Dear Rex,

-----Original Message----- 
From: Rex Allen
Sent: Thursday, November 04, 2010 12:40 AM
Subject: Re: Probability, Necessity, and Infinity

On Wed, Nov 3, 2010 at 5:50 PM, Stephen Paul King <> 
> On Tue, Nov 2, 2010 at 8:24 PM, Rex Allen <> wrote:
>> "if laws were contingent, they would change so frequently, so
>> frenetically, that we would never be able to grasp anything
>> whatsoever, because none of the conditions for the stable
>> representation of objects would ever obtain. In short, if causal
>> connection were contingent, we would know it so well that we
>> would no longer know anything. As can be seen, this
>> argument can only pass from the notion of contingency to the
>> notion of frequency given the presupposition that it is
>> extraordinarily  improbable  that the laws should remain
>> constant rather than being modified in every conceivable way
>> at every moment."
> Here we have what appears to be a well reasoned argument until
> we inquire as to the definition of the term "to know" that is
> used. If an entity exists in a universe subject to frequent
> and contingent change what is to allow the mind of that entity
> the ability to have the ability to know anything at all? The
> entity and its brain/mind would be subject to the very same
> capricious randomness that the rest of that universe undergoes
> and thus the notion of knowing becomes null and void.
If an entity exists in a universe that is subject to unchanging causal
laws, how can it have justified true beliefs (a.k.a. knowledge)

    I am not sure of what you mean by "unchanging causal laws" so I will offer 
a definition: A set of relations that given some sharp configuration of 
energy-momentum on a plane of simultaneity there will always be some other 
sharp configuration and none other. Is this a satisfactory definition of 
"unchanging causal laws" to you? Note that this definition is consistent with 
the classical "block universe" model of the universe.

If the entity's beliefs are the result of some more fundamental
underlying process, then those beliefs aren't held for reasons of
logic or rationality.

    What is "more fundamental"? If the block universe of classical physics is 
taken to be the totality of what can and does exist, there can be no "more 
fundamental" anything, not even a "process". I am amused to reread the 
occasionalist and epiphenomenalist theories of mind that have been offered to 
account for the notion of knowledge (a.k.a. justified true belief). I am sure 
that you agree that without a mind there can be no belief, justified or 
otherwise, not logic nor rationality. So if the universe does not allow for 
entities to have something that can be considered as mind then we can go no 
further down this line of reasoning as we have removed all possible means to 

Rather, the entity holds the beliefs that are necessitated by the
initial conditions and causal laws of it's universe.

Those initial conditions and causal laws *may* be such that the entity
holds true beliefs, but there is no requirement that this be the case
(for example, our own universe produces a fair number of delusional

    We have not established that an entity can have a mind in a universe that 
is subject to unchanging causal laws so until we do we can ask no further 
questions. OTOH, in the spirit of the discussion I will overlook this fatal 
flaw, but we are presented with another problem: How do we distinguish the 
schizophrenics, deluded or otherwise, from the non-schizophrenics? Following 
your reasoning, the same causal laws would generate both, so the difference is 
a set of initial conditions. What determined those to be such rather than some 
other? I see a crack opening here that allows us to recover many worlds... The 
point is that if there is any choice at all in the state of the universe and 
anything therein, then it is necessary that a multiplicity of prior possibility 
    The mere fact that I have a mind or some delusion that leads to a similar 
condition leads me also to the conclusion that there is something that is like 
the subjective sense of having a choice of action and that sense of having a 
choice extends to what ever reasons, rationalizations or delusions that I might 
have about the nature and origin of justified true beliefs. So if there is the 
possibility, encapsulated in the word *may*, as in "Those initial conditions 
and causal laws *may* be such that the entity 
holds true beliefs...", then it inescapably follows that there is a 
multiplicity of at least initial conditions that could have lead to this state 
of affairs. So we cannot coherently hold that unchanging causal laws disallow 
for justified true beliefs. You seem to agree with this conclusion below.

So holding true beliefs, even in a universe with causal laws, is
purely a matter of luck - i.e., is the entity in question lucky enough
to live in a universe with initial conditions and causal laws that
lead to it holding true beliefs.

    I agree, because there is no a priori restriction of the possible initial 
conditions that could, following those causal laws, generate the condition or 
state of having a mind that holds true beliefs. But your point about holding 
those beliefs is a "matter of luck" necessitates a prior spectrum of initial 
conditions from which a set of true beliefs can obtain and thus, at least in an 
a priori sense, a plurality of possible initial conditions. So if we have an a 
priori plurality of initial conditions, we left in a condition where we have at 
least have way recovered the notion of a plurality of possible worlds. Is this 
not countering your claim here, at least partly?

Further, if the initial conditions and causal laws don't cause the
entity to present and believe true rational arguments, there would be
no way for the entity to ever detect this, since there is no way to
step outside of the universe's control of one's beliefs to
independently verify the "reasonableness" of the beliefs it generates.

