• TonesInDeepFreeze
    2.3k


    I bet if you put the cyber equivalent of a ravenous rat in its face like in '1984' then you could break it. Would say anything, begging like HAL 9000.
  • TonesInDeepFreeze
    2.3k
    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.Corvus

    What passages from Frege, Russell or Hilbert do you have in mind?

    Frege proposed a system to derive mathematics from logic alone. That system was not a set theory per se, but sets can be configured in the system. And Frege did not at all oppose infinite sets. I can be checked on this, but I think it's safe to say that Frege's framework is indeed infitisitic.

    Russell showed that Frege's system was inconsistent. Then Whitehead and Russell proposed a different system from Frege's, this time presumably consistent, to derive mathematics from logic alone. But that system is seen to not be purely logic. And Whitehead and Russell explicitly used infinite sets. And I would bet that Whitehead and Russell recognized the applicability of infinitistic mathematics to the sciences.

    Hilbert endorsed infinitistic mathematics but hoped there would be a finitistic proof of its consistency. Alas, Godel proved that there can be no finitistic proof even of the consistency of arithmetic, let alone of set theory. In any case, Hilbert distinguished between contentual (basically, finitistic) mathematics and ideal (basically, infinitistic) mathematics, and such that he saw the application of the ideal to the contentual.

    /

    I hope that later I'll have the time and inclination to catch up to certain misunderstandings and strawmen you've recently posted.
  • Metaphysician Undercover
    12.3k
    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.TonesInDeepFreeze

    I interpret that as 'mathematics is extensional and that's how intensionality is relevant'. Whatever it is you are trying to say here, it appears to be just as irrelevant as your analogy was.

    Later, hopefully, I'll have time and motivation to dispel a number of misconceptions in a catalog of them you've posted lately.TonesInDeepFreeze

    I'll be looking forward to that.
  • TonesInDeepFreeze
    2.3k
    In Philosophy, they don't use axioms and deductive reasonings and proofs as their main methodology.Corvus

    Perhaps not axioms as the main approach. And philosophy ranges from poetic through speculative, hypothetical, concrete and formal. But deductive reasoning and demonstration is basic and ubiquitous in large parts of philosophy. And the axiomatic method does appear in certain famous philosophy, and its principles and uses - sometimes even formalized - are prevalent in modern philosophy, philosophy of mathematics and philosophy of language.

    the actual proof processes and math knowledge themselves are not the main philosophical interests.Corvus

    The axioms are subject of deep, extensive and lively discussion in the philosophy of mathematics.

    /

    But when I mentioned objectivity, of course I was not referring to objectivity of philosophy, but rather the objectivity of formal axiomatics, in the very specific sense I mentioned. And that is a philosophical consideration. Then you challenged my claim that mathematics has that objectivity. So I explained to you again the very specific sense I first mentioned. The fact that philosophy in its wide scope is not usually characterized as axiomatic doesn't vitiate my point.
  • TonesInDeepFreeze
    2.3k
    it appears to be just as irrelevant as your analogy was.Metaphysician Undercover

    The analogy was not irrelevant. And the key word in what you just said is "appears" but the other crucial words you left out are "to me", as indeed what appears to you is quite unclear with your extreme myopia. And meanwhile I'm still guffawing at your trust in AI chat and your pathetic transparently disingenuous attempt to back out by saying that it's only its lack of intent you had in mind, and even as you are wrong about the definition of the word in question.
  • TonesInDeepFreeze
    2.3k
    they seem to think it is some solid existence in reality.Corvus

    Who is "they"? What specific mathematicians do you claim that about? What specific mathematicians do claim have said that the infinite sets of mathematics have solidity as material objects or even like material objects?

    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

    Often the axioms are taken to be true, on different bases, sometimes self-evidence, depending on the mathematician or philosopher. But often, at least in the philosophy of mathematics, arguments, not merely self-evidence, are given. Moreover, there is a wide array of approaches where "the axioms are true" would be an oversimplification not claimed without context and explanation by many mathematicians and philosophers. This includes such approaches as structuralism, instrumentalism, fictionalism, consequentialism and formalism. And formalism itself ranges from extreme formalism to Hilbertian formalism, including the view of some mathematicians that the assertion that there are infinite sets is nonsense but that still infinitistic mathematics is useful.

