Merlin - Aug 19, 2013:

Other image processing items of potential interest:
http://www.phrogz.net/SVG/animation_on_a_curve.html
http://cs.brown.edu/~pff/segment/

That first demo is incredible in that it can be written with so little code, but then I suppose the guts of the program are in the referenced functions whose code isn’t shown. I “broke” their demo, though, when I moved both ends of the curve to outside the square, whereupon the handles disappeared so I couldn’t move the ends back. But at least the Back button on my browser still worked.

Thanks for the C code link for image segmentation. Given the direction I’m headed, I have little doubt that someday I’ll be digging into all the nuts and bolts of edge extraction, region detection, and all those other tedious issues of low-level image processing.

PART 6: HOW TO CREATE A TOP-LEVEL BRANCH OF MATH

In retrospect, looking at entire top-level *branches* of math for AI applicability makes a lot of sense since the individual math systems we’ve been using have always been frustratingly limited in some way when applied to AI. It’s also a very deep topic: I’ve *never* seen any tutorial for getting one up to speed on all the branches of mathematics at once, especially about how they are related, or especially how to create your own branch of math. Maybe authors figured that nobody was going to be that ambitious. But they figured wrong (http://www.divshare.com/download/24421267-7d4 ), so here’s my tutorial on how to create a new top-level branch of math, based on my own empirical observations. (By “top-level branch” I mean a major branch akin to group theory or topology, in contrast to a minor branch like shape theory, which is a subbranch of algebraic topology, which in turn is a subbranch of topology in general.)

My proposed procedure:

(1) Think of an *abstract* concept that relates disparate objects.
Examples of sufficiently abstract concepts: quantity, set, cardinality, connectivity, group, shape, complexity, effective procedure, (maybe) graph, (maybe) database
Examples of insufficiently abstract concepts: (probably) dimension, color, pitch of sound, waveform, atom, emotion
(1b) Come up with a name for your math/theory/system.

(2a) Determine the individual objects and the collections of those objects.
(2b) Decide if the objects involved tend to be solitary versus tend to need to be grouped into larger collections.
Examples of mostly solitary objects: surfaces, images
Examples of mostly collective objects: elements of a group, elements of a set, formulas
(2c) Come up with names for both the individual objects and the collections.
Examples of individual object names: scalar, element, state, surface, manifold, automaton, vector, algebraic structure, attribute
Examples of collection names: vector, array, list, set, bag, group, vector space, field, object, linked list

(3) Decide on one or more goals that you’d like to accomplish with your entities and/or the collections of those entities.
Examples of useful goals: determine factors of an integer, determine if two groups are equal, classify surfaces, identify any symmetry present, determine if two images are identical, measure the nearness of two points, produce coordinates that are not affected by chirality, production of canonical forms, count elements, order members, determine order of complexity, compute perimeter length, compute volume, compute dimension, enumeration of all permutations, truth value determination, determine the strongest contributions, determine reliability, determine probability, find the minimum or maximum, solve a puzzle, win a game, determine optimal strategy

(4a) Determine any transformations or operations that can be performed on either the individual entities, the collections, or both.
Examples: sum of scalars, product of scalars, eigenvectors of a matrix, face turns on Rubik’s Cube, tying of knots in a looped string, conversion from time domain to frequency domain, edge extraction, decomposition into components, combining groups, combining knots, concatentation, implication, temporally subsequent
(4b) Come up with names for such transformations or operations.
Examples of names: addition, multiplication, exponentiation, dot product, outer product, Fourier transform, Haar transform, affine transformation, convolution, rotation, translation, scaling, direct product, semidirect product, knit product, join, sort, concatenate, duplicate, paste, group together

(5) Endlessly refine all the above steps except for (1).
Examples of commonly useful refinements/discoveries:
Produce definitions for new concepts, such as: coset, subgroup, dihedral, regular, singular, nonsingular, symmetric, even, odd, normal, continuous, differentiable, perfect, prime
Determine if your system falls into a more general category, such as: set, vector space, algebra
Determine if transformations produce more than one entity, such as: multiple functions, multiple roots, mirror image negative frequency diagram, principal components
Determine if transformations or operations produce new *types* of entities, such as: a frequency diagram and a phase, average and distance in a Haar transform
Determine an identity element for each operation, such as: 0, 1, no-op, empty set, no action
Determine if any natural organization occurs, such as: natural coordinate systems (e.g., cartesian, polar), natural constants (e.g., e, pi, gamma)
Determine any alternative coordinate systems, such as: RGB versus HSV in color coordinates
If more than one operation exists, determine the rules for combining them, especially with respect to commutativity, associativity, transitivity.
Produce formulas that describe relationships between the different operations operating on different types of entities.
Develop and prove theorems that generalize observations.
Take statistics (count, min, max, average, etc.), especially if the domain is particularly complicated or unpredictable.
Find ways to represent all related information, especially visual methods, or methods suitable for digital computers.
Identify some very difficult problems in the field, such as: Poincare conjecture, Three Body Problem, Riemann Hypothesis, P=NP, irrationality of gamma
Determine metrics, such as: size, length, volume, distance, similarity, dimension, complexity
Determine comparison operators, such as: >, <, =, near to, congruent to, implied by

It is interesting to see what happens if you pick a concept that is *not* abstract enough: what seems to happen is that you still tend to get a theory that is very useful, but the theory doesn’t relate very disparate phenomena in the deep, surprising ways that are typical of a top-level branch of math. A few examples are listed below of insufficiently abstract but very useful attributes/concepts, each leading to an entire named theory…

