The excerpts are from After Finitude of Quentin Meillassoux (page numbers are from the ebook version.). Disclaimer: Fair use – In the purpose of critique /Answering a question: text analysis
(Not very clean formatting. Pardon me)
This absolute lies at the
furthest remove from the absolutization we sought: the one that
would allow mathematical science to describe the in-itself. We
claimed that our absolutization of mathematics would conform to
the Cartesian model and would proceed by identifying a primary
absolute (the analogue of God), from which we would derive a
secondary absolute, which is to say, a mathematical absolute (the
analogue of extended substance).
(p 105)
Note: Absoulutization of mathematics (derivation of a secondary mathematical absolute) is a requirement to establish a Cartesian in-itself. No problem, he does not say he did it.
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We are not claiming that the non-totalizing axiomatic is the
only possible (i.e. thinkable) one. Consequently, we are not
claiming that the possible is always untotalizable, even if this is
the case in the standard axiomatic of sets. For we cannot deny a
priori the possibility that it could also be thinkable that the
thinkable constitutes a totality. The fact that the untotalizable is
thinkable within a given axiomatic does not prevent anyone from
choosing another axiomatic, in which the frequentialist
implication would still be valid. There is always more than one
axiomatic, and despite its eminence, that of standard set-theory is
only one among many. Thus, we cannot rule out a priori the
possibility of selecting an axiomatic in which the realm of possible
worlds would constitute an ultimate and determinate numerical
totality. But this at least must be accorded to us: we have at our
disposal one axiomatic capable of providing us with the resources
for thinking that the possible is untotalizable. However, the mere
fact that we are able to assume the truth of this axiomatic enables
us to disqualify the necessitarian inference, and with it every
reason for continuing to believe in the existence of the necessity of
physical laws - a necessity that is mysteriously superimposed onto
the fact of the stability of these same laws.
(169)
Note:
Meillassoux does not necessitate the possible (or the physical/possible world) to be untotalizable. Ok. He knows that there are alternative axioms. In the case of physical world nobody has ruled out any of those axioms (untotalizable or other). Hence world may be untotalizable.
To prove stability of the world in this way, there are two steps.
Step 1. Prove that the physical world (possible worlds) is non-totalizable, in the means of deriving it from the thesis of factiality/contingency.
Step 2: Prove the stability of the world starting from non-totalizable.
Step 1 is yet to be done. Ok. Nobody has disproved it even. OK. (It is to be proved or disproved.) Step 1 is a conjecture.
Questions:
1. What is it about step 2, is it completed? (Is it completed when the step 1 is done?)
2. Is step 1 is required to get the conclusion of step 2? (Should one proves step 1 first to prove possibility of stability?)
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What the set-theoretical axiomatic demonstrates is at the very
least a fundamental uncertainty regarding the totalizability of the
possible.1 But this uncertainty alone enables us to carry out a
decisive critique of the necessitarian inference by destroying one
of the latter's fundamental postulates2: we can only3 move
immediately3 from the stability of laws to their necessity so long as
we do not question3 the notion that the possible is a priori
totalizable3. But since this totalization is at best only operative
within certain axiomatics, rather than in all of them, we can no
longer continue to claim that the frequentialist implication is
absolutely valid. We have no way of knowing whether the
possible can be totalized in the same way as the faces of a set of
dice can be totalized.
(170)
Note:
This is a part of Step 2.
1. Non-totalizability is a valid conjecture. OK.
2. Assuming Step 1 we can complete Step 2. (N.B. enables us, destroying)
3. Answer to Question 2. Step 1 is required: If this world is totalizable we can immediately conclude that the stability of laws guaranties the necessity of laws. Therefore we must question totalizability of the world. Best case we should this world or all possible physical worlds are non-totalizable. It is required to complete the proof. At least we should grasp about the validity of the conjecture. Hence, non-totalizability which is a requirement to grasp our experience of stable history in a factual/contingent world and which is also mathematical is an elegant and necessary secondary absolute to complete this project. But the derivation is yet to be done.
My opinion: This not OK. This is the part I disagree. One cannot move immediately from the stability to necessity even if the events are totalizable. Ensemble probabilities and ergodicity does not constitute a necessary relation. Zero probability does not mean impossibility and probability of one does not mean necessity. Other probabilities does not mean any requirement of actualization.
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But how is Kant able to determine the actual frequency of the modification of laws,
assuming the latter to be contingent? How does he know that this
frequency would be so extraordinarily significant as to destroy the
very possibility of science, and even of consciousness? By what
right does he rule out a priori the possibility that contingent laws
might only very rarely change - so rarely indeed that no one
would ever have had the opportunity to witness such a
modification? It can only be by the right which he derives from
applying the calculus of probability to our world as a whole,
rather than to any phenomenon given within the world, and
hence from an a priori totalization of the possible, which we
know, since Cantor, can no longer lay claim to any logical or
mathematical necessity - which is to say, to any sort of a priori
necessity
(172)
Note:
Not OK. A priori totalization is not needed for probability and there is no impact on actualization argument from the constraints of assigning numbers to events. Concept of probability is one thing, assigning numbers to probability is another.
