only leftists equate wanting border control with racism
— Judaka

I equate what might be described as ‘excessive or inefficient’ border control with an effort to mobilize political support by exploiting a natural conservative tendency. — praxis

I haven't read any of this thread and only looked at this one post. I happen to have an interest in the US/Mexico border, being a Californian who's made many trips to Mexico and followed border politics for decades. The excessive and inefficient border control you speak of is the result of decades of bipartisan hypocrisy. Indeed, Obama built the cages that he kept kids in as he separated them from families or turned them over to traffickers. It's all a matter of public record.

What the left does that's very disingenuous is to call Trump a bad person for enforcing the laws that Democrats have made. Look up the actual immigration policies of Hillary and DiFi and all the Clinton and Obama era Dem legislators and administrations. The Dems passed the Secure Fence act of 2006 giving Trump the legal authority to build his wall (which for the record I strongly oppose). Bill Clinton and Barack Obama were the ones who militarized the border and took a bad situation and made it worse.

If the Dems want open borders, let them pass a bill op;ening the borders. Or abolishing ICE (which they all voted for at the time). Or abolishing detaining kids in cages. And where would you keep them while their relationship to the adults who CLAIM to be their parents is sorted out? The cages are to keep violent sexual predators out. Separating the kids from their ALLEGED parents is how you prevent turning kids over to traffickers. What do you propose instead? What solutions have the Dem politicians proposed? NONE. Just insults that Trump's a bad person.

To be clear about where I'm coming from: In general I'm more of an open borders type. I favor good relations between the US and our friend, neighbor, and third largest trading partner Mexico. I think funds spent on the wall would be better spent staffing up the official border stations so that legitimate crossers can pass more quickly.

But I abhor the awful hypocrisy of the left when they say that Trump is a bad person for enforcing the laws that they passed and for doing exactly the same things on the border that Obama did when he had a massive refugee crisis in 2014. I've watched the Dems steadily make the border crisis worse decade by decade from Clinton onward and now trying to blame the whole mess on Trump. That's hypocritical and totally counterproductive. The Dems have no interest in solutions on the border. They never have. Neither have the GOP of course. The ongoing hypocrisy and humanitarian disaster on the southern border serves both their interests.

ps -- Wiki agrees with me.

In its formal representation, the law of identity is written "a = a" or "For all x: x = x", where a or x refer to a term rather than a proposition, and thus the law of identity is not used in propositional logic. It is that which is expressed by the equals sign "=", the notion of identity or equality. It can also be written less formally as A is A.

https://en.wikipedia.org/wiki/Law_of_identity

It is impossible that the numeral, the symbol "2" represents the same object every time it occurs. — Metaphysician Undercover

You've just swapped in the term numeral for number. That's a particularly low form of false argument. It's like saying that two isn't two because some people call it zwei or dos or deux.

If you deny that the number 4 is the same as the number 4 you are entitled to your opinion, but that kind of sophistry is of no interest to me.

I did want to add that earlier when you said that mathematical equality is not identity but rather only equality in a certain context, you are thinking of equivalence or isomorphism. Mathematical equality is identity, not mere equivalence or isomorphism. You're simply wrong about that.

The idea that equality means that two "different" things are "the same" is nonsense. Equality means that two distinct expressions or representations of a thing refer to the same thing. 2 + 2 = 4 is an identity. I can't help what your grade school teacher put in your young and uncomprehending head. It's tragic that by your own admission your mind is stuck in the third grade.

I've demonstrated how equality is different from identity. — Metaphysician Undercover

You have never done so, If you had we could talk about it. You have indeed expressed belief in the false claim that mathematical equality does not express identity. Repeating a false claim does not constitute a demonstration. On the contrary. Mathematical equality DOES express identity.

If you proposed an argument rather than just a repeated false claim, we could talk about it.

But in the end you have now said, and not for the first time, that you don't believe the number 4 is the same as the number 4. There is no conversation to be had (at least on this topic) with someone who professes such an obvious falsehood.

↪fishfry ↪alcontali Thank you. Sorry for being lazy about this but do you have an idea about what kind of problems Brahmagupta was dealing with when he needed to formalize zero/nothing? — TheMadFool

Me? Not my bailiwick I'm afraid.

