A Dialogue on Infinity

Most likely you have heard about HEFCE’s proposal that in the REF (a  replacement for the RAE) 25% of future research funding would be  allocated according to the ‘economic and social impact’ of submitted  research. Many of our colleagues believe that this ‘impact’ proposal  represents an attack on the knowledge process and constitutes a threat  to the existence of basic research activity in the UK.

What appears to be missing from the increasingly intensive discussion is that the REF proposal provides not just the poison to kill independent  academic research, it offers a syringe for injection, too. The latter is described in a few innocuous lines about the aims of the exercise:

“We will be able to use the REF to encourage desirable behaviours at three levels:

• THE BEHAVIOUR OF INDIVIDUAL RESEARCHERS WITHIN A SUBMITTED UNIT [...]“

[http://www.hefce.ac.uk/pubs/hefce/2009/09_38/09_38.pdf , page 8]

The emphasis on inducing change in the behaviour of “individual researchers” is the result of a slow evolution of the RAE/REF. In 1996 and in 2001, the RAE went to great lengths to ensure that individual researchers could not be identified in the panels’ responses. This changed in 2008, when the percentages of the submission with each number of stars were published. So it was possible, in the case of a small unit, to work out exactly how many papers were internationally excellent, etc., and make a fairly good guess which papers they were.

The passage in the REF proposal concerned with “individual researchers” is much more worrying, especially since this time “the overall excellence profile will combine three sub-profiles – one for each of output quality, impact and environment – which will also be published.”

If “behaviour” just meant “doing good/bad/no research”, it would not be so terrible, but since extraneous things like “impact” now loom large, HoDs will be able to use this to warn staff off doing their preferred research and onto more “impactful” projects. There is a danger that, at department level, the REF might be translated into unheard of levels of bullying and harassment.

Please sign the Number 10 Petition:

http://petitions.number10.gov.uk/REFandimpact/

Please also sign the UCU petition STAND UP FOR RESEARCH (even if you are not an UCU member; signing is open to everyone):

http://www.ucu.org.uk/standupforresearch

Israel Gelfand (Израиль Моисеевич Гельфанд) passed away yesterday. RIP.

L’infini
Les galaxies,Les rimes en* I*,
le pipi,Et puis………
Ne fait pas la comédie
Mais c’est joli l’infini…
Et puis c’est pas fini!!
Noémie Kantor
(18 Mai 2000)

I do not know if this is of interest to you or not but here is a thought-experiment of mine, probably around the age of 7-8, for sure after 6 and before 10.

I started thinking about death and wanted to convince myself I would never die, instead of thinking about life after death… So I started thinking about an infinity in this way: first, I assumed that my entire life was only one dream in one night in another life where I am still the same person but could not fully realize that a full life goes on in each dream (an interesting point about personal identity, I guess). Now, that other life would be finite and have only a finite number of nights. So, I thought further that in each night there must be a finite number of dreams, encapsulating a finite number of lives. This was still short of infinity, so I started thinking that in each of these finitely many dreams of the finitely many nights, I would live a life that would in turn contain finitely many nights, which would contain finitely many dreams, and so on. I was not so sure that I was safe that way (i.e. that I would go on living forever), but I convinced myself that these were enough lives to live, so that even if the process would end, I would still have lived enough, and stopped thinking about it.

[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.

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.

(more…)

act now!

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.

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.”