(1)
INSUFFICIENTLY ABSTRACT CONCEPT FOR A TOP-LEVEL MATH BRANCH: color
RESULTING THEORY NAME: “color theory” (http://en.wikipedia.org/wiki/Color_theory)
INDIVIDUAL OBJECTS: hue, saturation, value
COLLECTIVE OBJECT: color
SOME PRIMARY GOALS: mathematically describe any given color, determine which colors are not seen from prism output, determine how colors mix additively and subtractively, determine which colors look nice together and why
TRANSFORMATIONS/OPERATIONS: (additively and subtractively) superimposing colors, (additively and subtractively) determining component colors
EXAMPLES OF REFINEMENTS: use of HSV coordinates versus RGB coordinates and formulas to convert between them, definition of complementary colors, definition of warm versus cool colors, color sphere

(2)
INSUFFICIENTLY ABSTRACT CONCEPT FOR A TOP-LEVEL MATH BRANCH: pitch of sound
RESULTING THEORY NAME: “music theory” (http://en.wikipedia.org/wiki/Music_theory)
INDIVIDUAL OBJECTS: notes (primarily composed of pitch and duration)
COLLECTIVE OBJECT: musical composition
SOME PRIMARY GOALS: how to combine notes and chords in desirable ways, understand why a given composition is appealing, determine which notes sound similar and why, determine which notes sound good at the same time and why, determine where to place frets on a stringed instrument, where to find harmonics on a stringed instrument
TRANSFORMATIONS/OPERATIONS: superimposed notes, temporally ordered notes
EXAMPLES OF REFINEMENTS: timbre, overtones, tempered scales, definition of key, definition of chord, definition of scale, definition of mode, interplay of chords and scales, altered chord tones, categories of song structure, chord functions

In contrast, here are some top-level math branches that *were* based on sufficiently abstract concepts, and became major, important fields of study.

(1)
SUFFICIENTLY ABSTRACT CONCEPT FOR A TOP-LEVEL MATH BRANCH: set
RESULTING THEORY NAME: “set theory” (http://en.wikipedia.org/wiki/Set_theory)
INDIVIDUAL OBJECTS: elements
COLLECTIVE OBJECT: set
SOME PRIMARY GOALS: to group or classify objects, to represent visually represent logic, to count members of groups indirectly
TRANSFORMATIONS/OPERATIONS: union, intersection, difference
EXAMPLES OF REFINEMENTS: Venn diagram, open set, closed set, Zermelo–Fraenkel axioms, axiom of choice, fuzzy set

(2)
SUFFICIENTLY ABSTRACT CONCEPT FOR A TOP-LEVEL MATH BRANCH: group
RESULTING THEORY NAME: “group theory” (http://en.wikipedia.org/wiki/Group_theory)
INDIVIDUAL OBJECTS: elements
COLLECTIVE OBJECT: group
SOME PRIMARY GOALS: to model systems in nature, to recognize if a new group matches a known group, to understand complex system behavior, to solve certain puzzles
TRANSFORMATIONS/OPERATIONS: direct product, semidirect product, knit product
EXAMPLES OF REFINEMENTS: cyclic groups, permutation groups, Lie groups, Cayley diagrams, Cayley’s Theorem, Sylow Theorems

(3)
SUFFICIENTLY ABSTRACT CONCEPT FOR A TOP-LEVEL MATH BRANCH: connectedness
RESULTING THEORY NAME: “topology” (http://en.wikipedia.org/wiki/Topology)
INDIVIDUAL OBJECTS: surfaces
COLLECTIVE OBJECT: (none?)
SOME PRIMARY GOALS: to categorize surfaces or knots, to generalize the concept of convergence
TRANSFORMATIONS/OPERATIONS: tying a knot, adding a handle
EXAMPLES OF REFINEMENTS: orientability, Mobius band, Klein bottle, homology, Euler characteristic, hairy ball theorem, Poincare conjecture

If you choose a sufficiently abstract, foundational concept, and develop a branch of math from it, you will eventually find it useful for real-world applications, especially in unexpected ways in unexpected domains. Here is one such example:

(p. 198)
  The discovery was made quite by chance. Jones
was working on a problem in analysis that had ap-
plications in physics. The problem concerned math-
ematical structures known as von Neumann alge-
bras. In looking at the way these von Neumann
algebras were built up from simpler structures, he
discovered patterns to do with knots, discovered
by Emil Artin in the 1920s. Sensing that he had
stumbled onto an unexpected, hidden connection,
Jones consulted knot theorist Joan Birman, and the
rest, as they say, is history. Like the Alexander poly-
nomial, the Jones polynomial can be obtained from
the knot diagram. But, far from being a simple vari-
ant of the Alexander polynomial, as Jones himself
had at first thought, his polynomial was something
quite new.
  In particular, the Jones polynomial can distin-
guish between a reef knot and a granny. The differ-
ence between these two knots depends on the ori-
entation of the two trefoils relative to each other.

(“Mathematics: The Science of Patterns: The Search for Order in Life, Mind, and the Universe”, Keith Devlin, 1994)

Ordinarily I’d encourage everyone to try to create a new top-level branch of math for practice but there are probably very few useful, fundamental, abstract concepts that are suitable for the foundation of a top-level branch of math, and most of those have probably been used. I’m really surprised no one seems to have considered shape so far, at least not directly. Regardless, the procedure I gave above can be used to great effect even in subbranches of math, or even in more physical (less abstract) branches of science.

