Platonist Assertory Mathematics. Thought of the Day 88.0

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Traditional Platonism, according to which our mathematical theories are bodies of truths about a realm of mathematical objects, assumes that only some amongst consistent theory candidates succeed in correctly describing the mathematical realm. For platonists, while mathematicians may contemplate alternative consistent extensions of the axioms for ZF (Zermelo–Fraenkel) set theory, for example, at most one such extension can correctly describe how things really are with the universe of sets. Thus, according to Platonists such as Kurt Gödel, intuition together with quasi-empirical methods (such as the justification of axioms by appeal to their intuitively acceptable consequences) can guide us in discovering which amongst alternative axiom candidates for set theory has things right about set theoretic reality. Alternatively, according to empiricists such as Quine, who hold that our belief in the truth of mathematical theories is justified by their role in empirical science, empirical evidence can choose between alternative consistent set theories. In Quine’s view, we are justified in believing the truth of the minimal amount of set theory required by our most attractive scientific account of the world.

Despite their differences at the level of detail, both of these versions of Platonism share the assumption that mere consistency is not enough for a mathematical theory: For such a theory to be true, it must correctly describe a realm of objects, where the existence of these objects is not guaranteed by consistency alone. Such a view of mathematical theories requires that we must have some grasp of the intended interpretation of an axiomatic theory that is independent of our axiomatization – otherwise inquiry into whether our axioms “get things right” about this intended interpretation would be futile. Hence, it is natural to see these Platonist views of mathematics as following Frege in holding that axioms

. . . must not contain a word or sign whose sense and meaning, or whose contribution to the expression of a thought, was not already completely laid down, so that there is no doubt about the sense of the proposition and the thought it expresses. The only question can be whether this thought is true and what its truth rests on. (Frege to Hilbert Gottlob Frege The Philosophical and Mathematical Correspondence)

On such an account, our mathematical axioms express genuine assertions (thoughts), which may or may not succeed in asserting truths about their subject matter. These Platonist views are “assertory” views of mathematics. Assertory views of mathematics make room for a gap between our mathematical theories and their intended subject matter, and the possibility of such a gap leads to at least two difficulties for traditional Platonism. These difficulties are articulated by Paul Benacerraf (here and here) in his aforementioned papers. The first difficulty comes from the realization that our mathematical theories, even when axioms are supplemented with less formal characterizations of their subject matter, may be insufficient to choose between alternative interpretations. For example, assertory views hold that the Peano axioms for arithmetic aim to assert truths about the natural numbers. But there are many candidate interpretations of these axioms, and nothing in the axioms, or in our wider mathematical practices, seems to suffice to pin down one interpretation over any other as the correct one. The view of mathematical theories as assertions about a specific realm of objects seems to force there to be facts about the correct interpretation of our theories even if, so far as our mathematical practice goes (for example, in the case of arithmetic), any ω-sequence would do.

Benacerraf’s second worry is perhaps even more pressing for assertory views. The possibility of a gap between our mathematical theories and their intended subject matter raises the question, “How do we know that our mathematical theories have things right about their subject matter?”. To answer this, we need to consider the nature of the purported objects about which our theories are supposed to assert truths. It seems that our best characterization of mathematical objects is negative: to account for the extent of our mathematical theories, and the timelessness of mathematical truths, it seems reasonable to suppose that mathematical objects are non-physical, non- spatiotemporal (and, it is sometimes added, mind- and language-independent) objects – in short, mathematical objects are abstract. But this negative characterization makes it difficult to say anything positive about how we could know anything about how things are with these objects. Assertory, Platonist views of mathematics are thus challenged to explain just how we are meant to evaluate our mathematical assertions – just how do the kinds of evidence these Platonists present in support of their theories succeed in ensuring that these theories track the truth?

Carnap, c-notions. Thought of the Day 87.0

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A central distinction for Carnap is that between definite and indefinite notions. A definite notion is one that is recursive, such as “is a formula” and “is a proof of φ”. An indefinite notion is one that is non-recursive, such as “is an ω-consequence of PA” and “is true in Vω+ω”. This leads to a distinction between (i) the method of derivation (or d-method), which investigates the semi-definite (recursively enumerable) metamathematical notions, such as demonstrable, derivable, refutable, resoluble, and irresoluble, and (ii) the method of consequence (or c-method), which investigates the (typically) non-recursively enumerable metamathematical notions such as consequence, analytic, contradictory, determinate, and synthetic.

A language for Carnap is what we would today call a formal axiom system. The rules of the formal system are definite (recursive) and Carnap is fully aware that a given language cannot include its own c-notions. The logical syntax of a language is what we would today call metatheory. It is here that one formalizes the c-notions for the (object) language. From among the various c-notions Carnap singles out one as central, namely, the notion of (direct) consequence; from this c-notion all of the other c-notions can be defined in routine fashion.

