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

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.”

# Whitehead and Peirce’s Synchronicity with Hegel’s Capital Error. Thought of the Day 97.0

The focus on experience ensures that Whitehead’s metaphysics is grounded. Otherwise the narrowness of approach would only culminate in sterile measurement. This becomes especially evident with regard to the science of history. Whitehead gives a lucid example of such ‘sterile measurement’ lacking the immediacy of experience.

Consider, for example, the scientific notion of measurement. Can we elucidate the turmoil of Europe by weighing its dictators, its prime ministers, and its editors of newspapers? The idea is absurd, although some relevant information might be obtained. (Alfred North Whitehead – Modes of Thought)

The wealth of experience leaves us with the problem of how to cope with it. Selection of data is required. This selection is done by a value judgment – the judgment of importance. Although Whitehead opposes the dichotomy of the two notions ‘importance’ and ‘matter of fact’, it is still necessary to distinguish grades and types of importance, which enables us to structure our experience, to focus it. This is very similar to hermeneutical theories in Schleiermacher, Gadamer and Habermas: the horizon of understanding structures the data. Therefore, we not only need judgment but the process of concrescence implicitly requires an aim. Whitehead explains that

By this term ‘aim’ is meant the exclusion of the boundless wealth of alternative potentiality and the inclusion of that definite factor of novelty which constitutes the selected way of entertaining those data in that process of unification.

The other idea that underlies experience is “matter of fact.”

There are two contrasted ideas which seem inevitably to underlie all width of experience, one of them is the notion of importance, the sense of importance, the presupposition of importance. The other is the notion of matter of fact. There is no escape from sheer matter of fact. It is the basis of importance; and importance is important because of the inescapable character of matter of fact.

By stressing the “alien character” of feeling that enters into the privately felt feeling of an occasion, Whitehead is able to distinguish the responsive and the supplemental stages of concrescence. The responsive stage being a purely receptive phase, the latter integrating the former ‘alien elements’ into a unity of feeling. The alien factor in the experiencing subjects saves Whitehead’s concept from being pure Spirit (Geist) in a Hegelian sense. There are more similarities between Hegelian thinking and Whitehead’s thought than his own comments on Hegel may suggest. But, his major criticism could probably be stated with Peirce, who wrote that

The capital error of Hegel which permeates his whole system in every part of it is that he almost altogether ignores the Outward clash. (The Essential Peirce 1)

Whitehead refers to that clash as matter of fact. Although, even there, one has to keep in mind that matter-of-fact is an abstraction.

Matter of fact is an abstraction, arrived at by confining thought to purely formal relations which then masquerade as the final reality. This is why science, in its perfection, relapses into the study of differential equations. The concrete world has slipped through the meshes of the scientific net.

Whitehead clearly keeps the notion of prehension in his late writings as developed in Process and Reality. Just to give one example,

I have, in my recent writings, used the word ‘prehension’ to express this process of appropriation. Also I have termed each individual act of immediate self-enjoyment an ‘occasion of experience’. I hold that these unities of existence, these occasions of experience, are the really real things which in their collective unity compose the evolving universe, ever plunging into the creative advance.

Process needs an aim in Process and Reality as much as in Modes of Thought:

We must add yet another character to our description of life. This missing characteristic is ‘aim’. By this term ‘aim’ is meant the exclusion of the boundless wealth of alternative potentiality, and the inclusion of that definite factor of novelty which constitutes the selected way of entertaining those data in that process of unification. The aim is at that complex of feeling which is the enjoyment of those data in that way. ‘That way of enjoyment’ is selected from the boundless wealth of alternatives. It has been aimed at for actualization in that process.

# The Sibyl’s Prophecy/Nordic Creation. Note Quote.

The Prophecy of the Tenth Sibyl, a medieval best-seller, surviving in over 100 manuscripts from the 11th to the 16th century, predicts, among other things, the reign of evil despots, the return of the Antichrist and the sun turning to blood.

The Tenth or Tiburtine Sibyl was a pagan prophetess perhaps of Etruscan origin. To quote Lactantus in his general account of the ten sibyls in the introduction, ‘The Tiburtine Sibyl, by name Albunea, is worshiped at Tibur as a goddess, near the banks of the Anio in which stream her image is said to have been found, holding a book in her hand’.

