A Dialogue on Infinity

Our very first post here spoke about the Fraïssé amalgam, a way of constructing a universal object out of a countable set of similar objects satisfying certain conditions. For example the amalgam of the set of finite graphs is the infinite random graph.

Alexandre was asking there why the results of such amalgamation should be the kinds of entity we encounter through different routes. I should imagine that the answer to this has much in common with answers to the questions Michiel Hazewinkel is posing in Niceness Theorems:

Many things in mathematics seem almost unreasonably nice. This includes objects, counterexamples, proofs. In this preprint I discuss many examples of this phenomenon with emphasis on the ring of polynomials in a countably infinite number of variables in its many incarnations such as the representing object of the Witt vectors, the direct sum of the rings of representations of the symmetric groups, the free lambda ring on one generator, the homology and cohomology of the classifying space BU, … . In addition attention is paid to the phenomenon that solutions to universal problems (adjoint functors) tend to pick up extra structure.

Evidently Hazewinkel sees category theory as the right tool for the problem. So might Fraïssé amalgamation be given a category theoretic gloss? Here are a few attempts.

Trevor Irwin, Fraisse limits and colimits with applications to continua:

The classical Fraïssé construction is a method of taking a direct limit of a family of finite models of a language provided the family fulfills certain amalgamation conditions. The limit is a countable model of the same language which can be characterized by its (injective) homogeneity and universality with respect to the initial family of models. A standard example is the family of finite linear orders for which the Fraïssé limit is the rational numbers with the usual ordering.

We present this classical construction via category theory, and within this context we introduce the dual construction. This respectively constitutes the Fraïssé colimits and limits indicated in the title. We provide several examples.

We then present the projective Fraïssé limit as a special case of the dual construction, and as such it is the categorical dual to the classical (injective) Fraïssé limit. In this dualization we use a notion of model theoretic structure which has a topological ingredient. This results in the countable limit structures being replaced by structures which are zero-dimensional, compact, second countable spaces with the property that the relations are closed and the functions are continuous.

We apply the theory of projective Fraïssé limits to the pseudo-arc by first representing the pseudo-arc as a natural quotient of a projective Fraïssé limit. Using this representation we derive topological properties of the pseudo-arc as consequences of the properties of projective Fraïssé limits. We thereby obtain a new proof of Mioduszewski’s result that the pseudo-arc is surjectively universal among chainable continua, and also a homogeneity theorem for the pseudo-arc which is a strengthening of a result due to Lewis and Smith. We also find a new characterization of the pseudo-arc via the homogeneity property.

We continue with further applications of these methods to a class of continua known as pseudo-solenoids, and achieve analogous results for the universal pseudo-solenoid.

Wieslaw Kubiś, Fraisse sequences – a category-theoretic approach to universal homogeneous structures:

We present a category-theoretic approach to universal homogeneous objects, with applications in the theory of Banach spaces and in set-theoretic topology.

Olivia Caramello, Fraïssé’s construction from a topos-theoretic perspective:

We present a topos-theoretic interpretation of (a categorical generalization of) Fraisse’s construction in model theory, with applications to countably categorical theories.

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

I note a sighting of what we know as the symbol for infinity. The example accompanies a hexagram or Star of David. These symbols are to be seen on a stone in the garden of the archeological museum in Tire, Izmir province, Turkey (a couple of dolmuş rides away from the Nesin Mathematics Village).

infinity sign and hexagram

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

In an ordered Abelian group, one positive element can be described as infinite with respect to another if the former exceeds every integral multiple of the other. If there are no such elements, the group can be called Archimedean.

Non-Archimedean ordered Abelian groups exist: for example, the group of ordered pairs (x,y) of integers, with the left-lexicographic ordering, so that (x,y)<(a,b) if and only if either x<a, or else x=a and y<b.

The main point of this article is to observe that an ordered
Abelian group is Archimedean if and only if it has a completion.
I do not know of a reference for this result, though I can well imagine that a reference exists.

