Probabilities summed to more or less than unity

John Bjarne Grover

What is so ridiculous about this idea? I have suggested that an epidemiological theory with considerable explanatory force can be constructed on basis of the idea of the probabilities (of, say, an entropy valuation) summing up to more or less than unity (1,0), and my ideas of a 'poetic computer' likewise. See also the ideas added towards the end 'on 7 february 2021' under the Tintoretto discussion - as well as my analysis of Wittgenstein's 'Tractatus Logico-Philosophicus' - the idea that the symmetry of his truth table need not be symmetric after all. The idea is based on the assumption of the human psyche being full of redundancies which creates causalities and laws of nature which a larger metaphysical reality need not consider so very essential. The formula under this file would then be seen to model the interface between a larger metaphysical reality and the historic reality of humans wherein information takes shape.

Humans are seen as special with their language and the logic closely akin to that. The interpretation could be made suggestive by the idea that the functionality of language is formulaically explained as a series of words conjoined by syntactic structures which correlate with logical form. Each word or affix will be an element in a paradigm or subset of words or affixes. The entropy (information flux) is computed from the distribution in each paradigm and these can be combined by the addition and commutative laws of conditional subset entropies - which traditionally take it for granted and self-evident (by definition) that the sum of probabilities in each subset necessarily run to unity. But that is not really how language is, tells I: Those who would argue for the necessary sum of unity tell that a category has its semantic value by the sum of phonological elements therein, and if you change the set of phonological elements classified under a certain category, the semantic value of the category can change as well, but it doesnt half-change: It is either-or. My argument is that it does half-change and that is what syntactic structure is - simply because you cannot think a category without any sort of semantic value therein. Category is inherently semantic, at least a little. Natural language lingers (is that why it is called linguistics?) - like the following scheme:

Value 1
Value 2

Semantic value 1 correlates with phonological elements a,b,c while semantic value 2 correlates with phonological elements d,e,f. But sometimes element c correlates with semantic value 2. Then what? 'Unitarianism' (not to be confused with the religious movement) will claim that then it is just to reconsider the categories and re-compute the probabilities untill unity is regained. Of course, of course, but what about the state of the system in the mean time? That is probably just what History is - and that is where natural language exists: The inbetween time is represented by a slanting branch which goes from c over to value 2 and that is what a syntactic branching of natural language is. (What should the branchings otherwise be?) Even natural language strives towards logical form, that is true, but in the mean time History is not always logical.

Therefore the sum of probabilities by definition summing up to unity can be called abstract logic while the sum being more or less than unity can be called language. Both value 1 and 2 can be more or less than 1,0 - and that could make for the classic four types. The exact rules for which distributions make for equality in the scheme in this file is probably difficult math which I cannot solve, but the overall idea of 2 * over/under could be the general idea of grammatical types. If the criteria for equality when the sums are not 1,0 are not solvable, they could be innate.

The human psyche will naturally strive towards unity because that is when humans are in harmony or balance with the metaphysical reality - and the gates to paradise are ajar. If a serious misbalance between the state of human and metaphysical reality arises and is not easily resolved, or, like that equation which is not easily solved, then the categories that must strive towards unity can lead to a vacuum pump that causes viruses to arise untill unity is regained.

The immune system is a sort of memory which classifies and stores the acquired experience from an epidemic (disease) for solving the problem immediately if it arises again. A vaccine is an artificial immunity. But if the empirical phenomenon that must be understood is very difficult or counter-intuitive (like the present theory?), it may be that the quick briefing is not enough. The worst cases may be those where it is hard to find the source of an idea to credit.

If some fanatics has spread my ideas around but told that it is not permitted to credit Mr.Grover as the source (because of this PTRSIM PIK status - which now must be brought to an end as soon as possible by downgrading and publication of the entire history - the program of 'beast of blasphemy' is no good idea), that could well be a problem which the covid-19 pandemic could be about.

