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Achilles, Tortoise and Yessenin-Volpin February 16, 2009

Posted by Alexandre Borovik in Uncategorized.

[moved here from the old blog]

I quote a description of Zeno’s “Achilles and Tortoise” paradox from Wikipedia:

“In a race, the quickest runner can never overtake the slowest, since the pursuer must first reach the point whence the pursued started, so that the slower must always hold a lead.” (Aristotle Physics VI:9, 239b15)

In the paradox of Achilles and the Tortoise, we imagine the Greek hero Achilles in a footrace with the plodding reptile. Because he is so fast a runner, Achilles graciously allows the tortoise a head start of a hundred feet. If we suppose that each racer starts running at some constant speed (one very fast and one very slow), then after some finite time, Achilles will have run a hundred feet, bringing him to the tortoise’s starting point; during this time, the tortoise has “run” a (much shorter) distance, say one foot. It will then take Achilles some further period of time to run that distance, during which the tortoise will advance farther; and then another period of time to reach this third point, while the tortoise moves ahead. Thus, whenever Achilles reaches somewhere the tortoise has been, he still has farther to go. Therefore, Zeno says, swift Achilles can never overtake the tortoise. Thus, while common sense and common experience would hold that one runner can catch another, according to the above argument, he cannot; this is the paradox.

Scott Aaronson’s post in his blog Shtetl-Optimized “And they say complexity has no philosophical implications” (see more about it below) reminded me that the most natural approach to the paradox is complexity-theoretic. Indeed, we have two different timescales: the one, in which the motion of Achilles and the Tortoise takes place, and another one, in which we discuss their motion, repeating again and again

it will then take Achilles some further period of time to run that distance, during which the tortoise will advance farther“.

Clearly, each our utterance takes time bounded from below by a non-zero constant; therefore the sum of the lengths of our utterances diverges. However, our personal time flow has no relevance to the physical time of the motion!

Well, probably this explanation of the paradox is well-known, but the reason why I am writing this post is the next, even more fascinating story mentioned in Shtetl-Optimized. In a sense, it is dual to the Achilles and Tortoise paradox (and perhaps the duality could be made explicit). It is told in Harvey M. Friedman’s lectures Philosophical Problems in Logic. Friedman said:

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.

Yessenin-Volpin’s response makes it clear that the Achilles and the Tortoise paradox is not so much about the actual infinity as of a potential infinity (or just plain technical feasibility) of producing the sequence

1/2, 1/4, 1/8, 1/16, etc.

in real time. I agree with Scott Aaronson: and they say complexity has no philosophical implications!

However, there is yet another layer in this story. Anonymous said in a discussion in Shtetl-Optimized:

What a beautifully clever way to respond to such a line of questioning!





Well, one should remember that Alexander Yessenin-Volpin (listed in Wikipedia as Esenin-Volpin) was one of the founding fathers of the Soviet human rights movement and spent many years in prisons, exile and psychiatric hospitals. He knows a thing or two about interrogations; in 1968, he wrote and circulated via Samizdat the famous “Memo for those who expects to be interrogated“, much used by fellow dissidents.

It is remarkable how the personality of a mathematician can be imprinted on his work and his philosophical views.

Indeed, Alexander Sergeevich Yessenin-Volpin was also a pote of note. One of his poems, a very clever and bitterly ironic rendition of Edgar Alan Poe‘s The Raven, is quite revealing in the context of our discussion. I give here only the first two and the last three lines of the poem. (A full text of the poem (in Russian) can be found here and here.)


Как-то ночью, в час террора, я читал впервые Мора,
Чтоб Утопии незнанье мне не ставили в укор …


… Но зато как просто гаркнул чёрный ворон: «Nеvеrmоrе!»
И вожу, вожу я тачку, повторяя: «Nеvеrmоrе…»
Не подняться… «Nеvеrmore!»


To make these lines more friendly to the English speaking reader, I explain that the first two lines refer to Thomas More‘s Utopia: the protagonist reads Utopia to avoid an accusation that he has not familiarized himself with the utopian teachings promoted by the totalitarian system. The three exclamations “Nevermore!” which end the poem do not need translation.

The poem is written in 1948 (by a remarkable coincidence, the year when George Orwell wrote his 1984 — the title of the novel is just a permutation of digits; in 1949, when Orwel’s novel was published, Yessenin-Volpin started his first spell in prisons). As we can see, Yessenin-Volpin, who was 23 years old at the time, developed an ultrafinitist approach to utopian theories (and especially to the utopian practice) much earlier than to problems of mathematical logic.

