Mathematical Method • Discussion 6

Re: Ontolog ForumAlex Shkotin

Dear Alex,

Thanks for the very apt segue from Jon Barwise —

Modern mathematics might be described as the science of abstract objects, be they real numbers, functions, surfaces, algebraic structures or whatever.  Mathematical logic adds a new dimension to this science by paying attention to the language used in mathematics, to the ways abstract objects are defined, and to the laws of logic which govern us as we reason about these objects.  The logician undertakes this study with the hope of understanding the phenomena of mathematical experience and eventually contributing to mathematics, both in terms of important results that arise out of the subject itself (Gödel’s Second Incompleteness Theorem is the most famous example) and in terms of applications to other branches of mathematics.  (Barwise p. 6)

When it comes to mathematics as the science of abstract objects I have my personal favorite classes among its abstract gardens and zoos.  One order of particular interest in the great chain of abstract being descends from the family of mathematical relations to the genus of triadic relations to the species of triadic sign relations.

By a curious turn, but no real surprise when we stop to think about it, sign relations, with their object, sign, and interpretant sign domains, come into being whenever we reflect on the systems of signs we use to describe any universe of objects, abstract or otherwise, and thus they are just the tickets we need to enter that “new dimension” of mathematical logic.

References

  • Barwise, J. (1977), “An Introduction to First-Order Logic”, pp. 5–46 in Barwise, J. (1977, ed.), Handbook of Mathematical Logic, Elsevier (North Holland), Amsterdam.
  • Eisele, C. (1982), “Mathematical Methodology in the Thought of Charles S. Peirce”, Historia Mathematica 9, pp. 333–341.  OnlinePDF.

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Animated Logical Graphs • 34

Re: Ontolog ForumJohn Sowa
Re: Peirce ListJohn Sowa

Dear John,

I can’t imagine why anyone would bother with Peirce’s logic if it’s just Frege and Russell in a different syntax, which has been the opinion I usually get from FOL fans.  But the fact is Peirce’s 1870 “Logic of Relatives” is already far in advance of anything we’d see again for a century, in principle in most places, in practice in many others, chock full of revolutionary ideas, not all of which he developed fully in subsequent work.  Although I studied the 1870 Logic from early on I did not realize how far ahead of its time it was until I began reading approaches to logic from category-theoretic and computation-theoretic angles in the 1970s and 1980s.  An indication of Peirce’s innovations can be found in the series of selections and commentary I started on the 1870 Logic of Relatives.

Here’s the work in progress so far on the OEIS Wiki.

Here’s the overview for a parallel series of blog posts.

Resources

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Mathematical Method • Discussion 5

Re: Ontolog ForumPaul Tyson

Dear Paul,

“How We Think” is a topic for the descriptive science of psychology, and its ways are legion beyond definitive or exhaustive description.

“How We Ought To Think” if we wish to succeed at specified purposes is a topic for the normative science of logic, lumping together for the moment the evolving varieties of informal, formal, mathematical, and technologically augmented methods.

They’re all good questions and I see no reason not to pursue them all, aside from the limitations of our brief lives, but we have to keep the spectrum of different aims sorted.

John Dewey wrote the book How We Think in 1910.  Peirce had earlier summed up his “non-psychological conception of logic” in the pithy motto “Logic has nothing to do with how we think” and this led some scholars to suspect Dewey’s title was aimed as a poke in Peirce’s ribs.  But the book itself is a How-To guide devoted to improving our capacity for learning and reasoning, what we’d call today instruction in critical thinking.

All that is prologue to Vannevar Bush’s 1945 article, “As We May Think”, projecting the ways technology may amplify our capacity for inquiry going forward into the future.  I think this is where we came in …

Reference

  • Eisele, C. (1982), “Mathematical Methodology in the Thought of Charles S. Peirce”, Historia Mathematica 9, pp. 333–341.  OnlinePDF.

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Mathematical Method • Discussion 4

Re: Peirce ListHelmut Raulien

Dear Helmut,

It’s one of the occupational hazards of the classifying mind that one can start out consciously characterizing aspects of real situations and end up unwittingly thinking one’s gotten everything under the sun sorted into mutually exclusive bins.

