Skip to main content

"Torsion & Tension" [Chapter Two, PDF download]

Keeping with my promise to make my unpublished draft manuscript about dialectical materialism––Torsion & Tension––available as a serial, I have copy-edited and prepared the second chapter, turned it into a PDF, and provided the download link at the bottom of this post.  This chapter is called "Dialectical Logic Defined" and it is focused on looking at the simplest definition of dialectical logical, while relating it to analytical logic.

As noted in previous posts about this retired manuscript, there are aspects about this project that I'm now ambiguous about.  One of the things that bothers me here, though I did try to account for this bothersomeness in the draft a little bit, is the way in which I represent dialectics according to the formalization of analytical logic.  Although I did qualify that it was a crude analogy, and that you really can't formalize dialectics in the same way, I'm unsure now if I should have just dropped this attempt from the get-go… I don't know if it is helpful and, even worse, it opens the door to criticisms that I am uninterested in turning into my metaphorical hill to defend.  Still, keeping with my promise to provide the original manuscript as it was, and since I don't have the energy to make substantial edits for a retired manuscript (copy-editing is boring enough), I haven't changed that aspect.  From what I recall (but I'll have to see when I look at the later chapters) the analogical formalization comes up again, so if I wasted time killing it here then I would have to kill it elsewhere and the whole thing would become a mess.  (Dialectics in action: if I make changes without looking at the whole thing as a totality then I'll have problems!)

Also, for those interested in the general outline of the book––if you want to know how many chapters it possesses and where it's going––here is the table of contents:

Chapter One: The Meaning of Dialectical Materialism
Chapter Two: Dialectical Logic Defined
Chapter Three: Pseudo-dialectics
Chapter Four: Unity of Opposites
Chapter Five: The Fundamental Contradiction of Dialectical Materialism

So, yeah, a mini-manual: five chapters with an introduction and epilogue––it's not very long.  Think of it kind of as a non-fiction novella about materialist dialectics.  If you've been following it this far, and are interested in the next chapter, here it is:

Download Torsion & Tension Chapter 2


  1. Reading your stuff on dialectics, it doesn't look like anything new. It reminds me of a remark I read once about Stalin. It is said that a (unnamed) Western linguist once described Stalin's work on linguistics as “Trite but competent.” It kind of seems to me this could apply to your work. I mean this as a compliment, but I'm sure you won't see it that way.

    Your handling of the non-contradiction stuff seems common enough, but I believe I have stumbled upon a better method of talking about it, utilizing set theory. Rather than using an example where it would seem obvious enough to tell whether it is raining or not raining at any particular moment in time, I prefer the example of the Sun revolving around the Earth or not. Both have an aspect that you can't really blend the two statements together without changing their meaning (as you say, sleet is not rain, and is a B, not A or ~A). The Sun either revolves around the Earth or it does not. The difference between the example is speculation; merely looking outside and seeing rain falling on the ground should be enough to settle whether or not it is raining. As everyone knows, simply looking up at the sky isn't enough to be able to tell whether or not the Sun is revolving around the Earth.

    Call the proposition that the Sun revolves Earth to be R. It is the case that either R or ~R is true, as formal logic says. As this proposition is speculative, there will be arguments made for both R and ~R. Consider the arguments and evidence in favor of R to be belong to a set of objects attesting to the truth of R, let's call this set R'. R' = {a1, a2,} where each element an is some type of argument or observation which purports to establish the truth of R. Similarly, ~R has a corresponding set ~R' = {b1, b2,} where all the elements are arguments/observations purporting to establish the truth of ~R.

    Even though it is only the case that either R or ~R is true, the dialectical interplay between their corresponding sets R' and ~R' are added up together when the truth of R or ~R is determined. For instance, say one of the elements of R is “The Bible says the Sun revolves around the Earth, and the Bible is the literal word of God and can not be false.” Then this statement gets added as a negation to the corresponding set ~R' if R is false.

    This is important, I think, because it at once shows that dialectical thought grows in importance relative to the speculative nature of the propositions in question, but that it also is possible to utilize it even in the most trivial of cases.