Identity" applies to one thing, the same thing, its identity. So "identity" relates to what makes one specified thing other than everything else. "Equality" applies to two distinct things which are judged to be "the same" in a specific way. You might consider that "identity means "the same" in an absolute way, whereas "equal" means "the same" in a qualified, relative way. — Metaphysician Undercover

I understand what you're saying. You're wrong about mathematical objects. The number 2 is identical to the number 2. The number 2 + 2 is identical to the number 4. Identical as in your definition. There is only one thing, the Platonic number 4, which we may denote as 4 or 2 + 2 or the positive square root of 16 or 3.999... and many other representations.

I understand what you are saying and I deny it.

I do of course recognize contexts in which what you say is right. For example in group theory, two distinct groups that are isomorphic are often taken to be the same with respect to isomorphism. And in univalent foundations, we take as an axiom that isomorphism is equality (I'm paraphrasing greatly here but that's the essence as I understand it).

https://en.wikipedia.org/wiki/Univalent_foundations

So ok for that aspect of things.

But in ZFC, the domain of discourse in which you originally claimed that identity differs from equality, I tell you that you are incorrect. But I have said nothing new, I've written the same things over and over.

I hear you saying that in math the axioms speak of equality but not identity. I take that as profoundly insignificant, you take it as profound. This I believe is where we differ.

I actually find no logical connection between a dhow and mathematicians. Why would a dhow interest a mathematician? I'm genuinely interested. — TheMadFool

Can't answer that specifically but there are other examples of the same thing. The Latin word for pebble is calculi, which gives us calculation. When we calculate we are literally pushing pebbles around. Even if we we're pushing them really fast through semiconductor circuits, logically we are only pushing around pebbles. A lot of technical math terms have their root in non-related or metaphorical objects. In higher math they have sheaves and germs and stalks, evoking their meaning in nature.

↪fishfry I would disagree and say that any fiction is a compilation of prior reality, mixed up into something which only appears novel. — Razorback kitten

Yes of course. Moby Dick is based on the true story of the Essex, a whaling ship sunk by a whale. All fiction is based on reality. Even science fiction always has recognizable themes from our own lives.

That doesn't mean that Captain Ahab existed a moment before Melville conceived and wrote him into existence.

I apologize if I've forgotten exactly what point we were making. Perhaps we're in agreement on something of importance. Fictional entities are interesting to me because I like the idea of mathematical fictionalism. Math is fiction exactly like Moby Dick is. And like Moby Dick, math is based on reality; but it is not itself reality. Math, like a novel, has value because it's interesting and enjoyable and because it gives us insight into life. It need not be literally true to have value. That's a good way to look at math.

https://en.wikipedia.org/wiki/Essex_(whaleship)

https://plato.stanford.edu/entries/fictionalism-mathematics/

It couldn't have been a typo — Metaphysician Undercover

I see no good faith on your side (claiming my typo isn't a typo??) but I'll stipulate that you disagree. I'm done with this thread. You should, in the fullness of time, go back to the detailed proof from the Peano axioms that 1 + 1 and 2 are identical. You would learn something. The fact that you refuse to engage with that proof makes this thread irrelevant to my life. You asked for a proof, I gave you a proof and now you want to quibble that you have no obligation to read it? For weeks on end? Is that your idea of honest discussion?

Regarding the proof: That's how you show that 1 + 1 = 2 and that moreover, IF you believe that = means something other than identity, that 1 + 1 is identical to 2. I deny that mathematical equality differs from identity in set theory, except in a handful of casual conventions that can easily be rigorized on demand. You CLAIM they have different meanings but have not even attempted to defend or explain your claim but only seem to be avoiding the question. I use set theory because YOU are the one who invoked ZFC, claiming, and after all this time without evidence, that in ZFC two things that are not identical are asserted to be equal. I categorically deny that (except for as usual a handful of conventions such as embedding the integers in the rationals in the reals etc.) You made a claim about ZFC. I tell you that you are factually incorrect. You have failed to produce an example of your claim. You choose not to engage on the Peano proof, which contradicts your belief about 1 +1 and 2. I see no logical continuation of this dialog.

Again I do understand that you don't see things this way. That's what makes horse races.

Have a nice evening.

This is the false premise you stated:
1.1 We have the law of identity that says that for each natural number, it is equal to itself.
— fishfry

That is not the law of identity. The law of identity is the philosophical principle which states that a thing is the same as itself. — Metaphysician Undercover

You deliberately re-quoted exactly the line that I apologized for, explained as a typo, and corrected in my previous post. Why? You do know you're strenuously arguing against a typo for which a correction has already been issued, don't you?

You know this is a philosophy forum don't you? So it's likely that you should expect that we are discussing a philosophical issue. If you want to discuss a mathematical issue, maybe a different forum would be better. — Metaphysician Undercover

Weren't you the one who originally made a mathematical claim, which I am refuting?

