By the law of identity, two distinct sets cannot be the same. If they actually are the same, then they are necessarily one, the same set. It's contradictory to say that two things are the same. If it is the same, it is only one. — Metaphysician Undercover

Yes I do understand take this philosophical objection. If I say 2 + 2 = 4 then if they are the same object they're the same. I'm saying nothing! If I stand up and say, "I am me!" I am saying nothing. I've only affirmed the law of identity.

Yet ironically, we have a world full of people standing up and saying "I am me!" and this is of great psychological and sociological and political importance! Going off topic a little but noting the irony.

But yes I've seen this argument before. Math is meaningless because in the end it's all tautologies and saying that a thing is equal to itself.

I don't think that can really be true though. Math IS useful and meaningful because it takes human effort to determine whether two different representations of a thing are actually the same thing. Don't you agree? 2 + 2 = 4 is formally a tautology. But historically, it was a really big deal for humanity. Agree or no?

I seem to recall the old philosophical standby of the morning star and the evening star, which appear to be two different things but (upon astronomical research that took millennia) turn out to be the same thing, namely the planet Venus and not a star at all.

If you reduce everything to the law of identity, you are saying those millennia of observation and theory and hard work by humans means nothing. I don't accept that.

The categorical vocabulary itself, however, seems to be spreading like wildfire. — alcontali

I'm glad I could turn you on to this paper.

I saw a little category theory back in grad school many moons go, then left math. When the Internet appeared in the 90's I was amazed to discover that category theory was being used in loop quantum gravity in theoretical physics. (This was of course from John Baez's This Week in Mathematical Physics on Usenet and later on his blog). Now category theory is in economics, biology (also via Baez) and of course computer science. Functional languages are the big thing now and they have monads, so Youtube is full of CS lectures on category theory. I find it all quite amazing to have seen this mind virus grow over the decades. Then again it takes a long time. Category theory was invented/discovered/whatever in the 1940's and was confined to math till probably the 90's. Another reason that "useless abstract math" should be valued. You never know what's going to eventually be important.

This by the way is my annoyance with the "indispensability argument" for mathematical existence. Category theory was not indispensable for computer science in 1940 but today it is. Therefore it has to be retconned as retro-indispensable; and then by analogy, everything is. Because in math, not everything is indispensable; but everything is potentially indispensable.

It is almost literally what you will find mentioned in the page on the "Brouwer-Hilbert controversy" — alcontali

@Mephist and I had a monumental pages-long conversation about constructive math a while ago. You might find it interesting. All in all I learned far more about constructivism than I ever did before, and even read some technical papers on the subject. But in the end I never gained any affinity for the subject. I still regard constructive math as unnecessarily tying your hands behind your back. But of course neo-intuitionism is making a comeback via computer science and homotopy type theory. I call it Brouwer's revenge.

https://thephilosophyforum.com/discussion/5791/musings-on-infinity

I
Concerning "no bearing on the topic at hand", you undoubtedly say that, because you are not aware of that famous discussion between Hilbert and Weyl in 1927, which was exactly about this. Could that have something to do with "weaker" Wiki skills? ;-) — alcontali

LOL. Of course I was only damning with faint praise, complementing your Wiki research but not your understanding. In that particular matter, at least. I'm perfectly well aware of the history of intuitionism and the Hilbert-Brouwer debate. FWIW I honestly did not make the connection from what you wrote, to the subject of intuitionism and constructive math. Could just be me.

Arbitrary axioms are the hallmark of creativity! — alcontali

Well, I really don't agree, and I think you misunderstand creativity. Art is not a product of arbitrariness, there are reasons for what the artist does, purpose, so arbitrariness is not the hallmark of creativity.

I don't think that can really be true though. Math IS useful and meaningful because it takes human effort to determine whether two different representations of a thing are actually the same thing. Don't you agree? 2 + 2 = 4 is formally a tautology. But historically, it was a really big deal for humanity. Agree or no? — fishfry

Yes I agree, but the key is understanding the limitations of math. If some logician were to argue that '2+2' and '4' are both the very same thing, because they are equal, we'd have to correct that person, showing that these are symbols, and '2+2' clearly has a different meaning from '4'. But then we are at the position of needing to explain what it is that is signified by these symbols. If we take the Platonic route, we say that the numeral '2' represents the number 2, and we avoid the question of meaning altogether. There is now no problem of what '2' means, because '2' represents a mathematical object which is 2. But now we are totally lost, because we can have no idea what the number 2 is, it's just a mathematical object. We cannot turn to meaning, because then we might as well just go back to the symbol, the numeral '2', and ask what it means. At this point we cannot turn to Platonism and say it's a mathematical object, because we want to know what the symbol actually means, not just say that it stands for an object (the existence of which cannot be validated).

