"No rule without an exception, except this rule." by Stefan Weckbach

Branko, thanks for your comment, your question and that you cherish what you read.

As to your question, you cited a case where a prediction of a theory has been confirmed. Let's say this is the precession of Mercury's perihelion, what comes about in GR to be approximately the value of observation. Since GR was not designed to a posteriori fit these observational data (as far as I can know!:), I would say that GR predicted it.

I cannot exclude that there are different theories possible which also can predict this phenomenon, theories that rest on different postulates than GR does.

For the case of Newton's theory of gravity, I cannot see how it can hold to explain the precission of Mercury's perihelion. But that I cannot see it does not mean that you necessarily must be wrong. Tell me more about your view on Newton's theory of gravity if you like.

Andrew, you claim very much with your comment... You claim e.g. that

1. I have not read Gödel's work

2. I do not understand it

3. My essay is a loose paraphrase of what Gödel was bringing up

All three points you made are false.

Even for the case that Gödel's work would turn out to be - for whatever hitherto unknown reasons - false, my essay does not in any way depend on the correctness of Gödel's work. The latter just reassures my conclusions, it is not in opposition to my conclusions. I think you misunderstood what I intended to say with the realm of fundamental truth. This is not merely a realm of logical / mathematical tautologies, but an existential realm that incorporates consciousness as a fundamental ingredient. Not all tautologies are fundamental truths, but according to my essay, all fundamental truths are perceived (and are) as self-evident tautologies in the realm of fundamental truth. Self-evidently, this realm must be located beyond space and time, since space and time could turn out to be just temporary appearances. They are, in my opinion, for the reasons I layed out in my essay, not fundamental truths, since in my opinion, a fundamental truth should be timeless.

I have not found any hints in Gödel's official mathematical work that states that truth must be in union with consciousness, what my essay says is independent from what Gödel brought up with his work. Therefore the points you made are twice devoid of any meaning for the conclusions in my essay. Gödel's work is important, but for me, Gödel is not the one and only authority when it comes to the question "what is fundamental".

Good luck also for you!

Interesting. If you are right, then a former ‚less fundamental' (space in GR) could turn out to be 'more fundamental' than we thought - if your approach takes space and time as an independent background like Newton did. Does it do so?

Andrew, you claim very much with your comment... You claim e.g. that

1. I have not read Gödel's work

2. I do not understand it

3. My essay is a loose paraphrase of what Gödel was bringing up

All three points you made are false.

Even for the case that Gödel's work would turn out to be - for whatever hitherto unknown reasons - false, my essay does not in any way depend on the correctness of Gödel's work. The latter just reassures my conclusions, it is not in opposition to my conclusions. I think you misunderstood what I intended to say with the realm of fundamental truth. This is not merely a realm of logical / mathematical tautologies, but an existential realm that incorporates consciousness as a fundamental ingredient. Not all tautologies are fundamental truths, but according to my essay, all fundamental truths are perceived (and are) as self-evident tautologies in the realm of fundamental truth. Self-evidently, this realm must be located beyond space and time, since space and time could turn out to be just temporary appearances. They are, in my opinion, for the reasons I layed out in my essay, not fundamental truths, since in my opinion, a fundamental truth should be timeless.

I have not found any hints in Gödel's official mathematical work that states that truth must be in union with consciousness, what my essay says is independent from what Gödel brought up with his work. Therefore the points you made are twice devoid of any meaning for the conclusions in my essay. Gödel's work is important, but for me, Gödel is not the one and only authority when it comes to the question "what is fundamental".

Good luck also for you!

Hi Georgina, thanks for reading and of course for commenting! Your effort of commenting is appreachiated by me.

I indeed took the terms frog's view and bird's view from Max' paper. They encode a huge problem not yet solved, the dichotomy between the subject (consciousness) and the object (matter), between relative, subjective truths (in reference to what is fundamental) and the necessity for the existence of such fundamental truths, means the fact that there is an external world independent of relative, subjective truths (objective truths) and whether or not these objective truths come out of literally nothing in the sense I defined it in the essay.

Many accounts on these problems assume the realm of what is most fundamental to be infinite. I am not quite sure if Max does also, but I take both possibilities into account. If mathematics is infinite, then one cannot speak meaningfully of an 'outside' for the black box I described as gedankenexperiment, hence there cannot be a bird's view, since then, the mathematical landscape is infinitely infinite, so to speak. Otherwise, if what is most fundamental would be finitely describable (albeit in a coarse-grained manner), one could refer to an 'outside' meaningfully in the sense whether or not there are further objective reasons for this most fundamental thing (in the case of Max's paper it would again be mathematics) to be fundamental at all.

