Moneytheism

Posted on June 10, 2013 by Jon Awbrey

Re: Cathy O’Neil • Profit as Proxy for Value

There is a deep and pervasive analogy between systems of commerce and systems of communication, turning on their near-universal use of symbola (images, media, proxies, signs, symbols, tokens, etc.) to stand for pragmata (objects, objectives, the things we really care about — or would really care about if we examined our values in practice thoroughly enough).

Both types of sign-using systems are prey to the same sort of dysfunction or functional disease — it sets in when their users confuse signs with objects so badly that signs become ends instead of means.

There is a vast literature on this topic, once you think to go looking for it.  And it’s a perennial theme in fable and fiction.

Posted in Commerce, Communication, Economics, Moneytheism, Semiotics, Sign Relations | Tagged , , , , , | 6 Comments

Fourier Transforms of Boolean Functions • 2

Posted on June 4, 2013 by Jon Awbrey

Re: R.J. Lipton and K.W. Regan • Twin Primes Are Useful

Note.  Just another sheet of scratch paper, exploring possible alternatives to the Fourier transforms in the previous post.  As a rule, I like to keep Boolean problems in Boolean spaces, partly for aesthetic reasons and partly from a sense that it doesn’t reduce the computational complexity of Boolean problems to replace them with integer or real number problems.  I’ll begin by copying the previous post as a template and gradually transform it as I proceed.

Begin with a survey of concrete examples, perhaps in tabular form.

Notation

Boolean domain {\mathbb{B} = \{0, 1\}}.

Boolean function on {k} variables {f : \mathbb{B}^k \to \mathbb{B}}.

Boolean coordinate projections {\mathcal{X} = \{ x_1, \ldots, x_k \}},
where {x_j : \mathbb{B}^k \to \mathbb{B}} such that {x_j : (x_1, \ldots, x_j, \ldots, x_k) \mapsto x_j}.

Minimal negation operator {\nu_k : \mathbb{B}^k \to \mathbb{B}}. In contexts where the meaning is clear, {\nu_k (x_1, \ldots, x_k)} may be written as {\nu (x_1, \ldots, x_k)} or even, using a different style of parentheses, as \texttt{(} x_1 \texttt{,} \ldots \texttt{,} x_k \texttt{)}.

{k = 2}

For ease of reading formulas, let {x = (x_1, x_2) = (u, v)}.

Identify {x \in \mathbb{B}^2} with the corresponding singular proposition {x : \mathbb{B}^2 \to \mathbb{B}}.

Try some other bases, but with addition as in {\mathbb{F}_2 = \text{GF}(2)}.

Observation. The propositions {f_7, f_{11}, f_{13}, f_{14}} are pairwise orthogonal.

Let {\mathcal{G} = \{ f_7, f_{11}, f_{13}, f_{14} \}}. I’m thinking of calling these the cosingular or fenestral propositions. (I would have called them the lacunary or umbral propositions but those terms already have established meanings in mathematics.) On third thought, I think I’ll call them crenular propositions, a crenel being a notch at the top of a structure.

Definitions

Fourier coefficient of {f} on {g}

{\displaystyle \hat{f}(g) = \sum_{x \in \mathbb{B}^2} f(x) \cdot g(x)}

Fourier expansion of {f}

{\displaystyle f(x) = \sum_{g \in \mathcal{G}} \hat{f}(g) \cdot g(x)}

Tables

\begin{array}{|c||*{4}{c}|} \multicolumn{5}{c}{\text{Table 2.1. Values of}~ g(x)} \\[4pt] \hline g & f_{8} & f_{4} & f_{2} & f_{1} \\ & \texttt{ } u \texttt{ } v \texttt{ } & \texttt{ } u \texttt{ (} v \texttt{)} & \texttt{(} u \texttt{) } v \texttt{ } & \texttt{(} u \texttt{)(} v \texttt{)} \\ \hline\hline f_{7} & 0 & 1 & 1 & 1 \\ f_{11} & 1 & 0 & 1 & 1 \\ f_{13} & 1 & 1 & 0 & 1 \\ f_{14} & 1 & 1 & 1 & 0 \\ \hline \end{array}
 

