Archive for the ‘Mathematics’ Category

Fourier Analysis, V2

December 8, 2017

Here is another example of a fourfold, in the mathematics of Fourier Analysis. Here the four elements of our investigation resolve into Discrete Time, Continuous Time, the Fourier Series, and the Fourier Transform.

From the three dualities of Time – Frequency, Periodic – Aperiodic, and Discrete – Continuous, we obtain the four combinations Discrete Time/Periodic Frequency, Continuous Time/Aperiodic Frequency, the Fourier Series (Periodic Time/Discrete Frequency), and the Fourier Transform (Aperiodic Time/Continuous Frequency).

In the table below, T stands for Time and f for Frequency. The subscripts denote the attributes of each: D for Discrete, C for Continuous, P for Periodic, and A for Aperiodic. So T subscript C, f subscript A means that when Time is Continuous, Frequency is Aperiodic, etc. Please see Steve Tjoa’s web site for the equations for the Fourier Series and the Fourier Transform in Continuous and Discrete Time.References:

http://stevetjoa.com/633

http://en.wikipedia.org/wiki/Fourier_analysis

http://en.wikipedia.org/wiki/Fourier_series

http://en.wikipedia.org/wiki/Fourier_transform

[*7.74, *7.108]

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A Rosetta Stone

December 6, 2017

Abstract of Physics, Topology, Logic and Computation: A Rosetta Stone by John Baez and Michael Stay:

In physics, Feynman diagrams are used to reason about quantum processes. In the 1980s, it became clear that underlying these diagrams is a powerful analogy between quantum physics and topology: namely, a linear operator behaves very much like a “cobordism”. Similar diagrams can be used to reason about logic, where they represent proofs, and computation, where they represent programs. With the rise of interest in quantum cryptography and quantum computation, it became clear that there is extensive network of analogies between physics, topology, logic and computation. In this expository paper, we make some of these analogies precise using the concept of “closed symmetric monoidal category”. We assume no prior knowledge of category theory, proof theory or computer science.

  • Physics
  • Logic
  • Topology
  • Computation

Perhaps Category Theory is a “Fifth Essence”?

Further Reading:

http://math.ucr.edu/home/baez/rosetta/rose3.pdf

https://arxiv.org/abs/0903.0340

[*9.168, *10.50]

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The Duality of Time and Information, V3

October 1, 2017

 

The states of a computing system bear information and change time, while its events bear time and change information.

from The Duality of Time and Information by Vaughn Pratt

The most promising transformational logic seems to us to be Girard’s linear logic.

— from Rational Mechanics and Natural Mathematics by Vaughn Pratt

 

Here we have three duals:

  • Information – Time
  • States – Events
  • Bear – Change

Further Reading:

Vaughan Pratt / The Duality of Time and Information http://boole.stanford.edu/pub/dti.pdf

Vaughan Pratt / Time and Information in Sequential and Concurrent Computation http://boole.stanford.edu/pub/tppp.pdf

Vaughan Pratt / Rational Mechanics and Natural Mathematics http://chu.stanford.edu/guide.html#ratmech

[*5.170]

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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.

References:

https://en.wikipedia.org/wiki/Banach%E2%80%93Tarski_paradox

https://en.wikipedia.org/wiki/Paradoxical_set

https://en.wikipedia.org/wiki/Axiom_of_choice

http://faculty.mccneb.edu/jdlee3/thesisonline.pdf

[*9.124, *9.125]

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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.

References:

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

http://cacm.acm.org/magazines/2015/12/194626-propositions-as-types/fulltext

Preprint at

http://homepages.inf.ed.ac.uk/wadler/papers/propositions-as-types/propositions-as-types.pdf

Also see:

http://www.drdobbs.com/old-ideas-form-the-basis-of-advancements/184404384

https://en.wikipedia.org/wiki/Hindley%E2%80%93Milner_type_system

https://en.wikipedia.org/wiki/System_F

https://en.wikipedia.org/wiki/Curry%E2%80%93Howard_correspondence

[*9.92-9.94]

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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”.

References:

https://en.wikipedia.org/wiki/Dualism_%28philosophy_of_mind%29

http://boole.stanford.edu/pub/ratmech.pdf

http://chu.stanford.edu/

http://en.wikipedia.org/wiki/Chu_space

[*6.74, *6.75, *9.76]

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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)

References:

https://en.wikipedia.org/wiki/Bayes’_theorem

[*6.132, *9.48]

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King – Man + Woman = Queen

September 21, 2015

sq_king_man_woman_queen

Here is a fascinating article and paper on computational linguistics.

References:

http://www.technologyreview.com/view/541356/king-man-woman-queen-the-marvelous-mathematics-of-computational-linguistics/

http://arxiv.org/abs/1509.01692

[*9.46]

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The Collatz Conjecture

October 17, 2014

sq_collatz

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…

http://en.wikipedia.org/wiki/Collatz_conjecture

http://xkcd.com/710/

[*8.88]

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Four Dimensional Space-time

September 1, 2014

sq_4d_spacetime

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.

Links:

http://en.wikipedia.org/wiki/Special_relativity

http://en.wikipedia.org/wiki/Minkowski_space

http://en.wikipedia.org/wiki/Four-dimensional_space

http://en.wikipedia.org/wiki/Flatland

http://en.wikipedia.org/wiki/Flatland_%282007_film%29

http://en.wikipedia.org/wiki/Spaceland_%28novel%29

[*8.72]

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