Philosophie dürften man eigentlich nur dichten.

  The English translation of this is: “Philosophy ought really to be written only as a poetic composition.” The translation of dichten seems to me mistaken. I interpret this as:

  Philosophical writing really ought to be fiction.

  This is supported by Goethe's autobiographical work Wahrheit und Dichtung which is usually translated as “Truth and Fiction”. This was published already in 1833.

  6.2.2. Language

  Often it is claimed that the later Wittgenstein differs from the early one. This is only partially true, but some aspects are explicitly brought up by himself.

  Elementary propositions are not independent as claimed in the Tractatus [10] In fact, in a system the assignment of a property implies the denial of other properties of the system. Logical attributes have inner connections. To consider a “fact” as a “complex of objects” is misleading [82]. Logical products are not hidden in sentences

  It is wrong to think that analysis is to bring out what is hidden. What is “hidden” depends on the method of discovery. Elementary propositions are conditioned by each other.

  Expecting identity in form between thought and reality is an error. However, thought must have the same logical form as reality. An object cannot have two colors. This is the negation of an a priori proposition; hence it is a priori itself. In quantum physics, this may be compared to the complementary properties.

  Wittgenstein is indifferent to solutions of purely scientific problems. He is not concerned with facts of physics except in so far as they determine the laws governing how things appear. Physics strives to produce “truth” but our descriptions require sense.

  (p.173) Compare the giving of information by maps and drawings with the giving of information by sentences. The sentences are no more arbitrary than the drawings are; only the words are arbitrary. On the other hand the projection of the map is arbitrarily devised by us.

  If we want to study problems of truth and falsehood, agreement and disagreement of propositions with reality, the nature of assertions, assumptions, and questions, we should look at language, without the confusing background of any complicated processes of thought.

  In every serious philosophical question, uncertainty extends to the very roots of the problem. The rule‐governed nature of our language permeates our lives.

  Language contains the same traps for everyone as long as there are words like: “to be”, “identical”, “true”, “false”, “possible”, as well as “flow of time”.

  The terms used in discourse have the haziness of concept‐words in our language.

  We cannot achieve greater generality in philosophy than in life and science. When we talk about language one must use the language of practice. How strange that we can do with the one we have.

  A proposition is conceived as a picture of the state of affairs; it describes how things stand if it is true. Thus it shows the possibility of the asserted state. It can set forth what is not the case. It depends on our grammar what is possible and what is not, i.e. what grammar permits. “Possible” means conceivable. But possibility does not yet guarantee existence. How strange that we should be able to say that a state of affairs is inconceivable. Looking somehow presupposes that I know what I am looking for, without what I am looking for having to exist.

  What expectation has to do with reality is that it refers to another point in the same space. You can only search in a space, because only in a space do you stand in relation to where you are not. How do we describe that which does not exist?

  All forms of speech are taken from the ordinary physical language and cannot be used in epistemology or phenomenology without casting a distorting light on their objects. To say that a straight line can be drawn is to say that it makes sense to talk about this line.

  Anything can picture anything. Every projection must have something in common with what is projected. Wittgenstein uses the word “projection” for what physics terms “correspondence rule”.

  Language cannot express what belongs to the essence of the world, it can only say what we also could imagine differently. The self‐evidence of the world is expressed in the very fact that language refers only to itself and can mean only it. Wittgenstein uses early the term “systems of communication”: Does this give “language games”?

  Grammar is not accountable to any reality. It is grammatical rules that constitute meaning, and so they are not themselves answerable to any meaning and to that extent they are arbitrary. The rules of grammar are arbitrary in the same sense as the choice of unit of measurements. Mathematical texts are (p.174) not descriptions of “something”; they are the thing itself. Mathematics consists only in algebraic operations and derives no meaning from its outside. We cannot describe mathematics; we can only do it.

  To invent a language may be to invent an instrument for a particular purpose on the basis of the laws of nature. The concept of language is conditioned on the concept of communication.

  Laws of logic are arbitrary. We can have different rules of “all”, “some”, “not”, and “any”. They have different meaning if applied to cardinals or reals. Our world looks quite different if we surround it with different possibilities. The difficulty in philosophy is to find one's way about. There is no definite class of features characterizing all uses of the word “can”.

  Wittgenstein introduces the central concept of “logical multiplicity”: The negation of a proposition describes the same situation but with different polarity. When a negation is meaningful, we have to know what is negated.

  A negative proposition has the same multiplicity as its positive counterpart. According to Wittgenstein, they make just as much sense. To know what is the case, we need to know also what is implied not to be the case. The superfluous part of the language, Wittgenstein terms “wheels turning idly”.

