Introduction: Formalization and scientific systems
In Postmodern Philosophy and the Scientific Turn, I used the phrase ‘the scientific turn’ to characterize a philosophical paradigm widely utilized in twentieth-century Continental philosophy but which, I argue, originated primarily in analytic philosophy as a turn toward the language of logic, and toward discrete and formal computation in the philosophy of language and philosophy of mind.1 I made the claim that a number of significant postmodern philosophers and theorists embraced this turn, but that among these theorists it is referred to as the linguistic turn, and that the linguistic turn, like its correlate in analytic philosophy, finds its theoretical roots in a methodology broadly construed as mathematical or logical formalism. What I would like to do in this essay is to briefly discuss the nature of formalism, and then to focus on its relationship to materialism in order to ask if it is the case, as some newly emerging neomaterialist claims coming from Continental philosophy imply, that formal brain mechanisms and the matter of the brain will ultimately suffice to explain all psychologically described phenomena.2 I pursue this route in order to ask to how it is that postmodern philosophy seems to have given rise to materialism in contemporary Continental philosophy and because it appears to me that it is important to address the sudden rise of materialism in Continental thought, especially its negative relation to consciousness. Although poststructuralist and postmodern philosophies have been willing to set aside consciousness along with the idea of the subject, phenomenology in general has not. The impact of materialist critiques on phenomenology must then be taken into account.
The move toward formalist thinking had numerous originators, among them Bernhard Bolzano (1741–1848), who argued that science is a demonstrated theory that dispenses with verification. According to Jean Cavaillès, this means that science is independent of both the human mind and being in itself, and it becomes an ‘object sui generis’.3 This is not a trivial conclusion, for it indicates that science is not one object among others in a cultural milieu but that it is autonomous, capable of generating its own intelligible elements. Different sciences are unified by their common inclusion in this one system, ‘a self-enclosed dynamism’ without beginning or end that is therefore outside of time; in particular, it is outside of the ‘lived experience of a consciousness’.4 What this establishes is that science is seen to be an unending and unstoppable conceptual becoming, independent of what the scientist herself understands.
As Cavaillès states, for all the natural sciences, including the biological sciences, ‘growth occurs without external borrowing … [thus] there is a break between sensation or right opinion, and science’.5 This is why the structure of science is and only can be demonstration defined as logic, ‘the internal rule which directs it posits each of its steps’, as well as its essential traits: unity, necessary indefinite progression and closure upon itself.6 However, unlike the logical positivists, Cavaillès does not accept the idea that the theory of science is logic alone, for that position ends up abandoning even truth as correspondence and leaves only a coherence theory, which stipulates that so called ‘atomic’ statements or judgements of perception – the reported sensations of a particular observer at a specific moment in time, formerly said to be irrefutable and therefore foundational – are really only the result of syntactical commitments, that is, the arrangement of signs.7 The general problem with this approach is that syntactical formalization cannot complete itself by itself; it cannot help but refer to objects, so the system is not in fact self-enclosed.8 For philosophers like Cavaillès, the logic of a formal system requires an ontology to complete it; that is, in addition to the formal system it requires that there is a reference to an exteriority, to objects, and not just to other signs in the system.9 But for hardened formalists ‘all external questions are “metaphysical” and therefore nonsensical’, the ‘external’ referring only to systems of signs or, at most, to marks on paper and foregoing the necessity that signs are not objects and imply a reference to an external actuality.10
This is a problem to which postmodern philosophers sought to find a solution. Here, a bit of historical information may be revelatory. Cavaillès, a leader of the French Resistance, was captured and after several escapes recaptured, then murdered in 1944 by the Nazis.11 Georges Canguilhem, who was both a friend and colleague, was recruited into the Resistance by Cavaillès, where he served as a medical doctor. After Cavaillès’s death, Canguilhem wrote a book on the work and life of Cavaillès. In 1948, Canguilhem became the director of the Institut d’histoire des sciences at the Sorbonne. ‘Canguilhem also served from 1964 to 1968 as the President of the Jury d’Agrégation in philosophy, which provided him an institutional influence over the teaching of philosophy and which helped consolidate the future influence of students.’12 Among these students were Gilles Deleuze, Michel Foucault, Louis Althusser and Jacques Derrida. With this in mind, we might want to look more closely at ideas developed by Cavaillès that seem to have made their way into the work of postmodern philosophers.