    In the realm of pure mathematical abstraction, the dialectical proof is the proof by contradiction. You start by assuming the opposite of what you want to prove, and show that the reasoning leads to a contradiction; assuming ~A leads to proof of A. Your readers might be interested to know that there are many mathematicians who explicitly refuse to recognize the proof by contradiction, and construct alternative mathematical systems to get around utilizing it. This includes constructing alternative logical systems, where things like double negation (~~A → A) and implication statements don't apply or change their meaning entirely. Interested readers should google “constructivism” and “intuitionism.”

    Once dialectics is viewed in this way, it is enough to dispense with some of the things you have written, like (A v ~A) & (~A v A), which is not only gibberish, but repetitive gibberish. As I'm sure you're aware, the or/and statements are commutative, so you just wrote the equivalent of P & P.

    1. Well I never intended it to be anything more than a simple manual that gets away from the grand obfuscation that is usually brought to bear on any discussion of diamat, so I would agree with your trite/competent assessment. Which is also why I didn't care to go much further with this manuscript because, feeling so textbook, it also got rather boring for me.

      I do like your Sun example, and we can think of many different ones, but I was trying to keep it as simple sounding as possible. From what I remember (I'll know better when I do the copy-edits of later chapters) chapters four is an expansion and gets much more complex, with more complex examples, and then chapter five pushes it directly into what it's supposed to be about all along: social processes.

      While I agree that there are other significant mathematical systems (such as constructivism and intuitionism), I'm not sure it's helpful to bring this to bear. Yes the reduction of the formalization I used is a problem [I'm wondering if I made a typo, though, because the intent was to write (A v ~A) & (A & ~A)… still a problem but this would actually mean, in the way that logicians would take it (and the way I was trained in my graduate training to take it), A & B, which is indeed something formal logic can handle] but as i indicated it is kind of a useless gambit to formalize dialectics at all. There are a few folks out there who are trying, but most formal logicians who aren't dismissive of dialectics (like my logician colleagues) generally agree it's something that doesn't formalize like this. Hence one of the reasons I had a problem with this approach eventually and just said fuck it.

    2. I think the formalisation of the dialectic is one of those holy grails of Marxist theory. I had a colleague who was trying to work through that problematic for a number of years as his dissertation project, but unfortunately I think he has abandoned it. One key work he always pointed to as being pretty close, but still far away was the three volume attempt by Uwe Petersen (

    3. Yeah, I would say it was a holy grail quite literally: it's impossible in the same way (as an analogy) you can't formalize biology according to the language of physics, or physics according to the language of chemistry. There's nothing wrong with that, and the inability to do so doesn't mean that one is "more correct" than another. So with this in mind, while the Uwe Petersen attempt looks interesting, I'm not sure how it can escape the problem of false analogy or category mistake.

      Another person who is interested in finding a connection between analytical and dialectical logic, though with Hegel and formal logic, is Robert Brandom who was a Wittgensteinian. I'm not sure if he ever went so far as to try and formalize all of dialectics, but he did try to explain Hegel's logic in a way that made it coherent to those trained in the analytical tradition. That is, he's an analytic philosopher who doesn't dismiss dialectics as quackery and has been working on analytical appreciations of Hegel. Here's a link to his work on the *Phenomenology* (which he has chosen not to publish yet because he sees it, due to these translation problems, as a work in progress):

      He also has a paper, available on his university faculty homepage ( called "Understanding the Object/Property Structure in Terms of Negation: An Introduction to Hegelian Logic and Metaphysics" that is pretty neat. Meaning, contrary to those dialecticians who claim that formal logicians can't stand Hegel's dialectics and those formal logicians who think Hegel is the scourge of proper logic, there are people in the analytical camp who take him seriously and have tried to think through his logic.

    4. Don't know about Brandom's work, but Petersen's work is a serious attempt to formalise the dialectic. I am not exactly sure how category mistakes would apply in a pure formalisation, but you should check it out and see if he gets around it. Also, I would check out the work of Jean-Yves Girard and his approach to the problem of formulating a polytime logic. I mean both Petersen and Girard are working with some pretty dense and complicated formal logic and have been at it for close to 25-30 years.

    5. I wasn't saying that Petersen's work wasn't a serious attempt (in fact, it looks pretty cool) only that formalising the dialectic might be itself a kind of category mistake because it is something that shouldn't, by definition, be formalized according to a language system that belongs to a different logical lens.

  2. >While I agree that there are other significant mathematical systems (such as constructivism and intuitionism), I'm not sure it's helpful to bring this to bear.

