So Many Modes Of Mathematical Thought

So many modes of mathematical thought,
So many are learned, so few are taught.
There are streams that flow beneath the sea,
There are waves that crash upon the strand,
Lateral thoughts that spread and meander —
Who knows what springs run under the sand?

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{ Information = Comprehension × Extension } • Comment 4

There are still many things that puzzle me about Peirce’s account at this point.  I indicated a few of them by means of question marks at several places in the last two Figures.  There is nothing for it but returning to the text and trying once more to follow the reasoning.

Let’s go back to Peirce’s example of abductive inference and try to get a clearer picture of why he connects it with conjunctive terms and iconic signs.

Figure 1 shows the implication ordering of logical terms in the form of a lattice diagram.

Figure 1. Conjunctive Term z, Taken as Predicate

Figure 1. Conjunctive Term z, Taken as Predicate

The relationship between conjunctive terms and iconic signs may be understood as follows.  If there is anything that has all the properties described by the conjunctive term — spherical bright fragrant juicy tropical fruit — then sign users may use that thing as an icon of an orange, precisely by virtue of the fact that it shares those properties with an orange.  But the only natural examples of things that have all those properties are oranges themselves, so the only thing that can serve as a natural icon of an orange by virtue of those very properties is that orange itself or another orange.

Reference

  • Peirce, C.S. (1866), “The Logic of Science, or, Induction and Hypothesis”, Lowell Lectures of 1866, pp. 357–504 in Writings of Charles S. Peirce : A Chronological Edition, Volume 1, 1857–1866, Peirce Edition Project, Indiana University Press, Bloomington, IN, 1982.

Resources

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Iconoclast

Things thrown out from a center
Cooling signs of a central fire
Expiring cinders of life within
Sent from a star to start again
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{ Information = Comprehension × Extension } • Comment 3

Peirce identifies inference with a process he describes as symbolization.  Let us consider what that might imply.

I am going, next, to show that inference is symbolization and that the puzzle of the validity of scientific inference lies merely in this superfluous comprehension and is therefore entirely removed by a consideration of the laws of information(467).

Even if it were only a weaker analogy between inference and symbolization, a principle of logical continuity — what in physics is called a correspondence principle — would suggest parallels between steps of reasoning in the neighborhood of exact inferences and signs in the vicinity of genuine symbols.  This would lead us to expect a correspondence between degrees of inference and degrees of symbolization that extends from exact to approximate or non-demonstrative inferences and from genuine to approximate or degenerate symbols.

For this purpose, I must call your attention to the differences there are in the manner in which different representations stand for their objects.

In the first place there are likenesses or copies — such as statues, pictures, emblems, hieroglyphics, and the like.  Such representations stand for their objects only so far as they have an actual resemblance to them — that is agree with them in some characters.  The peculiarity of such representations is that they do not determine their objects — they stand for anything more or less;  for they stand for whatever they resemble and they resemble everything more or less.

The second kind of representations are such as are set up by a convention of men or a decree of God.  Such are tallies, proper names, &c.  The peculiarity of these conventional signs is that they represent no character of their objects.

Likenesses denote nothing in particular;  conventional signs connote nothing in particular.

The third and last kind of representations are symbols or general representations.  They connote attributes and so connote them as to determine what they denote.  To this class belong all words and all conceptions.  Most combinations of words are also symbols.  A proposition, an argument, even a whole book may be, and should be, a single symbol.  (467–468).

In addition to Aristotle, the influence of Kant on Peirce is very strongly marked in these earliest expositions.  The invocations of “conceptions of the understanding”, the “use of concepts” and thus of symbols in reducing the manifold of extension, and the not so subtle hint of the synthetic à priori in Peirce’s discussion, not only of natural kinds but also of the kinds of signs that lead up to genuine symbols, can all be recognized as pervasive Kantian themes.

In order to draw out these themes and see how Peirce was led to develop their leading ideas, let us bring together our previous Figures, abstracting from their concrete details, and see if we can figure out what is going on here.

Figure 3 shows an abductive step of inquiry, as taken on the cue of an iconic sign.

Figure 3. Conjunctive Predicate z, Abduction of Case x ⇒ y

Figure 3. Conjunctive Predicate z, Abduction of Case xy

Figure 4 shows an inductive step of inquiry, as taken on the cue of an indicial sign.

Figure 4. Disjunctive Subject u, Induction of Rule v ⇒ w

Figure 4. Disjunctive Subject u, Induction of Rule vw

To be continued …

Reference

  • Peirce, C.S. (1866), “The Logic of Science, or, Induction and Hypothesis”, Lowell Lectures of 1866, pp. 357–504 in Writings of Charles S. Peirce : A Chronological Edition, Volume 1, 1857–1866, Peirce Edition Project, Indiana University Press, Bloomington, IN, 1982.

Resources

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{ Information = Comprehension × Extension } • Comment 2

Let’s examine Peirce’s second example of a disjunctive term — neat, swine, sheep, deer — within the style of lattice framework that we used before.

Hence if we find out that neat are herbivorous, swine are herbivorous, sheep are herbivorous, and deer are herbivorous;  we may be sure that there is some class of animals which covers all these, all the members of which are herbivorous.  (468–469).

Accordingly, if we are engaged in symbolizing and we come to such a proposition as “Neat, swine, sheep, and deer are herbivorous”, we know firstly that the disjunctive term may be replaced by a true symbol.  But suppose we know of no symbol for neat, swine, sheep, and deer except cloven-hoofed animals.  (469).

