Husserl’s Flip-Flop on Arithmetic Axiomatics. Thought of the Day 118.0

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Husserl’s position in his Philosophy of Arithmetic (Psychological and Logical Investigations with Supplementary Texts) was resolutely anti-axiomatic. He attacked those who fell into remote, artificial constructions which, with the intent of building the elementary arithmetic concepts out of their ultimate definitional properties, interpret and change their meaning so much that totally strange, practically and scientifically useless conceptual formations finally result. Especially targeted was Frege’s ideal of the

founding of arithmetic on a sequence of formal definitions, out of which all the theorems of that science could be deduced purely syllogistically.

As soon as one comes to the ultimate, elemental concepts, Husserl reasoned, all defining has to come to an end. All one can then do is to point to the concrete phenomena from or through which the concepts are abstracted and show the nature of the abstractive process. A verbal explanation should place us in the proper state of mind for picking out, in inner or outer intuition, the abstract moments intended and for reproducing in ourselves the mental processes required for the formation of the concept. He said that his analyses had shown with incontestable clarity that the concepts of multiplicity and unity rest directly upon ultimate, elemental psychical data, and so belong among the indefinable concepts. Since the concept of number was so closely joined to them, one could scarcely speak of defining it either. All these points are made on the only pages of Philosophy of Arithmetic that Husserl ever explicitly retracted.

In On the Concept of Number, Husserl had set out to anchor arithmetical concepts in direct experience by analyzing the actual psychological processes to which he thought the concept of number owed its genesis. To obtain the concept of number of a concrete set of objects, say A, A, and A, he explained, one abstracts from the particular characteristics of the individual contents collected, only considering and retaining each one insofar as it is a something or a one. Regarding their collective combination, one thus obtains the general form of the set belonging to the set in question: one and one, etc. and. . . and one, to which a number name is assigned.

The enthusiastic espousal of psychologism of On the Concept of Number is not found in Philosophy of Arithmetic. Husserl later confessed that doubts about basic differences between the concept of number and the concept of collecting, which was all that could be obtained from reflection on acts, had troubled and tormented him from the very beginning and had eventually extended to all categorial concepts and to concepts of objectivities of any sort whatsoever, ultimately to include modern analysis and the theory of manifolds, and simultaneously to mathematical logic and the entire field of logic in general. He did not see how one could reconcile the objectivity of mathematics with psychological foundations for logic.

In sharp contrast to Brouwer who denounced logic as a source of truth, from the mid-1890s on, Husserl defended the view, which he attributed to Frege’s teacher Hermann Lotze, that pure arithmetic was basically no more than a branch of logic that had undergone independent development. He bid students not to be “scared” by that thought and to grow used to Lotze’s initially strange idea that arithmetic was only a particularly highly developed piece of logic.

Years later, Husserl would explain in Formal and Transcendental Logic that his

war against logical psychologism was meant to serve no other end than the supremely important one of making the specific province of analytic logic visible in its purity and ideal particularity, freeing it from the psychologizing confusions and misinterpretations in which it had remained enmeshed from the beginning.

He had come to see arithmetic truths as being analytic, as grounded in meanings independently of matters of fact. He had come to believe that the entire overthrowing of psychologism through phenomenology showed that his analyses in On the Concept of Number and Philosophy of Arithmetic had to be considered a pure a priori analysis of essence. For him, pure arithmetic, pure mathematics, and pure logic were a priori disciplines entirely grounded in conceptual essentialities, where truth was nothing other than the analysis of essences or concepts. Pure mathematics as pure arithmetic investigated what is grounded in the essence of number. Pure mathematical laws were laws of essence.

He is said to have told his students that it was to be stressed repeatedly and emphatically that the ideal entities so unpleasant for empiricistic logic, and so consistently disregarded by it, had not been artificially devised either by himself, or by Bolzano, but were given beforehand by the meaning of the universal talk of propositions and truths indispensable in all the sciences. This, he said, was an indubitable fact that had to be the starting point of all logic. All purely mathematical propositions, he taught, express something about the essence of what is mathematical. Their denial is consequently an absurdity. Denying a proposition of the natural sciences, a proposition about real matters of fact, never means an absurdity, a contradiction in terms. In denying the law of gravity, I cast experience to the wind. I violate the evident, extremely valuable probability that experience has established for the laws. But, I do not say anything “unthinkable,” absurd, something that nullifies the meaning of the word as I do when I say that 2 × 2 is not 4, but 5.