The work interprets the Sibyl’s dream in which she foresees the downfall and apocalyptic end of the world; 9 suns appear in the sky, each one more ugly and bloodstained than the last, representing the 9 generations of mankind and ending with Judgment Day. The original Greek version dates from the end of the 4th century and the earliest surviving manuscript in Latin is dated 1047. The Tiburtine Sibyl is often depicted with Emperor Augustus, who asks her if he should be worshipped as a god.

The foremost lay of the Elder Edda is called Voluspa (The Sibyl’s Prophecy). The volva, or sibyl, represents the indelible imprint of the past, wherein lie the seeds of the future. Odin, the Allfather, consults this record to learn of the beginning, life, and end of the world. In her response, she addresses Odin as a plurality of “holy beings,” indicating the omnipresence of the divine principle in all forms of life. This also hints at the growth of awareness gained by all living, learning entities during their evolutionary pilgrimage through spheres of existence.

Hear me, all ye holy beings, greater as lesser sons of Heimdal! You wish me to tell of Allfather’s works, tales of the origin, the oldest I know. Giants I remember, born in the foretime, they who long ago nurtured me. Nine worlds I remember, nine trees of life, before this world tree grew from the ground.

Paraphrased, this could be rendered as:

Learn, all ye living entities, imbued with the divine essence of Odin, ye more and less evolved sons of the solar divinity (Heimdal) who stands as guardian between the manifest worlds of the solar system and the realm of divine consciousness. You wish to learn of what has gone before. I am the record of long ages past (giants), that imprinted their experience on me. I remember nine periods of manifestation that preceded the present system of worlds.

Time being inextricably a phenomenon of manifestation, the giant ages refer to the matter-side of creation. Giants represent ages of such vast duration that, although their extent in space and time is limited, it is of a scope that can only be illustrated as gigantic. Smaller cycles within the greater are referred to in the Norse myths as daughters of their father-giant. Heimdal is the solar deity in the sign of Aries – of beginnings for our system – whose “sons” inhabit, in fact compose, his domain.

Before a new manifestation of a world, whether a cosmos or a lesser system, all its matter is frozen in a state of immobility, referred to in the Edda as a frost giant. The gods – consciousnesses – are withdrawn into their supernal, unimaginable abstraction of Nonbeing, called in Sanskrit “paranirvana.” Without a divine activating principle, space itself – the great container – is a purely theoretical abstraction where, for lack of any organizing energic impulse of consciousness, matter cannot exist.

This was the origin of ages when Ymer built. No soil was there, no sea, no cool waves. Earth was not, nor heaven above; Gaping Void alone, no growth. Until the sons of Bur raised the tables; they who created beautiful Midgard. The sun shone southerly on the stones of the court; then grew green herbs in fertile soil.

To paraphrase again:

Before time began, the frost giant (Ymer) prevailed. No elements existed for there were ‘no waves,’ no motion, hence no organized form nor any temporal events, until the creative divine forces emanated from Space (Bur — a principle, not a locality) and organized latent protosubstance into the celestial bodies (tables at which the gods feast on the mead of life-experience). Among these tables is Middle Court (Midgard), our own beautiful planet. The life-giving sun sheds its radiant energies to activate into life all the kingdoms of nature which compose it.

The Gaping Void (Ginnungagap) holds “no cool waves” throughout illimitable depths during the age of the frost giant. Substance has yet to be created. Utter wavelessness negates it, for all matter is the effect of organized, undulating motion. As the cosmic hour strikes for a new manifestation, the ice of Home of Nebulosity (Niflhem) is melted by the heat from Home of Fire (Muspellshem), resulting in vapor in the void. This is Ymer, protosubstance as yet unformed, the nebulae from which will evolve the matter components of a new universe, as the vital heat of the gods melts and vivifies the formless immobile “ice.”

When the great age of Ymer has run its course, the cow Audhumla, symbol of fertility, “licking the salt from the ice blocks,” uncovers the head of Buri, first divine principle. From this infinite, primal source emanates Bur, whose “sons” are the creative trinity: Divine Allfather, Will, and Sanctity (Odin, Vile, and Vi). This triune power “kills” the frost giant by transforming it into First Sound (Orgalmer), or keynote, whose overtones vibrate through the planes of sleeping space and organize latent protosubstance into the multifarious forms which will be used by all “holy beings” as vehicles for gaining experience in worlds of matter.

