Malicious Machine Learnings? Privacy Preservation and Computational Correctness Across Parties. Note Quote/Didactics.

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Multi-Party Computation deals with the following problem: There are n ≥ 2 parties P1, . . ., Pn where party Pi holds input ti, 1 ≤ i ≤ n, and they wish to compute together a functions = f (t1, . . . , tn) on their inputs. The goal is that each party will learn the output of the function, s, yet with the restriction that Pi will not learn any additional information about the input of the other parties aside from what can be deduced from the pair (ti, s). Clearly it is the secrecy restriction that adds complexity to the problem, as without it each party could announce its input to all other parties, and each party would locally compute the value of the function. Thus, the goal of Multi-Party Computation is to achieve the following two properties at the same time: correctness of the computation and privacy preservation of the inputs.

The following two generalizations are often useful:

(i) Probabilistic functions. Here the value of the function depends on some random string r chosen according to some distribution: s = f (t1, . . . , tn; r). An example of this is the coin-flipping functionality, which takes no inputs, and outputs an unbiased random bit. It is crucial that the value r is not controlled by any of the parties, but is somehow jointly generated during the computation.

(ii) Multioutput functions. It is not mandatory that there be a single output of the function. More generally there could be a unique output for each party, i.e., (s1, . . . , sn) = f(t1,…, tn). In this case, only party Pi learns the output si, and no other party learns any information about the other parties’ input and outputs aside from what can be derived from its own input and output.

One of the most interesting aspects of Multi-Party Computation is to reach the objective of computing the function value, but under the assumption that some of the parties may deviate from the protocol. In cryptography, the parties are usually divided into two types: honest and faulty. An honest party follows the protocol without any deviation. Otherwise, the party is considered to be faulty. The faulty behavior can exemplify itself in a wide range of possibilities. The most benign faulty behavior is where the parties follow the protocol, yet try to learn as much as possible about the inputs of the other parties. These parties are called honest-but-curious (or semihonest). At the other end of the spectrum, the parties may deviate from the prescribed protocol in any way that they desire, with the goal of either influencing the computed output value in some way, or of learning as much as possible about the inputs of the other parties. These parties are called malicious.

We envision an adversary A, who controls all the faulty parties and can coordinate their actions. Thus, in a sense we assume that the faulty parties are working together and can exert the most knowledge and influence over the computation out of this collusion. The adversary can corrupt any number of parties out of the n participating parties. Yet, in order to be able to achieve a solution to the problem, in many cases we would need to limit the number of corrupted parties. This limit is called the threshold k, indicating that the protocol remains secure as long as the number of corrupted parties is at most k.

Assume that there exists a trusted party who privately receives the inputs of all the participating parties, calculates the output value s, and then transmits this value to each one of the parties. This process clearly computes the correct output of f, and also does not enable the participating parties to learn any additional information about the inputs of others. We call this model the ideal model. The security of Multi-Party Computation then states that a protocol is secure if its execution satisfies the following: (1) the honest parties compute the same (correct) outputs as they would in the ideal model; and (2) the protocol does not expose more information than a comparable execution with the trusted party, in the ideal model.

Intuitively, the adversary’s interaction with the parties (on a vector of inputs) in the protocol generates a transcript. This transcript is a random variable that includes the outputs of all the honest parties, which is needed to ensure correctness, and the output of the adversary A. The latter output, without loss of generality, includes all the information that the adversary learned, including its inputs, private state, all the messages sent by the honest parties to A, and, depending on the model, maybe even include more information, such as public messages that the honest parties exchanged. If we show that exactly the same transcript distribution can be generated when interacting with the trusted party in the ideal model, then we are guaranteed that no information is leaked from the computation via the execution of the protocol, as we know that the ideal process does not expose any information about the inputs. More formally,

Let f be a function on n inputs and let π be a protocol that computes the function f. Given an adversary A, which controls some set of parties, we define REALA,π(t) to be the sequence of outputs of honest parties resulting from the execution of π on input vector t under the attack of A, in addition to the output of A. Similarly, given an adversary A′ which controls a set of parties, we define IDEALA′,f(t) to be the sequence of outputs of honest parties computed by the trusted party in the ideal model on input vector t, in addition to the output of A′. We say that π securely computes f if, for every adversary A as above, ∃ an adversary A′, which controls the same parties in the ideal model, such that, on any input vector t, we have that the distribution of REALA,π(t) is “indistinguishable” from the distribution of IDEALA′,f(t).

Intuitively, the task of the ideal adversary A′ is to generate (almost) the same output as A generates in the real execution or the real model. Thus, the attacker A′ is often called the simulator of A. The transcript value generated in the ideal model, IDEALA′,f(t), also includes the outputs of the honest parties (even though we do not give these outputs to A′), which we know were correctly computed by the trusted party. Thus, the real transcript REALA,π(t) should also include correct outputs of the honest parties in the real model.

We assumed that every party Pi has an input ti, which it enters into the computation. However, if Pi is faulty, nothing stops Pi from changing ti into some ti′. Thus, the notion of a “correct” input is defined only for honest parties. However, the “effective” input of a faulty party Pi could be defined as the value ti′ that the simulator A′ gives to the trusted party in the ideal model. Indeed, since the outputs of honest parties look the same in both models, for all effective purposes Pi must have “contributed” the same input ti′ in the real model.

Another possible misbehavior of Pi, even in the ideal model, might be a refusal to give any input at all to the trusted party. This can be handled in a variety of ways, ranging from aborting the entire computation to simply assigning ti some “default value.” For concreteness, we assume that the domain of f includes a special symbol ⊥ indicating this refusal to give the input, so that it is well defined how f should be computed on such missing inputs. What this requires is that in any real protocol we detect when a party does not enter its input and deal with it exactly in the same manner as if the party would input ⊥ in the ideal model.

As regards security, it is implicitly assumed that all honest parties receive the output of the computation. This is achieved by stating that IDEALA′,f(t) includes the outputs of all honest parties. We therefore say that our currency guarantees output delivery. A more relaxed property than output delivery is fairness. If fairness is achieved, then this means that if at least one (even faulty) party learns its outputs, then all (honest) parties eventually do too. A bit more formally, we allow the ideal model adversary A′ to instruct the trusted party not to compute any of the outputs. In this case, in the ideal model either all the parties learn the output, or none do. Since A’s transcript is indistinguishable from A′’s this guarantees that the same fairness guarantee must hold in the real model as well.

