Re: The Third Time is the Charm
For Alexander Grothendieck the number would be more like ten. At least.
Re: The Third Time is the Charm
Eilenberg and Mac Lane introduced the notion of natural transformation in their 1945 category theory paper with applications to algebraic topology in mind.
To define natural transformations they introduced functors.
To define functors they introduced categories.
2 out of 3 ain’t bad; Re: The Third Time is the Charm
Albert Einstein.
(1) Special Relativity;
(2) General Relativity;
(3) The Unified Field Theory of 1957 which, after Godel’s time travel solutions to the field equations, was used to extend his life and win his second Nobel Prize.
Whoops, (3) is that not on the branch of the multiverse where we reside…
classical mechanics; Re: The Third Time is the Charm
It was a community, not a single person, who established this sequence of three formulations of classical mechanics:
(1) Newtonian,
(2) Lagrangian, and
(3) Hamiltonian.
Re: The Third Time is the Charm
For a long time I’ve enjoyed this example:

categories,

functors,

natural transformations.
Saunders Mac Lane is said to have remarked, “I didn’t invent categories to study functors; I invented them to study natural transformations.” The reason is that mathematicians were comparing different cohomology theories, and needed a precise concept of when two such theories were ‘naturally isomorphic’.
One reason this example is noteworthy is that it means Eilenberg and Mac Lane were really inventing the 2category of categories. This illustrates the ‘upwards pull’ of categorification: to really understand $n$categories, you need the $(n+1)$category of all $n$categories, which means you need to understand $(n+1)$categories.
But yesterday, while teaching an undergraduate class, I noted another trio of concepts in which the ‘highest’ and least obvious of the three is really the one needed in the most glorious application of the theory.
What is it?
I like ‘special relativity, general relativity, unified field theory’ — but that’s not it, and unfortunately Einstein never really found the third item.
3 Math theories of Communications; Re: The Third Time is the Charm
Again, creation by a community:
Krippendorff, Klaus. “Information” Paper presented at the annual meeting of the International Communication Association, TBA, San Francisco, CA, . 20081211
Abstract:
Much contemporary communication research stands on the shoulders of three mathematical theories: the logical, statistical, and algorithmic theories of information. Logical theory concerns distinctions and selections among logical possibilities. As such, it addresses a semantic space of possible realities. Statistical theory concerns observed probabilities, and is familiar from Claude Shannon’s theory of communication. Algorithmic theory concerns the information that is required to compute a function or problem (i.e., some software requires more memory than others). Each of these theories center on the distinctions (possibilities, probabilities, and length of instructions) which, from the human perspective, give rise to meaning. The translation of distinguishable signals into informative messages depends on at least three aspects of their context: (i) temporality (e.g., a comprehensible sequence); (ii) rules (e.g., situational constraints); (iii) a history of relationships between senders and receivers; and (iv) the known intentions of senders and needs of receivers.
Re: The Third Time is the Charm
From mathematical finance:
1.) BlackScholes (Stock options from “point” prices)
2.) LIBOR Market Model (Bond options from “yield curve” prices)
3.) Not established yet, but I would like to :)
Kurt GĂ¶del again; Re: The Third Time is the Charm
Since I mentioned Kurt Gödel [28 April 1906  14 Jan 1978]:
(1) doctoral dissertation under Hahn’s supervision in 1929 submitting a thesis proving the completeness of the first order functional calculus;