Beautiful Midgard, our physical globe earth, is but one of the “tables” raised by the creative trinity, whereat the gods shall feast. The name Middle Court is suggestive, for the ancient traditions place our globe in a central position in the series of spheres that comprise the terrestrial being’s totality. All living entities, man included, comprise besides the visible body a number of principles and characteristics not cognized by the gross physical senses. In the Lay of Grimner (Grimnismal), wherein Odin in the guise of a tormented prisoner on earth instructs a human disciple, he enumerates twelve spheres or worlds, all but one of which are unseen by our organs of sight. As to the formation of Midgard, he relates:

Of Ymer’s flesh was the earth formed, the billows of his blood, the mountains of his bones, bushes of his hair, and of his brainpan heaven. With his eyebrows beneficent powers enclosed Midgard for man; but of his brain were surely all dark skies created.

The trinity of immanent powers organize Ymer into the forms wherein they dwell, shaping the chaos or frost giant into living globes on many planes of being. The “eyebrows” that gird the earth and protect it suggest the Van Allen belts that shield the planet from inimical radiation. The brain of Ymer – material thinking – is surely all too evident in the thought atmosphere wherein man participates.

The formation of the physical globe is described as the creation of “dwarfs” – elemental forces which shape the body of the earth-being and which include the mineral. vegetable, and animal kingdoms.

The mighty drew to their judgment seats, all holy gods to hold counsel: who should create a host of dwarfs from the blood of Brimer and the limbs of the dead. Modsogne there was, mightiest of all the dwarfs, Durin the next; there were created many humanoid dwarfs from the earth, as Durin said.

Brimer is the slain Ymer, a kenning for the waters of space. Modsogne is the Force-sucker, Durin the Sleeper, and later comes Dvalin the Entranced. They are “dwarf”-consciousnesses, beings that are miðr than human – the Icelandic miðr meaning both “smaller” and “less.” By selecting the former meaning, popular concepts have come to regard them as undersized mannikins, rather than as less evolved natural species that have not yet reached the human condition of intelligence and self-consciousness.

During the life period or manifestation of a universe, the governing giant or age is named Sound of Thor (Trudgalmer), the vital force which sustains activity throughout the cycle of existence. At the end of this age the worlds become Sound of Fruition (Bargalmer). This giant is “placed on a boat-keel and saved,” or “ground on the mill.” Either version suggests the karmic end product as the seed of future manifestation, which remains dormant throughout the ensuing frost giant of universal dissolution, when cosmic matter is ground into a formless condition of wavelessness, dissolved in the waters of space.

There is an inescapable duality of gods-giants in all phases of manifestation: gods seek experience in worlds of substance and feast on the mead at stellar and planetary tables; giants, formed into vehicles inspired with the divine impetus, rise through cycles of this association on the ladder of conscious awareness. All states being relative and bipolar, there is in endless evolution an inescapable link between the subjective and objective progress of beings. Odin as the “Opener” is paired with Orgalmer, the keynote on which a cosmos is constructed; Odin as the “Closer” is equally linked with Bargalmer, the fruitage of a life cycle. During the manifesting universe, Odin-Allfather corresponds to Trudgalmer, the sustainer of life.

A creative trinity plays an analogical part in the appearance of humanity. Odin remains the all-permeant divine essence, while on this level his brother-creators are named Honer and Lodur, divine counterparts of water or liquidity, and fire or vital heat and motion. They “find by the shore, of little power” the Ash and the Elm and infuse into these earth-beings their respective characteristics, making a human image or reflection of themselves. These protohumans, miniatures of the world tree, the cosmic Ash, Yggdrasil, in addition to their earth-born qualities of growth force and substance, receive the divine attributes of the gods. By Odin man is endowed with spirit, from Honer comes his mind, while Lodur gives him will and godlike form. The essentially human qualities are thus potentially divine. Man is capable of blending with the earth, whose substances form his body, yet is able to encompass in his consciousness the vision native to his divine source. He is in fact a minor world tree, part of the universal tree of life, Yggdrasil.