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We then noticed that at the root of this
presupposition lay an instance of probabilistic reasoning applied
to the laws of nature themselves; a piece of reasoning which there
was no reason to accept once its condition - the claim that
conceivable possibilities constitute a totality - was revealed to be
no more than a hypothesis, as opposed to an indubitable truth.
In doing so, although we have not positively demonstrated
that the possible is untotalizable, we have identified an alternative
between two options - viz., the possible either does or does not
constitute a totality -with regard to which we have every reason to
opt for the second - every reason, since it is precisely the second
option that allows us to follow what reason indicates - viz., that
there is no necessity to physical laws - without wasting further
energy trying to resolve the enigmas inherent in the first option.
(173)
Note :
Not Ok. Previously explained.
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But what is most fundamental in all this - and this was
already one of the guiding intuitions of Being and Event - is the
idea that the most powerful conception of the incalculable and
unpredictable event is provided by a thinking that continues to be
mathematical - rather than one which is artistic, poetic, or
religious. It is by way of mathematics that we will finally succeed
in thinking that which, through its power and beauty, vanquishes
quantities and sounds the end of play.
(175)
Note:
Not Ok. This answers Question 2. Here Meillassoux says it is required to complete step 1 to get the conclusion of stability. One has to go through mathematics to grasp it, not any other way. Meillassoux has not completed the Step 1 but that step is required. According to him his incomplete proof recruits mathematics rather than any other thought to establish stability. Remaining task is to derive it.
I disagree. This second step is flawed. There is no guarantee that first step is required.
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Most importantly:
However, we must return to our proposed resolution of
Hume's problem, for the former cannot wholly satisfy us. As we
saw, this resolution is fundamentally non-Kantian insofar as it
aims to establish the conceiv-ability of the actual contingency of
the laws of nature. Yet it would not be quite right to assert that
our resolution itself is unequivocally speculative, just because it is
anti-transcendental in intent. For although the thesis we advanced
was ontological, and did indeed assert something about the in-
itself rather than phenomena in maintaining the detotalization of
the possible, nevertheless, it was only advanced as an ontological
hypothesis. We have not established whether this non-totalization
actually obtains, we have merely supposed it, and drawn the
consequences of the fact that such a supposition is possible.4 In
other words, although our proposed resolution of Hume's
problem gives us grounds for not immediately giving up on the
idea of factial speculation, it has not itself been engendered as a
truth by means of speculative reasoning.5 For a properly factial
resolution of Hume's problem would require that we derive the
non-totalization of the possible from the principle of factiality itself.
In order to elaborate such a solution, we would have to derive the
non-whole as a figure of factiality6, just as we sketched the
derivation of consistency and of the 'there is'. This would entail
absolutizing the transfinite6 in the same way in which we
absolutized consistency - which is to say that we would have to
think the former as an explicit condition of contingent-being7, rather than merely construing it as a mathematically formulated
hypothesis7 that can be advantageously supported by the
speculative. But it is clear that such resolution of the problem
would require that we be in a position to do for mathematical
necessity what we tried to do for logical necessity. We would have
to be able to rediscover an in-itself that is Cartesian, and no
longer just Kantian - in other words, we would have to be able to
legitimate the absolute bearing of the mathematical - rather than
merely logical - restitution of a reality that is construed as
independent of the existence of thought. It would be a question
of establishing that the possibilities of which chaos - which is the
only in-itself - is actually capable cannot be measured by any
number, whether finite or infinite, and that it is precisely this
super-immensity of the chaotic virtual that allows the impeccable
stability of the visible world.
(178-179)
Note:
4. He is saying step 2 is completed.
5. He is saying step 1 is yet to be done.
6. He is saying step 1 is required to conclude that a factual universe can be stable.
7. He says it required to prove that transfinite is not just another mathematical axiom but the only mathematical axiom applicable to the physical world. Therefore some sort of mathematics absolutely govern the world.
My opinion: I disagree. Non-totalizability or any other axiom does not prevent ergodicity and simply we can reject applicability of ergodicity (which is mathematical). Simply we can move further saying there is no absolute mathematics. Step 2 is easily resolved. There is no requirement for step 1.
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But it is clear that such a derivation would not only have to be
far more complex, but also more adventurous than that of
consistency, since it would have to demonstrate how a specific
mathematical theorem, and not just a general rule of the logos, is
one of the absolute conditions of contingency8. Accordingly, it
might seem wiser to confine9 ourselves to our hypothetical
resolution9 of Hume's problem, since the latter seems sufficient to
vanquish the objection from physical stability10, which provided
the only 'rational' motive for not simply abandoning every variant
of the principle of reason. However, there is another problem that
rules out such caution, and it is precisely the problem of
ancestrality. For as we saw, the resolution of the latter demanded
an unequivocal demonstration of the absoluteness of mathematical
discourse. We begin to see then, albeit dimly, that there now seem to be two problems tied to the issue of the absolute scope of
mathematics - the problem of the arche-fossil and the problem of
Hume. It remains for us to connect them in such a way as to
provide a precise formulation of the task for non-metaphysical
speculation.
(179)
Note:
8. Step 1 is yet to prove and it is required.
9. Wow! Now he is saying it is wise to say this is THE PROOF although it is not completed. Why?
10. Meillassoux: “Why? I proved the second step, right! When stage 1 is done the game is over. Right!”