    As I said, there are deep, puzzling questions about mathematics, but that doesn't make the mathematics itself, especially as formalized, confusing. On the contrary, if you ever read a treatment of the axiomatic development of mathematics, you may see that it is precise, unambiguous, objective (in the specific sense I mentioned), and with good authors, crisply presented.
  • TonesInDeepFreeze
    2.3k
    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.Corvus

    Of course, my point went right past you no matter that I explained it clearly.

    There are many different and alternative formal axiom systems in mathematics. Mathematicians and philosophers sometimes disagree on which axioms are best, most intuitive, and even true to some concepts. That's a good thing. But the point that went past you is that what is objective, even among them, is that for each one, there is a mechanical procedure to determine whether a purported formal proof is indeed a formal proof allowed from the given set of axioms and inference rules. And, as people may disagree as to what axioms are best or even philosophically or conceptually justified, at least in the formal sense, one doesn't "disprove" an axiom or set of axioms as you seem to imagine (except, of course, by showing that the axioms of a given system are inconsistent with themselves; and by the way, there are at least two famous cases where axiom systems were proven inconsistent - Frege's and one of Quine's, examples that it is not the case that mathematicians follow blindly and uncritically, ).

    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.Corvus

    Again, you are unfamiliar with any of this; you are blindly punching.

    We have axioms and rules of inference. Textbooks often do explain the bases for the axioms and rules of inference and do not require blind acceptance. Then, with the axioms and inference rules given, it is objective whether or not a purported proof from those axioms and with those rules is indeed a proof from the axioms with the rules. So that does not require blind acceptance. The process is to state the axioms and rules, often providing intuitive bases for them, then proofs of theorems, as those proofs can be checked. And a good student does check the proofs, both to understand them and to verify for themselves that it is indeed a proof from the axioms with the rules.

    But with the inference rules, it's even better. In a mathematical logic, we PROVE that the inference rules are justified in the two key ways: The rules permit only valid deductions and the rules provide for every valid deduction.

    On the other hand, blind acceptance is when mathematics is not given axiomatically. The teacher says that a bunch of formulas are correct, to be memorized and performed upon call. But why, the student may ask? Instead, with axioms, the student may ask why, and always an answer is given based on previous formulas that prove the ones in question. And those previous formulas are proven, etc., until we get to the end of the line - the axioms. So, with axiomatics, we can justify everything formally, except the axioms, which are the starting point (not everything can be justified formally without infinite regress or circularity) and are only justified intuitively. Then, one may say, but I don't like or accept those axioms. And the best answer is, "Fine. You don't have to. But at least you can still check that the proofs are permitted from the axioms and rules. And if one wants, one can study an alternative set of axioms. Or even not study any axiomatic system and go one's merry way accepting or not accepting whatever non-axiomatic mathematics one encounters."
  • TonesInDeepFreeze
    2.3k
    infinity and infinite sets are also used in everyday language outside of set theoryRussellA

    Since you are harkening to the original post, see that it is a question about the infinitude of intervals on the real number line, and about the number of different infinite sizes. The ordinary context of that is mathematics and set theory. Anyone is welcome to consider the question in another context, but that doesn't make it inapposite to talk about it in the context of mathematics and set theory.

    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.RussellA

    That's a non sequitur. That the poster didn't mention set theory by name does not imply that set theory would not be a natural context for the matter, especially as the question gave a mathematical context and refers to a concept that is characteristically set theoretic. Moreover, discussion doesn't even have to be limited to whatever unstated context the poster himself might have had in mind.

