I've just had a discussion about whether or not the fiction-based axiomatic system approach to mathematics is fundamentally flawed. The points discussed were the same old points that I have argued about time and time again. Therefore I would be very interested to know how other visitors to my channel feel about these arguments.
This discussion is in a comment thread under my video "The 0.999...= 1 Controversy & Is Mathematics Fundamentally Flawed?" (link=https://youtu.be/DnLEoS590d0). The top-level comment is by '@teamacio9043' and it begins:
"Short explanation: 0.9999=1 is because"
After reading all of the comments under this top-level comment, please tell me your opinion about which of us has the most convincing arguments (overall):
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ChatGPT on the ambiguity and self-reference problems of limits
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Further info from ChatGPT about the popular 'proof' for '1=0.999...'.
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ChatGPT identifies the flaw in the most popular 'proof' for 1=0.999... (4 images)
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We can say expressions like "infinitely thin line", "colourless blue" and "unreal person". Arguably these are just grammatical constructions for which the individual words can be understood but where the things that the expressions refer to cannot be imagined in any coherent and consistent way. However, despite this argument the concept of an "infinitely thin line" remains as a much used core concept within mathematics. Do you believe that humans can 'imagine' or otherwise conceive of infinitely thin lines without resorting to self-delusion?
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Anyone can devise their own system of mathematics as long as the core definitions and rules of logic within that system don't lead to contradiction. This is easier than you might think because you also get to decide what constitutes a contradiction.
So you can even decide that 1=0 is not a contradiction in your system. Indeed, if you decide that all the apparently weird consequences of your definitions and rules are not contradictions then you can claim that your system of mathematics is consistent.
This means that a true statement in one system can be a false statement in another mathematical system. It means that all systems of mathematics are no more than fairytales in which 'valid statement' has no tangible real-world meaning.
If mathematics is classified under the sciences and is used widely within the sciences, then it might be considered problematic that it is not evidence-based as all sciences should be. It might also be considered problematic that 'valid statement' has no tangible real-world meaning.
But mathematicians strongly reject the criticism that this fairytale approach to mathematics is an absurd situation. They strongly reject that the rules and definitions of mathematical systems should make sense in our shared physical reality. They claim that the success of mathematics is all that matters. Do you agree with them?
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Do you agree that since the concept of 'not finite' so often produces problems and/or contradictions, then on the balance of evidence we should consider the concept of 'infinite' to be an invalid concept?
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Are you convinced by any of the arguments which claim that 0.999... = 1?
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Karma Peny is an anagram of my (very common) real name, Mark Payne. My aim is to challenge outrageous claims where I consider the logic behind the claims to be flawed.