(2) Gödel’s Incompleteness Theorems”. In 1931 he published these results in Über formal unentscheidbare Sätze der Principia Mathematica und verwandter Systeme. He proved fundamental results about axiomatic systems, showing in any axiomatic mathematical system there are propositions that cannot be proved or disproved within the axioms of the system. In particular the consistency of the axioms cannot be proved;
(3) His masterpiece Consistency of the axiom of choice and of the generalized continuumhypothesis with the axioms of set theory (1940) is a classic of modern mathematics. In this he proved that if an axiomatic system of set theory of the type proposed by Russell and Whitehead in Principia Mathematica is consistent, then it will remain so when the axiom of choice and the generalized continuumhypothesis are added to the system. This did not prove that these axioms were independent of the other axioms of set theory, but when this was finally established by Cohen in 1963 he built on these ideas of Gödel.
Reference for above is Kurt Gödel, by J J O’Connor and E F Robertson
Re: The Third Time is the Charm
1) Heavens, 2) Earth, 3) Light.
Re: The Third Time is the Charm
Given that John is teaching Differential Geometry:
(1) Vector bundle
(2) Vector bundle with connection
(3) The Riemannian connection on the tangent bundle
I know that I found Diff. Geom. got much easier when I started classifying results according to which of these three concepts they were really about. It doesn’t exactly roll off the tongue, though.
Re: The Third Time is the Charm
Hmm…
pointed homotopy of pointed paths, homotopy *class* of a pointed path, universal covering space?
Hamilton again; Re: The Third Time is the Charm
Okay, since I mentioned Hamilton:
(1) Reals;
(2) Complex;
(3) Quaternion
(4) bonus, by Cayley et al.: Octonions
“On a new Species of Imaginary Quantities connected with a theory of Quaternions”
By Sir William R. Hamilton
[Proceedings of the Royal Irish Academy, Nov. 13, 1843, vol. 2, 424434]
It is known to all students of algebra that an imaginary equation of the form i^2 = 1 has been employed so as to conduct to very varied and important results. Sir Wm. Hamilton proposes to consider some of the consequences which result from the following system of imaginary equations, or equations between a system of three different imaginary quantities:
i^2 = j^2 = k^2 = 1.
ij = k, jk = i, ki = j.
ji = k, kj = i, ik = j.
no linear relation between i, j, k being supposed to exist…
Sir W. Hamilton calls an expression of the form Q a quaternion; and the four real quantities w, x, y, z he calls the constituents thereof. Quaternions are added or subtracted by adding or subtracting their constituents….
Re: The Third Time is the Charm
Leray invented (1) spectral sequences to compute (2) sheaf cohomology and for that he had to invent (3) sheafs.
Re: The Third Time is the Charm
The first two don’t really go in order here, but maybe
 differentiation
 integration
 the fundamental theorem of calculus
Re: The Third Time is the Charm
A dark horse candidate, and almost certainly not what you had in mind… but it does roll of the tongue.
Claude Shannon
1) Information
2) Compression
3) Capacity
Re: The Third Time is the Charm
When someone finally guesses the trio I have in mind, everyone else will probably get angry. They’ll probably argue that the first — and maybe even the second — concept in this trio was not really invented by the person who put the third to such astoundingly effective use.
And in a sense they’ll be right. But surely, the whole formalism that makes this trio into a tightly connected sequence was first clearly understood by the person in question. And surely everyone here is familiar with what I’m referring to. So I don’t feel at all guilty for posing this puzzle.
Re: The Third Time is the Charm
Integers, sets, groups; Galois
Re: The Third Time is the Charm
Is it: 1) Tangent, 2) Normal, and 3) Frenet frame?
Re: The Third Time is the Charm
I feel sure this isn’t right, but I’ll throw it out there:

Vector field

Connection

Curvature
All three are important in both math and physics, with the third the most important, the crowning glory.
Re: The Third Time is the Charm
Position, velocity, acceleration; Newton?
Re: The Third Time is the Charm
I know these are not what you have in mind. I am not even sure of the accuracy as I am not sure what name to put to them. How about?
1) vector space
2) dual vector space
3) cohomology
or
1) Hilbert space
2) selfadjoint operator
3) eigenstate
Re: The Third Time is the Charm
You’re not thinking of something as simple as:
addition, multiplication, exponentiation
are you?
(Even if you’re not, this suggestion might get people to aim a bit lower on the sophistication scale!)
Re: The Third Time is the Charm
Several examples come to mind:
(1) Integers, rational, reals.
(2) Chart, atlas, manifold
(3) Subset, open set, continuous map.
Based on David’s detective work, I think JB had (2) in mind.
Re: The Third Time is the Charm
Todd wrote:
Position, velocity, acceleration; Newton?
YES!!!
Each one is clearly obtained from the previous one in a systematic way, and Newton invented this way — differentiation!
But, he needed differentiation not for first derivatives, but for second derivatives. His grand discovery was this: the laws of motion become clear not when we try to explain an object’s position, nor when we try to explain its velocity, but only when we try to explain its acceleration!
So, just as Eilenberg and Mac Lane needed categories to define functors, and needed functors to define natural transformations, so they could make precise sense of cohomology theories being ‘naturally isomorphic’…
… Newton needed position to define velocity, and velocity to define acceleration, so he could make precise sense of
$F = m a .$
There’s even a nice analogy between these two trios. An category is a place for an object to sit. A functor is a way of changing an object. A natural transformation changes a way of changing an object.
I haven’t quite figured out how to exploit this analogy. It’s interesting to note that an $n$category internal to $Vect$ is automatically a chain complex and is thus related to second derivatives by the formula
$d^2 = 0 .$
But somehow I should get natural transformations (chain homotopies) into the game.
Anyway, it might be a sign of genius for somebody to study a trio of concepts, each systematically derived from the previous one, where the third one is the charm.
Re: The Third Time is the Charm
Zeh, in writing about time, mentions
initial conditions, which may be
analogous to position:
“It has indeed proven appropriate to
divide the formal dynamical description
of Nature into laws and initial
conditions. Wigner (1972), in his Nobel
Prize lecture, called this conceptual
distinction Newton’s greatest discovery,
since it demonstrates that the laws by
themselves are far from determining
Nature.”
Roger Penrose?
I’ll take the risk of suggesting something I not only don’t understand in any depth but also something that may be too specialised to count as concepts rather than specific gadgets, but how about Roger Penrose with:
1. Spinors networks.
2. Twistors.
3. Twistor sheaf cohomology.
Re: The Third Time is the Charm
Not to be deterred…
How about: charge, current, magnetic flux: Maxwell?
And is there a functorial relationship between those and position, velocity, acceleration?
Of course, anything along these lines was already properly formulated and detailed long ago, but I’m just getting started, so I don’t know anything yet.
Re: The Third Time is the Charm
How about:
Fourier series
Convergence
Sets
Re: The Third Time is the Charm
My first thoughts were:
1. Paths and loops
2. Homotopies
3. Fundamental groups
(Poincare?)
All about B and Omega
Just posted and quite relevant
arXiv:0902.0177
Date: Mon, 2 Feb 2009 00:45:24 GMT (49kb)
Title: Operadic bar constructions, cylinder objects, and homotopy morphisms of
algebras over operads
Authors: Benoit Fresse
Categories: math.AT
Comments: 65 pages
MSCclass: 18D50; 18G55
\
The purpose of this paper is twofold. First, we review applications of the
bar duality of operads to the construction of explicit cofibrant replacements
in categories of algebras over an operad. In view toward applications, we check
that the constructions of the bar duality work properly for algebras over
operads in unbounded differential graded modules over a ring.
In a second part, we use the operadic cobar construction to define explicit
cyclinder objects in the category of operads. Then we apply this construction
to prove that certain homotopy morphisms of algebras over operads are
equivalent to left homotopies in the model category of operads.
\ ( http://arxiv.org/abs/0902.0177 , 49kb)
Re: The Third Time is the Charm
How about examples of questions where the third step is where things go wrong?
For example, in ordinary differential equations? In one dimension everything is “trivial”: everything boils down to monotone regions; in two dimensions, again the asymptotics are generically straightforward (because separatrices actually separate different dynamics). But in three dimensions, we typically get chaos!
Or how about simplicial cycles? In an ncycle, the interior of an ncell is a good manifold, and so is the interior of an (n1)cell and even (!) the interior of an (n2)cell; but in codimension 3, we can get singular loci. Or perhaps you might consider that something going wrong at the *fourth* step, I don’t know…
Has anyone any lessgeometric examples?
Explicit Construction of a Transport 2Functor via Clifford Algebra
Thanks John and thanks Urs.
My questions about $VectMod$ come from my scribblings here:
==================
Formulation
First, due to the symmetry of the diagram above, we can consider the simplified region
$\array{
{} & {} & {} & x_{\tau} & {} & {} & {} \\
{} & {} & {} & \bullet & {} & {} & {} \\
{} & {} & {}^{v_0}\nearrow & {}_F\Rightarrow & \searrow^{v_{\tau}} & {} & {} \\
x_0 & \bullet & {} & \stackrel{v}{\longrightarrow} & {} & \bullet & x_{2\tau}
}$
This diagram represents a very simple strict 2category we will denote by $P^2(\mathbb{D}^2)$ with
 Objects $\{x_0,x_\tau,x_{2\tau}\}$
 Morphisms $\{x_0\stackrel{v_0}{\to}x_{\tau},x_\tau\stackrel{v_\tau}{\to}x_{2\tau},x_0\stackrel{v}{\to}x_{2\tau}\}$
 2Morphism $\{v_0\stackrel{F}{\Rightarrow}v_\tau\}$
We will explicitly construct a 2transport functor
$tra: P^2(\mathbb{D}^2)\to VecMod,$
where
$x\mapsto tra(x)\in\mathcal{M}^2$
===============
I was initially going to write $\mathcal{M}^4$, but thought the simple example could be described by $\mathcal{M}^2$, but then thought you might describe $\mathcal{M}^2$ as $CL_{1,1}$, but then recognized that as $\mathbb{C}$.
As you can tell, I am floundering about, but not giving up. Thanks again for your help.
Re: The Third Time is the Charm
For Alexander Grothendieck the number would be more like ten. At least.