I seem to recall the old philosophical standby of the morning star and the evening star, which appear to be two different things but (upon astronomical research that took millennia) turn out to be the same thing, namely the planet Venus and not a star at all. — fishfry

OK, but this analogy assumes that there is a thing, an object which "morning star" refers to, and it turns out to be the same object that "evening star" refers to. We cannot do that here with mathematical objects, because as I described above, if the symbol refers to an object, then we deny that there is any real meaning. The symbol stands for a mathematical object, and this is the only meaning there is. The symbol stands for an object, and that's that. There is no meaning. This is pretty much the stance that alcontali takes, axiom s are arbitrary, so there is no such thing as the axiom's meaning, it's just an arbitrary thing.

If you reduce everything to the law of identity, you are saying those millennia of observation and theory and hard work by humans means nothing. I don't accept that. — fishfry

No, this is exactly the opposite to what I am arguing. When we adhere to the law of identity, then everything has an identity proper to itself, therefore its own meaning. This does not rob meaning from mathematics, it only establishes clear limits to the possibilities of mathematics, so that mathematicians will not believe themselves to have accomplished the impossible, like putting the infinite within a set.

Well, I really don't agree, and I think you misunderstand creativity. Art is not a product of arbitrariness, there are reasons for what the artist does, purpose, so arbitrariness is not the hallmark of creativity. — Metaphysician Undercover

That would almost amount to saying that an artist's design choices are exclusively rational, and could therefore even be expressed in formal language. My own take is that I do not believe that. I believe that artists make use of other mental faculties, that are not rationality, when making their design choices. I also do not believe that it is possible to express, even in natural language, the output of these other mental faculties.

Functional languages are the big thing now and they have monads — fishfry

When I look at the monad page, it says:

With a monad, a programmer can turn a complicated sequence of functions into a succinct pipeline that abstracts away auxiliary data management, control flow, or side-effects.

Now, when I think of succinct pipelines, I think of method calls that return the object itself, such as in:

var p=new Person("John Doe").age(24).height(160).weight(70);

One problem is that programmers who discuss pipelining rarely use categorical language in their discussions. So, I cannot determine if both things are related (monads versus typical pipelining practices).

I also do not really recognize the examples in the monad page. because they are mostly in Haskell, while Haskell is really a specific niche. Mainstream programming does not (yet) include Haskell.

Haskell is only number 49 in the Tiobe popularity index for 2019 with 0.174% usage. I only know one tool in widespread use that was built in Haskell: the pandoc markup format converter; which I certainly use, because in my experience, pandoc is much, much better at gracefully handle lousy input that is full of errors.

Still, I find Haskell code incomprehensible to read. It requires jumping over an enormous hurdle, without any visible payoff. It would be perfectly possible to write pandoc in a language that is more mainstream.

But then again, of the first 35 languages in the tiobe index, I only like 6:

like (c, javascript, php, sql, assembly, lua)

dislike (java, python, c++, c#, vb.net, objC, ruby, matlab, groovy, delphi, vb, go, swift, perl, R, D, sas, pl/sql, dart, abap, f#, logo, rust, scratch, t/sql, cobol, fortran, lisp)

One of the languages I like the best is Bash, ranked only number 48 (0.187%), which surprises me, because the bash shell comes pre-installed with approximately every linux system. So, I suspect that the Tiobe index drastically underestimates its use.

What is the difference between actual infinity and potential infinity?

Same as between doing sex and an ability to do sex. That's how I'd explain this to my six-year-old.

One problem is that programmers who discuss pipelining rarely use categorical language in their discussions. So, I cannot determine if both things are related (monads versus typical pipelining practices). — alcontali

I'ts okay. Philosophers use no categorical language; they say, "That's post-modernist regressivism" or something of the like, and they leave it at that. It's us, dilettante, who spell everything out for each other.

I'ts okay. Philosophers use no categorical language; they say, "That's post-modernist regressivism" or something of the like, and they leave it at that. It's us, dilettante, who spell everything out for each other. — god must be atheist

Yeah, but what is a philosopher?

There are obviously the grandees, and then there is everybody else who discusses the grandees. Still, you cannot become a grandee yourself merely by talking about the grandees. There are no grandees about the grandees, or grandees in grandee-hood.