You claimed that in ZFC they misuse the law of identity in some way. I challenged you on that and you have not produced evidence.

YOU are the one who made a strictly mathematical claim about ZFC. And who can't defend it with facts.

Beyond your factual incorrectness, I found this a very patronizing and hostile remark. Did I misread it?

You made a specific claim about ZFC, an abstract mathematical system. I challenged you on your mathematical claim. You then say I have no right to talk mathematics? What kind of low-end game is that?

I feel badly misunderstood, but hey, this is the internet... — bongo fury

I misunderstand many things. My apologies if that is the case in this instance. LOL at the Annie Hall reference, one of my faves.

He's been the captain ever since Melville made him so. — creativesoul

We're entirely in agreement. My mistake if I thought otherwise.

Really? With a child, discussing how the set of 2 pens here plus the set of 2 pens there makes a set of 4? — bongo fury

@Metaphysician Undercover mentioned teaching children earlier as well. For the record I'm not speaking of pedagogy, but rather of sophisticated mathematical thinking that has only arisen in the past century and a half. I don't expect to explain the Peano axioms to children; but as adults, we are free to use our most sophisticated mental frameworks.

Wouldn't you want to be ready to climb down from platonist notions or foundations ("2 on the number line", or "the class of all pairs" etc.) and agree that the two separate concrete pairs of objects were being compared and found "equal" in cardinality or size, just as two pens might be found equal in weight, or in length? In other words, equivalent, and in the same equivalence class by this or that mode of comparison (in this case cardinality)? But obviously not identical? — bongo fury

I'm a Platonist when it suits me, and a formalist when that suits me. My derivation of the theorem 2 + 2 = 4 in Peano arithmetic was purely a formal exercise. When doing math it's helpful to think like a Platonist. When doing mathematical philosophy it's often helpful to think like a formalist. I'm a conceptual pluralist in that way.

I'll surely grant your point that two oranges are different than two fish; and that each pair is an instantiation of the abstract concept of two. But I am being careful to not talk about the world at all. Frankly I am not trying to convince anyone that two pens plus two pens is four pens. I take no interest in such mundane applications!

My only hard claim in this thread at this moment is that 2 + 2 = 4 is a theorem of Peano arithmetic. I can base the rest of my argument on that. But in the end its a formalist argument. I have no idea how many pens is two pens plus two pens. Maybe they're four for the price of three. I have no knowledge of such things.

Or would you want to get them with the platonist program straight away, and make sure they understood that 2 on the number line "sends" with itself in a two argument function returning at 4? — bongo fury

We don't burden young minds with higher mathematical abstractions. In fact that was the great failure of the educational fad of "new math," which was coming into vogue around the time I was too old to be scarred by it. After that failed they tried "new new" math, and some other things, and now they've got Common Core about which I hear awful things. I simply am not discussing early math pedagogy. I'm not discussing the subject of what we should teach children about numbers.

Notice they will soon learn to equivocate anyway between identity and equivalence, like any good mathematician not presently embroiled in philosophical or foundational quandary. — bongo fury

No I don't agree at all. Most people will never care one way or another. And you have snuck in a word NOBODY is talking about, equivalence. In math there can be equivalences between very different things. So that's a red herring, a distortion of the argument. No mathematician obfuscates identity versus equivalence. On the contrary, mathematicians are very precise about the distinction.

Not that Metaphysician Undercover will be happy with any cavalier embrace of equivocation. — bongo fury

I am not equivocating anything. In set theory, 2 + 2 and 4 are strings of symbols that represent or point to the exact same abstract mathematical object. That's a fact.

Yes, the irony... that competence in maths should not only involve easy equivocation imputing (with the equals sign) absolute identity here and mere equivalence (identity merely in some respect) there, but then also involve an "identity" (e.g. site menu) sign meaning only a batch-load (for all values of a variable) of cases of "equals", the latter still (in each case) ambiguous between identity and mere equivalence! (The ambiguity removed only by a probably unnecessary commitment to a particular interpretation.) — bongo fury

You're making strawman attacks, assuming facts not in evidence and imputing error to others when in fact your own thinking, or at least your writing, is muddled. I could not track that last paragraph.

