A Dialogue on Infinity

The John Templeton Foundation serves as a philanthropic catalyst for discoveries relating to the Big Questions of human purpose and ultimate reality. We support research on subjects ranging from complexity, evolution, and infinity to creativity, forgiveness, love, and free will. We encourage civil, informed dialogue among scientists, philosophers, and theologians and between such experts and the public at large, for the purposes of definitional clarity and new insights.

Our vision is derived from the late Sir John Templeton’s optimism about the possibility of acquiring “new spiritual information” and from his commitment to rigorous scientific research and related scholarship. The Foundation’s motto, “How little we know, how eager to learn,” exemplifies our support for open-minded inquiry and our hope for advancing human progress through breakthrough discoveries.

Just a link to Terry Tao.

In his Systematic Theology, Vol. III, Wolfhart Pannenberg argues that God as eternal comprehends the different moments of time simultaneously and orders them to constitute a whole or totality. The author contends that this approach to time and eternity might solve the logical tension between the classical notion of divine sovereignty and the common sense belief in creaturely spontaneity/human freedom. For, if the existence of the events constituting a temporal sequence is primarily due to the spontaneous decisions of creatures, and if their being ordered into a totality or meaningful whole is primarily due to the superordinate activity of God, then both God and creatures play indispensable but nevertheless distinct roles in the cosmic process.

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.