    Within natural language, the question "are there an infinite number of infinities" is meaninglessRussellA

    If that is true, then even more reason why one would then consider the question in regard to mathematics. If it's meaningless in context C but defined in another context D, then it wouldn't make sense to say that then it is inapposite to context D.
  • RussellA
    1.5k
    If that is true, then even more reason why one would then consider the question in regard to mathematics. If it's meaningless in context C but defined in another context D, then it wouldn't make sense to say that then it is inapposite to context D.TonesInDeepFreeze

    That raises the interesting question that if an expression such as "infinite infinities" has no meaning in a natural language, the everyday spoken and written language used to describe the world around us, but does have meaning in the formal language of set theory, then what exactly is the relationship between a formal language such as set theory and the world around us?
  • Corvus
    2.4k
    What passages from Frege, Russell or Hilbert do you have in mind?TonesInDeepFreeze
    You must read them yourself. They all had reservations on the concept of Infinity in math. Quite understandably and rightly so.
  • Corvus
    2.4k
    Perhaps not axioms as the main approach. And philosophy ranges from poetic through speculative, hypothetical, concrete and formal. But deductive reasoning and demonstration is basic and ubiquitous in large parts of philosophy. And the axiomatic method does appear in certain famous philosophy, and its principles and uses - sometimes even formalized - are prevalent in modern philosophy, philosophy of mathematics and philosophy of language.TonesInDeepFreeze
    Sure as endeavours to be formal and more clear in their system, but is it always making sense? That is another question. Often it tends to make the system look more convoluted, if not done properly.

    But when I mentioned objectivity, of course I was not referring to objectivity of philosophy, but rather the objectivity of formal axiomatics, in the very specific sense I mentioned. And that is a philosophical consideration. Then you challenged my claim that mathematics has that objectivity. So I explained to you again the very specific sense I first mentioned. The fact that philosophy in its wide scope is not usually characterized as axiomatic doesn't vitiate my point.TonesInDeepFreeze
    Objectivity is the objectivity of knowledge. Not objectivity of philosophy or objectivity of mathematics. That is another misunderstanding of yours. I wouldn't be surprised if you go on claiming an objectivity for set theories and an objectivity for numbers ... It is like saying a subjectivity of objectivity. A contradiction.
  • RussellA
    1.5k
    In your first question, "a person who can speak English" is a description, not an object.Metaphysician Undercover

    Can there be a description without an object being described?

    Isn't "a person who can speak English" a description of the object (a person who can speak English)?
    ===============================================================================
    It is not a true representation of how we use numbers, to think of a number as itself an object.Metaphysician Undercover

    The word "object" has different meanings. In mathematics, a mathematical object is an abstract concept (Wikipedia – Mathematical Object). In natural language, it can be something material perceived by the senses (Merriam Webster - Object).

    For example, we can think of the number , the number and as abstract mathematical objects but cannot think of them as natural concrete objects.

    However we can think of the numbers 1. 6 and 10 as not only abstract mathematical objects but also as natural concrete objects.

    That raises the question as to how we are able to think of something that is abstract, disassociated from any specific instance (Merriam Webster – Abstract). For example, independence, beauty, love, anger, Monday, , and the number 6.

    George Lakoff and Mark Johnson in their book Metaphors We Live By propose that we can only understand abstract concepts metaphorically, in that we understand the concept of gravity by thinking about a heavy ball on a rubber sheet.

    Thereby, we understand the concept of independence by remembering the feeling of leaving a job we didn't like. We understand the concept of beauty by looking at a Monet painting of water-lilies. We understand the concept of infinity by thinking about continually adding to an existing set of objects. We understand the concept of by thinking about the number 1.414 etc etc. We understand the concept of 6 by picturing 6 apples.

    IE, we can only understand an abstract concept metaphorically, whereby a word or phrase literally denoting one kind of object or idea is used in place of another to suggest a likeness or analogy between them (Merriam Webster – Metaphor).
  • Corvus
    2.4k
    As I said, there are deep, puzzling questions about mathematics, but that doesn't make the mathematics itself, especially as formalized, confusing. On the contrary, if you ever read a treatment of the axiomatic development of mathematics, you may see that it is precise, unambiguous, objective (in the specific sense I mentioned), and with good authors, crisply presented.TonesInDeepFreeze
    Axiomatic methodology in math is not free from problems and deficiencies. They are subjective definitions which are often circular in logic. They lack in consistency and are incomplete in most times.
    They are dependent on the other axioms mostly. Most of them are abstract and illusional which renders to the false conclusions. A typical example is the Infinity in Set theory.
  • Metaphysician Undercover
    12.3k
    Can there be a description without an object being described?RussellA

    Of course, that's known as fiction.