Ygg in conjunction with other words has been variously translated as Eternal, Awesome or Terrible, and Old. Sometimes Odin is named Yggjung, meaning the Ever-Young, or Old-Young. Like the biblical “Ancient of Days” it is a concept that mind can grasp only in the wake of intuition. Yggdrasil is the “steed” or the “gallows” of Ygg, whereon Odin is mounted or crucified during any period of manifested life. The world tree is rooted in Nonbeing and ramifies through the planes of space, its branches adorned with globes wherein the gods imbody. The sibyl spoke of ours as the tenth in a series of such world trees, and Odin confirms this in The Song of the High One (Den Hoges Sang):

I know that I hung in the windtorn tree nine whole nights, spear-pierced, given to Odin, my self to my Self above me in the tree, whose root none knows whence it sprang. None brought me bread, none served me drink; I searched the depths, spied runes of wisdom, raised them with song, and fell once more from the tree. Nine powerful songs I learned from the wise son of Boltorn, Bestla’s father; a draught I drank of precious mead ladled from Odrorer. I began to grow, to grow wise, to grow greater and enjoy; for me words from words led to new words, for me deeds from deeds led to new deeds.

Numerous ancient tales relate the divine sacrifice and crucifixion of the Silent Watcher whose realm or protectorate is a world in manifestation. Each tree of life, of whatever scope, constitutes the cross whereon the compassionate deity inherent in that hierarchy remains transfixed for the duration of the cycle of life in matter. The pattern of repeated imbodiments for the purpose of gaining the precious mead is clear, as also the karmic law of cause and effect as words and deeds bring their results in new words and deeds.

Yggdrasil is said to have three roots. One extends into the land of the frost giants, whence flow twelve rivers of lives or twelve classes of beings; another springs from and is watered by the well of Origin (Urd), where the three Norns, or fates, spin the threads of destiny for all lives. “One is named Origin, the second Becoming. These two fashion the third, named Debt.” They represent the inescapable law of cause and effect. Though they have usually been roughly translated as Past, Present, and Future, the dynamic concept in the Edda is more complete and philosophically exact. The third root of the world tree reaches to the well of the “wise giant Mimer,” owner of the well of wisdom. Mimer represents material existence and supplies the wisdom gained from experience of life. Odin forfeited one eye for the privilege of partaking of these waters of life, hence he is represented in manifestation as one-eyed and named Half-Blind. Mimer, the matter-counterpart, at the same time receives partial access to divine vision.

The lays make it very clear that the purpose of existence is for the consciousness-aspect of all beings to gain wisdom through life, while inspiring the substantial side of itself to growth in inward awareness and spirituality. At the human level, self-consciousness and will are aroused, making it possible for man to progress willingly and purposefully toward his divine potential, aided by the gods who have passed that way before him, rather than to drift by slow degrees and many detours along the road of inevitable evolution. Odin’s instructions to a disciple, Loddfafner, the dwarf-nature in man, conclude with:

Now is sung the High One’s song in the High One’s hall. Useful to sons of men, useless to sons of giants. Hail Him who sang! Hail him who kens! Rejoice they who understand! Happy they who heed!

# Abstract Expressions of Time’s Modalities. Thought of the Day 21.0

According to Gregory Bateson,

What we mean by information — the elementary unit of information — is a difference which makes a difference, and it is able to make a difference because the neural pathways along which it travels and is continually transformed are themselves provided with energy. The pathways are ready to be triggered. We may even say that the question is already implicit in them.

In other words, we always need to know some second order logic, and presuppose a second order of “order” (cybernetics) usually shared within a distinct community, to realize what a certain claim, hypothesis or theory means. In Koichiro Matsuno’s opinion Bateson’s phrase

must be a prototypical example of second-order logic in that the difference appearing both in the subject and predicate can accept quantification. Most statements framed in second-order logic are not decidable. In order to make them decidable or meaningful, some qualifier needs to be used. A popular example of such a qualifier is a subjective observer. However, the point is that the subjective observer is not limited to Alice or Bob in the QBist parlance.

This is what is necessitated in order understand the different viewpoints in logic of mathematicians, physicists and philosophers in the dispute about the existence of time. An essential aspect of David Bohm‘s “implicate order” can be seen in the grammatical formulation of theses such as the law of motion:

While it is legitimate in its own light, the physical law of motion alone framed in eternal time referable in the present tense, whether in classical or quantum mechanics, is not competent enough to address how the now could be experienced. … Measurement differs from the physical law of motion as much as the now in experience differs from the present tense in description. The watershed separating between measurement and the law of motion is in the distinction between the now and the present tense. Measurement is thus subjective and agential in making a punctuation at the moment of now. (Matsuno)

The distinction between experiencing and capturing experience of time in terms of language is made explicit in Heidegger’s Being and Time

… by passing away constantly, time remains as time. To remain means: not to disappear, thus, to presence. Thus time is determined by a kind of Being. How, then, is Being supposed to be determined by time?