If maths is finite, I think this would be a surprise for everybody. But I conclude just this in my essay: mathematics is a finitely, physical construct, as physical as one assumes matter, enery, wave functions and laws of physics to be physical. I consider 'infinity' from a logical point of view as merely an alternative term to express that something is fundamentally undefined - and undefinable (at least in our limited world).

You raise the question of many worlds. Many worlds fall naturally out of a global wave function, the latter seen as fundamental. The question is whether or not such a global wave function does exist ontologically. I cannot exclude this, but I doubt it, due to the arguments I gave in my essay against the exclusiveness of the complete formalizability of all that exists.

I like your painting analogy. This is what we normally do by inferencing due to antivalent thinking. The point I wanted to make in my essay is that you never can picture such a painting objectively with only antivalent thinking at hand. The best example for this impossibility seems to me the very essay contest here, where different assumptions are hold about what is true and what is false.

My own approach stems from the considerations of what properties a realm must have that does not suffer from the dichotomy of true and false propositions. My conclusion is that falseness as an option should evaporate into 'thin air' for at all being able to meaningfully speak about a 'most fundamental' as the basis for objective reality. Just consider what an angel in a spiritual realm ('heaven') would experience: she wouldn't experience the possibility that her realm could be just a fake, a kind of computer animation (since then it wouldn't be heaven anymore but just like earth...). She wouldn't experience this possibility, but not due to an error in her perception, but due to the fact (the truth) that this realm refers not anymore to 'time', but to eternity. Eternity in this sense means eternal truth without falseness in it. So the reason why you can't objectively paint this realm is that there is only 'white' in it, but no black.

You state that "I think the truth can be arrived at by finding all of the falsehoods and putting them out of the way. Which is how the scientific method at its best works." Albeit there is some truth in this statement, personally I wouldn't fully agree, since obviously there are situations where you aren't able to unequivocally identify some falsehood in the sense of a decisive proof for a counterexample, or a decisive proof for a certain assumption to be true at all. The problem is not that we can't observe in many cases *how* nature behaves, the problem is to unequivocally prove for at least some cases *why* it does so.

Thanks again for your thoughtful comment and good luck in the constest!

Kind regards, Stefan

Hi Georgina, i found your article and am going to read it now with much eager and will reply here and at the other site. It takes much time for the second bunch of essays to go online. I hope your essay contribution will be published soon, so i can also read your 'official' contribution to the essays current theme.

Hi Georgina, thanks for leading me to your 2 page article, which i just read. I have to think about it at bit longer, but just want to mention my spontaneous thoughts about some issues you raised.

Your attempt to model human intelligence and scientific progress as a result of a hive mind of a swarm of intelligent beings is interesting. I think it has pro's and con's. the pro's are surely that the progress in sciences was due to a huge swarm of beings, working on the question how nature works, falsifying some options and finding out that other options obviously do work and have some truth in them. But I would not glorify such a 'swarm intelligence' - and do not automatically assume that you indeed do. For me, the con's about the 'swarm intelligence' are directly before our very eyes. Different to, say birds, or flies, humans have the ability to some (assumed) unlimited imagination. That's an advantage, but I also see a certain danger, in that certain 'memes', 'ideas', 'trends' become dominant, trends that are not healthy for mankind. I could mention for example the glorification of money, but moreover, the glorification of science, the glorification of certain hypothesis of science, and in general, the glorification of the *assumption* that science can solve all problems human kind will probably face in the future. I see a trend that many unproven things are taken at face value and that more and more scientists cannot anymore discriminate between logical and physical / ontological necessities. Surely, the latter is impossible in general, but that's exactly the reason why one should handle hypothesis' as hypothesis', and not as some discovered truths.