Husserl taught that every judgment either is a truth or cannot be a truth, that every presentation either accorded with a possible experience adequately redeeming it, or was in conflict with the experience, and that grounded in the essence of agreement was the fact that it was incompatible with the conflict, and grounded in the essence of conflict that it was incompatible with agreement. For him, that meant that truth ruled out falsehood and falsehood ruled out truth. And, likewise, existence and non-existence, correctness and incorrectness cancelled one another out in every sense. He believed that that became immediately apparent as soon as one had clarified the essence of existence and truth, of correctness and incorrectness, of Evidenz as consciousness of givenness, of being and not-being in fully redeeming intuition.

At the same time, Husserl contended, one grasps the “ultimate meaning” of the basic logical law of contradiction and of the excluded middle. When we state the law of validity that of any two contradictory propositions one holds and the other does not hold, when we say that for every proposition there is a contradictory one, Husserl explained, then we are continually speaking of the proposition in its ideal unity and not at all about mental experiences of individuals, not even in the most general way. With talk of truth it is always a matter of propositions in their ideal unity, of the meaning of statements, a matter of something identical and atemporal. What lies in the identically-ideal meaning of one’s words, what one cannot deny without invalidating the fixed meaning of one’s words has nothing at all to do with experience and induction. It has only to do with concepts. In sharp contrast to this, Brouwer saw intuitionistic mathematics as deviating from classical mathematics because the latter uses logic to generate theorems and in particular applies the principle of the excluded middle. He believed that Intuitionism had proven that no mathematical reality corresponds to the affirmation of the principle of the excluded middle and to conclusions derived by means of it. He reasoned that “since logic is based on mathematics – and not vice versa – the use of the Principle of the Excluded Middle is not permissible as part of a mathematical proof.”

Constructivism. Note Quote.

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Constructivism, as portrayed by its adherents, “is the idea that we construct our own world rather than it being determined by an outside reality”. Indeed, a common ground among constructivists of different persuasion lies in a commitment to the idea that knowledge is actively built up by the cognizing subject. But, whereas individualistic constructivism (which is most clearly enunciated by radical constructivism) focuses on the biological/psychological mechanisms that lead to knowledge construction, sociological constructivism focuses on the social factors that influence learning.

Let us briefly consider certain fundamental assumptions of individualistic constructivism. The first issue a constructivist theory of cognition ought to elucidate concerns of course the raw materials on which knowledge is constructed. On this issue, von Glaserfeld, an eminent representative of radical constructivism, gives a categorical answer: “from the constructivist point of view, the subject cannot transcend the limits of individual experience” (Michael R. Matthews Constructivism in Science Education_ A Philosophical Examination). This statement presents the keystone of constructivist epistemology, which conclusively asserts that “the only tools available to a ‘knower’ are the senses … [through which] the individual builds a picture of the world”. What is more, the so formed mental pictures do not shape an ‘external’ to the subject world, but the distinct personal reality of each individual. And this of course entails, in its turn, that the responsibility for the gained knowledge lies with the constructor; it cannot be shifted to a pre-existing world. As Ranulph Glanville confesses, “reality is what I sense, as I sense it, when I’m being honest about it” .

In this way, individualistic constructivism estranges the cognizing subject from the external world. Cognition is not considered as aiming at the discovery and investigation of an ‘independent’ world; it is viewed as a ‘tool’ that exclusively serves the adaptation of the subject to the world as it is experienced. From this perspective, ‘knowledge’ acquires an entirely new meaning. In the expression of von Glaserfeld,

the word ‘knowledge’ refers to conceptual structures that epistemic agents, given the range of present experience, within their tradition of thought and language, consider viable….[Furthermore] concepts have to be individually built up by reflective abstraction; and reflective abstraction is not a matter of looking closer but at operating mentally in a way that happens to be compatible with the perceptual material at hand.

To say it briefly, ‘knowledge’ signifies nothing more than an adequate organization of the experiential world, which makes the cognizing subject capable to effectively manipulate its perceptual experience.

It is evident that such insights, precluding any external point of reference, have impacts on knowledge evaluation. Indeed, the ascertainment that “for constructivists there are no structures other than those which the knower forms by its own activity” (Michael R. MatthewsConstructivism in Science Education A Philosophical Examination) yields unavoidably the conclusion drawn by Gerard De Zeeuw that “there is no mind-independent yardstick against which to measure the quality of any solution”. Hence, knowledge claims should not be evaluated by reference to a supposed ‘external’ world, but only by reference to their internal consistency and personal utility. This is precisely the reason that leads von Glaserfeld to suggest the substitution of the notion of “truth” by the notion of “viability” or “functional fit”: knowledge claims are appraised as “true”, if they “functionally fit” into the subject’s experiential world; and to find a “fit” simply means not to notice any discrepancies. This functional adaptation of ‘knowledge’ to experience is what finally secures the intended “viability”.