A further relaxation of the definition of security is to provide only correctness and privacy. This means that faulty parties can learn their outputs, and prevent the honest parties from learning theirs. Yet, at the same time the protocol will still guarantee that (1) if an honest party receives an output, then this is the correct value, and (2) the privacy of the inputs and outputs of the honest parties is preserved.

The basic security notions are universal and model-independent. However, specific implementations crucially depend on spelling out precisely the model where the computation will be carried out. In particular, the following issues must be specified:

  1. The faulty parties could be honest-but-curious or malicious, and there is usually an upper bound k on the number of parties that the adversary can corrupt.
  2. Distinguishing between the computational setting and the information theoretic setting, in the latter, the adversary is unlimited in its computing powers. Thus, the term “indistinguishable” is formalized by requiring the two transcript distributions to be either identical (so-called perfect security) or, at least, statistically close in their variation distance (so-called statistical security). On the other hand, in the computational, the power of the adversary (as well as that of the honest parties) is restricted. A bit more precisely, Multi-Party Computation problem is parameterized by the security parameter λ, in which case (a) all the computation and communication shall be done in time polynomial in λ; and (b) the misbehavior strategies of the faulty parties are also restricted to be run in time polynomial in λ. Furthermore, the term “indistinguishability” is formalized by computational indistinguishability: two distribution ensembles {Xλ}λ and {Yλ}λ are said to be computationally indistinguishable, if for any polynomial-time distinguisher D, the quantity ε, defined as |Pr[D(Xλ) = 1] − Pr[D(Yλ) = 1]|, is a “negligible” function of λ. This means that for any j > 0 and all sufficiently large λ, ε eventually becomes smaller than λ − j. This modeling helps us to build secure Multi-Party Computational protocols depending on plausible computational assumptions, such as the hardness of factoring large integers.
  3. The two common communication assumptions are the existence of a secure channel and the existence of a broadcast channel. Secure channels assume that every pair of parties Pi and Pj are connected via an authenticated, private channel. A broadcast channel is a channel with the following properties: if a party Pi (honest or faulty) broadcasts a message m, then m is correctly received by all the parties (who are also sure the message came from Pi). In particular, if an honest party receives m, then it knows that every other honest party also received m. A different communication assumption is the existence of envelopes. An envelope guarantees the following properties: a value m can be stored inside the envelope, it will be held without exposure for a given period of time, and then the value m will be revealed without modification. A ballot box is an enhancement of the envelope setting that also provides a random shuffling mechanism of the envelopes. These are, of course, idealized assumptions that allow for a clean description of a protocol, as they separate the communication issues from the computational ones. These idealized assumptions may be realized by a physical mechanisms, but in some settings such mechanisms may not be available. Then it is important to address the question if and under what circumstances we can remove a given communication assumption. For example, we know that the assumption of a secure channel can be substituted with a protocol, but under the introduction of a computational assumption and a public key infrastructure.
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Embedding Branes in Minkowski Space-Time Dimensions To Decipher Them As Particles Or Otherwise

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The physics treatment of Dirichlet branes in terms of boundary conditions is very analogous to that of the “bulk” quantum field theory, and the next step is again to study the renormalization group. This leads to equations of motion for the fields which arise from the open string, namely the data (M, E, ∇). In the supergravity limit, these equations are solved by taking the submanifold M to be volume minimizing in the metric on X, and the connection ∇ to satisfy the Yang-Mills equations.

Like the Einstein equations, the equations governing a submanifold of minimal volume are highly nonlinear, and their general theory is difficult. This is one motivation to look for special classes of solutions; the physical arguments favoring supersymmetry are another. Just as supersymmetric compactification manifolds correspond to a special class of Ricci-flat manifolds, those admitting a covariantly constant spinor, supersymmetry for a Dirichlet brane will correspond to embedding it into a special class of minimal volume submanifolds. Since the physical analysis is based on a covariantly constant spinor, this special class should be defined using the spinor, or else the covariantly constant forms which are bilinear in the spinor.

The standard physical arguments leading to this class are based on the kappa symmetry of the Green-Schwarz world-volume action, in which one finds that the subset of supersymmetry parameters ε which preserve supersymmetry, both of the metric and of the brane, must satisfy

φ ≡ Re εt Γε|M = Vol|M —– (1)

In words, the real part of one of the covariantly constant forms on M must equal the volume form when restricted to the brane.

Clearly dφ = 0, since it is covariantly constant. Thus,

Z(M) ≡ ∫φ —– (2)

depends only on the homology class of M. Thus, it is what physicists would call a “topological charge”, or a “central charge”.

If in addition the p-form φ is dominated by the volume form Vol upon restriction to any p-dimensional subspace V ⊂ Tx X, i.e.,

φ|V ≤ Vol|V —– (3)

then φ will be a calibration in the sense of implying the global statement

φ ≤ ∫Vol —– (4)

for any submanifold M . Thus, the central charge |Z (M)| is an absolute lower bound for Vol(M).

A calibrated submanifold M is now one satisfying (1), thereby attaining the lower bound and thus of minimal volume. Physically these are usually called “BPS branes,” after a prototypical argument of this type due, for magnetic monopole solutions in nonabelian gauge theory.

For a Calabi-Yau X, all of the forms ωp can be calibrations, and the corresponding calibrated submanifolds are p-dimensional holomorphic submanifolds. Furthermore, the n-form Re eΩ for any choice of real parameter θ is a calibration, and the corresponding calibrated submanifolds are called special Lagrangian.

This generalizes to the presence of a general connection on M, and leads to the following two types of BPS branes for a Calabi-Yau X. Let n = dimR M, and let F be the (End(E)-valued) curvature two-form of ∇.