  A symbolic system with the right multiplicity will render syntax superfluous. On the other hand, the syntax may render symbols superfluous. An incomplete symbolic system can always be amended by adding rules of syntax. Only physics can tell how many concepts are needed.

  The multiplicity corresponds not to one verification but to the law obeyed by verification. Two propositions must have the same sense if every possible experience confirming the one also confirms the other. Your knowledge about the general fact has the same multiplicity as your reason. In order to guide one's action, the negation must have the same multiplicity as the proposition it denies.

  The grammatical rule for the terms of a general proposition must contain the multiplicity of possible causes provided for by the proposition.

  The best comparison for every hypothesis is a body in relation to a systematic series of views from different angles.

  It is always single faces of a hypothesis that are verified. If that face is laid against reality the hypothesis becomes a proposition.

  A command must have the right logical multiplicity, i.e. the same as its actions. Controls must have the multiplicity of what the machine is capable of making. To expect more is to talk nonsense in language. Expectations and the verifications have to have the same multiplicity.

  Epistemology pays no attention to the truth or falsity of contingent propositions. Epistemology highlights everything that can be thought.

  6.2.3. On objects

  By correlating the names to facts, one knows how many entities there are. If I write “149” on the board, have I written one letter or two? Depending on whether “A” means this shape or the character.

  (p.175) if “objects” are elements of representation, there cannot be ∞ many of them. The formula for real numbers is not to say that it holds for all real numbers, but that given an arbitrary real number, it holds of this. I can say “Wipe the table!” but not “Wipe all the points of the table!”

  Points cannot be seen and singled out. Euclid never stated that lines consisted of points; they were logically independent entities. Two lines define a point, but no property defines all points. The point of intersection of two lines is not a common member of two classes of points, it is the meeting of two laws.

  According to Wittgenstein, the integers can be used to transfer a property through induction, which can be represented formally as

  (8)

  This does not, however, justify

  (9)

  Wittgenstein says that this assumption is one of his early mistakes eliminated later. The statement

  is, however, equivalent with

  (10)

  The form (10) allows that there are other elements x in addition to A, the form (9) does not. For the former it makes sense to add

  but if A as a set is complete, it tells everything. This opens the question of the conjugate set, is it possibly empty? Only if there exists a set E ≠ ∅ ≡ {} and A ∈ E, can we define a complement

  If we list A we do not need to say that “there are no others”. If à is nonempty the set

  has got “extension”, otherwise not.

  This relates to the meaning of negation. Namely, stating P, we can give meaning to ¬P only if we may construct a set

  It is not sufficient to state

  i.e. x does not imply that P is not true. We have to verify that P is false.

  (p.176) The meaning of a proposition is its mode of verification. The meaning of a word is its use.

  The subject constructs his formal space but the result is objective. In physics space is Space. Two elementary propositions cannot contradict each other. A classical point mass can have only one velocity at a time. There can be only one charge at a point in an electric field, only one temperature on a surface, the body can have only one temperature. No one can doubt that these are self‐evident and their denial is contradictory.

  It is not a proposition we put against reality as a meter stick, it is a system of propositions.

  6.2.4. Signs and symbols

  Wittgenstein introduces “shadow of a fact” = “proposition and its sense”.

  He also introduces the concepts of “sign”, which is only a (simple) figure, a scratch or sound, and “symbol”, which carries a meaning. This attributes an “intention” when it is used. The significance of a sign can only derive from its participation in a symbol. It is significant only within a system. The genuine criterion for a formal structure is precisely which propositions make sense for it—not which are true.

  The place of a symbol in language is given by its use. A map is a picture of reality only through its intention. Explanation or experience can only add to the meaning of the symbol. The word says nothing about the world; a proposition says something and thus can be false. We understand a symbol only as part of a system described by its grammar. The symbol is a sign together with all the conditions necessary to give it significance. Understanding the symbol does not tell if it is true or false, for this you need the fact too. There are no necessary facts. Everything that can be described must be capable of being symbolized. Grammar gives the rules for the combination of symbols.

  Imagine someone who knows his way around the city perfectly, i.e. finds the shortest way, but is unable to draw a map of the city. Understanding is effected by explanation but also by training. Knowledge by words is not a translation of something that was there before. What is the language game of “to know”?

  Does a border case mean that the word “plant”, in all cases, is infected by uncertainty? Would an exact definition clear up the ordinary use?

  The representation ought only to contain adequate concepts. If they are not needed for the use of the conceptual system, they are Wittgenstein's “wheels turning idly”.