We now turn to Carnap’s account of his fundamental notions, most notably, the analytic/synthetic distinction and the division of primitive terms into ‘logico-mathematical’ and ‘descriptive’. Carnap actually has two approaches. The first approach occurs in his discussion of specific languages – Languages I and II. Here he starts with a division of primitive terms into ‘logico-mathematical’ and ‘descriptive’ and upon this basis defines the c-notions, in particular the notions of being analytic and synthetic. The second approach occurs in the discussion of general syntax. Here Carnap reverses procedure: he starts with a specific c-notion – namely, the notion of direct consequence – and he uses it to define the other c-notions and draw the division of primitive terms into ‘logico-mathematical’ and ‘descriptive’.

In the first approach Carnap introduces two languages – Language I and Language II. The background languages (in the modern sense) of Language I and Language II are quite general – they include expressions that we would call ‘descriptive’. Carnap starts with a demarcation of primitive terms into ‘logico-mathematical’ and ‘descriptive’. The expressions he classifies as ‘logico-mathematical’ are exactly those included in the modern versions of these systems; the remaining expressions are classified as ‘descriptive’. Language I is a version of Primitive Recursive Arithmetic and Language II is a version of finite type theory built over Peano Arithmetic. The d-notions for these languages are the standard proof-theoretic ones.

For Language I Carnap starts with a consequence relation based on two rules – (i) the rule that allows one to infer φ if T \vdash \!\, φ (where T is some fixed ∑10-complete formal system) and (ii) the ω-rule. It is then easily seen that one has a complete theory for the logico-mathematical fragment, that is, for any logico-mathematical sentence φ, either φ or ¬φ is a consequence of the null set. The other c-notions are then defined in the standard fashion. For example, a sentence is analytic if it is a consequence of the null set; contradictory if its negation is analytic; and so on.

For Language II Carnap starts by defining analyticity. His definition is a notational variant of the Tarskian truth definition with one important difference – namely, it involves an asymmetric treatment of the logico-mathematical and descriptive expressions. For the logico-mathematical expressions his definition really just is a notational variant of the Tarskian truth definition. But descriptive expressions must pass a more stringent test to count as analytic – they must be such that if one replaces all descriptive expressions in them by variables of the appropriate type, then the resulting logico-mathematical expression is analytic, that is, true. In other words, to count as analytic a descriptive expression must be a substitution-instance of a general logico-mathematical truth. With this definition in place the other c-notions are defined in the standard fashion.

The content of a sentence is defined to be the set of its non-analytic consequences. It then follows immediately from the definitions that logico-mathematical sentences (of both Language I and Language II) are analytic or contradictory and (assuming consistency) that analytic sentences are without content.

In the second approach, for a given language, Carnap starts with an arbitrary notion of direct consequence and from this notion he defines the other c-notions in the standard fashion. More importantly, in addition to defining the other c-notion, Carnap also uses the primitive notion of direct consequence (along with the derived c-notions) to effect the classification of terms into ‘logico-mathematical’ and ‘descriptive’. The guiding idea is that “the formally expressible distinguishing peculiarity of logical symbols and expressions [consists] in the fact that each sentence constructed solely from them is determinate”. Howsoever the guiding idea is implemented the actual division between “logico-mathematical” and “descriptive” expressions that one obtains as output is sensitive to the scope of the direct consequence relation with which one starts.

With this basic division in place, Carnap can now draw various derivative divisions, most notably, the division between analytic and synthetic statements: Suppose φ is a consequence of Γ. Then φ is said to be an L-consequence of Γ if either (i) φ and the sentences in Γ are logico-mathematical, or (ii) letting φ’ and Γ’ be the result of unpacking all descriptive symbols, then for every result φ” and Γ” of replacing every (primitive) descriptive symbol by an expression of the same genus, maintaining equal expressions for equal symbols, we have that φ” is a consequence of Γ”. Otherwise φ is a P-consequence of Γ. This division of the notion of consequence into L-consequence and P-consequence induces a division of the notion of demonstrable into L-demonstrable and P-demonstrable and the notion of valid into L-valid and P-valid and likewise for all of the other d-notions and c-notions. The terms ‘analytic’, ‘contradictory’, and ‘synthetic’ are used for ‘L-valid’, ‘L-contravalid’, and ‘L-indeterminate’.

It follows immediately from the definitions that logico-mathematical sentences are analytic or contradictory and that analytic sentences are without content. The trouble with the first approach is that the definitions of analyticity that Carnap gives for Languages I and II are highly sensitive to the original classification of terms into ‘logico-mathematical’ and ‘descriptive’. And the trouble with the second approach is that the division between ‘logico-mathematical’ and ‘descriptive’ expressions (and hence division between ‘analytic’ and ‘synthetic’ truths) is sensitive to the scope of the direct consequence relation with which one starts. This threatens to undermine Carnap’s thesis that logico-mathematical truths are analytic and hence without content. 

