The latest issue of Red Pepper was guest edited by organisers of The World Transformed – Momentum’s political festival that took place as a complementary fringe event around Labour’s national conference in Liverpool, September 2016. I was one of those guest editors. And I was also an author, with Gabriel Bristow, of this article: ‘Momentum needs a strategy to win power in local elections‘. Enjoy!
This is my first remotely systematic attempt to reckon with Marx’s dialectical method and social science epistemology. It’s an expansion of a short peripheral piece I had to write for my PhD, so please excuse all the footnotes and the Harvard referencing. Maybe the extravagant metaphors can make up for it a little. I wouldn’t say it offered me any certainty (what else do you expect by now?), but it did help me begin to grasp dialectics in all its limited, contradictory glory. Enjoy.
Introduction: Thought and Reification
In one of Jorge Luis Borges’ shortest short stories, ‘On Exactitude in Science’, he charts the impossible futility of a perfect science, exemplified by a cartophilic Empire, whose ‘Unconscionable Maps’ grew in accuracy and size until, finally, ‘the Cartographers Guilds struck a Map of the Empire whose size was that of the Empire, and which coincided point for point with it’ (Borges 2000: 181). This useless map was quickly forgotten, left to fade in the glaring deserts, its dusty remnants an Ozymandian warning against pretentions of a coincident match between reality and the necessarily incomplete models of thought. For, ‘to think is to forget a difference, to generalize, to abstract’ (Borges 1993: 90). In another story Borges’ narrator encounters Ireneo Funes who, following brain trauma, gains infinitely infallible perception and memory, thereby forgetting how to forget. Slowly – sadly – the narrator comes to realise that, ‘In the overly replete world of Funes’, as in the great coincident map of that great Empire, ‘there were nothing but details, almost contiguous details,’ and that as such thought as we know it had become impossible (ibid).
In this gap between detail and abstraction, between reality and knowledge, sits epistemology. Beleaguered and bedraggled, a tired king enthroned on a bustling beach, the inexorable oceans of uncertainty lap at his feet, a cohort of desperate subjects, bent on knowledge, imploring him (against his Canutian piety – he knows the Sisyphusean nature of the task) to hold back the tide. Upon this beach sandcastles are built, from small bucket-shaped identities to intricately sculpted polylithic theories, and in between a cacophony of crumbling typologies and classifications. These sandcastles are, somewhat miraculously, immensely useful. But to focus myopically on them alone is to ignore the vast stretch of processes – constructions, destructions, reconstructions – of which the sandcastles are snapshots, reifications.
In my first post I wrote about this essentially obvious, but nevertheless important speculative disjunct of knowledge. For there is no small mischievous pleasure in reading Nietzsche describe ‘how wretched, how shadowy and flighty, how aimless and arbitrary, the human intellect appears in nature’ (1976: 42); or Foucault proclaiming from atop Nietzsche’s shoulders that ‘There is nothing in knowledge that enables it, by any right whatever, to know this world. It is not natural for nature to be known’ (2002: 9). But the even more interesting question is how, nevertheless, we do produce useful knowledge, against all the odds.
Reified, identitarian sandcastle thinking, despite its great achievements, increasingly finds itself beached on a field of complexity that it cannot adequately explain. For instance, how can a photon be both a wave and a particle (whether simultaneously or complementarily)? Of course we must be wary of drawing amateurish metaphors from natural science, but it seems that at least this can be logically inferred: the traditional concepts handed down to us by identitarian language prove at every turn ultimately insufficient to the complexity of sufficient detail. The answer, ultimately, is the great, useless, impossible coincident map. Rather we must proceed by questions. Are there ways of thinking whose sandcastles stand not in obstinate denial of their indeterminate relation to reality, but which rather try to flow with and elucidate something of the processes of identity formation?
Marx’s dialectical method is a mode of thought-abstraction that, in part, attempts such a task. Reaching towards an understanding of the processes that lay in excess of, and sometimes in contradiction to identity, Marx abstracts from the apparent material realities of the modern economy (such as the commodity) towards what he sees as contradictory unities within: dualities of process and thing, abstract and concrete. For example: the circulation of exchange and the fixity of use; the abstraction of exchange-value, which is both dependent and infringes upon use-value; or the internal unity of sale and purchase and the external spatio-temporal antithesis of sales becoming new purchases (to be further explained below). In this way Marx deconstructs the reifications of 19th century political economy, while also, he claims, building a picture of the social totality and evolutionary historical process they constitute.