Koichiro Matsuno’s comment on this is:

Time passing away is an abstraction from accepting the distinction of the grammatical tenses, while time remaining as time refers to the temporality of the durable now prior to the abstraction of the tenses.

Therefore, when trying to understand the “local logics/phenomenologies” of the individual disciplines (mathematics physics, philosophy, etc., including their fields), one should be aware of the fact that the capabilities of our scientific language are not limitless:

…the now of the present moment is movable and dynamic in updating the present perfect tense in the present progressive tense. That is to say, the now is prior and all of the grammatical tenses including the ubiquitous present tense are the abstract derivatives from the durable now. (Matsuno)

This presupposes the adequacy of mathematical abstractions specifically invented or adopted and elaborated for the expression of more sophisticated modalities of time’s now than those currently used in such formalisms as temporal logic.

# Geach and Relative Identity

The Theory of Relative Identity is a logical innovation due to Peter Thomas Geach  (P.T. Geach-Logic Matters) motivated by the same sort of mathematical examples as Frege’s definition by abstraction. Like Frege Geach seeks to give a logical sense to mathematical talk “up to” a given equivalence E through replacing E by identity but unlike Frege he purports, in doing so, to avoid the introduction of new abstract objects (which in his view causes unnecessary ontological inflation). The price for the ontological parsimony is Geach’s repudiation of Frege’s principle of a unique and absolute identity for the objects in the domain over which quantified variables range. According to Geach things can be same in one way while differing in others. For example two printed letters aa are same as a type but different as tokens. In Geach’s view this distinction does not commit us to a-tokens and a-types as entities but presents two different ways of describing the same reality. The unspecified (or “absolute” in Geach’s terminology) notion of identity so important for Frege is in Geach’s view is incoherent.

Geach’s proposal appears to account better for the way the notion of identity is employed in mathematics since it does not invoke “directions” or other mathematically redundant concepts. It captures particularly well the way the notion of identity is understood in Category theory. According to Baez & Dolan

In a category, two objects can be “the same in a way” while still being different.

So in Category theory the notion of identity is relative in exactly Geach’s sense. But from the logical point of view the notion of relative identity remains highly controversial. Let x,y be identical in one way but not in another, or in symbols: Id(x,y) & ¬Id'(x,y). The intended interpretation assumes that x in the left part of the formula and x in the right part have the same referent, where this last same apparently expresses absolute not relative identity. So talk of relative identity arguably smuggles in the usual absolute notion of identity anyway. If so, there seems good reason to take a standard line and reserve the term “identity” for absolute identity.

We see that Plato, Frege and Geach propose three different views of identity in mathematics. Plato notes that the sense of “the same” as applied to mathematical objects and to the ideas is different: properly speaking, sameness (identity) applies only to ideas while in mathematics sameness means equality or some other equivalence relation. Although Plato certainly recognizes essential links between mathematical objects and Ideas (recall the “ideal numbers”) he keeps the two domains apart. Unlike Plato Frege supposes that identity is a purely logical and domain-independent notion, which mathematicians must rely upon in order to talk about the sameness or difference of mathematical objects, or any other kind at all. Geach’s proposal has the opposite aim: to provide a logical justification for the way of thinking about the (relativized) notions of sameness and difference which he takes to be usual in mathematical contexts and then extend it to contexts outside mathematics (As Geach says):

Any equivalence relation … can be used to specify a criterion of relative identity. The procedure is common enough in mathematics: e.g. there is a certain equivalence relation between ordered pairs of integers by virtue of which we may say that x and y though distinct ordered pairs, are one and the same rational number. The absolute identity theorist regards this procedure as unrigorous but on a relative identity view it is fully rigorous.

