The word real is rather important and I don’t think the conversation can proceed without it unless it is replaced with something else. I would suggest ‘existent’ as such a replacement.
“What is meant by “follows” is not unconditional. Implication is itself conditional on logical axioms.”
In order for humans to understand implication, it is helpful, or even necessary, for us to denote and specify the logical axioms using symbols and manipulation rules, but this does not mean that they are invented by humans ( though I understand you didn’t claim that they were). Rather, I would claim that they were discovered to be ways to accurately represent pieces of Logic. Logic is simply the necessary structure of relations between things. So I would argue that logic itself is unconditionally existent.
Alternatively, you could describe logical deduction as a form of computation, which necessarily plays out in a particular way. It is unconditionally true that a particular algorithm produces a particular output given a particular input, therefore it is unconditionally true that certain theorems follow from certain axioms, because the ‘conditions’ , which here are both the input of the algorithm and the information describing the algorithm itself, have been incorporated into the algorithm+input ensemble. You might then object that information is also necessary to specify the computer running the algorithm, but this can itself be conceptualized as part of the algorithm.
“Our brains evolved in the reality and developed the kind of reasoning that is useful for it. No more explanation is required.” I am not sure that this is true, unless stated within the context of the ‘mathematical universe hypothesis’ and anthropic principle as I mentioned, in which case I don’t think we disagree with each other. Someone might object that we didn’t evolve to understand counterintuitive ideas like Relativity and Quantum mechanics because we live in a regime in which Newtonian mechanics describes what we observe very well, and is simpler. Therefore we evolved only to understand Newtonian mechanics.
Update for Ape in the coat if you read this thread again:
I am rate limited and so can’t currently respond directly to you. My reply would have been as follows:
“Of course the meaning is important, however not all words can be defined in terms of others, and this is exactly what providing a ‘gears level model’ would require. Have you thought of a way to do this in the case of concepts like reality and existence? ”
The word real is rather important and I don’t think the conversation can proceed without it unless it is replaced with something else. I would suggest ‘existent’ as such a replacement.
You are failing at the rationalist taboo game. Read this, please, and try again.
The word itself is just a label. The meaning is important. Try explaining the gears behind it, instead of comming up with a synonym.
Of course the meaning is important, however not all words can be defined in terms of others, and this is exactly what providing a ‘gears level model’ would require. Have you thought of a way to do this in the case of concepts like reality and existence?
I would guess that you distinguish between platonic reality, physical reality, perhaps simulated or thought reality, and maybe others. However I would consider all of these to be real, and it is clearly impossible for something to be real in the physical or other sense while not existing in the platonic sense. Therefore, when mathematicians claim that mathematics has platonic existence, I would say they’re justified in saying it exists (in the most general sense) . It then makes sense to say that they discover it when they become aware of it.
The word real is rather important and I don’t think the conversation can proceed without it unless it is replaced with something else. I would suggest ‘existent’ as such a replacement.
“What is meant by “follows” is not unconditional. Implication is itself conditional on logical axioms.”
In order for humans to understand implication, it is helpful, or even necessary, for us to denote and specify the logical axioms using symbols and manipulation rules, but this does not mean that they are invented by humans ( though I understand you didn’t claim that they were). Rather, I would claim that they were discovered to be ways to accurately represent pieces of Logic. Logic is simply the necessary structure of relations between things. So I would argue that logic itself is unconditionally existent.
Alternatively, you could describe logical deduction as a form of computation, which necessarily plays out in a particular way. It is unconditionally true that a particular algorithm produces a particular output given a particular input, therefore it is unconditionally true that certain theorems follow from certain axioms, because the ‘conditions’ , which here are both the input of the algorithm and the information describing the algorithm itself, have been incorporated into the algorithm+input ensemble. You might then object that information is also necessary to specify the computer running the algorithm, but this can itself be conceptualized as part of the algorithm.
“Our brains evolved in the reality and developed the kind of reasoning that is useful for it. No more explanation is required.” I am not sure that this is true, unless stated within the context of the ‘mathematical universe hypothesis’ and anthropic principle as I mentioned, in which case I don’t think we disagree with each other. Someone might object that we didn’t evolve to understand counterintuitive ideas like Relativity and Quantum mechanics because we live in a regime in which Newtonian mechanics describes what we observe very well, and is simpler. Therefore we evolved only to understand Newtonian mechanics.
Update for Ape in the coat if you read this thread again:
I am rate limited and so can’t currently respond directly to you. My reply would have been as follows:
“Of course the meaning is important, however not all words can be defined in terms of others, and this is exactly what providing a ‘gears level model’ would require. Have you thought of a way to do this in the case of concepts like reality and existence? ”
You are failing at the rationalist taboo game. Read this, please, and try again.
The word itself is just a label. The meaning is important. Try explaining the gears behind it, instead of comming up with a synonym.
Of course the meaning is important, however not all words can be defined in terms of others, and this is exactly what providing a ‘gears level model’ would require. Have you thought of a way to do this in the case of concepts like reality and existence?
I would guess that you distinguish between platonic reality, physical reality, perhaps simulated or thought reality, and maybe others. However I would consider all of these to be real, and it is clearly impossible for something to be real in the physical or other sense while not existing in the platonic sense. Therefore, when mathematicians claim that mathematics has platonic existence, I would say they’re justified in saying it exists (in the most general sense) . It then makes sense to say that they discover it when they become aware of it.