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Remarks on the Foundations of Mathematics (Bilingual Edition) (English and German Edition), by Ludwig Wittgenstein
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- Sales Rank: #2542123 in Books
- Published on: 1967-01
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- Binding: Paperback
- 408 pages
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THE PHILOSOPHER’S THOUGHTS ON LOGIC AND MATHEMATICAL THEORY
By Steven H Propp
Ludwig Josef Johann Wittgenstein (1889-1951) was an Austrian-British philosopher whose books such as Tractatus Logico-Philosophicus and Philosophical Investigations are among the acknowledged “classics” of 20th century philosophy. Born into a wealthy family, he gave all of his inheritance away, served in the Austrian Army during World War I, taught schoolchildren in remote Austrian villages, but ultimately taught at Cambridge for many years. The Tractatus was the only book he published during his lifetime, but his papers have been posthumously edited, and notes of lectures taken by his students have been transcribed, and have resulted in many published books, such as Lectures & Conversations on Aesthetics, Psychology, & Religious Belief, Philosophical Grammar, Philosophical Remarks, The Blue and Brown Books, Remarks on the Philosophy of Psychology, Remarks on Colour, Zettel, etc.
The Editor’s Preface states, “The remarks on the philosophy of mathematics and logic, which are published here, were written in the years 1937-1944. After that time Wittgenstein did not again return to this topic. He had written a great deal on this subject in the period 1929 to roughly 1932, part of which we hope to publish later… This earlier work belongs to a stage in Wittgenstein’s development which is still fairly close to the Tractatus Logico-Philosophicus. The remarks presented in THIS volume are of a piece with the thought of Philosophical Investigations.”
He says, “There is a transition from one proposition to another VIA other propositions, that is, a chain of inferences… There is nothing occult about this; it is s derivation of one sentence from another according to a rule… We call it a ‘conclusion’ when the inferred proposition CAN in fact be derived from the premise… Now what does it mean to say that one proposition CAN be derived from another by means of a rule? Can’t anything be derived from something by means of SOME rule---or even according to any rule, with a suitable interpretation? What does it mean for me to say e.g.: this number can bed got by multiplying these two numbers?” (I, �6 & 7)
He observes, “I might also say as a result of the proof: ‘From now on an H and a P are called ‘the same in number.’ Or: this proof doesn’t EXPLORE the essence of the two figures, but it does express what I am going to count as belonging to the essence of the figures from now on. I deposit what belongs to the essence among the paradigms of language. The mathematician creates ESSENCE.” (�32)
He argues, “How is it established which pattern is the multiplication of 13 X 13? Isn’t it DEFINED by the rules of multiplication? But what if, using these rules, you get different results today from what all the arithmetic books say? Isn’t that possible?---‘Not if you apply the rules as THEY do.’ Of course not! But that is a mere pleonasm… Well, it never in fact happens that somebody who has learnt to calculate goes on obstinately getting different results… But if it should happen, then we should declare him abnormal, and take no further account of his calculation.” (�112)
He comments, “The laws of logic are indeed the expression of ‘thinking habits’ but also of the habit of THINKING. That is to say that they can be said to shew: how human beings think, and also WHAT human beings call ‘thinking’… The propositions of logic are ‘laws of thought,’ ‘because they bring out the essence of human thinking’---to put it more correctly: because they bring out, or shew, the essence, the technique, of thinking. They shew what thinking is and also shew kinds of thinking.” (�131, 133)
He notes, “In philosophy it is always good to put a QUESTION instead of an answer to a question. For an answer to a philosophical question may easily be unfair; disposing of it by means of another question it is not. Then should I put a question here, for example, instead of the answer that the arithmetical proposition cannot be proved by Russell’s method?” (II, �5)
He argues, “I want to say: with the logic of Principia Mathematica it would be possible to justify an arithmetic in which 1000 + 1 = 1000; and all that would be necessary for this purpose would be to doubt the sensible correctness of calculations. But if we do not doubt it, then it is not our conviction of the truth of logic that is responsible. When we say in a proof” ‘This MUST come out’---then this is not for reasons that we do not SEE. What convinces us---THAT is the proof; a configuration that does not convince us is not the proof, even when it can be shewn to exemplify the proved proposition. That means: it must not be necessary to make a physical investigation of the proof-configuration in order to shew us what has been proved.” (�39)
He goes on: “We incline to the belief that LOGICAL proof has a peculiar, absolute cogency, deriving from the unconditional certainty in logic of the fundamental laws and the laws of inference. Whereas propositions proved in this way can after all not be more certain than is the correctness of the way those laws of inference are APPLIED. The logical certainty of proofs… does not extend beyond their geometrical certainty.” (�43)
He states, “EXPERIENCE teaches us that we all find this calculation correct. We start ourselves off and get the result of the calculation. But now… we aren’t interested in having---under such and such conditions say---actually produced this result, but in the pattern of our working; it interests us as a convincing, harmonious pattern---not, however, as the result of an experiment, but as a PATH.” (�69)
He notes, “We went sleepwalking along the road between abysses. But even if we now say: ‘Now we are awake’---can we be certain that we shall not wake up one day? (And then say: so we were asleep again.) Can we be certain that there are no abysses now that we do not see? But suppose I were to say: The abysses in a calculus are not there if I don’t see them! Is no demon deceiving us at present? Well, if he is, it doesn’t matter. What the eye doesn’t see the heart doesn’t grieve over.” (�78)
He comments: “‘Only the proof of consistency shews me that I can rely on the calculus.’ What sort of proposition is it, that only THEN can you rely on the calculus? But what if you do rely on it WITHOUT that proof! What sort of mistake have you made?” (�84)
He says, “Consider also the rule which forbids one digit in certain places, but otherwise leaves the choice open. Isn’t it like this? The concepts of infinite decimals in mathematical propositions are not concepts of series, but of the unlimited technique of expansion of series.” (IV, �19)
He asserts, “Everything I say really amounts to this, that one can know a proof thoroughly and follow it step by step, and yet at the same time not UNDERSTAND what it was that was proved. And this in turn is connected with the fact that one can form a mathematical proposition in a grammatically correct way without understanding its meaning. Now when does one understand it? I believe: when one can apply it.” (�25)
He contends, “The curse of the invasion of mathematics by mathematical logic is that now any proposition can be represented in a mathematical symbol, and this makes us feel obligated to understand it. Although of course this method of writing is nothing but the translation of vague ordinary prose.” (�46)
He concludes, “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.”(�53)
As always, Wittgenstein’s ideas are provocative, stimulating, and often profound. This book will be of great value to anyone studying his thought (particularly in its less-“linguistic” manifestations).
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