Added on 11 february 2021:

The derivation in this file goes as follows:

The entropy of a category A = a1, a2... an is computed as

H(A) = - ∑i p(ai)/p(A) log p(ai)/p(A)

The entropy of a compound aibj, when for example A = noun roots and B = grammatical morphs which can be attached to it, is consequently the simple

H(AB) = - ∑i,j p(aibj)/p(AB) log p(aibj)/p(AB)

Conditional entropy is the measure on probability distribution in a category B when it occurs dependently relative to a category A. The conditional entropy of B relative to A is H(BIA) and is computed by stopping at all A in the corpus and then counting how many times a B follows in the neighbourhood of A. The probability of the compound p(AB) globally in the corpus is normally defined as the probability of A = p(A) multiplied with the probability of B in the context of A = p(BIA). That is, p(AB) = p(A) p(BIA). One can count the number of AB's in a corpus and compute their probability, and one can count the number of B's in the neighbourhood of A's and compute the relative probability independently - and then counting discrete symbols in a finite corpus will prove that the mathematical relation H(A) + H(BIA) = H(AB) is valid. That is when the symbols are discrete and the corpus is finite.

This is the one out of the laws which apply to this - the addition and commutative laws of conditional entropies. The addition law is this H(A) + H(BIA) = H(AB) - which means that the entropy of category A plus the conditional entropy of category B in the context of A equals the entropy of the compound AB. The commutation law says that H(A) + H(BIA) = H(B) + H(AIB), which means that the addition law applies equally with the dependencies of the relation are imposed on either A or B. Now since

H(BIA) = - ∑i p(ai) ∑j p(bjIai)/p(BIai) log p(bjIai)/p(BIai)


H(A) = - ∑ p(ai)/p(A) log p(ai)/p(A)

and since ∑j p(bjIai)/p(BIai) = 1.0 for all ai, this can be inserted into the expression for H(A), which then gives

H(BIA) + H(A) = - ∑i,j p(ai)p(bjIai)/p(A)p(BIai) log p(bjIai)/p(BIai) - ∑i,j p(ai)p(bjIai)/p(A)p(BIai) log p(ai)/p(A)
= - ∑i,j p(aibj)/p(A)p(BIai) log p(aibj)/p(A)p(BIai)

while the entropy of the compound AB is given by

H(AB) = - ∑i,j p(aibj)/p(AB) log p(aibj)/p(AB)
= - ∑i,j p(aibj)/p(A)p(BIA) log p(aibj)/p(A)p(BIA)

This shows - if I have computed right - that the addition and hence commutative laws for conditional subset entropies are proven by way of the expansion

j p(bjIai)/p(BIai) = 1.0 for all ai

which is the summation which runs to 1 and which is not an allowed expansion if it is more or less than 1. (However, and that is the essential point: The equation could still obtain - but then one has to adjust the probabilities on each side of the equation).

The question is what all this is about. My guess is that the explanation is the following:

Conditionsal subset entropies measures the degree of dependency between two 'occurrences' - and hence is a measure on intuitive causality. The phenomenon of causality is a philosophical matter - but it can be taken intuitively to be about the phenomenon of redundancies in the human psyche on a general level. It can be postulated that this is what the human psyche is - a local field of concentrated redundancies in the universe of free (or potentially different) metaphysical distribution, and inside this bubble of causality and natural laws there are the humans chewing their apples. The big question is all the time: Is the human world comparable to the 'real world' outside the bubble? Do we know anything at all about the world or are we confined in a hermetic enclosure which only rarely opens up for a glimpse of the real world when the clouds briefly let some sunshine through or a Christ walks around on earth for a few years? The phenomenon of pandemics and epidemics tells us that we probably do not really know very much - there is probably a sort of difference which we cannot define so well. What is it?

The commutative law of conditional entropies is therefore probably defined in order to calm down the human nerves - by telling that the dependency between category A and category B relative to A is a measure on causality and hence the explanation to all sorts of things. When the commutative law tells that this entropy is the same as the dependency between category B and category A relative to B, that is, that there is a basis for a mutual understanding of the person who relates from A and the person who relates from B (and these exchange probability distributions in conversation), then it probably calms the human nerves by telling that this also applies for the relation between the human bubble and the world outside - by way of the restriction that the sum of probabilities of the occurrences in A always must run to 1. If it so does, and if we can assume that it does so also outside the human reality, then we can calm down and be convinced that we know something and hence the chances when we come to St.Peter are reasonably good - could be the church can be consulted on the details.