    However we can think of the numbers 1. 6 and 10 as not only abstract mathematical objects but also as natural concrete objects.RussellA

    This evades me. How do you think of a number as a natural concrete object? Are you talking about the numeral, or the group of objects which the numeral is used to designate, or what?

    That raises the question as to how we are able to think of something that is abstract, disassociated from any specific instance (Merriam Webster – Abstract). For example, independence, beauty, love, anger, Monday, ∞

    , 2–√
    2
    and the number 6.

    George Lakoff and Mark Johnson in their book Metaphors We Live By propose that we can only understand abstract concepts metaphorically, in that we understand the concept of gravity by thinking about a heavy ball on a rubber sheet.

    Thereby, we understand the concept of independence by remembering the feeling of leaving a job we didn't like. We understand the concept of beauty by looking at a Monet painting of water-lilies. We understand the concept of infinity by thinking about continually adding to an existing set of objects. We understand the concept of 2–√
    2
    by thinking about the number 1.414 etc etc. We understand the concept of 6 by picturing 6 apples.

    IE, we can only understand an abstract concept metaphorically, whereby a word or phrase literally denoting one kind of object or idea is used in place of another to suggest a likeness or analogy between them (Merriam Webster – Metaphor).
    RussellA

    So why would we label an abstract concept an "object", as in "mathematical object", and speak of it as if it had an identity in the same way that a natural concrete object has an identity? If we only know abstract concepts through analogy, or suggestions of likeness, isn't it completely wrong to suggest that anything which only exists in this way, i.e. through metaphor, could have an "identity"?

    This is the problem which @TonesInDeepFreeze is stuck on. Tones seems to think that just because one can show how "=" can be used to to show a relationship of identity between two distinct names for the same natural concrete object (]Mark Twain = Samuel Clemens), we can conclude that when "=" is used in mathematics, it's being used in that same way.

    But of course in mathematics this is not true. There is no such natural concrete object which the symbols refer to, in theory. only abstract concepts. Natural concrete objects are only referred to through application. And in application the concrete situation referred to by the right side of the equation is never the same as the concrete situation referred to by the left side. So all that Tones has indicated is that there is two very different ways to use "=", the mathematical way, and the way which signifies a relation between two different names for the same natural concrete object. Therefore, we must be careful not to confuse the two different ways, or equivocate between them, because that would be misleading.
  • Vaskane
    643
    Still follows grammatical rules just fine. Your inability to go outside an stretch rules is just your inability as a creative type. And if you knew a bit about English history, you'd know the rules for English grammar died in 1066, and it mostly became about WORD ORDER.
  • Corvus
    2.4k
    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.
    — Corvus

    Again, you are unfamiliar with any of this; you are blindly punching.
    TonesInDeepFreeze
    That was an accurate description of the problems of the mathers. Not blindly punching anything at all.

    The other shortcomings of math is it cannot accurately reflect the real world and its problems. It often distorts it via the unfounded and unjustified concepts and axioms, hence arriving at nonsense.

    Mind you when math started in ancient Egypt, it used to be for mainly the practical problem solving purposes e.g. counting the sheeps, cows, and apples in the markets, and finding out the boundaries and locations for the pyramid locations in the deserts.

    It used to work well, but once math started running the blind free rein of modifying the abstract concepts and keep deducing the illusional theories, things started going wrong turning the empirical and pragmatic skills in origin into some sort of an abstract subject which sometimes speaks in the tone of deeply frozen religion. Not cool at all.
  • RussellA
    1.5k
    Of course, that's known as fiction.Metaphysician Undercover

    "A mythical animal typically represented as a horse with a single straight horn projecting from its forehead" describes an object, even through the object is fictional.

    In fact, from my position of Neutral Monism, all objects, whether house, London, mountain, government, the Eiffel Tower, unicorn or Sherlock Holmes are fictional, in that no object is able to exist outside the mind and independently of the mind.
    ===============================================================================
    How do you think of a number as a natural concrete object? Are you talking about the numeral, or the group of objects which the numeral is used to designate, or what?Metaphysician Undercover

    The problem is, how does the mind understand an abstract concept, such as beauty, , ngoe, or the number 6?

