Archive for April, 2012

Kant’s Reflective Perspectives on Experience

April 27, 2012

The web site of Stephen R. Palmquist has a great wealth of material on fourfolds in relation to Kant’s as well as his own philosophy. From my own initial reading of his extensive material I have tried to choose a canonical Kantian fourfold which has the most relevance to my project.

The fourfold shown above Dr. Palmquist calls Kant’s “reflective perspectives on experience”. Consisting of the logical, the empirical, the transcendental, and the hypothetical, these facets bear a close analogical likeness to many of the fourfolds presented here.

Logical: Analytic a priori
Transcendental: Synthetic a priori
Hypothetical: Analytic a posteriori
Empirical: Synthetic a posteriori

Dr. Palmquist also has many of his own books available on his web site for the interested reader. I will certainly be returning to his web site in the future for much enjoyable study.

References:

http://www.hkbu.edu.hk/~ppp/

http://www.hkbu.edu.hk/~ppp/ksp2/KCR3.htm

[*7.68, *7.84]

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The Four Binary Operators of Linear Logic

April 17, 2012


The four binary operators for Linear Logic can be described by their logical sequents, or inference rules (shown above in the table). Note that in the rules for the operators, the operator appears below and not above the horizontal line. Thus the sequents can be considered in two ways: moving from top to bottom, the operator is introduced, and moving from bottom to top, the operator is eliminated. As operators are introduced the rules deduce a formula containing instances of them, and as operators are eliminated the rules construct a proof of that formula without them. Also note that in the sequents the symbols A and B denote formula in our logic, and the symbols Γ and Δ are finite (possibly empty) sequences of formulae (but the order of the formulae do not matter), called contexts.

The rule for the operator & (with), additive conjunction, says that if A obtains along with the context Γ, and if B obtains also along with Γ, then A & B obtains along with Γ.

The rule for the operator (plus), additive disjunction, says that if A obtains along with the context Γ, or if B obtains along with Γ, then A B obtains along with Γ.

The rule for the operator (par), multiplicative disjunction, says that if the combination A,B obtains along with the context Γ, then A B obtains along with Γ.

The rule for the operator (tensor), multiplicative conjunction, says that if A obtains along with context Γ, and if B obtains along with context Δ, then A B obtains along with the combined context Γ,Δ.

There are several immediate observations we can make about these rules.

First, note that for operators & and , they are reversible. That is, if one obtains A & B or A B, then one knows exactly what the previous step had to be to introduce the operator. In contrast, and are not reversible. If one has A B, one doesn’t know if we started with A or with B. If one has A B, one doesn’t know what was the context of A or what was the context of B.

For this reason, I will consider & and to be rational, and and to be empirical.

Second, note that all of the parts of the multiplicative sequents for and are the same above and below the horizontal line. In contrast, the parts of the additive sequents for & and are different above and below the line. For &, a duplicated context is eliminated even as the operator is introduced. For , a new formula is introduced along with the introduction of the operator.

For this reason, I will consider & and to be discrete, and and to be continuous.

References:

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

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

http://www.uni-obuda.hu/journal/Mihalyi_Novitzka_42.pdf

[*6.38, *6.40]

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A Digital Universe

April 12, 2012

A digital universe – whether 5 kilobytes or the entire Internet – consists of two species of bits: differences in space, and differences in time. Digital computers translate between these two forms of information – structure and sequence – according to definite rules. Bits that are embodied as structure (varying in space, invariant across time) we perceive as memory, and bits that are embodied as sequence (varying in time, invariant across space) we perceive as code. Gates are the intersections where bits span both worlds at the moments of transition from one instant to the next.

— George Dyson, from Turing’s Cathedral

George Dyson / Turing’s Cathedral: the origins of the digital universe

[*7.82, *7.83, *7.153]

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The Four Freedoms

April 9, 2012

From Wikipedia:

The Four Freedoms were goals articulated by US President Franklin D. Roosevelt on January 6, 1941. In an address known as the Four Freedoms speech (technically the 1941 State of the Union address), he proposed four fundamental freedoms that people “everywhere in the world” ought to enjoy:

  •     Freedom of speech
  •     Freedom of worship
  •     Freedom from want
  •     Freedom from fear

His inclusion of the latter two freedoms went beyond the traditional US Constitutional values protected by its First Amendment, and endorsed a right to economic security and an internationalist view of foreign policy. They also anticipated what would become known decades later as the “human security” paradigm in social science and economic development.

References:

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

http://www.americanrhetoric.com/speeches/fdrthefourfreedoms.htm

[*7.76]

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Systems Dynamics

April 2, 2012

Another interesting fourfold that I discovered while reading mathematical physicist John C. Baez’s blogs Azimuth and This Week’s Finds in Mathematical Physics concerns the notions of system dynamics and bond graphs. These concepts generalize the fourfold of the basic electronic components into other types of physical systems, such as mechanics, hydraulics, and to some extent even thermodynamics and chemistry.

The types of systems that can be modeled by system dynamics are described by two variables that vary functionally over time and their corresponding integrals. These four functions can be thought of as flow and effort and their respective integrals displacement and momentum.

  Displace-
ment
Flow Momentum Effort
Mechanics of translation Position Velocity Momentum Force
Mechanics of rotation Angle Angular velocity Angular momentum Torque
Electronics Charge Current Flux Voltage
Hydraulics Volume Flow Pressure momentum Pressure
Thermo-dynamics Entropy Entropy flow Temperature momentum Temperature
Chemistry Moles Molar flow Chemical momentum Chemical potential

References:

http://johncarlosbaez.wordpress.com/2012/02/02/quantizing-electrical-circuits/

http://math.ucr.edu/home/baez/week288.html

http://math.ucr.edu/home/baez/week292.html

http://ncatlab.org/johnbaez/show/Diagrams

http://magisciences.tuxfamily.org/BondGraph/co/03%20Passive%20elements.html

http://www.feedcontfinals.ewebsite.com/articles/chapter-5/chapter-5-page-2/

Dean C. Karnopp, Donald L. Margolis, Ronald C. Rosenberg / System Dynamics: modeling and simulation of mechatronic systems

[*7.60, *7.61]

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