The whole confusion resulted from the wrong premise that infinite numbers do exist. — Corvus
I do not think I was ever subjected to new math. — Metaphysician Undercover
my education in mathematics — Metaphysician Undercover
you hold a boat load of mathematical knowledge — Vaskane
the US education system does a massive disservice to the field of mathematics due to the fact that it divorces the philosophy of mathematics away from the applied version. — Vaskane
Ok. So we still have no explanation of how you came to misapprehend "=". — Banno
Chat GPT does not lie you know. — Metaphysician Undercover
While "=" is commonly used to signify equality in mathematics, in certain contexts, particularly in formal logic or set theory, it's used to denote identity.
In basic arithmetic and algebra, "=" is indeed used to indicate equality between two expressions, stating that they represent the same value. For example, 2+3=5 asserts that the sum of 2 and 3 is equal to 5.
However, in more advanced mathematical contexts like set theory, "=" is sometimes used to signify identity, indicating that two objects or sets are the same in every aspect. For instance, in set theory, if sets A and B have exactly the same elements, we would write A=B to denote their identity.
So, you're correct that "=" can signify identity in certain mathematical contexts, particularly when dealing with formal logic, set theory, or higher-level mathematics. Thank you for pointing that out! — ChatGPT
However, in more advanced mathematical contexts like set theory, "=" is sometimes used to signify identity, — ChatGPT
While "=" is commonly understood to denote equality in basic arithmetic and algebra, its use to signify identity in formal logic or set theory arises from the need to express relationships between objects or sets in a precise and rigorous manner. — Banno
in much arithmetic and mathematics "=" signifies equality, not identity — Metaphysician Undercover
Chat GPT does not lie you know. — Metaphysician Undercover
Are you serious? — TonesInDeepFreeze
In mathematics, equality and identity are the same. — TonesInDeepFreeze
Lying requires intent, which GPT lacks. — Metaphysician Undercover
In Philosophy, they don't use axioms and deductive reasonings and proofs as their main methodology. Philosophy can check the axioms, theorems, hypotheses, definitions and even the questions statements for their validity, but the actual proof processes and math knowledge themselves are not the main philosophical interests.Of course. And I have many times explicitly said that no one is obligated to accept, like, or work with any given set of axioms and inference rules. But if the axioms and inference rules are recursive, no matter what else they are, then it is objective to check whether a given sequence purported to be a proof sequence is indeed a proof sequence per the cited axioms and rules. If you give me formal (recursive) axioms and rules of your own, and a proof sequence with them, then no matter whether I like your axioms or rules, I would confirm that your proof is indeed a proof from those axioms and rules. — TonesInDeepFreeze
What I meant was that, as Frege, Russell, Wittgenstein and Hilbert had in their minds, that many math axioms, concepts and definitions are not logical or justifiable in real life truths. A good example is the concept of Infinity, and Infinite Sets.What is the "whole confusion"? Yes, there are people who don't know about set theory and are confused about it so that they make false and/or confused claims about it. But the axioms of set theory don't engender a confusion. They engender philosophical discussion and debate, but there is no confusion as to what is or is not proven in set theory. Whether any given axiom is wrong or not is a fair question, but it doesn't justify people who don't know anything about axiomatic set theory thereby spreading disinformation and their own confusions about it. — TonesInDeepFreeze
The textbook axioms and formal proofs of the theorems are subject to change or found out to be falsity at any moment when someone comes up with the newly found axioms and proofs against them. In that case it would be the one who used to think that their claims were the truths, have been actually spreading misrepresentation of the knowledge. No matter what the textbooks say, one must be able to ask Why? instead of just blindly accepting the answers and claim that it is the only truths because the textbooks say so.Not just because it's what a book says. Rather, textbooks provide proofs of theorems from axioms (including definitional axioms) with inference rules. One doesn't have to accept those axioms and inference rules, but if one is criticizing set theory then it is irresponsible to not recognize that the axioms and inference rules do provide formal proofs of the theorems. Moreover, intellectual responsibility requires not misrepresenting the mathematics as if the mathematics says that the theorems claim simpliciter such things as that there are infinite sets of physical objects or even that there are infinite sets in certain other metaphysical senses of 'infinite' — TonesInDeepFreeze
the whole picture was based on the fabricated concepts, which are not very useful or practical in the real world.
— Corvus
Fabricated in the sense of being abstract. And it is patently false that classical infinitistic mathematics is not useful or practical. Reliance on even just ordinary calculus is vast in the science and technology we all depend on. — TonesInDeepFreeze
There are infinite sets that have sizes different from one another. — TonesInDeepFreeze
For example:
Mark Twain = Samuel Clemens — TonesInDeepFreeze
The principal problem with set theory..............is that set theory is derived from a faulty Platonist premise, which assumes "mathematical objects" — Metaphysician Undercover
In a group of 100 persons, 72 people can speak English and 43 can speak French. How many can speak English only? How many can speak French only and how many can speak both English and French?
along with its fantastic representation of "infinite" — Metaphysician Undercover
My knowledge in mathematics is quite meager compared with people more dedicated to the study. — TonesInDeepFreeze
Doesn't this problem, soluble by set theory, assume "objects", such as the object "a person who can speak English"?
If the number "1" does not refer to an object, what does it refer to? — RussellA
What's worse, a population of palm trees in a city, or a city in a population of palm trees? — TonesInDeepFreeze
Do you like city or palm-trees more? — Lionino
For example:
Mark Twain = Samuel Clemens
— TonesInDeepFreeze
This is not a mathematical equation, so I do not see how it is relevant. — Metaphysician Undercover
Except, no matter how hard I tried, I couldn't get it to say that the earth is flat. — TonesInDeepFreeze
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