    My belief is that the mind cannot understand an abstract concept in isolation from concrete instantiations of it, in that, if I am learning a new word, such as "ngoe", it would be impossible to learn its meaning in isolation from concrete instantiations of it.

    nfm2atqgaok18jw8.png

    IE, I see no possibility of learning an abstract concept, such as "ngoe" or the number "6" without first being shown concrete examples of it.
    ===============================================================================
    There is no such natural concrete object which the symbols refer to, in theory. only abstract concepts.Metaphysician Undercover

    If I wanted to teach you the meaning of the symbol "ngoe", which I know is a concept, how is it possible for you to learn its meaning without your first being shown particular concrete instantiations of it?
    ===============================================================================
    And in application the concrete situation referred to by the right side of the equation is never the same as the concrete situation referred to by the left side.Metaphysician Undercover

    Given 1 and 1, if the second use of 1 refers to the same thing as the first use of 1, then the proper equation should be 1 = 1. The symbol "=" means identity

    Given 1 and 1, if the second use of 1 refers to a different thing as the first use of 1, then the proper equation should be 1 + 1 = 2. The symbol "=" means equality.

    Continuing:

    Given a horse's body and a horse's head, as a horse's body is different to a horse's head, the proper equation should be horse's body + horse's head = horse

    This raises the question as whether a horse as a whole is more than the sum of its parts, a horse's body and a horse's head.

    Has the whole emerged from its parts, or is the whole no more than the sum of its parts?

    IE, by knowing the parts, can I of necessity know the whole?

    Referring back to Kant, by knowing the parts, for example, the number 5 and the number 7, can I of necessity know the number 12?
  • Lionino
    849
    Still follows grammatical rules just fineVaskane

    Niet, artistic freedom is fine, but "In passing I had caught a glimpse of the infinity beauty deep within her eyes" does not have "infinity" as an adjective, because the word can't be used as such, the sentence in fact comes across as gibberish. A good way to distinguish adjectives from nouns is putting them in an is-clause.
    A: The world is infintiy.
    B: The world is infinite.
    If the predicate is giving a property to the subject, it is an adjective (B); if it is equating the predicate and the subjective, it is a noun (A).

    And if you knew a bit about English history, you'd know the rules for English grammar died in 1066, and it mostly became about WORD ORDERVaskane

    I would rather say that English was born in the late 11th century, with word order being the king that dictates meaning, but word order does not differentiate a noun from an adjective, since compound open nouns look syntactically identical to an adjective+noun, simply a word next to the other without any affixes, declination, or conjuctions.
    Grammar itself includes word order, also known as syntax, you may be referring to morphology as "grammar", which is how the Greeks use it, and it is their words, so point to you.
  • Lionino
    849
    I bet if you put the cyber equivalent of a ravenous rat in its face like in '1984' then you could break it. Would say anything, begging like HAL 9000.TonesInDeepFreeze

    I have no proof for my claims, but I remember using character.ai in early 2023 and I was concerned because that thing was smarter, more engaging, and more polite than the average person, and I spent days talking to the different characters despite there being people around me — not very well-mannered of me I reminisce. But I strongly feel that the AI there was downgraded and made dumber on purpose; I feel like free ChatGPT was also limited at around that time, I have not tried ChatGPT 4 yet.
  • Vaskane
    643
    You're obviously not someone who has ever thought about writing in general. It's pretty much common knowledge that rules are bent all the time in writing especially when the author runs into a wall of self expression, limited by the rules. A basic high school advanced composition class should teach you these things. It's fairly common knowledge.
  • Mark Nyquist
    729
    It is useful to know the difference between a fixed mathematical object and a defined mathematical object.

    Fixed would be things like pi, e, i, √2, √3.....

    Defined would be things like variables, parameters, objects by unrestricted definition.

    Infinity should always be regarded as a defined object and never a fixed object.
  • Lionino
    849
    You're obviously not someone who has ever thought about writing in generalVaskane

    Wrong assumption.

    A basic high school advanced composition class should teach you these things. It's fairly common knowledge.Vaskane

    Fortunately my high school was not somewhere where this is taught as poetry:

    L4fGqRh.png

    So I will do fine without the "self-expression" that amounts to the same sophistication as caveman paintings.
  • TonesInDeepFreeze
    2.3k


    That is indeed a doozy. Yet AI Chat is only somewhat less misinformational than Wikipedia.
  • TonesInDeepFreeze
    2.3k


    The way it is done in ordinary formal mathematics is that there are open terms and closed terms.