In the first approach, the original division of terms into ‘logico-mathematical’ and ‘descriptive’ is made by stipulation and if one alters this division one thereby alters the derivative division between analytic and synthetic sentences. For example, consider the case of Language II. If one calls only the primitive terms of first-order logic ‘logico-mathematical’ and then extends the language by adding the machinery of arithmetic and set theory, then, upon running the definition of ‘analytic’, one will have the result that true statements of first-order logic are without content while (the distinctive) statements of arithmetic and set theory have content. For another example, if one takes the language of arithmetic, calls the primitive terms ‘logico-mathematical’ and then extends the language by adding the machinery of finite type theory, calling the basic terms ‘descriptive’, then, upon running the definition of ‘analytic’, the result will be that statements of first-order arithmetic are analytic or contradictory while (the distinctive) statements of second- and higher-order arithmetic are synthetic and hence have content. In general, by altering the input, one alters the output, and Carnap adjusts the input to achieve his desired output.

In the second approach, there are no constraints on the scope of the direct consequence relation with which one starts and if one alters it one thereby alters the derivative division between ‘logico-mathematical’ and ‘descriptive’ expressions. Logical symbols and expressions have the feature that sentences composed solely of them are determinate. The trouble is that the resulting division of terms into ‘logico-mathematical’ and ‘descriptive’ will be highly sensitive to the scope of the direct consequence relation with which one starts. For example, let S be first-order PA and for the direct consequence relation take “provable in PA”. Under this assignment Fermat’s Last Theorem will be deemed descriptive, synthetic, and to have non-trivial content. For an example at the other extreme, let S be an extension of PA that contains a physical theory and let the notion of direct consequence be given by a Tarskian truth definition for the language. Since in the metalanguage one can prove that every sentence is true or false, every sentence will be either analytic (and so have null content) or contradictory (and so have total content). To overcome such counter-examples and get the classification that Carnap desires one must ensure that the consequence relation is (i) complete for the sublanguage consisting of expressions that one wants to come out as ‘logico-mathematical’ and (ii) not complete for the sublanguage consisting of expressions that one wants to come out as ‘descriptive’. Once again, by altering the input, one alters the output.

Carnap merely provides us with a flexible piece of technical machinery involving free parameters that can be adjusted to yield a variety of outcomes concerning the classifications of analytic/synthetic, contentful/non-contentful, and logico-mathematical/descriptive. In his own case, he has adjusted the parameters in such a way that the output is a formal articulation of his logicist view of mathematics that the truths of mathematics are analytic and without content. And one can adjust them differently to articulate a number of other views, for example, the view that the truths of first-order logic are without content while the truths of arithmetic and set theory have content. The point, however, is that we have been given no reason for fixing the parameters one way rather than another. The distinctions are thus not principled distinctions. It is trivial to prove that mathematics is trivial if one trivializes the claim.

Carnap is perfectly aware that to define c-notions like analyticity one must ascend to a stronger metalanguage. However, there is a distinction that he appears to overlook, namely, the distinction between (i) having a stronger system S that can define ‘analytic in S’ and (ii) having a stronger system S that can, in addition, evaluate a given statement of the form ‘φ is analytic in S’. It is an elementary fact that two systems S1 and S2 can employ the same definition (from an intensional point of view) of ‘analytic in S’ (using either the definition given for Language I or Language II) but differ on their evaluation of ‘φ is analytic in S’ (that is, differ on the extension of ‘analytic in S’). Thus, to determine whether ‘φ is analytic in S’ holds one needs to access much more than the “syntactic design” of φ – in addition to ascending to an essentially richer metalanguage one must move to a sufficiently strong system to evaluate ‘φ is analytic in S’.

In fact, to answer ‘Is φ analytic in Language I?’ is just to answer φ and, in the more general setting, to answer all questions of the form ‘Is φ analytic in S?’ (for various mathematical φ and S), where here ‘analytic’ is defined as Carnap defines it for Language II, just to answer all questions of mathematics. The same, of course, applies to the c-notion of consequence. So, when in first stating the Principle of Tolerance, Carnap tells us that we can choose our system S arbitrarily and that ‘no question of justification arises at all, but only the question of the syntactical consequences to which one or other of the choices leads’, where he means the c-notion of consequence.

Pluralist Mathematics, Minimalist Philosophy: Hans Reichenbach. Drunken Risibility.