The first kind of BPS D-brane, based on the ωp calibrations, is (for historical reasons) called a “B-type brane”. Here the BPS constraint is equivalent to the following three requirements:

  1. M is a p-dimensional complex submanifold of X.
  2. The 2-form F is of type (1, 1), i.e., (E, ∇) is a holomorphic vector bundle on M.
  3. In the supergravity limit, F satisfies the Hermitian Yang-Mills equation:ω|p−1M ∧ F = c · ω|pMfor some real constant c.
  4. F satisfies Im e(ω|M + ils2F)p = 0 for some real constant φ, where ls is the correction.

The second kind of BPS D-brane, based on the Re eΩ calibration, is called an “A-type” brane. The simplest examples of A-branes are the so-called special Lagrangian submanifolds (SLAGs), satisfying

(1) M is a Lagrangian submanifold of X with respect to ω.

(2) F = 0, i.e., the vector bundle E is flat.

(3) Im e Ω|M = 0 for some real constant α.

More generally, one also has the “coisotropic branes”. In the case when E is a line bundle, such A-branes satisfy the following four requirements:

(1)  M is a coisotropic submanifold of X with respect to ω, i.e., for any x ∈ M the skew-orthogonal complement of TxM ⊂ TxX is contained in TxM. Equivalently, one requires ker ωM to be an integrable distribution on M.

(2)  The 2-form F annihilates ker ωM.

(3)  Let F M be the vector bundle T M/ ker ωM. It follows from the first two conditions that ωM and F descend to a pair of skew-symmetric forms on FM, denoted by σ and f. Clearly, σ is nondegenerate. One requires the endomorphism σ−1f : FM → FM to be a complex structure on FM.

(4)  Let r be the complex dimension of FM. r is even and that r + n = dimR M. Let Ω be the holomorphic trivialization of KX. One requires that Im eΩ|M ∧ Fr/2 = 0 for some real constant α.

Coisotropic A-branes carrying vector bundles of higher rank are still not fully understood. Physically, one must also specify the embedding of the Dirichlet brane in the remaining (Minkowski) dimensions of space-time. The simplest possibility is to take this to be a time-like geodesic, so that the brane appears as a particle in the visible four dimensions. This is possible only for a subset of the branes, which depends on which string theory one is considering. Somewhat confusingly, in the type IIA theory, the B-branes are BPS particles, while in IIB theory, the A-branes are BPS particles.

Complete Manifolds’ Pure Logical Necessity as the Totality of Possible Formations. Thought of the Day 124.0

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In Logical Investigations, Husserl called his theory of complete manifolds the key to the only possible solution to how in the realm of numbers impossible, non-existent, meaningless concepts might be dealt with as real ones. In Ideas, he wrote that his chief purpose in developing his theory of manifolds had been to find a theoretical solution to the problem of imaginary quantities (Ideas Pertaining to a Pure Phenomenology and to a Phenomenological Philosophy).

Husserl saw how questions regarding imaginary numbers come up in mathematical contexts in which formalization yields constructions which arithmetically speaking are nonsense, but can be used in calculations. When formal reasoning is carried out mechanically as if these symbols have meaning, if the ordinary rules are observed, and the results do not contain any imaginary components, these symbols might be legitimately used. And this could be empirically verified (Philosophy of Arithmetic_ Psychological and Logical Investigations with Supplementary Texts).

In a letter to Carl Stumpf in the early 1890s, Husserl explained how, in trying to understand how operating with contradictory concepts could lead to correct theorems, he had found that for imaginary numbers like √2 and √-1, it was not a matter of the possibility or impossibility of concepts. Through the calculation itself and its rules, as defined for those fictive numbers, the impossible fell away, and a genuine equation remained. One could calculate again with the same signs, but referring to valid concepts, and the result was again correct. Even if one mistakenly imagined that what was contradictory existed, or held the most absurd theories about the content of the corresponding concepts of number, the calculation remained correct if it followed the rules. He concluded that this must be a result of the signs and their rules (Early Writings in the Philosophy of Logic and Mathematics). The fact that one can generalize, produce variations of formal arithmetic that lead outside the quantitative domain without essentially altering formal arithmetic’s theoretical nature and calculational methods brought Husserl to realize that there was more to the mathematical or formal sciences, or the mathematical method of calculation than could be captured in purely quantitative analyses.

Understanding the nature of theory forms, shows how reference to impossible objects can be justified. According to his theory of manifolds, one could operate freely within a manifold with imaginary concepts and be sure that what one deduced was correct when the axiomatic system completely and unequivocally determined the body of all the configurations possible in a domain by a purely analytical procedure. It was the completeness of the axiomatic system that gave one the right to operate in that free way. A domain was complete when each grammatically constructed proposition exclusively using the language of the domain was determined from the outset to be true or false in virtue of the axioms, i.e., necessarily followed from the axioms or did not. In that case, calculating with expressions without reference could never lead to contradictions. Complete manifolds have the

distinctive feature that a finite number of concepts and propositions – to be drawn as occasion requires from the essential nature of the domain under consideration –  determines completely and unambiguously on the lines of pure logical necessity the totality of all possible formations in the domain, so that in principle, therefore, nothing further remains open within it.

In such complete manifolds, he stressed, “the concepts true and formal implication of the axioms are equivalent (Ideas).

Husserl pointed out that there may be two valid discipline forms that stand in relation to one another in such a way that the axiom system of one may be a formal limitation of that of the other. It is then clear that everything deducible in the narrower axiom system is included in what is deducible in the expanded system, he explained. In the arithmetic of cardinal numbers, Husserl explained, there are no negative numbers, for the meaning of the axioms is so restrictive as to make subtracting 4 from 3 nonsense. Fractions are meaningless there. So are irrational numbers, √–1, and so on. Yet in practice, all the calculations of the arithmetic of cardinal numbers can be carried out as if the rules governing the operations are unrestrictedly valid and meaningful. One can disregard the limitations imposed in a narrower domain of deduction and act as if the axiom system were a more extended one. We cannot arbitrarily expand the concept of cardinal number, Husserl reasoned. But we can abandon it and define a new, pure formal concept of positive whole number with the formal system of definitions and operations valid for cardinal numbers. And, as set out in our definition, this formal concept of positive numbers can be expanded by new definitions while remaining free of contradiction. Fractions do not acquire any genuine meaning through our holding onto the concept of cardinal number and assuming that units are divisible, he theorized, but rather through our abandonment of the concept of cardinal number and our reliance on a new concept, that of divisible quantities. That leads to a system that partially coincides with that of cardinal numbers, but part of which is larger, meaning that it includes additional basic elements and axioms. And so in this way, with each new quantity, one also changes arithmetics. The different arithmetics do not have parts in common. They have totally different domains, but an analogous structure. They have forms of operation that are in part alike, but different concepts of operation.