\begin{array}{|*{9}{c|}} \multicolumn{9}{c}{\text{Table 2.2. Fourier Coefficients of Boolean Functions on Two Variables}}\\[4pt] \hline \text{~~~~~~~~} & \text{~~~~~~~~} & & \text{~~~~~~~~} & \text{~~~~~~~~} & \text{~~~~~~~~~} & \text{~~~~~~~~} & \text{~~~~~~~~} & \text{~~~~~~~~~} \\ L_1 & L_2 && L_3 & L_4 & \hat{f}(f_{7}) & \hat{f}(f_{11}) & \hat{f}(f_{13}) & \hat{f}(f_{14}) \\ ~&~&~&~&~&~&~&~&~\\ \hline &&u = & 1~1~0~0 &&&&& \\ &&v = & 1~0~1~0 &&&&& \\ \hline f_{0} & f_{0000} && 0~0~0~0 & (~) & 0 & 0 & 0 & 0 \\ f_{1} & f_{0001} && 0~0~0~1 & (u)(v) & 1 & 1 & 1 & 0 \\ f_{2} & f_{0010} && 0~0~1~0 & (u)~v~ & 1 & 1 & 0 & 1 \\ f_{3} & f_{0011} && 0~0~1~1 & (u) & 0 & 0 & 1 & 1 \\ f_{4} & f_{0100} && 0~1~0~0 & ~u~(v) & 1 & 0 & 1 & 1 \\ f_{5} & f_{0101} && 0~1~0~1 & (v) & 0 & 1 & 0 & 1 \\ f_{6} & f_{0110} && 0~1~1~0 & (u,~v) & 0 & 1 & 1 & 0 \\ f_{7} & f_{0111} && 0~1~1~1 & (u~~v) & 1 & 0 & 0 & 0 \\ \hline f_{8} & f_{1000} && 1~0~0~0 & ~u~~v~ & 0 & 1 & 1 & 1 \\ f_{9} & f_{1001} && 1~0~0~1 &((u,~v))& 1 & 0 & 0 & 1 \\ f_{10}& f_{1010} && 1~0~1~0 & v & 1 & 0 & 1 & 0 \\ f_{11}& f_{1011} && 1~0~1~1 &(~u~(v))& 0 & 1 & 0 & 0 \\ f_{12}& f_{1100} && 1~1~0~0 & u & 1 & 1 & 0 & 0 \\ f_{13}& f_{1101} && 1~1~0~1 &((u)~v~)& 0 & 0 & 1 & 0 \\ f_{14}& f_{1110} && 1~1~1~0 &((u)(v))& 0 & 0 & 0 & 1 \\ f_{15}& f_{1111} && 1~1~1~1 & ((~)) & 1 & 1 & 1 & 1 \\ \hline \end{array}
 

\begin{array}{|*{9}{c|}} \multicolumn{9}{c}{\text{Table 2.3. Fourier Coefficients of Boolean Functions on Two Variables}}\\[4pt] \hline \text{~~~~~~~~} & \text{~~~~~~~~} & & \text{~~~~~~~~} & \text{~~~~~~~~} & \text{~~~~~~~~~} & \text{~~~~~~~~} & \text{~~~~~~~~} & \text{~~~~~~~~~} \\ L_1 & L_2 && L_3 & L_4 & \hat{f}(f_{7}) & \hat{f}(f_{11}) & \hat{f}(f_{13}) & \hat{f}(f_{14}) \\ ~&~&~&~&~&~&~&~&~\\ \hline && u = & 1~1~0~0 &&&&& \\ && v = & 1~0~1~0 &&&&& \\ \hline f_{0} & f_{0000} && 0~0~0~0 & (~) & 0 & 0 & 0 & 0 \\ \hline f_{1} & f_{0001} && 0~0~0~1 & (u)(v) & 1 & 1 & 1 & 0 \\ f_{2} & f_{0010} && 0~0~1~0 & (u)~v~ & 1 & 1 & 0 & 1 \\ f_{4} & f_{0100} && 0~1~0~0 & ~u~(v) & 1 & 0 & 1 & 1 \\ f_{8} & f_{1000} && 1~0~0~0 & ~u~~v~ & 0 & 1 & 1 & 1 \\ \hline f_{3} & f_{0011} && 0~0~1~1 & (u) & 0 & 0 & 1 & 1 \\ f_{12}& f_{1100} && 1~1~0~0 & u & 1 & 1 & 0 & 0 \\ \hline f_{6} & f_{0110} && 0~1~1~0 & (u,~v) & 0 & 1 & 1 & 0 \\ f_{9} & f_{1001} && 1~0~0~1 &((u,~v))& 1 & 0 & 0 & 1 \\ \hline f_{5} & f_{0101} && 0~1~0~1 & (v) & 0 & 1 & 0 & 1 \\ f_{10}& f_{1010} && 1~0~1~0 & v & 1 & 0 & 1 & 0 \\ \hline f_{7} & f_{0111} && 0~1~1~1 & (u~~v) & 1 & 0 & 0 & 0 \\ f_{11}& f_{1011} && 1~0~1~1 &(~u~(v))& 0 & 1 & 0 & 0 \\ f_{13}& f_{1101} && 1~1~0~1 &((u)~v~)& 0 & 0 & 1 & 0 \\ f_{14}& f_{1110} && 1~1~1~0 &((u)(v))& 0 & 0 & 0 & 1 \\ \hline f_{15}& f_{1111} && 1~1~1~1 & ((~)) & 1 & 1 & 1 & 1 \\ \hline \end{array}