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Hans Reichenbach relativized the notion of the constitutive a priori. The key observation concerns the fundamental difference between definitions in pure geometry and definitions in physical geometry. In pure geometry there are two kinds of definition: first, there are the familiar explicit definitions; second, there are implicit definitions, that is the kind of definition whereby such fundamental terms as ‘point’, ‘line’, and ‘surface’ are to derive their meaning from the fundamental axioms governing them. But in physical geometry a new kind of definition emerges – that of a physical (or coordinative) definition:

The physical definition takes the meaning of the concept for granted and coordinates to it a physical thing; it is a coordinative definition. Physical definitions, therefore, consist in the coordination of a mathematical definition to a “piece of reality”; one might call them real definitions. (Reichenbach, 8)

Now there are two important points about physical definitions. First, some such correlation between a piece of mathematics and “a piece of physical reality” is necessary if one is to articulate the laws of physics (e.g. consider “force-free moving bodies travel in straight lines”). Second, given a piece of pure mathematics there is a great deal of freedom in choosing the coordinative definitions linking it to “a piece of physical reality”, since… coordinative definitions are arbitrary, and “truth” and “falsehood” are not applicable to them. So we have here a conception of the a priori which (by the first point) is constitutive (of the empirical significance of the laws of physics) and (by the second point) is relative. Moreover, on Reichenbach’s view, in choosing between two empirically equivalent theories that involve different coordinative definitions, there is no issue of “truth” – there is only the issue of simplicity. In his discussion of Einstein’s particular definition of simultaneity, after noting its simplicity, Reichenbach writes: “This simplicity has nothing to do with the truth of the theory. The truth of the axioms decides the empirical truth, and every theory compatible with them which does not add new empirical assumptions is equally true.” (p 11)

Now, Reichenbach went beyond this and he held a more radical thesis – in addition to advocating pluralism with respect to physical geometry (something made possible by the free element in coordinative definitions), he advocated pluralism with respect to pure mathematics (such as arithmetic and set theory). According to Reichenbach, this view is made possible by the axiomatic conception of Hilbert, wherein axioms are treated as “implicit definitions” of the fundamental terms:

The problem of the axioms of mathematics was solved by the discovery that they are definitions, that is, arbitrary stipulations which are neither true nor false, and that only the logical properties of a system – its consistency, independence, uniqueness, and completeness – can be subjects of critical investigation. (p 3)

It needs to be stressed here that Reichenbach is extending the Hilbertian thesis concerning implicit definitions since although Hilbert held this thesis with regard to formal geometry he did not hold it with regard to arithmetic.

On this view there is a plurality of consistent formal systems and the notions of “truth” and “falsehood” do not apply to these systems; the only issue in choosing one system over another is one of convenience for the purpose at hand and this is brought out by investigating their metamathematical properties, something that falls within the provenance of “critical investigation”, where there is a question of truth and falsehood. This radical form of pluralism came to be challenged by Gödel’s discovery of the incompleteness theorems. To begin with, through the arithmetization of syntax, the metamathematical notions that Reichenbach takes to fall within the provenance of “critical investigation” were themselves seen to be a part of arithmetic. Thus, one cannot, on pain of inconsistency, say that there is a question of truth and falsehood with regard to the former but not the latter. More importantly, the incompleteness theorems buttressed the view that truth outstrips consistency. This is most clearly seen using Rosser’s strengthening of the first incompleteness theorem as follows: Let T be an axiom system of arithmetic that (a) falls within the provenance of “critical investigation” and (b) is sufficiently strong to prove the incompleteness theorem. A natural choice for such an axiom system is Primitive Recursive Arithmetic (PRA) but much weaker systems suffice, for example, IΔ0 + exp. Either of these systems can be taken as T. Assuming that T is consistent (something which falls within the provenance of “critical investigation”), by Rosser’s strengthening of the first incompleteness theorem, there is a Π01-sentence φ such that (provably within T + Con(T )) both T + φ and T + ¬φ are consistent. However, not both systems are equally legitimate. For it is easily seen that if a Π01-sentence φ is independent from such a theory, then it must be true. The point being that T is ∑10-complete (provably so in T). So, although T + ¬φ is consistent, it proves a false arithmetical statement.

ε-calculus and Hilbert’s Contentual Number Theory: Proselytizing Intuitionism. Thought of the Day 67.0

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Hilbert came to reject Russell’s logicist solution to the consistency problem for arithmetic, mainly for the reason that the axiom of reducibility cannot be accepted as a purely logical axiom. He concluded that the aim of reducing set theory, and with it the usual methods of analysis, to logic, has not been achieved today and maybe cannot be achieved at all. At the same time, Brouwer’s intuitionist mathematics gained currency. In particular, Hilbert’s former student Hermann Weyl converted to intuitionism.

