Re: Science based on logic vs. Mathematics is not necessary
- --- In email@example.com, "Peter D Jones"
> Nothing has got very far in AI. I was talking about logic, anyway.
Logic, from Classical Greek ëüãïò (logos), originally meaning the
word, or what is spoken, (but coming to mean thought or reason) is
most often said to be the study of criteria for the evaluation of
arguments, although the exact definition of logic is a matter of
controversy among philosophers. However the subject is grounded, the
task of the logician is the same: to advance an account of valid and
fallacious inference to allow one to distinguish logical from flawed
I believe the original argument dealt with the inseparability of
grounding logic to physics. I still don't see how we could
possible "advance an account of valid and fallacious inference to
allow one to distinguish logical from flawed arguments" without
reference to physics and 100% of our knowledge of physics comes in
via the sensory interface.
IMO, the humble binary switch works very well for the purpose of
Q1. Do we need to contemplate tri-state or > switches to represent
our logic wrt AI?
(on, off, (on & off))?
and what would it possible mean for a switch to be both (on & off)?
How could we possible "advance an account of valid and fallacious
inference to allow one to distinguish logical from flawed arguments"
if a switch could be both?
Q2. Does the brain have the neural equivalent of these?
I think not. What would be the reason which couldn't be done by just
using more bits? The only difference gained would be time not ability
John J. Gagne
- On Thursday, August 03, 2006 6:44 AM, Eray Ozkural wrote in ai-
EO>> ... logic is simply a crude formalization of macro-level
causality, and locality that we observe in the world. There is
nothing else to it. But it's a cool abstraction nonetheless. <<EO
Your remark, and the various philosophical points raised on the
subject so far, prompt me to share some views on "What is logic, and
what is mathematics?" from a mathematicians point of view.
Without attempting to address the issue in its broader dimensions, we
may, without any loss of generality, consider mathematics simply as a
set of precise, symbolic, languages.
Any language of such a set, say Peano Arithmetic PA (or Russell and
Whitehead's Principia Mathematica, PM, or ZFC), is intended to
express - in a finite, unambiguous, and communicable manner -
relations between selected concepts that are external to the language
PA (or to PM, or to ZFC).
Each such language is, thus, in a Platonic sense, two-valued, if we
assume that the selected relations either hold or do not hold
externally (relative to the language).
Further, a selected, finite, number of primitive formal assertions
about a finite set of selected primitive relations of, say, a
language L are defined as axiomatically L-provable.
All other assertions about relations that can be effectively defined
in terms of the primitive relations are termed as L-provable if, and
only if, there is a finite sequence of assertions of L, each of which
is either a primitive assertion, or which can effectively be
determined in a finite number of steps as an immediate consequence of
any two assertions preceding it in the sequence by a finite set of
Logical rules of consequence.
Thus, each such language has an associated Logic that is specific to
it, and consists of the selection of the rules by which we
assign 'truth' and 'provability' values to the assertions of the
language under an intended interpretation.
So, changing the Logic of a language would yield an alternative mode
of expressing that which we intend to express and communicate by the
language. In other words, there is nothing sacrosant about the choice
of the Logical rules we opt to associate with a language.
An effective interpretation of a language L into another language,
say PM (or PA, or ZFC, or English, etc.), is essentially the
specification of an effective method by which any assertion of L is
translated unambiguously into a unique assertion of PM (or PA, or
ZFC, or English, etc.).
Clearly, if a relation is provable in L, then our Logic should be
chosen so that it should be effectively decidable as 'true' in any
interpretation of L that shares a common Logic - since a finite proof
sequence of L would, prima facie, translate as a finite proof
sequence in the interpretation.
The question arises: Is the converse true? In other words, if an
assertion is decidable as 'true' in an interpretation M of L, then
does such decidability translate into an effective method of
decidability in L?
Obviously, such a question can only be addressed unambiguously if
there is an effective method for determining whether an assertion is
decidable as 'true' in M. If there is no such effective method, then
we are faced with the following thesis:
Thesis: If there is no effective method for the unambiguous
decidability of the 'truth / falsity' of the assertions of a
mathematical language L under an interpretation M, then L can only be
considered a mathematical language of intuitive expression, but not a
mathematical language of effective, and unambiguous, communication.
What this means is that, in the absence of an effective method of
decidability of 'truth / falsity' in a mathematical language M that
is a model of, say, a mathematical language such as Peano Arithmetic,
any correlation of soundness between a PA-assertion and an assertion
in M is essentially arguable; so it is meaningless to ask whether, in
general, an assertion of PA is decidable under interpretation in M or
not (the question of whether the assertion is decidable in PA or not
is, then, an issue of secondary consequence).