One last note: Branches of math can be combined, and often are. For example, Galois theory is the combination of group theory and field theory, group theory and topology often mix well, and a mix of topology and probability has been proposed within shape theory.

Thank you Mark for this excellent topic.  Your right, this is a deep topic, so I’ll keep this short and simple, but I am interested in applying Math to A.I.  So,  I have a preliminary question about section (1) of your proposed procedure.  Following your tutorial, using a top level branch of math, as a model, makes A.I. an abstact concept, rather than an insufficiently abstract concept?

Please, may we next tackle “The Meaning of Life?”

42! Next?

Mark, that ‘how to create a top-level branch of math’ post was utterly fascinating!

Thinking about how all the ‘101’ math courses at uni tended to start at the first pages, I would say the big thing you should stress more is, once you have settled on your objects and their relations, to FORMALIZE it.

I.e. give formal definitions, taking advantage of ‘fundamental’ level math (almost always this will be first-order logic and set theory). This formalizing allows for your theory to develop a richness in a mathematically viable way. Any text books besides first-order logic and set theory will start this way

(because, think about it, as it stands your system could be applied to creating theories of any sort, not just mathematical ones. The math is in the formalizing, enriching your theory with a whole set of existing symbol-manipulation math to build on.

Note that formalizing is NOT just saying ‘put it in symbols’ - in essence, mathematical symbols are just shorthands for stuff that has a very specific meaning and rules surrounding this.

I am becoming aware as I am writing this that my equating ‘formalizing’ with relating to ‘fundamental’ math that I am a bit running into a circular problem, but still. Something in that direction.

Also, almost every math 101 book (and the ‘101’ does not mean ‘simple math’, we only started first looking into most of Mark’s examples here in grad level courses) will build as fast as possible to a first ‘fundamental theorem of [math branch]’, that’s kind of the ‘justification’ of the theory’s existence. The Big Names associated with math branches are often the ones that formulated and PROVED that theorem.

For analysis, it’s that integration and differentiation are each other’s inverse (‘fundamental theorem of calculus’ -> Newton)

For algebra, it’s that each polynomial equation has a number solution in the imaginary numbers equal to its highest power (‘fundamental theorem of algebra’ -> Gauss proved this one every other year while brushing his teeth, just because he was Gauss an he could)

For group theory, it’s that, if you build the group of extensions of a field, the subgroups (easy to find) of that group are homomorphic (is this proper English?) to the ‘intermediate groups’ (hard to find).

To get a bit closer to AI-related math stuffs, Chomsky’s formal grammars, I would say this one fits the bill, which I ironically can’t easily state in a simple phrase: http://tinyurl.com/np8o5kf

I am sure I am being WILDLY inaccurate in how I am stating these, but I do very distinctly remember all these as the ‘fundamental’ theorems of their theories.

Keep on going in your math journey!

One extra thought: Mark, if you want some inspiration what to create a branch of math for, how about your give ‘dialogue and conversation flow’ a shot - if you manage to inject some formality in that one, I’ll buy you a beer

(independent of the LANGUAGE aspect, mind you)

Elizabot - Sep 3, 2013:

So,  I have a preliminary question about section (1) of your proposed procedure.  Following your tutorial, using a top level branch of math, as a model, makes A.I. an abstact concept, rather than an insufficiently abstract concept?

Sorry, I don’t entirely understand your wording…

Are you asking what distinguishes sufficiently abstract from insufficiently abstract? I’ve been thinking about that. It seems that two different things can cause concepts to lose abstraction: (1) any *physically* manifested phenomenon (color, sound, taste, emotion, etc.) outright lacks sufficient abstraction immediately, (2) more detailed models. That latter explains why concepts like graphs (=> graph theory), databases (=> database theory), artificial neural networks, petri nets, and other abstract constructs made of multiple components connected in very specific ways don’t carry the same kind of universal applicability that top-level math does. Now I believe automata theory falls into that same category, since it is based on fairly specific types of devices made up of fairly specific types of components connected in fairly specific ways. The more specific the device, the less abstract the model. So, in general, those are two general ways I’ve found so far to go wrong in attempting to be sufficiently abstract.

Or are you asking if AI/intelligence is a sufficiently abstract concept for a math foundation? Other than my own acceptance of my own defintion of “intelligence,” there isn’t enough consensus on the definition of either concept to even begin to build a branch of math on it. The same is true of other vague AI-related words like “thought” or “consciousness”. My own definition doesn’t attempt to cross that hurdle: my own definition merely attempts to be more specific of what attributes are involved in the concept as people intend it to mean. Even if somebody tried to use my definition to create a branch of math, it is combining so many concepts (five, to be exact) at once that it is a little too specific to be a top-level branch of math. It would probably work as a branch of automata theory, though.

P.S.—I understand your wording now. As I mentioned in my previous paragraph, I would regard either of the concepts “AI” or “intelligence” as insufficiently abstract for a top-level branch of math.

Sorry everyone that I haven’t been on the Internet for about a week, and I’ve been extra busy lately, so I haven’t been able to finish my thread yet, though I still plan to post more later this week. So far it looks like it will be 11 posts long, with possibly 2-3 addenda in the future, if the mood strikes.