But, tells the panic, if the sum is not necessarily 1.0 every time, there is the chance that a yawning gap of chaos could come to open up.

Therefore it is forbidden to talk of summing probabilities to more or less than 1.0 even if this should be the essence of language.

Instead of touching this superhot forbidden theme, the very apple of modern times, (see Tintoretto's 'Pasqua degli Ebrei'?), science has constructed another theory for handling the same theme: That is Cantor's diagonal proof for the existence of realities outside the human bubble, and if we can compute reliably on those things, we can feel reasonably safe nevertheless.

The diagonal proof is a joke as far as this is concerned - and poor Cantor himself landed in psychiatric custody - but it seems to function as panic damper for modern science. Believe it or not, the famous proof tells that the ocean looks green but we can let down a drop of red fluid in it and then it is no longer green: Hence it is red, tells the proof.

It tells that the set of real numbers is countable since we can order them - but then we can construct a diagonal number which we thereby can prove is not in the countable set - and hence there exist uncountable sets, tells the proof. But, alas, this single uncountable number must be countable? It is just to count it: 1,0 is the amount of it.

Yes, tells the editor of the german periodical on logic and mathematics I spoke with on the phone (I do not remember who or which journal, it is probably this talk which is mentioned in my diary novel on 19.7.94 and 20.9.94), that is right, "aber man kann beliebig viele Diagonalen machen".

I also landed in psychiatric hospital some time later.

It is likely that this tells that 1,0 is the story and this 1,0 can be bigger or less than one. Then they sent Cantor to the psychiatry. There later arose a historic revenge called IDLE FITTER - that is the algorithmic ordering of the numbers by a FITTER who for the time being therefore is IDLE, that is the extra diagonal. This is the one and only threat which humans do not want to hear.

It is a hilarious comedy, this means.

You dont accept the logic - that one drop of red in the ocean of green turns the ocean red? Could be this question is the standard test for being admitted to universities. (In Norway I think the admittance used to be by a 'matriculation' ceremony called 'immatrikulering' of handshake with rector or a dean - 'himmel-trykk-ulering' could then resemble the handshake of Adam & Eve in Tintoretto's 'Peccato originale').

The fact is that the word COUNTABLE is an adjective - and an adjective is really a witness report: This means that we have witnesses to what is outside the human bubble - PTRSIM PIK or is it CHRIST - for the conclusion that we have reasons to believe that we know something about our chances at St.Peter. Doom is not the only probability.

If the two sides of the commutative law apply inside the human bubble (not bible!) to the left of the equals sign and outside to the right, then a summation to more or less than 1,0 can still lead to equality if we can get the variables under control. But it may be that 1,0 is the safest.

In Molde in the 60's I went to private lessons in modern math at 'Steinsland', as was his name. When we came to Fredrikstad in 1970, I was told to leave the class in the math lessons and sit in the Aula for reading the book by Walther R.Fuchs called 'Gyldendals bok om moderne matematikk'. This was at the same time as school dentist 'Aulie' drilled up all my teeth - and I have computed (on rather weak empirical basis, though) that sexual abuse of me could have been going on. When I took the book out again decades later, it opened by itself on Cantor's proof - which I remember that I studied with some interest. 'Walther R.Fuchs' = 'information flux'?