From our sponsors November 20, 2008

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You may have heard the news already, but I am delighted to let you know that the John Templeton Foundation was awarded the National Humanities Medal at the White House yesterday. The citation praised the Foundation “for opening new frontiers in the pursuit of answers to mankind’s oldest questions.” Dr. John M. Templeton, Jr. accepted on behalf of the Foundation. It is an award that belongs to everyone who has worked over the years to realize Sir John’s vision, and we thank all of you for your contributions to that effort.


Save The London Mathematical Society October 20, 2008

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act now!

A talk at Jodrell Bank September 24, 2008

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Yesterday I gave a talk “Social life of Infinity: from mathematics to Kitsch” at “Arts meet Science” event held at Jodrell Bank.  I mentioned in the talk that kisch is virtually unknown in mathematics and was presented with a counterexample: one of the speakers, sci-fi writer David  McIntee, gave me a Sudocube bought in the Visitor Centre shop. I proudly display a photo.

Back to Zeno, again and again… September 7, 2008

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A very interesting survey of the old conundrum written by a physicist, Zurab Silagadze.  Abstract:

“No one has ever touched Zeno without refuting him”. We will not refute Zeno in this paper. Instead we review some unexpected encounters of Zeno with modern science. The paper begins with a brief biography of Zeno of Elea followed by his famous paradoxes of motion. Reflections on continuity of space and time lead us to Banach and Tarski and to their celebrated paradox, which is in fact not a paradox at all but a strict mathematical theorem, although very counterintuitive. Quantum mechanics brings another flavour in Zeno paradoxes. Quantum Zeno and anti-Zeno effects are really paradoxical but now experimental facts. Then we discuss supertasks and bifurcated supertasks. The concept of localization leads us to Newton and Wigner and to interesting phenomenon of quantum revivals. At last we note that the paradoxical idea of timeless universe, defended by Zeno and Parmenides at ancient times, is still alive in quantum gravity. The list of references that follows is necessarily incomplete but we hope it will assist interested reader to fill in details.”

New Directions in Philosophy of Mathematics September 2, 2008

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MIMS Workshop on New Directions in Philosophy of Mathematics

Manchester, Saturday 4 October 2008.     

This workshop is part of the MIMS (Manchester Institute for Mathematical Sciences) New Directions series of workshops taking place in MIMS throughout 2008.

Mathematics and philosophy have a long history of involvement with each other. Profound changes to both disciplines have occurred through this interaction from the Greek exploration of the foundations of geometry, through the early modern philosopher-mathematicians, such as Descartes (analytic geometry) and Leibniz (calculus), to Frege and the beginnings of analytic philosophy under Russell. In recent times, however, this involvement has largely dwindled. For the most part philosophy’s interest in mathematics over the past half century has been of no interest to mathematicians. There has been a growing unrest with this state of affairs, and we are beginning to see encouraging signs of efforts to bridge the gulf between these great disciplines.

These new approaches range from studies which pay close attention to the cognitive, historical, or sociological aspects of mathematical practice, through to those which see developments within recent mathematics as being of philosophical importance, whether model theory, category theory, or the current intense interaction between mathematics and physics. The workshop will explore these themes, and will allow philosophers and mathematicians the chance to hear each other’s views on the direction forward for the philosophy of mathematics.

Organisers: Alexandre Borovik (School of Mathematics, University of Manchester), David Corfield (Department of Philosophy, University of Kent).

Programme, Saturday 4 October 2008:

10:00 Coffee

10:30 Mary Leng (Liverpool) Creation and discovery in mathematics

11:30 George Joseph (Manchester) History of Non-Western Mathematics: New Perspectives

12:30-13:00 Lunch, to be served on premises

13:00 Marcus Giaquinto (UCL) Curves in Proofs

14:00 Angus Macintyre (QMUL) The Impact of Incompleteness on Pure Mathematics

15:00 David Corfield (Kent) The Reality of Mathematics

16:00 Panel discussion, with some wine being served.

17:00 End

Venue: All talks will take place in Frank Adams Room in MIMS in the Alan Turing Building at the University of Manchester. The building is 20 minutes walk from the city centre and 15 minutes walk from Piccadilly or Oxford Road train stations.  

Directions to MIMS are available.

Night before, Friday 3 October: for those particiapants, who come on Friday to stay overnight, or who are local, we suggest an informal get-together at 19:00 at Lass O’Gowrie  with the aim of having dinner at 20:00 at East Z East Ibis Hotel on Princess street. Since an advanced booking for East Z East is needed, please notify me in advance that you are coming for dinner.