Wouter Smet - Sep 3, 2013:

if you want some inspiration what to create a branch of math for, how about your give ‘dialogue and conversation flow’ a shot - if you manage to inject some formality in that one, I’ll buy you a beer

Fortunately I can just resort to the lazy mathematician’s “out” to avoid having to think about the question: those are insufficiently abstract concepts!

(In a similar vein, ever hear one of those jokes about lazy mathematicians reducing a difficult problem to a previously solved problem by making the situation worse…?)

A mathematician and a physicist were asked the following question:

      Suppose you walked by a burning house and saw a hydrant and
      a hose not connected to the hydrant.  What would you do?

P: I would attach the hose to the hydrant, turn on the water, and put out
  the fire.

M: I would attach the hose to the hydrant, turn on the water, and put out
  the fire.

Then they were asked this question:

      Suppose you walked by a house and saw a hose connected to
      a hydrant.  What would you do?

P: I would keep walking, as there is no problem to solve.

M: I would disconnect the hose from the hydrant and set the house on fire,
  reducing the problem to a previously solved form.

http://www.boo.net/~jasonp/math

I’m going to have to think about your longer post about formality. My immediate reaction is that much of math, if not all, is ultimately based on undefined terms (such as the concept of a “point” in geometry), and that possibly what you are really saying is that each branch of math should have a fundamental theorem of some sort. Good observation. You might be very correct about that, and I may very well have left out a major piece of my suggested procedure. On the other hand, topology started with the Bridges of Konigsberg problem (http://en.wikipedia.org/wiki/Seven_Bridges_of_Königsberg), and group theory started with a theorem about solving the quintic equation where groups weren’t even called “groups” or considered a new branch of math, and in neither case was there a fundamental theorem in either branch at the time, so I question whether formality is required in the initial stages of a new branch of math.

(p. 36)
By now the reader is certainly convinced that group theory shows up in diverse situations.
But it would be a great disservice to the history of mathematics if I did not mention one
more application, the very reason that group theory was invented! In the nineteenth century,
two young mathematical prodigies, Neils Abel and Evariste Galois, solved a mathematical
problem that had stood unsolved for centuries. It has come to be called “the unsolvability
of the quintic.” It is one of the great discoveries in mathematics, and when you come to
Chapter 10 of this book, you will be ready to read about it in some detail. It rests on the
fact that the solutions to polynomial equations have a certain relationship to one another.
They form a group.
  Galois did not call the patterns he noticed groups; later mathematicians who extended
his work gave them that name. But he was the first to notice them and study them.

(“Visual Group Theory”, Nathan C. Carter, 2009)

(p. 93)
This semiformal version
still uses a lot of words in English with their usual meanings (words such as
“the”, “if”, “and”, “join”, “have”), although the everyday meaning has been
drained out of special words lke “POINT” and “LINE”, which are con-
sequently called undefined terms. Undefined terms, like the p and q of the
pq-system, do get defined in a sense: implicitly—by the totality of all proposi-
tions in which they occur, rather than explicitly, in a definition.

(“Gödel, Escher, Bach: an Eternal Golden Braid”, Douglas R. Hofstadter, 1979)

Anyway, this post gives me a chance to throw out another quote about the surprising and accidental discoveries that characterize math, in a book I came across not long ago, regarding the three constants of e, pi, and gamma:

(p. xix)
  The mid 1970s brought with it the hand-held, microchip-centred, battery-
powered, comparatively cheap calculator, thereby bringing to an end the role
(p. xx)
of logarithms and the slide rule as calculative aids. Yet the appearance of them
in a piece of mathematics is seldom a cause for surprise. Anyone who has
studied calculus would see them materialize time and again, quite probably in
the expression for the integral of some function or in their role as the inverse of
the exponential function, with e vying with pi for constant supremacy. They can
also arise without warning in situations that seem remote from their influence,
and when they do so they exercise a surprising control in unexpected places—as
we shall see: we will also see that the harmonic series, and others related to it,
enjoy an important existence of their own.

(“Gamma: Exploring Euler’s Constant”, Julian Havil, 2003)

Yeah, that’s true, it’s amazing in what weird places e and Pi (and i!) pop up, including the fact that they are bizarrely connected.

http://xkcd.com/179/

Fortunately I can just resort to the lazy mathematician’s “out” to avoid having to think about the question: those are insufficiently abstract concepts!

but HA! Luckily we are AI peeps first and foremost here, and mathematicians only second (well at least that’s how my current interests lie, and surely yours as well). Mathematics is, all in all, just a helpful tool in our quests to figure out how the mind works, and imitate it. I’m sure people said similar things to your resort here about such non-abstract topics as grammar, evolutionary biology and color theory - all subjects which have gained a heavy and crucial math component over time.

Loved the math/physics joke

Thanks again Mark.

Mark Atkins - Sep 5, 2013:

Are you asking what distinguishes sufficiently abstract from insufficiently abstract?

No.  That was my next question, which as a problem, you may have reduced to a previously solved form, before I even asked it.  So, then to have an emotional or cognitive impact, A.I. must lack sufficient abstraction.  This math foundation concept may coincide with the reasons the $100,000 Loebner prize Turing test requires A.I. to understand text, visual, and auditory.

Mark Atkins - Sep 5, 2013:

Or are you asking if AI/intelligence is a sufficiently abstract concept for a math foundation?

Yes, to begin my next question:  From the present state of A.I. which math foundation concept, “sufficiently abstract” or “insufficiently abstract”, should A.I. use to evolve, or change state to being more human?

Wouter Smet - Sep 5, 2013:

Luckily we are AI peeps first and foremost here, and mathematicians only second (well at least that’s how my current interests lie, and surely yours as well). Mathematics is, all in all, just a helpful tool in our quests to figure out how the mind works, and imitate it.

True and just adding, Wouter, at least in some ways, we may also use Mathematics to figure out how the mind works, and emulate it.

PART 7: SHAPE THEORY

In answer to my own two earlier questions about shape theory:

(1) Why was shape theory included under topology instead of being a standalone top-level branch of math in its own right?

Answer:

Shape theory is a branch of topology, generalizing the idea of homotopy theory to cases with unfavorable local properties. The overall goal of shape theory is adapt the methods and results from homotopy theory to more general spaces, such as compact metric spaces or compact Hausdorff spaces.

http://en.wikipedia.org/wiki/Shape_theory_(mathematics)

Homotopy is defined as “two continuous functions from one topological space to another” (http://en.wikipedia.org/wiki/Homotopy_theory), which is exactly what algebraic topology is all about: continuous deformation of one surface into another. Therefore by both its goals and foundations, shape theory is strongly based only on topology. The goal of shape theory is largely to generalize homotopy theory to more extreme cases—almost an intellectual exercise—whereas the goal of my own proposed math branch was mostly shape recognition for practical purposes in vision systems: very different goals, right from the start.

(2) Why did shape theory deal so much with approximations instead of the absolutely precise math that is standard in almost all other branches of math?

Answer: It doesn’t. I misunderstood some of the book’s terminology, especially the term “approximation”, which the book’s introduction itself later explained.

(p. 8)
  We have used the term ‘approximation’ but this clearly implies some means of
comparing objects. The convenient way to do this is by replacing the bare structure
of the classes of interest or of models by categories, thus enabling comparison of
objects and of models. Thus we suppose we have only two categories, A, of models and
B of objects of interest.

(“Shape Theory: Categorical Methods of Approximation”, J.-M. Cordier & T. Porter, 1989)

I believe the above description is merely a description of template matching (http://en.wikipedia.org/wiki/Pattern_recognition_(psychology)), where category A is the set of templates and category B is the set of observed objects to be classified, so the book’s use of the word “approximation” was misleading. Also the “class of objects on which one has a reasonably complete set of information” was referring to the templates, not the class of observed objects. Also, a metric space (http://en.wikipedia.org/wiki/Metric_space; http://www.kahany.com/mathematics/metricspace.html) does not imply the use of ad hoc metrics, as I initially feared. In fact, the textbook was almost entirely theoretical, to an extreme, rather than practical or ad hoc, and shape theory is definitely a precise branch of math, as I’d hoped, in constrast to an ad hoc engineering approach like artificial neural networks.

However, shape theory still barely applies to pattern recognition, therefore it barely applies to AI. For example, in the above mentioned shape theory book, the only mention of patttern recognition appeared in the book’s *appendix*, entitled “Categorical Shape Theory and Pattern Recognition, a possible link”, and their suggestion was to mix in probabilty to cover the parts of the theory that were too primitive for real-world applications. Awful!

(p. 198)
  The theory based on categorical shape theory is, of course, still primitive,
as it does not take into account the stochastic nature of many of the phenomena
involved in the ‘real-life’ situation which it hopes to model. However, this very
simplicity means that it should prove more accessible to workers in these areas who
(p. 199)
may then be able to develop more valid models for recognition systems, possibly
involving an enriched probabilistic version of this theory.

(“Shape Theory: Categorical Methods of Approximation”, J.-M. Cordier & T. Porter, 1989)

Introductions to shape theory usually show the Warsaw Circle (http://en.wikipedia.org/wiki/Warsaw_circle#Warsaw_Circle), which is a poorly behaved, almost fractal-like shape, due to one of its sides being bounded by a modified sine function that oscillates ever more rapidly as it approaches a limit, in this case an edge of the contour. However, “Borsuk’s original shape theory has been replaced by a more systematic approach by inverse systems, pioneered by Sibe Mardesic, and independently, by Timothy Porter.” (http://en.wikipedia.org/wiki/Shape_theory_(mathematics)). That modern approach is called “abstract shape theory” (http://ncatlab.org/nlab/show/shape theory). The idea there is that you take a subset of the objects you are trying to categorize, a projection functor records the approximating object for each approximation, and using methods known to work for that subset you can study the superset, though the drawback is that new types of maps are obtained that describe the superset. Some of the poorly behaved shapes used in modern shape theory are the Sierpinski gasket and p-adic solenoid (http://ncatlab.org/nlab/show/shape theory).

Yes, shape theory literature is very difficult material to read, with its own extensive and highly technical terminology. I still barely understand the gist of it, but its foundations in combination with the appendix quoted above is enough to convince me that this is definitely not the route to the mathematics of AI, if such a branch of math even exists. That strongly suggests to me that my mentioned idea of a top-level branch of math based on the abstract concept of shape is in fact original and is a much more promising direction of research. If this is true, then for the historical record the date of that idea was Friday, August 2, 2013 (at night while I was tossing and turning in bed!). I decided to name my new top-level branch of math “formology”, based on the Greek word “phorma”, which means “shape”.