We will first analyse the above contradictory unities as depicted in Capital Volume I, in order to exemplify Marx’s method in action, and especially its use of process to unsettle reified identities. Secondly, I will briefly discuss three key modes of abstraction in Marx as potential over-reifications: history, in relation to Cohen’s interpretation of Marx as a technological determinist; totality, in relation to Harvey’s analysis of footnote four of Capital Volume I’s fifteenth chapter; and contradiction, in relation to the Deleuzean charge that dialectics constitutes an overly reductive reification of difference. This understanding of the strengths, nuances and limitations of Marx’s dialectic will support the view that dialectics is by no means necessarily unscientific, but that it might not be best viewed as a science. Dialectics offers not a universal description of reality, but rather a tool, a conceptual framework for dereifying apparently unitary abstractions into potentially more nuanced and informative abstract dualisms, but dualisms which unavoidably constitute their own new reifications, and which must be critically understood as such, in both their utility and limitations.
Reification and Process in Capital
Marx certainly claimed that his was a thoroughly scientific endeavour. Beginning with the conceptual understandings already laid out by the political economists of his day (Smith, Ricardo, Malthus, Say, etc.), he used a revolutionised version of Hegel’s dialectic to unfold the inherent contradictions of the capital process that fundamentally undermine its stability. Marx famously described his materialist dialectic as Hegel’s idealist dialectic ‘turned right side up again’ (Marx 1873). Developing ‘from the actual, given relations of life the forms in which these have been apotheosized,’ Marx claimed, was ‘the only materialist, and therefore the only scientific’ method (Marx 1990: 494 n4).
In the below diagram (courtesy of Harvey 2010: 109) we see how Marx’s dialectical method in Capital often plays on contradictions between the abstract (the generally thinkable, but reductive) and the particular (the actual, the real, the uniquely qualitative), as well as between the fixed and the mobile.
The second duality is perhaps the clearest example; but we see the abstract and the concrete again in the form of money as measure of value and as medium of circulation, and in the first duality we see exchange-value: the abstract, quantitative, proportional relation ‘in which use-values of one kind exchange for usevalues of another kind’ (Marx 1990: 126), contrasted to use-value: ‘the physical body of the commodity itself’ (ibid), its unique, qualitative, subjective usefulness.
Of course all of these concepts, even those representing the concrete, are abstractions; but as Marx explains, ‘in the analysis of economic forms’, as in the general challenge of any social science, ‘neither microscopes nor chemical reagents are of assistance. The power of abstraction must replace both’ (1990: 90). By seeing the commodity as a contradictory unity of the conceptually concrete and the conceptually abstract – e.g. use-value and exchange-value – we can immediately begin to understand the contradictions of many social phenomena.
A house, for example, represents concrete use-values (shelter, comfort, security, etc.), but also exchange-value (an investment expected to rise in value). As an economy continues to shift emphasis from the former to the latter we can see increasing friction between these two antithetical (both partially autonomous and yet indivisible) characteristics. The more commodified housing becomes, the more its exchange-value is emphasised, the harder it becomes for the poorest to realise the simple use-value of an adequate house. Of course this principle reveals itself differently in different contexts. In the US we see it emerge through subprime mortgages and repossessions; in London through gentrification, luxury flats and exorbitant rents. But what unites both examples is the conflict between motion and fixity, process and thing. The valorisation of capital, the realisation of exchange-value, requires ever increasing speed. Just as the gadget economy relies on fashion, built in obsolescence and ever increasing software requirements to undermine any attempt to stand still in one’s technological consumption, the vibrancy of London’s housing market increasingly relies on the young middle class’s willingness to constantly move around, bringing their income from one place (where it has been outstripped by rent prices) to another (where they can still afford housing with satisfactory use-value). Middle class mobility drives up prices to the detriment of families embedded in local communities. The latter’s desire for fixity, the use-value they glean from stable, local social relations, stands in direct antagonism to the mobility required for exchange-value realisation.
Here dialectics helps us see behind the ideological veil that presents value realisation as the only valid measure of utility, and provides a more nuanced framework within which we can conceive of rebiasing an economy back towards the realisation of use-value, while understanding the systemic implications of any desire to tamper with a single aspect of the totality of the capital process. The dialectical mode of representation reveals the interrelatedness of the constitutive contradictions, and thus the potential for each to generate wholesale crises that reverberate across the entire system. This is why dialectics is an inherently revolutionary method, because it suggests profound limits to fiddling at the edges of the system; in the particular case of capital it implies that such tampering is only likely to exacerbate the procession of crises, and thus to reform is inevitably to lead towards revolution or destruction – socialism or barbarism.
Another example of Marx using the unfolding of unities into dualities to grasp capital as process comes when he is rambunctiously debunking ‘Say’s Law’: the ‘foolish … dogma that because every sale is a purchase, and every purchase a sale, the circulation of commodities necessarily implies an equilibrium between sales and purchases’ (1990: 208). Marx argues that the ‘internal unity’ of sale and purchase in fact ‘moves forward through external antitheses’ (ibid: 209). The ‘internal unity’ here is an agreement that indeed every purchase is a sale, and every sale a purchase. The external antithesis is that a sale does not necessarily proceed immediately to another purchase. If that money is kept out of circulation, ‘if the assertion of their external independence proceeds to a certain critical point, their unity violently makes itself felt by producing – a crisis’ (ibid). Again capital’s contradictory reliance on keeping concrete, material things (in this case money) in constant, ever accelerating motion is revealed through an appreciation of the possibilities of dialectical abstraction, along with the profound, (critical, no less) implications of this contradiction. The systemic effect of these contradictions is the constant threat of crisis. We might rephrase to say that, to the extent that capital is always struggling against its own contradictions, it is always in crisis, simply in varying qualities, quantities and locations. By investigating the spatio-temporal processes that interrupt the unity of exchange, dialectics opens up an appreciation of the limitations of taking reified identities at face value. Marx’s new categories, his new methods of abstraction, allow us to begin to deconstruct these processes of reification, but to what extent does his new method generate new problematic reifications of its own?
