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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 ...


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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 ...


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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 ...


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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 ...


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Well the question you pose indicates that you believed that the math definition was the end all be all and then reality struck. You found out the math definition was strictly in the context of math. Well why not start off teaching that way? The topic you study is NOT logic but strictly called "Mathematical Logic". There are other types of logic. Math is ...


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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.


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Philosophy aims at the logical clarification of thoughts. Philosophy is not a body of doctrine but an activity If Godel shows that "P = ΠP and ¬P = Π¬P" are not valid then any use of his ideas that also use "P = ΠP and ¬P = Π¬P" are, assuming we're talking philosophy, surely a misuse of "P = ΠP and ¬P = Π¬P". Any activity that misuses logic (or language) ...


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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 ⇒ ...


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Why would someone bring in an altogether different notion of their abstract reality, when the argument about numbers being a projection of reasoning (mental faculty/intelligence) is the most logical explanation? There is no such a thing as "the most logical explanation". An explanation is logical or it is not (and, most often, it is not). So, I will guess ...


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There's a lot to unpack here, so first I'm going to summarize my understanding of your question: You disagree with Platonism. Based on what you have written, you sound like a nominalist (i.e. you think mathematical entities are representations of "real" or physical entities). Some people who support Platonism make an argument of the form "If aliens (or ...


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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 ...


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@conifold- Here is a Spinoza response to a question which tangentially resembles the one here. It concerns ideas which appear to have no reference in reality, in this case a series of rectangles which can be 'pictured' or imagined in the minds-eye' but which have no existence. Not sure I can make any realtime sense of this, any thoughts. Cheers, Charles M ...


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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 ...


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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 ...


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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....


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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 ...


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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. When ...


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