The philosophical question, of critical interest to AI, then arises:
Can the Mind be considered as the standard - albeit intuitive - model
M of PA, with the brain as its domain, and can PA be considered a
formalisation of the Mind?
The first part of the question can, of course, be easily answered
affirmatively, since every proof sequence of PA can, intuitively, be
recognised by, and thus be seen as an effective method of
decidability in, M.
The second part, on the other hand, can be answered negatively just
as easily, if we accept that the Mind can be seen to experience
hallucinations (reflecting synaptic patterns that, by definition, we
assume correspond to some aspects of the physical state of the brain)
that cannot conceivably be effectively verified in any formal
However, if we adopt appropriate Theses, based on those proposed by
Church and Turing, then every instantiationally effective method, and
every algorithmically effective method, of decidability, or
computability, in the Mind corresponds to some effective method of
decidability, or computability, in the language that is used to
describe the Mind.
It follows that, if any two interpretations of such a language are
isomorphic, then the language can, in a sense, be taken as adequately
formalising that part of the activity of the brain that lends itself
to mathematical representation.
The contensious issues arise from, and reflect, the failure of
classical theory to distinguish between the representation, within a
language, of those abstract concepts (mental constructs of an
individual mind) that are individually significant within an
individual gestalt, and those of these abstract concepts that are,
further, communicable in an unambiguous and effective manner, and
which may, therefore, be termed as uniformly significant within a
collective of gestalts.
Prima facie, most of the challenges faced in unambiguously, and
effectively, communicating mathematical concepts, seem to involve
conclusions arrived at from debatable Realist premises.
Premises that, when introduced authoritatively into formal
mathematical reasoning, and into scientific discourse, permit us to
logically validate our subjective intuitive perceptions as being
reflective of some absolute Truth that - contrary to our intuitive
experience - must be of universal significance in a Utopian, Platonic
We could even go further, and consider whether the relation of a
particular language (whether mathematical or not), to that which the
language seeks to express, should be the preserve of the philosophers
of the language, and of those who study the nature of the mind and of
consciousness - not of the logicians or mathematicians who simply
provide the tools for such study.
In other words, the selection of the alphabet, selection of rules for
defining well-formed words, phrases, and declarative atomic and
compound sentences, selection of primitive objects (constants) and
primitive truths (axioms), selection of the Logical rules of
deduction for assigning truth values to non-axioms, etc., in a
language should be consequent to the adequate resolution of
philosophical issues that consider the question of what we visualise
as, and how we visualise, the properties and relations between
elements of an abstract, Platonically conceived, ontology - in our
individual gestalts - that a logician or mathematician is being asked
to represent within a formal language.
Whether the above properties and relations between elements of a
chosen abstract, Platonically conceived, ontology have limited,
individual significance, or have a wider, common significance, should
also be the rightful preserve of philosophy in general, along with
questions concerning the nature of such elements, their properties,
and their relationships.
However, once we have defined a language, we should not be at liberty
to use the rules of the language to deduce the existence of new,
unique, elements of the ontology (as opposed to elements of the
language) that are not explicitly specified by the definitions of the
Such deduction would amount to a creation by extraneous definition
that would contradict the premise that the language has already been
specifically defined to express selected pre-conceived, Platonic,
In other words, the aim of mathematics should not be to introduce
such extraneous definitions into a language, but to study the
consequences of a mathematical language that, beyond the
considerations involved in its definition, are independent of the
Platonic concepts that the language was designed to express.
By this yardstick, standard interpretations of Cantor's diagonal
argument , or of his power set theorem (as also of some of the
transfinite elements of set theory) seem to extravagantly violate
Thus, even 70 years after Gödel's seminal 1931 paper, where he
highlighted that there is an essential asymmetry between classically
true, and classically provable, arithmetical propositions, the roots
of such distinction continue, apparently, to elude effective
Now, prima facie, for any language to be termed as a language of
unambiguous, and effective, communication, the truth of its
propositions, under any interpretation, ought to be unambiguous, and
effectively verifiable independently of the domain of the
We are, thus, faced with the questions: Is mathematical truth
necessarily unverifiable effectively, and are there theoretically
absolute limitations on unambiguous, and effective, communication?