We have seen that Marx was deliberately setting up new abstractions as means of understanding – thus new reifications in the broadest sense of the word – but on the scale between useful reductions and what Whitehead (1997) called ‘fallacies of misplaced concreteness’, where do some of Marx’s key concepts fall? We will now discuss three such concepts via important interpretations, in order to get an idea of the useful-reductive balance allowed for in Marx’s dialectic: history, totality and contradiction. The first two are greatly softened by exegesis, but the charge of contradiction as reification better represents the limits of dialectics.
Many have reduced Marxism to something like a deterministic model of history, but few Marxists more so than G. A. Cohen. Believing that a ‘no-bullshit Marxism’ is only possible after dispensing with dialectics altogether, Cohen reduces Marx to a functionalist technological-determinist by taking Marx’s comments in his Preface to A Contribution to the Critique of Political Economy to be the preeminent statement of historical materialism:
The mode of production of material life conditions the social, political, and intellectual life process in general. It is not the consciousness of men that determines their being, but, on the contrary, their social being that determines their consciousness. At a certain stage of their development, the material productive forces of society come in conﬂict with the existing relations of production … From forms of development of the productive forces these relations turn into their fetters. Then begins an epoch of social revolution (Marx 2000: 425).
This reading is contestable by a number of exegetic paths. Firstly, a close reading of the passage itself shows a confusion of terms. It is one thing to say that it is men’s ‘social being that determines their consciousness’ – an immensely broad, dialectical statement, implying a kind of constructivist social ontology. But Marx does not say that ‘The mode of production’ determines ‘the social, political, and intellectual life process in general,’ he says that it conditions life processes. A much less causal and more dialectical formation, the nuances of which are completely lost when wilfully not read as such. ‘Then begins an epoch of social revolution’ is undoubtedly problematic, but mollified by the dialectical reading of the rest of the passage. Meanwhile in Capital itself, Marx barely touches strictly causal, deterministic terminology. He rather says that processes condition other processes, and that analysis of certain phenomena reveals other processes. For example:
Technology reveals the active relation of man to nature, the direct process of the production of his life, and thereby it also lays bare the process of the production of the social relations of his life, and of the mental conceptions that flow from those relations (Marx 1990: 494 n4, emphasis added).
To be clear, Cohen does not present an absolute, unidirectional determinism, as he allows feedback in his model between base and superstucture. But he does argue that ‘the productive forces strongly determine the character’ of the superstructure (Cohen 1978: 31), and this is still shown to be hopelessly reductionist when compared with the above footnote from Capital. David Harvey has done much to expound the importance of this footnote. In it he sees no less than six interrelated ‘conceptual elements’: technology, relation to nature, the actual process of production, the produ
nd reproduction of daily life, social relations, and mental conceptions (Harvey 2010: 192). Marx lays each element out in an ontologically flat coproductive network, as depicted by Harvey (2010: 195) in the figure to below. Marx does not claim here that any has ontological or deterministic priority over any others, merely that what is special about technology (for Capital) is that it has the capacity to ‘reveal’ the social processes of the production of the other areas of life. This does not even preclude the possibility that in other research any other of these nodes could function as a critical perspective from which to dialectically reveal the productive relations inherent in the rest. Capital can thus be seen not as a dogmatic statement of economism, but one work of many, the economic brick in the towering dialectical understanding of the social totality.
The idea of totality may also seem a totalitarian reification, and again it can easily become so if taken too seriously. However in Marx himself we once again find a much more nuanced picture than, say, an overbearing structuralism. Taking the above model as a guide we can see the modular nature of the dialectical method in relation to a ‘fluid and open’ totality, replete with contingency and ‘in perpetual transformation’ (Harvey 2010: 24). Marx’s totality exists as a conceptual horizon that allows us to think in terms of social processes rather than simplistically autonomous and stable entities, or deterministic historical processes. It structures our understanding and guides our learning, while simultaneously being reshaped by our own learning: revolutionary praxis. Harvey further explains Marx’s open totality as,
… not a Hegelian totality in which each moment tightly internalizes all the others. It is more like an ecological totality, what Lefebvre refers to as an ‘ensemble’ or Deleuze as an ‘assemblage,’ of moments coevolving in an open, dialectical manner (2010: 196).