For Husserl, formal constraints banning meaningless expressions, meaningless imaginary concepts, reference to non-existent and impossible objects restrict us in our theoretical, deductive work, but that resorting to the infinity of pure forms and transformations of forms frees us from such conditions and explains why having used imaginaries, what is meaningless, must lead, not to meaningless, but to true results.

Metaphysical Would-Be(s). Drunken Risibility.

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If one were to look at Quine’s commitment to similarity, natural kinds, dispositions, causal statements, etc., it is evident, that it takes him close to Peirce’s conception of Thirdness – even if Quine in an utopian vision imagines that all such concepts in a remote future will dissolve and vanish in favor of purely microstructural descriptions.

A crucial difference remains, however, which becomes evident when one looks at Quine’s brief formula for ontological commitment, the famous idea that ‘to be is to be the value of a bound variable’. For even if this motto is stated exactly to avoid commitment to several different types of being, it immediately prompts the question: the equation, in which the variable is presumably bound, which status does it have? Governing the behavior of existing variable values, is that not in some sense being real?

This will be Peirce’s realist idea – that regularities, tendencies, dispositions, patterns, may possess real existence, independent of any observer. In Peirce, this description of Thirdness is concentrated in the expression ‘real possibility’, and even it may sound exceedingly metaphysical at a first glance, it amounts, at a closer look, to regularities charted by science that are not mere shorthands for collections of single events but do possess reality status. In Peirce, the idea of real possibilities thus springs from his philosophy of science – he observes that science, contrary to philosophy, is spontaneously realist, and is right in being so. Real possibilities are thus counterposed to mere subjective possibilities due to lack of knowledge on the part of the subject speaking: the possibility of ‘not known not to be true’.

In a famous piece of self-critique from his late, realist period, Peirce attacks his earlier arguments (from ‘How to Make Our Ideas Clear’, in the late 1890s considered by himself the birth certificate of pragmatism after James’s reference to Peirce as pragmatism’s inventor). Then, he wrote

let us ask what we mean by calling a thing hard. Evidently that it will not be scratched by many other substances. The whole conception of this quality, as of every other, lies in its conceived effects. There is absolutely no difference between a hard thing and a soft thing so long as they are not brought to the test. Suppose, then, that a diamond could be crystallized in the midst of a cushion of soft cotton, and should remain there until it was finally burned up. Would it be false to say that that diamond was soft? […] Reflection will show that the reply is this: there would be no falsity in such modes of speech.

More than twenty-five years later, however, he attacks this argument as bearing witness to the nominalism of his youth. Now instead he supports the

scholastic doctrine of realism. This is usually defined as the opinion that there are real objects that are general, among the number being the modes of determination of existent singulars, if, indeed, these be not the only such objects. But the belief in this can hardly escape being accompanied by the acknowledgment that there are, besides, real vagues, and especially real possibilities. For possibility being the denial of a necessity, which is a kind of generality, is vague like any other contradiction of a general. Indeed, it is the reality of some possibilities that pragmaticism is most concerned to insist upon. The article of January 1878 endeavored to gloze over this point as unsuited to the exoteric public addressed; or perhaps the writer wavered in his own mind. He said that if a diamond were to be formed in a bed of cotton-wool, and were to be consumed there without ever having been pressed upon by any hard edge or point, it would be merely a question of nomenclature whether that diamond should be said to have been hard or not. No doubt this is true, except for the abominable falsehood in the word MERELY, implying that symbols are unreal. Nomenclature involves classification; and classification is true or false, and the generals to which it refers are either reals in the one case, or figments in the other. For if the reader will turn to the original maxim of pragmaticism at the beginning of this article, he will see that the question is, not what did happen, but whether it would have been well to engage in any line of conduct whose successful issue depended upon whether that diamond would resist an attempt to scratch it, or whether all other logical means of determining how it ought to be classed would lead to the conclusion which, to quote the very words of that article, would be ‘the belief which alone could be the result of investigation carried sufficiently far.’ Pragmaticism makes the ultimate intellectual purport of what you please to consist in conceived conditional resolutions, or their substance; and therefore, the conditional propositions, with their hypothetical antecedents, in which such resolutions consist, being of the ultimate nature of meaning, must be capable of being true, that is, of expressing whatever there be which is such as the proposition expresses, independently of being thought to be so in any judgment, or being represented to be so in any other symbol of any man or men. But that amounts to saying that possibility is sometimes of a real kind. (The Essential Peirce Selected Philosophical Writings, Volume 2)

In the same year, he states, in a letter to the Italian pragmatist Signor Calderoni:

I myself went too far in the direction of nominalism when I said that it was a mere question of the convenience of speech whether we say that a diamond is hard when it is not pressed upon, or whether we say that it is soft until it is pressed upon. I now say that experiment will prove that the diamond is hard, as a positive fact. That is, it is a real fact that it would resist pressure, which amounts to extreme scholastic realism. I deny that pragmaticism as originally defined by me made the intellectual purport of symbols to consist in our conduct. On the contrary, I was most careful to say that it consists in our concept of what our conduct would be upon conceivable occasions. For I had long before declared that absolute individuals were entia rationis, and not realities. A concept determinate in all respects is as fictitious as a concept definite in all respects. I do not think we can ever have a logical right to infer, even as probable, the existence of anything entirely contrary in its nature to all that we can experience or imagine. 

