If I understand your question correctly, you are asking in effect how do we distinguish logic from non-logic? Logical expressions give rise to valid arguments and logical truths, that is, arguments where if the premises are true it is impossible for the conclusion to be false, and truths such that there is no way for them to come out as false. But this ...
The "rules of logic" are the object of study of formal logic and mathematical logic.
They define languages and proof systems, like e.g. predicate calculus, that are "applicable" to any topics whatever.
The so-called "laws of logic" are formulas that are true irrespective of any possible interpretation, i.e. they hold in every interpretation.
In this ...
The issue is with the exprsssion "number line".
Consider the cartesian plane with axis x and y; take the point P of coordinates (1,1) and with a compass draw a circle centered at (0,0) and through P.
This circle will intersect the x-axis exactly in one point : for sure on the line there is one point : what are its coordinates ?
All amounts to this : the ...
Alfred North Whitehead started his academic career in 1884 as a mathematician. Only towards the '20s Whitehead turned his attention to philosophy.
See Alfred North Whitehead :
Whitehead’s intellectual life is often divided into three main periods. The first corresponds roughly to his time at Cambridge from 1884 to 1910. It was during these years that he ...
It is simple.
We have two approaches to "measuring" the size of two sets. The first one is based on counting and it is the usual one we are accustomed to with finite sets : if the number of elements of set A is lesser than the numbe rof elements of set B (and this amounts to saying that A is a proper subset of B) then we say that B is Greater than A.
Here are the questions:
What is the most general way to define and separate "the rules of logic" from "the things to which the rules are applied" ?
The following definition of "theory" from Wikipedia may help clarify the separation:
A theory about a topic is usually a first-order logic together with a specified domain of discourse over which the ...
There are two major issues here: the connection between existence and exact representability, and the notion of exact representability itself.
There is no justification given for the implicit claim that only numbers which have exact representations can exist. The OP merely makes the claim that numbers which can't be "marked" don't exist. This is not ...
1) What is the most general way to define and separate "the rules of logic" from "the things to which the rules are applied" ?
The basic distinction is obtained by the definition of propositions and of the logical operations done on them. "Separation" is achieved in the sense that only the truth of the propositions is relevant to the way the operations work....
Wittgensteins early prose was notoriously obscure and difficult to parse. Thus to call it ‘obviously’ anything seems very wrong-headed.
This is possibly why he gave up on his former philosophy calling it not philosophy at all and completely wrong.
His later philosophy is far more human in that it’s concerned with human values.
At the root of all these asymmetries is the fact that not all relations are commutative/symmetric.
Here are some non-commutative/asymmetric relations involved in numbers
1: Factors ⇒ Product
2,3,6: Premises ⇒ Conclusion
4: Plaintext message ⇒ Encrypted message (even in symmetric cryptography)
5: Matter ⇒ Form
7: Logic ⇒ Math ⇒ Physics ⇒ Morality ⇒ ...
I will interpret the question as suggested by Philip: whether an argument to this effect has been discussed in philosophy. It was. A philosophizing mathematician/computer scientist/physicist David Wolpert formalized just such an argument in Physical limits of inference. Wolpert formalizes measurement, observation of a phenomenon, memory of past information ...
Disclaimer: I am not familiar with advanced set theory, which you have talked about a lot in your question, and have no advanced knowledge of mathematics.
However, I would be happy to provide a deductive argument for the abstractness of numbers.
Firstly, I admire your knowledge and research, but your question is much too complicated and unstructured to ...