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Constructive Mathematics in St. Petersburg, Russia: A (Somewhat Subjective) View from Within

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Modern Logic 1850-1950, East and West

Part of the book series: Studies in Universal Logic ((SUL))

Abstract

In the 1970 and 1980s, logic and constructive mathematics were an important part of my life; it’s what I defended in my Master’s thesis, it was an important part of my PhD dissertation. I was privileged to work with the giants. I visited them in their homes. They were who I went to for advice. And this is my story.

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Notes

  1. 1.

    To be more precise, we are shown several possible recommendations, and told that none of them is perfect.

  2. 2.

    Moreover, many important mathematical theorems establish exactly such equivalences: when we know necessary and sufficient conditions for some property, this brings a sense of completion and satisfaction.

  3. 3.

    It looks like this ideas was up in the intellectual air, since at that same time, in addition to Pimenov, similar ideas were proposed by the famous geometer Busemann [12] and by physicists Kronheimer and Penrose [77].

  4. 4.

    the son of A. A. Markov Sr., of the Markov processes and the Markov chains fame.

  5. 5.

    The fact that we can represent sequences of symbols by natural numbers was first discovered by Gödel and is therefore called Gödelization. This idea was new in the 1930s, but with the computers, it is so trivial that we feel that over-using this term to describe an otherwise clear idea may only confuse readers. Besides, the original Gödelization algorithm involved exponentiation \(2^{a} \cdot 3^{b}\cdot \ldots\); in the 1930s, this was a reasonable idea but now, with the clear distinction between feasible (polynomial-time) and exponential-time (non-feasible) algorithms, it does not make sense to introduce an unnecessary exponential time into something as trivial as representing strings in a computer.

  6. 6.

    It should be mentioned that constructive functions can only be applied to mathematically constructible real numbers—moreover, to compute the value f(x), we must know the exact code of the program that generates the original number x.

  7. 7.

    It is worth mentioning that the algorithm SH is known to be equivalent (under a suitable coding in Heyting’s formalized intuitionistic arithmetic) to recursive realizability introduced by Kleene [42].

  8. 8.

    From the classical viewpoint, the constructive logic of Markov’s school can be completely described using the three above-described basic principles: recursive realizability, the Markov principle, and classical logic for sentences containing no constructive problems, i.e., \(\exists,\vee\)-free sentences [94, 113].

  9. 9.

    While I truly appreciate what Yuri did, I want to add that this was an example of the attitude that was prevalent (and actively cultivated) in our department in general, and among logicians in particular: paraphrasing Rudyard Kipling’s Mowgli, we all had a strong feeling that we are all “of one blood”, that we are all brothers and sisters in mathematics and in science.

  10. 10.

    This result makes physical sense: in real life, if we process real values which are obtained with a higher and higher degree of accuracy by performing more and more accurate measurements, then we should be able to return the result at some point, before we know the detailed value of the inputs x—which is exactly what continuity is about.

  11. 11.

    It is worth mentioning that the resulting approach turned out be similar to the approach proposed in a somewhat different context by Yuri Ershov (see, e.g., [20]).

  12. 12.

    Almost periodic functions were invented by Harald Bohr, a mathematician brother of the Nobelist physicist Niels Bohr.

  13. 13.

    Shortly after that, another version of constructive set theory—this time based on type theory—was proposed by Per Martin-Löf [91] (see also [29, 99, 112]). Since Martin-Löf did not need to deal with the more complex axioms of ZF, his theory is much clearer and simpler than the Gelfond and Lifschitz’s version—which is probably one of the reasons the reason why they never published their version.

  14. 14.

    In [47], it is shown that such non-algorithmic sequences are intuitively justified. Without them, discrete transition processes (e.g., radioactive decay) would potentially lead to devices checking whether a given Turing machine halts or not.

  15. 15.

    It is worth mentioning that when he presented this work in St. Petersburg, he drew a target on his flyer—expecting that in this center of constructive mathematics, he would be attacked for suggesting that non-constructive sequences are possible.

  16. 16.

    Fixed point and floating point formats have to be treated separately, since the transition from floating point to fixed point requires, in general, exponential time.

  17. 17.

    Luckily, our reasons for boom were different from Pushkin’s: he got stuck in the village of Boldino due to the quarantine caused by the deadly cholera epidemic.

  18. 18.

    It is not that everyone else willingly supported the Communist regime: when the first reasonably free elections where held in St. Petersburg in 1989, most communist candidates convincingly lost. However, many logicians went further than many others in their resistance.

  19. 19.

    Pimenov, by the way, taught me to not be afraid of the KGB-installed electronic bugs in our homes: they already know, he said, that we are mostly against them, so they do not gain anything by hearing us say it one more time.

  20. 20.

    This was even more appalling to me, since Xerox services were highly rationed, I could rarely get a copy of needed papers, but the KGB seemed to have an unlimited ability to copy everything we sent.

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Acknowledgements

This work was supported in part by the National Science Foundation grants HRD-0734825 and HRD-1242122 (Cyber-ShARE Center of Excellence) and DUE-0926721.The author is greatly thankful to all his colleagues for valuable discussions. My special thanks to Irving Anellis, who tirelessly kept alive interest in foundations and history of logic and foundations of mathematics, especially history of login in Eastern Europe. My sincere thanks to Francine F. Abeles and Mark E. Fuller for their great idea to have a book published in Irving’s memory, and for their support, encouragement, and editing help. Thank you all.

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  1. Department of Computer Science, University of Texas at El Paso, 500 W. University, El Paso, TX, 79968, USA

    Vladik Kreinovich

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  1. Vladik Kreinovich
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Correspondence to Vladik Kreinovich .

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  1. DepartmentsofMathematicsandComputerSci, Kean University, Union, New Jersey, USA

    Francine F. Abeles

  2. University of Wisconsin, Janesville, Wisconsin, USA

    Mark E. Fuller

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Kreinovich, V. (2016). Constructive Mathematics in St. Petersburg, Russia: A (Somewhat Subjective) View from Within. In: Abeles, F., Fuller, M. (eds) Modern Logic 1850-1950, East and West. Studies in Universal Logic. Birkhäuser, Cham. https://doi.org/10.1007/978-3-319-24756-4_11

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