# Kōjin Karatani versus Moishe Postone. Architectonics of Capitalism.

Kōjin Karatani’s theory of different modes of intercourse criticizes architectonic metaphor thinking that the logic of mods of production in terms of base and superstructure without ceding grounds on the centrality of the critique of political economy. the obvious question is what remains of theory when there is a departure not from the objective towards the subjective, but rather the simultaneous constitution of the subjective and the objective dimensions of the social under capitalism. One way of addressing the dilemma is to take recourse to the lesson of commodity form, where capitalism begets a uniform mode of mediation rather than disparate. The language of modes of production according to Moishe Postone happens to be a transhistorical language allowing for a transhistorical epistemology to sneak in through the backdoor thus outlining the necessity of critical theory’s existence only in so far as the object of critique stays in existence. Karatani’s first critique concerns a crude base-superstructure concept, in which nation and nationalism are viewed merely as phenomena of the ideological superstructure, which could be overcome by reason (enlightenment) or would disappear together with the state. But the nation functions autonomously, independent of the state, and as the imaginative return of community or reciprocal mode of exchange A, it is egalitarian in nature. As is the case with universal religions, the nation thus holds a moment of protest, of opposition, of emancipatory imagination. The second critique concerns the conception of the proletariat, which Marxism reduced to the process of production, in which its labor force is turned into a commodity. Production (i.e., consumption of labor power) as a fundamental basis to gain and to increase surplus value remains unchanged. Nonetheless, according to Karatani surplus value is only achieved by selling commodities, in the process of circulation, which does not generate surplus value itself, but without which there cannot be any surplus value. Understanding the proletariat as producer-consumer opens up new possibilities for resistance against the system. In late capitalism, in which capital and company are often separated, workers (in the broadest sense of wage and salary earners) are usually not able to resist their dependency and inferiority in the production process. By contrast, however, in the site of consumption, capital is dependent on the worker as consumer. Whereas capital can thus control the proletariat in the production process and force them to work, it loses its power over them in the process of circulation. If, says Karatani, we would view consumers as workers in the site of circulation, consumer movements could be seen as proletariat movements. They can, for example, resort to the legal means of boycott, which capital is unable to resist directly. Karatani bases his critique of capitalism not on the perspectives of globalization, but rather on what he terms neo-imperialism meaning state-based attempt of capital to subject the entire world to its logic of exploitation, and thus any logic to overcoming the modern world system of capital-nation-state by means of a world revolution and its sublation in a system is to be possible by justice based on exchange. For Postone Capital generates a system characteristically by the opposition of abstract universality, the value form, and particularistic specificity, the use value dimension. It seems to me that rather than viewing a socialist or an emancipatory movement as the heirs to the Enlightenment, as the classic working class movement did, critical movements today should be striving for a new form of universalism that encompasses the particular, rather than existing in opposition to the particular. This will not be easy, because a good part of the Left today has swung to particularity rather than trying to and a new form of universalism. I think this is a fatal mistake.

# Moishe Postone: Capitalism, Temporality, and the Crisis of Labor. Note Quote.

Moishe Postone’s work establishes a crucial distinction between the critique of capitalism from the standpoint of labour and the critique of labor in capitalism.The former implies a transhistorical account of work, while the latter situates labor as a consistent category – capable of “social synthesis” – within the capitalist mode of production. But, does this distinction require us to abandon any form of ontological account of labour? As Postone would say,

It depends what you mean by an ontological account of labour. It does force us to abandon the idea that transhistorically there is an on-going development of humanity which is effected by labour, that human interaction with nature as mediated by labour is a continuous process which is led to continuous change. And that labour is in that sense a central historical category. That position is closer actually to Adam Smith than it is to Marx. I think that the centrality of labour to something called historical development can be posited only for capitalism and not for any other form of human social life. On the other hand, I think one can retain the idea that humanity’s interaction with nature is a process of self-constitution.

