Volpin

The Beautiful Consistency of Mathematics — Alexander Yessenin-Volpin

Jan Gronwald

Dec 16, 2020 — 10 min read

Mathematics is often believed to bring people to madness. We hear many stories like those about Gödel, Cantor, Nash, and Grothendieck, describing geniuses haunted by insanity that is developing along with their mathematics. And there is something to it. A certain psychologist said that

A paranoid person is irrationally rational. . . . Paranoid thinking is characterized not by illogic, but by a misguided logic, by logic run wild

Mathematics is the paradigm of rationality and maybe if the rationality takes over all of the aspects of life, we can talk of a mental issue. But this time I want to bring to light an opposite example. This time I want to share a story about a mathematician who was the voice of reason and sanity in the world that has run wild. And one whose mathematics was the model of his approach in social life. Meet Alexander Yessenin-Volpin (1924–2016).

Alexander Sergeyevich with his mother in 1928

The Son of a Poet

Born in the era of power struggle in the USSR and raised under Stalin’s rule, young Alexander experienced the birth of one of the most oppressive political systems on earth. But this was not obvious from the beginning: many Russian intellectuals strongly supported Bolshevik ideas of overthrowing the rotten tsarism and bringing the power to people. Volpin’s father, Sergei Yesenin (1895–1925), was surely one of them. As one of the most influential Russian poets of the 20th century, he stood up for the revolutionists. Although he never met his son, the atmosphere of alliance of Russian intelligentsia with the communist government must have been accompanying the growing Alexander. It must have contributed to the shock of discovering how the idyllic idea met the real life in the soviet Russia. Volpin recounts in Free Philosophical Tractate, which he wrote when two decades old, his “adolescent crisis” in April 1939 when he pledged himself to reason over the mundanely-understood “emotion”. The latter was propagated by the Russian communist ideologues of the era as the antidote to the bourgeois abstract non-materialistic attitude of the anti-Marxist philosophies. Volpin however felt the need to free oneself form the ties of the down-to-earth pragmatism. In his early writings he repeated again and again that

Life is an old prostitute whom I refused to take as my governess.

He believed that the liberation should emerge through authenticity and precision of language, understood ideally as mathamatically-inspired formalisation of the language of areas closest to the practical and social life: ethics and jurisprudence. Without a language that is transparent and unambiguous we will not be able, he believed, to trust our thoughts.

He sought for a tool for making the legal languege more exact in applying modal calculi to the juristic dictionary. Today we know that various deontic logics turned out to be quite handy instruments in legal theory (but not practice). Indeed, they allow to make legal inferences more transparent, but only on a rather superficial level: problems arise always when it comes to specification of good and life-fitting definitions of “permitted” and “obligatory”, two basic operators in deontic logics, along with formalisation of other legal terms.

His Law

Volpin did not give up. He was one of the first initiators of the civil rights movement in USSR. And his approach was quite exceptional given the system he lived in.

He would explain to anyone who cared to listen a simple but unfamiliar idea [...]: all laws ought to be understood in exactly the way they are written and not as they are interpreted by the government, and the government ought to fulfill those laws to the letter.

Yessenin-Volpin in fact praised the 1936 “Stalinist” constitution for various civic rights it granted. Vladimir Bukovskii, a friend of Volpin and later dissident criticizing Soviet abuse of psychiatry for political purposes, recounted that Volpin was

the first person in our life who spoke seriously about Soviet laws. [ . . . ] We laughed at him: ‘what kind of laws can there be in this country? Who cares?’ ‘That’s the problem,’ replied Alik, ‘Nobody cares. We ourselves are to blame for not demanding fulfillment of the laws.’

He rebuked Russians for acting as if they had no rights. Surreal as it might sound, it was this “literal” approach of Volpin’s that forced Soviet authorities to let the political opposition meet at Maiakovsky Square in Moscow to publicly read (it’s obvious which kind of) poetry. And it was Volpin who convinced the court guards to let him in the courtroom during the trial of writers Bakshtein, Osipov and Kuznetsov by pointing to applicable paragraphs in his copy of the Soviet Criminal Code he always carried with him.

