An interesting consequence of combining the logical space of hypotheses, Bayes’ theorem, and taking priors from Kolmogorov complexity is that any hypothesis of a certain level of complexity will have at least two opposite child hypotheses, which are obtained by adding one more bit of complexity to the parent hypothesis in one of two possible states.
And, conversely, you can remove one bit from the hypothesis, make it ambiguous with respect to which child hypothesis will fit your world, but then you will need fewer bits of evidence to accept it a priori.
And accordingly, there will be one hypothesis that you will have to accept without proof at all, a program of zero length, an empty string, or rather an indefinite string, because it contains all its child hypotheses inside itself, it simply does not say which one of them is true.
And this absolutely a priori hypothesis will be that you exist within at least one world, the world of the space of hypotheses itself, it is not specified simply in which particular part of this world you are.
And this makes us look differently at the Tegmark multiverse. Because, let’s say, if we know that we are on Earth, and not on Mars, then this does not mean that Mars does not exist. We just aren’t there. If we are in a certain branch of Everett, then this does not mean that other branches of Everett do not exist, we just ourselves are not there. And continuing this series, if we are within one program of the physical laws of the universe, then this does not mean that other programs of universes with other laws do not exist, it’s just that we ourselves are not there.
If we bring this pattern to a logical result, then this means that the fact that we are inside one sequence of bits does not mean that others do not exist. In other words, based on an absolutely a priori hypothesis, all possible sequences of bits exist, absolutely everything, no restrictions, neither on computability, nor on infinity, or even on consistency. Just exist absolutely all mathematics in general.
In general, all mathematics is summed up in hypotheses of zero length and complexity, because it does not limit our expectations in any way. That is, this is not a request to accept some new hypothesis, it is, on the contrary, a request to remove unreasonable restrictions. In the broadest sense, there are no limits on either computability or infinity, and if you claim otherwise, then you are proposing a hypothesis that narrows our expectations, and then it is on you that the burden of proof lies.
However, this can also be called a rethinking of the concept of “exists”, people say it intuitively, but they absolutely cannot give it any definition. And if you refine this concept to something related to controlling your expectations, then you can talk about questions like how likely you are to see contradictions, infinity or something else like that, then you can specifically answer.
For example, one could say that you will never see inconsistent territory because it is a delusion of mind projection, “contradictory” is a characteristic of a map that does not actually have the territory it is supposed to describe.
Seeing contradictions in a territory is like seeing a “you are here” icon or an indication of the scale, or asking what is the error of the territory to itself, the answer is “none”, because the territory is not a map, not a model created from an object, it is the object itself , he cannot be in error with himself, and he cannot contradict himself.
It can all be called something like Tegmark V, mathematical realism and Platonism without limits. And personally, these arguments convince me quite well that all mathematics exists and only mathematics exists, or, to paraphrase, all that exists is mathematics or a specific part of it.
Edited: in some strange way I forgot to clarify, to indicate this obvious point in terms of the space of hypotheses and the narrowing of expectations.
Usually, any narrowing is done by one bit, that is, twice, so you will not reach zero in any finite number of steps, this is because you remove one of the two bits each time in a perpendicular to all previous direction, however, you can also do otherwise, you can not delete any of the two bits, or, after deleting the first bit, delete the second one, delete both, cut not one half, but * (1-1/2 = 1⁄2) = a * 1⁄2, and both halves, a*(1-1/2-1/2=0)=a*0, with such a removal that is not perpendicular to the previous ones, but parallel, we remove both possible options, we narrow the space to zero in some direction, and as a result, the whole hypothesis becomes automatically narrowing our entire space of hypotheses to zero, so that now it corresponds not to half the number of territories, but to zero, excluding both alternatives, it excludes all options.
This looks like a much better, much more technical definition of a contradiction than trying to proceed from its etymology, and thus it is clear that there is no “contradiction” exactly, just any card is obliged to narrow your expectations, leaving only one option for each choice, indefinite leaves both, therefore useless, but the contradictory leaves neither, excludes both options, therefore even more useless.
If there is a contradiction not on the map itself, but in the general system of the map and the territory, there is no such problem, it only means that the map is incorrect for this territory, but may be correct for another, it happens that we already have there is an a priori exclusion of one of the options, but the data received from the territory exclude the second option, if we draw up a second map based on these data, then it will not supplement and clarify ours, but will exclude both possible alternatives along with it, therefore the maps are incompatible, their cannot be combined without getting the exclusion of both sides of reality, the narrowing of space to zero.
