Complex and imaginary numbers have grown organically within mathematics, and they have proved their mettle in scientific applications, but pseudo-irrationals are inorganic creations invented solely for the sake of mistaken foundationalist aims. In his middle period, Wittgenstein begins a full-out assault on set theory that never abates. Late in the middle period, Wittgenstein seems to become more aware of the unbearable conflict between his strong formalism PG and his denigration of set theory as a purely formal, non -mathematical calculus Rodych — , which, as we shall see in Section 3.
Set theory attempts to grasp the infinite at a more general level than the investigation of the laws of the real numbers. Let the infinite accommodate itself in this box as best it can. PG ; cf. Only after does he provide concrete arguments purporting to show, e. Nonetheless, the intermediate Wittgenstein clearly rejects the notion that a non-denumerably infinite set is greater in cardinality than a denumerably infinite set.
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The set is of a different kind. It is nonsense, he says, to go from the warranted conclusion that these numbers are not, in principle, enumerable to the conclusion that the set of transcendental numbers is greater in cardinality than the set of algebraic numbers, which is recursively enumerable. What we have here are two very different conceptions of a number-type.
In the case of transcendental numbers, on the other hand, we have proofs that some numbers are transcendental i. Though the intermediate Wittgenstein certainly seems highly critical of the alleged proof that some infinite sets e. As we shall see in Section 3. For this reason and because some manuscripts containing much material on mathematics e. It must be emphasized, therefore, that this Encyclopedia article is being written during a transitional period.
Nothing exists mathematically unless and until we have invented it. In arguing against mathematical discovery, Wittgenstein is not just rejecting Platonism, he is also rejecting a rather standard philosophical view according to which human beings invent mathematical calculi, but once a calculus has been invented, we thereafter discover finitely many of its infinitely many provable and true theorems. At MS 3v; Oct.
MS , ; March 15, There are no mathematical facts just as there are no genuine mathematical propositions. Repeating his intermediate view, the later Wittgenstein says MS , 71v; 27 Dec. Platonism is dangerously misleading , according to Wittgenstein, because it suggests a picture of pre -existence, pre determination and discovery that is completely at odds with what we find if we actually examine and describe mathematics and mathematical activity.
Wittgenstein, however, does not endeavour to refute Platonism. His aim, instead, is to clarify what Platonism is and what it says, implicitly and explicitly including variants of Platonism that claim, e. The first, and perhaps most definitive, indication that the later Wittgenstein maintains his finitism is his continued and consistent insistence that irrational numbers are rules for constructing finite expansions, not infinite mathematical extensions. When we say, e. What harm is done e. Or: that they are already there, even though we only know certain of them?
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Why are these pictures not harmless? But can his omniscience decide whether they would have reached it after the end of the world? It cannot. I want to say: Even God can determine something mathematical only by mathematics.
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Even for him the mere rule of expansion cannot decide anything that it does not decide for us. Could God have known this, without the calculation, purely from the rule of expansion? I want to say: No. MS , pp. Weyl [ 97]. Like us, with our modest minds, an omniscient mind i. On this interpretation, the later Wittgenstein precludes undecidable mathematical propositions, but he allows that some undecided expressions are propositions of a calculus because they are decidable in principle i.
There is considerable evidence, however, that the later Wittgenstein maintains his intermediate position that an expression is a meaningful mathematical proposition only within a given calculus and iff we knowingly have in hand an applicable and effective decision procedure by means of which we can decide it. Furthermore, as we have just seen, Wittgenstein rejects PIC as a non-proposition on the grounds that it is not algorithmically decidable, while admitting finitistic versions of PIC because they are algorithmically decidable.
Since set theory was invented to provide mathematics with a foundation, it is, minimally, unnecessary. Even if set theory is unnecessary, it still might constitute a solid foundation for mathematics. The difference in kind between the two conceptions is represented, by a skew form of expression, as difference of extension. I believe, and hope, that a future generation will laugh at this hocus pocus. That is, the proof proves non-enumerability : it proves that for any given definite real number concept e. I call number-concept X non-denumerable if it has been stipulated that, whatever numbers falling under this concept you arrange in a series, the diagonal number of this series is also to fall under that concept.
As Wittgenstein says at MS , 71r; Dec.