Notes

References

21 May 2013 Twin Primes Are Useful
08 Nov 2012 The Power Of Guessing
05 Jan 2011 Fourier Complexity Of Symmetric Boolean Functions
19 Nov 2010 Is Complexity Theory On The Brink?
18 Sep 2009 Why Believe That P=NP Is Impossible?
04 Jun 2009 The Junta Problem
Posted in Boolean Functions, Computational Complexity, Fourier Transforms, Harmonic Analysis, Logic, Mathematics, Propositional Calculus | Tagged , , , , , , | Leave a comment

Wherefore Aught?

Posted on June 1, 2013 by Jon Awbrey

Re: R.J. Lipton and K.W. Regan • Why Is There Something?

Here is another one of those eternally recurring ideas echoed inimitably by C.S. Peirce in his sketch of a Cosmogonic Philosophy.

It would suppose that in the beginning,—infinitely remote,—there was a chaos of unpersonalized feeling, which being without connection or regularity would properly be without existence.  This feeling, sporting here and there in pure arbitrariness, would have started the germ of a generalizing tendency.  Its other sportings would be evanescent, but this would have a growing virtue.  Thus, the tendency to habit would be started;  and from this with the other principles of evolution all the regularities of the universe would be evolved.  At any time, however, an element of pure chance survives and will remain until the world becomes an absolutely perfect, rational, and symmetrical system, in which mind is at last crystallized in the infinitely distant future.  (Peirce, 1890/2010, p. 110).

The above quotation is taken from one of several discussions where Peirce introduces his idea that natural laws themselves evolve.  That idea has enjoyed yet another revival in recent days, notably by Lee Smolin in Time Reborn.

Reference

Charles S. Peirce (30 August 1890), “The Architecture of Theories”, pp. 98–110 in Peirce Edition Project (2010), Writings of Charles S. Peirce : A Chronological Edition, Volume 8, 1890–1892, Indiana University Press, Bloomington, IN. Published version, The Monist, vol. 1, no. 2 (January 1891), pp. 161–176.

Posted in C.S. Peirce, Cosmogony, Evolution, Existence, Natural Law, Peirce, Philosophy, References, Sources | Tagged , , , , , , , , | 2 Comments

Special Classes of Propositions

Posted on May 31, 2013 by Jon Awbrey

Adapted from Differential Propositional Calculus • Special Classes of Propositions

A basic proposition, coordinate proposition, or simple proposition in the universe of discourse \mathcal{X}^\bullet = \lbrack x_1, \ldots, x_k \rbrack is one of the propositions in the set \mathcal{X} = \lbrace x_1, \ldots, x_k \rbrace.

Among the 2^{2^k} propositions in \lbrack x_1, \ldots, x_k \rbrack are several families of 2^k propositions each that take on special forms with respect to the logical basis \lbrace x_1, \ldots, x_k \rbrace. Three of these families are especially prominent in the present context, the linear, the positive, and the singular propositions. Each family is naturally parameterized by the coordinate k-tuples in \mathbb{B}^k and falls into k + 1 ranks, with a binomial coefficient \dbinom{k}{j} giving the number of propositions that have rank or weight j.