In accordance with the constructivist view, the notion of ‘object’, far from indicating any kind of ‘existence’, it explicitly refers to a strictly personal construction of the cognizing subject. Specifically, “any item of the furniture of someone’s experiential world can be called an ‘object’” (von Glaserfeld). From this point of view, the supposition that “the objects one has isolated in his experience are identical with those others have formed … is an illusion”. This of course deprives language of any rigorous criterion of objectivity; its physical-object statements, being dependent upon elements that are derived from personal experience, cannot be considered to reveal attributes of the objects as they factually are. Incorporating concepts whose meaning is highly associated with the individual experience of the cognizing subject, these statements form at the end a personal-specific description of the world. Conclusively, for constructivists the term ‘objectivity’ “shows no more than a relative compatibility of concepts” in situations where individuals have had occasion to compare their “individual uses of the particular words”.

From the viewpoint of radical constructivism, science, being a human enterprise, is amenable, by its very nature, to human limitations. It is then naturally inferred on constructivist grounds that “science cannot transcend [just as individuals cannot] the domain of experience” (von Glaserfeld). This statement, indicating that there is no essential differentiation between personal and scientific knowledge, permits, for instance, John Staver to assert that “for constructivists, observations, objects, events, data, laws and theory do not exist independent of observers. The lawful and certain nature of natural phenomena is a property of us, those who describe, not of nature, what is described”. Accordingly, by virtue of the preceding premise, one may argue that “scientific theories are derived from human experience and formulated in terms of human concepts” (von Glaserfeld).

In the framework now of social constructivism, if one accepts that the term ‘knowledge’ means no more than “what is collectively endorsed” (David Bloor Knowledge and Social Imagery), he will probably come to the conclusion that “the natural world has a small or non-existent role in the construction of scientific knowledge” (Collins). Or, in a weaker form, one can postulate that “scientific knowledge is symbolic in nature and socially negotiated. The objects of science are not the phenomena of nature but constructs advanced by the scientific community to interpret nature” (Rosalind Driver et al.). It is worth remarking that both views of constructivism eliminate, or at least downplay, the role of the natural world in the construction of scientific knowledge.

It is evident that the foregoing considerations lead most versions of constructivism to ultimately conclude that the very word ‘existence’ has no meaning in itself. It does acquire meaning only by referring to individuals or human communities. The acknowledgement of this fact renders subsequently the notion of ‘external’ physical reality useless and therefore redundant. As Riegler puts it, within the constructivist framework, “an external reality is neither rejected nor confirmed, it must be irrelevant”.

The Mystery of Modality. Thought of the Day 78.0

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The ‘metaphysical’ notion of what would have been no matter what (the necessary) was conflated with the epistemological notion of what independently of sense-experience can be known to be (the a priori), which in turn was identified with the semantical notion of what is true by virtue of meaning (the analytic), which in turn was reduced to a mere product of human convention. And what motivated these reductions?

The mystery of modality, for early modern philosophers, was how we can have any knowledge of it. Here is how the question arises. We think that when things are some way, in some cases they could have been otherwise, and in other cases they couldn’t. That is the modal distinction between the contingent and the necessary.

How do we know that the examples are examples of that of which they are supposed to be examples? And why should this question be considered a difficult problem, a kind of mystery? Well, that is because, on the one hand, when we ask about most other items of purported knowledge how it is we can know them, sense-experience seems to be the source, or anyhow the chief source of our knowledge, but, on the other hand, sense-experience seems able only to provide knowledge about what is or isn’t, not what could have been or couldn’t have been. How do we bridge the gap between ‘is’ and ‘could’? The classic statement of the problem was given by Immanuel Kant, in the introduction to the second or B edition of his first critique, The Critique of Pure Reason: ‘Experience teaches us that a thing is so, but not that it cannot be otherwise.’