Although Cavaillès stood in opposition to logical positivism, he is thought to have been influenced by the mathematician David Hilbert’s formalism.13 Both argue that the truth of mathematics is in the demonstration, in the method of mathematics, so that science cannot be the product of the intentions of scientists. It is rather science itself that demonstrates what is true or not, so ‘the credit should and does go to science itself’.14 Thus, mathematical objects – such as the square root of -1 – are merely the product of the mathematical system that produces them and outside of this formal-linguistic context they are meaningless. They represent no idea until and unless such formulations become objects of study.15
If mathematical language extends itself, introducing its own formal idealizations, then the universe of mathematical objects is always in the process of formation, a conceptual becoming that cannot be stopped, that will always be beyond the reach of individuals.16 As the mathematician Vladimir Tasić points out, the same thing can be said about ‘truth’, especially the claim that ‘all truth changes all the time’, a statement that cannot itself be proven to be true since no formal language can formulate its own theory of truth, and even higher concepts of truth are needed to do this.17 The implication is that if mathematics is always and endlessly formulating its own object, mathematical truth cannot possibly be formulated by finite human understanding. Gödel’s famous incompleteness theorem expresses this by claiming that higher concepts will have to be continued into the transfinite, part of a conceptual continuum that never ends.18 Yet even Gödel notes that such concepts are put to the test, judged in human practice, in the lifeworld, in the cultural, social and intellectual milieu. Let us now turn to an examination of how this is done in mathematical formalism, and how this might have led to neomaterialist philosophical positions.
Cavaillès’s chosen task seems to have been to reconcile formal logic with worldly applications. ‘Through the detour of abstract axiomatics, the formalist elevates himself to the general theory of formal systems and succeeds in constituting systems in which the structure has completely eliminated the content.’19 For this reason Cavaillès rejected at least some aspects of Edmund Husserl’s account of the relation between mathematics and the physical world which, for Cavaillès, remained too much embedded in such a logical empiricism. Essentially, if it is the case that ‘physical theory is simply an empty mathematical form applied to the invariant intuitive contents of the lifeworld’, then mathematics does not truly augment our knowledge of the lifeworld, but rather merely idealizes our power to predict.20 In other words, this is the old problem of the Kantian schematism that is supposed to bring together an empirical intuition and a radically heterogeneous concept.
Cavaillès thought he could do better. Thus, he proposed that matter or content is unintelligible without its concept, so the condition and the conditioned must be within one system, and the connection between them is realized anew in a ‘conceptual becoming that cannot be halted’.21 As Cavaillès argues, science cannot be the intermediary between the human mind and the empirical world. Science is not a cultural object. ‘Now science is regarded as an object sui generis, original in its essence, autonomous in its movement.’22 Furthermore, science is unified and its unity is the ‘self-enclosed dynamism’ that is outside of the lived experience of consciousness, a conceptual becoming that cannot be stopped. Experience is thereby the incorporation of the world into the scientific universe.23 This means that the assertions of science appear as the ‘self-illumination of the scientific movement’, and that the structure of science simply ‘speaks about itself’. It is demonstration; a unified, necessary and indefinite progression.24
Although at this point I can see significant resonances between the position of Husserl, that of Gilles Deleuze, Cavaillès, and that of Michel Foucault, it is more productive for my purposes here to turn directly to the neomaterialist position of Quentin Meillassoux who, without mentioning Cavaillès in his texts or placing his work in the bibliography of his primary book, After Finitude, nevertheless seems to have followed Cavaillès to a rather high degree. Like Cavaillès, Meillassoux formulates his thesis in opposition to Kant. He proclaims the existence of worldly manifestations that arose prior to human existence, and that may continue after human extinction, so as to reveal a temporal discrepancy between thinking and being, between what is known and what exists.25
This is a discrepancy that arises with modern science because it is only insofar as the formalist mathematization of nature has come to define modern science that this question has even been raised.26 This discrepancy rests originally on the fact that science is not based on simple observations but on data that has been produced, processed and quantified by increasingly elaborate measuring instruments.27 Meillassoux renders this statement in the context of another distinction, the distinction between inorganic matter and life, and this distinction reinstates an older one, the distinction between primary and secondary qualities. Primary qualities are inseparable from the object and belong to things. The modern modification is that primary qualities can be formulated in mathematical terms.28 Secondary qualities are affective or perceptual, and so exist only as a relation between things and living beings.29
The division between life and matter, the organic and inorganic is hard and fast for Meillassoux, and does not take into account either the physics of energy and matter or evolutionary biology. Matter or the inorganic appears to be anything that is not life, such as the luminous emission of a star or an isotope undergoing radioactive decay.30 Scientific statements about matter can be formulated as mathematical data, for example, that the earth began occupying a certain volume that varied through time, starting 4.56 billion years ago. The claim is that it is safe to assume that statements backed by mathematical data are true unless and until the theory that produced this data is replaced by more elegant or accurate theories.31 This ‘realist attitude’ is set forth as normal and natural for the scientist, as well as the ultimate regime of meaning for understanding matter.32 Like Cavaillès, who opposes the Kantian a priori, Meillassoux objects to what he calls the correlationalist position in Kant and Husserl. The correlationist is a philosopher who maintains that it is impossible to hold the realms of subjectivity and objectivity as independent of one another.33 In other words, correlationalism is when events take place for thought, for a thinker.
The first objection Meillassoux anticipates from the correlationalist, whom he characterizes as an idealist, is that if there had been a witness to the origin and emergence of the planet earth, this occurrence would have been perceived in the manner the data tells us it did occur. The origin of the earth is no different, in fact, from a vase that falls off a shelf and breaks in a room where no one sees or hears it.34 What is different, however, for Meillassoux is that the origin of the earth is an event anterior to human terrestrial life, and hence, anterior to givenness, to perception or thought. It is a nongiven, an unwitnessed occurrence, that is not of the time of consciousness but another time, the time of ‘science’ which engenders – that is gives rise to or makes possible – the time of consciousness and of life.35
Meillassoux’s position leads him to ask how is it that ‘mathematical discourse [is] able to describe a world where humanity is absent, a world crammed with the things and events that are not the correlates of any manifestation … not the correlate of a relation to the world’?36 Mathematical discourse is described as a temporal discrepancy between thinking and being with respect to statements about events prior to and after the existence of human beings.37 It is a question, then, about scientific discourse, about which statements can be verified or falsified, and how throughout his discussion Meillassoux, like Cavaillès, refers to what ‘science’ has discovered or said: ‘Science could have discovered a synchronicity between humanity and the world’; or, ‘if science had discovered this synchronicity it would still have been a discovery’; or, that it is ‘the capacity of scientific discourse which concerns us’.38
Modern science arose when its statements became part of a cognitive process and when these statements became hypotheses able to be corroborated by experiments as instances of knowledge. All of this is part of the discourse of empirical science and its rational debate is what ‘science made it meaningful to debate’ and ‘to disagree about’.39 This argument continues until there is eventually an acknowledgement that even though science gives the means to rationally favour one hypothesis over another to humans it is, in the end, human beings who do the rational favouring.40 In other words, this argument inevitably collapses because something is, in the end, being thought, some choice is made, and so speculative materialism is not the inevitable remainder. Except possibly along some narrow cultural extremes, it seems largely indisputable that whatever is mathematically conceivable is certainly possible, but declaring it absolutely possible has the effect, in this case, of using this claim to an absolute status to rule out theories of consciousness without positing an alternative to how something can be thought without consciousness. Moreover, the claim that whatever is mathematizable can be posited hypothetically as existing independently of humans seems to be uncontroversial and is generally acknowledged.
Given this, what we might wish to pay attention to here is that, in spite of Meillassoux’s admission that Kant radically promoted science over metaphysics, too many philosophers have veered away from speculative materialism toward transcendental idealism.41 It seems that speculative materialism presents us with a dichotomy: either speculative materialism or correlationalism. That is, either thought now thinks the events that occurred prior to all thought (because science gives it) or the events occurred prior to the existence of thought but for thought through the necessary a priori forms by which a thinking being thinks. The question is: what would be necessary with respect to human thinking for Meillassoux’s thesis to hold? The apparent claim that something occurred for thought before there was any thought just seems to be exaggerated, and surely no Kantian or Husserlian would make this sort of claim. So, we can ask, is Meillassoux asserting a false dichotomy by exaggerating the claims of philosophers whose position he opposes?
We might also ask if these positions leave Meillassoux subject to the same unanswered questions as the analytic philosophers of mind. Like those materialists Meillassoux has no coherent account of psychological processes of the mind including thoughts, beliefs, desires and sensory experience, as he refers only to cognitive processes and, like the philosophers of mind, he offers no convenient explanation for how psychological phenomena are related to physical systems, either causally or functionally. Nor does Meillassoux tell us how it is possible for material phenomena to give rise to conscious states understood as subjective qualities. In fact, these considerations are precisely the ones that Meillassoux is eager to eliminate by claiming that aspects of objects formulable in mathematical terms – such as length, width, movement, depth, figure and size – are purely properties of the object in itself and have nothing to do with any subject’s relation to the world.42 Similarly, as he states, the truth or falsity of a physical law is not established with regard to human existence – a conclusion that is indicative of why philosophy is defined by him primarily, if not exclusively, as the invention by philosophy of strange forms of argumentation, making use of internal mechanisms for regulating its own inferences and eliminating the inadequacies of reasoning.43 This is particularly evidenced in his extensive argument against the Christian dogmatist and the agnostic, which leads him to the conclusion that there are no necessary entities, only necessary contingency; and – perhaps more telling – that following the example of paraconsistent logic, at least some contradictory statements are necessary.44
In choosing this route, Meillassoux has accomplished at least one thing in particular: he has released philosophy from dependence on any and all versions of subjectivity insofar as they all treat something abstract, something conceptual, as if it were concrete reality. Included in this list are Leibniz’s monad, Schelling’s Nature, Hegel’s Mind, Schopenhauer’s Will, Nietzsche’s Will to Power, Bergson’s ontological memory and Deleuze’s conception of Life. For each of these concepts, the contention is that there is no separation between the act of thinking and its content, a position contrary to the speculative realist position, that absolute reality is an entity without thought. Thus, thought is not necessary insofar as we can think a given reality by abstracting from the fact that we are thinking it but because thought, on this account, is random and immanent to ‘contingent atomic compounds’.45 In propositional logic, an atomic statement such as ‘A’ is said to affirm its own truth, and all atomic sentences are purely contingent. Compound atomic sentences are built from atomic sentences using sentence connectives.46 As Ken C. Klement put it,
propositional logic does not study those logical properties and relations that depend upon parts of statements that are not themselves statements on their own. This would include parts such as the subject and predicate of a statement, because simple statements are considered to be indivisible wholes, thus are not divisible into subject and predicate.47
With this, we enter the realm of axiomatic systems, a set of properties that are consistent, thus contain no contradiction, from which other properties may be derived.48 Nevertheless, given Cantor’s theorem, that the cardinal number of any set is lower than the cardinal number of the set of all its subsets, these systems or sets are never complete, never able to be totalized.49 Cardinality refers to the size of a set, not the order of its members, and if two sets each have an infinite number of elements, one may have a greater cardinality, which is to say that one may have a ‘more infinite’ number of elements and is therefore called the transfinite or an aleph, without reaching an absolute infinity.50 One may construct an unlimited series of infinite sets, each of which is some quantity superior to that of the set whose parts it collects together, but the series can never be totalized, never brought to an end as this or that ultimate quantity. Using Cantor’s concept of the transfinite, the quantifiable totality of what is thinkable cannot be thought without falling into contradiction and therefore cannot be said to even exist.51
Meillassoux admits that other systems are entirely possible, that standard set-theory is only one among many, but we have no way of knowing if any of them would guarantee necessity. Nevertheless, set-theory thinks the transfinite, the possible as untotalizable. Meillassoux thinks or assumes the truth of this system, which allows him to cease believing in the existence of necessary physical laws as opposed to mere stability. Thus Kant is scolded for claiming that, absent the transcendental ground of unity or, in other words, the claim that conceivable possibilities constitute a totality, there would be no knowledge. Kant bases this on a pre-Cantorian application of the calculus of probability to the world as a whole. If it is the case that the possible either does or does not constitute a totality, there is nothing to prevent one from choosing the latter and concluding that physical laws carry no necessity but are merely stable.52
Mind-brain interaction
Meillassoux states repeatedly that science gives the discourse of empirical science to cognition. Science relies on self-evident axioms that are consistent but seem not to originate with the human mind since they are given to cognition to think. How is it possible for science to be the structure that gives cognition its thoughts? If it is possible, what are the implications and what does this imply? Contemporary neuropsychological research frequently puts forth the thesis that the brain consists of material particles and fields that ultimately explain all mental phenomena. Thus terms like feeling, knowing and effort play no role; they are not ‘primary causal factors’.53 This view has been amplified by brain-imaging technology, which has correlated areas of the brain with a large number of mental activities including learning, memory and symbol manipulation. This has led many scientists to conclude that such measurable properties of physical brain mechanisms are all that is needed to explain mental or psychological events.54 In part, this is due to the classical physical models that have been used to understand the functional activity of the brain.
According to these models, all causal connections between observables are explainable in terms of mechanical interactions between material realities, an effect of experimental paradigms that focus primarily on changes in brain activation as primary variables used to explain observable behavioural changes in subjects who are primarily passive in the experimental situation.55 These situations reinforce the supposition set out above, that science gives the discourse of empirical science to cognition, and subsequently the material activation of the limbic system, the hypothalamus, the thalamus and subthalamus structures generates emotion and memory, while activation of the cerebral cortex generates thought, language and consciousness.
Although Meillassoux argues that nature’s laws are merely stable, and not absolute, this does not mean that they are not deterministic, meaning that the state of the physical world of matter at one time determines the state of the physical world at a future time although not necessarily absolutely. Of course physics must make use of mathematical models for the sake of intelligibility. Thus, a physical system defined by deterministic chaos is relatively indeterminate because even though the laws governing the field do not alter, there is an indeterminacy regarding which elements in the field will interact with one another, making predictability more difficult. It is true that cosmologists such as Lee Smolin have posited that when working on ‘the theory of spacetime and quantum space … we draw pictures which are networks of relations and how they change in time and our pictures look just like pictures of ecological networks that these people study’.56 Smolin hypothesizes that there is a deep relation between Einstein’s notion that everything is just a network of relations and Darwin’s notion of an ecological community as a network of individuals and species in relationships which evolve. But the question remains: is such a model necessarily materialist?
Perhaps of greater importance in Meillassoux’s model is that brains seem to be mechanical, the effect of material interactions governed by physical laws or forces. Emotion and cognition are redundant, and intentions are misleading illusions. They are either epiphenomenal by-products of matter or they are identical with the patterns taking place in the brain; they are so-called emergent properties.57 However, if it is possible within the conceptual framework of classical physics to take away consciousness while leaving intact the properties that enter into the materialist construct, namely the locations and motions of the tiny physical parts of the brain and its physical environment, then it is possible that materialism is either incomplete or simply incorrect.58 As we can see, the problem remains that of the connection between the mathematical system, consciousness and the world; and, for us, that is a question of the relation between physics and philosophy.
It has been pointed out by neuroscientists that understanding the connections between phenomena in terms of the mechanical interactions of material entities conforms to the conception of the world developed by classical physics. However, ‘terms such as “feeling”, “knowing” and “effort”, because they are intrinsically mentalistic and experiential, cannot be described exclusively in terms of material structure’.59 The claim has been made, however, that human choices and intentions can be described more accurately by means of quantum-based theories. This, I would argue, is an approach that was signalled in Maurice Merleau-Ponty’s early work, The Structure of Behavior.60 There, Merleau-Ponty raises the question we have been asking here: how to understand the relation between behaviour and physical events? He takes up the question by introducing the concept of form: form for Merleau-Ponty is not a material, physical reality but an object of perception, a perceived whole, the ‘empty x’. ‘This unity is the unity of perceived objects …. It is encountered in physics only to the extent that physics refers us back to perceived things as to that which it is the function of science to express and determine.’61
In other words, for phenomenology physical form is not the foundation or cause of the structure of behaviour. It is rather an object of perception, which in this case is expressed as an idea. The physical form is an object of knowledge (the empty x) of the fields of force and the dynamic unities of perception: just as perceived objects change properties when they change place, so in the physical structure of, for example, system wave mechanics the wave associated with the entire system propagates itself in an abstract configurational space.62
A wave function is the mathematics that accounts for how a wave varies in space and time. It describes the probable values of the attributes of quantum objects, but the equation for calculating the quantum wave function ‘has defied all attempts to give it an interpretation in terms of physically observable entities’.63 Once an actual physical observation/measurement is made, the wave collapses into a single determinate value, yet no one seems to know why. Unlike system wave mechanics, classical science appears to have constructed the image of an absolute physical reality and then proposed that perceptual structures are simply manifestations or projections of this fundamental ontological foundation. But the phenomenological concept of form indicates that this is not so: although the laws of physical reality conceptualize the perceived world, reference to the perceived world is nevertheless essential to knowledge of the physical world.64
Werner Heisenberg discovered the first successful mathematical quantum theory in 1925. He postulated that quantum behaviour, which represents the unobserved world, must consist of ‘possibility waves’ – that is, when not observed, the world might exist as waves of possibility.65 Atoms and elementary particles might form a world of potentialities or possibilities with numerous tendencies and not a world of determinate things. ‘As long as they remain unobserved, events in the atomic world are strictly in the realm of possibility … [but] because certain facts have become actual in our world, not everything is equally possible in the quantum world.’66 It is only in the act of measurement, an act chosen and carried out by human beings with consciousness, that quantum possibility becomes an actual event.
But in addressing quantum physics, Merleau-Ponty wisely asks: ‘what can one say, in a serious way, when one lacks technical competence?’67 The philosopher, he concludes, can best address that moment where science connects with prescientific being, at the point where science requires an image of reality and a language that gives meaning to its formalist structures.68 Merleau-Ponty therefore turns his attention to the question of probability in the standard model of quantum physical behaviour, stating that existing things such as particles or waves might best be taken not as individual but as generic or species behaviour.69 More specifically, in the prequantum theory classical model, apparatuses utilized to measure the movement of atoms can still be understood to be ‘prolongations of our senses … a more precise sensoriality’.70 That this is no longer the case for phenomena is nontrivial. The quantum measuring apparatus collapses the quantum wave into a particle provoking the appearance of a subatomic particle, fixing or sampling it in relation to the wave and leaving in question the gap between what is ‘perceived’ and what can be known, so that it appears to be the case that ‘known nature is artificial nature’ – that it is an effect of the measuring apparatus.71
But we must also be aware of the fact that the quantum measuring process involves the object to be measured, the measuring apparatus – both of which belong to the external world – and the observer herself who does not, who has a relationship with herself and whose observation and thought makes possible the emergence of an individual existence.72 Merleau-Ponty recognizes that formalist accounts of physics allow a lot of freedom but signify no reality because as formalist systems they are in actuality a radical nominalism. Likewise, against what would amount to a materialist position like Meillassoux’s, he argues that relations between reality and measurement must be conceptualized outside of the in-itself/representation dichotomy, thus outside of so-called correlationalism that Meillassoux attributes to phenomenology.73
What Merleau-Ponty calls for from a philosophy that corresponds to quantum physics is something both more realist – specifically, a philosophy not definable in transcendental terms – and more subjectivist, in the sense of a situated incarnate physicist who does not claim to be a universal and transcendental ‘I think’.74 Thus we may distinguish a plane of reality in which physical systems from a second plane of reality exist, that of the process of taking the measurement, and also a third plane, that of the structure independent of the measurement process and relative to the species being studied. Because the structural relations refer to the mathematical forms needed to describe the relations of the subject to the object and also to the theory in which they intervene as the schematism of the relation between observers and objects, this structure is for Merleau-Ponty ‘comparable to the Platonic objectivity of the idea vis-à-vis its sensible realizations’.75
Once again, this conception of structure takes us back to the perceived thing, not as a finished product but in its full ambiguity, which would affirm that there are ambiguous beings that are neither waves nor particles. If we accept, for example, that the perceived wind is ‘a continuation of movement without mobiles, of behaviors without subjects’ then what is perceived are beings that are probable, indeterminate, negative (defined by their absence) and neither infinite nor finite.76 Why is this the case? Because nothing about the scale of particles or waves can be understood without the existence of an incarnate subject whose perceptual experience includes the experience of space as ambiguous and thus not as an immediate given.77
Although Merleau-Ponty does not express his conceptualization of quantum theory in terms of what I will clarify as Intuitionist logic, his position nevertheless appears to me to able to be commensurate with that of physicist Fotini Markopolou, whose work I utilized in both The Universal (In the Realm of the Sensible) and in Postmodern Philosophy and the Scientific Turn. Markopolou characterizes her work as the effort to describe what the universe looks like from inside, eschewing formalist mathematical logic in favour of Intuitionism. She proposes a causal structure of space-time, the view from inside, meaning, what an observer inside the universe can observe.78 Arguing against classical models precisely because they lead to uncertainty, Markopoulou suggests utilizing causal sets, that is, large collections of events in discrete space-time partially ordered by temporal causal relations. Moreover, Markopoulou proposes to work with evolving sets that bring the causal past of each event as well as the causal structure of each event into a causal set. She further suggests that evolving sets satisfy a particular algebra called Heyting algebra, which utilizes a nonstandard logic whose historical development has been related to understanding the passage of time. Intuitionistic logic does not adhere to the Law of Excluded Middle. Whereas the classical Boolean mathematician believes that a statement x is true or false whether or not she has proof for it, Intuitionism does not allow proof by contradiction. From the onset, it does not consider x to be true or false unless there is a proof for it. In other words, without a proof the option is open as to whether x may be true tomorrow or false tomorrow. Intuitionistic logic is thus suited to time evolution, where certain physical statements become true at a certain time.79
This logic appears to correspond to Merleau-Ponty’s critique of the logic of classical physics. Quantum physics is probabilistic – that is, for objects with tiny masses and sizes such as atoms and molecules the statistical distribution of energy microstates in a system can be predicted but not that of individual particles. Thus quantum can be said to be ambiguous or indeterminate where this means uncertain, revealing not realities, but two phantoms, the particle and the wave.80 The problem lies in thinking that classical Boolean logic is the only valid logic and that the physical incompossibility of particle and wave is equivalent to and determined by logical incompossibility in a logical discourse in which the law is that of existence and nonexistence, and the passage from one to the other, the movement of time and change, the open future, the nonexcluded middle, is forgotten. It is the limitation of classical logic to consider only positive determinations and to be blind to the temporal movement that is change.81 The latter is a logic of ambiguity; it is still a logic but temporal and probabilistic, as we have defined this above.
What becomes crucial here for the reconceptualization of quantum physics as prescribed by Merleau-Ponty is that ‘a theory with internal observables is fundamentally different than a theory describing a system external to the observers’ insofar as this theory refers to observations made from ‘inside’.82 For physicists, this means inside the universe, a point of view from which such observations can only be partial. That is, they contain information that is in the causal past of an observer in a particular region of spacetime but, significantly, they do not contain predictions, meaning information about the future, information that should be obtainable from a classical dynamical perspective. When this partiality is represented in terms of light cones – light rays that form the outer boundary of the past in roughly the shape of a cone – information that constitutes a particular point of view is shown to be the effect of mutually influencing and overlapping light cones. If the causal past of an event consists of all the events that could have influenced it, these influences travel from some state in the past at the speed of light or less. The light rays arriving at an event form the outer boundary of the past of an event and make up the past light cone of an event. Under these conditions, the causal structure of states evolves and the motion of matter is a consequence of that evolution. Here we have the Intuitionist conception according to which intuition is a process of building or constructing, a time-bound process beginning in the past, existing in the present and evolving into an open future.
Because the information from the past evolving as the present into an open future occurs on the quantum scale, the scale of photons, the world can be said to be composed of discrete states that may be on a very small scale but are nevertheless discrete in space and time. Under such conditions, what might be observed? Called spin network graphs, representations of mutually influencing and overlapping light cones have been used to model spatial geometry in quantum physics as well as evolving events in a causal set, yielding a quantum causal history.83 Unlike other models, one significant implication of these graphs is that the manifold of space-time is not pregiven. Rather than a dynamically changing form of content and form of expression taking place in a preestablished space-time manifold yet produced or assembled from outside by the elements of that manifold, the model of quantum causal histories specifies that space-time and the states that evolve – the stage and the actors – evolve together.84 This is particularly useful for the exploration of states that occur at the Planck scale of quantum states where classical physics fails; in addition, it allows for the construction of a point of view that is not that of an atomistic individual but of a network. It is a point of view according to which different observers ‘see’ or ‘live’ partly different, partial views of the universe – partial views that nonetheless overlap.
