Even if all that one wishes were to happen, this could still only be called the Grace of God. There is no logical association between one's will and the world to guarantee it. The physical association one presumes to exist between any and every thing is surely nothing we could will ourselves. Just as only logical necessity exists, so too only logical impossibility exists. For example: that two colors are simultaneously present at the same place in the visual field is in fact logically impossible, since it is ruled out by the logical structure of color.
In physics, this contradiction appears like this: a particle cannot have two velocities at the same time; it cannot be in two places at the same time; particles that are in different places at the same time cannot be identical.
It is clear that the logical product of two elemental propositions can neither be a tautology nor a contradiction. The assertion that a point in the visual field has two different colors at the same time is a contradiction.
Monday, April 7, 2008
The relative position of logic and science.
That an image can be described using a grid with a given form tells us nothing about the image. (A grid works for all such images.) But what does characterize the image is that it can be completely described by a particular grid with a particular mesh size. So too, it tells us nothing about the world that it can be described by Newtonian mechanics or whatever. That it can be described at all, and in a particular way, does tell us something indeed. That one method of theoretical description is simpler than another also tells us something about the world.
Theoretical physics is an attempt to construct all the true propositions that we need to describe the world using a single plan. Throughout their whole logical apparatus, the laws of physics still speak about the objects of the world. We ought not forget that any theoretical description of the world will always be completely general. In mechanics, for example, one never speaks of particular point-masses, but only about any whatsoever.
Although the spots in our image are geometrical figures, it is obvious that geometry can say nothing at all about their actual form and position. The grid, however, is purely geometrical; all its properties can be given a priori. Laws like the principle of sufficient reason, etc. deal with the grid and not with what the grid describes.
If there were a law of causality, one might state it as: "There are laws of nature." But of course that cannot be said: it can be seen. Using Hertz's terminology, one might say: "Only regular correlations are thinkable. Hence the only way we can describe the lapse of time is to rely on some process such as the movement of a chronometer."
Something entirely analogous applies to space. Wherever one says that neither of two exclusive events can occur because there is no reason one should occur rather than the other, one is really dealing with the fact that one cannot describe either without some sort of asymmetry between them. And if such an asymmetry is found, we can regard it as the cause that made one occur and not the other.
Kant's problem about the right hand and the left hand, which cannot be made to coincide, exists already in two dimensions; indeed, even in one-dimensional space. 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.
The procedure of induction consists in accepting as true the simplest law that can be reconciled with our experiences. But this procedure has no logical, only a psychological, justification. There is no reason to believe that the simplest case will in fact be realized. That the sun will rise tomorrow is a hypothesis; we do not know whether it will rise. There is nothing to compel one thing to happen because something else has. There is only logical necessity.
The whole modernist world view is based on the illusion that the laws of nature actually explain natural phenomena. Thus they stand before the laws of nature as something inviolable, just as the ancients did before God and Fate. Both, in fact, are both right and wrong. Nevertheless, the view of the ancients is clearer in so far as they acknowledge it as closure, while the modern system tries to make it seem as if everything were explained.
Theoretical physics is an attempt to construct all the true propositions that we need to describe the world using a single plan. Throughout their whole logical apparatus, the laws of physics still speak about the objects of the world. We ought not forget that any theoretical description of the world will always be completely general. In mechanics, for example, one never speaks of particular point-masses, but only about any whatsoever.
Although the spots in our image are geometrical figures, it is obvious that geometry can say nothing at all about their actual form and position. The grid, however, is purely geometrical; all its properties can be given a priori. Laws like the principle of sufficient reason, etc. deal with the grid and not with what the grid describes.
If there were a law of causality, one might state it as: "There are laws of nature." But of course that cannot be said: it can be seen. Using Hertz's terminology, one might say: "Only regular correlations are thinkable. Hence the only way we can describe the lapse of time is to rely on some process such as the movement of a chronometer."
Something entirely analogous applies to space. Wherever one says that neither of two exclusive events can occur because there is no reason one should occur rather than the other, one is really dealing with the fact that one cannot describe either without some sort of asymmetry between them. And if such an asymmetry is found, we can regard it as the cause that made one occur and not the other.
Kant's problem about the right hand and the left hand, which cannot be made to coincide, exists already in two dimensions; indeed, even in one-dimensional space. 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.