Note that this formulation allows that experience can teach us that a necessary truth is true; what it is not supposed to be able to teach is that it is necessary. The problem becomes more vivid if one adopts the language that was once used by Leibniz, and much later re-popularized by Saul Kripke in his famous work on model theory for formal modal systems, the usage according to which the necessary is that which is ‘true in all possible worlds’. In these terms the problem is that the senses only show us this world, the world we live in, the actual world as it is called, whereas when we claim to know about what could or couldn’t have been, we are claiming knowledge of what is going on in some or all other worlds. For that kind of knowledge, it seems, we would need a kind of sixth sense, or extrasensory perception, or nonperceptual mode of apprehension, to see beyond the world in which we live to these various other worlds.

Kant concludes, that our knowledge of necessity must be what he calls a priori knowledge or knowledge that is ‘prior to’ or before or independent of experience, rather than what he calls a posteriori knowledge or knowledge that is ‘posterior to’ or after or dependant on experience. And so the problem of the origin of our knowledge of necessity becomes for Kant the problem of the origin of our a priori knowledge.

Well, that is not quite the right way to describe Kant’s position, since there is one special class of cases where Kant thinks it isn’t really so hard to understand how we can have a priori knowledge. He doesn’t think all of our a priori knowledge is mysterious, but only most of it. He distinguishes what he calls analytic from what he calls synthetic judgments, and holds that a priori knowledge of the former is unproblematic, since it is not really knowledge of external objects, but only knowledge of the content of our own concepts, a form of self-knowledge.

We can generate any number of examples of analytic truths by the following three-step process. First, take a simple logical truth of the form ‘Anything that is both an A and a B is a B’, for instance, ‘Anyone who is both a man and unmarried is unmarried’. Second, find a synonym C for the phrase ‘thing that is both an A and a B’, for instance, ‘bachelor’ for ‘one who is both a man and unmarried’. Third, substitute the shorter synonym for the longer phrase in the original logical truth to get the truth ‘Any C is a B’, or in our example, the truth ‘Any bachelor is unmarried’. Our knowledge of such a truth seems unproblematic because it seems to reduce to our knowledge of the meanings of our own words.

So the problem for Kant is not exactly how knowledge a priori is possible, but more precisely how synthetic knowledge a priori is possible. Kant thought we do have examples of such knowledge. Arithmetic, according to Kant, was supposed to be synthetic a priori, and geometry, too – all of pure mathematics. In his Prolegomena to Any Future Metaphysics, Kant listed ‘How is pure mathematics possible?’ as the first question for metaphysics, for the branch of philosophy concerned with space, time, substance, cause, and other grand general concepts – including modality.

Kant offered an elaborate explanation of how synthetic a priori knowledge is supposed to be possible, an explanation reducing it to a form of self-knowledge, but later philosophers questioned whether there really were any examples of the synthetic a priori. Geometry, so far as it is about the physical space in which we live and move – and that was the original conception, and the one still prevailing in Kant’s day – came to be seen as, not synthetic a priori, but rather a posteriori. The mathematician Carl Friedrich Gauß had already come to suspect that geometry is a posteriori, like the rest of physics. Since the time of Einstein in the early twentieth century the a posteriori character of physical geometry has been the received view (whence the need for border-crossing from mathematics into physics if one is to pursue the original aim of geometry).

As for arithmetic, the logician Gottlob Frege in the late nineteenth century claimed that it was not synthetic a priori, but analytic – of the same status as ‘Any bachelor is unmarried’, except that to obtain something like ‘29 is a prime number’ one needs to substitute synonyms in a logical truth of a form much more complicated than ‘Anything that is both an A and a B is a B’. This view was subsequently adopted by many philosophers in the analytic tradition of which Frege was a forerunner, whether or not they immersed themselves in the details of Frege’s program for the reduction of arithmetic to logic.

Once Kant’s synthetic a priori has been rejected, the question of how we have knowledge of necessity reduces to the question of how we have knowledge of analyticity, which in turn resolves into a pair of questions: On the one hand, how do we have knowledge of synonymy, which is to say, how do we have knowledge of meaning? On the other hand how do we have knowledge of logical truths? As to the first question, presumably we acquire knowledge, explicit or implicit, conscious or unconscious, of meaning as we learn to speak, by the time we are able to ask the question whether this is a synonym of that, we have the answer. But what about knowledge of logic? That question didn’t loom large in Kant’s day, when only a very rudimentary logic existed, but after Frege vastly expanded the realm of logic – only by doing so could he find any prospect of reducing arithmetic to logic – the question loomed larger.

