Archive for the ‘Mathematics’ Category

Buy One Get One Free

May 6, 2016

sq_free_group2Speaking of paradoxes, the Banach-Tarski Paradox is an interesting theorem in mathematics that claims that a solid 3-dimensional ball can be decomposed into a finite number of parts, which can then be reassembled in a different way (by using translation and rotation of the parts but no scaling is needed) to create two identical copies of the original structure. The theorem works by allowing the parts of the decomposition to be rather strange.

One of the important ingredients of the theorem’s proof is finding a “paradoxical decomposition” of the free group on two generators. If F is such a free group with generators a and b, and S(a) is the infinite set of all finite strings that start with a but without any adjacency of a and its inverse (a^-1) or similarly b and b^-1, and 1 is the empty (identity) string, then

F = 1 + S(a) + S(b) + S(a^-1) + S(b^-1)

But also note that

F = aS(a^-1) + S(a)

F = bS(b^-1) + S(b)

So F can be paradoxically decomposed into two copies of itself by using just two of the four S()’s for each copy. Both aS(a^-1) and bS(b^-1) contain the empty string, so I’m not sure what happens to the original one. One might think that aS(a^-1) is “bigger” than S(a) but they are actually both countably infinite and so are the same “size”.

The generators a and b are then set to be certain 3-dimensional isometries (distance preserving transformations which include translation and rotation). The rest of the theorem requires further constructions that may be of interest, as well as needing the Axiom of Choice or something like it. It is also curious that the paradox fails to work in dimensions of 1 or 2.

The diagram above tries to list the beginning of each of the sets S(a), S(b), S(a^-1), and S(b^-1). The empty string can be thought to occupy the center of the diagram but it is either not shown because it is empty, or it is shown as being empty. Alternatively one could create a more general fourfold with the aspects of structure, (paradoxical) decomposition, parts, and reassembly.


[*9.124, *9.125]


Propositions as Types

February 14, 2016

sq_propositions_as_types3For almost 100 years, there have been linkages forged between certain notions of logic and of computation. As more associations have been discovered, the bonds between the two have grown stronger and richer.

  • Propositions in logic can be considered equivalent to types in programming languages.
  • Proofs of propositions in logic can be considered equivalent to programs of given type in computation.
  • The simplification of proofs of propositions in logic can be considered equivalent to the evaluation of programs of types in computation.

The separate work of various logicians and computer scientists (and their precursors) can be paired:

  • Gerhard Gertzen’s work on proofs in intuitionistic natural deduction and Alonzo Church’s work on the simply typed lambda calculus.
  • J. Roger Hindley and Robin Milner’s work on type systems for combinatory logic and programming languages, respectively.
  • J. Y. Girard and John Reynold’s work on the second order lambda calculus and parametric polymorphic programs, respectively.
  • Haskell Curry’s and W. A. Howard’s work on the overall correspondence between these notions of proofs as programs or positions as types.

Logic and computation are the sequential chains of efficient causation and actions. Propositions and types are the abstract grids of formal causation and structures. Proofs and programs are the normative cycles of final causation and functions. Simplification and evaluation are the reductive solids of material causation and parts.


Philip Wadler / Propositions as Types, in Communications of the ACM, Vol. 58 No. 12 (Dec 2015) Pages 75-85.

Preprint at

Also see:




Four Transformations of Chu Spaces

November 29, 2015

sq_four_transformationsCan mathematics help us reformulate Cartesian Dualism? I have previously tried to diagram some of computer scientist Vaughn Pratt’s notions, such as a Duality of Time and Information and the Stone Gamut. Another recent attempt is the diagram above of four transformations that issue out of his analysis of Chu Spaces. Pratt’s conceptualization of these generalized topological spaces led him to propose a mathematization of mind and body dualism.

The duality of time and information was actually an interplay of several dualities, such as the aforementioned time and information, plus states and events, and changing and bearing (or dynamic and static). The philosophical mathematization in his paper “Rational Mechanics and Natural Mathematics” leads to additional but somewhat different dualities, shown in the following table:

Mind Body
Mental Physical
States Events
Anti-functions Functions
Anti-sets Sets
Operational Denotational
Infers Impresses
Logical Causal
Against time With time
Menu Object
Contingent Necessary

Pratt reveals two transformations that are “mental”: delete and copy, and two that are “physical”: adjoin and identify.

These four transformations are functions and their converses which:

  • Identify when the function is not injective.
  • Adjoin when the function is not surjective.
  • Copy when the converse is not injective.
  • Delete when the converse is not surjective.

Ordinarily we think of mind and body as being radically different in kind, but perhaps they are the same but merely viewed from a different perspective or direction. Recall what Heraclitus says, “the road up and the road down are the same thing”.


[*6.74, *6.75, *9.76]


Bayes’ Rule

October 10, 2015

sq_bayes_ruleBayes’ Rule or Theorem or Law. Because, why not?

P(B) P(A|B) = P(A) P(B|A)


[*6.132, *9.48]


King – Man + Woman = Queen

September 21, 2015


Here is a fascinating article and paper on computational linguistics.




The Collatz Conjecture

October 17, 2014


I thought I might be continuing my apparent effort to convert all triads into fourfolds, but I’ll take a little break to bring you this mathematical fourfold for the Collatz Conjecture. The Collatz algorithm starts with any specific nonzero integer, then iterates in the following way to turn the number into the next number to operate on, and so on. If X is even, then replace X with X/2. If X is odd, then replace X with 3*X + 1. If you get to 4, then you get 2, then 1, but you next go back to 4. In fact, if you have any power of 2, you quickly drop to 4 and so cycle. The Collatz Conjecture is that if you start with any number, then you will eventually reach 4. (Actually the conjecture is that you will reach 1, but since I’m partial to 4, I’ll state it in this way since it’s pretty much the same thing.)

