Ludwig Wittgenstein, Philosophical Remarks
If a mechanism is meant to act as a brake, but for some reason accelerates the machine, then the purpose of the mechanism cannot be found out from it alone.
If you were to say, "That's a brake lever but it doesn't work," you would be talking about intention. It's just the same as when we still call a broken clock a clock.
- Section 31, p. 69
To understand the sense of a proposition means to know how the issue of its truth or falsity is to be decided.
- Section 43, p. 77
You cannot compare a picture with reality, unless you can set it against a yardstick. You must be able to fit the proposition on to reality.
- Section 43, p. 77
What I wanted to say is it's strange that those who ascribe reality only to things and not to our ideas (the world as an idea) move about so unquestioningly in the world as an idea and never long to escape from it.
- Section 47, p. 80
...if a body has length, there can be no length without a body.
- Section 48, p. 81
Perhaps this whole difficulty stems from taking the time concept from time in physics and applying it to the course of immediate experience. It's a confusion of the time of the film strip with the time of the picture it projects. For "time" has one meaning when we regard memory as the source of time, and another when we regard it as a picture preserved from a past event.
- Section 49, p. 81
If, for instance, you ask, "Does the box still exist when I'm not looking at it?", the only right answer would be, "Of course, unless someone has taken it away or destroyed it." Naturally, a philosopher would be dissatisfied with this answer, but it woudl quite rightly reduce his way of formulating the question ad absurdum.
- Section 57, p. 88
We could adopt the following way of representing matters: if I, LW, have toothache, then that is expressed by means of the proposition, "There is toothache." But if that is so, what we now express by the proposition "A has toothache" is put as follows: "A is behaving as LW does when there is a toothache." Similarly we shall say "It is thinking" and "A is behaving as LW does when it is thinking." (You could imagine a despotic oriental state where the language is formed with the despot at the center and his name instead of LW.) It is evident that this way of speaking is equivalent to ours when it comes to questions of intelligibility and freedom from ambiguity. But it's equally clear that this language could have anyone at all at its centre.
- Section 50, p. 88-89
Not only does epistemology pay no attention to the truth or falsity of genuine propositions, it's even a philosophical method of focusing on precisely those propositions whose content seems to us as physically impossible as can be imagined (e.g., that someone has an ache in someone else's tooth). In this way, epistemology highlights the fact that its domain includes everything that can possibly be thought.
- Section 60, p. 90
With our language we find ourselves, so to speak, in the domain of the film, not of the projected picture. And if I want to make music to accompany what is happening on the screen, whatever produces the music must again happen in the sphere of the film.
- Section 70, p. 98
It could, e.g., be practical under certain circumstances to give proper names to my hands and to those of other people, so that you wouldn't have to mention their relation to somebody when talking about them, since the relation isn't essential to the hands themselves; and the usual way of speaking could create the impression that its relation to its owner was something belonging to the essence of the hand itself.
- Section 71, p. 100
Does all this mean then that a visual image does essentially contain or pre-suppose a subject after all? Or isn't it rather that those experiments give me nothing but purely geometrical information? That is to say, information that constantly only concerns the object. Objective information about reality.
- Section 73, p. 102
Think of such "objects" as: a flash of lightning, the simultaneous occurrence of 2 events, the point at which a line cuts a circle, etc.: the three circles in the visual field are an example for all of these cases.
- Section 115, p. 136
A cardinal number is an internal property of a list.
- Section 118, p. 140
We may also say: there is no path to infinity, not even an endless one.
- Section 123, p. 146
The situation would be something like this: we have an infinitely long row of trees, and so as to inspect them, I make a path beside them. All right, the path must be endless, But if it is endless, that means precisely that you can't walk to the end of it. That is, it does not put me in a position to survey the row. (Ex hypothesi not.)
That is to say, the endless path doesn't have an end "infinitely far away," it has no end.
- Section 123, p. 146
If it is objected that "if I run through the number series I either eventually come to the number with the required property, or I never do," we need only reply that it makes no sense to say that you eventually come to the number, and just as little that you never do. Certainly it is correct to say 101 is or is not the number in question. But you can't talk about all numbers, because there's no such thing as all numbers.