Many philosophers, however, convinced themselves that knowledge of logic also reduces to knowledge of meaning, namely, of the meanings of logical particles, words like ‘not’ and ‘and’ and ‘or’ and ‘all’ and ‘some’. To be sure, there are infinitely many logical truths, in Frege’s expanded logic. But they all follow from or are generated by a finite list of logical rules, and philosophers were tempted to identify knowledge of the meanings of logical particles with knowledge of rules for using them: Knowing the meaning of ‘or’, for instance, would be knowing that ‘A or B’ follows from A and follows from B, and that anything that follows both from A and from B follows from ‘A or B’. So in the end, knowledge of necessity reduces to conscious or unconscious knowledge of explicit or implicit semantical rules or linguistics conventions or whatever.

Such is the sort of picture that had become the received wisdom in philosophy departments in the English speaking world by the middle decades of the last century. For instance, A. J. Ayer, the notorious logical positivist, and P. F. Strawson, the notorious ordinary-language philosopher, disagreed with each other across a whole range of issues, and for many mid-century analytic philosophers such disagreements were considered the main issues in philosophy (though some observers would speak of the ‘narcissism of small differences’ here). And people like Ayer and Strawson in the 1920s through 1960s would sometimes go on to speak as if linguistic convention were the source not only of our knowledge of modality, but of modality itself, and go on further to speak of the source of language lying in ourselves. Individually, as children growing up in a linguistic community, or foreigners seeking to enter one, we must consciously or unconsciously learn the explicit or implicit rules of the communal language as something with a source outside us to which we must conform. But by contrast, collectively, as a speech community, we do not so much learn as create the language with its rules. And so if the origin of modality, of necessity and its distinction from contingency, lies in language, it therefore lies in a creation of ours, and so in us. ‘We, the makers and users of language’ are the ground and source and origin of necessity. Well, this is not a literal quotation from any one philosophical writer of the last century, but a pastiche of paraphrases of several.

Tantric Reality

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आत्मा त्वं गिरिजा मतिः सहचराः प्राणाः शरीरं गृहं पूजा ते विषयोपभोगरचना निद्रा समाधिस्थितिः।
सञ्चारः पदयोः प्रदक्षिणविधिः स्तोत्राणि सर्वा गिरो यद्यत्कर्म करोमि तत्तदखिलं शम्भो तवाराधनम्॥

Ātmā tvaṃ Girijā matiḥ sahacarāḥ prāṇāḥ śarīraṃ gṛham
Pūjā te viṣayopabhoga-racanā nidrā samādhi-sthitiḥ |
Sañcāraḥ padayoḥ pradakṣiṇa-vidhiḥ stotrāṇi sarvā giraḥ
Yad-yat karma karomi tat-tad-akhilaṁ Śambho tavārādhanam ||

You (tvam) (are) the Self (ātmā) and Girijā –an epithet of Pārvatī, Śiva’s wife, meaning “mountain-born”– (girijā) (is) the intelligence (matiḥ). The vital energies (prāṇāḥ) (are Your)companions (sahacarāḥ). The body (śarīram) (is Your) house (gṛham). Worship (pūjā) of You (te) is prepared (racanā) with the objects (viṣaya) (known as sensual) enjoyments (upabhoga). Sleep (nidrā) (is Your) state (sthitiḥ) of Samādhi –i.e. perfect concentration or absorption– (samādhi). (My) wandering (sañcāraḥ) (is) the ceremony (vidhiḥ) of circumambulation from left to right (pradakṣiṇa) of (Your) feet (padayoḥ) –this act is generally done as a token of respect–. All (sarvāḥ) (my) words (giraḥ) (are) hymns of praise (of You) (stotrāṇi). Whatever (yad yad) action (karma) I do (karomi), all (akhilam) that (tad tad) is adoration (ārādhanam) of You (tava), oh Śambhu — an epithet of Śiva meaning “beneficent, benevolent”.

This Self is an embodiment of the Light of Consciousness; it is Śiva, free and autonomous. As an independent play of intense joy, the Divine conceals its own true nature [by manifesting plurality], and may also choose to reveal its fullness once again at any time. All that exists, throughout all time and beyond, is one infinite divine Consciousness, free and blissful, which projects within the field of its awareness a vast multiplicity of apparently differentiated subjects and objects: each object an actualization of a timeless potentiality inherent in the Light of Consciousness, and each subject the same plus a contracted locus of self-awareness. This creation, a divine play, is the result of the natural impulse within Consciousness to express the totality of its self-knowledge in action, an impulse arising from love. The unbounded Light of Consciousness contracts into finite embodied loci of awareness out of its own free will. When those finite subjects then identify with the limited and circumscribed cognitions and circumstances that make up this phase of their existence, instead of identifying with the transindividual overarching pulsation of pure Awareness that is their true nature, they experience what they call “suffering.” To rectify this, some feel an inner urge to take up the path of spiritual gnosis and yogic practice, the purpose of which is to undermine their misidentification and directly reveal within the immediacy of awareness the fact that the divine powers of Consciousness, Bliss, Willing, Knowing, and Acting comprise the totality of individual experience as well – thereby triggering a recognition that one’s real identity is that of the highest Divinity, the Whole in every part. This experiential gnosis is repeated and reinforced through various means until it becomes the nonconceptual ground of every moment of experience, and one’s contracted sense of self and separation from the Whole is finally annihilated in the incandescent radiance of the complete expansion into perfect wholeness. Then one’s perception fully encompasses the reality of a universe dancing ecstatically in the animation of its completely perfect divinity.”