The reference to Deleuze brings us to a third reification: beyond the historical and totalitarian reconstructions of Marx’s dialectic, to its constituent contradictory unities themselves. Deleuze’s anti-dialectics is of course a huge topic, but Michael Hardt summarises for us how it took shape in one of Deleuze’s earliest works:
Deleuze’s primary charge against dialectical thinking is that, despite its claims, dialectics mystifies and destroys difference and is thus incapable of recognizing multiplicities. The dialectic pushes all differences to the extreme of contradiction so that it then can subsume them back into a unity. Real differences, according to Deleuze, are more subtle and nuanced than dialectical oppositions (Hardt 2006: xi).
In this sense we can see the limitations of dialectics and its new reifications. Though Marx’s unities and binary polar contradictions continuously unfold, arguably (an extremely tentative argument) amounting to a kind of Deleuzian ‘assemblage’ and a vague conception of a total multiplicity, they still obscure that beneath each reified contradiction lies a concrete spectrum of difference, and that beneath each contradictory fixity lie processes of individuation.
What remains of dialectics from this both critical and generous – pragmatic, perhaps – reading of Marx? We are left with dialectics as a tool. As an anti-positivist corrective. As the screwdriver set we use to open up the chassis and to separate some of the main components, before reaching for something more specifically and complexly suited to the local task – a soldering iron, a multimeter, an oscillograph, a software application, or even a hacksaw or explosive. Marx’s dialectic helps us understand the capital process for two key reasons. First, because the mainstream economic discourse was, and largely remains, so positivistic, so extremely reificatory, that dialectics presents a great leap of nuance and complexity. Second, to the extent that Capital has explanatory power it is in the ways in which capital does tend to operate dialectically, or at least in ways that lend themselves to dialectical presentation.
Is dialectics scientific? We might look to the hoops Althusser (2005) jumps through to try to prove that it is, and wonder if it even matters. More radically, we might resort to Foucault and consider ‘the aspiration to power that is inherent in the claim to being a science’ (2003: 10). In the ‘theoretico-political vanguard’ that attempts to exorcise itself ‘from all the massive, circulating, and discontinuous forms that knowledge can take,’ we can see that dialectics’ critical instrumental utility might in fact depend on its being anti-scientific, in the same way that Foucauldian genealogy attempts to ‘desubjugate’ knowledge, ‘to oppose and struggle against the coercion of a unitary, formal, and scientific theoretical discourse’ (ibid).
Marx’s nuances negate any claim to a true Marxist science, if conceived as anything more than ‘ruthless criticism of all that exists’ (Marx 1843). His dialectic is both much more useful and much more unwieldy than it may seem in the hands of many Marxists. As Marx himself warns, ‘to bring a science by criticism to the point where it can be dialectically presented is an altogether different thing from applying an abstract ready-made system of logic to mere inklings of such a system’ (cited in Frisby 1976: xvi). Marx allows us to grasp the ways in which capital behaves dialectically, he reveals useful interpretations through a dialectical mode of presentation, but dialectics is far from the universal law of nature that Engels, Lenin or Stalin envisaged. A continued ‘defetishization of Marx’s own concepts’ (Cleaver 2000: 12) can surely continue to reap new insurrectionary knowledge, but only where the subject matter is apt to the method. Dialectics must at every stage be weighed also against the state of existing knowledge (in that is a necessarily critical method), the political aims of the endeavour, the possibilities of other forms of knowledge (other tools), but also the necessary utility of abstraction. In the land of the blind, perhaps the best we can do is walk, ask, and, without the overbearing pretence of scientific elegance, fumble as best we can to find the best tools for the job at hand.
Althusser, L. (2005) For Marx. Brewster, B. tr. Verso.
Borges, J. L. (1993) ‘Funes, the Memorious’ in Ficciones. Everyman’s Library.
— (1999) ‘John Wilkins’ Analytical Language’ in Selected Non-Fictions. Viking Penguin.
— (2000) ‘On Exactitude in Science’ The Aleph. Penguin.
Cleaver, H. (2000) Reading Capital Politically. 2nd edn. AK Press.
Cohen, G. A. (1978) Karl Marx’s Theory of History: A Defence. Princeton University Press.
Foucault, M. (2002) ‘Truth and Juridical Forms’ in Power: Essential Works of Foucault 1954-1984. Faubion, J. D. ed. Penguin.
— (2003) Society Must Be Defended: Lectures at the Collège de France, 1975-76. Macey, D. tr. Picador.
Frisby, D. (1976) ‘Introduction to the English Translation’ in Adorno et al, The Positivist Dispute in German Sociology, Heinemann.
Hardt, M. (2006) ‘Foreword’, in Deleuze, G. Nietzsche and Philosophy. Columbia University Press.
Harvey, D. (2010) A Companion to Marx’s Capital. Verso.
— (2011) The Enigma of Capital and the Crises of Capitalism. Profile Books.