- Section 124, p. 147
Can I know that a number satisfies the equation without a finite section of the infinite series being marked out as one within which it occurs?
"Can God know all the places of the expansion of Pi?" would have been a good question for the schoolteacher to ask. In all such cases, the answer runs, "The question is senseless."
- Section 138, p. 156
We all know what it means to say there is an infinite possibility and a finite reality... but from where, then, do I derive any knowledge of the infinite at all? In some sense or another, I must have 2 kinds of experience: one which is of the finite, and which cannot transcend the finite, and one of the infinite. And that's how it is.
Experience as experience of the facts gives me the finite; the objects contain the infinite...
Where the nonsense starts is with our habit of thinking of a large number as closer to infinity than a small one.
- Section 138, p. 157
How do we construct an infinite hypothesis, such as that there are infinitely many fixed stars? Once more it can only be given through a law. Let's think of an infinite series of red spheres. Let's think of an infinite film strip. This is a typical case of an hypothesis reaching out to infinity. It's clear to us that no experience corresponds to it. It only exists in "the second system," that is in language; but how is it expressed there? (If a man can imagine an infinite film strip, then as far as he's concerned there is an infinite reality, and also the "actual infinite" of mathematics.)
It is expressed by a proposition of the form:
Everything relating to the infinite possibility (every infinite assertion about the film) is reproduced in the expression in the first bracket, and the reality corresponding to it in the second.
- Section 139, p. 159
Is it possible to imagine time with an end, or with two ends?
- Section 140, p. 159
The rules for a number system - say, the decimal system - contain everything that is infinite about the numbers. That, e.g., these rules set no limits on the left or right hand to the numerals; this is what contains the expression of infinity.
- Section 141, p. 160-161
What's more, it all hangs on the syntax of reality and possibility. contains the possibility of correlating any number with another, but doesn't correlate all numbers with others.
- Section 141, p. 161
That is to say, the propositions "Three things can lie in this direction" and "infinitely many things can lie in this direction" are only apparently formed the same way, but are in fact different in structure; the "infinitely many" of the second proposition doesn't play the same role as the "three" of the first.
- Section 142, p. 162
We might also ask: What is it that goes on when, while we've as yet no idea how a certain proposition is to be proved, we still ask, "Can it be proved or not?" and proceed to look for a proof? If we "try to prove it," what do we do? ... How we answer this question is a pointer as to whether the as yet unproved - or unprovable - proposition is senseless or not... every significant proposition must teach us through its sense how we are to convince ourselves whether it is true or false.
- Section 148, p. 170
I said: Where you can't look for an answer, you can't ask, either, and that means where there's no logical method for finding a solution, the question doesn't make sense either.
- Section 149, p. 172
I cannot draw the limits of my world, but I can draw limits within my world. I cannot ask whether the proposition p belongs to the system S, but I can ask whether it belongs to the part s of the system S. So that I can locate the problem of the trisection of an angle within the larger system, but can't ask, within the Euclidian system, whether it is soluble. In what language should I ask this? In the Euclidian? But neither can I ask in Euclidian language about the possibility of bisecting an angle within the Euclidian system. For in this language that would boil down to a question about absolute possibility, and such a question is always nonsense...
In mathematics, we cannot talk of systems in general, but only within systems. They are just what we can't talk about. And so, too, what we can't search for.
- Section 152, p. 179
There is no number outside a system.
The expansion of pi is simultaneously an expression of the nature of pi, and of the nature of the decimal system.
- Section 188, p. 231
Indeed, the way the irrationals are introduced in text books always makes it sound as if what is being said is: Look, that isn't a rational number, but still there is a number there. But why then do we still call what is there a number? And the answer must be, because there is a definite way for comparing it with the rational numbers.
- Section 192, p. 236
If we say the molecules of a gas move in accordance with the laws of probability, it creates the impression that they move in accordance with some a priori laws or other. Naturally, that's nonsense. The laws of probability, on which calculus is based, are hypothetical assumptions, which are then rehashed by the calculation and then in some another form empirically confirmed or refuted.
- Section 233, p. 290-291
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