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kant
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One stone plus 5 stones equals 6 stones. Is that a unique property of our universe, or it is conceivable that in a different universe with a different set of physical laws, arithmetic is impossible?
You make two very different errors in your question. First "one stone plus 5 stones equals 6 stones", is a statement extracted from a mathematical construct called arithmetic. Arithmetic is a collection of statements and deductions presumed to be an internally self consistent system (many people have thought about it for many years and they have come to agree that the system does not invalidate itself via contradiction). The possibility that someone can conceive of an internally self consistent set of definitions and actually set one up seems relatively clear.kant said:One stone plus 5 stones equals 6 stones. Is that a unique property of our universe, or it is conceivable that in a different universe with a different set of physical laws, arithmetic is impossible?
You make two very different errors in your question. First "one stone plus 5 stones equals 6 stones", is a statement extracted from a mathematical construct called arithmetic. Arithmetic is a collection of statements and deductions presumed to be an internally self consistent system (many people have thought about it for many years and they have come to agree that the system does not invalidate itself via contradiction). The possibility that someone can conceive of an internally self consistent set of definitions and actually set one up seems relatively clear.
That brings us to your question: "is it conceivable that ... ". What is and is not conceivable certainly cannot be judged via your (or anyone else's) ability to conceive of it.Speaking of conceiving of an internally self consistent set of definitions, you might consider your ability to conceive of a wholly different (and internally consistent) interpretation of the English language. The volume of information which needs be reinterpreted to accomplish that result is certainly a trivial fraction of the information represented by the idea "the universe". So it pretty well follows that our ability to conceive of things is clearly somewhat limited
I'll leave that to the mathematicians.kant said:hmm... can you tell me how you show the arithmetic statement "arithmetic is consistent" to be true?
No, I was just pointing something out to you that you find inconceivable. Relabeling the concepts in a different symbolic system (French, Chinese, etc.) has nothing to do with the validity of the concepts. What I was pointing out was that you can not comprehend the collection of concepts you attach to those words being the wrong collection of concepts (that possibility is thus something you cannot conceive of).kant said:i can reinterpret english using french or chinese, or change the name "english" to "chinese" etc... You are make a underlying assumption that are ability to conceive something is based on our ability to reinterpreted some thing.
Whether the proposition is true or not depends on one's premises (the definitions of "one", "5", "6", "plus" and "equals"). If you can provide definitions of these terms, then we can judge the truth of the proposition.kant said:One stone plus 5 stones equals 6 stones. Is that a unique property of our universe, or it is conceivable that in a different universe with a different set of physical laws, arithmetic is impossible?
-Job- said:The concept of a "unit" is necessary in addition. Without units there isn't anything to add. Humans experience a "quantized" Universe where everything that is perceived is automatically grouped into units. From our perspective of the Universe addition arises as a basic relationship.
I can identify two units of rocks on the left and four units on the right or 6 in front of me, all by the same process. This leads to a relationship between 2, 4 and 6.
Addition is more of an observation than a property, and not independent of a mind.
I don't agree that these are simply "constructs of the human perspective". The fact that 2 + 4 = 6 is an analytic truth (a truth by definition) - the same way that "all bachelors are unmarried men" is an analytic truth. Given the definitions of 2, 4, 6, + and =, it logically follows that 2 + 4 = 6 (in all logically possible worlds).oldman said:I fully agree with what you so straightforwardly say. A concept like "2", or "4", together with the concept of addition and indeed the rest of mathematics, are constructs of the human perspective which are not independent of a mind. In my view such entities have no physical existence. Useful and fun for us, though!
moving finger said:I don't agree that these are simply "constructs of the human perspective". The fact that 2 + 4 = 6 is an analytic truth (a truth by definition) - the same way that "all bachelors are unmarried men" is an analytic truth. Given the definitions of 2, 4, 6, + and =, it logically follows that 2 + 4 = 6 (in all logically possible worlds).
Best Regards
Are humans the only agents capable of producing hot air?oldman said:And what is "an analytic truth (a truth by definition)" , if not a "construct of the human perspective" (a.k.a. words, or put more crudely; hot air)?
Darn it. And I thought I disagreed with Penrose on most things.oldman said:3. You are not alone in your opinion. Sir Roger Penrose seems to share it. But I have yet to be argued into agreement on this point.
The laws of arithmetic are a set of principles that govern the basic operations of addition, subtraction, multiplication, and division. These laws include the commutative, associative, and distributive properties, as well as the identity and inverse properties for each operation.
Yes, the laws of arithmetic are universal and apply to all cultures. They are based on fundamental principles of logic and are not influenced by cultural or societal norms.
Yes, the laws of arithmetic can be proven through mathematical proofs and logical reasoning. These laws have been extensively studied and tested, and have been shown to be consistent and true.
No, the laws of arithmetic are not derived from empirical evidence. They are based on abstract concepts and logical principles, rather than observations or experiments.
The laws of arithmetic are essential for performing basic mathematical operations, such as calculating expenses, determining quantities, and solving problems. They also provide a foundation for more advanced mathematical concepts and real-world applications, such as engineering and finance.