A Dialogue on Infinity

[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.

Following Alexandre’s two posts on ultraproducts of fields (here and here), I was wondering about the category theoretic view on ultraproducts. From Michael Barr’s Models of Sketches we read

Unlike limits and colimits, an ultraproduct is not defined by any universal mapping property. Of course, if the category has limits and (filtered) colimits, then it has ultraproducts constructed as colimits of products…But usually the category of models of a coherent theory (such as the theory of fields) lacks products and hence does not have categorical ultraproducts.

In fact the category theoretic definition is slightly different from the usual one, and comes with an advantage, according to Barr:

The only argument for banning the empty model that has any force comes from the observation that if is a collection of models and is a nonprincipal ultraproduct of the , then one wants and expects that will be empty if and only if the set of  for which is null belongs to the ultrafilter. If one takes the traditional definition of an ultraproduct as a quotient of the product, the ultraproduct will be empty as soon as one factor is.

Defined in terms of a colimit of products this problem goes away.

But how to cope with the problem Barr mentions with fields?

Continuing our discussion of free entities, from Fiore and Leinster’s A simple description of Thompson’s group F we read

…many entities of interest can be described as free categories with structure. For instance, the braided monoidal category freely generated by one object is the sequence of Artin braid groups; the monoidal category freely generated by a monoid consists of the finite ordinals; the symmetric monoidal category freely generated by a commutative monoid consists of the finite cardinals; the symmetric monoidal category freely generated by a commutative Frobenius algebra consists of 1-dimensional smooth oriented manifolds and diffeomorphism classes of 2-dimensional cobordisms.

You can find definitions of many terms used in higher category theory at the exciting new wiki nLab. For example, see braided monoidal category.

Fiore and Leinster continue,

In this vein, our result is that the monoidal category freely generated by an object and an isomorphism is equivalent to the groupoid , where is the trivial group and is coproduct of groupoids.

and for the related Thompson’s group V,

just replace ‘monoidal category’ by ‘symmetric monoidal category’, or equally ‘finite-product category’.

I posted on coalgebra over at my other blog. I won’t rehash the material of the discussion over here. But I’m still searching for an answer to the question I had of whether coalgebra has been formalised slowly due to algebraic blinkers.

A highlight for me was this comment from Dan Piponi, where he explains about how one needs to employ guarded recursion to work with ‘codata’. He says

In the real world of open ended loops like OSes and word processors, it’s often not computability we need, but productivity (absolutely no pun intended). And that makes coalgebraic reasoning an important topic.

Operating systems don’t require the totality of input in order to calculate a response by recursively breaking it into its atomic parts and deriving a function’s value by recomposition. They respond as each piece of new input arrives.

Coalgebra throws up a whole load of infinitely large entities.

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!

To celebrate the founding of MIMS, the mathematics department of the recently unified Manchester University, it was proposed that various workshops named ‘New Directions in…’ be run. They kindly agreed to allow Alexandre Borovik and me to organise one of these workshops on the Philosophy of Mathematics.

So, on Saturday 4 October, we began with Mary Leng, a philosopher at Liverpool, talking about whether the creation of mathematical theories, e.g., Hamilton’s quaternions, gives us any more reason to think mathematical entities exist than does the discovery of new consequences within existing theories. She concluded that it does not — both concern the drawing of consequences from suppositions, e.g., “Were there to be a 3 or 4-dimensional number system sharing specified properties with the complex numbers, then…”.

(more…)

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.”