According to Hilbert, there is a privileged part of mathematics, contentual elementary number theory, which relies only on a “purely intuitive basis of concrete signs.” Whereas the operating with abstract concepts was considered “inadequate and uncertain,” there is a realm of extra-logical discrete objects, which exist intuitively as immediate experience before all thought. If logical inference is to be certain, then these objects must be capable of being completely surveyed in all their parts, and their presentation, their difference, their succession (like the objects themselves) must exist for us immediately, intuitively, as something which cannot be reduced to something else.

The objects in questions are signs, both numerals and the signs that make up formulas a formal proofs. The domain of contentual number theory consists in the finitary numerals, i.e., sequences of strokes. These have no meaning, i.e., they do not stand for abstract objects, but they can be operated on (e.g., concatenated) and compared. Knowledge of their properties and relations is intuitive and unmediated by logical inference. Contentual number theory developed this way is secure, according to Hilbert: no contradictions can arise simply because there is no logical structure in the propositions of contentual number theory. The intuitive-contentual operations with signs form the basis of Hilbert’s meta-mathematics. Just as contentual number theory operates with sequences of strokes, so meta-mathematics operates with sequences of symbols (formulas, proofs). Formulas and proofs can be syntactically manipulated, and the properties and relationships of formulas and proofs are similarly based in a logic-free intuitive capacity which guarantees certainty of knowledge about formulas and proofs arrived at by such syntactic operations. Mathematics itself, however, operates with abstract concepts, e.g., quantifiers, sets, functions, and uses logical inference based on principles such as mathematical induction or the principle of the excluded middle. These “concept-formations” and modes of reasoning had been criticized by Brouwer and others on grounds that they presuppose infinite totalities as given, or that they involve impredicative definitions. Hilbert’s aim was to justify their use. To this end, he pointed out that they can be formalized in axiomatic systems (such as that of Principia or those developed by Hilbert himself), and mathematical propositions and proofs thus turn into formulas and derivations from axioms according to strictly circumscribed rules of derivation. Mathematics, to Hilbert, “becomes an inventory of provable formulas.” In this way the proofs of mathematics are subject to metamathematical, contentual investigation. The goal of Hilbert is then to give a contentual, meta-mathematical proof that there can be no derivation of a contradiction, i.e., no formal derivation of a formula A and of its negation ¬A.

Hilbert and Bernays developed the ε-calculus as their definitive formalism for axiom systems for arithmetic and analysis, and the so-called ε-substitution method as the preferred approach to giving consistency proofs. Briefly, the ε-calculus is a formalism that includes ε as a term-forming operator. If A(x) is a formula, then εxA(x) is a term, which intuitively stands for a witness for A(x). In a logical formalism containing the ε-operator, the quantifiers can be defined by: ∃x A(x) ≡ A(εxA(x)) and ∀x A(x) ≡ A(εx¬A(x)). The only additional axiom necessary is the so-called “transfinite axiom,” A(t) → A(εxA(x)). Based on this idea, Hilbert and his collaborators developed axiomatizations of number theory and analysis. Consistency proofs for these systems were then given using the ε-substitution method. The idea of this method is, roughly, that the ε-terms εxA(x) occurring in a formal proof are replaced by actual numerals, resulting in a quantifier-free proof. Suppose we had a (suitably normalized) derivation of 0 = 1 that contains only one ε-term εxA(x). Replace all occurrences of εxA(x) by 0. The instances of the transfinite axiom then are all of the form A(t) → A(0). Since no other ε-terms occur in the proof, A(t) and A(0) are basic numerical formulas without quantifiers and, we may assume, also without free variables. So they can be evaluated by finitary calculation. If all such instances turn out to be true numerical formulas, we are done. If not, this must be because A(t) is true for some t, and A(0) is false. Then replace εxA(x) instead by n, where n is the numerical value of the term t. The resulting proof is then seen to be a derivation of 0 = 1 from true, purely numerical formulas using only modus ponens, and this is impossible. Indeed, the procedure works with only slight modifications even in the presence of the induction axiom, which in the ε-calculus takes the form of a least number principle: A(t) → εxA(x) ≤ t, which intuitively requires εxA(x) to be the least witness for A(x).

Tarski, Wittgenstein and Undecidable Sentences in Affine Relation to Gödel’s. Thought of the Day 65.0

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I imagine someone asking my advice; he says: “I have constructed a proposition (I will use ‘P’ to designate it) in Russell’s symbolism, and by means of certain definitions and transformations it can be so interpreted that it says: ‘P is not provable in Russell’s system.’ Must I not say that this proposition on the one hand is true, and on the other hand is unprovable? 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.” — Wittgenstein

Any language of such a set, say Peano Arithmetic PA (or Russell and Whitehead’s Principia Mathematica, or ZFC), expresses – in a finite, unambiguous, and communicable manner – relations between concepts that are external to the language PA (or to Principia, or to ZFC). Each such language is, thus, essentially two-valued, since a relation either holds or does not hold externally (relative to the language).