Some mathematicians are incredibly careful and thoughtful about these issues. See Barry Mazur's famous essay, When is One Thing Equal to Another Thing?

http://www.math.harvard.edu/~mazur/preprints/when_is_one.pdf

The novel existed in it's entirety prior to the first report of it. Melville reported upon something that existed in it's entirety while writing the novel as well. Prior to the report, Ahab and the Pequod was a collection of Melville's own thoughts, beliefs, and ideas. — creativesoul

Wait, what? Did Ahab and the Pequod exist before Melville existed?

Of course it's an interesting fact that Moby Dick is based on a true story. The whaling ship Essex was attacked and sunk by a whale. But of course all fiction is based on or inspired by some aspect of reality. That doesn't mean the characters of the novel existed before the author conceived of them.

Is there a point in time, in your opinion, in which Ahab did not exist? Or perhaps you mean to regard him as an archetype? The charismatic fanatic luring others to their doom? That's an eternal theme in human affairs.

But I'm not sure how to take your remark literally. The sun has always been a flaming ball of ga; but Ahab has not always been the captain of the Pequod. That's the thesis I am putting forth.

https://en.wikipedia.org/wiki/Essex_(whaleship)

I think we're making some progress, — Metaphysician Undercover

Good news. I'm working on a reply in case it takes a while. I do think you're failing to distinguish between:

* The philosophical question; and

* The mathematical question.

When you send me to SEP and make subtle (and interesting!) points about the nature of identity, that is part of the philosophical problem. About which I have already stipulated that I'm ignorant and open to learning.

But on the mathematical point, you still won't engage and that's still frustrating to me. You reiterate that my first premise is false but I thought I clarified that. Could you please repeat exactly what I said that you think is false? I'd like to respond but I am actually not clear on what you're referring to. I already acknowledged using the word equal when I meant to write identical at some point.

I'll spend some more time reading your interesting post. You make a lot of good points about symbology. I'll give it all some thought.

All things exist in their entirety prior to the first report of them.

I like that much better. Seems odd. I'm willing to defend the assertion.

Any takers? — creativesoul

Fiction.

Facts are true even before we know them. The sun was a flaming ball of gas long before we discovered that fact.

But fiction comes into existence at the moment a human conceives it. Ahab became captain of the Pequod when Melville decreed it. Before Melville wrote the novel, there was no Pequod and there was no Ahab.

Perhaps we can refine your idea to: Truths about actual things were always true long before we discovered them. But truths about fictional things become true when someone says so.

If it's true that all we need to know is axioms, then the question becomes, which axioms? We know that in math, at least, axioms are insufficient to characterize mathematical truth. Modern set theory is about the search for intuitively appealing axioms that resolve the questions we care about (Continuum hypothesis for example). In the end the choice of axioms is social and political, not just a matter of rationality. Do you want more restrictive axioms or more permissive ones? There's no right or wrong answer.

No, that's the point you cannot validly substitute "same" for "equal". It will produce equivocation. You don't seem to understand this — Metaphysician Undercover

It was a typo, a Freudian one if you like since I agree that this particular error makes you totally right and me totally wrong. I get that.

From my end it was a totally honest error. As a math person I'm so used to talking about equality that I never think about logical identity. I'm learning a lot from this thread. I typed "equal" because I've done it a million times in math; and the only time I talk about an identity is something like $e^{i \pi} + 1 = 0$. a specific numeric and/or algebraic identity meaning it's true by virtue of syntax. It's a logical truth. But as I think of it, that's logical identity too.

More than ever I see that mathematical equality is the same thing as logical identity. The same morally and the same technically in any mathematical framework you like.

By the way if called on to do so, I could drill that symbology down to an identity of sets. The thing on the left and the thing on the right are the same thing. It was true even before Euler discovered that fact in 1740. At that moment it became new knowledge; but it did not suddently become a new truth about the world. It's a mathematical truth. It was always true.

And I do acknowledge that my fingers typed equality when in the context of the discussion I should have used the word identity; and that this purely random or perhaps Freudianly determined; either way, it was one hell of a bad typo. I see that it generated a lot of confusion. Please just substitute "identity" or "logical identity" in my argument. My apologies.

To sum up my view I would say that

* On the philosophical aspect, I suspect I have much to learn. I'd be surprised to find out that logical identity is somehow different than mathematical equality; but I would not be surprised to be surprised.

* On the math aspect, you're just wrong. But neither of us has said anything new for quite some time, and I have nothing to add. Only that I'm disappointed at a personal level that I took the trouble to work out an immaculate technical proof; and you are just totally disinterested in actually following and engaging with the argument. It's your privilege not to engage, but that is definitely a disappointment at my end. I'd say to myself "Pearls before swine" but I'd never say such an uncharitable thing in public. But the phrase did pop into my head, and that does about sum up how I feel about the mathematical aspect.

