I didn't say that indirect proofs were the only methods of inference, ...
You did say that:
And if RAA is not a tautology then clearly indirect proof is not an inference method.
Do I really have to explain the implication of that?
I don't need to tell you that proofs are fundamental to mathematics - coming well before such niceties as you are proposing.
No proofs, no mathematics.
In other words, you conclude that if there are no indirect proofs, then there is no proof at all, which is a false conclusion.
Robin said:
In any case, proofs definitely are fundamental to mathematics.
Again, you clearly talked about proofs only in terms of indirect proofs.
Robin said:
Removing contradictions has implications to the rest of logic as well as you should be able to see.
For example what happens to "x implies y and not y implies not x"?
It is not "p that is both true and false" in any system of logic, so your point is invalid.
If "x implies y and not y implies not x" is equivalent to "p AND ~p" where ~p is the negation of p, then you are still using indirect proofs, which have no impact on direct proofs. Furthermore, in constructive mathematics, ~p is not necessarily the negation of p, and in this case "p AND ~p" is not necessarily a contradiction,
Robin said:
But "P AND NOT P" is a contradiction in constructive mathematics.
Only in the case that ~p is the negation of p, but this is not the only alternative about p and ~p, since by constructive mathematics ~p is not necessarily the negation of p.
Robin said:
There is an error in this sentence which makes it hard to parse. Did you mean "neither are"?
No one of the considered frameworks (direct proofs, indirect proofs, proofs of constructive mathematics, where ~p is not necessarily the negation of p) is the whole mathematical science.
Robin said:
You could say that constructive mathematics is at the root of the tree to, dealing as it does with metamathematics.
I can't say it, because constructive mathematics, is one of the branches of this science.
Robin said:
What do they have to do with the price of tea?
Nothing. Do you think that mathematics deals only with problems that are related to "the price of Tea" ( you ignored Qbits
http://en.wikipedia.org/wiki/Qubit )?
Robin said:
In any case, you did not answer my question. Are there theorems and proofs in your mathematics?
Let us start by 0 <
x <
∞.
It is derive from the concept of "magnitude of existence" as follows:
Let us use the empty set in order to define the concept "magnitude of existence".
Definition A: That has no successor has "the maximal magnitude of existence".
Definition B: That has no predecessor has "the minimal magnitude of existence".
The current scientific method, which was developed since the 17th century, states that the researcher must be omitted form the research environment, in order to avoid results that are influenced by subjective tendencies of the researcher. It must be stressed that Aristotle determined 4 causes that sands at the basis of any existing thing, which are:
1) The material cause (from what material a given thing is made?).
2) The efficient cause (what are the natural forces that change a given thing?)
3) The formal cause (what is the "blueprint" in once mind that has an influence on a given thing?)
4) The final cause (what is the final goal in once mind, that has an influence on a given thing?)
The current scientific method uses only causes (1) and (2), in order to avoid any researcher's subjective influence on the result.
A question: Is it possible to return the researcher to the research environment and also avoid his\her subjective influence of the results?
My answer: I think that it is can be done if the "researcher" returns to the research environment as a general concept.
Let us use definitions A and B in order to demonstrate this notion.
{} describes the "researcher" in terms of definition A where what is between {} is the "researched" in terms of definition B (known also as "emptiness").
The outer "{""}" of {{}} describes the "researcher" in terms of definition A, where the inner "{""}" of {{}} describes the "researched" in terms of the model of the "researcher" that his\her "researched" subject is "emptiness".
At {{}} case "the magnitude of existence" of the "researched" is greater than "emptiness" (it has a predecessor) and smaller than the "researcher" (it has a successor).
Whether a non-empty collection is finite or infinite, it has both predecessor ("emptiness") AND successor ("researcher").
Anything that has both predecessor ("emptiness") AND successor ("researcher"), can't be reduced into "emptiness" AND can't be extended into "researcher". Because of this reason any given infinite collection can't have an exact Magnitude that is described by Cardinality.
If we symbolize these notions then: 0 <
x <
∞, where
∞ represents the "the maximal magnitude of existence"("researcher") , 0 represents "the minimal magnitude of existence" ("emptiness"), and
x represents "the magnitude of existence" that is > "emptiness" AND < "researcher" (any non-empty collection, whether it is finite or infinite).
By carefully research "the magnitude of existence" of an infinite collection, it is concluded that the universal quantifier
"for all" has no meaning, because the strict "magnitude of existence" of any given infinite collection can't be satisfied (
x <
∞). On the contrary, the universal quantifier
"for all" has a meaning in the case of finite collections, because given any member of a finite collection it is defined as its final element, which in turn provides the strict "magnitude of existence" that is described by a strict Cardinality (A finite collection does not converge and does not diverge, which is a property that enables to determine its strict cardinality).
In other words, if
x is the cardinality of an infinite collection then
x <
∞ prevents its strict value.
Since the cardinality of any given infinite collection is non-strict, then the 1-1 correspondence technique can't be used to determine any meaningful thing about the cardinality of such collections.
Furthermore, the Contor's diagonal upon decimal representation of irrational numbers, actually proves that it is impossible to determine the strict value of
x if
x is related to an infinite collection.
Moreover, the ability to define a 1-1 correspondence between the natural numbers and a proper subset of them is a direct result of the impossibility to define the strict cardinality of an infinite collection.
I wish to add that Russell's paradox does not hold in this case since that has cardinality
∞ can't be identical to any of its members, because any given member has at most
x "magnitude of existence" and
x <
∞ .
- - - - - - - - - - - - - - - - - - - - - - - - -
The minimal observation is the result of the linkage among the total "magnitudes of existence", such that the "researcher" observes "emptiness".
A non-minimal observation, is an observation of "non-emptiness", where a non-minimal observation can be finite or infinite.
Some examples:
{} is the minimal observation ("that has no successor" observes "that has no predecessor").
{1} is an example of a finite observation.
{1,2,3,4,5,…} is an example of an infinite observation.
Code:
{
{1,2,3,4,…}
↕ ↕ ↕ ↕ is an observation between an infinite observed thing
{2,4,6,8,…} and its infinite proper subset.
}
etc … etc … infinitely many observations, where "that has no successor" (the non-personal aspect of the concept of "researcher", which is notated by the outer "{""}", is always presented.
- - - - - - - - - - - - - - - - - - - - - - - - -
0 <
x <
∞
X is a placeholder for 0,
x or
∞
Definition C: "That has no predecessor is not a successor".
Example: X+0=X
Details:
5+0=5, where 0 is not a successor.
|infinite collection|+0=|infinite collection|, where 0 is not a successor.
∞+0=
∞, where 0 is not a successor.
Definition D: "That has no successor is not a predecessor".
Example:
∞-X=
∞
Details:
∞-5=
∞, where
∞ is not a predecessor.
∞-|infinite collection|=
∞, where
∞ is not a predecessor.
∞-
∞=
∞, where
∞ is not a predecessor.