One of the most important contributions of Time, Labour and Social Domination is a novel theory of impersonal domination in capitalist society. To him “traditional Marxism” is a criticism of capitalism from the standpoint of labor. Postone’s Marxism, by contrast, is a critique of labor in capitalism. Since Marx’s theory refers to capitalism, not society in general, labor cannot be a transhistorical category. Instead, it must be understood as an integrated part of capitalism. This means that labor cannot provide a standpoint from which to criticize capitalism, and neither can the proletariat: “the working class is integral to capitalism, rather than the embodiment of its negation”. The struggle, then, should not be a struggle of labor against capital, as traditional Marxists thought, but a struggle against labor seen as an integral part of the valorization of capital.  This conclusion has implications for Postone’s understanding of domination in capitalism. Rather than being a matter of class relations, it takes the form of domination by impersonal and quasi-objective mechanisms such as fetishism, in the construction of which labor is deeply implicated. The benefit of this reinterpretation, according to Postone, is that it shows the usefulness of Marx’s theory not only in a criticism of liberal nineteenth-century capitalism but also in a criticism of contemporary welfare-state capitalism or Soviet-style state-capitalism. The latter forms of capitalism are just as capitalist as the former since they all build on the valorization of capital built on labor. Abolishing private ownership or rearranging the distribution of goods is not enough to escape capitalism. Postone both builds on and criticizes the approaches of Lukács and the Frankfurt School. There is much in his book that shows his affinities especially to the latter – such as the criticism of welfare state capitalism or the stress on fetishism – but he nevertheless criticizes these earlier thinkers for being bound to a transhistorical conception of labor. Lukács in particular is singled out for heavy criticism since he saw the proletariat as the Subject of history, as capable of grasping totality and hence offering the standpoint of critique. Engaging with the Hegelian legacy, or should I quip lunacy in Lukács, Postone arrives at one of his most important and provocative arguments. “Marx suggests that a historical Subject in the Hegelian sense does indeed exist in capitalism, yet he does not identify it with… the proletariat”. Instead it is capital that is portrayed as a Hegelian Geist – as a subject and self-moving substance, following its own immanent historical logic. Hegelian dialectics, then, is specific to capitalism and is not a tool for grasping history in general. Thus, to Marx, the “totality” was not the whole in general, and certainly not a standpoint which he affirmed. Instead, he identified totality with the capitalist system and made it the object of his critique: “the historical negation of capitalism would not involve the realization, but the abolition, of the totality”, Postone argues. The working class cannot lead history towards this negation. In fact, it is only by breaking with the logic consitutive of this totality, in which the working class forms part, that a different, post-capitalist society can be born.

The abolition of the totality would, then, allow for the possible constitution of very different, non-totalizing, forms of the political coordination and regulation of society.

In a question asked about if the capitalist form of domination not better defined as the appearance of truly abstract relations as if they were concrete, personal relations? Furthermore, does this inversion, or at least the recognition of the crucial role of abstraction in capitalism, render a definition of class struggle untenable, or are we rather in need of a concept of class that takes this distance from the concrete into consideration? Postone says,

I am not sure that I would fully agree with the attempted reformulation. First of all, with regard to the quote “relations between people appearing as relations between things” what is left out of this version of what Marx said is that he adds that relations among people appear as they are, as social relations between things and thingly relations between people. Marx only explicitly elaborated the notion of fetishism with the fetishism of commodity. All three volumes of Capital, are [our change] in many respects, however, a study on fetishism even when he doesn’t use that word. And fetishism means that because of the peculiar, double character of the structuring social forms of capitalism, social relations disappears from view. What we get are thingly relations: we also get abstractions. However, one dimension of the fetish is, as you put it, that abstract relations appear concrete.They appear in the form of the concrete. So, for example, the process of creating surplus value appears to be a material process, the labour process. It appears to be material-technical, rather than moulded by social forms. And yet there are also abstract dimensions and regularities that don’t appear in the form of the concrete. I am emphasising this is because certain reactionary forms of thought only view capitalism in terms of those abstract regularities and refuse to see that the concrete itself is moulded by, and is really drenched with, the abstract. I think a lot of forms of populism and anti-Semitism can be characterised that way. Now I am not sure that this appropriation of the categories of Marx’s critique of political economy renders a definition of class struggle untenable, but it does indicate that class struggle occurs within and is moulded by the structuring social forms.This position rejects the ontological centrality or the primacy of class struggle, as that which is truly social and real behind the veil of capitalist forms. Class struggle rather is moulded by the capitalist relations expressed by the categories of value, commodity, surplus value, and capital.