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If you now call the Cantorian procedure one for producing a new real number, you will now no longer be inclined to speak of a system of all real numbers. As we have seen, this criterion was present in the Tractatus 6. The reason for this absence is probably that the intermediate Wittgenstein wanted to stress that in mathematics everything is syntax and nothing is meaning. There seem to be two reasons why the later Wittgenstein reintroduces extra-mathematical application as a necessary condition of a mathematical language-game.
But if we do say it—what are we to do next? In what practice is this proposition anchored? It is for the time being a piece of mathematical architecture which hangs in the air, and looks as if it were, let us say, an architrave, but not supported by anything and supporting nothing. It must be emphasized, however, that the later Wittgenstein still maintains that the operations within a mathematical calculus are purely formal, syntactical operations governed by rules of syntax i. To say mathematics is a game is supposed to mean: in proving, we need never appeal to the meaning [ Bedeutung ] of the signs, that is to their extra-mathematical application.
Hence, the question whether a concatenation of signs is a proposition of a given mathematical calculus i. The first thing to note, therefore, about RFM App.
Thus, at RFM App. Must I not say that this proposition on the one hand is true, and on the other hand is unprovable?
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For suppose it were false; then it is true that it is provable. And that surely cannot be! And if it is proved, then it is proved that it is not provable. Thus it can only be true, but unprovable. The reasoning here is a double reductio. On this natural interpretation of RFM App. Given his syntactical conception of mathematics, even with the extra-mathematical application criterion, he would simply say that P , qua expression syntactically independent of PM , is not a proposition of PM , and if it is syntactically independent of all existent mathematical language-games, it is not a mathematical proposition.
Moreover, there seem to be no compelling non-semantical reasons—either intra-systemic or extra-mathematical—for Wittgenstein to accommodate P by including it in PM or by adopting a non-syntactical conception of mathematical truth such as Tarski-truth Steiner What will distinguish the mathematicians of the future from those of today will really be a greater sensitivity, and that will—as it were—prune mathematics; since people will then be more intent on absolute clarity than on the discovery of new games.
Philosophical clarity will have the same effect on the growth of mathematics as sunlight has on the growth of potato shoots. In a dark cellar they grow yards long. A mathematician is bound to be horrified by my mathematical comments, since he has always been trained to avoid indulging in thoughts and doubts of the kind I develop.
He has learned to regard them as something contemptible and… he has acquired a revulsion from them as infantile. That is to say, I trot out all the problems that a child learning arithmetic, etc. I say to those repressed doubts: you are quite correct, go on asking, demand clarification! PG , In his middle and later periods, Wittgenstein believes he is providing philosophical clarity on aspects and parts of mathematics, on mathematical conceptions, and on philosophical conceptions of mathematics.
Lacking such clarity and not aiming for absolute clarity, mathematicians construct new games, sometimes because of a misconception of the meaning of their mathematical propositions and mathematical terms.
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Education and especially advanced education in mathematics does not encourage clarity but rather represses it—questions that deserve answers are either not asked or are dismissed. Mathematicians of the future, however, will be more sensitive and this will repeatedly prune mathematical extensions and inventions, since mathematicians will come to recognize that new extensions and creations e.
Brouwer, Luitzen Egbertus Jan Frege, Gottlob mathematics, philosophy of: formalism mathematics, philosophy of: intuitionism mathematics: constructive private language Russell, Bertrand Wittgenstein, Ludwig Wittgenstein, Ludwig: logical atomism. Wittgenstein on Mathematics in the Tractatus 2. Non-Denumerability 3.
According to Wittgenstein, we ascertain the truth of both mathematical and logical propositions by the symbol alone i. Another crucial aspect of the Tractarian theory of mathematics is captured in 6.
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Mathematics as Human Invention: According to the middle Wittgenstein, we invent mathematics, from which it follows that mathematics and so-called mathematical objects do not exist independently of our inventions. Whatever is mathematical is fundamentally a product of human activity. Mathematical Calculi Consist Exclusively of Intensions and Extensions: Given that we have invented only mathematical extensions e.
Put succinctly, Wittgenstein thinks that the extension of this notion of concept-and-extension from the domain of existent i. See 1 just below. An infinite mathematical extension i. Algorithmic Decidability vs. Moreover, since mathematics is essentially what we have and what we know, Wittgenstein restricts algorithmic decidability to knowing how to decide a proposition with a known decision procedure.