The procedure of induction consists in accepting as true the simplest law that can be reconciled with our experiences. But this procedure has no logical, only a psychological, justification. There is no reason to believe that the simplest case will in fact be realized. That the sun will rise tomorrow is a hypothesis; we do not know whether it will rise. There is nothing to compel one thing to happen because something else has. There is only logical necessity.
The whole modernist world view is based on the illusion that the laws of nature actually explain natural phenomena. Thus they stand before the laws of nature as something inviolable, just as the ancients did before God and Fate. Both, in fact, are both right and wrong. Nevertheless, the view of the ancients is clearer in so far as they acknowledge it as closure, while the modern system tries to make it seem as if everything were explained.
Sunday, April 6, 2008
Mathematics is a method of logic.
Using equations characterizes the essence of the mathematical method. Because this so, every proposition of mathematics must be self explanatory. Mathematics arrives at equations by the method of substitution. Equations express that one can substitute an expression for another and so, starting from a number of equations, we advance to new equations by substituting different expressions as we go along in accordance with them. Thus the proof of the proposition 2 + 2 = 4 runs as follows:
Ω^(2×2')x=(Ω^2)^(2')x=(Ω^2)^(1+1')x=Ω^2'Ω^2'x=Ω^(1+1')Ω^(1+1')x
=(Ω'Ω)'(Ω'Ω)'x=Ω'Ω'Ω'Ω'x=Ω^(1+1+1+1')x=Ω4'x.
Exploring logic exploring everything that is subject to regularity, so everything outside of logic is happenstance. The law of induction cannot possibly be a law of logic, norcan it be an a priori law, since it is obviously a proposition that makes sense. The law of causality is not a law but the form of a law. 'Law of causality' that is the name of a type. Just as mechanics has minimizing principles, such as the law of least action, so too does physics have causal laws, laws of the causal form. In fact, one even surmised that there must be a 'law of least action' before know ing exactly how it went. (As always, what is a priori proves to be purely logical.)
We do not believe the law of conservation a priori, but rather know a priori that such a logical form is possible. 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 which the propositions of science can take.
Newtonian mechanics, for example, provides a unified form for describing the world. Let us imagine a white surface with irregular black spots on it. Whatever kind of image these make, one can always approximate its description as closely as one wishes by covering the surface with a sufficiently fine square grid, and then declaring every square black or white. This grid provided a unified form for describing the surface. That form is optional, since I using a triangular or hexagonal grid would have achieved the same result. It could be that a triangular grid would have been simpler; that is to say, that we could describe the surface more accurately with a coarse triangular grid than with a fine square grid (or conversely), and so on. The different grids correspond to different systems for describing the world.
Mechanics determines one particular form of description of the world by saying: All propositions describing the world must be obtained in a given way from a given set of propositions, the axioms of mechanics. It supplies the bricks for building the edifice of science, and says: 'Any building that you want to erect, it must be with these and only these components.' (Just as we must be able to write down any amount we wish using the number system, so we must be able to write down any proposition of physics that we wish using the system of mechanics.)
(Ω^ν)^μ'x=Ω^(ν×μ')x Def.,
Ω^(2×2')x=(Ω^2)^(2')x=(Ω^2)^(1+1')x=Ω^2'Ω^2'x=Ω^(1+1')Ω^(1+1')x
=(Ω'Ω)'(Ω'Ω)'x=Ω'Ω'Ω'Ω'x=Ω^(1+1+1+1')x=Ω4'x.
Exploring logic exploring everything that is subject to regularity, so everything outside of logic is happenstance. The law of induction cannot possibly be a law of logic, norcan it be an a priori law, since it is obviously a proposition that makes sense. The law of causality is not a law but the form of a law. 'Law of causality' that is the name of a type. Just as mechanics has minimizing principles, such as the law of least action, so too does physics have causal laws, laws of the causal form. In fact, one even surmised that there must be a 'law of least action' before know ing exactly how it went. (As always, what is a priori proves to be purely logical.)
We do not believe the law of conservation a priori, but rather know a priori that such a logical form is possible. 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 which the propositions of science can take.
Newtonian mechanics, for example, provides a unified form for describing the world. Let us imagine a white surface with irregular black spots on it. Whatever kind of image these make, one can always approximate its description as closely as one wishes by covering the surface with a sufficiently fine square grid, and then declaring every square black or white. This grid provided a unified form for describing the surface. That form is optional, since I using a triangular or hexagonal grid would have achieved the same result. It could be that a triangular grid would have been simpler; that is to say, that we could describe the surface more accurately with a coarse triangular grid than with a fine square grid (or conversely), and so on. The different grids correspond to different systems for describing the world.