The observation about completeness arises from considering that the field of real numbers is complete in two ways:

1. It is complete as an ordered field, because every nonempty
   subset with an upper bound has a least upper bound.

2. It is complete as a valued field, because every Cauchy
   sequence (with respect to the absolute value function) converges.

In sense (2), the field of real numbers is just one example of a
complete field. The complex numbers compose another such field,
as do the p-adic completions of the field of rational numbers.
Every valued field has a completion.

In sense (1) however, the field of real numbers is unique.
I have encountered at least one mathematician who seemed not to
be aware of this, or to have forgotten it, having apparently
confused completeness of valued fields with completeness of
ordered fields.

Possibly the distinction between ordered fields and valued fields
is like that between induction and completion: a distinction that
may be overlooked in one’s early education and then never
returned to.

At the end of his book Calculus, Michael Spivak constructs
the field of real numbers and proves its uniqueness (up to
isomorphism). It was from Spivak’s book that, as a student, I
first learned of the uniqueness of the real field. Spivak
praises the “one truly first-rate idea” in its construction:
Dedekind’s notion of a cut. Yet Spivak is disparaging of
the “drudgery” of going through the details of the construction.

I revisited the construction recently, in a course on
non-standard analysis at the Nesin Mathematics Village. If the
construction of the real numbers was going to be drudgery, I
wanted to see what more general results could be found in the
process.

The Dedekind cut construction gives a completion to every
ordered set (that is, totally ordered set). Indeed, let A be
such a set, and if x is in A, let (x) be the set of elements of A
that are less than or equal to x. Such sets compose a basis for
a topology on A. A cut of A can be defined as a nonempty closed
set in this topology, except A itself, unless this has a greatest
element. Let c(A) be the set of cuts of A. Then:

1. The set c(A) is ordered by inclusion
   and is complete with
   respect to this ordering.
2. The ordered set A embeds in c(A) under the map taking x to (x).

Again, an ordered Abelian group is Archimedean if, for any two
positive elements a and b, some multiple of a exceeds b. In
other words, a and b have a ratio in the sense of
Definition 4 of Book V of Euclid’s Elements. Indeed, the
positive part of an Archimedean ordered Abelian group would seem
to be just the set of magnitudes that have a ratio to some given
magnitude: the set is closed under addition, and under
subtraction of a lesser magnitude from a greater.

For any ordered Abelian group A, Archimedean or not, one can take
the completion of the underlying ordered set, and then extend the
definition of addition to the completion. One way to do this is
to define the sum of non-empty proper closed subsets X and Y of A
as the closure of the set of sums x+y, where x is in X and y is
in Y. Then c(A) becomes an Abelian monoid.

However, if A is a not Archimedean, then A cannot be complete,
since if the set of integral multiples of a positive element a is
bounded above by b, then b-a is also an upper bound. In c(A),
the set of multiples of a does have a supremum, c; but then c+a =
c.

Among Archimedean ordered Abelian groups, the group of integers
is the only discrete example, and this is complete. If A is a
dense Archimedean ordered Abelian group, then c(A) is the same.
If A and B are complete dense ordered Abelian groups, with
positive elements a and b respectively, then there is a unique
isomorphism from A to B taking a to b. The idea is that the
ordered field of rational numbers embeds in A under the map
taking 1 to a; then this map extends uniquely to the completion
of the rational field, which is the real field.

Consequently, for every real number a greater than 1, the
additive group of real numbers is isomorphic to the
multiplicative group of positive real numbers under a map taking
1 to a: this is the map commonly denoted by x |—> a^x. One
shows that multiplication distributes over addition on the
positive reals, then on all reals, and so the reals compose a
complete ordered field, which must be unique, because the
underlying group is unique as a dense complete ordered Abelian
group.

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.

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.

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.

So

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.

Evidence for the change from the Ancients is provided by Kvasz in his book Patterns of Change where he quotes Nicholas of Cusa on page 77:

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.

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.