  • The linear propositions, \lbrace \ell : \mathbb{B}^k \to \mathbb{B} \rbrace = (\mathbb{B}^k \xrightarrow{\ell} \mathbb{B}), may be written as sums:

    \begin{array}{llll} \displaystyle\sum_{i=1}^k e_i ~=~ e_1 + \ldots + e_k & \text{where} & \left\{ \begin{matrix} e_i = x_i \\ \text{or} \\ e_i = 0 \end{matrix} \right\} & \text{for}~ i=1 ~\text{to}~ k. \end{array}

  • The positive propositions, \lbrace p : \mathbb{B}^k \to \mathbb{B} \rbrace = (\mathbb{B}^k \xrightarrow{p} \mathbb{B}), may be written as products:

    \begin{array}{llll} \displaystyle\prod_{i=1}^k e_i ~=~ e_1 \cdot \ldots \cdot e_k & \text{where} & \left\{ \begin{matrix} e_i = x_i \\ \text{or} \\ e_i = 1 \end{matrix} \right\} & \text{for}~ i=1 ~\text{to}~ k. \end{array}

  • The singular propositions, \lbrace \mathbf{x} : \mathbb{B}^k \to \mathbb{B} \rbrace = (\mathbb{B}^k \xrightarrow{s} \mathbb{B}), may be written as products:

    \begin{array}{llll} \displaystyle\prod_{i=1}^k e_i ~=~ e_1 \cdot \ldots \cdot e_k & \text{where} & \left\{ \begin{matrix} e_i = x_i \\ \text{or} \\ e_i = \texttt{(}x_i\texttt{)} \end{matrix} \right\} & \text{for}~ i=1 ~\text{to}~ k. \end{array}

In each case the rank j ranges from 0 to k and counts the number of positive appearances of the coordinate propositions x_1, \ldots, x_k in the resulting expression. For example, for k = 3 the linear proposition of rank 0 is 0, the positive proposition of rank 0 is 1, and the singular proposition of rank 0 is \texttt{(} x_1 \texttt{)(} x_2 \texttt{)(} x_3 \texttt{)}.

The basic propositions x_i : \mathbb{B}^k \to \mathbb{B} are both linear and positive. So these two kinds of propositions, the linear and the positive, may be viewed as two different ways of generalizing the class of basic propositions.

Finally, it is important to note that all of the above distinctions are relative to the choice of a particular logical basis \mathcal{X} = \lbrace x_1, \ldots, x_k \rbrace. For example, a singular proposition with respect to the basis \mathcal{X} will not remain singular if \mathcal{X} is extended by a number of new and independent features. Even if one keeps to the original set of pairwise options \lbrace x_i \rbrace \cup \lbrace \texttt{(} x_i \texttt{)} \rbrace to pick out a new basis, the sets of linear propositions and positive propositions are both determined by the choice of basic propositions, and this whole determination is tantamount to the purely conventional choice of a cell as origin.

Posted in Boolean Functions, Computational Complexity, Differential Logic, Equational Inference, Functional Logic, Indication, Logic, Logical Graphs, Mathematics, Minimal Negation Operators, Propositional Calculus, Visualization | Tagged , , , , , , , , , , , | 2 Comments

Fourier Transforms of Boolean Functions • 1

Posted on May 30, 2013 by Jon Awbrey

Re: R.J. Lipton and K.W. Regan • Twin Primes Are Useful

The problem is concretely about Boolean functions {f} of {k} variables, and seems not to involve prime numbers at all. For any subset {S} of the coordinate [indices], the corresponding Fourier coefficient is given by:

\displaystyle \hat{f}(S) = \frac{1}{2^k} \sum_{x \in \mathbb{Z}_2^k} f(x) \chi_S(x)

where {\chi_S(x)} is {-1} if {\sum_{i \in S} x_i} is odd, and {+1} otherwise.

Note to Self. I need to play around with this concept a while.

Notation (from KLMMV • On the Fourier Spectrum of Symmetric Boolean Functions)

{f : \{0,1\}^k \to \{0,1\}}

{S \subseteq [k] = \{1, \ldots, k\}}

{\chi_S(x) = (-1)^{\sum_{i \in S} x_i}}

The order of a Fourier coefficient {\hat{f}(S)} is {|S|}.