Further, a selected, finite, number of primitive formal assertions about a finite set of selected primitive relations of, say, PA are defined as axiomatically PA-provable; all other assertions about relations that can be effectively defined in terms of the primitive relations are termed as PA-provable if, and only if, there is a finite sequence of assertions of PA, each of which is either a primitive assertion, or which can effectively be determined in a finite number of steps as an immediate consequence of any two assertions preceding it in the sequence by a finite set of rules of consequence.

The philosophical dimensions of this emerges if we take M as the standard, arithmetical, interpretation of PA, where:

(a)  the set of non-negative integers is the domain,

(b)  the integer 0 is the interpretation of the symbol “0” of PA,

(c)  the successor operation (addition of 1) is the interpretation of the “ ‘ ” function,

(d)  ordinary addition and multiplication are the interpretations of “+” and “.“,

(e) the interpretation of the predicate letter “=” is the equality relation.

Now, post-Gödel, the standard interpretation of classical theory seems to be that:

(f) PA can, indeed, be interpreted in M;

(g) assertions in M are decidable by Tarski’s definitions of satisfiability and truth;

(h) Tarskian truth and satisfiability are, however, not effectively verifiable in M.

Tarski made clear his indebtedness to Gödel’s methods,

We owe the method used here to Gödel who employed it for other purposes in his recently published work Gödel. This exceedingly important and interesting article is not directly connected with the theme of our work it deals with strictly methodological problems the consistency and completeness of deductive systems, nevertheless we shall be able to use the methods and in part also the results of Gödel’s investigations for our purpose.

On the other hand Tarski strongly emphasized the fact that his results were obtained independently, even though Tarski’s theorem on the undefinability of truth implies the existence of undecidable sentences, and hence Gödel’s first incompleteness theorem. Shifting gears here, how far was the Wittgensteinian quote really close to Gödel’s? However, the question, implicit in Wittgenstein’s argument regarding the possibility of a semantic contradiction in Gödel’s reasoning, then arises: How can we assert that a PA-assertion (whether such an assertion is PA-provable or not) is true under interpretation in M, so long as such truth remains effectively unverifiable in M? Since the issue is not resolved unambiguously by Gödel in his paper (nor, apparently, by subsequent standard interpretations of his formal reasoning and conclusions), Wittgenstein’s quote can be taken to argue that, although we may validly draw various conclusions from Gödel’s formal reasoning and conclusions, the existence of a true or false assertion of M cannot be amongst them.

Mappings, Manifolds and Kantian Abstract Properties of Synthesis

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An inverse system is a collection of sets which are connected by mappings. We start off with the definitions before relating these to abstract properties of synthesis.

Definition: A directed set is a set T together with an ordering relation ≤ such that

(1) ≤ is a partial order, i.e. transitive, reflexive, anti-symmetric

(2) ≤ is directed, i.e. for any s, t ∈ T there is r ∈ T with s, t ≤ r

Definition: An inverse system indexed by T is a set D = {Ds|s ∈ T} together with a family of mappings F = {hst|s ≥ t, hst : Ds → Dt}. The mappings in F must satisfy the coherence requirement that if s ≥ t ≥ r, htr ◦ hst = hsr.

Interpretation of the index set: The index set represents some abstract properties of synthesis. The ‘synthesis of apprehension in intuition’ proceeds by a ’running through and holding together of the manifold’ and is thus a process that takes place in time. We may now think of an index s ∈ T as an interval of time available for the process of ’running through and holding together’. More formally, s can be taken to be a set of instants or events, ordered by a ‘precedes’ relation; the relation t ≤ s then stands for: t is a substructure of s. It is immediate that on this interpretation ≤ is a partial order. The directedness is related to what Kant called ‘the formal unity of the consciousness in the synthesis of the manifold of representations’ or ‘the necessary unity of self-consciousness, thus also of the synthesis of the manifold, through a common function of the mind for combining it in one representation’ – the requirement that ‘for any s, t ∈ T there is r ∈ T with s, t ≤ r’ creates the formal conditions for combining the syntheses executed during s and t in one representation, coded by r.

Interpretation of the Ds and the mappings hst : Ds → Dt. An object in Ds can thought of as a possible ‘indeterminate object of empirical intuition’ synthesised in the interval s. If s ≥ t, the mapping hst : Ds → Dt expresses a consistency requirement: if d ∈ Ds represents an indeterminate object of empirical intuition synthesised in interval s, so that a particular manifold of features can be ‘run through and held together’ during s, some indeterminate object of empirical intuition must already be synthesisable by ‘running through and holding together’ in interval t, e.g. by combining a subset of the features characaterising d. This interpretation justifies the coherence condition s ≥ t ≥ r, htr ◦ hst = hsr: the synthesis obtained from first restricting the interval available for ‘running through and holding together’ to interval t, and then to interval r should not differ from the synthesis obtained by restricting to r directly.

