On 06/01/2018 13:38, Michael Cree wrote:
On Sat, Jan 06, 2018 at 11:21:14AM +1300, Lawrence D'Oliveiro wrote:
On Sat, 6 Jan 2018 10:50:55 +1300, Wolf wrote:
The rub is that in order to exclude membership from a set that set has to be finite. Which is a fallacy. Indeed, it is. A counterexample would help to see that. The set of integers is an infinite set. We note that there is an ordering operator on the integers, that is, we can establish unambigously whether any integer is bigger or smaller than any other integer, thus we can arrange the members of the infinite set of integers such that all integers listed to the right of a chosen integer in the set are larger than the chosen integer and all integers listed to the left of the chosen integer are smaller than the chosen integer. We can repeat that by choosing every integer in the set in turn, and it establishes a unique list (i.e. sorting) of the integers.
Now consider the rational number 3/2. We can see that is is not a member of the set of integers because it is bigger by the ordering operator than the integer 1 thus 3/2 should land to the right of 1 and it is smaller than the integer 2 thus it should land to the left of 2, but the unique sorting of the integers established that there are no elements in the set of integers that land both to the right of 1 and to the left of 2, thus 3/2 cannot be a member of the set of integers.
We have therefore excluded membership of 3/2 from the infinite set of integers, and we have been able to do that even though the set of integers is not finite.
Sorry, Michael, you have not, because of what you have hidden from your argument - data conversion rules if you restrict the argument to smallish numbers only (i.e. what the storage capabilities available to you allows you to work on), and no side-bands if all physical phenomena are included. The problem is that logic has been created at a time (about 800-400 BC, in the Greek speaking world, as a tool to win verbal arguments) not knowing that side-bands do exist. Physics, on the other hand, came about when Nicolaus Copernicus required that logical arguments also meet the realities of the real world (1543, in his book /De revolutionibus orbium coelestium). /The first inkling that side-bands do exist is due to Joseph Fourier, who published in 1822 (in "/Théorie analytique de la chaleur/") the first steps towards what is today known as Fourier Analysis. Since Fourier Analysis interprets a discontinuous function (i.e. a function describing the border of a discrete entity) as a superposition of continuous functions, any of these component functions constitutes a side-channel. Any of these side-channels can, of course, be isolated using an appropriate filter. And that physical entities are all discrete entities we can know, at the latest, when we apply General Relativity (for large entities) or Quantum Physics (for small entities). Now in detail to the fallacy of your argument. It was Joseph Fourier, again, who first disproved it, by showing that the dimension of the right hand side of an equation must be the same as the dimension on the left hand side. To give an example: Is 5 liter equal to 5 meter, or is it not? The answer is, as you know, that the question is undecidable, because you cannot compare apples with pears. Equally, when you compare 3/2 with 2, the comparison is undecidable if your comparison operator works on integers only. But if you expand the operator's capability by converting first the integer 2 into the rational number 2/1, expanding that to 4/2, you can indeed compare 3/2 with 2 and decide that 3/2 is less than 2. You can even expand your "proof" to include real numbers such as sqrt(2), by converting 2 into the real number 2.0, and comparing digit by digit. But you cannot expand your "proof" to a transcendental number like pi. How are you going to distinguish pi from a similar number that differs from pi only in its millionth digit when available storage forces you to abandon comparison after half a million digits? Infinite recursion depth is possible in the fictitious world of mathematics, but not in the real world. Your argument works because you have hidden from yourself some of the conditions that must be met before a comparison operator can be employed. Now reverse the argument, and convert 3/2 into an integer before applying the comparison. If the conversion is done by trunc(3/2), 3/2 becomes equal to 1. If round(3.2) is used, 3/2 becomes equal to 2. The result of the comparison becomes dependent upon the construction of the comparison operator. Or, consider this example: 1+1=2 Is that true? If you restrict yourself to the world of logic, it is. On the other hand, if you consider the numbers as measured quantities, then round(1.4)+round(1.4)=2, but round(1.4+1.4)=3 or, in general 1+1=1..3 if the numbers representing physical quantities are rounded to the nearest integer before/after addition takes place. That is where we differ. You restrict yourself to living in a one-dimensional world that can be divided into smaller and smaller quantities, ad infinitum, which is what logic does and where side bands cannot exist. I have been trained as a scientist and thus accept that all observed (and observable) quantities are of finite accuracy. You hide from yourself that the statement 1+1=2 is false in the real world (the world of Copernicus, which requires of logic that it matches observations in the real world, to the limit of measuring accuracy), by failing to consider the abstractions from reality that went into creating Logic, and the consequences this has when logic is applied to the real world. In other words, you have been lying to yourself. For me, 1 plus 1 is equal to any number between 1 and 3, because rounding, and its consequences, is part of my world. Intel, AMD and others are only now discovering that the reasoning they relied upon cannot be trusted. Since cumulative rounding errors can make any number grow out of bounds in a recursion, my original statement "The rub is that in order to exclude membership from a set that set has to be finite." still stands. Recursion is permissible in some situations, but not in all. The fallacy lies in ascribing trustworthiness to a system of reasoning (logic) that this system does not possess. Mathematicians have learned that, to their sorrow. So why not try to change the question? wolf