This “concrete” approach to law was a surprisingly effective method of opposition as it openly demanded that the authorities observe their own laws. But Volpin took consistency and transparency to the next level. He applied the same hard-core concretist reasoning in the most exact of sciences.

His Mathematics

Yessenin-Volpin believed that the traditional style of making mathematics is similarly hypocritical to the style of handling legal issues in the Soviet Union. He claimed that the unreasonable and careless inclusion of the concept of infinity into mathematical discourse is the culprit of depriving it of exactness it was actually to grant.

Therefore he urged for a radical revision of foundations of mathematics, based on the claim that the concept of infinity, both potential and actual, is utterly nonsensical. He repudiated the existence of the infinite and so confined the domain of mathematical objects only to the finite ones. Such approach might ring a bell when we think of the finitists or finitistically-inspired mathematicians like Hilbert or Skolem. But Volpin went much further: recall that Hilbert’s Program did not reject the existence of the “infinitary” part of mathematics, but only strived to found it on the more concrete “finitary” part. Apart from that, Hilbert allowed for what we now call recursive algorithms ranging over infinite domains whilst for Volpin operations involving them were meaningless.

The expression f(n) (for any n) was completely senseless for Volpin, since it involved an unspecified number n, when one cannot be sure whether f is applicable to all numbers or whether what mathematicians call “all numbers” even exists for that matter. Note that it is not even real numbers and continuum that we talk about. Volpin rejected even the idea of the set of natural numbers, hence he called his stance “ultrafinitism”, where he assumed that one can only operate on specific numeric symbols expressing finite numbers and those only. And so the conventional (especially real) analysis, irrational numbers, calculus, traditional number theory along with other fields get annihilated. Not even mentioning topology or set theory. Such approach is possibly even more heretical to a mathematician than the idea of allowing assemblies and free press was to Soviet authorities. But Volpin did not create it out of mere whim.

Like in ethics, he struggled for conceptual precission. If he was shown a symbol, he wanted to be given its exact meaning — and not the metaphorical or unspecified “any” or “some”. For, and I believe that we have to grant him at least that, when we talk about transfinite numbers, beginning with ω, we do take their meanings as metaphors of some sort and we do make a leap of faith that one can operate on infinity as if it was a number. Volpin wanted to achieve his required level of exactness by founding the mathematical endeavor on the more concrete and down-to-earth, physical world. Hence he even contested the existence of numbers too big to occur in the sensible physical description of the universe. Harvey Friedman in his lectures related that

I have seen some ultrafinitists go so far as to challenge the existence of 2¹⁰⁰ as a natural number, in the sense of there being a series of “points” of that length. There is the obvious “draw the line” objection, asking where in 2¹, 2², 2³, … , 2¹⁰⁰ do we stop having “Platonistic reality”? Here this is totally innocent, in that it can be easily be replaced by 100 items (names) separated by commas. I raised just this objection with the (extreme) ultrafinitist Yessenin-Volpin during a lecture of his. He asked me to be more specific. I then proceeded to start with 2¹ and asked him whether this is “real” or something to that effect. He virtually immediately said yes. Then I asked about 2², and he again said yes, but with a perceptible delay. Then 2³, and yes, but with more delay. This continued for a couple of more times, till it was obvious how he was handling this objection. Sure, he was prepared to always answer yes, but he was going to take 2¹⁰⁰ times as long to answer yes to 2¹⁰⁰ then he would to answering 2¹. There is no way that I could get very far with this.

