Logic in the early Soviet Union

(I changed the title of this a few months after writing it — all my titles are terrible. If this one jars with the content — I don’t think it does, much — ignore it and imagine instead a better title. In fact, go ahead and imagine a really great, perfect title, thanks.)

People today are interested in ideology. What exactly ideology means is a bit unclear, but I’ll understand it as a set of background beliefs imposed upon us (not necessarily arrived at ourselves), debatable, and action- and belief- shaping.

Imposed upon us: they’re not things that we reason to (the way I reasoned to the contents of this blog, for example) but are instead things given to us by our social and political position. We might say they are presupposed of us: we’re assumed to instantiate a given ideology, and to reject this we have to come out and explicitly say it. So, demographically, it’s a pretty good guess that I’d support the welfare state, and so I do.

Debatable: unlike some of the things imposed on us, such as common mathematical and scientific truths, people differ in ideology. Thus some people don’t think the welfare state is worth supporting, and they’ve reasoned to that conclusion.

Action- and belief- shaping: this is a bit vague, but it’s meant to get at the idea that ideology is important. It provides a set of concepts in terms of which we understand the world, in a way that my belief that, say, Russell is better than Wittgenstein (which is also unimposed and also debatable) doesn’t. I vote for leftist parties who think tax is good.

A reason to think that people are interested in ideology is that we can view many contemporary debates as clashes of ideology. Consider the role of science in society. John von Neumann seemed to think of science as something that floated free from moral concerns. Science is its own thing and its pursuit shouldn’t be affected by morality (see my last post for details, if interested). Many today think this is both factually and morally wrong. Science is always socially-bound and needs to be morally answerable. Thus anyone working on AI needs to face up to the problem that the datasets they use might encode human bias.

I would say these lead to two ideologies of science, two deep ways of understanding science that at least many of us come to inherit from others. Roughly, more humanities-oriented people (my friends, at least my Twitter mutuals, for example) opt for the latter, morally-laden view, and maybe Mark Zuckerberg the former (although he wouldn’t admit it). We see this further with things like the role of critical race theory in schools. The case is harder to make precisely here, but a bunch of people think that things like Nikole Hannah-Jones’s 1619 project is an attempt to smuggle in ideology into history, and thus that its being taught in schools is a bad thing. In Newt Gingrich’s words, it is “left-wing propaganda masquerading as ‘the truth’”. While Gingrich is a gigantic arsehole (this isn’t ideology — it’s just truth — as Terry Eagleton says “ideology, like halitosis, is what the other person has”) whom I’m loath to quote, that viewpoint is one hard to avoid in many corners of social media. And a third, more trivial (but pertinent for us) example, you might have seen people arguing whether the truths of mathematics are socially constructed, with people giving — tongue in cheek — arguments in favour of the claim that 2+2=5, as well as more serious ones that the practice and thus surely the results, of mathematics, as of any institution, has a certain power structure that could be more or less right.

So: important, timely topic. My aim here is to explore the varieties of ways that ideology can manifest and impose itself by presenting an intrinsically interesting and, as far as I know, underdiscussed example: that of a particular philosophical logician in the early Soviet Union, Ivan Orlov. My hope from the blog is i) that the background, both in terms of the intellectual culture of the Soviet Union and in terms of the logic is, interesting and underknown ii) that attending to the example can shed light on the interplay between ideology and belief. In order to begin our story, we must begin with Lenin’s Materialism and Empiro-Criticism.


Materialism and Empiro-criticism is possibly the most influential work of philosophy in the 20th century. It became the party line, the core philosophical text, “an obligatory subject of study in all institutions of higher education in the Soviet Union” (wikipedia), a description of which was included in the Short Course of the History of the Communist Party that everyone had to know. Literally tens of millions learned, or at least read the tenets of ‘diamat’:

1) The world is by its nature material;
2) matter is primary, an objective reality existing outside and independent of our minds, while mind is secondary, being a reflection of matter;
2) the world and its laws are fuly knowable, authentic knowledge being tested by experience and practice

The text itself reads like a subtweet (“not so much an argument as a howl”, Simon Ings says, “a mixture of rage, sarcasm, and personal abuse”, per Leslie Chamberlain). The philosophical target is Machianism as relayed by Alexander Bogdanov which is basically a pretty tame empiricist philosophy that attempts to make sense of the fact that there is a world, and we as experiencers of the world, and somehow our experiences track how the world is. For reasons that I struggle to keep in my head, Bogdanov and Lenin were initially political allies but drifted apart at the turn of the century, the former becoming a Menshevik and the latter, of course, a Bolshevik. Lenin was pretty pissed off at their falling out, and dealt with the situation like any mature adult, by storming off to the British library for a few months to write a 400 page epistemological-metaphysical treatise.