 

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).

My Appresentations Rest in Protention.

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The ego often originally feels the pull of an object in the case of great contrast, where a unified object stands out from its background and from other objects. While contrast is not a necessary contributor to affectivity in an object, it does often accompany an object’s affective pull. An object that is not the focus of my attention cannot pull me toward it, however, unless I am able to perceive beyond what is in focus at this moment. Apperception is my ability to extend beyond my currently intended object to other objects and meanings and beyond what is now. Only if an object which has pulled me to it were at least partially constituted in the background, attracting my attention, could there have been any pull at all. Thus we discover a link between affectivity and apperception, because an object can only call me to it if my consciousness is able to extend beyond that which is in my focus now. And, because apperception must rest in a protentional temporality in order to allow for my ability to extend beyond the zone of actualization, we also find an indirect link between affectivity and protention. Therefore, affectivity requires a temporal structure that extends my consciousness beyond the immediate present and what is currently fulfilled so that an object in the periphery can attract my attention. In other words, affectivity is related to apperception, and both function through the protentional aspect of my temporality.

This relation also reminds us of the relation between protention and appresentation, where appresentation, the concept that any presentation of an object necessarily goes beyond itself to presentations of the object not currently in view – like the back side or the inside of the building across the street – clearly requires protention. As we explained earlier, protention is the condition of possibility of my going beyond the presentation at hand to other presentations or experiences. Thus the possibility of my viewing an object as having other sides, even though I am only perceiving one side at any moment, rests in a protentional temporality; my appresentations rest in protention. The transformed affectivity that draws me to learn more about an object after it has attracted my attention, then, also resides in protention; it always calls me to experience more, to move beyond what is currently presented.

The New Husserl A Critical Reader 

Orgies of the Atheistic Materialism: Barthes Contra Sade. Drunken Risibility.

The language and style of Justine are inextricably tied to sexual pleasure. Sade makes it impossible for the reader to ignore this aspect of the text. Roland Barthes, whose essays in Sade, Fourier, Loyola describe the innovative language of each author, underscores the importance of pleasure when discussing the Sadian voyage:

If the Sadian novel is excluded from our literature, it is because in it novelistic peregrination is never a quest for the Unique (temporal essence, truth, happiness), but a repetition of pleasure; Sadian errancy is unseemly, not because it is vicious and criminal, but because it is dull and somehow insignificant, withdrawn from transcendency, void of term: it does not re­veal, does not transform, does not develop, does not edu­cate, does not sublimate, does not accomplish, recuperates nothing, save for the present itself, cut up, glittering, repeated; no patience, no experience; everything is carried immediately to the acme of knowledge, of power, of ejacula­tion; time does not arrange or derange it, it repeats, recalls, recommences, there is no scansion other than that which al­ternates the formation and the expenditure of sperm.

Barthes’s observation reflects La Mettrie’s influence on Sade, whose libertine characters parrot in both speech and action the philosopher’s view that the pursuit of pleasure is man’s raison d’être. Sexuality permeates a great many linguistic and stylistic features of Justine, for example, names of characters (onomastics), literal and figurative language, grammatical structures, cultural and class references, dramatic effects, repetition and exaggeration, and use of parody and caricature. Justine is traditionally the name of a female domestic (soubrette), connoting a person of the lower classes, who falls prey to promiscuous behavior. Near the beginning of Justine, Sade renames the heroine the moment she accepts employment at the home of the miserly Monsieur Du Harpin, surname evocative of Molière’s Harpagon. Sophie, the wise example of womanly Christian virtue in the first version, becomes Thérèse, the anti- philosophe in the second, who chooses to ignore the brutally realistic counsel of her libertine persecutors. Sade’s Thérèse recalls the heroine of Thérèse philosophe who, unlike his protagonist, profited from an erotic lifestyle.