Postone’s approach only seems far-fetched if we continue to equate capitalism with the economy. Not if we think of it as a form of life. For example, after Darwin wrote, natural processes, such as adaptation or sexual selection, came to be seen as operating within history. This gave us the naturalist novel of Zola or Norris. “Nature” was seen to structure history (the Rougon-Marcquet saga, the strike in Germinal etc.) as well compel individuals from within. For Postone, it is not “natural” Darwinian processes that do this but an historic process, capitalism. Another example of the same idea is Max Weber’s “spirit of capitalism.” Weber can be read, and wanted to be read, as saying that there are forces outside capitalism on which capitalism depends, such as religious ethics. However, Postone is suggesting that such “spiritual” Weberian forces as asceticism, compulsivity and hypocrisy (Weber’s famous triad) are internal to capitalism, structuring its motion. But, there are ambivalences to his theory, and especially ones concerning science and technology. Postone rejects the view, associated with traditional Marxism, that sees industrial production as a neutral, purely technical process that could be salvaged from capitalism and carried on in similar form in socialism. To criticize capitalism, he argues that we also need to criticize industrial production, or at least the form it has assumed in capitalism. The problem is that he simultaneously argues – based primarily on a famous passage in Grundrisse – that science and technology creates the preconditions for an overcoming of capitalism, since they enable human beings to create unprecedented “material wealth” in a way that relies less and less on human labor. Since in capitalism “value” can only be created by labor, capitalism increasingly comes to be characterized by a contradiction between the processes generating “wealth” and “value”. Unlike “value”, Postone appears to think that “wealth” is a category that it is fine to apply transhistorically. “Wealth” existed in precapitalist societies and must also be imagined as something that can exist in post-capitalist, socialist societies. What happens with capitalism is that the creation of “wealth” can only take place through the production of “value”, i.e. through the exploitation of labor and valorization of capital. However, by showing that “wealth” can be produced in abundance without relying on labor, science and technology open up possibilities of overcoming capitalism. Here Postone portrays science and technology, not as irremediably implicated in capitalism, but as potentially liberating forces that point beyond capitalism. That is of course fine, but the question then becomes how to distinguish the good and bad moments of science and technology. Postone calls for a transformation of not only of “relations of production” but also of the “mode of production”, but without giving us much in the way of explaining how much or how radically the latter needs to be changed.

Then there is the problem of dialectics. As mentioned, Postone confines Geist and totality to capitalism. This claim has some antecedents in earlier critical theory. Adorno, for instance, claims that the role of Spirit in capitalism is taken by “value”: “The objective and ultimately absolute Hegelian spirit [is] the Marxist law of value that comes into force without men being conscious of it” (Adorno). The posture of taking up arms against “totality” itself is of course also familiar from older critical theory. Adorno, however, never confined dialectics in toto to capitalism. Although Postone does allow for some forms of dialectical interaction (e.g. people changing their own nature reflexively through acting on nature or the reciprocal constitution of social practice and social structure), he argues that such interaction only becomes “directionally dynamic” in capitalism. In other words, dialectics in the sense of a historical logic or necessity only exists in capitalism. This raises the question of how capitalism can be overcome. If there is no Geist but capital, then dialectics cannot point the way out of capitalism. Liberation can only mean liberating oneself from dialectics, by creating a world in which it is no longer dominant.

The indication of the historicity of the object, the essential social forms of capitalism, implies the historicity of the critical consciousness that grasps it; the historical overcoming of capitalism would also entail the negation of its dialectical critique.

However, sometimes Postone himself seems to grasp the relation between capitalism and its outside dialectically, as when he uses the term “determinate negation” for the movement whereby capitalism is transcended. But if the overcoming of capitalism is a determinate negation, doesn’t that require the premise of a totality transcending the capitalist system, as Lukács thought?  Sometimes Postone writes as if the totality of capitalism were driven towards its own abolition by its inner contradictions. However, apart from the discussion of technology and wealth referred to above, it is hard to see that he specifies anywhere what kind of contraditions might bring about this self-abolition.

# Emergentic Philosophy or Defining Complexity

If the potential of emergence is not pregnant with what emerges from it, then emergence becomes just a gobbledygook (generally unintelligible) of abstraction and obscurity. What is this differentiation all about? The origin of differentiation is to be located in what has already been actualized. Thus, potential is not only abstract, but relative. Abstract, since, potential could come to mean a host of other things than that what it is meant for, and relative, since it is dependent on intertwinings within which it could unfold. Potentiality is creative for philosophy, through an expansive notion of unity through assemblages of multiple singularities helping dislodge anthropocentric worldviews that insist on rationale of the world as a solid and stable structure. A way out is to think in terms of liquid structures, where power to self-organize and untouched by any human static control allows for an existence at the edge of creative and flowing chaos. Such a position is tangible in history as a confluence of infinite variations, and rooted in materialism of a revived form. Emergence is a diachronic construction of functional structures in complex systems attaining a synchronic coherence of systemic behavior during the process of arresting the individual component’s behavior, so very crucial in ramifications for addressing burning questions in the philosophy of science, especially the ones concerning reductionism. Complexity investigates emergent properties, certain regularities of behavior that somehow transcend the ingredients that make them up. Complexity argues against reductionism, against reducing the whole to the parts. And in doing so, it transforms scientific understanding of far-from-equilibrium structures of irreversible times and of non-Euclidean spaces.