Anomaly
Before modern science existed means before Leonardo, Galileo and Descartes and before modern mathematics existed means before Euler, Mozart and Boole. So before those people the school of invention existed. And one of the schools of invention that existed in the same places as Leonardo, Galileo, Descartes, Euler, Mozart and Boole was based on three principles. And each of the three principles had three elements. The three principles were induction, analogy and anomaly. And the three elements of induction were constancy, duration and generality. The three elements of analogy were similarity, commonality and familiarity. And the three elements of anomaly were difference, peculiarity and exceptionality.
The idea of induction was the study of the truth of propositions. That is to say, where some sensible thing called phenomena was noticed to be the same in this place as that place it was called constant. And where it was noticed to be the same in this time as that time it was called duration. And where it was noticed to be the same throughout its examples it was called generality.
The idea of analogy was the study of the relationship between different things. That is to say, where some two different things were noticed to coincide with each other in some way they were called similars. And where the two different things were noticed to belong to the same larger thing it was called commonality. And where they were noticed to be connected to other things of like nature it was called familiarity.
The idea of anomaly was the study of those things not included. That is to say, where induction and analogy were used it was noticed that any thing excluded was called different. And where some things that were noticed would not be explained by induction or analogy they were called peculiar. And where some things were noticed to be unusually advanced beyond what was normal they were called exceptions.
Since Leonardo, Galileo, Descartes, Euler, Mozart and Boole things are much easier because modern science now exists. Modern science does not have to worry about three principles each containing three elements. Because modern science has only to deal with four functions, three processes, two rules and one exception. The four functions are called hypothesis, experiment, validation and model. The three processes are called generalisation, deletion and distortion. The two rules are called prediction and falsifiability. And the one exception is called anomaly.
In regard to the four functions: Hypothesis is to affirm a proposition which may or may not be true, such as "It is the case that stable things remain in place". Experiment is to find examples of where the propostion is true by pushing on stable things to see what happens. Validation is to confirm the correctness of the hypothesis in more not less instances. Model is the use of a method called deduction to provide good argumentation from the proposition to its proof.
In regard to the three processes: Generalisation is the requirement to demonstrate what is normally the case. Deletion is the requirement to eliminate any peculiar examples from the result that contradict what is normally the case. Distortion is the requirement to adjust the explanation of the result so that what is normally the case can be presented as a law of nature.
In regard to the two rules: Prediction is the rule that we must be able to determine beforehand in regard to any experiment that one of two most probable outcomes always happens and that possible improbable outcomes do not happen. Falsifiability is the rule that we must be permitted to check our results for correctness and accuracy or incorrectness and inaccuracy.
In regard to the one exception: Anomaly is the understanding that the use of hypothesis, experiment, validation, model and generalisation, deletion, distortion and predictability and falsifiability has a necessary outcome in terms of peculiar, particular exceptions that are not detailed in any of our work and only show up as blanks, missing links and connections that don't exist.
The anomaly pattern here is:
(+1) x (+1) = (+1) Plus one multiplied by plus one is plus one.
(+1) / (+1) = (+1) Plus one divided by plus one is plus one.
(-1) x (+1) = (-1) Minus one multiplied by plus one is minus one.
(-1) / (+1) = (-1) Minus one divided by plus one is minus one.
(+1) x (-1) = (0) Plus one multiplied by minus one is zero.
(+1) / (-1) = (0) Plus one divided by minus one is zero.
(-1) x (-1) = (0) Minus one multiplied by minus one is zero.
(-1) / (-1) = (0) Minus one divided by minus one is zero.
Given the pattern as stated, that means:
(+1) x (-1) =/= (-1) x (+1) Plus one multiplied by minus one does not equal minus one multiplied by plus one.
(+1) / (-1) =/= (-1) / (+1) Plus one divided by minus one does not equal minus one divided by plus one.
(+1) x (-1) = (+1) / (-1) = (-1) x (-1) = (-1) / (-1) Plus one multiplied by minus one equals plus one divided by minus one equals minus one multiplied by minus one equals minus one divided by minus one.
The definition of the rules for the above pattern is:
1. Positive number multiplied by plus one is original positive number.
2. Positive number divided by plus one is orginal positive number.
3. Negative number multiplied by plus one is original negative number.
4. Negative number divided by plus one is original negative number.
5. Positive number multiplied by minus one is zero.
6. Positive number divided by minus one is zero.
7. Negative number multiplied by same negative number as itself is zero.
8. Negative number divided by same negative number as itself is zero.
The question raised by the anomaly pattern is whether the Q function, which we could call the quantum factor or the indefinite variable or neither the unity set nor the empty set is a necessary element of the given terms. And the fact that it proves the deep structure involves an either or switch. That is, because we can detail two different rules at any one time for “minus a divided by minus a” such that it can give either zero or minus one. And dependent whether the rule we are using gives zero determines the coherent rule base that must be followed by the other rules. Since if we say the rule is that “minus a divided by minus a gives minus one” then the other rules must conform to that. What it means is, that “for any minus a divided by minus a, the answer may at any stage be either minus a or zero, which is the quantum variable or Q. Since we have built the quantum variable into our prime pattern it is useful to have the anomaly pattern clearly stated in order to prove Q.