To extend my template format from my previous post, the differences between shape theory and formology become clear:

ABSTRACT FOUNDATIONAL CONCEPT: connectedness, as described by numerocentric algebra
SUBBRANCH OF: algebraic topology (http://en.wikipedia.org/wiki/Algebraic_topology) <= topology
RESULTING THEORY NAME: “shape theory” (http://en.wikipedia.org/wiki/Shape_theory_(mathematics))
INDIVIDUAL OBJECTS: poorly behaved surfaces
COLLECTIVE OBJECT: (none?)
SOME PRIMARY GOALS: to generalize the idea of homotopy theory to cases with unfavorable local properties
TRANSFORMATIONS/OPERATIONS: shape maps
EXAMPLES OF REFINEMENTS: Warsaw Circle, absolute neighbourhood retracts, comma category, projection functor, CW-complex, Cech homotopy, Sierpinski gasket

ABSTRACT FOUNDATIONAL CONCEPT: shape
SUBBRANCH OF: (none)
RESULTING THEORY NAME: “formology”
INDIVIDUAL OBJECTS: (possibly) features of shape countours
COLLECTIVE OBJECT: shapes
SOME PRIMARY GOALS: to recognize shapes, to categorize shapes, to measure the similarity between shapes
TRANSFORMATIONS/OPERATIONS: (unknown so far)
EXAMPLES OF REFINEMENTS: (none so far)

Wouter Smet - Sep 3, 2013:

how about your give ‘dialogue and conversation flow’ a shot - if you manage to inject some formality in that one, I’ll buy you a beer

Now you’ve done it. You’ve gone and planted the seed of a challenge in my brain and it won’t go away now. Thanks a lot.

I remember this topic appearing sporadically elsewhere in this forum, so it must be important. Would you be interested in starting a thread about this problem for the purpose of people here brainstorming solutions to it? I’m not very familiar with the problem’s details and how it arises, but it sounds interesting. People’s proposed solutions would be publicly visible, though. I suspect imagery would be involved in my solutions, such as representing the flow of a conversation as a trajectory in space. It may also be desirable to analyze why conversations go off-track (boredom, sensitivity, lack of undersanding, sudden new ideas, real-world interrupts, psychosis, etc.).

Wouter Smet - Sep 3, 2013:

Yeah, that’s true, it’s amazing in what weird places e and Pi (and i!) pop up, including the fact that they are bizarrely connected.

This is getting off-topic, but I’d really love to know what the “fourth math constant” is, or if there is one, and if not, why not. If you browse through a book of math formulas, especially integrals, you’ll see only those three constants, no others. I’ve tried looking for a fourth constant but all other candidates just don’t have the same character…

()
The golden ratio is just based on the mundane sqrt(5). (http://en.wikipedia.org/wiki/Golden_ratio)
()
Khinchin’s constant is based on an ad hoc statistical measure (geometric mean) and never seems to occur naturally. (http://en.wikipedia.org/wiki/Khinchin’s_constant)
()
Apéry’s constant is just one of an infinite number of function outputs: zeta(3). We might as well invest interest and effort into sin(42) if we’re going to go that route.
(http://en.wikipedia.org/wiki/Apéry’s_constant)
()
Feigenbaum’s two constants look like the best candidates to me, but I don’t believe a formula for them exists, and they haven’t yet popped up outside of narrow contexts within chaos theory. (http://en.wikipedia.org/wiki/Feigenbaum_constants)

Anyway, my pi example of course was just one example of many where an abstract idea in math pops up unexpectedly in a totally different domain. That demonstrates the power of abstraction. Here’s another example:

(p. 159)
  It should be clear by now that a mathematical
study of sphere packing can contribute to our un-
derstanding of certain phenomena in the world
around us. This was, after all, Kepler’s reason for
commencing such a study. What he most certainly
did not anticipate was that the study of sphere pack-
ings would have application in the twentieth-cen-
tury technology of digital communications! More-
over, that application would arise as a result of
generalizing the sphere-packing problem to spaces
of dimension 4 and more. This surprising, and re-
cent, development is yet another example of a prac-
tical application arising from the pure mathemati-
cian’s search for abstract patterns that exist only in
the human mind.

(“Mathematics: The Science of Patterns: The Search for Order in Life, Mind, and the Universe”, Keith Devlin, 1994)

PART 8: TRANSFINITE NUMBERS APPLIED TO SHAPE

I like to use transfinite numbers as a quick way to approximate relative sizes of objects and architectures related to AI. A more formal way to do that would probably be to use big “O” notation, but that’s more complicated and less obvious. I’ll first do a quick overview of transfinite numbers before illustrating why I believe shape is provably a particularly profound concept, which in turn means shape is a particularly difficult concept for computers to handle, despite the concept of shape being intuitively obvious to humans. This topic has obvious relevance to AI since a truly intelligent machine would need to interpret the real world, especially through vision, therefore such a machine would need to comprehend and use shape.

In case you didn’t know, there are different sizes of infinity. Also, you can perform simple arithmetic on these various sizes of infinity. We’ve all seen the symbol for infinity that looks like a sideways “8”. Judging from my own past and from people I’ve tutored in math, performing arithmetic with this symbol seems to hold unusually high fascination for adolescent boys. For example, if you add element “0” to the set {1, 2, 3, ...}, a set that has an infinite number of elements, you get the set {0, 1, 2, 3, ...}, which also has an infinite number of elements. The hasty conclusion, especially by adolescent boys: infinity + 1 = infinity.