In Nassim Taleb's lingo, success in philosophy has "extremistan" characteristics. It is certainly not a normal, Gaussian distribution. It is the same in music or movies. There are just a few grandees, and everybody else is pretty much a nobody. Now, young people had better be aware of the fact that having a degree in philosophy, or any subject for that matter, will not elevate them from the status of a merely nobody. If all you can bring to the table, is a degree, then you are not needed, and also irrelevant, just like millions of others.

Furthermore, none of the actors we see on a TV screen or singers on music channels have a degree in film or in music. It is the same in philosophy. Not one grandee became one by rehashing from memory what other grandees had said.

Of course, people with a degree in philosophy would want to see an intermediate level in the hierarchy, i.e. of "grandee in discussing grandees", since the whole point of getting that degree was to get some recognition. Well, no. There is no recognition whatsoever. A degree signals nothing meaningful. It does not elevate you above the populace. On the contrary, it is just a worthless piece of paper. Get over it.

On the other hand, slagging off degrees is only fun if you can bring something else to the table, but that was exactly the point I was trying to make.

That would almost amount to saying that an artist's design choices are exclusively rational, and could therefore even be expressed in formal language. My own take is that I do not believe that. I believe that artists make use of other mental faculties, that are not rationality, when making their design choices. I also do not believe that it is possible to express, even in natural language, the output of these other mental faculties. — alcontali

Well, that would depend on how you define "arbitrary". Use of mental faculties in one's decisions negates randomness. If such decisions are arbitrary, then how do you understand "arbitrary"?

Well, that would depend on how you define "arbitrary". Use of mental faculties in one's decisions negates randomness. If such decisions are arbitrary, then how do you understand "arbitrary"? — Metaphysician Undercover

I see arbitrary in this context as "further unjustified".

For example, I do not trust what a literary critic says about why Charles Dickens wrote a particular passage. I strongly suspect that even Dickens himself did not really "know" it. It just came up to him, sourced from other mental faculties than mere rationality. So, from the outside it looks "arbitrary".

It is the same situation as with a sequence generated by a Mersenne Twister. From the outside, it looks random. From the inside, we can see that you will always get the same sequence depending on the seed that you use. Is the sequence random? For outsiders, yes. For insiders, no.

You can find an online demo for a Mersenne Twister here.

In that sense, if a true random number generator does not exist -- which is an unsettled question -- then randomness is always a subjective perception. Mutatis mutandis, if we have no clue as to how these other mental faculties work, then their output will appear as arbitrary to us.

It is the same situation as with a sequence generated by a Mersenne Twister. From the outside, it looks random. From the inside, we can see that you will always get the same sequence depending on the seed that you use. Is the sequence random? For outsiders, yes. For insiders, no. — alcontali

It's not arbitrary then, it just looks arbitrary, in appearance, but it really is not. That it is arbitrary is an illusion. Would you see mathematical axioms in the same way? They look arbitrary, but they really are not. What is required to get beyond the illusion of arbitrariness is to get inside of the head of the artist. This does not mean to literally get inside, but to learn how to think in the same way as the artist. Then you will no longer be an outsider who sees mathematical axioms as arbitrary.

It's not arbitrary then, it just looks arbitrary, in appearance, but it really is not. That it is arbitrary is an illusion. Would you see mathematical axioms in the same way? They look arbitrary, but they really are not. What is required to get beyond the illusion of arbitrariness is to get inside of the head of the artist. This does not mean to literally get inside, but to learn how to think in the same way as the artist. Then you will no longer be an outsider who sees mathematical axioms as arbitrary. — Metaphysician Undercover

Agreed, but that is exactly what is not possible. These other, unknown mental faculties are out of reach of any ability to inquire them rationally. Furthermore, their assumed input could still truly be random, because there is no method available to distinguish between the output of unknown mental faculties and sheer randomness.

Yes, I see mathematical axioms in the same way. They look arbitrary but they probably aren't.

There is something uncannily recognizable to Plato's theory of forms, but the purported link between the forms and the real, physical world is unfortunately out of scope for the instrument of rationality. Still, the uncanny sensation of recognition suggests that this link is not necessarily, completely out of scope for other, unknown mental faculties.

Furthermore, their assumed input could still truly be random, because there is no method available to distinguish between the output of unknown mental faculties and sheer randomness. — alcontali

I think we've been through this all before, you and I. I don't think that just because there is no method available, the input is out of reach. Methods come into existence, and evolve, so things which are reached by existing methods were at one time out of reach. Therefore it's reasonable to believe that a method could be developed to reach the things which presently cannot be reached. So if someone reaches a conclusion through a mental process which you consider to be by your standards, not rational, this does not mean that it is impossible to reach that conclusion through a rational process. It is possible that the required rational process could be developed.