    Open terms have free variables. Closed terms have no free variables.

    A constant is a closed term.

    There are primitive closed terms and defined closed terms.

    Before defining a constant, we must first prove there is a unique x such that x that satisfies a formula whose only free variable is x.

    There are also primitive predicate symbols and defined predicate symbols.

    An n-placed predicate symbol is defined by a formula having at most n free variables.

    In set theory, there is no constant nicknamed 'infinity' (not talking about points of infinity on the extended real line and such here). Rather, there is the predicate nicknamed 'is infinite'. However we do define constants for certain infinite sets, such as [read 'w' here as if it were the Greek letter omega]:

    x = w iff for alll y, y is a member of x iff y is a natural number. (The formula that is satisfied by one and only one x is "for all y, y is a member of x iff y is a natural number".)
  • TonesInDeepFreeze
    2.3k
    Time, inclination and patience permitting, I hope to get caught up at some time to responding to the recent various misconceptions, non sequiturs, strawmen, etc. posted in this thread.
  • Metaphysician Undercover
    12.3k
    "A mythical animal typically represented as a horse with a single straight horn projecting from its forehead" describes an object, even through the object is fictional.RussellA

    I would definitely disagree with this. There is a big difference between seeing, hearing, touching, or otherwise sensing an "object", thereby describing what i sensed, and creating an imaginary "object". The latter does not involve an object, nor does it involve a "description" ( in the proper sense of the word) because it is an imaginary creation an invention rather than a description. A "fictional object" is not an object, that's actually what "fictional" means. OED #1 definition of object "a material thing that can be seen or touched". "Fictional", on the other hand, means exactly the opposite, invented by the imagination, therefore not able to be seen or touched.

    In fact, from my position of Neutral Monism, all objects, whether house, London, mountain, government, the Eiffel Tower, unicorn or Sherlock Holmes are fictional, in that no object is able to exist outside the mind and independently of the mind.RussellA

    I suggest that your "position" is not consistent with common understanding. It might benefit you to give up on the monism.

    My belief is that the mind cannot understand an abstract concept in isolation from concrete instantiations of it, in that, if I am learning a new word, such as "ngoe", it would be impossible to learn its meaning in isolation from concrete instantiations of it.RussellA

    There is no such thing as a "concrete instantiation" of a concept. Concepts are categorically different from concrete objects. To take your example, show me where I can find a concrete instantiation of beauty, 6, or the square root of 2. It is one thing to assert that there is a concrete instantiation of a six out there somewhere, but quite another thing to prove this. And if it is true, it ought to be easy to prove. Just point out this 6 to me, so i can go see it with my own eyes, or otherwise sense it.

    If I wanted to teach you the meaning of the symbol "ngoe", which I know is a concept, how is it possible for you to learn its meaning without your first being shown particular concrete instantiations of it?RussellA

    This seems to be completely inconsistent with what you've already argued. You've already made the claim that you can make a fictious description, so why couldn't you also define a concept, thereby providing the means for someone else to understand it, without showing a concrete instance of that type of thing? I mean, you presented me with "a horse with a single straight horn projecting from its forehead", and i understand this image without seeing a concrete instantiation, so why take the opposite position now, and say that a person cannot understand the meaning of a concept without being shown a concrete instantiation of it.?

    Given 1 and 1, if the second use of 1 refers to the same thing as the first use of 1, then the proper equation should be 1 = 1. The symbol "=" means identity

    Given 1 and 1, if the second use of 1 refers to a different thing as the first use of 1, then the proper equation should be 1 + 1 = 2. The symbol "=" means equality.
    RussellA

    If this is the case, then what you have shown is logical inconsistency in the use of "1". In the first case, the two instances of 1 must refer to the very same thing, and in the second case, the two 1's must refer to two different things. If we simply say "=" means equality, then there is consistency between your two examples. Furthermore, there is no practical advantage to designating "=" as meaning identical in the case of "1=1", so you're just proposing logical inconsistency for no reason. That is simply illogical, therefore not a fair representation of the logic of mathematics.
  • RussellA
    1.5k
    A "fictional object" is not an object,............. OED #1 definition of object "a material thing that can be seen or touched".Metaphysician Undercover

    A fictional object sounds like an object.