Added on 14 feburary 2021:

The discussion can be completed with the notice that Cantor's transfinite numbers (the 'existence' of which is proven with his diagonal proof - that the green ocean is red) include the idea of the continuum hypothesis: This tells that the socalled ℵ0 = the normal infinity of natural or real numbers, that is, the set of countable numbers, relates to the set of transfinite (first-level uncountable) numbers ℵ1, by the equation

20 = ℵ1

It has counted as one of the riddles of mathematics to prove this equation, this hypothesis of Cantor. In light of the entropy formula, the commutative law, it is easy to see what it really means: It means that the 1,0 to the left of the equals sign plus the 1,0 to the right sum up to 2,0 exactly, and Cantor's hypothesis is that this real number 2,0 is the same as the integer number 2 - anything else leads to chaos and despair, could be depressions and wars. Enormous amounts of money (such as the billions spent by the research councils and universities) are invested into keeping the two sides of the equals sign stable and balanced - for the sake of our nerves. Probability theory is to pull the truth by its nose the one way, and Cantor's proof is to pull it by the nose the other way (you have to accept the red ocean on a logical basis) - the hypothesis tells that the noses must be pulled equally on both sides for reaching a balance. It may be essential that the theory balances what is rounded or 'butted' off in probabilities against what is rounded or 'butted' off in logical rigidity.

The role of Norway as 'the noses' - including those theories of vikings when the noses ruled the oceans of the world - could be just this. Could be 'the oceans' is not much more than what you can blow into a handkerchief - and hence that the theories of truth which rule the societies of 'homo sapiens' are not so far-reaching. Cantor's daring hypothesis could have been there somewhere - telling also that it is not impossible that the human theories of truth - if only peace and stability can arrived at - can come to be taken to reach a little further - beyond own nosetip.

Consulting the internet and not my old knowledge (I was used to think of the transfinite numbers as being beyond the real numbers), I cannot really find out of what things currently count as - a quote from this article goes:

The statement that there is no subset of the reals with cardinality strictly greater than ℵ0 and strictly smaller than 'c' is known as the continuum hypothesis (CH). It is known to be neither provable nor refutable using the axioms of Zermelo-Fraenkel set theory including the axiom of choice (ZFC) - the standard foundation of modern mathematics. In fact, some models of ZFC satisfy CH, while others violate it.

This sounds like pulling up the sleeve ('the erme-lo') for a frankel setting of the theory that the choice of axes is yours - halberd, double-edge, classic woodcutter or whatever - for a 'holmgang' which was the viking style of duel. It sounds more like power than truth - and many pages on the internet are bristling with jargon and notation. (Jargon is normally a sign of power more than truth). It could tell that the sum of probabilities must run to 1,0 - neither more nor less - summed to 2,0 for both participants in the 'conversation', and it is the best option for both that no part of the integers are removed from the 'corpus of distribution', sth like that.

In my view, this very unpleasant idea of oldfashioned axes in 'holmgang' (Valhalla transfinity?) could serve to scare people off the idea that a syntactic branching arises from a part of the probability paradigm entering into a biased summation to more or less than 1,0. The confusion of such entropies against the continuum hypothesis could be seen in the status of transfinity in number theory.

The exponential ℵ0 to the integer 2 suggests that this is the same as the role of the logarithm in the entropy or 'information flux' valuation.

In my former conceptualization, the numbers used to be the following:

Natural numbers = 1,2,3...
Integer numbers = 0,1,2,3....
Rational number = fraction of integers = infinite extension after comma with cyclicity/redundancy
Complex number = two or more rationals or reals in union
Real number = infinite extension after comma without cyclicity/redundancy
Tranfinite number = beyond any normal enumerable infinity

Do strange numbers exist that wedge in between the decimals of real numbers? Could be, and I would be inclined to believe so, but then the field of study is getting more semiotic than mathematic.

Clearly since all integers are easily enumerated on an x-axis, and all rationals are enumerated on an infinite two-dimensional (for the two parts of the fraction) chessboard a1,a2,b1,c1,b2,a3,a4,b3,c2,d1,e1,d2..., which can be called diagonal enumeration, all reals can then be enumerated (ordered for counting) in a similar way when what is to the left of the comma is written backwards (or vice versa) with infinite trail of zeros and similarly for what is right of the comma in the other dimension. (This enumeration will not follow the ascending 'size' of the number but will jump-around according to symbol of notation). I used to think that ℵ0 was the cardinality of these enumerable infinities while ℵ1 was the cardinality of the first uncountably infinite set - an infinity bigger than the normal one - but maybe this pulls the sleeves (for a Zermelo-Fraenkel choice is yours) more than the noses nowadays. The existence of the infinity bigger than the infinity of the natural numbers (ℵ0) is then proven or defined by Cantor's diagonal proof - which, though, boils down to a variant of the diagonal enumeration, like the T/F truth table of Wittgenstein: This corresponds to the diagonal which crosses through origo at a1 - and by changing all its digits (here ...000,000...) into something else there is the diagonal number which then must be 'non-existent' among the reals but exist in a higher 'transfinity'. The clue is probably that this is the same as the enumeration diagonal a1,b2,c3,d4.... with values 'True' for all entries in the enumeration - but when turned into 'False' in the diagonal it tells that the number ...111,111... does not exist among the infinity of reals.

