I posted the following in the Yahoo Adult Development forum and am cross-posting here. I'll keep you apprised of some key responses, provided I get any: 

Building on the post below* regarding Lakoff's embodied reason, he seems to call into question the type of abstract reasoning usually found at the formal operational level. This appears to be false reasoning based on the idea that reason is abstract, literal, conscious, can fit the world directly and works by logic (also see for example this article ). If formal reasoning is false wouldn't this call into question some of the assumptions of the MHC? That perhaps this "stage" is a dysfunction instead of a step toward post-formal reasoning? 

Now Lakoff has his own hierarchy of how embodied reason develops: image-schematic, propositional, metaphoric, metonymic, symbolic. (See for example "Metaphor, cognitive models and language" by Steve Howell.) So I'm wondering how the MHC takes into account Lakoff's work here and how it answers his charge of false reason? Terri Robinett noted in his Ph.D. dissertation (at the Dare Association site) that "work has already begun by Commons and Robinett (2006) on a hierarchically designed instrument to measure Lakoff’s (2002) theory of political worldview." So perhaps you can shed some light on this? 

* This is the referenced post: 

Since Michael brought up Lakoff as perhaps being "at right angles to the stage dimension" I read this by Lakoff this evening: "Why 'rational reason' doesn't work in contemporary politics." He distinguishes between real and false reason, the former being bodily based and the latter existing in some sort of objective, abstract realm. Very interesting indeed. Here are a few excerpts: 

"Real reason is embodied in two ways. It is physical, in our brain circuitry. And it is based on our bodies as the function in the everyday world, using thought that arises from embodied metaphors. And it is mostly unconscious. False reason sees reason as fully conscious, as  literal, disembodied, yet somehow fitting the world directly, and working not via frame-based, metaphorical, narrative and emotional logic, but via the logic of logicians alone."
"Real reason is inexplicably tied up with emotion; you cannot be rational without being emotional. False reason thinks that emotion is the enemy of reason, that it is unscrupulous to call on emotion. Yet people with brain damage who cannot feel emotion cannot make rational  decisions because they do not know what to want, since like and not like mean nothing. 'Rational' decisions are based on a long history of emotional responses by oneself and others. Real reason requires emotion."

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From chapter 9 of The Number Sense (link above), the section "Platonists, formalists, intuitionists":

"For some, traditionally labeled Platonists, mathematical reality exists in a abstract plane, and its objects are as real as those in everyday life.... For an epistemologist, a neurobiologist, or a neuropsychologist, the Platonist position seems hard to defend--as unacceptable, in fact, as Cartesian dualism is as a scientific study of the brain.... Platonism leaves in the dark how a mathematician in the flesh could ever explore the abstract realm of mathematical objects....in what extrasensory extrasensory ways does a mathematician perceive them?" (242).

As we already know from the thread above, through false reason, itself divorced from embodiment in the Platonic realm of ideas.

Nunez was co-author with Lakoff of Where Mathematics Comes From. Here is a site with his publications including books, book chapters and articles. Therein one can download the Preface, Introduction, Table of Contents and first four chapters of WMCF. From the Preface about the romance of mathematics, which they then meticulously debunk. Quote:

• Mathematics is abstract and disembodied—yet it is real.
• Mathematics has an objective existence, providing structure to this universe and any possible universe, independent of and transcending the existence of human beings or any beings at all.
• Human mathematics is just a part of abstract, transcendent mathematics.
• Hence, mathematical proof allows us to discover transcendent truths of the universe.
• Mathematics is part of the physical universe and provides rational structure to it. There are Fibonacci series in flowers, logarithmic spirals in snails, fractals in mountain ranges, parabolas in home runs, and pi in the spherical shape of stars and planets and bubbles.
• Mathematics even characterizes logic, and hence structures reason itself—any form of reason by any possible being.
• To learn mathematics is therefore to learn the language of nature, a mode of thought that would have to be shared by any highly intelligent beings anywhere in the universe.
• Because mathematics is disembodied and reason is a form of mathematical logic, reason itself is disembodied. Hence, machines can, in principle, think.

More from The Number Sense:

"The discoveries of the last few years....these empirical results tend to confirm...that number belongs to the 'natural objects of thought,' the innate categories according to which we apprehend the world" (244).

Here again are the basic categories and image schemata of L&J.

At a website called Edge* there was a discussion of Dehaene's paper, "What are numbers, really?" The discussion included Dehaene, Lakoff, Pinker and Nunez, among others. Quite interesting and along the lines of the above.

* Quite an interesting site. Much to explore there.