— (2014) Seventeen Contradictions and the End of Capitalism. Oxford University Press.
Holloway, J. (2005) Change the World without Taking Power. New edn. Pluto Press.
Lukács, G. (1971) History and Class Consciousness. Livingstone, R. (tr). Merlin Press.
Marx, K. (1843) ‘Marx to Ruge’, Letters from the Deutsch-Französische Jahrbücher. Available online at <https://www.marxists.org/archive/marx/works/1843/letters/43_09.htm> accessed 20 January 2016.
— (1873) ‘Afterword to the Second German Edition’, Capital Volume I. Available online at <https://www.marxists.org/archive/marx/works/1867-c1/p3.htm> accessed 14 January 2016.
— (1990) Capital Volume I. Fowkes, B. (tr). Penguin.
— (2000) Select Writings. 2nd edn. Oxford University Press.
Nietzsche, F. (1976) ‘On Truth and Lie in an Extra-Moral Sense,’ in The Portable Nietzsche. Kaufman, W. tr & ed. Penguin.
van Wyk, K. (2015) ‘Evidence of Hindu Religion on the Theory of Chomsky’s Transformational Grammar’ 2(5) International Journal of Language and Linguistics 27.
Whitehead, A. N. (1922) The Principle of Relativity with Applications to Physical Science. Cambridge University Press
— (1929) Process and Reality: An Essay in Cosmology. Cambridge University Press.
— (1997 ) Science and the Modern World. Simon & Schuster.
 It might be noted however that, rightly or wrongly, it is not only naïve social scientists who draw metaphysical implications from the natural sciences. It was also very common among physicists, at one time. For example: discussion of determinism in the Bohr–Einstein debates; Schilpp (1970); Heisenberg (1979); much of the work of Erwin Schrödinger; Whitehead’s move from mathematics (most importantly Russell & Whitehead 1910) to physics (e.g. Whitehead 1922) to philosophy (most importantly Whitehead 1929); and more recently Stenger et al (2015). Other physicists have felt it right to go beyond even metaphysics to make normative claims based on their scientific understandings. For example, Max Born: ‘loosening of thinking seems to me to be the greatest blessing which modern science has given to us. For the belief in a single truth and in being the possessor thereof is the root cause of all evil in the world’ (Born 2002: 261); and David Bohm (1980), whose dialectical theories of physics led him to argue that the usual linguistic bias towards the noun is both epistemologically beguiling, and a direct cause of confusion and conflict in the world, such that language should be reoriented towards the verb – a linguistic modality he named the ‘rheomode’.
 ‘Asking we walk’, as the Zapatistas say (Holloway 2005: 215).
 While there is much evidence that Marx saw dialectical processes in nature, he also speaks of dialectics as a means of presentation, which seems a more epistemologically defensible position. For example, when describing his project as bringing ‘a science by criticism to the point where it can be dialectically presented’ (correspondence to Engels, cited in Frisby 1976: xvi). Elsewhere, Ernest Mandel cites ‘Marx’s wish to present Capital as a “dialectically articulated artistic whole”’ (‘Appendix’, in Marx 1990: 944).
 One of David Harvey’s (2011) key arguments, for instance, is that capitalist crises are never fully resolved, but are rather shifted to other areas of the circulation process.
 It should be noted that Marx only seems to use the word ‘reification’ once in the Fowkes translation of Capital (Engels also inserts a footnote reiterating ‘the conversion of persons into things’ as the ‘reification of persons’ (Marx 1990: 209). But it is essentially the idea of reification that echoes when Marx speaks of fetishism (ibid: 163), and ‘all the mystifications of the capitalist mode of production’ (ibid: 680).
 Nomology also presents an interesting example of Marxist reification (possibly ameliorated by the ‘method of the tendency’), but alas we cannot cover everything.
 For corroboration on this point, see Harvey (2010: 193).
 It seems significant that, in the sentence preceding that quoted from footnote four, Marx cites Giambattista Vico, a very early epistemological constructivist known for the statement verum esse ipsum factum, or ‘the truth is precisely what is made’ (Costelloe 2014).
 ‘The coincidence of the changing of circumstances and of human activity or self-changing can be conceived and rationally understood only as revolutionary practice’ (Marx 2000: 172).
 The human as tool making animal (Benjamin Franklin, attributed by Marx 1990: 444) also reminds us of the human as political animal (Aristotle, Politics 1.1253a4), and thus another possible avenue of investigation: politics as epistemological axiomatic. Whitehead is reported to have said that ‘Every scientific man in order to preserve his reputation has to say he dislikes metaphysics. What he means is he dislikes having his metaphysics criticized’ (cited in van Wyk 2015: 36). We might say the same about politics. But that is for another day, and another reading of Capital (starting, probably, with Cleaver 2000).
Is all knowledge necessarily incomplete? We might intuit as much (in my first post you’ll see I’m vaguely of that persuasion). But more profoundly, is all knowledge demonstrably incomplete?
