L’infini vu par Noémie July 17, 2009Posted by Alexandre Borovik in Uncategorized.
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!!
(18 Mai 2000)
From Mathieu Marion July 17, 2009Posted by Alexandre Borovik in Uncategorized.
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.
Two Streams in Hatfield July 6, 2009Posted by dcorfield in Uncategorized.
Brendan Larvor and I ran a conference – Two Streams in the Philosophy of Mathematics from 1-3 July. I thought I’d put up a post here for post-conference discussion.
In accordance with one stream’s policy of encouraging dialogue with real mathematicians, we invited Yehuda Rav (Paris-Sud), Michael Harris (Jussieu) and this blog’s very own Alexandre Borovik. For me two of the most interesting issues to emerge during the conference was Borovik’s ‘phantoms’ and Harris’s ‘avatars’. The first of these may occur when there is a question as to whether a certain entity exists. Even if it does not, it may transpire that some counterpart of this nonexistent entity exists elsewhere. The setting of finite simple groups is a rich environment for this phenomenon.
In the case of avatars, on the other hand, they all exist, but they indicate the existence of a not yet expressible universal object. Grothendieck’s theory of motives is the classic example, and indeed it was here that he coined the term ‘avatar’ to describe an instantiation of a motive in a particular cohomological setting.
What I’d like to know is what can be said about these phenomena. What is the right language to formulate them? Do we have earlier cases of avatars or phantoms which we now know how to express? Might it be possible to understand both phenomena in the same framwork? I.e., perhaps there may be avatars which happen not to exist, but for which existing fellow avatars act as phantoms.
So that’s a small taste of two of the talks. There were fourteen others. Personally, I was very pleased to hear Ivor Grattan-Guinness speak about ‘notions’, such as symmetry, convexity, and linearity, continually reappearing in mathematics. My own talk focused on duality, but I gave it a Cassirerian gloss as a ‘principle’.
If anyone would like to share their thoughts on the conference, please feel free.
The Invisible Dialog Between Mathematics and Theology April 29, 2009Posted by dcorfield in Uncategorized.
An interesting paper by Ladislav Kvasz — The Invisible Dialog Between Mathematics and Theology, in Perspectives on Science and Christian Faith, Vol. 56, pp. 111-116.
The thesis of the paper is that monotheistic theology with its idea of the omniscient and omnipotent God, who created the world, influenced in an indirect way the process of this mathematicization. In separating ontology from epistemology, monotheistic theology opened the possibility to explain all the ambiguity connected to these phenomena as a result of human finitude and so to understand the phenomena themselves as unambiguous, and therefore accessible to mathematical description.
This thesis is explored through five notions: infinity, chance, the unknown, space and motion.
What we refer to today as infinite was in Antiquity subsumed under the notion of apeiron (). Nevertheless, compared with our modern notion of infinity, the notion of apeiron had a much broader meaning. It applied not only to that which was infinite, but also to everything that had no boundary (i.e. no peras), that was indefinite, vague or blurred. According to ancient scholars apeiron was something lacking boundaries, lacking determination, and therefore uncertain. Mathematical study of apeiron was impossible, mathematics being the science of the determined, definite and certain knowledge. That which had no peras, could not be studied using the clear and sharp notions of mathematics.
Modern mathematics, in contrast to Antiquity, makes a distinction between infinite and indefinite. We consider the infinite, despite the fact that it has no end (finis), to be determined and unequivocal, and thus accessible to mathematical investigation. Be it an infinitely extended geometrical figure, an infinitely small quantity or an infinite set, we consider them as belonging to mathematics. The ancient notion of apeiron was thus divided into two notions: the notion of the infinite in a narrow sense, which became a part of mathematics, and the notion of the indefinite, which, as previously, has no place in mathematics.
While for the Ancients apeiron was a negative notion, associated with going astray and losing the way, for the medieval scholar the road to infinity became the road to God. God is an infinite being, but despite His infiniteness, He is absolutely perfect. As soon as the notion of infinity was applied to God, it lost its obscurity and ambiguity. Theology made the notion of infinity positive, luminous and unequivocal. All ambiguity and obscurity encountered in the notion of infinity was interpreted as the consequence of human finitude and imperfection alone. Infinity itself was interpreted as an absolutely clear and sharp notion, and therefore an ideal subject of mathematical investigation.
It is already evident that there can be only one maximum and infinite thing. Moreover, since any two sides of any triangle cannot, if conjoined, be shorter than the third: it is evident that in the case of a triangle whose one side is infinite, the other two sides are not shorter. And because each part of what is infinite is infinite: for any triangle whose one side is infinite, the other sides must also be infinite. And since there cannot be more than one infinite thing, you understand transcendently that an infinite triangle cannot be composed of a plurality of lines, even though it is the greatest and truest triangle, incomposite and most simple… (Nicholas of Cusa 1440, p. 22) De Docta Ignorantia, trans. J. Hopkins.
It may seem odd to us that Nicholas could not imagine the limit as an isosceles triangle of fixed base is extended, but the point is that such a discussion of an infinitely large object would have been unthinkable for the Greeks.
Infinity symbol April 8, 2009Posted by David Pierce in Uncategorized.
Why is infinity denoted by a lemniscate?
In a recent talk in Ankara, Sasha Borovik used a photograph like those in his post Manifestation of Infinity. How does infinity appear in a picture of a ferry approaching a dock? One person in the audience suggested that a pair of tires on the side of the ferry formed the infinity symbol.
I speculate that the lemniscate is the simplest shape suggesting endlessness that will not be confused with the symbol for emptiness: the zero. That the zero is naturally a symbol for emptiness is suggested in the eighth Oxherding picture, Bull and Self Transcended:
Whip, rope, person, and bull — all merge in No-Thing.
This heaven is so vast no message can stain it.
How may a snowflake exist in a raging fire?
Here are the footprints of the patriarchs.
Comment: Mediocrity is gone. Mind is clear of limitation. I seek no state of enlightenment. Neither do I remain where no enlightenment exists. Since I linger in neither condition, eyes cannot see me. If hundreds of birds strew my path with flowers, such praise would be meaningless.
(English text by Reps and Senzaki.)
How significant the lemniscate may be in the East, I do not know. I was able to find, on one yoga website, a suggestion to visualize a figure-eight while practicing Spinal Breath:
There are a variety of practices with awareness moving up and down the spine with the breath. One may do this practice between particular energy centers (chakras) or form different shapes of the visualized flow, including elliptical or a figure-eight…
The most straight forward, and yet completely effective method is to:
- Imagine the breath flowing from the top of the head, down to the base of the spine on exhalation, and to
- Imagine the flow coming from the base of the spine to the top of the head on inhalation.
- This may be done lying down, or in a seated meditation posture.
One may simply experience the breath, or may be aware of a thin, milky white stream flowing in a straight line, up and down. This practice is very subtle when experienced at its depth, and can turn into a profoundly deep part of meditation practice.
Achilles, Tortoise and Yessenin-Volpin February 16, 2009Posted 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.
Ultraproducts the Category Theoretic Way January 21, 2009Posted by dcorfield in Uncategorized.
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?
Born Free January 12, 2009Posted by dcorfield in Uncategorized.
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…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.
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’.
Coalgebra December 8, 2008Posted by dcorfield in Uncategorized.
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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.