An interesting consequence of combining the logical space of hypotheses, Bayes’ theorem, and taking priors from Kolmogorov complexity is that any hypothesis of a certain level of complexity will have at least two opposite child hypotheses, which are obtained by adding one more bit of complexity to the parent hypothesis in one of two possible states.
And, conversely, you can remove one bit from the hypothesis, make it ambiguous with respect to which child hypothesis will fit your world, but then you will need fewer bits of evidence to accept it a priori.
And accordingly, there will be one hypothesis that you will have to accept without proof at all, a program of zero length, an empty string, or rather an indefinite string, because it contains all its child hypotheses inside itself, it simply does not say which one of them is true.
And this absolutely a priori hypothesis will be that you exist within at least one world, the world of the space of hypotheses itself, it is not specified simply in which particular part of this world you are.
And this makes us look differently at the Tegmark multiverse. Because, let’s say, if we know that we are on Earth, and not on Mars, then this does not mean that Mars does not exist. We just aren’t there. If we are in a certain branch of Everett, then this does not mean that other branches of Everett do not exist, we just ourselves are not there. And continuing this series, if we are within one program of the physical laws of the universe, then this does not mean that other programs of universes with other laws do not exist, it’s just that we ourselves are not there.
If we bring this pattern to a logical result, then this means that the fact that we are inside one sequence of bits does not mean that others do not exist. In other words, based on an absolutely a priori hypothesis, all possible sequences of bits exist, absolutely everything, no restrictions, neither on computability, nor on infinity, or even on consistency. Just exist absolutely all mathematics in general.
In general, all mathematics is summed up in hypotheses of zero length and complexity, because it does not limit our expectations in any way. That is, this is not a request to accept some new hypothesis, it is, on the contrary, a request to remove unreasonable restrictions. In the broadest sense, there are no limits on either computability or infinity, and if you claim otherwise, then you are proposing a hypothesis that narrows our expectations, and then it is on you that the burden of proof lies.
However, this can also be called a rethinking of the concept of “exists”, people say it intuitively, but they absolutely cannot give it any definition. And if you refine this concept to something related to controlling your expectations, then you can talk about questions like how likely you are to see contradictions, infinity or something else like that, then you can specifically answer.
For example, one could say that you will never see inconsistent territory because it is a delusion of mind projection, “contradictory” is a characteristic of a map that does not actually have the territory it is supposed to describe.
Seeing contradictions in a territory is like seeing a “you are here” icon or an indication of the scale, or asking what is the error of the territory to itself, the answer is “none”, because the territory is not a map, not a model created from an object, it is the object itself , he cannot be in error with himself, and he cannot contradict himself.
It can all be called something like Tegmark V, mathematical realism and Platonism without limits. And personally, these arguments convince me quite well that all mathematics exists and only mathematics exists, or, to paraphrase, all that exists is mathematics or a specific part of it.
Edited: in some strange way I forgot to clarify, to indicate this obvious point in terms of the space of hypotheses and the narrowing of expectations.
Usually, any narrowing is done by one bit, that is, twice, so you will not reach zero in any finite number of steps, this is because you remove one of the two bits each time in a perpendicular to all previous direction, however, you can also do otherwise, you can not delete any of the two bits, or, after deleting the first bit, delete the second one, delete both, cut not one half, but * (1-1/2 = 1⁄2) = a * 1⁄2, and both halves, a*(1-1/2-1/2=0)=a*0, with such a removal that is not perpendicular to the previous ones, but parallel, we remove both possible options, we narrow the space to zero in some direction, and as a result, the whole hypothesis becomes automatically narrowing our entire space of hypotheses to zero, so that now it corresponds not to half the number of territories, but to zero, excluding both alternatives, it excludes all options.
This looks like a much better, much more technical definition of a contradiction than trying to proceed from its etymology, and thus it is clear that there is no “contradiction” exactly, just any card is obliged to narrow your expectations, leaving only one option for each choice, indefinite leaves both, therefore useless, but the contradictory leaves neither, excludes both options, therefore even more useless.
If there is a contradiction not on the map itself, but in the general system of the map and the territory, there is no such problem, it only means that the map is incorrect for this territory, but may be correct for another, it happens that we already have there is an a priori exclusion of one of the options, but the data received from the territory exclude the second option, if we draw up a second map based on these data, then it will not supplement and clarify ours, but will exclude both possible alternatives along with it, therefore the maps are incompatible, their cannot be combined without getting the exclusion of both sides of reality, the narrowing of space to zero.