The main problem with using that common infinity symbol in arithmetic is that the infinity symbol essentially means “some infinite number”. My mini-equation above is therefore analogous to saying “some finite number + 1 = some finite number”. That is a true statement, but it isn’t very specific: it doesn’t specify exactly *which* finite numbers are involved, which is the whole idea of doing arithmetic, and therefore the “equation” technically doesn’t contain quantities as it should, so it is incorrectly formed. In this case, it doesn’t say which transfinite (= infinite) numbers are involved. Georg Cantor developed a sequence of infinite numbers that I’ll represent as aleph(0), aleph(1), aleph(2), etc., where aleph(0) is provably the smallest infinite number, aleph(1) is provably the next largest, and so on. (The standard notation actually used for those infinite numbers is a funky script Hebrew letter “A” with a numerical subscript, but have mercy: I’m using a QWERTY keyboard. I also have a longstanding complaint about the way Cantor considered and used these transfinite numbers, but I won’t get into that issue here.)

In the above example, the sets {1, 2, 3, ...} and {0, 1, 2, 3, ...} are both said to contain aleph(0) elements, so that the specific and proper way to write my above infinity equation is aleph(0) + 1 = aleph(0). The important fact here is that any set whose elements can be listed, whether in a row, table, or other means, contains aleph(0) elements. Therefore any concept of infinity that contains discrete, listable members of a set, such as the number of stars in an infinite universe, or the number of clones of yourself when standing between two mirrors, is aleph(0). Since it is possible to include all the negative integers almost as easily as the positive integers, in such a list, namely {-1, 1, -2, 2, ...}, then the number of all nonzero integers = 2 * the number of nonzero positive integers = aleph(0), which gives rise to correct mini-equations such as 2 * aleph(0) = aleph(0). Interestingly, the number of rational numbers is also only aleph(0), since that set can also be listed if you arrange the elements cleverly enough so that any given ratio can be reached in a finite amount of time. The key concept throughout here is that the notion of discrete objects in any infinite list is associated with aleph(0).

The next size of infinity is aleph(1), which is the quantity of real numbers, say the number of points between 0 and 1 on the number line. (I’ll overlook the theoretical caveat that you first must assume that the Generalized Continuum Hypothesis is true (http://en.wikipedia.org/wiki/Cardinal_arithmetic), though that assumption causes no mathematical problems.) The key concept here is that the notion of densely packed objects, each so small and packed so densely that the result is continuous/smooth, is associated with aleph(1). Since the number of points between 0 and 2 is conceptually the same as the number of points between 0 and 1, that gives rise to the correct mini-equation 2 * aleph(1) = aleph(1). It is important to note that, unlike aleph(0) patterns, there exist within any collection of aleph(1) many entities whose patterns or exact locations literally can *never* even be *described*!

There is an important pattern beginning to emerge here: note it is not possible to produce the next larger aleph by addition or multiplication of finite numbers or of even the same transfinite number to itself. For example, as seen above, n + aleph(0) = n * aleph(0) = aleph(0)^n, and the same is true for aleph(1), aleph(2), and so on. To get to any next larger aleph, exponentiation to an aleph power must be used: 2^aleph(0) = aleph(1), 2^aleph(1) = aleph(2), and so on. Fortunately, such exponentiation occurs naturally in sets since the “power set” of any set of n elements contains 2^n elements, which becomes the key to generating an arbitrarily large aleph. (http://www.math.utah.edu/~pa/math/sets.html)

The next size of infinity is aleph(2). According to George Gamow in a book I read in high school, aleph(2) is the number of possible curves, which equates to the number of all possible functions of real numbers:

(p. 22)
  According to Georg Cantor, the creator of the “arithmetics of
infinity,” infinite numbers are denoted by the Hebrew letter aleph
(aleph) with a little number in the lower right corner that indi-
cates the order of the infinity. The sequence of numbers (in-
cluding the infinite ones!) now runs:

    1. 2. 3. 4. 5. .  .  .  .  .  . aleph(1) aleph(2) aleph(3) . . . . . .

and we say “there are aleph(1) points on a line” or “there are aleph(2)
(p. 23)
different curves,” just as we say that “there are 7 parts of the
world” or “52 cards in a pack.”
  In concluding our talk about infinite numbers we point out
that these numbers very quickly outrun any thinkable collection
to which they can possibly be applied. We know that aleph repre-
sents the number of all integers, aleph(1) represents the number of all
geometrical points, and aleph(2) the number of all curves, but nobody
as yet has been able to conceive any definite infinite collection
of objects that should be described by aleph(3). It seems that the
three first infinite numbers are enough to count anything we can
think of, and we find ourselves here in a position exactly opposite
to that of our old friend the Hottentot who had many sons but
could not count beyond three!

(“One Two Three ... Infinity: Facts & Speculations of Science”, George Gamow, 1974)

Now we’re into the realm of vision and therefore AI, since curves are equivalent to shapes.

As for aleph(3), in 2012 I thought of a way to visualize it: think of a red coral necklace. More specifically, think of those red coral necklaces that look like a cluster of red sticks drilled transversely through their stems (e.g., http://4infashion.files.wordpress.com/2011/10/fb_coral-necklace.jpg; http://image0-rubylane.s3.amazonaws.com/shops/bdazzled/J9143.1L.jpg?40), rather than the kind that use identical beads. Like snowflakes, each stem of coral must in theory be slightly different from every other coral stem that has ever existed. Think of each coral stem as a curved path through space, therefore each coral stem represents aleph(2) many possibilities. Since the number of ways to order n objects on a string is n!, which is greater than 2^n for all integers greater than 3, the number of ways to string a necklace of such curves (ignoring the fact that aleph(2) many objects will not fit into a line segment) is at least 2^n, which in this case is 2^aleph(2), or aleph(3).