Sade may manipulate language to enhance erotic description but he also relies upon his observation of everyday life and class division of the ancien régime to provide him with models for his libertine characters, their mores, and their lifestyles. In Justine, he presents a socio-cultural microcosm of France during the reign of Louis XV. The power brokers of Sade’s youth who, for the most part, enriched themselves in his Majesty’s wars by means of corruption and influence, resurface in print as Justine’s exploiters. The noblemen, the financiers, the legal and medical professionals, the clergymen, and the thieves-robber barons representative of each social class-sexually maneuver their subjects to establish control. While we learn what the classes of mid-eighteenth-century France ate, how they dressed, where they lived, we also witness the ongoing struggle between victim and victimizer, the former personified by Justine, an ordinary bourgeois individual who can never vanquish the tyrant who maintains authority through sexual prowess rather than through wealth.

Barthes tells us that Sade’s passion was not erotic but theatrical. The marquis’s infatuation with the theater was inspired early on by the lavish productions staged by the Jesuits during his three and a half years at the Collège Louis-le-Grand. Later, his romantic dalliances with actresses and his own involvements in acting, writing, and production attest to his enormous attraction to the theater. In his libertine works, Sade incorporates theatricality, especially in his orgiastic scenes; in his own way, he creates the necessary horror and suspense to first seduce the reader and then to maintain his/her attention. Like a spectator in the audience, the reader observes well-rehearsed productions whose decor, script, and players have been predetermined, and where they are shown her various props in the form of “sadistic” paraphernalia.

Sade makes certain that the lesson given by her libertine victimizers following her forced participation in their orgies is not forgotten. Once again, Sade relies on man’s innate need for sexual pleasure to intellectualize the universe in a manner similar to his own. By using sexual desire as a ploy, Sade inculcates the atheistic materialism he so strongly proclaims into both an attentive Justine and reader. Justine cooperates with her depraved persecutors but refuses to adopt their way of thinking and thus continues to suffer at the hands of society’s exploiters. Sade, however, seizes the opportunity to convince his invisible readership that his concept of the universe is the right one. No matter how monotonous it may seem, repetition, whether in the form of licentious behavior or pseudo-philosophical diatribe, serves as a time-tested, powerful didactic tool.

Discontinuous Reality. Thought of the Day 61.0

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Convention is an invention that plays a distinctive role in Poincaré’s philosophy of science. In terms of how they contribute to the framework of science, conventions are not empirical. They are presupposed in certain empirical tests, so they are (relatively) isolated from doubt. Yet they are not pure stipulations, or analytic, since conventional choices are guided by, and modified in the light of, experience. Finally they have a different character from genuine mathematical intuitions, which provide a fixed, a priori synthetic foundation for mathematics. Conventions are thus distinct from the synthetic a posteriori (empirical), the synthetic a priori and the analytic a priori.

The importance of Poincaré’s invention lies in the recognition of a new category of proposition and its centrality in scientific judgment. This is more important than the special place Poincaré gives Euclidean geometry. Nevertheless, it’s possible to accommodate some of what he says about the priority of Euclidean geometry with the use of non-Euclidean geometry in science, including the inapplicability of any geometry of constant curvature in physical theories of global space. Poincaré’s insistence on Euclidean geometry is based on criteria of simplicity and convenience. But these criteria surely entail that if giving up Euclidean geometry somehow results in an overall gain in simplicity then that would be condoned by conventionalism.

The a priori conditions on geometry – in particular the group concept, and the hypothesis of rigid body motion it encourages – might seem a lingering obstacle to a more flexible attitude towards applied geometry, or an empirical approach to physical space. However, just as the apriority of the intuitive continuum does not restrict physical theories to the continuous; so the apriority of the group concept does not mean that all possible theories of space must allow free mobility. This, too, can be “corrected”, or overruled, by new theories and new data, just as, Poincaré comes to admit, the new quantum theory might overrule our intuitive assumption that nature is continuous. That is, he acknowledges that reality might actually be discontinuous – despite the apriority of the intuitive continuum.