I actually just popped in tonight to mention that someone, possibly a member of this forum, posted the following to philosophy.SE today:

https://philosophy.stackexchange.com/questions/67227/is-there-a-difference-between-equality-and-identity

Some interesting thoughts there.

The law of identity states that a thing is the same as itself, not that a natural number is equal to itself. This is the problem, you keep asserting that the law of identity says something about equality, when it does not. It says something about identity. — Metaphysician Undercover

If I use the word same instead of equal does that satisfy you?

The fact that you say you read my post and this is your complaint means we're done. You have no substantive reply? I showed a proof from first principles that 2 + 2 and 4 are identical. You ignore it?

You're not debating in good faith.

You could have said, "Oh I categorically reject the work of Giuseppe Peano and everything he stands for." Or, "I didn't understand the chain of equalities." Or SOMETHING. Anything. But you simply will not engage substantively.

You're asserting a falsehood. There is no case in math of an equality meaning anything other than identity; whether of abstract objects (logical identity) or sets (set identity or equality). Set identity is the same as set equality.

If you don't agree that's your right, even if you haven't and can't show a single example to support your claim. But in all this time you have not presented an argument. And you have never engaged substantively. And from me to you, you're factually wrong. All the best.

ps -- I apologize if this comes off rude or confrontational. I'm genuinely frustrated that you won't engage on the substantive technical points I made. I presented a proof from the Peano axioms that 2 + 2 and 4 are the same thing. Same as in same. Same as in equal. Equal is the same as same. You simply chose not to engage. I find that too frustrating to continue the convo. It's not you, it's me.

We went through this. You provided no such proof, it was just an assertion. — Metaphysician Undercover

Is it possible you missed this?

https://thephilosophyforum.com/discussion/comment/328116

Also you claimed that the law of identity does not apply to numbers; that for example 3 is not the same as 3. Please clarify or retract. Thank you.

ps -- Originally you claimed that in ZFC there are equalities that are not identities. I know of no such instance nor have you presented a single such example. Clarify or retract please. The only examples I know are natural injections, such as identifying the integers with the copy of the integers contained in the real numbers; and in casual contexts where we call two isomorphic groups "the same" when we know that we mean isomorphic. Other than those two contexts, I know of no instance in which mathematical equality is anything other than set identity and logical identity. I have challenged you on this point and found your responses lacking in specificity.

We went through this. You provided no such proof, it was just an assertion. — Metaphysician Undercover

I provided a direct proof from first principles. You might have questions or specific objections. But to claim I did not provide a proof means you happened to not see my lengthy post, or ... well I just don't know. It would be helpful if you'd specifically engage with the proof I gave; not claim I didn't give one.

Like, I’m thinking for example of how the euphoria regarding the potential for future human space exploration following the Apollo 11 mission – almost Dan Dare whizzing between the planets and all that - has since been superseded by a more sober recognition of the extreme limits which are in fact imposed on the potential for such activity by theoretical physics - rather than merely engineering knowledge — Robert Lockhart

It was neither physics or engineering. It was politics. Even at the time people asked why should we send men into space when we had so many social problems here on earth. That point of view won out and we lost fifty years of progress. In the fullness of time it won't matter; but it's sad that the promise of manned space exploration has stagnated due to politics within the lifetimes of those of us who saw the moon landing on tv.

You remember in the film The Right Stuff, a NASA bureaucrat asked the astronauts if they knew what makes the rockets go up. "Funding. No bucks, no Buck Rodgers."

Or is it just carry on as we are and if it happens it happens. — Malcolm Parry

Now I did not say that and I hope you can see that I did not say that. I will forgive your rhetorical excess but I will most definitely respond.

I am old enough to attest that the air is cleaner now than it was in the 1970's. I like whales and stuff. Everything in moderation. The opposite of rabid, hysterical environmentalism is sensibly balancing respect for the environment, one the one hand, and the needs of 300 million people to have a functioning economy, on the other.

The negation of fanaticism is sensible progress; not fanaticism in the opposite direction.

Sad that our modern politics involves two groups of fanatics yelling at each other while the rest of us look on in horror and hope for the best.

A curious case from the Netherland’s raises questions about euthanasia. — NOS4A2

Interesting case. It reminds me of the days of Dr. Kevorkian. He became the face of assisted suicide. But he operated in a very gray area. Some of his patience changed their mind but he killed them anyway.

Very often the actual real-life examples of general principles are far messier than the principles. That's why abstract political ideas must be tempered by reality.