To Prove Q: Given minus one divided by minus one.
Minus one divided by minus one equals minus one If the rule we are following is any negative number divided by itself is minus one or alternatively if the rule we are following is any negative number divided by minus one is minus one. Minus one divided by minus one equals zero If the rule we are following is any negative number divided by itself is zero or alternatively if the rule we are following is any negative number divided by minus one is zero.
Then since there is an indefinite variable built in to our choice of rule at that precise point we have to use the Q function in order to notify ourselves that a more abstract universal level does not make the decision either way. Instead, at that more abstract level neither the empty set nor the unity set is quite acceptable.
Given the anomaly pattern raises the matter and does not solve it, then this is one possible direction towards its solution.
We can state specific matters in their most clear formulation at this stage, with the understanding that our workings out may enable us to better state later.
One. The rule that any number whether positive or negative when multiplied by the unit number called plus one, then the result will be the original positive or negative number.
Two. The rule that any number whether positive or negative when multiplied by the unit number called minus one, then the result will be zero.
Three. Therefore minus one multiplied by plus one must always give the result called Q, that is, the quantum indefinite or neither one nor zero or both minus one and zero.
To clarify: according to rule one, 10 x 1 = 10 ; -10 x 1 = -10 ; -1 x 1 = -1. To clarify: according to rule two, 10 x -1 = 0 ; -10 x -1 = 0 ; 1 x -1 = 0. To clarify: according to rule three -1 x +1 = -1 ; +1 x -1 = 0 therefore Q.
Now the only thing we can do at this stage is to challenge our rules, that is to say, either rule one or rule two is false, or both rule one and rule two are false.
Since rule one is held to fit the condition "most certain" we would tend to maintain that as correct. Then we must challenge rule two. What we must do here is ask ourselves why rule two is the given rule rather than some other rule, and we do that so that we do not immediately replace rule two with a different wrong answer.
The reason rule two is as given is because there is an earlier conditional statement. The earlier conditional statement is that "multiplication by minus one" must provide a different answer to "multiplication by plus one".
Using our sign labels that means:
a . +1 = a and ~a . +1 = ~a
a . -1 =/= a and ~a . -1 =/= ~a
Then in order to meet the earlier conditional statement we determine whether we can find out what "multiplication by minus one" must give. The way we do so is to first ask what we mean by "multiplication by plus one". And we say, any positive number multiplied by plus one gives the original positive number because multiplied by plus one means to say : One example of the original number. And we then say: multiplication by plus two means two examples of the given number combined. And multiplication by plus three means three examples of the given number combined.
Following the same rule through to any negative number multiplied by plus one gives the original negative number because multiplied by plus one means to say: One example of the original negative number. And multiplication by plus two means two examples of the given negative number combined. And multiplication by plus three means three examples of the given negative number combined.
Given that, then what must "multiplication by minus one" give? And our answer is any positive number multiplied by minus one gives zero because multiplied by minus one means to say: one example of the given number subtracted. So any positive number subtracted from itself must give the answer zero.
What we cannot say is that any positive number multiplied by minus one gives the negative of the same number, because that is what we would mean by multiplied by minus two.
The proof is as follows:
If the original number is plus ten, then:
+10 x -1 = +10 - (+1 x +10) = +10 - 10 = 0. +10 x -2 = +10 - (+2 x +10) = +10 - 20 = -10. +10 x -3 = +10 - (+3 x +10) = +10 - 30 = -20.
Which makes most sense because the difference between plus ten and zero is ten units, the difference between plus ten and minus ten is twenty units, the difference between plus ten and minus twenty is thirty units.
If we decide that is not acceptable then the rule we would replace it with is that any positive number multiplied by minus one gives the same number in the negative. That would be:
+1 x -1 = -1 +2 x -1 = -2 +3 x -1 = -3 +10 x -1 = -10
In that case we would have plus one multiplied by minus one is minus one.
And we would have minus one multiplied by plus one is minus one.
Which would seem to be okay, so we go back to our original pattern and see how it fits. We are required to make all the other changes demanded to maintain coherency within the pattern, and then to restate our rules for the different version of the anomaly pattern.
The anomaly pattern here is:
(+1) x (+1) = (+1) Plus one multiplied by plus one is plus one.