What I have been pursuing throughout my work is the possibility that this structure and these processes might involve the participation of vulnerable and sensitive beings in an ontological spatiotemporalization, an ever-changing perspective made up of a crowd of perspectives in the heterogeneity of space and time. Such a perspective, if it is thinkable, if it is real, could manifest itself as a sort of history; but is more like a complex causality – layers and layers of states always susceptible to realignment, its patterns and particles resolving in a point of view that is the effect of a crowd of influences and itself contributing to a crowd of influences. These light rays, conical flows of information that are often imperceptible, influence one another and in this they influence the sensibility of all things.
As models of quantum consciousness proliferate, it is my hope that they will supplant the rough materialism favoured today in philosophy as well as in some quarters of natural science. If it is the case, as John von Neumann posited in the 1930s, that the quantum physical world consists of nothing but possibility waves, then the collapse of the wave function requires something outside of the physical realm. If that something truly is, as von Neumann concluded, consciousness, then the world remains in a state of possibilities except wherever a conscious mind takes the measure of that world and actualizes it.
Notes
1.Vladamir Tasić, Mathematics and the Roots of Postmodern Thought (Oxford: Oxford University Press, 2001), pp. 27, 31.
2.Jeffrey M Schwartz, Henry P. Stapp and Mario Beauregard, ‘Quantum Physics in Neuroscience and Psychology: A Neurophysical Model of Mind–Brain Interaction’, Philosophical Transactions of the Royal Society B 360 (2005): 1309.
3.Jean Cavaillès, ‘On Logic and the Theory of Science’, trans. Theodore J. Kisiel, in Theodore J. Kisiel and Joseph J. Kockelmans (eds), Phenomenology and the Natural Sciences: Essays and Translations (Evanston: Northwestern University Press, 1970), pp. 370–1.
4.Ibid., pp. 371–2.
5.Ibid., p. 372.
6.Ibid., p. 373.
7.Ibid., p. 350. As Kirk Ludwig remarks, such statements are called ‘protocol sentences’:
If we want to know whether a given sentence is meaningful or not, we must decide whether or not we associate with it a method of verification, for the meaning of a sentence lies in the method that we would employ to verify or falsify it. This means that we must specify the conditions under which it would be possible to verify the sentence. In stating what those conditions are, of course, we must use sentences. Unless we want to be involved in an infinite regress (or a circle), there must be some sentences that we can verify directly, which will then form the foundation for verifying other sentences. Those sentences are the protocol sentences. Protocol sentences were taken (initially at least) to express conditions whose obtaining or not is directly verifiable. (‘Carnap, Neurath, and Schlick on protocol sentences’. Available from http://www.clas.ufl.edu/users/ludwig/PHP5785/set8__2009.pdf [accessed 7 January 2013].)
8.Cavaillès, ‘On Logic’, p. 350.
9.Ibid., p. 350.
10.Ibid., p. 350.
11.Peter Hallward, Concept and Form: The Cahiers pour l’Analyse and Contemporary French Thought. Available from http://cahiers.kingston.ac.uk/names/cavailles.html [accessed 28 January 2015].
12.Ibid.
13.Tasić, Mathematics, p. 85.
14.Ibid., pp. 87–8. Emphasis added.
15.Ibid., pp. 86, 87
16.Ibid., pp. 88, 89.
17.Ibid., p. 88.
18.Ibid., p. 89, citing Kurt Gödel.
19.Jean Ladrière, ‘Mathematics in a Philosophy of the Sciences’, trans. Theodore J. Kisiel, in Kisiel and Kockelmans (eds), Phenomenology, p. 472.
20.Cavaillès, ‘On Logic’, p. 351.
21.Ibid., p. 360. See also Tasić, Mathematics, p. 86.
22.Cavaillès, ‘On Logic’, p. 371.
23.Ibid., p. 372.
24.Ibid., p. 373.
25.Quentin Meillassoux, After Finitude: An Essay on the Necessity of Contingency, trans. Ray Brassier (London: Continuum Books, 2008), p. 112.
26.Ibid., p. 113.
27.Ibid., p. 114.
28.Ibid., pp. 2–3. The distinction is explicitly that of John Locke’s but Meillassoux traces it back to René Descartes as well.
29.Ibid., p. 2.
30.Ibid., p. 9.
31.Ibid., p. 12.
32.Ibid., p. 14.
33.Ibid., p. 5.
34.Ibid., p.19; I have simplified the argument for the sake of clarity.
35.Ibid., pp. 20–1.
36.Ibid., p. 26.
37.Ibid., p. 112.