Again...schizophrenics are generally pretty convinced of the truth of
their delusions.

    Sure, but are we not arguing against the direct evidence that we have at 
hand, evidence that contradicts our premise? What is the point of doing that? 
Even if our ideas are delusions the very fact that we have something that is 
like having those delusions contradicts the mere possibility that we cannot 
detect such a situation! Are we restricted in only being able to step outside 
of the universe's control to verify reasonableness? That looks far too similar 
to a false choice argument to be considered seriously.

Even in a lawful universe how do you justify your beliefs?  And then
how do you justify your justifications of your beliefs?  And then how
do you justify the justifications of the justifications of your
beliefs?  And so on.  Agrippa's Trilemma.

So.  Given the capricious randomness involved in the selection of the
entity's universe's initial conditions and causal laws (of which the
vast majority of conceivable combinations would result in false
beliefs) the notion of knowing becomes null and void.

Neither Meillassoux's scenario nor the "lawful universe" scenario
allow for knowledge.  In both cases, holding true beliefs is a matter
of luck, and no belief can be justified (not even the belief that no
belief can be justified).

    OK, so maybe there is something wrong with the premises and assumptions 
that we are using in our reasoning here. In my study of philosophy I have found 
that a lot of the problems, such as Agrippa's Trilemma, etc. rest on the 
assumption of well foundedness and much smarter people than me have found at 
least one solution to this mess. I respectfully request that you look into 
Non-wellfounded set theory and its logic and see for yourself have we can avoid 
this conundrum that you find yourself in.  That Meillassoux was unable to grasp 
this solution was no fault of mine or yours, but a failure to allow for the 
possibility that an alternative exists does fall on our shoulders since it is 
our duty to perform the due diligence that research and study requires.

> [SPK]
> Does Meillassoux not understand anything about
> calculus, analysis, computational complexity
> theory or other higher mathematics?
Perhaps you could be a little more specific in exactly how you feel he
exposed his ignorance?


    For example, in analysis there is a notion of a Borel set  that is used to define a measure on the 
set of Real numbers that allows for a coherent notion of probability, among 
other things. This is just one example of how infinities are used in a coherent 
way that does not fall prey to a supposed infinite regress or runaway 
condition.A philosopher making an argument that professes to use logic and that 
is oblivious to well established principles in mathematics is not helping his 

    We also have the which 
Meillassoux is using a crude version to make his argument. It would be helpful 
for use to understand the solution to this paradox.

>From the Wiki page reference we find: 

begin cut/paste
“Statement and proof 
In order to state the paradox it is necessary to understand that the cardinal 
numbers admit an ordering, so that one can speak about one being greater or 
less than another. Then Cantor's paradox is:

  Theorem: There is no greatest cardinal number.
This fact is a direct consequence of Cantor's theorem on the cardinality of the 
power set of a set.

  Proof: Assume the contrary, and let C be the largest cardinal number. Then 
(in the von Neumann formulation of cardinality) C is a set and therefore has a 
power set 2C which, by Cantor's theorem, has cardinality strictly larger than 
that of C. But the cardinality of C is C itself, by definition, and therefore 
we have exhibited a cardinality (namely that of 2C) larger than C, which was 
assumed to be the greatest cardinal number. This contradiction establishes that 
such a cardinal cannot exist.
Discussion and consequences
Since the cardinal numbers are well-ordered by indexing with the ordinal 
numbers (see Cardinal number, formal definition), this also establishes that 
there is no greatest ordinal number; conversely, the latter statement implies 
Cantor's paradox. By applying this indexing to the Burali-Forti paradox we also 
conclude that the cardinal numbers are a proper class rather than a set, and 
(at least in ZFC or in von Neumann–Bernays–Gödel set theory) it follows from 
this that there is a bijection between the class of cardinals and the class of 
all sets. Since every set is a subset of this latter class, and every 
cardinality is the cardinality of a set (by definition!) this intuitively means 
that the "cardinality" of the collection of cardinals is greater than the 
cardinality of any set: it is more infinite than any true infinity. This is the 
paradoxical nature of Cantor's "paradox".”

end cut/paste

We can start down this path of finding a solution by studying Non-well-founded 

> OTOH, to wonder which infinity the set of all
> possible worlds belongs to is not trivial matter
I think Meillassoux's main point with this digression into Cantorian
set theory is that just as there can be no end to the process of set
formation and thus no such thing as the totality of all sets, there is
also no absolute totality of all possible cases.

In other words, there is no "set of all possible worlds".  And thus
"we cannot legitimately construct any set within which the foregoing
probabilistic reasoning could make sense."

    Sorry Rex, no joy. The reasoning is not sound so the proof does not follow. 
Let’s try harder.



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