Now, following Gödel's interpretation of his own reasoning, and
conclusions, in his 1931 paper, standard interpretations of classical
mathematical theory seem to implicitly imply that, even in a
mathematical language as basic as formal Peano Arithmetic, the most
fundamental of our intuitive mathematical concepts cannot be
expressed, and communicated in a complete, unambiguous, and effective
The extent to which such interpretations are accepted as setting
implicit, and, apparently absolute, limitations on unambiguous, and
effective, communication, depends, no doubt, on individual
psychological, and philosophical, predilections.
Thus, although professional mathematicians and mathematical logicians
might succeed in placing such limitations in comfortably abstract,
albeit counter-intuitive, perspective, such comfort may be denied to
other disciplines that primarily need to express their fundamental
concepts and observations in an unambiguous, and effectively
For instance, seekers of extra-terrestrial intelligence, or those
striving to replicate human intelligence in controllable artifacts,
could find such acceptance disturbingly constricting, and
This is an issue of increasing interest to scientific disciplines
that look to mathematics for providing a language of reliable, and
verifiable, external expression and communication. Ignoring the issue
seems to obscure the raison d'etre of a mathematical language - to
communicate unambiguously, and effectively.
Thus, if mathematics is to serve both as a language that expresses
all possible abstract (including Platonic) mathematical concepts in
formal languages, and also to serve as a language of precise
expression and unambiguous communication, then, particularly in the
latter case, we may need to specify effective decision procedures for
determining whether, or not, a proposition of a formal language of
mathematical communication, L, is to be termed as true or not under
each given interpretation M.
In other words, we need to reflect deeper upon the way we choose to
perceive the nature of Intuitive Knowledge, and more particularly the
nature of factual, or intuitive truth.
By Intuitive Knowledge we refer loosely to that body of pro-active
knowledge that stems directly from our conscious states, in contrast
to our reactive Instinctive Knowledge, which stems from, and lies
within, our sub-conscious and unconscious states.
Now one may, when attempting to express mental concepts within a
language, argue reasonably - as Gödel does - in Platonistic terms,
and define intuitive truths as characteristics of relationships that
are assumed to exist in some absolute sense (that is, even in the
absence of any perceiver) between the objects of an external ontology
(both of which are also taken to exist in some absolute sense).
However, for effective communication, we may need an alternative view
of relationships as belonging to individual perceptions that we
consciously construct, and selectively assign, to abstract objects
(that themselves are individual conceptual constructs) of an abstract
ontology (that is similarly an individual conceptual construct).
In other words, each individual perception can, reasonably, be
assumed to be a subjective, abstract, construct, which is based on a
unique, one-of-a-kind, never-to-be-repeated, consciousness of an
An intuitive truth is, then, essentially, a constructed, space-time
localised, individual factual truth (we shall, henceforth, use the
terms synonymously). It corresponds to a subjectively constructed
characteristic of the expression of an individual perception.
Loosely speaking, it corresponds to a characteristic of the way we
construct an expression for that which we select as common to a
series of subjective perceptions, rather than to a characteristic
that we discover of an objectively observed something.
The distinction seems significant. Platonic concepts, when introduced
into the interpretations of a formal language of communication, could
permit us to misleadingly validate our subjective intuitive
perceptions of individual factual truth as being reflective of some
absolute Truth that must be of universal significance in a Utopian,
Platonistic world. The Biblical Tower of Babel can, arguably, be seen
as illustrating the extreme possibilities of such Platonistic beliefs.
If we accept that, in formal languages, a selected set of axiomatic
truths is expressed as a set of Axioms (or Axiom schemas), then the
selection criteria should reflect that the Axioms are readily
accepted by any perceiver as faithfully referring to some significant
factual truths, pertaining to the expression of abstract constructed
elements of the constructed ontology under consideration, as
perceived and conceived by the perceiver.
For the most basic, and intuitive, of our scientific languages,
namely Number Theory or our Arithmetic of the natural numbers, we
take the commonly accepted selection of such axiomatic truths as the
classical set of Peano's Postulates, first expressed in semi-
axiomatic format by Dedekind in 1901.
The challenge, then, is to express these in a formal language such as
Peano Arithmetic, PA, along with a suitable set of Logical Rules of
Inference by which we can assign unique formal truth-values
algorithmically to as many well-formed propositions of the theory as
possible that are not Axioms.
The concept of formal algorithmic truths is, thus, merely the result
of the application of a set of Rules of Inference for effectively
assigning such formal truth-values algorithmically to various logical
permutations and combinations of axiomatic truths as (finite and
infinite) compound assertions (which, ideally, should not introduce
any new axiomatic truths that are not already implicit within the