The Fourier expansion of {f} is:

{\displaystyle f(x) = \sum_{S \subseteq [k]} \hat{f}(S) \chi_S(x)}

Added Notation (for shifting between coordinate indices and actual coordinates)

{\mathcal{X} = \{x_1, \ldots, x_k\}}

{\mathcal{S} = \mathcal{X}_S = \{x_i : i \in S\}}

{\boldsymbol{\chi}_\mathcal{S}(x) = \chi_S(x)}

{\hat{f}(\mathcal{S}) = \hat{f}(S)}

Additional Formulas

{\displaystyle \boldsymbol{\chi}_\mathcal{S}(x) = \prod_{x_i \in \mathcal{S}} (-1)^{x_i}}

Begin with a survey of concrete examples, perhaps in tabular form.

{k = 1}

…

{k = 2}

For ease of reading formulas, let {x = (x_1, x_2) = (u, v)}.

Tables

\begin{array}{|c||*{4}{c}|} \multicolumn{5}{c}{\text{Table 2.1. Values of}~ \boldsymbol{\chi}_\mathcal{S}(x)} \\[4pt] \hline \mathcal{S} & (1, 1) & (1, 0) & (0, 1) & (0, 0) \\ \hline\hline \varnothing & +1 & +1 & +1 & +1 \\ \{ u \} & -1 & -1 & +1 & +1 \\ \{ v \} & -1 & +1 & -1 & +1 \\ \{ u, v \} & +1 & -1 & -1 & +1 \\ \hline \end{array}
 

\begin{array}{|*{5}{c|}*{4}{r|}} \multicolumn{9}{c}{\text{Table 2.2. Fourier Coefficients of Boolean Functions on Two Variables}}\\[4pt] \hline \text{~~~~~~~~} & \text{~~~~~~~~} & & \text{~~~~~~~~} & \text{~~~~~~~~} & \text{~~~~~~~~~} & \text{~~~~~~~~} & \text{~~~~~~~~} & \text{~~~~~~~~~} \\ L_1 & L_2 && L_3 & L_4 & \hat{f}(\varnothing) & \hat{f}(\{u\}) & \hat{f}(\{v\}) & \hat{f}(\{u,v\}) \\ ~&~&~&~&~&~&~&~&~\\ \hline && u = & 1~1~0~0 &&&&& \\ && v = & 1~0~1~0 &&&&& \\ \hline f_{0} & f_{0000} && 0~0~0~0 & (~) & 0 & 0 & 0 & 0 \\ f_{1} & f_{0001} && 0~0~0~1 & (u)(v) & 1/4 & 1/4 & 1/4 & 1/4 \\ f_{2} & f_{0010} && 0~0~1~0 & (u)~v~ & 1/4 & 1/4 &-1/4 &-1/4 \\ f_{3} & f_{0011} && 0~0~1~1 & (u) & 1/2 & 1/2 & 0 & 0 \\ f_{4} & f_{0100} && 0~1~0~0 & ~u~(v) & 1/4 &-1/4 & 1/4 &-1/4 \\ f_{5} & f_{0101} && 0~1~0~1 & (v) & 1/2 & 0 & 1/2 & 0 \\ f_{6} & f_{0110} && 0~1~1~0 & (u,~v) & 1/2 & 0 & 0 &-1/2 \\ f_{7} & f_{0111} && 0~1~1~1 & (u~~v) & 3/4 & 1/4 & 1/4 &-1/4 \\ \hline f_{8} & f_{1000} && 1~0~0~0 & ~u~~v~ & 1/4 &-1/4 &-1/4 & 1/4 \\ f_{9} & f_{1001} && 1~0~0~1 &((u,~v))& 1/2 & 0 & 0 & 1/2 \\ f_{10}& f_{1010} && 1~0~1~0 & v & 1/2 & 0 &-1/2 & 0 \\ f_{11}& f_{1011} && 1~0~1~1 &(~u~(v))& 3/4 & 1/4 &-1/4 & 1/4 \\ f_{12}& f_{1100} && 1~1~0~0 & u & 1/2 &-1/2 & 0 & 0 \\ f_{13}& f_{1101} && 1~1~0~1 &((u)~v~)& 3/4 &-1/4 & 1/4 & 1/4 \\ f_{14}& f_{1110} && 1~1~1~0 &((u)(v))& 3/4 &-1/4 &-1/4 &-1/4 \\ f_{15}& f_{1111} && 1~1~1~1 & ((~)) & 1 & 0 & 0 & 0 \\ \hline \end{array}
 

