On 30 Nov 2012, at 13:33, Roger Clough wrote:

Hi everything-list

Perhaps Penrose's emphasis of intution, and the noncomputability thereof,
is that intuition is is closely related to meaning, to semantics.
I think that a necessary feature of any machine to emulate human thought is to be able to understand meaning, the science of which is called semantics.

My limited understanding of current semantics is that meaning is represented
syntactically (sentence diagramming).


In computer science, semantics is often represented by infinite sets, which describe in extenso the meaning of some syntactical construct ("number").

For example the semantics of "x + 1", might be given by the infinite set {(0,1), (1,2), (2, 3), ...}. The semantic of "factorial" can be given by the set {(0,1), (1,1), (2, 2), (3,6), (4,24), ...}
Or the semantics of "number" can be {0, 1, 2, 3, ...}.

You can see computation as being an association from syntax to semantics.
Like from "x+1" to "{(0,1), (1,2), (2, 3), ...}".

You can see learning as being the inverse association, where you give the "reality" (semantics) to a machine, like a sequence of finite portions of "{(0,1), (1,2), (2, 3), ...}", and the learning machine will build a syntactical construct (program) like "x + 1". A "learning machine" is given (portion of a) reality, or meaning, semantics, and the machine proceeds to the synthesis of program (finite syntactical construct) accounting for the data given.

Roughly speaking, for those who reminds what the phi_i are: computation, abstractly is in the i -> phi_i, or (i, x) -> phi_i(x) type of arrows. Learning is in the phi_i -> i direction.

Likewise the semantics of a theory (syntactical) is given by a mathematical structure (usually infinite), called "model" which satisfies (in some sense) the axioms and theorems of the theory. Usually theories have many models.

With this, it should be clear that a semantics is not typically a finite syntactical object, but an infinite structure, plausibly an object "in the mind".

Bruno



Peirce seems to have abandoned arithmetic
reasoning (I may have overstated that) in favor of a new, semiotic or graphics-based reasoning (semiotics), which is as vital to understand as it is difficult.
One can even conceive of an iconic- or  graphics-based
computer.

See below for an alternate discussion of this topic:
http://jeannicod.ccsd.cnrs.fr/docs/00/05/33/40/HTML/index.html

"The central idea developed in Peirce's account of necessary deductive reasoning is that it proceeds by constructions of diagrams, which are a species of icons. This is as true for logical reasoning as it is for mathematical reasoning, which is in fact the paradigm of deduction. Such a conception has important bearings not only for a conception of iconic logic, but for certain peculiarities that are attached to mathematical deduction as well.

 3.1 The main characters of the icon

An icon is a sign which 'refers to the Object that it denotes merely by virtue of characters of its own, and which it possesses just the same, whether any such Object actually exists or not' (2. 247). "

<SNIP>

"The first things I found out were that all mathematical reasoning is diagrammatic and that all necessary reasoning is mathematical reasoning, no matter how simple it may be. By diagrammatic reasoning, I mean reasoning which constructs a diagram according to a precept expressed in general terms, performs experiments upon this diagram, notes their results, and expresses them in general terms. This was a discovery of no little importance, showing as it does, that all knowledge comes from observation. "
[Roger Clough], [rclo...@verizon.net]
11/30/2012
"Forever is a long time, especially near the end." -Woody Allen


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