Once the idols of compartmentality and the illusions of autonomous abstraction get their hold on our minds it is almost impossible to reconstitute or synthesize what we’ve torn asunder, if only in our own minds.  The ounce of prevention here is always keeping in mind that from which all abstractions are abstracted, living experience.

Reference

  • Eisele, C. (1982), “Mathematical Methodology in the Thought of Charles S. Peirce”, Historia Mathematica 9, pp. 333–341.  OnlinePDF.

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Mathematical Method • Discussion 3

Re: Peirce ListGary Fuhrman

Dear Gary,

Auke van Breemen wrote:

AvB:
It seems to come down to:  never consider the textual production of a scientist only in itself, but also look at the reality the text tries to explain.

I took this as an admirably succinct statement of the difference between (1) scriptural hermeneutics — I’d call it “corpus hermeneutics” except for the risk of confusion with Corpus Hermeneticum — and (2) scientific interpretation, that is, any development of interpretant texts in relation to an independent object domain with the aim of forming true descriptions or gaining knowledge of that domain.

What I wrote in response to Auke was this:

JA:
Exactly!
We interpret texts
in relation to
the object in view.

All I did there was mention the three roles in a sign relation.  We take in texts or whole bodies of work as signs of an object domain and we form interpretive texts as signs of the same domain.  For my part, I interpret Peirce’s work as signs of an object world, one with respect to which other writers, artists, signifactors of all sorts have generated signs worthy of our interpretation.

I wouldn’t take the words “in view” too literally.  I just as easily could have said “at hand” or “in mind” but I went with “object in view” on account of the fondness one of my old teachers had for Dewey’s signature “end in view”.  As far as indicial signs are concerned, we’re all embroiled in concernful situations all the time, making our selves the initial signs of those pragmata, from which we derive all the remainder.

Peirce’s distinction between theorematic and corollarial reasoning has come up before.  From what I recall of previous discussions, we should not read the word “theorematic” in too reductive or purely deductive a sense.  Years ago it was something of a commonplace, even outside Peircean circles, to call attention to the etymology of “theorem” as having an observational, even “visionary” sense, cognate with “theatre”, and some would even point to the sacred origins of theatre, though maybe that’s a bridge too far …

As far as the iconic aspects of mathematics go, or even our knowledge representations in general, they are nice when we can get them, but I’m careful not to stress them too far — it’s too easy to “fall victim to a picture”, in Wittgenstein’s phrase, or succumb to the short-sightedness of Russell’s isomorphism theory of knowledge.  Icons are specializations of symbols and thus fall short of symbols’ full potential.  There is more to science than serving as a mirror of nature.

Reference

  • Eisele, C. (1982), “Mathematical Methodology in the Thought of Charles S. Peirce”, Historia Mathematica 9, pp. 333–341.  OnlinePDF.

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Mathematical Method • Discussion 2

Re: Peirce ListAuke van Breemen

AvB:
It seems to come down to:  never consider the textual production of a scientist only in itself, but also look at the reality the text tries to explain.

Dear Auke,

Exactly!

We interpret texts
in relation to
the object in view.

Reference

  • Eisele, C. (1982), “Mathematical Methodology in the Thought of Charles S. Peirce”, Historia Mathematica 9, pp. 333–341.  OnlinePDF.

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Mathematical Method • Discussion 1

Re: Peirce ListJohn Sowa

Dear John,

Thanks for the notice of Carolyn Eisele’s article — it’s always worth reading what she has to say.  We’ve had discussions of Peirce’s distinction between theorematic and corollarial reasoning before and I know there’s a respectable amount of literature out there about it.  The subject has curiously enough come up just recently in discussions on Facebook and Academia.edu, mostly on account of points brought up by John Corcoran.  It’s also related to a number of discussions I’ve had over the years about the difference between “insight” proofs and “routine” proofs, partly in connection with theorem proving apps and Peirce’s logical graphs.  Usually these discussions take off into the stratosphere of high-sounding blue-skying about Gödel incompleteness and all that — but I want to keep my focus on more nuts and bolts issues at the moment and I’ll try to avoid going off on those planes.