  If there are several languages; what they have in common is then what they depict. What is invariant under change of representation is a fact. This suggests the “invariants” of Eino Kaila and Felix Klein.

  Wittgenstein also discusses “external” and “internal” relations. Internal relations derive from the grammar of a concept, whereas external ones relate (p.177) to pairs of objects (or larger collections). A concept is not essentially a predicate. An internal relation lies in the character of things. It is never a relation between two objects but a relation between two concepts. The electron's internal relations are correlations of charge, spin, and rest mass. Their interactions are external.

  The method of answering a question tells what the question is about. Where you can ask a question, look for an answer; where you cannot look for an answer, you cannot ask either. Nor can you find an answer. Only where there is a method of solution is there a question. Do not ask for a definition, get clear about the grammar. To construct a language that covers all situations is a conceptual absurdity.

  A philosophical problem is of the form: “I do not know my way out.” Such a philosophical problem must be completely solvable. “Realism” and “Idealism” are metaphysical terms; they try to say something about the essence of the world.

  The solution to the riddle of life may not make life easier. It was possible to live even before it was recognized. If there were a result of logic, even before this was discovered it was possible to think.

  6.2.5. Procedures of mathematics

  Wittgenstein tries to show that a mathematical discovery is rather a mathematical invention. Solution of a mathematical problem never helps in philosophy.

  Russel's calculus is a bit of mathematics. It is possible as operations even without the formulation in terms of classes. Philosophy may not interfere with the actual use of any language; it can only describe it. It cannot give it any foundations either, A “leading problem of mathematical logic” is a problem of mathematics like any other.

  Tables, ostensive definitions and similar instruments we may call “rules”. The use of a rule is explained by a further rule. In mathematics everything is algoritm and nothing is meaning. Set theory preserves the picture of something discrete but fails to carry the argument to the finish:

  Number is an attribute of a function defining a class, not a property of the extension.

  Fitting the number of primes to a logarithm is like an experiment in physics. We must think of “number” as we think of “length”, or “weight” and of counting or correlating as we think of weighing or measuring. We may say that the body does not have any weight at all except when being registered, or no definite weight except when being weighted.

  By the means of our intellect we look into a certain realm and see that the propositions of logic are true. This makes logic the science of the intellectual realm. Number is an external property of a concept and an internal property of its extension (the list of objects that fall under it). Extension of a concept is a list.

  (p.178) Knowing how to go on must be based on prior information. Complete induction is not expressible as a proposition:

  Let us define the scheme A by the operations:

  In words: the property tested for by f (n) is true for all values of n. The scheme A is not a proposition because it cannot be negated; what would be the meaning of ¬A?

  It is impossible to extend a system; many propositions in the extended system lack meaning in the original system. Thus it is not just a “completion”. We know as much about our formalisms as God does. What we do not have, we do not have. There are surprises in reality but not in language. Language can expand. Is it an accident that in order to complete the sign system we have to go outside the written and spoken language? Wittgenstein is presumably thinking of extensions of the number system. The integer “2” is not the same entity as the real number denoted in the same way.

  The actual infinity is a mere word. What can it mean to say that a straight line can be arbitrarily prolonged? Further: there is no such thing as an infinite list.

  The idea of an infinite chain of reasons follows from a confusion like: A line of certain length consists of an infinite number of points because it is indefinitely divisible. But no algorithm can produce all its points, including every irrational one.

  Proofs of equal cardinality by matching figures are like experiments. If the patterns do not match, we seem to have learned something about reality.

  A contradiction is manifest only if encountered. Hidden contradictions are not to be worried about as long as the game can go on. A contradiction does no harm until it is made to operate. Language games are languages complete in themselves. We have not encountered contradictions in mathematics. A mathematical statement has no sense before being proved true or false within some system. What is not proved is not known.

  A proof implies only what it proves. In mathematics different proofs prove different things. A proof of existence gives the meaning of “existence” in that theorem. Of course, we may not and cannot deny the essential existence of anything. Some philosophizing mathematicians are not aware of the many different usage of the word “proof”. All our collected knowledge fails to prove God's existence or lack thereof.

  Mathematics is a game. Since mathematics is a calculus, and hence is not really about anything, there is no metamathematics. Metamathematics is mathematics in disguise. One cannot compare two classes of consequences unless one forms a new system in which both classes occur.

  (p.179) To someone saying that he has discovered that between the rational points of a line there are more points, we reply that he has not discovered new points but he has constructed the new points. How do we know that a theorem proved by transfinite methods cannot be violated by a concrete example? This is the mathematical problem of consistency.

 

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