We do not put any further requirements on the mappings hst : Ds → Dt, such as surjectivity or injectivity. Some indeterminate object of experience in Dt may have disappeared in Ds: more time for ‘running through and holding together’ may actually yield fewer features that can be combined. Thus we do not require the mappings to be surjective. It may also happen that an indeterminate object of experience in Dt corresponds to two or more of such objects in Ds, as when a building viewed from afar upon closer inspection turns out to be composed of two spatially separated buildings; thus the mappings need not be injective.

The interaction of the directedness of the index set and the mappings hst is of some interest. If r ≥ s, t there are mappings hrs : Dr → Ds and hrt : Ds → Dt. Each ‘indeterminate object of empirical intuition’ in d ∈ Dr can be seen as a synthesis of such objects hrs(d) ∈ Ds and hrt(d) ∈ Dt. For example, the ‘manifold of a house’ can be viewed as synthesised from a ‘manifold of the front’ and a ‘manifold of the back’. The operation just described has some of the characteristics of the synthesis of reproduction in imagination: the fact that the front of the house can be unified with the back to produce a coherent object presupposes that the front can be reproduced as it is while we are staring at the back. The mappings hrs : Dr → Ds and hrt : Ds → Dt capture the idea that d ∈ Dr arises from reproductions of hrs(d) and hrt(d) in r.

Hilbert’s Walking the Rope Between Real and Ideal Propositions. Note Quote.

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If the atomic sentences of S have a finitistic meaning, which is the case, for instance, when they are decidable, then so have all sentences of S built up by truth-functional connectives and quantifiers restricted to finite domains.

Quantifiers over infinite domains can be looked upon in two ways. One of them may be hinted at as follows. Let x range over the natural numbers, and let A(x) be a formula such that A(n) expresses a finitary proposition for every number n. Then a sentence ∀xA(x) expresses a transfinite proposition, if it is understood as a kind of infinite conjunction which is true when all of the infinitely many sentences A(n), where n is a natural number, hold.

Similarly, a sentence ∃xA(x) expresses a transfinite proposition, if it is understood as a kind of infinite disjunction which is true when, of all the infinitely many sentences A(n), where n is a natural number, there is one that holds. There is a certain ambiguity here, however, depending on what is meant by ‘all’ and ‘there is one’. To indicate the transfinite interpretation one should also add that the sentences are understood in such a way that it is determined, regardless of whether this can be proved or not, whether all of the sentences A(n) hold or there is some one that does not hold.

If instead an assertion of ∀xA(x) is understood as asserting that there is a method which, given a specific natural number n, yields a proof of A(n), then we have to do with a finitary proposition. Similarly, we have a case of a finitary proposition, when to assert ∃xA(x) is the same as to assert that A(n) can be proved for some natural number n.

It is to be noted that the ‘statement’ “In the real part of mathematics, either in the real part of S or in some extension of it, that for each A ∈ R, if ⌈S A, then A is true” is a universal sentence. Hence, the possibility of giving a finitary interpretation of the universal quantifier is a prerequisite for Hilbert’s program. Does the possibility of interpreting the quantifiers in a finitary way also mean that one may hope for a solution of the problem stated in the above ‘statement’ when all quantified sentences interpreted in that way are included in R?

A little reflection shows that the answer is no, but that R may always be taken as closed under universal quantification. For it can be seen (uniformly in A) that if we have established the ‘statement’ when R contains all instances of a sentence ∀xA(x), then the ‘statement’ also holds for R+ = R U {∀xA(x)}. To see this let ∀xA(x) be a formula provable in S whose instances belong to R, and let a method be given which applied to any formula in R and a proof of it in S yields a proof of its truth. We want to show that ∀xA(x) is true when interpreted in a finitistic way, i.e. that we have a method which applied to any natural number n yields a proof of A(n). The existence of such a method is obvious, because, from the proof given of ∀xA(x), we get a proof of A(n), for any n, and hence by specialization of the given method, we have a method which yields the required proof of the truth of A(n), for any n.

Having included universal sentences ∀xA(x) in R such that all A(n) are decidable, it is easy to see that one cannot in general also let existentially quantified sentences be included in R, if the ‘statement’ is still to be possible. For let S contain classical logic and assume that R contains undecidable sentences ∀xA(x) with A(n) decidable; by Gödel’s theorem there are such sentences if S is sufficiently rich. Then one cannot allow R to be closed under existential quantification. In particular, one cannot allow formulas ∃y(∀xA(x) V ¬ A(y)) to belong to R ∀ A: the formulas are provable in S but all of them cannot be expected to be true when interpreted in a finitistic way, because then, for any A, we would get a proof of ∀xA(x) V ¬ A(n) for some n, which would let us decide ∀xA(x).