Expressivity of Bodies: The Synesthetic Affinity Between Deleuze and Merleau-Ponty. Thought of the Day 54.0

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It is in the description of the synesthetic experience that Deleuze finds resources for his own theory of sensation. And it is in this context that Deleuze and Merleau-Ponty are closest. For Deleuze sees each sensation as a dynamic evolution, sensation is that which passes from one ‘order’ to another, from one ‘level’ to another. This means that each sensation is at diverse levels, of different orders, or in several domains….it is characteristic of sensation to encompass a constitutive difference of level and a plurality of constituting domains. What this means for Deleuze is that sensations cannot be isolated in a particular field of sense; these fields interpenetrate, so that sensation jumps from one domain to another, becoming-color in the visual field or becoming-music on the auditory level. For Deleuze (and this goes beyond what Merleau-Ponty explicitly says), sensation can flow from one field to another, because it belongs to a vital rhythm which subtends these fields, or more precisely, which gives rise to the different fields of sense as it contracts and expands, as it moves between different levels of tension and dilation.

If, as Merleau-Ponty says (and Deleuze concurs), synesthetic perception is the rule, then the act of recognition that identifies each sensation with a determinate quality or sense and operates their synthesis within the unity of an object, hides from us the complexity of perception, and the heterogeneity of the perceiving body. Synesthesia shows that the unity of the body is constituted in the transversal communication of the senses. But these senses are not pre given in the body; they correspond to sensations that move between levels of bodily energy – finding different expression in each other. To each of these levels corresponds a particular way of living space and time; hence the simultaneity in depth that is experienced in vision is not the lateral coexistence of touch, and the continuous, sensuous and overlapping extension of touch is lost in the expansion of vision. This heterogenous multiplicity of levels, or senses, is open to communication; each expresses its embodiment in its own way, and each expresses differently the contents of the other senses.

Thus sensation is not the causal process, but the communication and synchronization of senses within my body, and of my body with the sensible world; it is, as Merleau-Ponty says, a communion. And despite frequent appeal in the Phenomenology of Perception to the sameness of the body and to the common world to ground the diversity of experience, the appeal here goes in a different direction. It is the differences of rhythm and of becoming, which characterize the sensible world, that open it up to my experience. For the expressive body is itself such a rhythm, capable of synchronizing and coexisting with the others. And Merleau-Ponty refers to this relationship between the body and the world as one of sympathy. He is close here to identifying the lived body with the temporization of existence, with a particular rhythm of duration; and he is close to perceiving the world as the coexistence of such temporalizations, such rhythms. The expressivity of the lived body implies a singular relation to others, and a different kind of intercorporeity than would be the case for two merely physical bodies. This intercorporeity should be understood as inter-temporality. Merleau-Ponty proposes this at the end of the chapter on perception in his Phenomenology of Perception, when he says,

But two temporalities are not mutually exclusive as are two consciousnesses, because each one knows itself only by projecting itself into the present where they can interweave.

Thus our bodies as different rhythms of duration can coexist and communicate, can synchronize to each other – in the same way that my body vibrated to the colors of the sensible world. But, in the case of two lived bodies, the synchronization occurs on both sides – with the result that I can experience an internal resonance with the other when the experiences harmonize, or the shattering disappointment of a  miscommunication when the attempt fails. The experience of coexistence is hence not a guarantee of communication or understanding, for this communication must ultimately be based on our differences as expressive bodies and singular durations. Our coexistence calls forth an attempt, which is the intuition.

The Concern for Historical Materialism. Thought of the Day 53.0

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The concern for historical materialism, in spite of Marx’s differentiation between history and pre-history, is that totalisation might not be historically groundable after all, and must instead be constituted in other ways: whether logically, transcendentally or naturally. The ‘Consciousness’ chapter of the Phenomenology, a blend of all three, becomes a transcendent(al) logic of phenomena – individual, universal, particular – and ceases to provide any genuine phenomenology of ‘the experience of consciousness’. Natural consciousness is not strictly speaking a standpoint (no real opposition), so it can offer no critical grounds of itself to confer synthetic unity upon the universal, that which is taken to a higher level in ‘Self-Consciousness’ (only to be retrospectively confirmed). Yet Hegel does just this from the outset. In ‘Perception’, we read that, ‘[o]n account of the universality [Allgemeinheit] of the property, I must … take the objective essence to be on the whole a community [Gemeinschaft]’. Universality always sides with community, the Allgemeine with the Gemeinschaft, as if the synthetic operation had taken place prior to its very operability. Unfortunately for Hegel, the ‘free matters’ of all possible properties paves the way for the ‘interchange of forces’ in ‘Force and the Understanding’, and hence infinity, life and – spirit. In the midst of the master-slave dialectic, Hegel admits that, ‘[i]n this movement we see repeated the process which represented itself as the play of forces, but repeated now in consciousness [sic].