Here lies the core of Peirce’s metaphysical insistence on the reality of ‘would-be’s. Real possibilities, or would-bes, are vague to the extent that they describe certain tendential, conditional behaviors only, while they do not prescribe any other aspect of the single objects they subsume. They are, furthermore, represented in rationally interrelated clusters of concepts: the fact that the diamond is in fact hard, no matter if it scratches anything or not, lies in the fact that the diamond’s carbon structure displays a certain spatial arrangement – so it is an aspect of the very concept of diamond. And this is why the old pragmatic maxim may not work without real possibilities: it is they that the very maxim rests upon, because it is they that provide us with the ‘conceived consequences’ of accepting a concept. The maxim remains a test to weed out empty concepts with no conceived consequences – that is, empty a priori reasoning and superfluous metaphysical assumptions. But what remains after the maxim has been put to use, is real possibilities. Real possibilities thus connect epistemology, expressed in the pragmatic maxim, to ontology: real possibilities are what science may grasp in conditional hypotheses.

The question is whether Peirce’s revision of his old ‘nominalist’ beliefs form part of a more general development in Peirce from nominalism to realism. The locus classicus of this idea is Max Fisch (Peirce, Semeiotic and Pragmatism) where Fisch outlines a development from an initial nominalism (albeit of a strange kind, refusing, as always in Peirce, the existence of individuals determinate in all respects) via a series of steps towards realism, culminating after the turn of the century. Fisch’s first step is then Peirce’s theory of the real as that which reasoning would finally have as its result; the second step his Berkeley review with its anti-nominalism and the idea that the real is what is unaffected by what we may think of it; the third step is his pragmatist idea that beliefs are conceived habits of action, even if he here clings to the idea that the conditionals in which habits are expressed are material implications only – like the definition of ‘hard’; the fourth step his reading of Abbott’s realist Scientific Theism (which later influenced his conception of scientific universals) and his introduction of the index in his theory of signs; the fifth step his acceptance of the reality of continuity; the sixth the introduction of real possibilities, accompanied by the development of existential graphs, topology and Peirce’s changing view of Hegelianism; the seventh, the identification of pragmatism with realism; the eighth ‘his last stronghold, that of Philonian or material implication’. A further realist development exchanging Peirce’s early frequentist idea of probability for a dispositional theory of probability was, according to Fisch, never finished.

The issue of implication concerns the old discussion quoted by Cicero between the Hellenistic logicians Philo and Diodorus. The former formulated what we know today as material implication, while the latter objected on common-sense ground that material implication does not capture implication in everyday language and thought and another implication type should be sought. As is well known, material implication says that p ⇒ q is equivalent to the claim that either p is false or q is true – so that p ⇒ q is false only when p is true and q is false. The problems arise when p is false, for any false p makes the implication true, and this leads to strange possibilities of true inferences. The two parts of the implication have no connection with each other at all, such as would be the spontaneous idea in everyday thought. It is true that Peirce as a logician generally supports material (‘Philonian’) implication – but it is also true that he does express some second thoughts at around the same time as the afterthoughts on the diamond example.

Peirce is a forerunner of the attempts to construct alternatives such as strict implication, and the reason why is, of course, that real possibilities are not adequately depicted by material implication. Peirce is in need of an implication which may somehow picture the causal dependency of q on p. The basic reason for the mature Peirce’s problems with the representation of real possibilities is not primarily logical, however. It is scientific. Peirce realizes that the scientific charting of anything but singular, actual events necessitates the real existence of tendencies and relations connecting singular events. Now, what kinds are those tendencies and relations? The hard diamond example seems to emphasize causality, but this probably depends on the point of view chosen. The ‘conceived consequences’ of the pragmatic maxim may be causal indeed: if we accept gravity as a real concept, then masses will attract one another – but they may all the same be structural: if we accept horse riders as a real concept, then we should expect horses, persons, the taming of horses, etc. to exist, or they may be teleological. In any case, the interpretation of the pragmatic maxim in terms of real possibilities paves the way for a distinction between empty a priori suppositions and real a priori structures.

The Third Trichotomy. Thought of the Day 121.0

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The decisive logical role is played by continuity in the third trichotomy which is Peirce’s generalization of the old distinction between term, proposition and argument in logic. In him, the technical notions are rhema, dicent and argument, and all of them may be represented by symbols. A crucial step in Peirce’s logic of relations (parallel to Frege) is the extension of the predicate from having only one possible subject in a proposition – to the possibility for a predicate to take potentially infinitely many subjects. Predicates so complicated may be reduced, however, to combination of (at most) three-subject predicates, according to Peirce’s reduction hypothesis. Let us consider the definitions from ‘Syllabus (The Essential Peirce Selected Philosophical Writings, Volume 2)’ in continuation of the earlier trichotomies:

According to the third trichotomy, a Sign may be termed a Rheme, a Dicisign or Dicent Sign (that is, a proposition or quasi-proposition), or an Argument.

A Rheme is a Sign which, for its Interpretant, is a Sign of qualitative possibility, that is, is understood as representing such and such a kind of possible Object. Any Rheme, perhaps, will afford some information; but it is not interpreted as doing so.

A Dicent Sign is a Sign, which, for its Interpretant, is a Sign of actual existence. It cannot, therefore, be an Icon, which affords no ground for an interpretation of it as referring to actual existence. A Dicisign necessarily involves, as a part of it, a Rheme, to describe the fact which it is interpreted as indicating. But this is a peculiar kind of Rheme; and while it is essential to the Dicisign, it by no means constitutes it.

An Argument is a Sign which, for its Interpretant, is a Sign of a law. Or we may say that a Rheme is a sign which is understood to represent its object in its characters merely; that a Dicisign is a sign which is understood to represent its object in respect to actual existence; and that an Argument is a Sign which is understood to represent its Object in its character as Sign. ( ) The proposition need not be asserted or judged. It may be contemplated as a sign capable of being asserted or denied. This sign itself retains its full meaning whether it be actually asserted or not. ( ) The proposition professes to be really affected by the actual existent or real law to which it refers. The argument makes the same pretension, but that is not the principal pretension of the argument. The rheme makes no such pretension.

The interpretant of the Argument represents it as an instance of a general class of Arguments, which class on the whole will always tend to the truth. It is this law, in some shape, which the argument urges; and this ‘urging’ is the mode of representation proper to Arguments.

