Sleeping Beauty is put to sleep and coin is tossed. If the coin comes up heads, Beauty will be awakened and interviewed only on Monday. If the coin comes up tails, she will be awakened and interviewed on both Monday and Tuesday. On Monday she will be put into sleep by amnesia inducing drug. In either case, she will be awakened on Wednesday without interview and the experiment ends. Any time Sleeping Beauty is awakened and interviewed, she is asked, "What is your belief now for the proposition that the coin landed heads?" No other communications are allowed so that the Beauty does not know whether it is Monday or Tuesday.
The question is about the belief of the Sleeping Beauty on basis of the information she has, not about the actual probability that the coined landed heads. If one wants to debate one imagine oneself to the position of Sleeping Beauty. There are two basic debating camps, halfers and thirders.
- Halfers argue that the outcome of coin tossing cannot in any manner depend on future events and one has have P(Heads)= P(Tails)=1/2 just from the fact that that the coin is fair. To me this view is obvious. Lubos has also this view. I however vaguely remember that years ago, when first encountering this problem, I was ready to take the thirder view seriously.
- Thirders argue in the following manner using conditional probabilities. The conditional probability P(Tails|Monday) =P(Head|Monday) (P(X/Y) denotes probability for X assuming Y) and from the basic formula for the conditional probabilities stating P(X|Y)= P(X and Y)P(Y) and from P(Monday)= P(Tuesday)=1/2 (this actually follows from P(Heads)= P(Tail)=1/2 in the experiment considered!) , one obtains P(Tails and Tuesday)= P(Tails and Monday).
Furthermore, one also has P(Tails and Monday)= P(Heads and Monday) (again from P(Heads)= P(Tails)=1/2!) giving
P(Tails and Tuesday)= P(Tails and Monday)=P(Heads and Monday). Since these events are independent for one trial and one of them must occur, each probability must equal to 1/3. Since "Heads" implies that the day is Monday, one has P(Heads and Monday)= P(Heads)=1/3 in conflict with P(Heads)=1/2 used in the argument. To me this looks like a paradox telling that some implicit assumption about probabilities in relation to time is wrong.
When one speaks about independent events and their probabilities in physics they are must be causally independent and occur at the same moment of time. This is crucial in the application of probability theory in quantum theory and also classical theory. If time would not matter, one should be able to replace time-line with space-like line - say x-axis. The counterparts of Monday, Tuesday, and Wednesday can be located to x-axis with a mutual distance of say one meter. One cannot however realize the experimental situation since the notion of space-like amnesia does not make sense! Or crystallizing it: independent events must have space-like separation. The arrow of time is also essential. For the conditional probabilitys P(X|Y) used above X occurs before Y and this breaks the standard arrow of time.
This clearly demonstrates that philosophy and mathematics cannot be separated from physics and that the notion of time should be fundamental issued both in philosophy, mathematics and physics!
9 comments:
What's with the quantum theorists and their sadistic thought experiments? As if Schrödinger's can was not enough, now they are putting damsell in distress and torturing also Sleeping Beauty!!!
Philosophy indeed. How much energy, time and effort is used for trying cage quantum holonomy outside physicist's box? Where genuine cat with nine lives claws or quantum jumps outside the box and Sleaping beauty gives the researcher finger on Monday and double finger on Tuesday until Prince Valiant rides to rescue and gives a kiss. :)
But in terms of physics, the argument that causally independent events have to occur at the same moment of time - or that causally dependent events have to occur before and after - is not valid, at least in relativity. For a relativistic-time-space observer observing events and their relations all options are observable. Einstein wanted to save causality, and in modern physics linear causality is not logically derived from anything, just defined tautologically as basic axiom. Hume's scepticism of linear causality is fundamental issue of philosophy, as is everyday psychological acting AS-IF there was linear causality with single arrow of time.
That said, there are also many kinds of psychological states - and narratives - where a teleological Purpose from future affects now and past.
I mentioned with causality relativistic causality. Proper time distance is the time that is in question in the framework or special relativity. Events with space-like separation are not independent not only those in time=constant hyper-surface.
Logical causality, causality of conscious experience, and the causation of classical physics must be distinguished from each. For the causality of free will the causality corresponds to experienced time order.
Causality in Einstein's sense is precisely defined and follows from Lorentz invariance when one speaks of classical fields. In quantum theory the formulations states the absence of tachyons and states that energies are positive.
I did not say anything about the possibility that the arrow of geometric causality can vary as it indeed does in zero energy ontology. It is an additional finesse.
My point was that one cannot do anymore mathematics and philosophy without taking into account the world view of quantum physics.
Does Einstein causality follow from Lorentz invariance or the other way around? Which is cause and which is effect and in what time?
Stenger lends additional light on the matter at hand: "When you read, "Einstein proved that particles cannot go faster than the speed of light" you have to understand that this was not a consequence of the basic axioms of the theory of special relativity. To prove this he introduced an additional assumption now called the "principle of Einstein causality": cause must always precede effect. In that case, it then follows that we can't have superluminal motion."
http://www.huffingtonpost.com/victor-stenger/no-cause-to-dispute-einst_b_982429.html
And even reversible notion of alone is not enough if you take delayed choice experiment seriously and I believe you do. If I'm not mistaken, one of the key notions of TGD is (was?) part-whole principle of various scales of 'now'.