Reference

  • Eisele, C. (1982), “Mathematical Methodology in the Thought of Charles S. Peirce”, Historia Mathematica 9, pp. 333–341.  OnlinePDF.

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Differential Analytic Turing Automata • Discussion 2

Re: Scott AaronsonThe Busy Beaver Frontier

Dear Scott,

This discussion inspired me to go back and look at some of the work I did in the late 80s when I was trying to understand Cook’s Theorem.  One of the programs I wrote to explore the integration of sequential learning and propositional reasoning had a propositional calculus module based on an extension of C.S. Peirce’s logical graphs, so I used that syntax to write out the clauses for finite approximations to Turing machines, taking the 4-state parity machine from Herbert S. Wilf’s Algorithms and Complexity as an object example.  It was 1989 and all I had was a 289 PC with 600K heap, but I did manage to emulate a parity machine capable of 1 bit of computation.  Here’s a link to an exposition of that.

It may be quicker to skip to Part 2 and refer to Part 1 only as needed.

I’ll work up the case of a 2-state Busy Beaver when I get a chance.
I always learned a lot just from looking at the propositional form.

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Riffs and Rotes • 5

Rote 123456789

Re: Scott AaronsonThe Busy Beaver Frontier

All my favorite integer sequences, some very fast growing, spring from the “lambda point” where graph theory, logic, and number theory meet.  My fascination with them goes back to a time when I was playing around with Gödel numbers of graph-theoretic structures and thinking about computational complexity.  I’m busy as a beaver with other business at the moment so I’ll leave it now with just a few links to chew on till whenever.

Resources

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Differential Logic, Dynamic Systems, Tangent Functors • Discussion 9

Re: FB | Systems SciencesKenneth Lloyd

Dear Kenneth,

Mulling over recent discussions put me in a pensive frame of mind and my thoughts led me back to my first encounter with category theory.  I came across the term while reading and I didn’t fully understand it.  But I distinctly remember a short time later catching up with my math TA — it was on the path by the tennis courts behind Spartan Stadium — and asking him about it.

The instruction I received that day was roughly along the following lines.

“Actually . . . we’re already doing a little category theory, without quite calling it that.  Think about the different types of spaces we’ve been discussing in class, the real line \mathbb{R}, the various dimensions of real-value spaces, \mathbb{R}^n, \mathbb{R}^m, and so on, along with the various types of mappings between those spaces.  There are mappings from the real line \mathbb{R} into an n-dimensional space \mathbb{R}^n — we think of those as curves, paths, or trajectories.  There are mappings from the plane \mathbb{R}^2 to values in \mathbb{R} — we picture those as potential surfaces over the plane.  More generally, there are mappings from an n-dimensional space \mathbb{R}^n to values in \mathbb{R} — we think of those as scalar fields over \mathbb{R}^n — say, the temperature at each point of an n-dimensional volume.  There are mappings from \mathbb{R}^n to \mathbb{R}^n and mappings from \mathbb{R}^n to \mathbb{R}^m where n and m are different, all of which we call transformations or vector fields, depending on the use we have in mind.”

All that was pretty familiar to me, though I had to admire the panoramic sweep of his survey, so my mind’s eye naturally supplied all the arrows for the maps he rolled out.  A curve \gamma through an n-dimensional space would be typed as a function \gamma : \mathbb{R} \to \mathbb{R}^n, where the functional domain \mathbb{R} would ordinarily be regarded as a time dimension.  A mapping \alpha from the plane to a real value would be typed as a function \alpha : \mathbb{R}^2 \to \mathbb{R}, where we might be thinking of \alpha(x, y) as the altitude of a topographic map above each point (x, y) of the plane.  A scalar field \beta defined on an n-dimensional space would be typed as a function \beta : \mathbb{R}^n \to \mathbb{R}, where \beta(x_1, \ldots, x_n) is something like the pressure, the temperature, or the value of some other dependent variable at each point (x_1, \ldots, x_n) of the n-dimensional volume.  And rounding out the story, if only the basement and ground floor of a towering abstraction still under construction, we come to the general case of a mapping f from an n-dimensional space to an m-dimensional space, typed as a function f : \mathbb{R}^n \to \mathbb{R}^m.

To be continued …

Resources

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