Predicates being general is of course a standard logical notion; in Peirce’s version this generality is further emphasized by the fact that the simple predicate is seen as relational and containing up to three subject slots to be filled in; each of them may be occupied by a continuum of possible subjects. The predicate itself refers to a possible property, a possible relation between subjects; the empty – or partly satiated – predicate does not in itself constitute any claim that this relation does in fact hold. The information it contains is potential, because no single or general indication has yet been chosen to indicate which subjects among the continuum of possible subjects it refers to. The proposition, on the contrary, the dicisign, is a predicate where some of the empty slots have been filled in with indices (proper names, demonstrative pronomina, deixis, gesture, etc.), and is, in fact, asserted. It thus consists of an indexical part and an iconical part, corresponding to the usual distinction between subject and predicate, with its indexical part connecting it to some level of reference reality. This reality needs not, of course, be actual reality; the subject slots may be filled in with general subjects thus importing pieces of continuity into it – but the reality status of such subjects may vary, so it may equally be filled in with fictitious references of all sorts. Even if the dicisign, the proposition, is not an icon, it contains, via its rhematic core, iconical properties. Elsewhere, Peirce simply defines the dicisign as a sign making explicit its reference. Thus a portrait equipped with a sign indicating the portraitee will be a dicisign, just like a charicature draft with a pointing gesture towards the person it depicts will be a dicisign. Even such dicisigns may be general; the pointing gesture could single out a group or a representative for a whole class of objects. While the dicisign specifies its object, the argument is a sign specifying its interpretant – which is what is normally called the conclusion. The argument thus consists of two dicisigns, a premiss (which may be, in turn, composed of several dicisigns and is traditionally seen as consisting of two dicisigns) and a conclusion – a dicisign represented as ensuing from the premiss due to the power of some law. The argument is thus – just like the other thirdness signs in the trichotomies – in itself general. It is a legisign and a symbol – but adds to them the explicit specification of a general, lawlike interpretant. In the full-blown sign, the argument, the more primitive degenerate sign types are orchestrated together in a threefold generality where no less than three continua are evoked: first, the argument itself is a legisign with a halo of possible instantions of itself as a sign; second, it is a symbol referring to a general object, in turn with a halo of possible instantiations around it; third, the argument implies a general law which is represented by one instantiation (the premiss and the rule of inference) but which has a halo of other, related inferences as possible instantiations. As Peirce says, the argument persuades us that this lawlike connection holds for all other cases being of the same type.

Utopia Banished. Thought of the Day 103.0

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In its essence, utopia has nothing to do with imagining an impossible ideal society; what characterizes utopia is literally the construction of a u-topic space, a space outside the existing parameters, the parameters of what appears to be “possible” in the existing social universe. The “utopian” gesture is the gesture that changes the coordinates of the possible. — (Slavoj Žižek- Iraq The Borrowed Kettle)

Here, Žižek discusses Leninist utopia, juxtaposing it with the current utopia of the end of utopia, the end of history. How propitious is the current anti-utopian aura for future political action? If society lies in impossibility, as Laclau and Mouffe (Hegemony and Socialist Strategy Towards a Radical Democratic Politics) argued, the field of politics is also marked by the impossible. Failing to fabricate an ideological discourse and incapable of historicizing, psychoanalysis appears as “politically impotent” and unable to encumber the way for other ideological narratives to breed the expectation of making the impossible possible, by promising to cover the fissure of the real in socio-political relations. This means that psychoanalysis can interminably unveil the impossible, only for a recycling of ideologies (outside the psychoanalytic discourse) to attempt to veil it.

Juxtaposing the possibility of a “post-fantasmatic” or “less fantasmatic” politics accepts the irreducible ambiguity of democracy and thus fosters the prospect of a radical democratic project. Yet, such a conception is not uncomplicated, given that one cannot totally go beyond fantasy and still maintain one’s subjectivity (even when one traverses it, another fantasy eventually grows), precisely because fantasy is required for the coherence of the subject and the upholding of her desire. Furthermore, fantasy is either there or not; we cannot have “more” or “less” fantasy. Fantasy, in itself, is absolute and totalizing par excellence. It is the real and the symbolic that always make it “less fantasmatic”, as they impose a limit in its operation.

So, where does “perversion” fit within this frame? The encounter with the extra-ordinary is an encounter with the real that reveals the contradiction that lies at the heart of the political. Extra-ordinariness suggests the embodiment of the real within the socio-political milieu; this is where the extra-ordinary subject incarnates the impossible object. Nonetheless, it suggests a fantasmatic strategy of incorporating the real in the symbolic, as an alternative to the encircling of the real through sublimation. In sublimation we still have an (artistic) object standing for the object a, so the lack in the subject is still there, whereas in extra-ordinariness the subject occupies the locus of the object a, in an ephemeral eradication of his/her lack. Extra-ordinariness may not be a condition that subverts or transforms socio-political relations, yet it can have a certain political significance. Rather than a direct confrontation with the impossible, it suggests a fantasmatic embracing of the impossible in its inexpressible totality, which can be perceived as a utopian aspiration.

Following Žižek or Badiou’s contemporary views, the extra-ordinary gesture is not qualified as an authentic utopian act, because it does not traverse fantasy, it does not rewrite social conditions. It is well known that Žižek prioritizes the negativeness of the real in his rhetoric, something that outstrips any positive imaginary or symbolic reflection in his work. But this entails the risk of neglecting the equal importance of all three registers for subjectivity. The imaginary constitutes an essential motive force for any drastic action to take place, as long as the symbolic limit is not thwarted. It is also what keeps us humane and sustains our relation to the other.

It is possible to touch the real, through imaginary means, without becoming a post-human figure (such as Antigone, who remains the figurative conception of Žižek’s traversing of the fantasy). Fantasy (and, therefore, ideology) can be a source of optimism and motivation and it should not be bound exclusively to the static character of compensatory utopia, according to Bloch’s distinction. In as much as fantasy infuses the subject’s effort to grasp the impossible, recognizing it as such and not breeding the futile expectation of turning the impossible into possible (regaining the object, meeting happiness), the imaginary can form the pedestal for an anticipatory utopia.

The imaginary does not operate only as a force that disavows difference for the sake of an impossible unity and completeness. It also suggests an apparatus that soothes the realization of the symbolic fissure, breeding hope and fascination, that is to say, it stirs up emotional states that encircle the lack of the subject. Moreover, it must be noted that the object a, apart from real properties, also has an imaginary hypostasis, as it is screened in fantasies that cover lack. If our image’s coherence is an illusion, it is this illusion that motivates us as individual and social subjects and help us relate to each other.