Against that presumably deeper and more fundamental background, is there logical and mathematical necessity for Einstein-causality (and Lorentz invariance), or are we speaking about metaphysical belief and wishful thinking?
PS: Lubos post and discussion was purely about mathematical problem that, as said, is not clearly defined and formulated, hence only intelligent position mentioned was that there is no logical reason to lend support to either position.
PPS: I'm not at all certain that quantum physics has definite "the" world view. ;)
Einstein causality is Lorentz invariance + fixed arrow of time. In ZEO one weakens Einstein causality since selves can have have both arrows of time: in re-incarnation of self the arrow of geometric time changes. Each self is irreversible by NMP - I do not assume reversibility at quantum level.
This allows to consider effective super-luminality since signals can be reflected in time direction. Say from my brain in geometric future or past: during sleep we remember future events- I have somewhere in my bookshelf a book about documents memories of future! Do not remember the author.
Signals can be reflected also from brain of some alien in some distant galaxy: this I proposed as a possible interpretation for the experiences of meeting ETs induced by psychedelics. These events would always involve re-incarnation of subself representing signal to the geometric past of self (geometric past is relative notion- future for self which is sleeping!).
The most interesting application in neuroscience is new view about memory. Also sensory-motor cycle could be understood as a sequence in which sensory mental image dies and re-incarnates as motor mental image.
I am not sure about what you mean with part-whole principle - Wikipedia did not help. There is hierarchy of selves, maybe you mean that.
I take Lorentz invariance as a fact. In TGD framework it has deep roots: the geometry of infinite-D WCW exists if it has infinite-D group of isometries. This is the case under very restricted conditions. 4+4 dimensionally follows from
general number theoretical reasons. Lightone boundary for M^4 has gigantic conformal symmetries and 4-D space
time surfaces have light-like partonic orbits with similar huge symmetries. In 8-D M^4xCP_2 one has huge super-symplectic symmetries for WCW.
The existence of twistor structure relating to Yangian invariance is one further prerequisite. M^4xCP_2 is completely unique in the sense that its factors are the only 4-geometries for which twistor space has Kaehler structure. This was discovered at the same time I discovered M^4xCP_2 but no one told about this to me!
It would be interesting to know, whether some colleagues knew about this but did not bother to tell for some reason;-). In any case, once you have M^4 you have Lorentz invariance.
Arrow of time follows from quantum measurement theory generalised to ZEO and accepting NMP (and already in its ordinary form). Einstein causality follows both as an experimental fact (even in its weakened for) and as a mathematical necessity.
I restate my view - and also that of Lubos although we might have different justifications. Thirders are wrong because they bring in conditional probability although it is not needed at all. Even worse, they also apply it in wrong manner. It might be that probability theorists outside physics circles do not realize that causal independence in physical sense is highly relevant factor when one talks about probabilities.
I saw term "part-whole principle" together with "numerosities" as attempt to save set theory of infinite sets, and search gave e.g. this link:
http://www.math.uni-hamburg.de/home/loewe/HiPhI/Slides/parker.pdf
The connection to hierarchy of selves is obvious, but the notion of part-whole principle is at least implied already in the ordinal foundational level of number theory of ordered fields and their arithmetics, ie. e.g. 1<2 is a part - whole relation given that 1+1=2. Hence, it can be said that p-adic numbers are by definition part-whole relations, but same does not seem to apply with equal clarity on the atomistic-reductionistic real side. When three dots refer to "larger than", part-whole relation works, but when three dots refer to "smaller than", what is the whole and what is the part?
I do not see any reason for saving set theory from infinite sets: if something is saved it is the naive belief that the world is what our sensory limitations tell it must be. Both geometric and number theoretic aspects are important and one ends up with difficulties when on accepts only the other one. Modifying merciselly Einstein's well-known statement about religion and science: Mathematics without geometry is blind and Mathematics without number theory is lame;).
Part-whole principle states is geometry based and states that if set is subset of another set, it is "smaller". This is ok but one should not confuse the size as the number of elements of set used by Cantor and defined in terms of 1-1 correspondences with the metric size used in part-whole principle.
Reals from 0/1 to infinity is a good example. Size of set as number of elements is something different from the size of set defined by metric, since metric brings in structure.
The notion of infinite prime involves power sets or their subsets.
One considers second quantisation of supersymmetric arithmetic QFT. Should one allow states with literally infinite number of fermions and bosons or only states with finite but unbounded number? Depending on the choice each step in the hierarchy of quantisations gives/does gives power set/subset of power set having larger cardinality/same cardinality. If one demands finite energy /particle numbers for the states, each quantisation gives same enumerable number of states.
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Matti, have you considered continuing your blog over at wordpress.com? It has the ability to generate pictures from LaTeX. for instance, see https://almostsure.wordpress.com/2009/11/08/filtrations-and-adapted-processes/
- --Stephen
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Thank you, could you elaborate a little bit. Is the signature below all that is needed to this continuation?
https://en.support.wordpress.com/latex/
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