Mechanics determines one particular form of description of the world by saying: All propositions describing the world must be obtained in a given way from a given set of propositions, the axioms of mechanics. It supplies the bricks for building the edifice of science, and says: 'Any building that you want to erect, it must be with these and only these components.' (Just as we must be able to write down any amount we wish using the number system, so we must be able to write down any proposition of physics that we wish using the system of mechanics.)
Logic is transcendental.
Logic is not a body of doctrine, but a mirror-image of the world. Logic is transcendental. Mathematics is a logical method. The propositions of mathematics are equations, and therefore pseudo-propositions. A proposition of mathematics does not express a thought.
Indeed in real life a mathematical proposition is never what one needs. Rather, one uses mathematical propositions only to make inferences from propositions that do not belong to mathematics to others that likewise do not belong to mathematics. (In philosophy the question, 'What do we actually use this word or this proposition for?' repeatedly leads to valuable insights.)
The logic of the world, which is shown in tautologies by the propositions of logic, is shown in equations by mathematics. If two expressions are joined by the sign of equality, they can be substituted for one another. But it must be manifest in the two expressions themselves whether this is the case or not. When two expressions can be substituted for one another, that characterizes their logical form.
It is a property of affirmation that it can be construed as double negation. It is a property of '1 + 1 + 1 + 1' that it can be construed as '(1 + 1) + (1 + 1)'.
Frege says that the two expressions have the same meaning but different senses. But the essential point about an equation is that it is not necessary in order to show that the two expressions connected by the sign of equality have the same meaning, since this can be seen from the two expressions themselves. And the possibility of proving the propositions of mathematics means simply that their correctness can be perceived without its being necessary that what they express should itself be compared with the facts in order to determine its correctness.
It is impossible to assert the identity of meaning of two expressions. For in order to be able to assert anything about their meaning, I must know their meaning, and I cannot know their meaning without knowing whether what they mean is the same or different. An equation merely marks the point of view from which I consider the two expressions, it marks their equivalence in meaning.
Intuition is needed to solve mathematical, but language itself provides the necessary intuition. The process of calculating serves to bring about that intuition. Calculation is not an experiment.
Indeed in real life a mathematical proposition is never what one needs. Rather, one uses mathematical propositions only to make inferences from propositions that do not belong to mathematics to others that likewise do not belong to mathematics. (In philosophy the question, 'What do we actually use this word or this proposition for?' repeatedly leads to valuable insights.)
The logic of the world, which is shown in tautologies by the propositions of logic, is shown in equations by mathematics. If two expressions are joined by the sign of equality, they can be substituted for one another. But it must be manifest in the two expressions themselves whether this is the case or not. When two expressions can be substituted for one another, that characterizes their logical form.
It is a property of affirmation that it can be construed as double negation. It is a property of '1 + 1 + 1 + 1' that it can be construed as '(1 + 1) + (1 + 1)'.
Frege says that the two expressions have the same meaning but different senses. But the essential point about an equation is that it is not necessary in order to show that the two expressions connected by the sign of equality have the same meaning, since this can be seen from the two expressions themselves. And the possibility of proving the propositions of mathematics means simply that their correctness can be perceived without its being necessary that what they express should itself be compared with the facts in order to determine its correctness.
It is impossible to assert the identity of meaning of two expressions. For in order to be able to assert anything about their meaning, I must know their meaning, and I cannot know their meaning without knowing whether what they mean is the same or different. An equation merely marks the point of view from which I consider the two expressions, it marks their equivalence in meaning.
Intuition is needed to solve mathematical, but language itself provides the necessary intuition. The process of calculating serves to bring about that intuition. Calculation is not an experiment.
One can describe all true logical propositions in advance.
It is possible, even according to the old conception of logic, to describe all true logical propositions in advance. Hence there can never be surprises in logic.
One can calculate whether a proposition belongs to logic, by calculating the logical properties of the symbol. This is how one proves a logical proposition. For, without bothering about sense or meaning, we construct the logical proposition out of others using only rules that deal with signs. One proves logical propositions by generating them from logical propositions by successively applying certain operations that always generate further tautologies out of the initial ones. (And in fact only tautologies follow from a tautology.) Of course this way of showing that the propositions of logic are tautologies is not at all essential to logic, if only because the propositions from which the proof starts must show without any proof that they are tautologies.