Reading Logicomix I was reminded of Gödel’s ‘incompleteness theorems’, which seem to claim that ‘there will always be unanswerable questions.’ Let’s see how this was depicted in Logicomix:
Bertrand Russell wasn’t actually at that lecture – the authors of Logicomix explain at the end that this was a narrative device. John von Neumann, however, really was, and really did say afterwards, ‘It’s all over.’
So it’s tempting for some (continental philosophers, say, and other anti-positivist types) to see this, jump up and shout Hah! In your face logic and positivism! But reading more about the incompleteness theorems you quickly realise that they’re (unsurprisingly) hugely complex and, more importantly perhaps, quite specific in what they claim. They concern the interrelations of axioms within a particularly defined system of logic – not all logic necessarily. For instance, it only applies to second order logic, while Gödel himself actually proved an earlier completeness theorem for first order logic.
As the Stanford Encyclopedia of Philosophy explains, ‘These results have had a great impact on the philosophy of mathematics and logic. There have been attempts to apply the results also in other areas of philosophy such as the philosophy of mind, but these attempted applications are more controversial.’ And indeed my own sporadic reading quickly threw up the idea that it is widely seen as simplistic and ill-conceived to draw broad, or even any philosophical implications from the theorems.
But that didn’t stop Gödel. One of the main analytic-philosophical debates around the theorems seems to be that of human mechanism – is the human brain a finite, deterministic machine, or does it have some irreducible quality, what we might call free will? According to Stanford, people have claimed that Gödel’s incompleteness theorem ‘proves that Mechanism is false, that is, that minds cannot be explained as machines (J.R. Lucas 1961); but ‘These Gödelian anti-mechanist arguments are, however, problematic, and there is wide consensus that they fail.’ For his part, Gödel believed the theorems did prove that ‘either … the human mind (even within the realm of pure mathematics) infinitely surpasses the power of any finite machine, or else there exist absolutely unsolvable diophantine problems.’ And furthermore, that the latter possibility above ‘seems to disprove the view that mathematics is only our own creation … that mathematical objects and facts … exist objectively and independently of our mental acts and decisions,’ i.e. would prove mathematical Platonism!
With all this uncertainty I was content to leave incompleteness be. But then I stumbled across this episode of the Radiolab podcast: ‘Loops‘.
From 38mins in they speak with mathematician Steve Strogatz about incompleteness – its history, its claims and its implications. And it’s a great story, so I’ll give it some illustration below.
We start with Gottlob Frege’s quest to find the foundations of mathematics (Bertrand Russell shared this quest throughout his early career). All the way back in the late 19th century, Frege dreamed of a rational machine that could, if programmed with the foundations of mathematics, begin autonomously discovering all the truths of the universe. But what are the foundations of mathematics? He thought that it isn’t number, but that number rather expresses the common foundational quality of ‘sets’. And this is wonderfully explained by Strogatz using this sesame street sketch:
Humphrey takes a food order from some penguin guests: fish, fish, fish, fish, fish, fish. Ingrid tries to remember the order, but makes a mistake: ‘fish, fish, fish, fish, fish?’ ‘No, fish, fish, fish, fish, fish, fish.’ Ernie comes to the rescue to explain that it’s much easier to count the fish and remember that there are 6 of them. ‘Does it work on other stuff??’ asks Humphrey, blown away by this profound breakthrough; making one wonder what it was like before hominids had words for numbers, and how people reacted to their invention. What’s this got to do with anything? Well Strogatz explains that this shows how sets exist deeper than numbers themselves, as the quality of six-ness that’s shared by sets of 6 things. This can be represented by the number 6, but the number 6 is not necessarily the same thing as the foundational quality of six-ness per se.
So Frege built set theory, a logical system that would found mathematics on grounds of certainty. This was the same quest as Bertrand Russell’s, so let’s see how he conceived of it in Logicomix:
Russell studied set theory and thought it had great potential:
And then the bugger discovered the Barber Paradox:
In set theoretical terms, this is the set of all sets that do not contain themselves. Like the liar’s paradox (‘this statement is false’), if it’s true it’s false, and if it’s false it’s true. It’s unprovable, and thus scuppered set theory’s claim to a certain logical foundation. Even Frege (perhaps Frege more than anyone) immediately saw the implications, and sought to stop the imminent publication of Volume 2 of his Foundations of Arithmetic. It did get published, but with an astonishing Addendum. I think every PhD student understands the terror associated with this kind of cruel ‘breakthrough’ in your field.