38.Ibid., p. 113.
39.Ibid., p. 114.
40.Ibid., p. 114.
41.Ibid., pp. 120–1.
42.Ibid., p. 3.
43.Ibid., pp. 76–7.
44.Ibid., pp. 55–7, 64, 76–9. Full analysis of these arguments requires an essay in itself. We will therefore restrict our analysis to the more familiar arguments against causal necessity.
45.Ibid., pp. 36–7. What is here called ‘Life’ on Deleuze’s behalf is presumably more usually referred to in Deleuze’s own work as ‘desire’.
46.Nancy A. Stanlick, ‘Logic: Clarification of Terms and Concepts in Sentence Logic’. Available from http://pegasus.cc.ucf.edu/~stanlick/slterms.htm [accessed 28 January 2015].
47.Kevin C. Klement, ‘Propositional Logic’, Internet Encyclopedia of Philosophy. Available from http://www.iep.utm.edu/prop-log/ [accessed 28 January 2015].
48.Carl Lee, ‘Axiomatic Systems’, Topics in Geometry. Available from http://www.ms.uky.edu/~lee/ma341/chap1.pdf [accessed 10 January 2013].
49.Eric W. Weisstein, ‘Cantor’s Theorem’, MathWorld – A Wolfram Web Resource. Available from http://mathworld.wolfram.com/CantorsTheorem.html [accessed 28 January 2015].
50.Aaron Krowne, ‘Cardinality’ (version 21), PlanetMath.org. Available from http://planetmath.org/ [accessed 10 January 2013].
51.Meillassoux, After Finitude, p. 104.
52.Ibid., p. 107.
53.Schwartz, Stapp and Beauregard, ‘Neuroscience’, p. 1.
54.Ibid., p. 2.
55.Ibid., pp. 3–4.
56.Lee Smolin, ‘Cosmological Evolution’, Edge. Available from http://edge.org/conversation/-cosmological-evolution [accessed 28 January 2015].
57.Schwartz, Stapp and Beauregard, ‘Neuroscience’, pp. 7–8.
58.Ibid., ‘Neuroscience’, p. 9.
59.Ibid., ‘Neuroscience’, p. 2.
60.Maurice Merleau-Ponty, The Structure of Behavior, trans. Alden L. Fisher (Boston: Beacon Press, 1967). Originally published in French as La Structure du Comportment (Paris: Presses Universitaires de France, 1942). Page references are given first to the English translation then the French original.
61.Ibid., p. 144/156; emphasis added.
62.Ibid., p. 144/156. This appears to be a reference to what mathematicians call ‘state space’.
63.John Casti, Complexification: Explaining a Paradoxical World Through the Science of Surprise (New York: HarperCollins, 1994), pp. 205–6.
64.Merleau-Ponty, The Structure of Behavior, p. 145/157. This concept of form reverses the view of the natural sciences that perceptual reality is only a projection of an absolute physical reality.
65.Nick Herbert, Elemental Mind, Human Consciousness, and the New Physics (New York: Penguin Books, 1993), p. 157.
66.Herbert, Elemental Mind, pp. 158–9.
67.Maurice Merleau-Ponty, Nature. Course Notes from the College de France, trans. Robert Vallier, compiled by Dominique Séglard (Evanston: Northwestern University Press), p. 89. Originally published in French as La Nature, Notes, Cours du Collège de France, établi et annoté par Dominique Séglard (Paris: Éditions du Seuil, 1995), p. 125. Page references are given first to the English translation then the French original.
68.Ibid., p. 90/125.
69.Ibid., p. 92/128.
70.Ibid., p. 93/129–30.
71.Ibid., p. 93/130.
72.Ibid., p. 94/130–1.
73.Ibid., p. 96/133.
74.Ibid., p. 97/134.
75.Ibid., p. 98/136.
76.Ibid., p. 99/137.
77.Ibid., pp. 99–100/137.
78.Fotini Markopoulou, ‘The internal description of a causal set: What the universe looks like from inside’, Communications in Mathematical Physics, 211, (2000): 559–83. Available from http://arxiv.org/pdf/gr-qc/9811053.pdf [accessed 28 January 2015].
79.Ibid.
80.Merleau-Ponty, Nature, p. 91/127.
81.Ibid., p. 92/128.
82.Fotini Markopoulou, ‘An insider’s guide to quantum causal histories’, Nuclear Physics – Proceedings Supplements, 88 (2000): 308–13. Available from http://arxiv.org/abs/hep-th/9912137 [accessed 28 January 2015].
83.Ibid.
84.Fotini Markopoulou, ‘Planck scale models of the universe’, in John D. Barrow, Paul C. W. Davies and Charles L. Harper, Jr (eds), Science and Ultimate Reality: Quantum Theory, Cosmology, and Complexity (Cambridge: Cambridge University Press, 2002), pp. 550–63. Available from http://arxiv.org/abs/gr-qc/0210086 [accessed 28 January 2015].