This is apparently a stock example of inductive reasoning that Peirce borrows from traditional discussions, so let us pass over the circumstance that modern taxonomies may classify swine as omnivores.

In view of the analogical symmetries that the disjunctive term shares with the conjunctive case, I think that we can run through this example in fairly short order.  We have an aggregation over four terms:

\begin{array}{lll}  s_1 & = & \mathrm{neat}  \\  s_2 & = & \mathrm{swine}  \\  s_3 & = & \mathrm{sheep}  \\  s_4 & = & \mathrm{deer}  \end{array}

Suppose that u is the logical disjunction of the above four terms:

\begin{array}{lll}  u & = &  \texttt{((} s_1 \texttt{)(} s_2 \texttt{)(} s_3 \texttt{)(} s_4 \texttt{))}  \end{array}

Figure 2 depicts the situation that we have before us.

Figure 2. Disjunctive Term u, Taken as Subject

Figure 2. Disjunctive Term u, Taken as Subject

Here we have a situation that is dual to the structure of the conjunctive example.  There is a gap between the logical disjunction u, in lattice terminology, the least upper bound (lub) of the disjoined terms, u = \mathrm{lub} \{ s_1, s_2, s_3, s_4 \}, and what we might regard as the natural disjunction or natural lub of these terms, namely, v, cloven-hoofed.

Once again, the sheer implausibility of imagining that the disjunctive term u would ever be embedded exactly as such in a lattice of natural kinds leads to the evident naturalness of the induction to v \Rightarrow w, namely, the rule that cloven-hoofed animals are herbivorous.

Reference

  • Peirce, C.S. (1866), “The Logic of Science, or, Induction and Hypothesis”, Lowell Lectures of 1866, pp. 357–504 in Writings of Charles S. Peirce : A Chronological Edition, Volume 1, 1857–1866, Peirce Edition Project, Indiana University Press, Bloomington, IN, 1982.

Resources

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Pragmatic Cosmos • 1

Re: Michael HarrisNot About Fibonacci

I have often reflected on the interminglings of the main three normative sciences.  In one of my earliest meditations I saw Beauty, Goodness, and Truth as the intersecting circles of a Venn diagram, with the summum bonum the central cell.  As far as our ability to approach our object from our origin without, perfect knowledge of the Good would require us to know all the consequences of our contemplated actions while perfect knowledge of the True would require us to know all the axiom sets that never beget a contradiction.  As far as I could tell, and as far as I could see deciding with the empirical tests and theorem provers I could morally and mathematically envision devising, these two tasks exceed the talents of mortal humans and all their technological extensions.

But when it comes to Beauty, our form of being appears to have an inborn sense to guide us on our quest to the highest good.  That way through beauty to our ultimate goal I called the human-hearted path.

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{ Information = Comprehension × Extension } • Comment 1

At this point in his inventory of scientific reasoning, Peirce is relating the nature of inference, information, and inquiry to the character of the signs mediating the process in question, a process he is presently describing as symbolization.

In the interests clarity let’s draw from Peirce’s account a couple of quick sketches, designed to show how the examples he gives of conjunctive terms and disjunctive terms might look if they were cast within a lattice-theoretic frame.

Let’s examine Peirce’s example of a conjunctive term — spherical, bright, fragrant, juicy, tropical fruit — within a lattice framework.  We have these six terms:

\begin{array}{lll}  t_1 & = & \mathrm{spherical}  \\  t_2 & = & \mathrm{bright}  \\  t_3 & = & \mathrm{fragrant}  \\  t_4 & = & \mathrm{juicy}  \\  t_5 & = & \mathrm{tropical}  \\  t_6 & = & \mathrm{fruit}  \end{array}

Suppose that z is the logical conjunction of the above six terms:

\begin{array}{lll}  z & = & t_1 \cdot t_2 \cdot t_3 \cdot t_4 \cdot t_5 \cdot t_6  \end{array}

What on earth could Peirce mean by saying that such a term is not a true symbol or that it is of no use whatever?

In particular, consider the following statement:

If it occurs in the predicate and something is said to be a spherical bright fragrant juicy tropical fruit, since there is nothing which is all this which is not an orange, we may say that this is an orange at once.

In other words, if something x is said to be z, then we may guess fairly surely that x is really an orange, in other words, that x has all of the additional features that would be summed up quite succinctly in the much more constrained term y, where y means an orange.

Figure 1 shows the implication ordering of logical terms in the form of a lattice diagram.

Figure 1. Conjunctive Term z, Taken as Predicate

Figure 1. Conjunctive Term z, Taken as Predicate

What Peirce is saying about z not being a genuinely useful symbol can be explained in terms of the gap between the logical conjunction z, in lattice terms, the greatest lower bound (glb) of the conjoined terms, z = \mathrm{glb} \{ t_1, t_2, t_3, t_4, t_5, t_6 \}, and what we might regard as the natural conjunction or natural glb of these terms, namely, y, an orange.  That is to say, there is an extra measure of constraint that goes into forming the natural kinds lattice from the free lattice that logic and set theory would otherwise impose.  The local manifestations of this global information are meted out over the structure of the natural lattice by just such abductive gaps as the one between z and y.

Reference

  • Peirce, C.S. (1866), “The Logic of Science, or, Induction and Hypothesis”, Lowell Lectures of 1866, pp. 357–504 in Writings of Charles S. Peirce : A Chronological Edition, Volume 1, 1857–1866, Peirce Edition Project, Indiana University Press, Bloomington, IN, 1982.

Resources

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