This anecdote perfectly captures Volpin’s consistent approach: if 4 is twice as much as 2, we should need twice as much time to realize the “shape” or the four-ness of this number. The tacit idea here is that numbers are not all cognized in the same kind of mental act, but are composed of other numbers, so in order to come to grips with the idea of a bigger number, one firstly has to grasp the idea of a smaller one. The procedure of answering “yes” to each of Friedman’s question with respective delays aptly pictures the ultrafinitistic stance on the mathematical reality. The latter is understood as a structure build up the most concrete “atoms” of mathematics — units. And this is the concretist, anti-metaphorical approach that made Volpin interpret mathematics in this manner. We can imagine him saying ‘Look, here are the “bricks” of mathematics — the starting point of mathematical reflection. One can operate on them in various ways: add and multiply them and do all sorts of operations on them, but without external presumptions about their nature or other metaphysics.’

As in ethics, Volpin wanted to free mathematics from what he believed to be unjustified dogmatism, from which originated all murky considerations about the infinite. He wrote in 1959 that the fallacy lies in the deceptive dogma that what is useful is also true:

We desire some kind of practical result, and we divide the sphere of all possible assumptions into two parts. One corresponds to “yes”; the other to “no”.

We explore reality and also divide the sphere of possible assumptions into two parts corresponding to “yes” and “no”. … We very often forget that these two divisions differ from one another, and as a result we adopt as reality that which is favorable.

We can note that such link is repeatedly occurrent in statements of various Platonists, regardless of it being G.H. Hardy connecting the beauty of mathematics with its truthfulness, or W.V.O. Quine stating that the usefulness of mathematics in explaining the nature necessitates its truthfulness. Yessenin-Volpin dubbed this fallacy ignoratio elenchi (ignoration of refutation) and believed that it was “the intellectual basis for every kind of demagogy.”

Yessenin-Volpin’s most renowned work in mathematics may be found in the following proceedings:

Ésénine-Volpine, A. S. (1961). “Le programme ultra-intuitionniste des fondements des mathématiques” in Infinistic Methods (Proc.) Oxford. pp. 201–223.
Yessenin-Volpin, A. S. (1970). “The ultra-intuitionistic criticism and the antitraditional program for foundations of mathematics” in Intutionism and proof theory (Proc.) Amsterdam. pp. 3–45.
Yessenin-Volpin, A. S. (1981). “About infinity, finiteness and finitization (in connection with the foundations of mathematics)” in Constructive mathematics (Berlin-New York. pp. 274–313.

His Fight

Thus we see Alexander Yessenin-Volpin struggling against demagogy in two most fundamental realms of human’s intellectual activity, the pure and practical reason. His efforts for civic rights in USSR earned him a number of periods in psychiatric asylums and even a 5-year exile in Syberia. Most interestingly, the official “diagnosis” that put him into asylum in 1968 was, as Vladimir Bukovskii reported, pathological honesty. Whether being honest with others and oneself could cause a mental issue is a topic for psychiatrists, but it is sure that Volpin, with his independence and simple sincerity, did not fit into the oppressive society he lived in. And he inspired others with his inner freedom: he stood behind the famous Glasnost (transparency) demonstration in 1965, was called the intellectual godfather of the civil rights movement in Russia, and contributed to awakening of the generation of political dissidents a decade before Solzhenitsyn. When he was incarcerated in the asylum in 1968, 99 Soviet mathematicians sent an open letter to the authorities requesting his release. After the case became international, Volpin was set free. He emigrated to USA in 1972.

Ironically, he was similarly alienated for his mathematical ideas in the free world as he was in the Soviet Union for political reasons. I believe this says something about traditional mathematics. The upshot is that either Volpin mistakenly interpreted the philosophical and foundational assumptions at the underpinning of mathematical practice, or his thought aptly pictures the intellectual inconsistencies in the so-called free society. It is certainly valuable to study his ultrafinitism in search for misconceptions, whether it be for recovering the philosophical justification for mathematics, or for sole development in scholarship.

But regardless of the question whether there is some point to Yessenin-Volpin’s heresies, what is exceptional in this figure is the intention of overarching struggle for independence and unity of thought. To me, the story of his life and fight is the realization of a deep message about the abstract and the practical being not so distant from each other. I interpret it as the manifestation of Weininger’s words that