Bogdanov’s view is basically (as I understand it — I haven’t read him, so I’m going on secondary sources including Lenin) that the building block of reality is sensations, and that objects are collections of sensations. A rose is a red sensation, a whatever-shape-roses-are sensation, a spiky sensation, a smell. The objective world is characterized by a progression of sensations that has a sort of order; it’s that that distinguishes subjective experiences (say hallucinations or illusions) from objective experience.

For turn of the century philosophy, this is a respectable view: it would be called phenomenalism a bit later and have famous defenders. Lenin isn’t a fan. He thinks that the Machian, whose goal is to create an epistemology that fits with our scientific theory of the world, ends up with solipism, idealism, Berkleanstvo. He defends:

“the opposite theory [to Mach’s] which holds that sensations are “symbols” of things (it would be more correct to say images or reflections of things). The latter theory is philosophical materialism.”

This view hasn’t gone down too well in the secondary literature (it has “has never been attacked enough” (Chamberlain), it is “against evidence, reason, and every serious thinker since Plato” (Ings)), but they’re wrong: it’s again a perfectly reasonable view (albeit one with plenty of well-known problems), indeed one not far from what Betrand Russell describes in his Problems of Philosophy, published a few years after Lenin’s book:

“if we take any common object of the sort that is supposed to be known by the senses, what the senses immediately tell us is not the truth about the object as it is apart from us, but only the truth about certain sense-data which, so far as we can see, depend upon the relations between us and the object. Thus what we directly see and feel is merely ‘appearance’, which we believe to be a sign of some ‘reality’ behind.”

The problem with Lenin, we’re tempted to say, is not what he believes, but why he believes it. I’ve already said it’s because he disliked Bogdanov, but there’s more: his whole book is basically an appeal to authority. The above passage immediately continues, with some admittedly good parenthetical snark:

“For instance, the materialist Frederick Engels — the not unknown collaborator of Marx and a founder of Marxism — constantly and without exception speaks in his works of things and their mental pictures or images (Gedanken-Abbilder), and it is obvious that these mental images arise exclusively from sensations. It would seem that this fundamental standpoint of the “philosophy of Marxism” ought to be known to everyone who speaks of it, and especially to anyone who comes out in print in the name of this philosophy.”

And it continues like this, on and on (and on). Lots of quotations from friends and enemies, lots of polemic, few arguments. Perhaps the most read work of 20th century philosophy isn’t really philosophy at all, in the sense of an argument for a conclusion; but, despite this, and strangely, it still ends up, as if accidentally, defending a defensible position. The important thing is that this materialist Leninism will function as the ideology shaping Soviet thinkers, perhaps the most notorious example of which is


Without getting into too many details — which are better told by others, because the story is complicated and nuanced and in certain respects the jury is out on parts of the science (I’m thinking of epigenetics and related things) — people ended up the view that Lamarckian theories of inherited acquired characteristics were right, because they were viewed as more in line with Marxist materialism. For the Lamarckian, if one generation acquires a characteristic it previously lacked — imagine we all had to go out fishing and hunting all day, leaving our computers behind, becoming athletic, no longer sweating when we go upstairs — its offspring could inherit that characteristic. This fit well with Marxism, as

“Marx had claimed that socialism could reform and improve people’s well-being within a single generation (at least, that was how Lenin and Bogdanov read him), and the inheritance of acquired characteristics seemed the only possible mechanism by which this miracle might be achieved” (Ings)

And so in the Soviet Union Lamarckian theories were to some extent party-line, defended by one Troifim Lysenko. And this of course had effects, because it compelled scientists, notably agriculturalists, to hew to wrong theories, with real-world effects (less good crop yields; I don’t know quite how disastrous this was, so this might be a gigantic understatement).

This is a clear example of ideology disrupting those who held it — indeed, it’s probably the most famous example of which we know. And its lesson seems clear: appealing to authorities in this way and rejecting truths that don’t fit with it leaves you, literally, unable to live, unable to feed yourself.

What I want to do now is present a less well-known example from the same society. In order to do so, I need to lay some groundwork by discussing the revolutions in logic at the turn of the twentieth century.


