review in time.