Enter Gödel; and Janna Levin to the Radiolab show. She claims the incompleteness theorems stemmed from Gödel thinking about paradoxes like the barber’s paradox or the liar’s paradox, which ultimately results in the mathematisation of the statement ‘this statement is unprovable.’ After a lot of mathematical jiggery pokery, says Levin, the statement ‘this statement is unprovable’ turns out to be… true. The upshot being, says Strogatz, that Gödel felt profoundly freed and liberated, that (as discussed above) humans couldn’t be reduced to certain mechanisms, that ‘there was profound mystery forever.’ Strogatz uses Goldbach’s conjecture as an example of Gödelian undecidability. It hypothesises that ‘Every even integer greater than 2 can be expressed as the sum of two primes.’ Despite huge efforts for over three centuries, no one has yet found a way to either prove or disprove this conjecture. We can test it on every possible number, and we’re doing that, and so far every even integer up to 4 × 1018 complies to the hypothesis. But we’ll never be able test every even integer, as there are infinitely many of them. So without an abstract mathematical proof, this conjecture remains neither provable nor disprovable: undecidable. And it looks like there are plenty of other undecidable problems.
Since Gödel there has been a huge proliferation of logical systems. Axiomatic set theory, for example, apparently solves or somehow negates Russell’s barber paradox. And as discussed, trying to draw conclusions from Gödel’s theorems beyond their narrow intended application in (certain kinds of) mathematical logic is extremely fraught.
But there’s actually another segment to that Radiolab episode (from 7m7s), which is interesting to throw into the mix. They discuss transient global amnesia (TGA), and how when the sufferer’s memory resets, every 90 seconds in the early stages of onset, they tend to begin almost exactly the same conversation all over again. And again. And again. And again. (listen from 10m20s to hear this happening; it’s creepy as hell). They then speak with a doctor who’s had experience with a number of cases of this rare condition, and he confirms that the same thing happened in all of those cases. He muses about the implications of how predictable the human mind seems to become when confronted with highly similar sensory input: ‘it makes the brain seem a little bit more like a machine. You give the machine the exact same set of inputs, and see if the output ever varies, and (it doesn’t), it almost seems like the patient has no free will.’
And then consider David Eagleman’s recent book and tv show The Brain, in which he describes (among many other interesting things) the extent to which many so-called ‘decisions’ happen prior or exterior to what we would think of as our actual consciousness or free will.
So I suppose there are two main threads to this post: free will, and epistemology. Is the quest to prove all of the axioms of maths and logic a wild goose chase, and does this have anything meaningful to say about the human mind, or any kind of philosophy outside the philosophy of mathematical logic? Does this fundamentally undermine the whole project of analytic philosophy, and support the validity of interpretive epistemologies of social science? What about dialectics?
It’s probably undecidable.
But maybe once I’ve read more about all this I’ll have more to say than that. See my next post for a discussion of dialectics, in a (very) vaguely relevant manner.
 This was of course a dream of many of the most starry eyed mathematicians and logicians. E.g. Leibniz would surely have dreamt of a combination of his Calculus Ratiocinator (universal logical calculation framework) and his step reckoner (a mechanical calculator too advanced for the manufacturing technology of time, but which became the basis of Thomas de Colmar’s Arithmometer around 150 years later, the first mass-produced calculating machine.
Welcome, dear reader, to my new blog. I suppose convention calls for a brief introduction. It’s a blog for blog’s sake kind of blog, a spewing forth of reflections on politics, philosophy and the like, largely from an anarchical perspective (I prefer anarchical to anarchist etc for reasons that I’ll surely go into one day), in a manner that stems from a deep and universal skepticism. The thread of continuity will be my PhD research in law and politics, but hopefully there’ll be something for everyone; though it’s explicitly for me first, so forgive me an occasional self-indulgent wordplay or the odd bad poem and we’ll get along just fine. To brighten up the place I’ll have a regular slot for Left Good News, thus laying the necessary optimistic emotional groundwork for full social revolution.
To kick things off here’s a reflection on the precarity of knowledge, ergo everything. It’s a joyfully self-indulgent scenic landscape painting, an abstract epistemological starting block, and a titular defence; but don’t worry, it won’t all be like this.
This blog-shaped conurbation of disparate thoughts will represent my attempts to see the world through its obfuscatory manifestations, occasionally even venturing the obscenity of trying to understand it. This is in spite of the fact that such attempts are always precarious, temporary, fictional, false, and none more so than here.
He who takes action fails.
He who grasps things loses them.
But beauty and utility have, on occasion, been found by mortal beings among the scattered ruins of our precarious imaginations, new truths are everyday forged, so maybe (just maybe) everything is not pointless and illusory. All is insufficient, and that must be our motivation.
A tree as big as a man’s embrace grows from a tiny shoot.
A tower of nine storeys begins with a heap of earth.
The journey of a thousand li starts from where one stands.
That things get done, made, finished, defined – in naive antagonism to the infinite maelstrom of beginnings, destructions, cycles and disagreements – is an utter confusion (a miracle; legerdemain). Yet they do. We define processes like shoots becoming trees becoming fossils, earth becoming brick becoming towers becoming civilisation becoming dust… Our fictional narratives have meaning – are the source of meaning – they generate and structure reality, despite always being wrong. How can this work? How is this sustainable? Is it even?