Still, the uncanny sensation of recognition suggests that this link is not necessarily, completely out of scope for other, unknown mental faculties. — alcontali

Yes, my point is that it requires effort to distinguish the good from the bad, but these "unknown mental faculties" may be brought into the realm of the known.

No, this is exactly the opposite to what I am arguing. When we adhere to the law of identity, then everything has an identity proper to itself, therefore its own meaning. This does not rob meaning from mathematics, it only establishes clear limits to the possibilities of mathematics, so that mathematicians will not believe themselves to have accomplished the impossible, like putting the infinite within a set. — Metaphysician Undercover

You made the statement that ZFC allows two different things to be equal. I said I know of no such example and you have not backed up your claim or put it in any context that I can understand. You must be thinking of something, I'm just curious to know what.

2 + 2 and 4 represent the exact same mathematical set. '2+ 2" and '4' are distinct strings of symbols. I don't know any mathematicians confused about this. And, as you agree, the discovery that these two strings of symbols represent the same set, is a nontrivial accomplishment of humanity and is meaningful.

I really don't understand your remark that ZFC allows distinct things to be regarded as the same. Unless you mean colloquially, as in the integer 1 and the real number 1 being identified via a natural injection.

Here is your quote.

ZFC theory allows that two distinct things are the same, contrary to the law of identity. — Metaphysician Undercover

I categorically deny that claim. Please put it in context for me. As stated it's flat out false as far as I know. Of course one thing may have multiple representations; and it may have taken years, decades, or centuries to discover that fact.

You made the statement that ZFC allows two different things to be equal. I said I know of no such example and you have not backed up your claim or put it in any context that I can understand. You must be thinking of something, I'm just curious to know what. — fishfry

I was in discussion with alcontali, referring to what was said by alcontali:

S1 and S2 describe the same set. Therefore, S1 = S2. — alcontali

2 + 2 and 4 represent the exact same mathematical set. '2+ 2" and '4' are distinct strings of symbols. I don't know any mathematicians confused about this. And, as you agree, the discovery that these two strings of symbols represent the same set, is a nontrivial accomplishment of humanity and is meaningful. — fishfry

The point I made is that 2+2 is not the same as 4. So if set theory treats them as the same, it is in violation of the law of identity.

I really don't understand your remark that ZFC allows distinct things to be regarded as the same. Unless you mean colloquially, as in the integer 1 and the real number 1 being identified via a natural injection. — fishfry

'2+2' is clearly different from '4'. Each of those two expressions are composed of different symbols, having different meaning. Despite the fact that they are said to be equal, in no way are they the same. If ZFC allows that they are the same, as you say above, then ZFC allows two distinct things to be regarded as the same.

Since they are not the same according to the law of identity, by what principle of identity does ZFC claim that '2+2' is the same as '4'?

The point I made is that 2+2 is not the same as 4. So if set theory treats them as the same, it is in violation of the law of identity. — Metaphysician Undercover

Of course 2 + 2 is the same thing as 4. I cannot imagine the contrary nor what you might mean by that claim.

But more importantly, they are the same set in ZFC. So it's not an example of your claim that ZFC allows two distinct things to be regarded as the same.

But you hold that 2 + 2 and 4 are not the same? How so? Without quotes around them they are not strings of symbols, they are the abstract concept they represent. And they represent the same abstract concept, namely the number 4. You deny this? I do confess to bafflement.

ps -- I didn't read the S1 and S2 parts of the thread, if it might help I'll go back and review them.

Of course 2 + 2 is the same thing as 4. I cannot imagine the contrary nor what you might mean by that claim. — fishfry

Explain to me then, how this set '2+2', is the same thing as this set, '4'. They look very different to me, and also have a completely different meaning. By what principle do you say that they are the same?

But more importantly, they are the same set in ZFC. So it's not an example of your claim that ZFC allows two distinct things to be regarded as the same. — fishfry

Yes, that's exactly the point. ZFC says that they are "the same" set, when they are clearly not the same by any intelligent reading of the law of identity. Therefore ZFC must employ some other principle of identity in order to say that they are the same. Can you state ZFC's law of identity?