(+1) / (+1) = (+1) Plus one divided by plus one is plus one.
(-1) x (-1) = (+1) Minus one multiplied by minus one is plus one.
(-1) / (-1) = (+1) Minus one divided by minus one is plus one.
(-1) x (+1) = (-1) Minus one multiplied by plus one is minus one.
(-1) / (+1) = (-1) Minus one divided by plus one is minus one.
(+1) x (-1) = (-1) Plus one multiplied by minus one is minus one.
(+1) / (-1) = (-1) Plus one divided by minus one is minus one.
The definition for the rules for the above pattern is:
1. Positive number multiplied by plus one is original positive number.
2. Positive number divided by plus one is orginal positive number.
3. Negative number multiplied by minus one is positive of original number.
4. Negative number divided by minus one is positive of original number.
5. Negative number multiplied by plus one is original negative number.
6. Negative number divided by plus one is original negative number.
7. Positive number multiplied by minus one is negative of original number.
8. Positive number divided by minus one is negative of original number.
The position we are now in is where we have two versions of the anomaly pattern, each of which is coherent with the rules associated to its own pattern, and each of which provides a different answer to the other.
Given that we have two versions of the anomaly pattern where we would prefer to only have one, and we have two sets of rules where we would prefer to only have one, then the idea may be to determine whether one of the two patterns is clearly correct and the other clearly wrong.
However, what we are prohibited from doing is making an arbitrary choice to affirm one pattern and negate the other simply because we only want one pattern to deal with. By which I mean, we can do so, and if we do so it is just as likely that a different decision maker will make the contradictory choice. Which would mean that given both versions of the anomaly pattern and two different decision makers it is quite possible for both decision makers to take a contradictory line to the other in regard to assigning a truth value to either pattern.
Since that is reasoned, we are in a position where we would like to be confident that one version of all the possible versions of the anomaly pattern is more correct than any other possible version. To obtain that more correct version necessitates we consider the higher order pattern through universal abstraction. And we already know that pattern will require the use of the quantum indefinite, neither one nor zero, or Q.
What that gives us is the answer that either rule base for the anomaly pattern is correct and at the same time notifies us that more than one rule base exists for the particular set of terms. What it will not do is specify that one answer is more correct than the other. Essentially what the higher order universal abstraction says is: Multiplication of any number by minus one gives minus one or zero or plus one or the negative of the original postive number or the positive of the original negative number. That is the nature of Q in the universal abstract. And all it means is that particular instances of Q will give only one of the possible answers, dependent on being coherent with the rule base being applied in that instance.
For that reason we may decide to further detail the anomaly pattern at the level of mathematics without moving up a logical level to the universal abstraction. Because what we would like to do is determine what are the conditions that enable both versions of the anomaly pattern to exist. And as well, to consider whether it may be possible to detail how the patterns may relate one to the other.
And the problem with that line will be that what it will show is that one set of rules for the anomaly pattern is referring to the result as a processing number and the other set of rules for the anomaly pattern is referring to the result as a final object. Which will then mean that both answers exist at the same time. Using this example provided earlier to clarify what I mean:
+10 x -1 = [+10 - (+1 x +10) = +10 - 10] = 0. +10 x -2 = [+10 - (+2 x +10) = +10 - 20] = -10. +10 x -3 = [+10 - (+3 x +10) = +10 - 30] = -20.
The information contained in the square brackets is a processing number and the information on either side of the brackets is a final object number.
It means that when we say +1 x -1 = -1 we mean that the result "-1" is a processing number that must then be applied to the original object number of "+1" which will give "+1 -1" as a processing calculation which then works out as "0" as a final object number.
In that way it may be that the second anomaly pattern details in whole or in part the processing number in regard to the stated terms, which when then worked out against the original object number will give the object result detailed in the first anomaly pattern. And the difficulty is that if we accept this theory the second anomaly pattern strengthens the first anomaly pattern because it says it provides a processing number result for the final object result. And if that is correct then we are back to the original anomaly.
What the previous explanation does is direct attention towards a simple pattern that can be detailed in two ways, both different and either of which could be considered correct. It easily may be that a third or fourth version of the same patterns can be detailed. The only requirement in this matter is that the basic format is adhered to and that the rules being applied in any particular version are properly described. In that way it may be possible that one pattern is shown to be clearly correct making the other versions less necessary. Since all that is desired in this particular is that we can detail one correct version of the given variables.
For a more detailed discussion of some of the ideas that lead towards the anomaly pattern and perhaps some alternative methods that could help towards its solution, please follow the hyperlink: http://en.wikiversity.org/wiki/Boolean_algebra
For a complete index to the various articles I have used to introduce these and related patterns, please follow the hyperlink:
