• TonesInDeepFreeze
    2.3k
    The whole confusion resulted from the wrong premise that infinite numbers do exist.Corvus

    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
    2.3k
    I do not think I was ever subjected to new math.Metaphysician Undercover

    Virtually any student is subjected to certain instruction whether they like it or not. It would be fair to say that New Math is not good only if one at least knows what it is.

    my education in mathematicsMetaphysician Undercover

    You have virtually no education or self-education in the mathematics you so obdurately opinionate about.
  • TonesInDeepFreeze
    2.3k


    What's worse, a population of palm trees in a city, or a city in a population of palm trees?
  • TonesInDeepFreeze
    2.3k
    you hold a boat load of mathematical knowledgeVaskane

    My knowledge in mathematics is quite meager compared with people more dedicated to the study.

    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

    That might be true. Also, the fact that formal logic is not conveyed so that students could see how the mathematics is derived logically rather than simply decreed.

    One can always learn vastly more about logic, mathematics, philosophy and the philosophy of mathematics, but my first interest in logic (which led to mathematics) came from my interest in philosophy, and as I learned logic and mathematics, I was learning about the philosophy of mathematics right alongside.
  • Metaphysician Undercover
    12.3k
    Ok. So we still have no explanation of how you came to misapprehend "=".Banno

    Did you not understand the example I gave you in the other thread? I suggest you go back and read that post you made for me when you fed that example to Chat GPT. It totally agreed with me. It said, in much arithmetic and mathematics "=" signifies equality, not identity. Chat GPT does not lie you know. The simple fact, as my example shows, an "equation" would be completely useless if the left side signified the very same thing as the right side.
  • Banno
    22.9k
    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

    :wink:
  • Metaphysician Undercover
    12.3k
    However, in more advanced mathematical contexts like set theory, "=" is sometimes used to signify identity, — ChatGPT


    As I argue, there is much inconsistency in mathematics. The use of "sometimes" here is very telling.
  • Banno
    22.9k



    ChatGPT
    The use of "=" to signify both equality and identity in different mathematical contexts doesn't necessarily imply an inconsistency in mathematics. Instead, it reflects the flexibility and versatility of mathematical notation to adapt to various branches and subfields within mathematics.

    Mathematics is a vast and multifaceted discipline, encompassing diverse areas such as arithmetic, algebra, calculus, geometry, logic, and more. Each of these areas may have its own conventions, definitions, and notational systems tailored to the specific concepts and structures being studied.

    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.

    Inconsistencies in mathematics would arise if there were contradictions or logical paradoxes within a particular mathematical system. However, the use of "=" in different contexts doesn't inherently introduce inconsistencies; rather, it reflects the richness and diversity of mathematical language and notation.
  • Metaphysician Undercover
    12.3k
    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

    That mathematics consists of "objects" with identity is Platonist metaphysics. In this metaphysical theory, mathematical ideas like numbers are objects, rather than quantitative values. Set theory is nothing but Platonist based mathematical theory. Notice that it is "theory", not mathematics in practise.

    In the actual application of mathematics, values are assigned, and the left side of an equation must represent something different from the right side, or the equation would be useless, as I explained.

    The conclusion we can make is that set theory does not represent mathematics, as mathematics is actually used. That's the problem, We can define terms, or in this case symbols, for theory, in a way which doesn't actually represent how they are used in practise. That's an idealist folly. I think Wittgenstein made a similar point.
  • TonesInDeepFreeze
    2.3k
    in much arithmetic and mathematics "=" signifies equality, not identityMetaphysician Undercover

    Chat GPT got it wrong. As is common.

    In mathematics, equality and identity are the same.

    Chat GPT does not lie you know.Metaphysician Undercover

    Are you serious?
  • TonesInDeepFreeze
    2.3k
    After catching Chat GPT in what seems to be a conflation of equivalence with equality (indeed equivalence and identity are not the same, while equality and identity are the same), Chat GPT wrote this:

    " "=" typically denotes identity, meaning the left side is considered the same as the right side."

    Though that is correct, it's worthless coming from Chat GPT, which is not even remotely an authority on mathematics, and famously known to fabricate on all kinds of subjects.

    Anyone who thinks Chat GPT doesn't lie and can be relied upon for accurate information is grossly uninformed about Chat GPT.
  • Metaphysician Undercover
    12.3k
    Are you serious?TonesInDeepFreeze

    Lying requires intent, which GPT lacks.

    In mathematics, equality and identity are the same.TonesInDeepFreeze

    Here's the example I gave Banno in the other thread. You and I are each one. Together we are two. We can symbolize this as 1+1=2. The two 1's here each represent something different, one represents you, the other I. Because the two each represent something different, the two together as 1+1 can make 2, meaning two distinct things. Also, we can say 1=1. But if the two 1's here both represent the same thing, then 1+1 could not make 2, because we'd still just have two different representations of the very same thing.
  • Banno
    22.9k
    Yeah, I recall that. Still can't make sense of it.

    Just to be clear, my use of ChatGPT here is purely rhetorical, intended for amusement.
  • TonesInDeepFreeze
    2.3k
    Lying requires intent, which GPT lacks.Metaphysician Undercover

    Oh puhleeze! The point is not about the definition of 'lie' but rather that there would not be any point in you saying that it doesn't lie if you didn't mean that it is a reliable source. (The word used most commonly for AI making false statements is 'hallucinating'.) Moreover, lying does not always require intent, as false statement made from negligence, especially repeated negligence may also be considered lies. And that is the case with Chat GPT, as its designers are negligent in allowing it to spew falsehoods. Indeed, the makers of such AI will say themselves that its main purpose is for composition of prose and not always to be relied upon for information.

    Hopefully, now it's agreed that Chat GPT is not a reliable source. Indeed, it is worse than not reliable. So your quote of it is worthless.

    I'll explain it to you again as I did years ago:

    Let T and S be any terms.

    T = S

    means that what 'T' denotes is the same thing as what 'S' denotes.

    That is not vitiated by the fact that aside from denotation there is also sense.

    For example:

    Mark Twain = Samuel Clemens

    means that 'Mark Twain' and 'Samuel Clemens' denote the same person

    But the names 'Mark Twain' and 'Samuel Clemens' are different names and have different senses, such as 'Mark Twain' is a pen name and 'Samuel Clemens' is a birth name.

    Now, denotation is extensional and sense is intensional. Ordinary mathematics handles only the extensional. So, again:

    S = T

    means that S and T stand for the same thing, though, of course, S and T may be very different terms.
  • TonesInDeepFreeze
    2.3k


    Got it.

    One can get Chat GPT to claim just about anything you want it to claim. I've gotten it to make all kinds of ridiculously false claims. I've even got it to make a claim, then retract that claim, then retract the retraction. Except, no matter how hard I tried, I couldn't get it to say that the earth is flat.
  • Corvus
    2.4k
    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
    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.
  • Corvus
    2.4k
    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
    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.

    Infinity is not numeric, but a property of motions, operations and actions. But they seem to think it is some solid existence in reality. When they talk about the concepts like infinite sets and claim this or that as if there are self-evident truths for them, it sounds confused.
  • Corvus
    2.4k
    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 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.

    Bottom line is that, truth speaks for itself. One doesn't need to say to the others, they are wrong unless when it is absolutely necessary. But just tell the arguments and conclusions, which are true. If in any case of doubt, ask why and how so.
  • Corvus
    2.4k
    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

    "A careful reader will find that literature of mathematics is glutted with inanities and absurdities which have had their source in the infinite. " - David Hilbert, On the Infinite, pp.184 Philosophy of Mathematics Selected Readings, Edited by H. Putnam and P. Benacerraf 1982
  • RussellA
    1.5k
    There are infinite sets that have sizes different from one another.TonesInDeepFreeze

    I take the OP as asking the question "are there an infinite number of infinities?"

    The answer would depend on whether looked at from set theory or natural language.

    Set Theory is a specific field of knowledge with its own rules, and as the Scientific American noted: As German mathematician Georg Cantor demonstrated in the late 19th century, there exists a variety of infinities—and some are simply larger than others.

    However the terms infinity and infinite sets are also used in everyday language outside of set theory, such as "I have an infinity of problems" and "I have an infinite set of problems".

    As the OP doesn't refer to the very specific field of "set theory", having its own particular rules, I think the OP should be considered as a problem of natural language.

    Within natural language, the question "are there an infinite number of infinities" is meaningless, as not only is "an infinite number" unknowable, it follows that whether there is one or more infinite numbers must also be unknowable.

    On the assumption that the OP refers to a problem in natural language, otherwise it would have specifically referred to "set theory", as it refers to that which is unknowable, although syntactically correct is semantically meaningless.
  • Metaphysician Undercover
    12.3k

    The issue is not whether or not some mathematicians define "=" as meaning 'is identical to', as a premise for a mathematical theory, or some other purpose, like debate or discussion. We've seen very much evidence here that some actually do this. So there is no question concerning that.

    The question is how "=" is actually used in the application of mathematics. And anyone who takes a critical look at an equation in the application of mathematics will see that the right side never signifies the very same thing as the left side. In fact, it's quite obvious that if the right side did signify the same thing as the left, the equation would be completely useless. That is why many philosophers will argue that the law of identity is a useless tautology.

    Since this is the case, we can clearly see that those mathematicians who define "=" as meaning 'is identical to' do not properly represent the meaning of "=" with that definition. Therefore we can say that they are wrong with that definition.

    For example:

    Mark Twain = Samuel Clemens
    TonesInDeepFreeze

    This is not a mathematical equation, so I do not see how it is relevant. You are trying to compare apples with oranges, as if they are the same thing, but the requirement that "Mark Twain = Samuel Clemens" is a representation of a mathematical equation renders your analogy as useless.

    Please consider a real mathematical equation as an example, like how the circumference of a circle "is equal to" the diameter times pi, or the square of the hypotenuse of a right triangle "is equal to" the sum of the squares of the two perpendicular sides, for example. Be my guest, pick an equation, any equation, and we'll see if the right side signifies the very same thing as the left side. I think that an intelligent mathematician such as yourself, ought to know better than to argue the ridiculous claim that you have taken up.


    The principal problem with set theory, as I indicated in my reply to @Banno above, which is evident from Chat GPT's statement, is that set theory is derived from a faulty Platonist premise, which assumes "mathematical objects". If we recognize as fact, that mathematics does not consist of objects, we must reject the whole enterprise of set theory, along with its fantastic representation of "infinite" and "transfinite", as completely unsound, i.e. based in a false premise.
  • RussellA
    1.5k
    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 random web site is set a problem that can be solved by set theory:
    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?

    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?

    along with its fantastic representation of "infinite"Metaphysician Undercover

    I would say that "infinite number" does not refer to an object, because unknowable by a finite mind, but does refer to a process along the lines of addition, which is knowable by a finite mind.
  • Vaskane
    643
    My knowledge in mathematics is quite meager compared with people more dedicated to the study.TonesInDeepFreeze

    My knowledge of the philosophy of mathematics is nil. I grew up in the USA, where they divorce the philosophy from applied use. I was trained to decrypt radio signals by hand using the US Navy's RASIN Manual. After coming out of the Navy, during my college years, I even ventured to attempt to apply Claude Shannon's Theories in Signals Communication to the body's biochemical signaling system. Lo and behold I stumbled across, after months of digging, the 1 person, who shared my idea, and had been making it his life's work since a little before I was even born. In the same manner that I knew Infinities have different sizes, is the same manner I understood that Claude Shannon's communication theory can be applied to the way our biochemical signaling system functions: https://www.ncbi.nlm.nih.gov/pmc/articles/PMC3220916/

    I get you're trying to be all like "you think this is a lot, bruh you don't know shit." But bruh, I can guarantee you I've done more shit than you, with math, other than philosophy.
  • Vaskane
    643
    Any place you use infinite, you can replace with infinity, it's an artistic effect like I said: "In passing I had caught a glimpse of the infinity beauty deep within her eyes." This way you're free to use "infinite," later without repetition, but rather alliteration... "It made her modesty infinitely more attractive."
  • Metaphysician Undercover
    12.3k
    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

    The issue is a little more complex than how you represent here, but this is a good indication of why "set theory" is not applicable to mathematics. In your first question, "a person who can speak English" is a description, not an object. It represents a category by which we could sort objects. In the second sentence, the numeral "1" represents a specific concept, which can be described as a quantitative value. It is not a true representation of how we use numbers, to think of a number as itself an object. Set theory may represent a number as an object, but that's the false premise of set theory.
  • Lionino
    849
    By that logic every adjective can be used as a noun.
    Why call for Grammar in Artistic License's house?

    What's worse, a population of palm trees in a city, or a city in a population of palm trees?TonesInDeepFreeze

    Depends. Do you like city or palm-trees more?
  • Banno
    22.9k


    When first I played with ChatGPT I had it "prove" 999983 is not a prime - it just baldly asserted that it was the product of two integers. Then correct itself. Regretfully, I was using the playground so the record is lost.



    They are coming out of the woodwork now.
  • TonesInDeepFreeze
    2.3k
    Do you like city or palm-trees more?Lionino

    I like cities as grim and forbidding as can be, thus without palm trees.
  • TonesInDeepFreeze
    2.3k
    For example:

    Mark Twain = Samuel Clemens
    — TonesInDeepFreeze

    This is not a mathematical equation, so I do not see how it is relevant.
    Metaphysician Undercover

    It is exactly the point that it is not a mathematical expression, so mathematics is not called on to account for its intensionality. More generally that ordinary mathematics is extensional, and we don't require that it also accommodate intensioncality. That is how it is relevant.

    /

    Later, hopefully, I'll have time and motivation to dispel a number of misconceptions in a catalog of them you've posted lately.
  • Lionino
    849
    Except, no matter how hard I tried, I couldn't get it to say that the earth is flat.TonesInDeepFreeze

    Those are hard-coded, just like anything revolving sensitive western politics.
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