The anti-imaginary undercurrent in psychoanalysis is also what accounts for renunciation of idealism in the democratic discourse. The point de capiton is not just a common point of reference; it is a master signifier, which means it constitutes an ideal par excellence. The master signifier relies on fantasy and imaginary certainty about its supreme status. The ideal embodied by the master is what motivates action, not only in politics, but also in sciences, and arts. Is there a democratic prospect for the prevalence of an ideal that does not promise impossible jouissance, but possible jouissance, without confining it to the phallus? Since it is possible to touch jouissance, but not to represent it, the encounter with jouissance could endorse an ideal of incompleteness, an ideal of confronting the limits of human experience vis-à-vis unutterable enjoyment.

We need an extra-ordinary utopianism to the extent that it provokes pre-fixed phallic and normative access to enjoyment. The extra-ordinary himself does not go so far as to demand another master signifier, but his act is sufficiently provocative in divulging the futility of the master’s imaginary superiority. However, the limits of the extra-ordinary utopian logic is that its fantasy of embodying the impossible never stops in its embodiment (precisely because it is still a fantasy), and instead it continues to make attempts to grasp it, without accepting that the impossible remains impossible.

An alternative utopia could probably maintain the fantasy of embodying the impossible, acknowledging it as such. So, any time fantasy collapses, violence does not emerge as a response, but we continue the effort to symbolically speculate and represent the impossible, precisely because in this effort resides hope that sustains our reason to live and desire. As some historians say, myths distort “truth”, yet we cannot live without them; myths can form the only tolerable approximation of “truth”. One should see them as “colourful” disguises of the achromous core of his/her existence, and the truth is we need more “colour”.

Žižek’s Dialectical Coincidentia Oppositorium. Thought of the 98.0

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Without doubt, the cogent interlacing of Lacanian theorization with Hegelianism manifests Žižek’s prowess in articulating a highly pertinent critique of ideology for our epoch, but whether this comes from a position of Marxist orthodoxy or a position of a Lacanian doctrinaire who monitors Marxist politics is an open question.

Through this Lacanian prism, Žižek sees subjectivity as fragmented and decentred, considering its subordinate status to the unsurpassable realm of the signifiers. The acquisition of a consummate identity dwells in impossibility, in as much as it is bound to desire, provoked by a lacuna which is impossible to fill up. Thus, for Žižek, socio-political relations evolve from states of lack, linguistic fluidity, and contingency. What temporarily arrests this fluid state of the subject’s slithering in the realm of the signifiers, giving rise to her self-identity, is what Lacan calls point de capiton. The term refers to certain fundamental “anchoring” points in the signifying chain where the signifier is tied to the signified, providing an illusionary stability in signification. Laclau and Mouffe (Hegemony and Socialist Strategy Towards a Radical Democratic Politics) were the first to make use of the idea of the point de capiton in relation to hegemony and the formation of identities. In this context, ideology is conceptualized as a terrain of firm meanings, determined and comprised by numerous points de capiton (Zizek The Sublime Object of Ideology).

The real is the central Lacanian concept that Žižek implements in his rhetoric. He associates the real with antagonism (e.g., class conflict) as the unsymbolizable and irreducible gap that lies in the heart of the socio-symbolic order and around which society is formed. As Žižek argues, “class struggle designates the very antagonism that prevents the objective (social) reality from constituting itself as a self-enclosed whole” (Renata Salecl, Slavoj Zizek-Gaze and Voice As Love Objects). This logic is indebted to Laclau and Mouffe, who were the first to postulate that social antagonism is what impedes the closure of society, marking thus its impossibility. Žižek expanded this view and associated antagonism with the notion of the real.

Functioning as a hegemonic fantasmatic veil, ideology covers the lacuna of the symbolic, in the form of a fantasy, so that it protracts desire and hence subjectivity. On the imaginary level, ideology functions as the “mirror” that reflects antagonisms, that is to say, the real unrepresentable kernel that undermines the political. Around this emptiness of representation, the fictional narrative of ideology, its meaning, is to unfurl. The role of socio-ideological fantasy is to provide consistency to the symbolic order by veiling its void, and to foster the illusion of a coherent social unity.

Nevertheless, fantasy has both unifying and disjunctive features, as its role is to fill the void of the symbolic, but also to circumscribe this void. According to Žižek, “the notion of fantasy offers an exemplary case of the dialectical coincidentia oppositorium”. On the one side, it provides a “hallucinatory realisation of desire” and on the other side, it evokes disturbing images about the Other’s jouissance to which the subject has no (symbolic or imaginary) access. In so reasoning, ideology promises unity and, at the same time, creates another fantasy, where the failure of acquiring the anticipated ideological unity is ascribed.

Pertaining to Jacques Derrida’s work Specters of Marx (Specters of Marx The State of the Debt, The Work of Mourning; the New International), where the typical ontological conception of the living is seen to be incomplete and inseparable from the spectre, namely, a ghostly embodiment that haunts the living present (Derrida introduces the notion of hauntology to refer to this pseudo-material incarnation of the spirit that haunts and challenges ontological present), Žižek elaborates the spectral apparitions of the real in the politico–ideological domain. He makes a distinction between this “spectre” and “symbolic fiction”, that is, reality per se. Both have a common fantasmatic hypostasis, yet they perform antithetical functions. Symbolic fiction forecloses the real antagonism at the crux of reality, only to return as a spectre, as another fantasy.

Rants of the Undead God: Instrumentalism. Thought of the Day 68.1

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Hilbert’s program has often been interpreted as an instrumentalist account of mathematics. This reading relies on the distinction Hilbert makes between the finitary part of mathematics and the non-finitary rest which is in need of grounding (via finitary meta-mathematics). The finitary part Hilbert calls “contentual,” i.e., its propositions and proofs have content. The infinitary part, on the other hand, is “not meaningful from a finitary point of view.” This distinction corresponds to a distinction between formulas of the axiomatic systems of mathematics for which consistency proofs are being sought. Some of the formulas correspond to contentual, finitary propositions: they are the “real” formulas. The rest are called “ideal.” They are added to the real part of our mathematical theories in order to preserve classical inferences such as the principle of the excluded middle for infinite totalities, i.e., the principle that either all numbers have a given property or there is a number which does not have it.