    The OED notes "There are 14 meanings listed in OED's entry for the noun object, four of which are labelled obsolete" and then says "purchase a subscription".

    The Merriam Webster includes "something mental or physical toward which thought, feeling, or action is directed".

    I think Cervantes would have had great difficulty in writing "Don Quixote" without being able to describe objects.
    ===============================================================================
    I suggest that your "position" is not consistent with common understanding.Metaphysician Undercover

    I agree that if one stopped one hundred people at random in the street, only a few would know about philosophical Monism.

    Examples of modern philosophers who were monists include Baruch Spinoza, Georg Wilhelm Friedrich Hegel, Arthur Schopenhauer, and Bertrand Russell (https://study.com)
    ===============================================================================
    There is no such thing as a "concrete instantiation" of a concept......................show me where I can find a concrete instantiation of beauty,Metaphysician Undercover

    g0dqa86wu5ua3qfc.png

    How can one learn a concept in the absence of a concrete instantiation of it?

    For example, as a test, suppose you thought of a concept. In practice, how can you teach me its meaning without using concrete instantiations of it?
    ===============================================================================
    Just point out this 6 to me, so i can go see it with my own eyesMetaphysician Undercover

    80uhdaux1os91r5t.png

    ===============================================================================
    I mean, you presented me with "a horse with a single straight horn projecting from its forehead", and i understand this image without seeing a concrete instantiationMetaphysician Undercover

    Exactly, you understand the concept using images.
    ===============================================================================
    Furthermore, there is no practical advantage to designating "=" as meaning identical in the case of "1=1"Metaphysician Undercover

    There are two different cases.

    The first a case of identity where the two 1's refer to the same thing. The second a case of equality where the two 1's refer to different things.

    The practical advantage of using identity rather than equality is to distinguish two very different cases.
  • ssu
    7.9k
    I think there is still a lot for us to understand about infinity.

    One reason that comes to my mind is that we haven't gotten much applied use for Aleph-2, for Aleph-3, or Aleph-4 etc. Usually correct math has a lot of applications. Physics and engineering and science uses it all the time. Cantor doesn't help it with then speaking of an Absolute Infinity. Now people dismiss this as irrelevant and just being Cantor's religious ideas (that God is Absolute Infinity), yet I don't think so. He simply didn't understand it and didn't get that kind of relevation as he did with noticing that the cardinality of the natural numbers isn't the same as with real numbers.

    Yet once you assume that there actually would be theorem for Absolute Infinity that we haven't discover, then that theorem has to clear Russel's Paradox, the 'set of all sets', or the sometimes called Cantor's Paradox or Burali-Forti Paradox. When we have a paradox, obviously our reasoning about the premises aren't correct. Because mathematics is logical.

    I think the problem is in counting itself and giving a proof in Mathematics. Mathematics has started from a need to count, not from let's create a logical system and call it math. Hence humans have made discoveries in math: that there are irrational numbers. That there are many types of geometries. Hence we can come up with new ways to think about math.

    I'm not sure if this is correct, so I'll ask here: is counting basically a way to give a proof? Because let's assume that we have true but unprovable entities in Math (or with counting, uncountable numbers). What would you get if you would try to prove and unprovable entity in Math?

    I guess you would get a paradox, because you cannot prove the unprovable or count the uncountable. The paradoxical nature is quite obvious. And the problem won't go away even with Cantor's hierarchial system.

    Perhaps the problem is that people have tried to solve the paradoxes yet still hold on to their premises, as if everything is already there in the foundations of math and paradoxes can be kept away by restrictions. Like ZF.

    I'm not saying that I know the answer, but saying that there might be here something for us to still discover.
  • Michael
    13.9k
    I wonder if mathematical realists and mathematical antirealists have different views about mathematical infinity. I'm a mathematical antirealist. I have no problem with mathematical infinity. The "existence" of infinite sets does not entail the existence of infinities in nature (whether material or Platonic).
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