An axiom is a sentence or proposition that is taken for granted as true, and serves as a starting point for deducing other truths. In many usage axiom and postulate are used as synonyms.
In certain epistemological theories, an axiom is a self-evident truth upon which other knowledge must rest, and from which other knowledge is built up. An axiom in this sense can be known before one knows any of these other propostions. Not all epistemologists agree that any axioms, understood in that sense, exist.
In logic and mathematics, an axiom is not necessarily a self-evident truth, but rather a formal logical _expression_ used in a deduction to yield further results. To axiomatize a system of knowledge is to show that all of its claims can be derived from a small set of sentences that are independent of one another. This does not imply that they could have been known independently; and there are typically multiple ways to axiomatize a given system of knowledge (such as arithmetic). Mathematics distinguishes two types of axioms: logical axioms and non-logical axioms.
It speaks for itself. "We" (not you and me) create axioms to make 'our' theories work. Then we consider the 'system' in question based on such axioms. I try to scrutinize them, to find alternates and scrutinize those also.
The other one is an 'a priori (physical?) theory' - sounds in physics similar to 'your' numbers which you may consider 'a priori' existing. If I may ask: what 'natural' senses may detect numbers? Unless. of course, you consider our mind a 'natural sense' (what may be true). As I 'believe': anything recognized by our 'senses' are our mental interpretations of the unattainable 'reality' (if we condone its validity). "My world" is a posteriori.
Le 23-août-06, à 03:58, Brent Meeker a écrit :
>> People who believes that inputs (being either absolute-material or
>> relative-platonical) are needed for consciousness should not believe
>> that we can be conscious in a dream, given the evidence that the brain
>> is almost completely cut out from the environment during rem sleep.
> Almost is not completely.
I am glad you don't insist.
> In any case, I don't think consciousness is maintained
> indefinitely with no inputs. I think a "brain-in-a-vat" would go into
> an endless
> loop without external stimulus.
OK, but for our reasoning it is enough consciousness is maintained a
nanosecond (relatively to us).
>> guess they have no problem with comatose people either.
> Comatose people are generally referred to as "unconscious".
? ? ?
I mean this *is* the question. In mind'sI (Dennett Hofstadter) we learn
that a woman has been in comatose state during 50 years (if I remember
correctly), and said she never stop to be conscious.
They are more than one form of comatose state. To say they are
"unconscious" is debatable at the least. And then there is the case of
dreams. And for those who does not like dream, what about the following
question: take a child and enclose him/her in a box completely isolated
from the environement. Would that fact suppress his/her consciousness?
Some parents will appreciate and feel less guilty with such ideas ...
>> Of course they cannot be even just troubled by the UD, which is a
>> program without inputs and without outputs.
> As I understood the UD the program itself was not conscious, but
> rather that some
> parts are supposed to be, relative to a simulated environment.
Yes. some "person" attached to (infinity) of special computations,
>> Now, without digging in the movie-graph, I would still be interested
>> someone accepting "standard comp" (Peter's _expression_) could explain
>> how a digital machine could correctly decide that her environment is
> "Decide" is ambiguous. She could very well form that hypothesis and
> find much
> confirming and no contrary evidence. What are you asking for? a
> proof from some
> axioms? Which axioms?
Sorry, I have used the word "decide" in the logician sense (like in
undecidable). To decide = to proof, or to test, or to solve, in some
Which axioms? Indeed, good question, that's makes my point. Well, I was
thinking about some physical theory the "someone" would argue for.
Anyone a priori.
>> If such machine and reasoning exist, it will be done
>> in Platonia, and, worst, assuming comp, it will be done as correctly
>> the real machine argument. This would lead to the fact that in
>> Platonia, there are (many) immaterial machines proving *correctly*
>> they are immaterial. Contradiction.
> Suppose a physical machine implements computation and proves relative
> to some axioms
> that physical machines don't exist. Contradiction?
If by "physical" you mean what Peter Jones means, then indeed the
"physical machine" is in contradiction. This means that her axioms are
indeed contradictory. If moreover, the physical machine gives a
"correct" proof, as as I say in the quotes, then we get a total
contradiction, like a proof that PI is an integer, for example. That we
are in contradiction.
As far as we are consistent, this just means that no X-machine can
correctly proof that X-machine does not exist.
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