As the title of a famous anthology has it, from Frege, who resolved long-standing problems in logic, proposed a novel philosophy of maths, and laid the ground for 20th century linguistic semantics, to Goedel, who presented astonishing results about the newly rigorous formal systems people like Frege (and Peano, and Russell, and Hilbert, and many more) discovered, the period from roughly 1870 to 1930 was a golden age for logic and the foundations of maths.

One of the motivations behind the new logicians was certain unclarities in the foundations of maths, especially concerning the infinite. One such uncertainty was owing to calculus which was a thoroughly indispensable tool for mathematics and science, but one whose nature was mysterious. In taking a derivative, it’s sometimes said, one is summing infinitesimal changes in the value of a function, but the nature (and indeed existence of) these infinitesimal changes in a function’s value were shrouded in mystery. After all, the infinite is a pretty mysterious, weird concept, and various mathematical trolls in the 19th century created messed up functions that didn’t behave nicely and confused people, but which forced thinkers to get serious about what they were talking about when talking about derivatives and functions and such.

Various famous mathematicians doing various complicated maths things that I won’t pretend to properly understand made plenty of headway here (I think Weierstrass is important), but the important one for our purposes is Cantor and his work on the nature of infinity.

Thankfully, the part of it I need for my purposes is super straightforward and intuitive. Among other things, Cantor is famous for his diagonal argument that shows that there is at least two different types of infinity, in the sense that some infinities are bigger than others. See, doesn’t that already sound mysterious and weird?

To see what we need to see, we can begin by asking: what is infinity? We might think that the natural numbers are infinite, and that what that means is that we can start counting with the first one, 1, and continue counting for infinite time. On this view, infinity would be something that was never ‘complete’, never actual, because we could never finish the counting — that’s what makes it infinite!

With this mere idea of counting, of pairing off numbers with something else (here, time), Cantor showed a lot. He showed the counterintuitive idea that infinity minus one is still infinity. To see this, consider the natural numbers, and the natural numbers apart from one. Here’s something you can do: you can pair off the natural numbers with the natural numbers minus one, and it’s obvious you’ll still be counting forever. The set of numbers, and the set of numbers minus 1, or 10, or 100,000,000, is the same size!

But more. If some infinities that seem like they should be different sized are same sized, he also showed there were bigger infinities. Imagine we list the natural numbers like this:


Now, if we’ve helped ourselves to the infinity of natural numbers, we have conceptual space for numbers consisting of infinite digits, which we would depict as so d1d2d3. For simplicity, let’s say that the only digits are zero and one. Then we’ll have numbers like 010100... or 101010..., where I’ll use dots to indicate infinite continuation.

Okay, so we’ve got our natural numbers; 1,2,3, …, of which we agree there’s an infinity. Then we have our d-numbers, infinite numbers like 001010…. We can then ask: how many such d-numbers are there?

And the answer is: there are more of them than the natural numbers. But the natural numbers are infinite. So the d-numbers must be bigger than infinite — they must be a bigger infinity! To see this, imagine we pair off the d-numbers with the natural numbers. We would probably use some sensible function to assign a a d-number to a natural number, but if we didn’t and just dumped them willynilly, we might have something like

1 101010101…
2 111110010…
3 101010111…
4 000111011…
5 011011010…
6 100000000..
7 011000000..
8 001111100..
9 111111111..

Now here’s the thing: there must be a d-number missing from that list. Indeed, we know which one. It’s the number got by for any natural number n, switching the n-th location in the d-number assigned to that n. It’s easier to look at the example:


So it turns out that that number must be missing from our enumeration, and thus that we can’t count all the d-numbers with the natural numbers, thus the d-numbers must be greater than the naturals, thus, given the latter are infinite, the former must be a bigger infinity! Pretty wild stuff. What did our Leninists make of it?