I believe transfinite numbers explain best why shape is such a profound concept. No shape exists in 0D other than a single point—a trivial case. No real shape exists in 1D since the contour of a 1D shape would merely correspond to two endpoints, which are representable by two points, of which there exist only 2 * aleph(1) = aleph(1) possibilities: not very interesting. But in 2D, 3D, and all higher dimensions, shape does inherently exist. Also, unlike some topological concepts like knots, which exist only in 3D, shape continues to exist in all higher dimensions.

Now let’s combine all the above observations about transfinite numbers into some practical implications:

()
Knowledge of the basics of transfinite numbers can prevent inventors from making some foolish mistakes. Around 1991 I was finally able to obtain a copy of an interesting-sounding journal article by E. Bribiesca and A. Guzman from 1978 (http://turing.iimas.unam.mx/~ernesto/shape.htm) that I’d been seeking for 1-2 years, about an invention called “shape numbers” where the author devised a way to represent images with numbers. I was sure disappointed after all that wait, since I found out his creation was just an ad hoc method that—if I recall correctly—used consecutive sections of digits within an integer to represent certain attributes of the shape. Later I mentally kicked myself for having been so foolish: I’d read Gamow’s book in high school and was well aware that there existed aleph(2) many curves but only aleph(1) many real numbers, so I should have known that any such invention would have been impossible to implement in a fully general case without losing information. A similar and more recent proposal was by Huang, Dai, and Lin, where they used integers to “define a shape number as a permutation of an angle string such that this permutation forms an integer of minimum magnitude” (http://imdb.cs.nchu.edu.tw/teacherclass/image database design/2-shape representation and recognition.pdf).
()
Intelligent machines might very well need to have the equivalent of a continuous 2D memory—something like an electrified plate containing shifting images—because only in 2D or higher does the concept of shape arise, and without an ability to efficiently represent shape, machines will not be able to interpret real-world data efficiently, since real-world visual data consists of 2D data streams (possibly synchronized to produce stereoscopic information). While digital computers can in theory simulate any real-world phenomenon with arbitrary accuracy, such Turing machine-based computers are based on aleph(0) models (e.g., input tapes of length aleph(0)), which is a step removed from the aleph(1) objects needed that exist in a profusion numbering aleph(2). For example, some problems that no Turing machine could ever solve are: move two arbitrary, congruent, perfect circles side by side so that their perimeters touch exactly, find the coordinates of a point that was plotted on a perfect inputted graph, determine if a curve on a perfect inputted graph is continuous.

There also exist implications for my proposed branch of math called formology:

()
Formology will never be able to list all possible shapes because any list inherently contains only aleph(0) many elements whereas there exist aleph(2) many shapes.
()
Formology will never even be able to approximate the location or configuration of all possible shapes visually because shapes cannot be ordered like points on a number line.
()
Formology will never be able to represent even all the underlying features of shapes, since such features number aleph(2) many: the same quantity as the shapes themeselves.

aleph(2) and therefore shape is in a world of complexity by itself that is going to be difficult to deal with, so formology will constantly have to be careful to sidestep such problems, especially while it is being developed.

ERRATA ON TRANSFINITE NUMBERS

(1)
I believe the quote from George Gamow should start the list of alephs with a plain “aleph”, as follows:

1. 2. 3. 4. 5. .  .  .  .  .  . aleph aleph(1) aleph(2) . . . . . .

Gamow is using a plain “aleph” to mean “aleph(0)”. Most authors use “aleph(0)”. I botched up his quoted list while attempting to make the notation consistent throughout my post. (I don’t have a copy of that page on hand to verify what I remember about how he wrote that list, though.)

(2)
I accidentally omitted the main reason I believe shape is a profound concept: It is because of shape that associated cardinality makes a sudden jump from aleph(1) to aleph(2) upon moving from 1D to 2D, which is a type of jump that doesn’t naturally occur in any lower dimensions, and (to my knowledge) never occurs again, no matter how many dimensions are added. (If it did, we would have a simpler example of aleph(3), which we don’t have.)

(3)
“It is important to note that, unlike aleph(0) patterns, there exist within any collection of aleph(1) many entities whose patterns or exact locations literally can *never* even be *described*!”
...should read…
“It is important to note that, unlike aleph(0) patterns, there exist within any collection of aleph(1) many entities, entities whose patterns or exact locations literally can *never* even be *described*!”

At least this errata post gives me a chance to post a cool, applicable quote by David Marr, which I didn’t have space to post in my previous post:

(p. 16)
  The first great revelation was that the problems are difficult. Of course,
these days this fact is commonplace. But in the 1960s almost no one
realized that machine vision was difficult. The field had to go through the
same experience as the machine translation field did in its fiascoes of the
1950s before it was at last realized that here were some problems that had
to be taken seriously. The reason for this misperception is that we humans
are ourselves so good at vision.

(“Vision: A Computational Investigation into the Human Representation and Processing of Visual Information”, David Marr, 1982)

P.S.—By the way, regarding shape theory, here’s what a Sierpinski gasket looks like (it’s also called a Sierpinski sieve):

http://mathworld.wolfram.com/SierpinskiSieve.html

And here’s a 2-adic solenoid:

http://www.math.iupui.edu/~kitchens/

The “p” in “p-adic solenoid” means a prime number.