It's the human ability to imaginate a full blown version of a wishfull future that is also a danger to mankind. Since most of us easily accept rose gardens and science-fiction like versions of ultimate reality, most of us tend to forget (or not even investigate) how such versions came about in the first place. A prime example for me is Max's mathematical universe - but to apologize to him I have to say that his new book (life 2.0) is a good piece of responsible science when it comes to probable future challenges. But let's briefly check what the mathematical universe is at its core. For me, it is a hypothesis that is able to camouflage itself as an established fact. It does so by using Curry's paradox. The latter are propositional statements of the form 'if this sentence is true, then Santa Claus does exist'. Obviously this sentence is true - hence Santa Claus *has* to exist. Examining the logical structure of Curry's paradox a bit closer (what I will not do here) reveals that it is just a kind of tautology of the form "if Santa Claus exists, then Santa Claus exists". Max's mathematical universe hypothesis uses (I think not deliberately) an instant of Turing's halting problem to appear as a proven fact. It states that all existent things are completely captureable by mathematics, even consciousness. This may all be true, but *until* we have captured consciousness in this manner, it remains an instant of the halting problem. Nonetheless, Curry's paradox enables that many recipients of Max's hypothesis to switch from "if this sentence is true, than all things that exist are exlusively and completely describable by mathematics" to the 'truth' of the second part of such a conditional proposition. Worse, you can also switch from "if this sentence is true, then all things that exist *are* exclusively only mathematics." to the 'truth' that "all things that exist *are* exlusively only mathematics". As I have mentioned, I see a tendency in theoretical physics to overlook such kinds of mechanics by not carefully discriminating between logical possibilities and ontological necessities. But I think I move on heretical grounds, since most professionals as well as non-professionals are convinced that mathematics is a kind of 'God-like' entity in a platonic heaven.

Your comments on entanglement and variables are also interesting. In this field, a similar self-confirming pattern as I described in my essay about the 'paradigma' of 'strict determinism' can be observed for a probabilistic interpretation of quantum mechanics. Once the premise is established that this theory is not about universal determinism, but rather about contingent single events in nature which are in most cases not predictable ('explainable'), but nonetheless statistically well defined, one must come to the following conclusion about the statistical method of the theory itself: if a huge trial of measurement outcomes is analyzed for possible deviations from the well defined standards of the theory, the theory should be capable of discriminating between random errors whose probability distributions are known and systematic errors, which can be excluded. The latter however, is not guaranteed by any probabilistic theory, because the theory concludes from observable events to counterfactual events and after that the other way round. This means, it presupposes to have taken into account all possible parameters and therefore assumes to know all governing laws of the counterfactual events. Moreover, even estimating the size of the expected random errors of huge trials of measurement outcomes on the basis of such a theory depends on what one assumes to be part of the theory. Putting it differently, the estimation of expected errors was put into the theory in the first place (necessarily without being falsifiable by 'further' measurements) on the basis of necessarily only measurable facts, and therefore the estimation of expected errors in the measured data base also becomes part of the output of the measurements when it comes to analyze the gathered data. This is again an extrapolation from known or assumed-to-be-known things to things that cannot be known for sure due to Moore's theorem.

Hi Lawrence, thanks for your excellent comments and your thoughts envoled in them. I will give my answers subsequently in this reply according to the points you made.

According to Russell's barber, this paradox necessarily presupposes that *none* of the other men (wich means the group of men in the village *except* the barber) are allowed to shave those other men of the village. Otherwise the barber wouldn't be a barber - *according to Russell's definition*. So, Russell's whole definition of a barber is *ill-defined*, since this is not what a barber must be defined, even for the case that there is only one barber in the village. The point is, that Russell's definition of the barber as "no men in the village are allowed to shave other men - except the barber" (this is the real fixed-point that can be extracted from the context in which Russell's definition seems to be true) does *not imply* that people are *not* allowed to shave themselves. Therefore the barber can shave himself.

What you say about Gödel's theorem and provability is interesting. It all crucially relies on whether or not logics is able to capture some fundamental truths. If logic and with it the mathematical systems which produced Gödel's results in the first place are inconsistent, then of course everything is provable with the help of these systems. So, the presupposition that logics is consistent demands that there are some true statements within those systems which are not provable with the help of these systems.

Otherwise one potentially could prove a system to be inconsistent and incomplete, but could also prove that a system is *not* inconsistent and incomplete. So would the latter proof be a stronger one, in the sense that the system under consideration should be considered as consistent, but incomplete? This is a senseless question, since we observed from the very start that the system generated a contradiction, leading to *everything* at the end of the day. If you can't make anymore a reliable distinction with a certain system, it is then senseless to further use this system.

Gödel's results are only *fundamentally* true under the following two presuppositions:

1. logics is consistent

2. Mathematics is eternal and infinite

If one of these two presuppositions is false, Gödel's results have no fundamental impact whatsoever. In my essay I argue that the second point may be false in the sense that our traditional view of mathematics as an eternal platonic realm is difficult to reconsile with Gödel's results, since every extension of a mathematical system critically hinges on what one considers to be a necessary additional axiom - for making such an extension not only *consistent*, but *eternally true*.