In accordance with these observations, the line between real and ideal propositions was drawn in Hilbert’s program in such a way as to include among the real ones decidable propositions and universal generalizations of them but nothing more; in other words, the set R in the ‘statement’ is to consist of atomic sentences (assuming that they are decidable), sentences obtained from them by using truth-functional connectives, and finally universal generalizations of such sentences.

Given that R is determined in this way and that the atomic sentences in the language of S are decidable and provable in S if true (and hence that the same holds for truth-functional compounds of atomic sentences in S), which is normally the case, the consistency of S is easily seen to imply the statement in the ‘statement’ as follows. Assume consistency and let A be a sentence without quantifiers that is provable in S. Then A must be true, because, if it were not, then ¬ A would be true and hence provable in S by the assumption made about S, contradicting the consistency. Furthermore, a sentence ∀xA(x) provable in S must also be true, because there is a method such that for any given natural number n, the method yields a proof of A(n). By applying the decision method to A(n); by the consistency and the assumption on S, it must yield a proof of A(n) and not of ¬A(n).

Infinitesimal and Differential Philosophy. Note Quote.

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If difference is the ground of being qua becoming, it is not difference as contradiction (Hegel), but as infinitesimal difference (Leibniz). Accordingly, the world is an ideal continuum or transfinite totality (Fold: Leibniz and the Baroque) of compossibilities and incompossibilities analyzable into an infinity of differential relations (Desert Islands and Other Texts). As the physical world is merely composed of contiguous parts that actually divide until infinity, it finds its sufficient reason in the reciprocal determination of evanescent differences (dy/dx, i.e. the perfectly determinable ratio or intensive magnitude between indeterminate and unassignable differences that relate virtually but never actually). But what is an evanescent difference if not a speculation or fiction? Leibniz refuses to make a distinction between the ontological nature and the practical effectiveness of infinitesimals. For even if they have no actuality of their own, they are nonetheless the genetic requisites of actual things.

Moreover, infinitesimals are precisely those paradoxical means through which the finite understanding is capable of probing into the infinite. They are the elements of a logic of sense, that great logical dream of a combinatory or calculus of problems (Difference and Repetition). On the one hand, intensive magnitudes are entities that cannot be determined logically, i.e. in extension, even if they appear or are determined in sensation only in connection with already extended physical bodies. This is because in themselves they are determined at infinite speed. Is not the differential precisely this problematic entity at the limit of sensibility that exists only virtually, formally, in the realm of thought? Isn’t the differential precisely a minimum of time, which refers only to the swiftness of its fictional apprehension in thought, since it is synthesized in Aion, i.e. in a time smaller than the minimum of continuous time and hence in the interstitial realm where time takes thought instead of thought taking time?

Contrary to the Kantian critique that seeks to eliminate the duality between finite understanding and infinite understanding in order to avoid the contradictions of reason, Deleuze thus agrees with Maïmon that we shouldn’t speak of differentials as mere fictions unless they require the status of a fully actual reality in that infinite understanding. The alternative between mere fictions and actual reality is a false problem that hides the paradoxical reality of the virtual as such: real but not actual, ideal but not abstract. If Deleuze is interested in the esoteric history of differential philosophy, this is as a speculative alternative to the exoteric history of the extensional science of actual differences and to Kantian critical philosophy. It is precisely through conceptualizing intensive, differential relations that finite thought is capable of acquiring consistency without losing the infinite in which it plunges. This brings us back to Leibniz and Spinoza. As Deleuze writes about the former: no one has gone further than Leibniz in the exploration of sufficient reason [and] the element of difference and therefore [o]nly Leibniz approached the conditions of a logic of thought. Or as he argues of the latter, fictional abstractions are only a preliminary stage for thought to become more real, i.e. to produce an expressive or progressive synthesis: The introduction of a fiction may indeed help us to reach the idea of God as quickly as possible without falling into the traps of infinite regression. In Maïmon’s reinvention of the Kantian schematism as well as in the Deleuzian system of nature, the differentials are the immanent noumena that are dramatized by reciprocal determination in the complete determination of the phenomenal. Even the Kantian concept of the straight line, Deleuze emphasizes, is a dramatic synthesis or integration of an infinity of differential relations. In this way, infinitesimals constitute the distinct but obscure grounds enveloped by clear but confused effects. They are not empirical objects but objects of thought. Even if they are only known as already developed within the extensional becomings of the sensible and covered over by representational qualities, as differences they are problems that do not resemble their solutions and as such continue to insist in an enveloped, quasi-causal state.

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