READ STUDY GUIDE: 6.3–6.3751 |
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Propositions 6.3-6.3751
| 6.3 The |
| exploration of logic means the exploration of everything that is subject to |
| law . And outside logic everything is accidental. |
6.31 The so-called law of induction cannot possibly be a law of logic, since it is obviously a proposition with sense.—-Nor, therefore, can it be an a priori law.
6.32 The law of causality is not a law but the form of a law.
6.321 'Law of causality'—that is a general name. And just as in mechanics, for example, there are 'minimum-principles', such as the law of least action, so too in physics there are causal laws, laws of the causal form.
6.3211 Indeed people even surmised that there must be a 'law of least action' before they knew exactly how it went. (Here, as always, what is certain a priori proves to be something purely logical.)
6.33 We do not have an a priori belief in a law of conservation, but rather a priori knowledge of the possibility of a logical form.
6.34 All such propositions, including the principle of sufficient reason, tile laws of continuity in nature and of least effort in nature, etc. etc.—all these are a priori insights about the forms in which the propositions of science can be cast.
6.341 Newtonian mechanics, for example, imposes a unified form on the description of the world. Let us imagine a white surface with irregular black spots on it. We then say that whatever kind of picture these make, I can always approximate as closely as I wish to the description of it by covering the surface with a sufficiently fine square mesh, and then saying of every square whether it is black or white. In this way I shall have imposed a unified form on the description of the surface. The form is optional, since I could have achieved the same result by using a net with a triangular or hexagonal mesh. Possibly the use of a triangular mesh would have made the description simpler: that is to say, it might be that we could describe the surface more accurately with a coarse triangular mesh than with a fine square mesh (or conversely), and so on. The different nets correspond to different systems for describing the world. Mechanics determines one form of description of the world by saying that all propositions used in the description of the world must be obtained in a given way from a given set of propositions—the axioms of mechanics. It thus supplies the bricks for building the edifice of science, and it says, 'Any building that you want to erect, whatever it may be, must somehow be constructed with these bricks, and with these alone.' (Just as with the number-system we must be able to write down any number we wish, so with the system of mechanics we must be able to write down any proposition of physics that we wish.)
6.342 And now we can see the relative position of logic and mechanics. (The net might also consist of more than one kind of mesh: e.g. we could use both triangles and hexagons.) The possibility of describing a picture like the one mentioned above with a net of a given form tells us nothing about the picture. (For that is true of all such pictures.) But what does characterize the picture is that it can be described completely by a particular net with a particular size of mesh. Similarly the possibility of describing the world by means of Newtonian mechanics tells us nothing about the world: but what does tell us something about it is the precise way in which it is possible to describe it by these means. We are also told something about the world by the fact that it can be described more simply with one system of mechanics than with another.
6.343 Mechanics is an attempt to construct according to a single plan all the true propositions that we need for the description of the world.
6.3431 The laws of physics, with all their logical apparatus, still speak, however indirectly, about the objects of the world.
6.3432 We ought not to forget that any description of the world by means of mechanics will be of the completely general kind. For example, it will never mention particular point-masses: it will only talk about any whatsoever.
6.35 Although the spots in our picture are geometrical figures, nevertheless geometry can obviously say nothing at all about their actual form and position. The network, however, is purely geometrical; all its properties can be given a priori. Laws like the principle of sufficient reason, etc. are about the net and not about what the net describes.
6.36 If there were a law of causality, it might be put in the following way: There are laws of nature. But of course that cannot be said: it makes itself manifest.
6.361 One might say, using Hertt:'s terminology, that only connexions that are subject to law are thinkable.
6.3611 We cannot compare a process with 'the passage of time'—there is no such thing—but only with another process (such as the working of a chronometer). Hence we can describe the lapse of time only by relying on some other process. Something exactly analogous applies to space: e.g. when people say that neither of two events (which exclude one another) can occur, because there is nothing to cause the one to occur rather than the other, it is really a matter of our being unable to describe one of the two events unless there is some sort of asymmetry to be found. And if such an asymmetry is to be found, we can regard it as the cause of the occurrence of the one and the non-occurrence of the other.
6.36111 Kant's problem about the right hand and the left hand, which cannot be made to coincide, exists even in two dimensions. Indeed, it exists in one-dimensional space in which the two congruent figures, a and b, cannot be made to coincide unless they are moved out of this space. The right hand and the left hand are in fact completely congruent. It is quite irrelevant that they cannot be made to coincide. A right-hand glove could be put on the left hand, if it could be turned round in four-dimensional space.
6.362 What can be described can happen too: and what the law of causality is meant to exclude cannot even be described.
6.363 The procedure of induction consists in accepting as true the simplest law that can be reconciled with our experiences.
6.3631 This procedure, however, has no logical justification but only a psychological one. It is clear that there are no grounds for believing that the simplest eventuality will in fact be realized.
6.36311 It is an hypothesis that the sun will rise tomorrow: and this means that we do not know whether it will rise.
6.37 There is no compulsion making one thing happen because another has happened. The only necessity that exists is logical necessity.
6.371 The whole modern conception of the world is founded on the illusion that the so-called laws of nature are the explanations of natural phenomena.
6.372 Thus people today stop at the laws of nature, treating them as something inviolable, just as God and Fate were treated in past ages. And in fact both are right and both wrong: though the view of the ancients is clearer in so far as they have a clear and acknowledged terminus, while the modern system tries to make it look as if everything were explained.
6.373 The world is independent of my will.
6.374 Even if all that we wish for were to happen, still this would only be a favour granted by fate, so to speak: for there is no logical connexion between the will and the world, which would guarantee it, and the supposed physical connexion itself is surely not something that we could will.
6.375 Just as the only necessity that exists is logical necessity, so too the only impossibility that exists is logical impossibility.
6.3751 For example, the simultaneous presence of two colours at the same place in the visual field is impossible, in fact logically impossible, since it is ruled out by the logical structure of colour. Let us think how this contradiction appears in physics: more or less as follows—a particle cannot have two velocities at the same time; that is to say, it cannot be in two places at the same time; that is to say, particles that are in different places at the same time cannot be identical. (It is clear that the logical product of two elementary propositions can neither be a tautology nor a contradiction. The statement that a point in the visual field has two different colours at the same time is a contradiction.)