The problem is, if there exists an infinite, non-denumerable number of truths within this mathematical landscape, then also Moore's theorem should hold and every physically or mathematically defined final theory of everything would be a final theory of nothing - when refered to the question why our universe is what it is - and why mathematics is what it is. The suggested advantage of an eternal landscape of mathematics is that it seems to justify that there exists something at all, rather than nothing. What brings me back to the very beginning, since even the quest for the existence of God is in some form an instance of unprovability - in the same sense the eternal mathematical landscape is. In this sense, people knew all along, long before Gödel's results, that there may be things which cannot be logically proven nor disproven, but nonetheless could be true. Surely, the mathematical universe hypothesis, for example, does rest on empirically gathered data, means, on the truth that nature indeed incorporates a certain amount of mathematics. But it is also true that it incorporates a certain amount of consciousness. Max's (the latter I appreciate in very many respects) claim of the MUH rests (beneath others) on the assumption that even consciousness is fully formalizable by mathematics. I doubt this by saying that mathematics can at the maximum merely establish correlations between some brain actions and some mathematical patterns.

The question of how to properly interpret such correlations is a fundamental one. I am currently working on this, but cannot present yet any robust results. Trying to describe a strictly deterministic system in terms of axioms seems to be impossible to me other than taking it at face value and therefore as a true axiom that all that exists is indeed 'merely' a strictly deterministically acting system. It could turn out that a superposition of states yields *less* information about the system then the parts of the system themselves. Therefore it is crucial for me to look how one can incorporate an observer into quantum mechanics, the latter being independent of a strict determinism, but nonetheless being able to have some limited free-will at hand to decide between two mutually exclusive options. And you are right about closed systems. As I described in my essay, it may turn out that defining ultimate reality solely in terms of formal systems may itself be just a closed system. I am working on stepping out of such a system in a logically and meaningful manner in terms of how to properly interpret a global wave function. If I succeed, I surely will publish what I found.

Oh, of course it should be

"Otherwise one potentially could prove a system to be inconsistent and incomplete, but could also prove that *this* system is *not* inconsistent and incomplete."

Instead of

"Otherwise one potentially could prove a system to be inconsistent and incomplete, but could also prove that a system is *not* inconsistent and incomplete."

"If you can't make anymore a reliable distinction with a certain system, it is then senseless to further use this system."

The latter statement is, for avoiding misunderstandings, of course in reference to what the system can and has to say about 'more or less fundamental' things. For example non-euclidean geometry us indeed useful for examining the consequences of GR, but it doesn't (and cannot) say anything about whether or not gravity is merely a statistical description of something else (as annotated in the Contest Guidelines).

I hope to have clarified some rather uncouthed formulations in my previous post. If not - just ask.

Hi Lawrence... and of course I have once more forgotten to answer an important part of your comment, because there are so many issues involved in your comment. So here is my answer to your remarks concerning rule F:

This rule is perfectly consistent and not paradoxical. Try to imagine it like this:

There are 1000 doors. Behind every door, there is a certain rule. Behind the first 999 doors, there are arbitrary rules, with each an exception. Behind door number 1000, there is a rule that captures a regularity common to all the other rules behind the 999 doors, namely that each of these rules have an exception.

The paradoxical character of Rule F only comes about due to the semantical issue of the usage of the word "exception" in the second part of rule F. If you imagine door number 1000 (behind where rule F resides) and compare it to the other rules, it has no exception, since all rule F is about is a compressed commonality all the 999 other rules have. For rule F to have an exception itself would depend on at least one of the 999 other rules having *no exception*, right? Then rule F could indeed no more considered as truly describing a commonality all the 999 other rules have.

In this sense, the fact that *no* rule of those 999 rules has *no* exception (caution, double negation!), necessitates that rule number 1000 (rule F) *has* indeed *no* exception. Remember, Rule F is exclusively only about the 999 other rules' exceptions. In the form I wrote it down in my essay, it may seem that it is also about some exceptions for Rule F itself, because Rule F is as well as all the other 999 rules a rule. Despite the undeniable fact that rule F is indeed a rule, it is clear that it refers the whole lot only to the 999 other rules' exceptions, not to an exception that it has itself - albeit rule F indeed does refer to itself in the form I wrote it down in my essay. But this self-reference is harmless and non-paradoxical.

Replace one of the 999 rules with a rule that has *no exception* - and rule F is forced *to have an exception*. Globally seen, in this system, when just one of the 999 rules are replaced in the manner described, there will always be one rule without an exception. Surely it is not guaranteed that such a replacement results again in the original rule F. This depends on what one does insert as a new rule behind the one of those 999 doors.