Personally I hope I get hit from behind by a meteor that I never knew was coming. Fast and painless. All you can ask for.

Is democracy capable of changing the course of inevitable disaster? — Malcolm Parry

The world has been in a state of inevitable environmental disaster since Malthus. The predictions never come true. Some think the notion that there even is an "inevitable disaster" is itself a form of mass hysteria. As recently as the 1980s Paul Erlich made a bunch of doomsday predictions, none of which came true. He bet that a basket of commodities would be far higher in price, and they turned out to be far lower. We crawled out of caves and built all this. I would not bet against the human race.

Your question reveals the unspoken, evil truth about environmentalism. You ask if democracy is capable. Because if it isn't ... you advocate authoritarianism.

That's what underlies environmentalism. The desire for YOU to control the world; because only YOU know what's best. Just the other day Bernie Sanders said he wanted a massive program of birth control implemented in third world countries. Bernie doesn't want the poor people of the world to reproduce. That's socialism. Abstract principles over actual human beings; by force of law and force of arms.

I'll take democracy, the voice of the people; and free markets, the voice of people spending their own money on things that give them value.

In summary, Newton's laws boil down to f=ma.

f = ma is essentially a definition. A very clarifying definition to be sure, but it's not a fact or a theorem. It's a definition. That is my understanding.

The law of identity doesn't say that a thing is equal to itself, it says that a thing is the same as itself. In formal logic, "the same as" is represented by "=". So when the law of identity is expressed in formal logic as "a=a" or some such thing, the "=" represents "the same as". Zuhair is arguing that all mathematical axioms can be interpreted as "=" representing "the same as", but this is equivocation plain and simple. — Metaphysician Undercover

I take no responsibility for and neither endorse nor necessarily agree with anything written by anyone on this site but myself; nor do I necessarily disagree. I have no idea why you are quoting some other poster's thoughts to me on this subject. I've written plenty to you already that you haven't engaged with, including a proof directly from first principles that 2 + 2 and 4 are identical.

You claimed the other day that the law of identity does not apply to the number 3, or numbers in general. At that point I assumed you've simply given up rational debate and/or recognized the impossibility of your own position. Talk me down please.

Newton's laws have not had examples in real life that would nullify his laws, — god must be atheist

What do you make of the 1919 eclipse that confirmed Einstein and disproved Newton?

https://en.wikipedia.org/wiki/Eddington_experiment

1.1 We have the law of identity that says that for each natural number, it is equal to itself.
— fishfry

This is our point of disagreement. The law of identity does not say this, you are claiming this. — Metaphysician Undercover

Perhaps you can clarify this point for me then. The law of identity is that a thing is equal to itself. Why wouldn't this apply to numbers? A rock is identical to itself, the number 3 is identical to itself. You are claiming the former and denying the latter? Perhaps this is a clue to why we disagree. How can a number not be identical with itself by virtue of the law of identity?

Every child who asks "Why?" sometimes gets caught in the whys and keeps asking long after the efficacy of the question has been exhausted. — tim wood

Newton took a lot of flack at the time. His law of gravity told us what gravity does; but not what it is. He famously said that "I frame no hypotheses." Newton well understood that science is descriptive and not explanatory; a point that modern scientists and philosophers of science would do well to understand.

But at the time, Newton's contemporaries did NOT understand. Descartes had a theory of vortices that said what gravity WAS, not just how it acted. That was, by the scientific ethos of the time, better science than what Newton did. It took people a while to come around to Newton's point of view. We see this echoed today, when people want to "interpret" quantum mechanics rather than be satisfied that it describes what the universe does; and not necessarily why.

https://en.wikipedia.org/wiki/Hypotheses_non_fingo

I just wanted to add, that we can actually have a very simple system in which 2 + 2 = 4, that of first order logic and add to it primitives of identity (equality) symbolized as "=" which is a binary relation symbol, and of "+" denoting addition which is a two place function symbol, and of "1" denoting what we customarily know as one, which is a constant symbol. I'll try to coin a system in which 1 is the first number, i.e. doesn't have zero in it. — Zuhair

Yes thanks for making that point. In fact Russell and Whitehead famously took 400 or whatever pages to prove that 1 + 1 = 2 directly from logic; and presumably they could do 2 + 2 = 4. The only reason I didn't mention it is that I'm not familiar with the development of numbers as in R&W. My knowledge is mostly in the math domain which means I need to start with the Peano axioms. But Russell and Whitehead is probably the right answer to how you show that 2 + 2 = 4 is a logical identity.