In logic process and result are equivalent. (Hence no surprise.) Proof in logic is merely a mechanical expedient to facilitate the recognition of tautologies in complicated cases. It would be altogether too remarkable if a proposition that had sense could be proved logically from others, and so too could a logical proposition. It is clear from the start that a logical proof of a proposition that makes sense and a proof in logic must be two entirely different things.
A proposition that makes sense states something, which is shown by its proof to be so. In logic every proposition is the form of a proof. Every proposition of logic is a modus ponens represented in signs. (And one cannot express the modus ponens by means of a proposition.) It is always possible to construe logic so that every proposition is its own proof.
All the propositions of logic are of equal status: it is not the case that some of them are essentially derived propositions. Every tautology itself shows that it is a tautology.
The number of primitive propositions of logic is arbitrary, since one could derive logic from a single primitive proposition, e.g. by simply constructing the logical product of Frege's primitive propositions. (Frege would perhaps say that we should then no longer have an immediately self-evident primitive proposition. But it is remarkable that a thinker as rigorous as Frege appealed to the degree of self-evidence as the criterion of a logical proposition.)
One can calculate whether a proposition belongs to logic, by calculating the logical properties of the symbol. This is how one proves a logical proposition. For, without bothering about sense or meaning, we construct the logical proposition out of others using only rules that deal with signs. One proves logical propositions by generating them from logical propositions by successively applying certain operations that always generate further tautologies out of the initial ones. (And in fact only tautologies follow from a tautology.) Of course this way of showing that the propositions of logic are tautologies is not at all essential to logic, if only because the propositions from which the proof starts must show without any proof that they are tautologies.
In logic process and result are equivalent. (Hence no surprise.) Proof in logic is merely a mechanical expedient to facilitate the recognition of tautologies in complicated cases. It would be altogether too remarkable if a proposition that had sense could be proved logically from others, and so too could a logical proposition. It is clear from the start that a logical proof of a proposition that makes sense and a proof in logic must be two entirely different things.
A proposition that makes sense states something, which is shown by its proof to be so. In logic every proposition is the form of a proof. Every proposition of logic is a modus ponens represented in signs. (And one cannot express the modus ponens by means of a proposition.) It is always possible to construe logic so that every proposition is its own proof.
All the propositions of logic are of equal status: it is not the case that some of them are essentially derived propositions. Every tautology itself shows that it is a tautology.
The number of primitive propositions of logic is arbitrary, since one could derive logic from a single primitive proposition, e.g. by simply constructing the logical product of Frege's primitive propositions. (Frege would perhaps say that we should then no longer have an immediately self-evident primitive proposition. But it is remarkable that a thinker as rigorous as Frege appealed to the degree of self-evidence as the criterion of a logical proposition.)
We can do without logical propositions.
The propositions of logic demonstrate the logical properties of propositions in that they combine them to form propositions without content. This could also be called a null method. In a logical proposition, propositions are brought into equilibrium with one another, and that equilibrium then indicates what the logical constitution of these propositions must be.
It follows from this that we can even do without logical propositions because a suitable notation enables one to recognize the formal properties of propositions by inspection. If, for example, two propositions (p) and (q) in the compound proposition (p⊃q) yield a tautology, then it is clear that (q) follows from (p). That (q) follows from (p⊃q.p) is seen from the two propositions themselves, but it is also possible to show it by combining them to form (p⊃q.p:⊃:q), and then showing that this is a tautology.
Logical propositions cannot be confirmed by experience any more than they can be refuted by it. Not only must a proposition of logic be irrefutable by any possible experience, but it must also be unconfirmable by any possible experience. We can postulate the truths of logic in sthat we can postulate an adequate notation. Clearly: the laws of logic cannot themselves be subject to laws of logic. (There is not, as Russell thought, a special law of contradiction for each 'type'; one law is enough instead, since it is not applied to itself.)
The mark of a logical proposition is not general validity, since that only means to be valid for all things by happenstance. An ungeneralized proposition can just as well be tautological as a generalized one.
We could call logical generality essential, in contrast with the accidental generality of such propositions as 'All men are mortal'. Propositions like Russell's 'axiom of reducibility' are not logical propositions, and this explains our feeling that, even if they were true, their truth could only be the result of a fortunate accident.