But you hold that 2 + 2 and 4 are not the same? How so? Without quotes around them they are not strings of symbols, they are the abstract concept they represent. And they represent the same abstract concept, namely the number 4. You deny this? I do confess to bafflement. — fishfry

This is absolutely false. The symbol 2 has a meaning, the symbol 4 has a meaning, and the symbol + has a meaning. Clearly 2+2 is not the same concept as 4. Otherwise there would be no point in writing the exact same concept in two different ways, and the symbol =, which is commonly used to express the relationship between these two different concepts, would be meaningless. Do you see that 4=4 would be a meaningless equation in mathematics? Therefore it is very evident that 2+2 is not the same concept as 4, and the = sign expresses a meaningful relationship between these two distinct concepts, a relationship which is quite different from the useless expression of 4=4. The usefulness of an equation is due to the fact that something different from what is expressed on the right side, is expressed on the left side

Explain to me then, how this set '2+2', is the same thing as this set, '4'. They look very different to me, and also have a completely different meaning. By what principle do you say that they are the same? — Metaphysician Undercover

I walked through this in detail a few posts ago. In the Peano axioms they are both the number SSSS0. In ZF they are both the set {0, 1, 2, 3}. = { ∅, {∅}, {∅, {∅}}, {∅, {∅}, {∅, {∅}}} }.

See for example

https://en.wikipedia.org/wiki/Ordinal_number#Von_Neumann_definition_of_ordinals

This is absolutely false. The symbol 2 has a meaning, the symbol 4 has a meaning, and the symbol + has a meaning. Clearly 2+2 is not the same concept as 4. — Metaphysician Undercover

We must be talking past each other in some way. I cannot conceive of anyone claiming 2 + 2 and 4 are not the same thing. I can't respond because from where I sit you're talking nonsense. If you have some subtle philosophical point it eludes me. I just can't respond. Perhaps you have a reference to support your point of view.

Here's my earlier post where I described the Peano and ZF constructions.

https://thephilosophyforum.com/discussion/comment/323461

I acknowledge that there might well be some philosophical point of view that allows you to claim that 2 + 2 and 4 are not the same thing. I've never heard of it and I don't understand what you mean, but that could just be due to my own ignorance.

But you claim that 2 + 2 and 4 are not the same object in ZFC. And THAT is an area where I am not ignorant. You're just wrong. 2 + 2 and 4 represent the same set in ZFC.

But forget set theory. You claim that 2 + 2 is not 4? The last time I heard that it was in the novel 1984 when the protagonist Winston Smith is being tortured to obtain his submission to Big Brother. He's ordered to believe that 2 + 2 = 5; and in the end, he does.

You tell me how 2 + 2 is not 4. If it's not, what is it? And have we always been at war with eastasia?

The point I made is that 2+2 is not the same as 4. So if set theory treats them as the same, it is in violation of the law of identity. — Metaphysician Undercover

Well, first there is the understanding that the "=" symbol pretty much never means "identical". The symbol is much more permissive than that. It usually means something along the lines of "extensional" or "equivalent", depending on the axioms in use, but not necessarily "identical".

In arithmetic, "2+2=4" means that the equality is provable from number theory -- or from the larger, encompassing theory such as set theory -- by using the inference/rewrite rules of arithmetic.

From there the notion that two different, non-identical symbol streams can be "equal" to each other, with "equal" meaning that their extensionality or equivalence is provable from the main theory in which we happen to be operating.

In abstract, Platonic worlds, we almost always assume that computation does not require "effort". This view may very well fall apart in virtual worlds, generated by running computer processes. There will be some calculation effort involved for a computer process to derive that "2+2" resolves to "4". So, the more computing intensive the resolution process, the less the equality will be automatic. In the real, physical world, the problem is further exacerbated by the fact that humans tend to be slow and error prone at carrying out calculations in arithmetic.

The meaning of the "=" symbol is not only very context-sensitive, but I actually do not know of even one context in which it means "identity".

For example, in Javascript, the single "=" symbol is already taken up to express assignment. For example, "a=5" means store value "5" in variable "a". It does not want to express that "a" and 5 would be equal or so. It is rather an instruction that seeks the side effect of changing what value is stored in a.

"a === b" means "a == b" and "typeof(a)==typeof(b)".

So, '3' == 3 resolves to "true" but '3' === 3 does not, because the string '3' and the integer 3 are of a different type.

They are considered equal in the expression " '3' == 3 " because the "==" operator will carry out enough work to convert both types to a permissive common denominator, and if it can then declare a match somewhere, it will return true.

The permissiveness of the "==" operator is generally considered questionable. The practice creates dangerous corner cases. For example, 0 is considered a falsy value while "0" a truthy one. That can go surprisingly wrong in the context of permissive equality.