If you've read my other posts then you know that the Theory of Relativity was derived from measurements from instruments that followed Newtonian laws. What do you have to say about that? — TheMadFool

Me? Wasn't sure about the quoting. Newton's instruments only followed Newton's laws to a certain degree of approximation. They follow Feynman's laws -- quantum laws -- to a far greater degree of approximation. Newton didn't happen to know that, but it was true nevertheless. How is this point not perfectly obvious to everyone?

There's an even bigger point here which the OP is hinting at - that the theory of relativity which has supplanted Newton's was proven by instruments that had to follow Newton's laws. — TheMadFool

Of course this is false. Newton looked through a telescope that he himself had made. He was a master lens grinder. That's peripheral to the discussion but just an interesting factoid. Now the point is that Feynman taught us exactly how light passes through lenses. The laws of optics were known to Newton or rather discovered by him. But optics are a quantum phenomenon and Feynman (and others) won the Nobel prize for elucidating this fact.

Everything in the world is a quantum phenomenon. A rock doesn't fly apart because of quantum theory. I think someone made this point earlier but it bears repeating. Newton didn't know his telescope worked on quantum principles, but eventually we discovered that it does.

It does not state that the sets are the same, it states that if the members are the same, then the sets are equal. Therefore the sets remain distinct, as two equal sets, not one and the same set. — Metaphysician Undercover

I can't respond to this. You're factually wrong. There's only one set {0,1,2, pi}. There isn't "another" set that happens to have the same elements and is therefore equal. Any set with the same elements is identical to this set.

If you don't get that or you don't want to get that or you think I'm completely wrong, that's your privilege. You're making mathematical claims that are false. It's ok. A lot of people do that and I can't fix them all. I've said my piece here. All the best.

Your post was great but I don't think it would've satisfied the OP who said: — TheMadFool

I have already confessed to not reading the post and only lazily making one of my standard hobby horse points. Did you want me to confess again? Is there a particular punishment you have in mind? I watched the entire Democratic debate tonight, that was punishment enough!

Let's not get too technical. The problem for the OP is how an instrument that is Newtonian can ever prove that some other event is NOT Newtonian in nature. If I only have red paint, whatever I paint will surely be red.

A better analogy is the biased judge adjudicating a case. The trial wouldn't be fair. — TheMadFool

I responded primarily to the title of the OP without reading much of the post, and without reading the other responses in the thread. If I misconstrued the question, so be it. It's kind of a reflex on my part to point out that the laws of physics are historically contingent approximations; and that whether there are laws of the universe at all, let alone ones accessible to humans, is an open question. But if that wasn't the question, then ... well, what was the question. I didn't understand the bit about how Newtonian instruments could prove an event is not Newtonian. In fact perfectly conventional Newtonian telescopes were used to observe the solar eclipse in 1919 that confirmed Einstein's theory of relativity.

https://en.wikipedia.org/wiki/Solar_eclipse_of_May_29,_1919

Note that by Newtonian telescope, I meant not only the OP's sense of a telescope operating by Newtonian physics; but also that the Newton invented the reflecting telescope! So there's a double meaning there.

I would like to know how can you prove these laws, but not using devices that use the the same laws. — Fernando Rios

You could never prove them. For one thing they're not "true," if by true you mean that the universe actually works that way. We know that for objects moving at very high speeds and/or objects with extremely large mass, Newton's laws are superseded by Einstein's; and that even for everyday object like bowling balls, Newton's laws are only an approximation.

Secondly, NO law of physics can ever be proved; because every law of physics is a historically contingent approximation, good to a few decimal places, to the results of the experiments we're capable of doing at any given level of technology. Our very best physical theory, quantum electrodynamics, gives the magnetic moment of the electron to "a few parts in $10^{13}$. That's great by physics standards but there are a lot more decimal places out there and a lot of physics we still don't know.

https://physics.stackexchange.com/questions/497087/what-is-the-most-precise-physical-measurement-ever-performed

No physical theory is true. Or rather, "true" is defined in physics as our best physical theory! So Newton's laws were true in 1687 with the publication of the Principia, and became false in 1915 with Einstein's publication of general relativity.

Pick your definition of true. The ultimate truth of the universe -- if there even is any such thing? Or just the latest widely agreed on theory from the physicists?

You seem to have left something out. You've taken the '+' for granted. You've shown me what '2' represents, and you've shown me what '4' represents. Then you claim that '2+2' magically represents the same thing as '4'. — Metaphysician Undercover

Let me remedy that omission.