Orlov published in this journal, which can be read on archive.org. The photocopying isn’t great, but it seems churlish to complain


With all that finally said I can now turn to the point of this post. It turns out that in the first few decades of the 20th century, Russia produced some interesting work in logic, despite the fact that it increasingly became necessary to tow the Lenin party line concerning dialectical materialism. For example, Nikolai Vasiliev, already in the 1910s, was — through a glass, very darkly — anticipating trends in non-classical logic. In particular, just as his countryman Lobachevsky developed non-Euclidean geometry, he wondered whether he could create a sort of “curved” logic as thinkers had created geometries of curved space, and so he wondered about, just as the non-Euclidian rejected things like Euclid’s fifth postulate, he could reject fundamental laws of logic, like the law of non-contradiction. While he arguably didn’t get too far (see this article for an English account of one of his papers; I’m relying on Valentin Bazhanov’s 2001 Essays on the Social History of Logic in Russia; all details in this section are from this book; all (not very good) translations are mine), it’s still fascinating to see these modern ideas presented back then, and it makes one wonder what Russian philosophical logic could have been had it not been for the repressive time.

The figure I want mainly to talk about, Ivan Orlov, has the interesting twin properties of being a hardcore Leninist but, at the same time, advancing, indeed anticipating by decades, an important trend in logic, namely relevance logic. Orlov didn’t like the hierarchy of infinities Cantor introduced. He had technical reasons for this, but also ideological ones. I quote from Bazhanov:

“He explained the popularity of Cantor’s theory by the inclination mathematicians had for its abstract, supersensible character. Mathematicians were inclined to relativism and mathematical fetishism in so far as they don’t have to deal with immediate experience, and as science moves away from experience and towards pure speculative reason it produces “soap bubble thoughts” [can’t think of the appropriate English idiom but I think there is one, quotes in original] like actual infinity.”

This, especially the talk of ‘relativism’ sounds Leniny, and immediately after in the book cited we read him criticising a book by probability theorist for not containing ‘a trace of dialectics’ and being too abstract, while a few pages later, Bazhanov quotes him as saying that even in the domain of mathematics

“There’s one unified front for fighting against idealists and peddlers of the official “truths” of bourgeois universities”

So, seems pretty conclusive: our Orlov is just a less famous Lysenko, ideologically captured and spouting the party line.

Yes, but. In 1928, he wrote the article called “The Logic of Compatibility of Propositions” in which he can be seen to anticipate a range of important work in philosophical logic known as relevant (or relevance) logic (he’s even credited as such on relevant logic’s wiki). This somewhat recondite-sounding topic actually speaks to a crucial and still discussed feature of logic and semantics, namely the proper treatments of conditionals. In classical logic, sentences like “if the moon is made of cheese, I’m a millionaire” come out true, because conditionals are always true if the first sentence in them (called the antecedent) is true.

It is not much of an exaggeration to say that an awful lot of 20th century philosophy has been concerned with coming up with better theories of conditionals. Modal logic is based on the idea of strengthening it; some pragmatic theories of language use aim to explain how it might be that such a sentence could seem true while being false. One idea, from the former domain, for example, is that conditionals are strict — they are only true provided it’s necessary that if the first bit is true the second is. This avoids some problems, but only goes a certain way. Relevant logic proposes that such sentences are true provided there’s some sort of connection in meaning between the two sentences, and Orlov presented a pretty neat formal theory of them, albeit one certainly of its time, and that only goes so far.

The key point is this: despite being in an ideologically cramped environment, Orlov was able to produce important work. Indeed, I wonder — but here I go beyond anything for which I have citations — whether that very crampedness might have helped him. Perhaps the attempt to be a somewhat commonsense thinker, one not lost in mathematical abstraction and tethered to the world, like a good materialist, encouraged him to explore these avenues. If that were so, then it could be that even bad ideologies can lead to good effects. And indeed I think we see this. I think I can learn things from ideologically diverse people: from libertarians I learn about cryptocurrency, from Christians I learn about forgiveness. It might be — -though this is way underdeveloped — that a certain ideological diversity is epistemically good, as suggesting ideas you wouldn’t otherwise come across, even if you think the diverse ideologies are wrong ones.

What should we take from all this? Well, I don’t know. Hopefully it’s just an interesting story, for the most part. Personally, I think the what if question about how Russian philosophy could have turned out, given the obvious talent around (there are other chapters in Bazhanov’s book I could have discussed). Orlov gave up philosophy and became a chemist. No one took up Vasiliev’s gropings towards non-classical logics. In an alternative history, we could easily imagine Russians leading the philosophical world (I made the case for some other Russian thinkers in previous posts). I think the Lenin material is just intrinsically neat: that the philosophical edifice of the Soviet Union was trolly point-scoring is funny in a dark sort of way. Finally, and most importantly, I think our Orlov shows us how truth can out against ideological challenges, something perhaps useful when assessing people coming at the world from places other than ours.



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