Our knowledges do not precede ‘discovery’, lying hidden, pressed beneath the sediment of ignorance; they are invented, in perhaps the crudest sense of the word – made up, speculated, wagered. Our scientific narratives begin as wild dreams and do not cease to be such simply by being repeated and believed – nor even by being useful. This is exemplified charmingly (if you like this sort of thing) on the individual scale by the numerous scientists whose most groundbreaking work materialised under the influence of psychedelic drugs, perhaps most famously Francis Crick, supposedly high on LSD when he envisaged the double helix structure of DNA.
Knowledge-as-invention at the macro scale requires (I reckon) a little attention to Foucault. Dissatisfied with the outlook of scientific authorship he described scientific knowledge at the social level as ‘a matter of a collective and complex transformation of … understanding in its practice and its rules’. But within this structural view of knowledge there is no determinism; it too is a process of invention, and this was, we could say, trying to annoy Foucault, ‘discovered’ by Nietzsche, who Foucault claims instigated ‘a great break with the prior tradition of Western philosophy’ by claiming a ‘discontinuity’ both between the subject and knowledge, and between knowledge and the object.
In some remote corner of the universe, poured out and glittering in innumerable solar systems, there once was a star on which clever animals invented knowledge. That was the haughtiest and most mendacious minute of ‘world history’ – yet only a minute. After nature had drawn a few breaths the star grew cold, and the clever animals had to die. One might invent such a fable and still not have illustrated sufficiently how wretched, how shadowy and flighty, how aimless and arbitrary, the human intellect appears in nature. There have been eternities when it did not exist; and when it is done for again, nothing will have happened.
Building on this superb Nietzsche quote, Foucault is able to make some wondrously poetic epistemological claims himself:
Knowledge must struggle against a world without order, without connectedness, without form, without beauty, without wisdom, without harmony, and without law. That is the world that knowledge deals with. There is nothing in knowledge that enables it, by any right whatever, to know this world. It is not natural for nature to be known.
Out of the invention of knowledge flows logically its falsity. As the great David Thomas puts it, ‘By simplifying, lumping dissimilar things together as the same, knowledge is always a misconstruction.’ Hence (if my rudimentary understandings hold any water) the incompatibility of quantum physics and general relativity; hence particles being waves being strings being hyperdimensional ripples (or something); hence Heisenberg’s uncertainty principle; hence Zeno’s dichotomy paradox; hence the infinity of the British coastline; hence the improvable axioms of mathematics; hence A ≠ not A, and yet dialectics; hence A = A, and yet quantum superposition; hence the impossibility of politics; hence the irreducibility of antagonism; hence the fractured subject; hence religion; hence poetry; hence beauty as the only meaningful category? I’ll leave that for another day. Hence I might be digressing into nonsense, but poetic nonsense of a sort, I can only hope.
Drowning in skepticism, I find myself with a foot on the bottom rung of an academic career that will require me to define, contrast, state and conclude, armed with all the surety of a rabbit caught in headlights: there’s sex, and there’s death, and everything else is up for grabs (as too are sex and death if one is brave enough to stop and think about them).
So watch, dear reader, as I magically, before your very eyes, turn water into wine, base elements into purest gold, and purest doubt into vaguest truth. Look forward to posts on Dalí’s Tête Raphaëlesque Éclatée and quantum mysticism, ‘Capitalism. Or Not?’, anarchical social movements and social theories, ‘Law. Or Not?’, Modernity and the Holocaust, posthuman rights, law and determinism, human nature, The Grain of the Voice, addiction, remedial economics, utopia, noise. This is my precarious imagination, and I’m amazed you’ve stuck with it this far.
 ‘I am well aware that I have never written anything but fictions. I do not mean to say, however, that truth is therefore absent. It seems to me that the possibility exists for fiction to function in truth, for a fictional discourse to induce effects of truth, and for bringing it about that a true discourse engenders or “manufactures” something that does not as yet exist, that is, “fictions” it. One “fictions” history on the basis of a political reality that makes it true, one “fictions” a politics not yet in existence on the basis of a historical truth.’ Michel Foucault, Power/Knowledge: Selected Interviews and Other Writings 1972–1977 (Pantheon Books 1980) 193.
 Tao Te Ching Ch.64, from one of the endless procession of English translations (Chan).
 Michel Foucault, The Chomsky-Foucault Debate on Human Nature (The New Press 2006) 18.
 Friedrich Nietzsche, ‘On Truth and Lie in an Extra-Moral Sense,’ in The Portable Nietzsche (Walter Kaufman tr & ed, Penguin 1976) 42.
 Michel Foucault, ‘Truth and Juridical Forms,’ in Power: Essential Works of Foucault 1954-1984 (James D Faubion ed, Penguin 2002) 9.
 David Thomas, ‘Lecture 1 of Foucault’s Truth and Juridical Forms’ (Unpublished 2014).