The same scheme can be used to account for rational numbers but then the numbers in both dimensions are finite and not infinite, their fractions being the rationals.

The sum of all this confusion is therefore probably that the theory tries to account for the mystery of swap of LEXIS = the finite symbolism with LOGOS = the infinite mystery which seems (by my theory) to be the essence of the conflict between catholicism and lutheran (script-based) protestantism. The happy conclusion at the end of all this confusion seems to be that there is no real conflict between catholicism and lutheranism except for the difference between semiotics and symbol-manipulating mathematics. (I always believed that this is why it is called 'continuum hypothesis' - that the continuum is what is beyond the discrete symbols of mathematics). The conclusion that these are the same nevertheless tells that there is a safe basis for the information technology - that it balances in a 1,0 = 1,0 identity with the metaphysical reality. It takes (according to the interface between my blue PEB = POLAKK English Bloggi and the historic reality) about 1000 years to develop a new information technology - Luther was about half way - and that is the time it takes to balance the human reality up against the surrounding ocean of metyphysics - be that red or green.

This explains also the mystery that if the number 2 in Cantor's continuum hypothesis is the same as the identity 1.0 = 1.0, it means that you move one of the 1's over to the other side for 2 = 0. How come? It is because LEXIS swaps with LOGOS that 1 is added and not subtracted as usual.

Riddle: The continuum hypothesis is the first of Hilbert's famous 23 problems set forth in 1900. The riddle is now: Are these 23 problems the same as my 23 black-and-white photos from the Danube Island? For this mystery, see also the very interesting parallelism with the chinese film from 1963. I notice e.g. the mystery of photo 23 apparently showing alphabetic letters taking shape in a landscape: Does it tell 'φωσ-STA' or is it 'STAu-ροσ'? The current 'continuum hypothesis' could be the first = the second photo - the 'map of Europe', so to speak - as if the status of 'real numbers' were moved from France-Normandie and the PTRSIM PIK of England over onto the 'iron curtain' with the origins of Adolf Hitler - the very diagonal IDLE FITLER?

PS 15/2-21: The observation on Hilbert's 23 problems in parallel with my photos and the chinese film could be of some interest. If you should want to use it, if, say, you should come to suggest e.g. that the 'imaginary quadratic field' of Hilbert's 12th problem has affinities with the rolling board of the male archetype in a 'man-on-wagon' format of my photo #12 from the Danube Island (the scene is seen from the other side of the street compared with a normal viewpoint - from behind the boy with the lifted hand behind his hat and the waiting board under and in front of him), do not hesitate with crediting my work. If there should come suggestions that 'it is not permitted to mention the name of the PTRSIM PIK', that is of course a lie - on the contrary, the law tells that the source should be credited. All that old political PTRSIM PIK nonsense will be downgraded and the whole political abuse circus which it seems to be will be terminated and brought to daylight and it will then probably be easily shown that it was a bunch of criminals who ran the program of abuse. I suppose you also want to be paid for the work you do. To claim that this is something very different from everything else - a sort of quasi Jesus or Messiah who is supposed to carry the burden for other people - would be only an attempt to engineer a 'beast of blasphemy' from the revelation and it would be an example of a violation of human rights only.

18/2-21: See also 'The clotheshorse' for Hilbert's 13th problem.

© John Bjarne Grover
On the web 10 february 2021
Last updated 18 february 2021