The following is an interesting article about the difference between === and == in javascript.

I walked through this in detail a few posts ago. In the Peano axioms they are both the number SSSS0. In ZF they are both the set {0, 1, 2, 3}. = { ∅, {∅}, {∅, {∅}}, {∅, {∅}, {∅, {∅}}} }. — fishfry

This does not show me the principle of identity. Saying that two things are the same does not make them the same. It's a hollow assertion without a principle. And I don't see any reference to sameness in your reference.

We must be talking past each other in some way. I cannot conceive of anyone claiming 2 + 2 and 4 are not the same thing. — fishfry

I can't believe there is a person who does not see a difference between 2+2 and 4. The two are equivalent. And, as I explained equivalence would be meaningless, and equations useless, if there was not a difference between the left side of an equation and the right. The Wikipedia page on "equation provides an analogy. "An equation is analogous to a scale into which weights are placed." Do you see that the things on the two separate sides of a balance are not "the same"? They are said to have the same weight, but this does not make them the same thing.

So, in the case of ZFC, by what property are the two sets said to be "the same"? It's not the same weight, as in the scale analogy, nor is it the same numerical value, as is the case in the equation (what alcontali refers to as "number theory" above). What is the principle of sameness?

Perhaps you have a reference to support your point of view. — fishfry

If you have no idea of what equivalent means, or of how equations are used, then I don't think I can help you. If you are simply asserting 2+2 is the same as 4 without thinking about what you are saying, because it supports your metaphysics, then why don't you smarten up?

But you claim that 2 + 2 and 4 are not the same object in ZFC. And THAT is an area where I am not ignorant. You're just wrong. 2 + 2 and 4 represent the same set in ZFC. — fishfry

I fully acknowledge, that in ZFC 2+2 is "the same" as 4. I am not denying this. I am saying that it is wrong, because it violates the law of identity, without any justification. If one wants to establish a principle in violation of a fundamental law of logic like the law of identity, then that person ought to provide justification for the proposed principle. Without justification, use of that principle is mere sophistry.

Well, first there is the understanding that the "=" symbol pretty much never means "identical". — alcontali

Tell that to fishfry, who is arguing the opposite, that the left and right of the equation actually are the same.

You tell me how 2 + 2 is not 4. If it's not, what is it? — fishfry

Refer to alcontali's post above. Thanks al.

The senses in which 1+1=2 is said to be an analytic sentence or tautology, are not the same as the senses in which 1+1=2 is said to be an empirically contingent proposition.

With Actual infinity you have a set that contains EVERY natural number as an element, while with potential infinity you can only have finite sets of naturals but without having a limit to formation of bigger finite sets of naturals, so when you have a set of say x,y,...,z naturals i.e. when you have the set {x,y,..,z} which is finite you can simply form the still FINITE set {x,y,...,z, k} where k is not an element of the first set, this process of always being able to add an extra-element to a finite set to get a bigger finite set is what is called as Potential Infinity. The difference is that Actual infinity poses an additional claim that is the existence of a set that contains ALL natural numbers as elements of it and that that set is an infinite set of naturals. And so this set would witness actual infinity.

I fully acknowledge, that in ZFC 2+2 is "the same" as 4. I am not denying this. I am saying that it is wrong, because it violates the law of identity, without any justification. — Metaphysician Undercover

I was skimming your reply looking for a point of reference, something I could understand. I came to this. I think it's a point of irreconcilable difference. I don't agree with your judgment, but I am incapable of rational response, because I cannot fathom the point of view.

This is my own personal limitation, I'm certain of that. My ignorance of philosophy is profound. I have no doubt that you have a point to make that, from your point of view, is a valid point.

I myself do not ever think I could agree with or even understand such a point of view. Within ZFC, at least, the statement 2 + 2 = 4 is a theorem that can be proven according to strict logical principles that are clearly expressed; and using assumptions that are clearly stated. Within this framework. 2 + 2 = 4 expresses an identity of sets. This is a technical fact that is beyond dispute. And set equality is defined directly in terms of the logical law of identity. I thought I explained it. The axiom of extensionality leverages the logical law of identity. So 2 + 2 = 4 is a perfect expression of the law of identity.

I acknowledge that you feel differently about this but I don't regard myself as being capable of ever understanding such a point of view. And I would not want to be able to understand such a point of view even if I could!

I'll read whatever you write on the topic in the hopes I might learn something, but I don't think it will be productive for me to engage on this.