So let's recall this from earlier in the thread, from "Introduction to the model of hierarchical complexity." This is an extended version from the previous quote:

"But if one does not understand the difference between the ideal and the real one can get into trouble. The failure of the Pythagorean school rested with its need to make its assertions absolute. How could one conduct science or have knowledge in general without the possibility that this knowledge corresponds with reality? Later, Plato handled this problem by rejecting the correspondence account of truth. We cannot ever know the truth in its complete and pure form. Anything we can say about reality is only a likely story of the ideal truth.

Here, the ideal truth is the mathematical forms of Platonic ideal. An essential element of science is direct observation and interaction with the world. But Plato set forth a very different doctrine, to the effect that knowledge cannot be derived from the senses; real knowledge only has to do with concepts. The senses can only deceive us; hence we should, in acquiring knowledge, ignore sense impressions and develop reason. In codifying such logical reasoning, Aristotle (384–322 BC) set down rules of inference and recognized the importance of axioms for logic, postulates for the subject at hand, definitions of terms, and the importance of giving logical arguments that start with the postulates. By combining Aristotle’s precise formulation of logic with Thales’s method, the main elements of modern science were then in place. Most philosophic analyses of the philosophy of Thales come from Aristotle. Thales is credited as the first person about whom we know to propose explanations of natural phenomena that were materialistic rather than mythological or theological. Because his views of nature gave no role to mythical beings, Thales’s theories could be refuted by evidence. Arguments could be put forward in attempts to discredit them. Thales’s hypotheses were rational and scientific.

The Model of Hierarchical Complexity (MHC) follows in that tradition.... The MHC is a mathematical theory of the ideal. It is a perfect form as Plato would have described it. It is like a circle. A circle is an ideal form that exists. Once one draws a circle, something additional and different has been created. The new creation is a representation of a circle, but it is not, itself, a perfect ideal circle. The lines have width whereas a circle does not, and thus cannot perfectly represent the perfect form itself. The representation is not perfect nor can a drawn circle be perfectly round. This distinction between the ideal form and representations of the ideal is important for understanding the MHC and its relationship to stage of performance" (312-15).

The following earlier section in the article adds some context to the above, on terminology:

"Four basic terms are essential in discussing the Model: orders, tasks, stage, and performance. The orders are the ideal forms prescribed by the theory’s axioms. They are the constructs used to refer to the Model’s levels of complexity. The orders of hierarchical complexity are objective because they are grounded in the hierarchical complexity criteria of mathematical models (Coombs, Dawes, and Tversky, 1970) and information science (Commons and Richards, 1984a, 1984b; Commons and Rodriguez, 1990, 1993; Lindsay and Norman, 1977). Tasks are quantal in nature. They are either completed correctly—and thus meet the definition of task—or not completed at all. There is no intermediate state. An example is the adding of two numbers: it can be done only correctly or not at all. Tasks differ in their degree of complexity. The MHC measures the performance of tasks in terms of distinct stages, and it characterizes all stages as distinct. The term stage is used to refer to an actual task performed at an order of hierarchical complexity: order is the ideal form, stage is the performed form. Performance is understood as the organization of information. Performance, like the tasks themselves, is quantal in nature. That is, there are no intermediate performances. Tasks are understood as the activity of organizing information. Each task’s difficulty has an order of hierarchical complexity required to complete it correctly. For example, the task of adding numbers correctly is the necessary condition before performing the task of multiplying numbers. The successful completion of the tasks of adding and of multiplying are examples of two different stages of performance that can be quantified using the MHC. These stages vary only in their degrees of hierarchical complexity. This objective, quantal feature of tasks and stages means that discrete ordinal scores can be assigned to them" (306-07).

In re-reading yet again this fine thread I was struck today by this post on Badiou's version of mathematical set theory, reminding me of Byrant's withdrawn.

"As the name implies, a subtractive ontology is to be distinguished from a discourse which pretends to convey being as something present and substantial, something accessible to a sort of direct experience or articulation. By subtractive ontology Badiou means a discourse which accepts that its referent is not accessible in this sense. As Badiou conceives it, being is not something that shows itself in a sort of primordial revelation; still less is it the object of some divine or quasi-divine act of creation.... The ontologist knows that the ground of being eludes direct articulation, that it is thinkable only as the non-being upon which pivots the whole discourse on being....in mathematical set theory (the theory of consistent multiplicity) the ultimate ‘stuff’ presumed and manipulated by the theory is itself, as we shall see in a moment, inconsistent – it can be presented only as no-thing. In other words, ontology does not speak being or participate in its revelation; it articulates, on the basis of a conceptual framework indifferent to poetry or intuition, the precise way in which being is withdrawn or subtracted from articulation."