 If you wish to attend, please, send an e-mail by Friday 26 September to

Internal Set Theory August 19, 2008

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From Edward Nelson’s introduction to his book:

Ordinarily in mathematics, when one introduces a new concept one defines it. For example, if this were a book on “blobs” I would begin with a definition of this new predicate: x is a blob in case x is a topological space such that no uncountable subset is Hausdorff. Then we would be all set to study blobs. Fortunately, this is not a book about blobs, and I want to do something different. I want to begin by introducing a new predicate “standard” to ordinary mathematics without defining it.

The reason for not defining “standard” is that it plays a syntactical, rather than a semantic, role in the theory. It is similar to the use of “fixed” in informal mathematical discourse. One does not define this notion, nor consider the set of all fixed natural numbers. The statement “there is a natural number bigger than any fixed natural number” does not appear paradoxical. The predicate “standard” will be used in much the same way, so that we shall assert “there is a natural number bigger than any standard natural number.” But the predicate “standard”— unlike “fixed”—will be part of the formal language of our theory, and this will allow us to take the further step of saying, “call such a natural number, one that is bigger than any standard natural number, unlimited.”

We shall introduce axioms for handling this new predicate “standard” in a consistent way. In doing so, we do not enlarge the world of mathematical objects in any way, we merely construct a richer language to discuss the same objects as before. In this way we construct a theory extending ordinary mathematics, called Internal Set Theory that axiomatizes a portion of Abraham Robinson’s nonstandard analysis. In this construction, nothing in ordinary mathematics is changed.

Edward Nelson, “Internal set theory: A new approach to nonstandard analysis,” Bulletin American Mathematical Society 83 (1977), 1165–1198.

Alain Robert, “Analyse non standard,” Presses polytechniques romandes, EPFL Centre Midi, CH–1015 Lausanne, 1985; translated by the author as “Nonstandard Analysis,” Wiley, New York, 1988.

Internal set theory and non-standard analysis August 18, 2008

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I am reading a fantastically beautiful book:

Robert, Alain (1985). NonStandard Analysis. John Wiley & Sons. ISBN 0-471-91703-6

A timely reminder August 16, 2008

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A leaflet from a local pizza parlour landed at my doorstep, a timely reminder that I have to write in this blog more regularly: I have more material than I manage to type up (the delay is mostly caused by my damaged hands).

Same but multifaceted July 11, 2008

Posted by Alexandre Borovik in Uncategorized.

Continuing the topic of “sameness”, it is interesting to compare behaviour of two familiar objects: the field of real numbers \mathbb{R} and the field of complex numbers \mathbb{C}.

\mathbb{C} is uncountably categorical, that is, it is uniquely described in a language of first order logic among the fields of the same cardinality.

In case of \mathbb{R}, its elementary theory, that is, the set of all closed first order formulae that are true in \mathbb{R}, has infinitely many models of cardinality continuum 2^{\aleph_0}.

In naive terms, \mathbb{C} is rigid, while \mathbb{R} is soft and spongy and shape-shifting. However, \mathbb{R} has only trivial automorphisms (an easy exercise), while \mathbb{C} has huge automorphism group, of cardinality 2^{2^{\aleph_0}} (this also follows with relative ease from basic properties of algebraically closed fields). In naive terms, this means that there is only one way to look at \mathbb{R}, while \mathbb{C} can be viewed from an incomprehensible variety of different point of view, most of them absolutely transcendental. Actually, there are just two comprehensible automorphisms of \mathbb{C}: the identity automorphism and complex conjugation. It looks like construction of all other automorphisms involves the Axiom of Choice. When one looks at what happens at model-theoretic level, it appears that “uniqueness” and “canonicity” of a uncountable structure is directly linked to its multifacetedness. I am still hunting appropriate references for this fact. Meanwhile, I got the following e-mail from a model theorist colleague, Zoe Chatzidakis:

Models of uncountably categorical theories behave really like vector spaces: if inside a model M you take a maximal independent set X of elements realizing the generic type, and take any permutation of X, it extends to an automorphism of the model. So, if M is of size \kappa > \aleph_0, then any basis has size \kappa, and its automorphism group has size 2^\kappa.

I don’t know a reference, but it should be in any model theory book which talks about strongly minimal sets. Or maybe in the paper by ??? Morley ??? which shows that you have a notion of dimension and so on? I.e., that \aleph_1 categorical theories and strongly minimal sets are the same.

It is really a well-known result, so you probably don’t need a reference if you cite it in a paper.