Before I start I hope we're agreed that there are two levels to this discussion:

1) The philosophical point that 2 + 2 is not identical to 4 because the former conveys the information that a thing, namely 2, that is manifestly different than 4, is being combined with itself to produce something entirely different, namely 4. This I take to be your viewpoint.

I might argue that point with you, but I would not be on firm footing. There are subtle philosophical issues that I'm ignorant of; but that at the very least I can see I'm ignorant of them. So I'm not entirely conceding your point; but I must depart the field. I haven't the capacity to defend my side.

2) But on the mathematical side, I claim that 2 + 2 and 4 designate identical numbers and identical sets and of that I have not got the slightest doubt. I regard this as simply a technical matter that I'm educated about and that you are about to be educated about. You may disagree but at least you know where I'm coming from.

So: I say that when in math we write $x = y$ we are asserting that x and y are identically the same. I shall now state my case. (* See note at the end).

1.1 We have the law of identity that says that for each natural number, it is equal to itself.

1) We assume we have the natural numbers as given to us by the Peano axioms. These are denoted by the symbols 0, 1, 2, 3, ... I don't know if you regard this as an objectionable premise. We have to start somewhere.

PA says:

(1) There is an undefined symbol $0$, which we call a "number."

(2) There is a function $S$, called the successor function, that inputs a number and outputs a number.

(3) If $n$, $Sn$ is a number.

There are some other axioms to make sure numbers are suitably well behaved.

https://en.wikipedia.org/wiki/Peano_axioms

With these three axioms we have an endless sequence of numbers: $0, S0, SS0, SSS0, SSSS0, \dots$. As a matter of convention we introduce the following names: $S0 = 1, S2 = 2, S2 = 3, S3 = 4, \dots$. I hope these are not unfamiliar.

Now we need to define the arithmetic operations. We define $+$ inductively as follows:

(*) $n + 0 = 0$

(*) $n + Sm = S(n + m)$

With these definitions in hand we may now evaluate $2 + 2$.

$2 + 2 = 2 + SS0 = S(2 + S0) = SS(2 + 0) = SS2 = SSSS0 = 4$

This puts the matter to rest. The expressions $2 + 2$ and $4$ refer to the same number. It's practically a definition, following so easily from the Peano axioms and the definitions of the symbols $2$, $+$, and $4$.

If you think it means something else, you are mistaken. You may have some intuitions that $+$ means "combining two things to make some other thing," but nothing in the math supports that point of view. I can't help what they told you in first grade.

These are strings of symbols manipulated by formal rules. A computer could implement the rules. The symbols are devoid of meaning except for what we bring to them with our intuitions. And our intuitions are part of our philosophy. They are not in the math itself.

As I say that's the mathematical story. I will concede that you may have a point if you overload the symbols with your intuitions about what they mean. That's the philosophical question. If you think $2 + 2 = 4$ "means" something that the math doesn't say, then you are making a metaphysical point, not a mathematical one. If you got your intuitions in first grade, I'd ask you to update them in the light of how the math actually works.

On the math there is no question. $2 + 2 = 4$ is an identity derived directly from the law of identity, the Peano axioms, and the definitions of the numbers and of $+$. As I say it's practically a definition.

For completeness this is only half the story. The Peano axioms aren't strong enough to develop a theory of the real numbers, for example. For that we need set theory, including the axiom of infinity. With the appropriate assignment of sets to numbers, and a definition of the successor function, the universe of sets contains a model of the Peano axioms; and the proof I gave can be lifted directly to a proof that the sets designated by $2 + 2$ and $4$ are the same set.

(*) Note -- I did not prove the starred claim that every mathematical equality is a statement of set identity. That would be part of the extension of the discussion to set theory, and I did not want to add those details to this post.

I've asked fishfry for this principle of identity, to no avail. — Metaphysician Undercover

I was going to reply to you later but just ran across this, which could not be more false.

I have repeatedly explained to you that the axiom of extensionality is directly derived from the logical law of identity. I thought that had already been mentioned by someone even before I joined this thread. So if you don't understand what it means, or want to see more detail, just ask.

If you are claiming that equality isn't identity in natural language, you might have a philosophical point.

But if you are making a mathematical claim, you're just factually wrong. Mathematical equality is identity of sets. A mathematical equality states that the sets on either side of the equation are the same set.

it seems to me that the quality of discussion on these prolific religious threads falls far short of 'philosophical debate' or even 'coherence' for participants — fresco

You oughta see the political forums.

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?