    A couple of things I believe are helpful. Firstly, it shows that the juxtaposition between dialectics and formal logic isn't as straightforward as it seems, and there is no unified thing called formal logic that everyone who studies it agrees upon.

    Secondly, the issue of intuitionism and constructivism in mathematics is about the nature of proof by contradiction. There are people who would deny things which are proved by contradiction. One of the more famous proofs by contradiction involves proving that the square root of 2 is irrational. Assume sqr(2) is rational. Then it can be expressed as a fraction a/b, where atleast one of the terms must be odd (otherwise the fraction could be reduced further). a/b = sqr(2) yields a = b{sqr(2)}, and squaring both sides gives a^2 = 2b^2. Since 2 times anything is even, a^2 is even. This means b is odd. However, any even number squared is a multiple of 4, and since a^2 is a multiple of 4, then 2b^2 is a multiple of 4, and b is even! Therefore, sqr(2) can not be expressed as the fraction a/b.

    The constructivist would reject this argument, and the constructive proof of that the square root of 2 is irrational is a little bizarre, and these terms are defined differently. In the mind of the constructivist, not being rational is not the same thing as being iirational, so they just construct a proof that the sqr(2) is somehow apart from the rational numbers.

    That there exists this type of division amongst mathematicians and logicians, precisely over the nature of proof by contradiction, is not a coincidence. It is in essence a rejection of dialectical thinking mathematics, and can easily been seen as attempting to do mathematics without one of the best tools available for it.

    Moreover, the way dialectical thinking is used outside of mathematics isn't very different. It only appears different, by the method is similar, and can yield just as powerful of insights.

    1. Okay, in these ways I agree. My issue was more that it might be worth stepping outside of analytical versus dialectical conception of logic altogether rather than trying to unify them into a single practice. But yes, if the point is to demonstrate to someone who is completely convinced that the two are worlds apart, then your point is definitely salient.

      You don't even have to go too far outside of "traditional" formal logic to demonstrate the lack of unity in its practice. There's a reason why there's a bunch of books on the conditional arguing with each other over its "proper" understanding. On some level you have a basic and unified arithmetic, but once it starts getting complex then you often end up with that non-euclidean to euclidean geometry analogy.

      As an aside, and connected to this, what bothers me quite a bit about the typical Trotskyist understanding of materialist dialectics is that, following his *ABCs* they just assert that formal logic was a simple and completed study that hasn't developed anything new since the ancients. That in and of itself should make them embarrassed to talk about logic, but nope…

  3. Don't denigrate yourself so much about the usefulness (or lack thereof) of this manuscript. It might not be anything new and your stuffy academic peers might scoff at it, but you have a reader base consisting of plenty of non-academics and shitty internet "communists" who would probably find this useful. I've enjoyed these installments so far -- thank you.

    One other thing:

    On the 9th page of this chapter (page 32 of the manuscript) in the first full paragraph, I believe you erroneously reverse contradiction and non-contradiction in the last sentence. It's either that or I misunderstood something. RS

    1. Thanks for the comments. But to be clear the problem isn't that I have stuffy academic peers who would scoff at this (the academics I consider my colleagues are not the stuffy types, who I generally avoid and try not to be peers with, lol), because I'm generally disinterested in publishing for a primarily academic audience. My problem with this manuscript is that I dislike parts of it, and started losing my excitement for writing and reworking it around the time the draft was nearing completion. There's some things with which I disagree here and there, or am less certain about, so it's more of my own subjective feelings about my own work. The idea of it not being new of course never bothered me from the beginning, as I said to a previous commentator who was giving what might have been an underhanded compliment, the whole idea wasn't to provide something new but something precise that possessed a certain level of clarity. As someone who thinks it's philosophy's role to do that, if we're talking about philosophy of the Marxist type, then I'm pleased if I'm clarifying not new things––that's pretty much all I care about doing, hahaha.

      The stuffy academics I know (even the "Marxist" ones) but don't consider my peers (in the sense that I don't socialize with them) probably hate 90% of my output: I doubt they would like The Communist Necessity, and I'm sure they will be horrified with a new manuscript that just got picked up by a publishing company [more on that on this blog at a later date], but I don't denigrate these projects because I like them more.

      I'll have to look it over, but you're probably correct about it being a mistake. You know how it goes when you copy-edit your own work: you read what you think you wrote at certain places rather than what you actually wrote.


Post a Comment