\begin{array}{|*{5}{c|}*{4}{r|}} \multicolumn{9}{c}{\text{Table 2.3. Fourier Coefficients of Boolean Functions on Two Variables}}\\[4pt] \hline \text{~~~~~~~~} & \text{~~~~~~~~} & & \text{~~~~~~~~} & \text{~~~~~~~~} & \text{~~~~~~~~~} & \text{~~~~~~~~} & \text{~~~~~~~~} & \text{~~~~~~~~~} \\ L_1 & L_2 && L_3 & L_4 & \hat{f}(\varnothing) & \hat{f}(\{u\}) & \hat{f}(\{v\}) & \hat{f}(\{u,v\}) \\ ~&~&~&~&~&~&~&~&~\\ \hline && u = & 1~1~0~0 &&&&& \\ && v = & 1~0~1~0 &&&&& \\ \hline f_{0} & f_{0000} && 0~0~0~0 & (~) & 0 & 0 & 0 & 0 \\ \hline f_{1} & f_{0001} && 0~0~0~1 & (u)(v) & 1/4 & 1/4 & 1/4 & 1/4 \\ f_{2} & f_{0010} && 0~0~1~0 & (u)~v~ & 1/4 & 1/4 &-1/4 &-1/4 \\ f_{4} & f_{0100} && 0~1~0~0 & ~u~(v) & 1/4 &-1/4 & 1/4 &-1/4 \\ f_{8} & f_{1000} && 1~0~0~0 & ~u~~v~ & 1/4 &-1/4 &-1/4 & 1/4 \\ \hline f_{3} & f_{0011} && 0~0~1~1 & (u) & 1/2 & 1/2 & 0 & 0 \\ f_{12}& f_{1100} && 1~1~0~0 & u & 1/2 &-1/2 & 0 & 0 \\ \hline f_{6} & f_{0110} && 0~1~1~0 & (u,~v) & 1/2 & 0 & 0 &-1/2 \\ f_{9} & f_{1001} && 1~0~0~1 &((u,~v))& 1/2 & 0 & 0 & 1/2 \\ \hline f_{5} & f_{0101} && 0~1~0~1 & (v) & 1/2 & 0 & 1/2 & 0 \\ f_{10}& f_{1010} && 1~0~1~0 & v & 1/2 & 0 &-1/2 & 0 \\ \hline f_{7} & f_{0111} && 0~1~1~1 & (u~~v) & 3/4 & 1/4 & 1/4 &-1/4 \\ f_{11}& f_{1011} && 1~0~1~1 &(~u~(v))& 3/4 & 1/4 &-1/4 & 1/4 \\ f_{13}& f_{1101} && 1~1~0~1 &((u)~v~)& 3/4 &-1/4 & 1/4 & 1/4 \\ f_{14}& f_{1110} && 1~1~1~0 &((u)(v))& 3/4 &-1/4 &-1/4 &-1/4 \\ \hline f_{15}& f_{1111} && 1~1~1~1 & ((~)) & 1 & 0 & 0 & 0 \\ \hline \end{array}

To be continued …

Notes

References

21 May 2013 Twin Primes Are Useful
08 Nov 2012 The Power Of Guessing
05 Jan 2011 Fourier Complexity Of Symmetric Boolean Functions
19 Nov 2010 Is Complexity Theory On The Brink?
18 Sep 2009 Why Believe That P=NP Is Impossible?
04 Jun 2009 The Junta Problem
Posted in Boolean Functions, Computational Complexity, Fourier Transforms, Harmonic Analysis, Logic, Mathematics, Propositional Calculus | Tagged , , , , , , | 1 Comment

Strangers In Paradise

Posted on May 22, 2013 by Jon Awbrey

Re: Kilvington’s Sophismata

Comment 1

On the one hand Aristotle gives us the logic of analogy (παραδειγμα).  On the other hand he cautions us that different paradigms may have no common measure.  It seems these Immortals are always getting ahead of their time❢

Comment 2

How much drama in Plato’s Heaven when Heraclitus and Parmenides are reconciled❢

Or does it trail off to the anticlimax that Sisyphus gradually wears down the mountain?

Comment 3

Dealing with qualitative change in logical terms has long been of interest to me.  It became a hot topic in Artificial Intelligence Research during the 1980s — work by Ben Kuipers and Ken Forbus especially comes to mind.  Many of the settings where I worked at the time required me to find bridges between qualitative (logical) and quantitative (statistical) research methods.  I recall describing my efforts in that vein to one of my Master’s thesis advisers under the following rubric.