The above is not like Hegelian types of 'consistent' set theory which cannot handle void sets, the withdrawn or dialetheisms of the kind Morton describes in this post and following (using Graham and Priest), more typical of real reason.

I'm also reminded of this post from Women, Fire and Dangerous Things on non-Badiouian set theory:

"The classical theory of categories provides a link between objectivist metaphysics and and set-theoretical models.... Objectivist metaphysics goes beyond the metaphysics of basic realism...[which] merely assumes that there is a reality of some sort.... It additionally assumes that reality is correctly and completely structured in a way that can be modeled by set-theoretic models" (159).

He argues that this arises from the correspondence-representation model.

Along the lines of the above I just read an article by Christopher Norris in the Speculations III, “Diagonals: Truth procedures in Derrida and Badiou.” Therein he explores Badiou's reading of Derrida, and how they are akin yet different. A few excerpts follow of relevance to the discussion above.

“Badiou exempts Derrida from his otherwise sweeping condemnation of the linguistic turn in its sundry current guises as merely an update on old sophistical or cultural-relativist themes.... What is crucially different about Derrida’s commentaries on canonical texts from Plato to Husserl is his relentless teasing-out of aporetic or contradictory chains of logical implication” (157).

“Badiou is attracted not only by the rigour of Derrida’s work but also...by its quest for alternative, less sharply polarised terms of address or some means to shift argumentative ground from a downright clash of contradictory logics (within the text or amongst its commentators) to a 'space of flight,' as Badiou describes it, beyond all those vexing antinomies” (158).

“In Derrida it is chiefly a matter of revealing the various deviant, non-classical, or paraconsistent logics that can be shown to inhabit their texts and produce those moments of undecidability—aporias, in the strict sense of the term—which call into question certain of the author’s leading premises or presuppositions.... In Badiou, it is a chiefly a matter of showing how certain overt ontological commitments—those that endorse some version of a plenist or changeless, timeless, and wholly determinate ontology—are fissured by the need to introduce an anomalous term that implicitly concedes the problematical status of any such doctrine and its covert reliance on that which it has striven to keep off bounds. This is why Badiou devotes a large portion of his commentary in the early sections of Being and Event to a detailed rehearsal of the issue of the one and the many.... What emerges here is the conceptual impossibility of thinking an absolute plenitude of being—an absolute dominion of the one over the many, or of the timeless and unchanging over everything subject to time and change...so deeply repugnant to Plato’s idealist mind-set” (163).

“Thus Derridean deconstruction, as distinct from its various spin-offs or derivatives, necessarily maintains a due respect for those axioms or precepts of classical logic (such as bivalence and excluded middle) that have to be applied right up to the limit—the point where they encounter some instance of strictly irresolvable aporia—if such reading is to muster any kind of demonstrative force” (175).

“It will soon strike any attentive reader that when Derrida writes about the logic of the pharmakon in Plato, or supplementarity in Rousseau, or the parergon in Kant, or différance in Husserl (etc.) he is certainly out to discredit the...idealist conception but by no means seeking to undermine the very notions of truth and reference.... it gives rise to a truth-procedure that may for some time—like Cantor’s proposals—come up against strong doxastic or institutional resistance, but which thereafter acts as a periodic spur to the activity of thought by which paradox is turned into concept” (177).

“Derrida’s classic essays must involve...a strong analytical grasp of the logical or logicosemantic
structures that are thereby subject to a dislocating torsion beyond their power to contain or control. After all, this could be the case—or register as such—only on condition that the reader is able and willing to apply the most rigorous standards of logical accountability (including the axioms of classical or bivalent true/false reasoning) and thereby locate those moments of aporia or logico-semantic breakdown that signal the limits of any such reckoning” (179).

“Here again he agrees with Badiou that thought can make progress...only so long as it persists in the effort to work its way through and beyond those dilemmas that periodically emerge in the course of enquiry and can later be seen to have supplied the stimulus to some otherwise (quite literally) unthinkable stage of advance” (180).

“Such is the requirement even, or especially, where this leads up to an aporetic juncture or moment of strictly unresolvable impasse so that the logical necessity arises to deploy a non-classical, i.e., a deviant, paraconsistent, non-bivalent, or (in Derrida’s parlance) a 'supplementary' logic.... it is revisionism only under pressure, that is, as the upshot of a logically meticulous reading that must be undertaken if deconstruction is not to take refuge in irrationality or even—as with certain of its US literary variants—in some specially (often theologically) sanctioned realm of supra-rational ambiguity or paradox” (185).

This post in the OOO thread is also of relevance here in showing the difference between Hegel's and Derrida's 'dialectic.'