The conjecture is not proven, but has been shown to be true for every number up to some very large numbers. Some numbers can jump around from larger to smaller to larger again, or up and down and back up, for quite a few steps before being reduced to the 4-2-1-4 sequence. That’s why these sequences are also called hailstone numbers.

One might think of the Collatz algorithm as a model of a toy universe in the following way. The procedure starts with a given number, which for the toy universe would be its initial state. As time moves from instant to instant, the procedure operates to halve the number, or multiply by three and add one. So instant after instant the procedure operates, making the number smaller, or larger as required. The conjecture, if true, would mean that no matter how it starts, our toy universe would always wind down and cycle endlessly through the sequence of 4-2-1-4-2-1-4-2-1-4…



Four Dimensional Space-time

September 1, 2014


Here’s a simple fourfold I’ve been ignoring just because it’s so trivial, but that triviality can be deceiving. Space-time as formulated in special relativity has four dimensions: three of space and one of time. Our everyday experience shows us the three dimensions of space: length, width (or breadth), and depth (or height), but time is a different kind of thing because we cannot see or move forward and backward through time with our eyes or body, like we can along the axes of space.

Personally, only our memory and imagination can let us range through time. Of course, after the invention of language and more recent technologies, the spoken word, writings, photographs, audio recordings, and videos can also be used. But it’s not the same as shifting one’s gaze along the length of something or moving one’s body across a width.

So, we can move semi-freely through the three spatial dimensions but our movement in time seems to be fixed into a relentless forward motion that we have no control over. And because gravity pulls us down onto the surface of the world, one of the spatial dimensions (depth or height) is more limiting than the other two.

sq_ll2Thus another interesting comparison to this fourfold is to that of linear logic. One observation is that length and width can be considered reversible but depth and time can be considered somewhat irreversible. That’s not true of course, but because of gravity it is easier to descend than to ascend, and it’s far easier to move into the future than into the past. But we can see into the distant past, just not our own, as we turn our telescopes to the heavens.

Space without time could have four or even higher dimensions, but we have no empirical evidence that it is so. Mathematically, however, we can easily construct multidimensional spaces. One representation of four dimensional space is by using quaternions, which have four dimensions to the complex numbers’ two. Tuples of real numbers or even vector spaces can also be used. However, the geometry of space-time is not Euclidean; it is described by the Minkowski metric.

Novels about characters living in different numbers of spatial dimensions are an interesting way to learn and think about them. The very first was Flatland by Edwin Abbott Abbott, about a being limited to two dimensions that learns about a third outside his experience when a three dimensional being comes to visit. Just recently I’ve finished reading Spaceland by Rudy Rucker, about an ordinary human person limited to the three dimensions of space that learns about the fourth dimension by similar reasons.




Complex Numbers and Quaternions

November 29, 2013

complex_numbersTwo conceptualizations of four directions in mathematics are Complex numbers and Quaternions. For complex numbers, two of the directions are the opposite or negative of the other two. Complex numbers are like Cartesian coordinates in that they combine an (x,y) coordinate pair of real numbers into one complex number x + yi. Plus, complex numbers can by manipulated by extensions of arithmetic operations, like addition, subtraction, multiplication, and division.
quaternionsQuaternions are an extension of the complex numbers where the directions are all different, and each direction is perpendicular to the other three. Like complex numbers are a notion of numbers that cover a plane, quaternions are a notion of number that fill four dimensional space. Thus they combine a 4-tuple (w,x,y,z) into one quaternion number w + xi + yj + zk. Like complex numbers, quaternions can be manipulated by further extensions of arithmetic operations.



The Stone Gamut

May 17, 2013


Our thesis is that the category Set is the ultimate abstraction of body, and that Set^op, equivalent to the category of complete atomic Boolean algebras (i.e. power sets), which we shall advocate thinking of as antisets, is dually the ultimate abstraction of mind.

— From Chu Spaces: automata with quantum aspects by Vaughn Pratt

Reflecting an era of reduced expectations for the superiority of humans, we have implemented causal interaction not with the pineal gland but with machinery freely available to all classical entities, whether newt, pet rock, electron, or theorem (but not quantum mechanical wavefunction, which is sibling to if not an actual instance of our machinery).

— From Rational Mechanics and Natural Mathematics by Vaughn Pratt




My Dear Aunt Sally

March 5, 2013

sq_my_dear_aunt_sallyWhen I took first year algebra in school, I learned the rule “My Dear Aunt Sally” as a mnemonic for the order of applying binary operations in algebraic expressions. “My Dear” meant to perform multiplication and division first. “Aunt Sally” meant to perform addition and subtraction next and last. Most of us have learned some variation of this rule. I see that it has now been enlarged to “Please Excuse My Dear Aunt Sally” to include parentheses and exponentiation, and to perform these two first before the original and now last four.

Why remark about this simplistic and even obsolete rule? Note the similarity between this fourfold of binary arithmetic operators and the four binary linear logic operators. In each there are two operators for combining: addition and multiplication in arithmetic, and the conjunctive operators with and tensor in linear logic. In each there are two operators for separating: subtraction and division in arithmetic, versus the disjunctive operators plus and par in linear logic. In each there are two rules for attraction and two rules for repulsion.

In addition, the double duality of the four arithmetic operators is revealed, as in arithmetic addition and subtraction are duals, and multiplication and division are duals. In linear logic, with and plus are duals, and tensor and par are duals. Can arithmetic be simulated by linear logic, or vice versa? Is linear logic equivalently exchangable with arithmetic? I don’t think so but perhaps some expert can tell us.