I will depart from this thread, feeling on my side that I can't talk to someone who is claiming that 2 + 2 is something other than 4. And also feeling deep down that I must be missing something really profound, but I don't think it's something I'd want to get even if I could. — fishfry

I suppose the feeling is mutual. I really cannot believe that there is a rational human being who truly believes that 2+2 is the same thing as 4. Isn't this what we learn in basic math, first grade? You take two things, add to them another two things, and you have four things. Very good. But we can get four by adding three to one, or by subtracting two from six, and an infinite number of 'different' ways. So it is impossible that 2+2 is the same as 4, because there would be infinitely many different things which are the same as four. Does it make any sense to you, to believe that there is an infinite number of different things which are all the same? Or can you see that 2+2 is not the same as 8-4?

In certain contexts, we might treat the expressions 5+7 and 12 identically, as for example when discussing what our shared formal convention says concerning our customs of numeric substitution. Yet in mathematics applications, these expressions cannot be treated the same, as simply demonstrated by children who must sum with their fingers.

It is wrong to insist that our mathematical convention grounds the meaning and truth of mathematics in application. For instance, it is incorrect to claim that "the conventions of logic a priori determine that 5+7=12, whereas the physical calculation merely confirms it".

Consider, for example the addition of two summands that are so large that their summation cannot be precisely determined in any individual physical experiment, let alone by hand. Here there isn't a clear distinction between the truth of the summation according to convention, versus the confirmation of the summation bu physical demonstration.

And in visual psychology, it should not be regarded as an error if a test subject reports that he saw 5+7 as 13. It simply means that visual phenomena are not a good model of ordinary arithmetic and vice versa.

And in visual psychology, it should not be regarded as an error if a test subject reports that he saw 5+7 as 13. It simply means that visual phenomena are not a good model of ordinary arithmetic and vice versa. — sime

I think my eye doctor would prescribe glasses if I saw 5+7 as 13.

" {|||||,|||||||} 'equals' {||||||||||||} " isn't visually acceptable to me. The left side looks too short.

I suppose the feeling is mutual. I really cannot believe that there is a rational human being who truly believes that 2+2 is the same thing as 4. — Metaphysician Undercover

I think I understand your point but I have some counterpoints. I believe you are saying that when we say 2 + 2 = 4 we are saying two things: One, that they represent the same natural number; and two, that 2 + 2 is a legal decomposition of 4, which is not necessarily known beforehand. So 2 + 2 = 4 asserts something more than merely saying 2 + 2 or 4 by themselves. And you're right about that.

However it's not an ontological fact, it's an epistemological fact. That is, the partition of 4 into 2 + 2 is literally a matter of definition. It's an immediate consequence of the way we define the symbols. So it was true before we knew it. If you believe that math has Platonic existence, it was true even before there were humans.

You're right that before someone told us that 2 + 2 = 4, we may not have known it. So we have learned something new; but we have not made something formerly false be true. So 2 + 2 = 4 imparts knowledge of that particular partition; but it was true before we knew it. Ontologically it's an identity as I have been saying all along. But I will grant that epistemologically it is new information above and beyond the mere fact that they're the same number.

Isn't this what we learn in basic math, first grade? You take two things, add to them another two things, and you have four things. — Metaphysician Undercover

Yes, we LEARN that. But it was always true. It was always an identity, even before we learned it. But I agree with you that it's new information that we have learned. It's the morning star and the evening star. They were always the same object, the planet Venus. We LEARNED that they are the same, and that was new. But it was true -- that is, it was an identity -- even before we learned it.

Very good. But we can get four by adding three to one, or by subtracting two from six, and an infinite number of 'different' ways. — Metaphysician Undercover

You have actually hit on some very deep math. Subtraction isn't useful, since as you note there are infinitely many ways of expressing 4 as the difference of two integers. But if we restrict our attention to positive integers, it's a very interesting question. 4 = 1 + 3 = 2 + 2 = 2 + 1 + 1 = 1 + 1 + 1 + 1. so there are 5 partitions, as they're called. We say that "5 is the partition number of 4."

https://en.wikipedia.org/wiki/Partition_(number_theory)

If you happen to have seen the movie The Man Who Knew Infinity, it was this partition problem that was solved by Ramanujan. He found a formula that gives the partition number for any positive integer. It's a big deal in number theory.

But these are discoveries. God, or the Platonic universe, already knows the partition number of every integer. No new information is created by the discovery; rather, only new KNOWLEDGE is created.

So I would say that 2 + 2 = 4 is an expression of the law of identity; but we did not always KNOW that until someone discovered it and taught it to others. Is this a distinction you find meaningful?

