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], [[email protected]]
11/30/2012
"Forever is a long time, especially near the end." -Woody Allen
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