  • Approaching a Qualitative Theory of Differential Equations (QTDE)
    By Means of a Differential Theory of Qualitative Equations (DTQE)

Another slogan for the approach might as follows.

  • Exchanging a Change of Quality (CQ) for a Quality of Change (QC)

Here’s another piece I wrote in that line:

Posted in Albert Camus, Analogy, Aristotle, Differential Logic, Eleatic Stranger, Heraclitus, Incommensurability, Logic, Metabasis, Paradigmata, Paradox, Parmenides, Plato, Richard Kilvington, Sisyphus, Sophismata, Thomas Kuhn, Zeno | Tagged , , , , , , , , , , , , , , , , , | Leave a comment

⚠ It’s A Trap ⚠

Posted on May 18, 2013 by Jon Awbrey

Re: Kenneth W. Regan • Graduate Student Traps

The most common mathematical trap I run across has to do with Triadic Relation Irreducibility, as noted and treated by the polymath C.S. Peirce.

This trap lies in the mistaken belief that every 3-place (triadic or ternary) relation can be analyzed purely in terms of 2-place (dyadic or binary) relations — “purely” here meaning without resorting to any 3-place relations in the process.

A notable thinker who not only fell but led many others into this trap is none other than René Descartes, whose problematic maxim I noted in the following post.

As mathematical traps go, this one is hydra-headed.

I don’t know if it’s possible to put a prior restraint on the varieties of relational reduction that might be considered, but usually we are talking about either one of two types of reducibility.

Compositional Reducibility.  All triadic relations are irreducible under relational composition, since the composition of two dyadic relations is a dyadic relation, by the definition of relational composition.

Projective Reducibility.  Consider the projections of a triadic relation L \subseteq X \times Y \times Z on the 3 coordinate planes X \times Y, ~ X \times Z, ~ Y \times Z and ask whether these dyadic relations uniquely determine L.  If so, we say L is projectively reducible, otherwise it is projectively irreducible.

Et Sic Deinceps …

  • More Discussion of Relation Reduction • OEIS Wiki • PlanetMath
  • Previous Posts on Triadic Relation Irreducibility • (1) • (2) • (3)
Posted in C.S. Peirce, Category Theory, Descartes, Error, Fallibility, Logic, Logic of Relatives, Mathematical Traps, Mathematics, Peirce, Pragmatism, Reductionism, Relation Theory, Semiotics, Sign Relations, Triadic Relations | Tagged , , , , , , , , , , , , , , , | 5 Comments

Triadic Relation Irreducibility • 3

Posted on May 16, 2013 by Jon Awbrey

References

Related Readings

Posted in C.S. Peirce, Category Theory, Inquiry, Logic, Logic of Relatives, Mathematics, Peirce, Pragmatism, Relation Theory, Semiosis, Semiotics, Sign Relational Manifolds, Sign Relations, Teridentity, Thirdness, Triadic Relations | Tagged , , , , , , , , , , , , , , , | 1 Comment

grabitational singularity

Posted on May 16, 2013 by Jon Awbrey
the trouble with a bubble
on a pyramid top
is the point
when it
pop
⚠
⚠⚠
⚠⚠⚠
⚠⚠⚠⚠
⚠⚠⚠⚠⚠
⚠⚠⚠⚠⚠⚠
Posted in Grabitational Singularity, Singularity, Verse | Tagged , , | 2 Comments

What part do arguments from authority play in mathematical reasoning?

Posted on May 14, 2013 by Jon Awbrey

In forming your answer you may choose to address any or all of the following aspects of the question:

Descriptive
What part do arguments from authority actually play in mathematical reasoning?
Normative
What part do arguments from authority ideally play in mathematical reasoning?
Regulative
What if any discrepancies exist between the actual and the ideal?
What if anything should be done about the discrepancies that exist?

Recycled from a question I asked on MathOverFlow.

Posted in Artificial Intelligence, Authority, Control, Control Theory, Cybernetics, Fixation of Belief, History of Mathematics, History of Science, Information, Information Theory, Inquiry, Inquiry Driven Systems, Intelligent Systems, Intuition, Logic, Logic of Science, Mathematical Intuition, Mathematical Reasoning, Operations Research, Optimal Control, Optimization, Philosophy of Science, Scientific Method | Tagged , , , , , , , , , , , , , , , , , , , , , , | 2 Comments