It is the extension of the real part of the theory by the ideal, infinitary part that is in need of justification by a consistency proof – for there is a condition, a single but absolutely necessary one, to which the use of the method of ideal elements is subject, and that is the proof of consistency; for, extension by the addition of ideals is legitimate only if no contradiction is thereby brought about in the old, narrower domain, that is, if the relations that result for the old objects whenever the ideal objects are eliminated are valid in the old domain. Weyl described Hilbert’s project as replacing meaningful mathematics by a meaningless game of formulas. He noted that Hilbert wanted to “secure not truth, but the consistency of analysis” and suggested a criticism that echoes an earlier one by Frege – why should we take consistency of a formal system of mathematics as a reason to believe in the truth of the pre-formal mathematics it codifies? Is Hilbert’s meaningless inventory of formulas not just “the bloodless ghost of analysis? Weyl suggested that if mathematics is to remain a serious cultural concern, then some sense must be attached to Hilbert’s game of formulae. In theoretical physics we have before us the great example of a [kind of] knowledge of completely different character than the common or phenomenal knowledge that expresses purely what is given in intuition. While in this case every judgment has its own sense that is completely realizable within intuition, this is by no means the case for the statements of theoretical physics. Hilbert suggested that consistency is not the only virtue ideal mathematics has –  transfinite inference simplifies and abbreviates proofs, brevity and economy of thought are the raison d’être of existence proofs.

Hilbert’s treatment of philosophical questions is not meant as a kind of instrumentalist agnosticism about existence and truth and so forth. On the contrary, it is meant to provide a non-skeptical and positive solution to such problems, a solution couched in cognitively accessible terms. And, it appears, the same solution holds for both mathematical and physical theories. Once new concepts or “ideal elements” or new theoretical terms have been accepted, then they exist in the sense in which any theoretical entities exist. When Weyl eventually turned away from intuitionism, he emphasized the purpose of Hilbert’s proof theory, not to turn mathematics into a meaningless game of symbols, but to turn it into a theoretical science which codifies scientific (mathematical) practice. The reading of Hilbert as an instrumentalist goes hand in hand with a reading of the proof-theoretic program as a reductionist project. The instrumentalist reading interprets ideal mathematics as a meaningless formalism, which simplifies and “rounds out” mathematical reasoning. But a consistency proof of ideal mathematics by itself does not explain what ideal mathematics is an instrument for.

On this picture, classical mathematics is to be formalized in a system which includes formalizations of all the directly verifiable (by calculation) propositions of contentual finite number theory. The consistency proof should show that all real propositions which can be proved by ideal methods are true, i.e., can be directly verified by finite calculation. Actual proofs such as the ε-substitution procedure are of such a kind: they provide finitary procedures which eliminate transfinite elements from proofs of real statements. In particular, they turn putative ideal derivations of 0 = 1 into derivations in the real part of the theory; the impossibility of such a derivation establishes consistency of the theory. Indeed, Hilbert saw that something stronger is true: not only does a consistency proof establish truth of real formulas provable by ideal methods, but it yields finitary proofs of finitary general propositions if the corresponding free-variable formula is derivable by ideal methods.

Being Mediatized: How 3 Realms and 8 Dimensions Explain ‘Being’ by Peter Blank.

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Experience of Reflection: ‘Self itself is an empty word’
Leary – The neuroatomic winner: “In the province of the mind, what is believed true is true, or becomes true within limits to be learned by experience and experiment.” (Dr. John Lilly)

Media theory had noted the shoring up or even annihilation of the subject due to technologies that were used to reconfigure oneself and to see oneself as what one was: pictures, screens. Depersonalization was an often observed, reflective state of being that stood for the experience of anxiety dueto watching a ‘movie of one’s own life’ or experiencing a malfunction or anomaly in one’s self-awareness.

To look at one’s scaffolded media identity meant in some ways to look at the redactionary product of an extreme introspective process. Questioning what one interpreted oneself to be doing in shaping one’s media identities enhanced endogenous viewpoints and experience, similar to focusing on what made a car move instead of deciding whether it should stay on the paved road or drive across a field. This enabled the individual to see the formation of identity from the ‘engine perspective’.

Experience of the Hyperreal: ‘I am (my own) God’
Leary – The metaprogramming winner: “I make my own coincidences, synchronities, luck, and Destiny.”

Meta-analysis of distinctions – seeing a bird fly by, then seeing oneself seeing a bird fly by, then thinking the self that thought that – becomes routine in hyperreality. Media represent the opposite: a humongous distraction from Heidegger’s goal of the search for ‘Thinking’: capturing at present the most alarming of what occupies the mind. Hyperreal experiences could not be traced back to a person’s ‘real’ identities behind their aliases. The most questionable therefore related to dismantled privacy: a privacy that only existed because all aliases were constituting a false privacy realm. There was nothing personal about the conversations, no facts that led back to any person, no real change achieved, no political influence asserted.

From there it led to the difference between networked relations and other relations, call these other relations ‘single’ relations, or relations that remained solemnly silent. They were relations that could not be disclosed against their will because they were either too vague, absent, depressing, shifty, or dangerous to make the effort worthwhile to outsiders.

The privacy of hyperreal being became the ability to hide itself from being sensed by others through channels of information (sight, touch, hearing), but also to hide more private other selves, stored away in different, more private networks from others in more open social networks.

Choosing ‘true’ privacy, then, was throwing away distinctions one experienced between several identities. As identities were space the meaning of time became the capacity for introspection. The hyperreal being’s overall identity to the inside as lived history attained an extra meaning – indeed: as alter- or hyper-ego. With Nietzsche, the physical body within its materiality occasioned a performance that subjected its own subjectivity. Then and only then could it become its own freedom.

With Foucault one could say that the body was not so much subjected but still there functioning on its own premises. Therefore the sensitory systems lived the body’s life in connection with (not separated from) a language based in a mediated faraway from the body. If language and our sensitory systems were inseparable, beings and God may as well be.

Being Mediatized

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).