Re-reading this post and following from the OOO thread I decided to cut-and-paste a few excerpts relevant here:

Even accepting that human brain anatomy and consciousness is the most complex structure known, and partially accepting developmental psychological research into stages of enaction like the MHC, the key point of contention remains to be how such stages are formulated, as well as just what (meta)paradigms enact those stages. I've argued that the likes of integral pluralists like Bryant (and others) are indeed indicative of those mereological stages, with the likes of kenninligus and the MHC itself being just more complex formal operations due to the very way mereology itself is approached.

I'm reminded on the following from Protevi in The Speculative Turn on mereology, which is also akin to my latest tangent in the Rifkin thread:

"DG operationalize the notion of affect as the ability of bodies to form ‘assemblages’ with other bodies, that is, to form emergent functional structures that conserve the heterogeneity of their components. For DG, then, ‘affect’ is physiological, psychological, and machinic: it imbricates the social and the somatic in forming a ‘body politic’ which feels its power or potential to act increasing or decreasing as it encounters other bodies politic and forms assemblages with them (or indeed fails to do so). In this notion of assemblage as emergent functional structure, that is, a dispersed system that enables focused behaviour at the system level as it constrains component action, we find parallels with novel positions in contemporary cognitive science (the ‘embodied’ or ‘extended’ mind schools), which maintain that cognition operates in loops among brain, body, and environment. In noting this parallel, we should note that DG emphasizes the affective dimension of assemblages, while the embodied-embedded school focuses on cognition" (394).

And the following from the SEP entry:

"All the theories examined above...appear to assume that parthood is a perfectly determinate relation: given any two entities x and y, there is always an objective, determinate fact of the matter as to whether or not x is part of y. However, in some cases this seems problematic. Perhaps there is no room for indeterminacy in the idealized mereology of space and time as such; but when it comes to the mereology of ordinary spatio-temporal particulars (for instance) the picture looks different. Think of objects such as clouds, forests, heaps of sand. What exactly are their constitutive parts? What are the mereological boundaries of a desert, a river, a mountain? Some stuff is positively part of Mount Everest and some stuff is positively not part of it, but there is borderline stuff whose mereological relationship to Everest seems indeterminate. Even living organisms may, on closer look, give rise to indeterminacy issues.... And what goes for material bodies goes for everything. What are the mereological boundaries of a neighborhood, a college, a social organization? What about the boundaries of events such as promenades, concerts, wars? What about the extensions of such ordinary concepts as baldness, wisdom, personhood? These worries are of no little import, and it might be thought that some of the principles discussed above would have to be revisited accordingly—not because of their ontological import but because of their classical, bivalent presuppositions."

Also I've been pondering the process of iteration, and how it is formulated in the likes of set theory and the MHC versus the likes of Badiou and Derrida. And also how fractals are viewed in complexity theory between Mandlebrot and Prigogine. Both see something novel emerging from an interaction of the parts. I.e., something is retained yet something novel emerges in the iterative process. But it appears the mathematical formula for that process is itself the same for each level, itself just iterating or repeating the same algebraic pattern.

What I'm suggesting is that that very mathematical formula must itself undergo the kind of iteration that retains something and something novel emerges. The math itself must go postformal, which is what I've been getting at from the beginning of this thread, where Deleuze uses a calculus of uncertainty. Or Badiou infuses the empty set into the formula and changes the entire dynamic of set theory. Or Priest's paraconsistent mathematical logic. And that the math changes at each stage, both for itself and for the objects if represents, for it is not a unchanging Platonic form but itself a contingent construct that must undergo iteration. Hence the actual math for postformal stages is different.

I've come across this open access book from Re-Press, The Mathematics of Novelty: Badiou’s Minimalist Metaphysics by Sam Gillespie. First I want to thank Re-Press for providing cutting edge books (Bryant, Morton etc.) via open access as well as selling them in the traditional way. They've proven that they can still make a healthy profit with sales and give away knowledge to those not able to pay. Their hybrid P2P or distributed capitalism is the next wave in socio-economics, so well articulated by Rifkin.

And now a few relevant excerpts from the book:

"Unlike an ontological realist who would assert that numbers exist as ideal entities independently of the mind that thinks them, Badiou's thought is aligned with mathematical formalism.... Being does not derive from thought.... There must be a point of departure where being is posited. And this point is not, as some may imagine, the number one, but rather zero.... A mathematical thought of being, apart from its instantiation in symbols and manipulations, is nothing independently of these symbols and manipulations. And this nothing that is deemed to exist outside mathematical formulation is rudimentary for ontology as a whole. Zero exists.... Badious's thesis is that ontology is a discourse of being; it pronounces what is expressible of being" (46-7).

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