One can imagine a world in which the axiom of reducibility is not valid. But it is clear that logic has nothing to do with the question of whether our world is really so or not.
Logical propositions describe the structural skeleton of the world. They have no content on their own, but presuppose that names have meaning and elementary propositions make sense; and that connects them to the world. Clearly, it must show something about the world that certain conjunctions of symbols—that in essence have a specific character—are tautologies. This is a decisive point.
Some things are arbitrary in the symbols that we use and some things are not. In logic, only the latter express. That does not mean we express what we wish with the help of signs, but rather that one in which the nature of the absolutely necessary signs speaks for itself. If we know the logical syntax of any symbolic language, then we have already been given all the propositions of logic.
It follows from this that we can even do without logical propositions because a suitable notation enables one to recognize the formal properties of propositions by inspection. If, for example, two propositions (p) and (q) in the compound proposition (p⊃q) yield a tautology, then it is clear that (q) follows from (p). That (q) follows from (p⊃q.p) is seen from the two propositions themselves, but it is also possible to show it by combining them to form (p⊃q.p:⊃:q), and then showing that this is a tautology.
Logical propositions cannot be confirmed by experience any more than they can be refuted by it. Not only must a proposition of logic be irrefutable by any possible experience, but it must also be unconfirmable by any possible experience. We can postulate the truths of logic in sthat we can postulate an adequate notation. Clearly: the laws of logic cannot themselves be subject to laws of logic. (There is not, as Russell thought, a special law of contradiction for each 'type'; one law is enough instead, since it is not applied to itself.)
The mark of a logical proposition is not general validity, since that only means to be valid for all things by happenstance. An ungeneralized proposition can just as well be tautological as a generalized one.
We could call logical generality essential, in contrast with the accidental generality of such propositions as 'All men are mortal'. Propositions like Russell's 'axiom of reducibility' are not logical propositions, and this explains our feeling that, even if they were true, their truth could only be the result of a fortunate accident.
One can imagine a world in which the axiom of reducibility is not valid. But it is clear that logic has nothing to do with the question of whether our world is really so or not.
Logical propositions describe the structural skeleton of the world. They have no content on their own, but presuppose that names have meaning and elementary propositions make sense; and that connects them to the world. Clearly, it must show something about the world that certain conjunctions of symbols—that in essence have a specific character—are tautologies. This is a decisive point.
Some things are arbitrary in the symbols that we use and some things are not. In logic, only the latter express. That does not mean we express what we wish with the help of signs, but rather that one in which the nature of the absolutely necessary signs speaks for itself. If we know the logical syntax of any symbolic language, then we have already been given all the propositions of logic.
Recognizing a Tautology.
In order to recognize a tautology in cases where no generality-sign occurs in it, one can employ the following method.
Instead of 'p', 'q', 'r', etc. one writes elemental propositions as 'WpF', 'WqF', 'WrF', etc. ('Wahr' is 'True' in German.)
One expresses combinations using brackets, e.g.
Lines connect the truth or falsity of the compound proposition with the truth value combinations that are its arguments.
This sign, for instance, represents the proposition (p⊃q) 'p implies q'.
Now, one can examine the proposition ~(p .~p) (the law of contradiction) in order to determine whether it is a tautology.
In our notation the form '~ξ' is written as
and the form 'ξ.η' as:
Hence, one writes the form ~(~p .q) as
When one substitutes 'p' for 'q' and examines how the outermost T and F connect to the innermost, the result will be that the truth of the whole proposition is correlated with all the combinations of its argument, and its falsity with none of them.
Instead of 'p', 'q', 'r', etc. one writes elemental propositions as 'WpF', 'WqF', 'WrF', etc. ('Wahr' is 'True' in German.)
One expresses combinations using brackets, e.g.
Lines connect the truth or falsity of the compound proposition with the truth value combinations that are its arguments.
This sign, for instance, represents the proposition (p⊃q) 'p implies q'.
Now, one can examine the proposition ~(p .~p) (the law of contradiction) in order to determine whether it is a tautology.
In our notation the form '~ξ' is written as
and the form 'ξ.η' as:
Hence, one writes the form ~(~p .q) as
When one substitutes 'p' for 'q' and examines how the outermost T and F connect to the innermost, the result will be that the truth of the whole proposition is correlated with all the combinations of its argument, and its falsity with none of them.
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