So it is impossible that 2+2 is the same as 4, because there would be infinitely many different things which are the same as four. — Metaphysician Undercover

But there aren't. There are infinitely many different representations of the concept of 4, just as schnee and snow are two representations of the white stuff that falls from the sky in the winter. And you are right that it may sometimes take hundreds or thousands of years for us to discover that two representations represent the same thing. But they were always the same even before we knew that.

Does it make any sense to you, to believe that there is an infinite number of different things which are all the same? — Metaphysician Undercover

No of course that doesn't make sense to me. What makes sense to me is that there may be infinitely many distinct representations of the same thing; and that it sometimes takes hard work to discover that fact. But when we discovered that the world was round, it wasn't flat the day before. We created new knowledge; but we did not create a new reality. The world was round and then we discovered the world was round. 2 + 2 = 4 and then we discovered that fact. Two representations of the same thing.

Or can you see that 2+2 is not the same as 8-4? — Metaphysician Undercover

They're two representations of the exact same identical thing. If they weren't they would not deserve the equal sign. Do you agree that schnee and snow are identical, even though one has to pick up a little German (or English as the case may be) in order to discover that?

I think I understand your point but I have some counterpoints. I believe you are saying that when we say 2 + 2 = 4 we are saying two things: One, that they represent the same natural number; and two, that 2 + 2 is a legal decomposition of 4, which is not necessarily known beforehand. So 2 + 2 = 4 asserts something more than merely saying 2 + 2 or 4 by themselves. And you're right about that. — fishfry

I wouldn't even say that. '2+2' represents two distinct quantities of two, being added together. So there are two distinct units, a unit of two, and another unit of two represent here with '2' and '2'. On the other hand '4' represents one unit of four, so there is only one unit represented, a unit of four. Notice there are two symbols of 2, so two distinct things represented on the left side, and only one symbol.'4', therefore one thing represented on the right side.

However it's not an ontological fact, it's an epistemological fact. — fishfry

What I am talking about is ontological, because it is the objects which are represented by the symbols. We need to first clarify what is represented by the symbols before we can proceed to epistemological principles.

That is, the partition of 4 into 2 + 2 is literally a matter of definition. — fishfry

It's not a partition which is represented, that would be division, four divided by two. What we have in 2+2 is two distinct units of two being unified with the symbol '+'. Conversely, we could take a unit of four, and divide it into two distinct units of two. That would be a partitioning.

So I think you have things reversed. Ontologically, 2+2 is clearly distinct from 4, but epistemologically we might say that they have the same value. They are equivalent by an epistemic principle, but distinct by ontological principles.

It was always an identity, even before we learned it. — fishfry

It's not an identity though, it's an epistemic principle. '2' Identifies one thing, '4' identifies another thing. That two '2's has the same value as one '4' is not an identity it is a conclusion drawn from an epistemic principle, what alcontali called number theory.

So I would say that 2 + 2 = 4 is an expression of the law of identity; but we did not always KNOW that until someone discovered it and taught it to others. — fishfry

That's not an expression of the law of identity at all. That's an expression of an equation. As alcontali explained, it's a conclusion drawn from the principles of number theory. Do you know the law of identity? A thing is the same as itself.

But there aren't. There are infinitely many different representations of the concept of 4, just as schnee and snow are two representations of the white stuff that falls from the sky in the winter. And you are right that it may sometimes take hundreds or thousands of years for us to discover that two representations represent the same thing. But they were always the same even before we knew that. — fishfry

What I've been trying to explain to you, is that '2+2' does not represent the concept of four, '4' does. As I explained already, if both sides of an equation represented the exact same concept the equation would be useless. But equations are not useless, they are very useful for many different purposes, and that is because they express an equality between two distinct concepts. It's nonsense to say that the right side and left side of an equation each represent the exact same concept. What could an equation do for us if all it expressed was '4=4', or '2=2', or '50=50'? If both sides represented the same concept, that's all we'd have. It's only because the one side represents something different from the other, that the equation is useful.

Do you agree that schnee and snow are identical, even though one has to pick up a little German (or English as the case may be) in order to discover that? — fishfry

This is not relevant. It's a simple fact that '2+2' does not say the same thing as '4'. There is no language in which '2+2' would be translated as '4'. Each of these says something different, and they maintain their difference in all languages, so '2+2' is never translated into another language as '4'. That would be a mistaken translation. So it's very clearly a mistake on your part, to say that '2=2', and '4' "are "two representations of the exact same identical thing". It seems so basic that I can't believe you actually believe that.