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:

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.

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.

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.

in much arithmetic and mathematics "=" signifies equality, not identity — Metaphysician 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?

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.

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.

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.

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.

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.

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.

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.

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.

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

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.

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.

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.

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.

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?

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.

Do you like city or palm-trees more? — Lionino

I like cities as grim and forbidding as can be, thus without palm trees.

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.

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.

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.

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.

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.

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.

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.

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.

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."