The first time I argued with a philosophy professor some 40 years ago, after he taught us to trace all premises back to a point of agreement before moving forward, he posited that very equation of 2+2=4. I asked "Two what plus two what; and what do you mean by 'plus' and what do you mean by 'equal.' After all, two people plus two people could equal five people if one couple had a single child. Likewise two drops of water plus two drops of water could equal one drop of water." He agreed and took a step back to set definitions. That was my first exposure to the "gentlemen's agreement" which subsequently fell apart on the burden of proof. LOL! We had fun but I think there was another kid in the room, looking for a grade, who hated digressions. Carry on. — James Riley

@Metaphysician Undercover is on record stating that he does not believe that 2 + 2 and 4 denote the same mathematical object. He's wrong but confirmed in his belief. I did at one point present to him a clean proof from the Peano axioms in which I defined "2", "4", "+", and "=", and proved that 2 + 2 = 4. Of course the truth of any symbolic expression depends on the interpretation given to the symbols; but it is NOT in dispute that 2 + 2 and 4, with their standard interpretations, denote the same mathematical object.

It was wrong of me to bait @Meta when he asked a simple factual question regarding this recent possible discovery in physics. But if baiting @Meta is wrong, I don't wanna be right.

Of course the truth of any symbolic expression depends on the interpretation given to the symbols; — fishfry

And hence the gentlemen's agreement.

I'm over my head here but some wag once said, if you are the smartest person in the room, find another room. So I left a political conservative echo chamber and came to tread water in yuse guys, including @Meta and you and everyone else. I'll watch you bait, and learn.

I did at one point present to him a clean proof from the Peano axioms in which I defined "2", "4", "+", and "=", and proved that 2 + 2 = 4. — fishfry

I never denied that 2+2=4. That would be stupid. What I deny is that "=" indicates is the same as. I think that to believe such a thing would be stupid as well. So your proof that 2+2=4 really does nothing for your claim that "2+2" denotes the same object as "4".

I never denied that 2+2=4. That would be stupid. What I deny is that "=" indicates is the same as. I think that to believe such a thing would be stupid as well. So your proof that 2+2=4 really does nothing for your claim that "2+2" denotes the same object as "4" — Metaphysician Undercover

I don't believe I have ever said that you deny 2 + 2 = 4. I am always careful to note that you deny that 2 + 2 and 4 denote the same mathematical object. Can you please point me to an instance where I failed to make that distinction?

Metaphysician Undercover is on record stating that he does not believe that 2 + 2 and 4 denote the same mathematical object. — fishfry

This is what I said and this is what I always say. Will you agree that I characterized your position fairly and that I did NOT say what you claim I said?

I don't believe I have ever said that you deny 2 + 2 = 4. I am always careful to note that you deny that 2 + 2 and 4 denote the same mathematical object. Can you please point me to an instance where I failed to make that distinction? — fishfry

You just said:
"I did at one point present to him a clean proof from the Peano axioms in which I defined "2", "4", "+", and "=", and proved that 2 + 2 = 4."
If you knew that I didn't dispute 2+2=4, then your so-called proof is an intentional strawman.

If your definition of "=" is "denotes the same object as", then you're begging the question with a false premise.

"I did at one point present to him a clean proof from the Peano axioms in which I defined "2", "4", "+", and "=", and proved that 2 + 2 = 4."
If you knew that I didn't dispute 2+2=4, then your so-called proof is an intentional strawman. — Metaphysician Undercover

The proof shows that the two expressions denote the same mathematical object. But we're making progress. For three years (has it been that long?) you totally ignored the proof. Now at least you're acknowledging it.

The proof shows that the two expressions denote the same mathematical object. But we're making progress. For three years (has it been that long?) you totally ignored the proof. Now at least you're acknowledging it. — fishfry

I didn't ignore the proof, I showed you how it was not a proof of what you claimed it was. But we could go through it again if you want.

I didn't ignore the proof, I showed you how it was not a proof of what you claimed it was. But we could go through it again if you want. — Metaphysician Undercover

It took you three years to even acknowledge its existence. I don't wish to go through it again.

I've just been ignoring those claims because you ignored my reply to your proof.. And you continue to ignore this. If I remember correctly, your proposed proof violated the law of identity, and you refused to acknowledge this. And that violation of the law of identity was what I was already discussing in the first place, so your proposed proof was completely irrelevant because it did nothing to mitigate this violation..

I've just been ignoring those claims because you ignored my reply to your proof.. And you continue to ignore this. If I remember correctly, your proposed proof violated the law of identity, and you refused to acknowledge this. And that violation of the law of identity was what I was already discussing in the first place, so your proposed proof was completely irrelevant because it did nothing to mitigate this violation.. — Metaphysician Undercover

You never even bothered to acknowledge my proof. I asked you repeatedly to criticize it or disagree with it and you just ignored those posts too. And now you're making claims contrary to facts. Your recent objections to the proof are three years after the fact. This is a silly conversation. I'm not playing anymore.

You never even bothered to acknowledge my proof. I asked you repeatedly to criticize it or disagree with it and you just ignored those posts too. And now you're making claims contrary to facts. Your recent objections to the proof are three years after the fact. This is a silly conversation. I'm not playing anymore. — fishfry

I really don't remember the specifics of your so-called proof. I remember that you produced something you called a proof, and it was very easy for me to show that it did not prove what you intended it to prove, through reference to the law of identity. So I demonstrated this and moved along. You did not seem to have a firm grasp of the law of identity at the time, so you did not seem to understand how your supposed proof failed. Then you kept referring back to this supposed "proof", as if it really proved what it didn't.

If you really think that you have a proof that "2+2" denotes the same mathematical object as "4", when "same" is held to the rigorous definition of the law of identity, then produce it again, and I'll show you how it fails, again. Maybe this time you'll pay respect to the law of identity.

I really don't remember the specifics of your so-called proof. I remember that you produced something you called a proof, and it was very easy for me to show that it did not prove what you intended it to prove, through reference to the law of identity. So I demonstrated this and moved along. You did not seem to have a firm grasp of the law of identity at the time, so you did not seem to understand how your supposed proof failed. Then you kept referring back to this supposed "proof", as if it really proved what it didn't. — Metaphysician Undercover

You pointedly ignored it for three years. It's a basic proof that 2 + 2 = 4 from the Peano axioms. I had a perfectly good grasp of the law of identity at the time and still do. You didn't move along. You kept engaging with my posts but refused to acknowledge the proof I showed you.

If you really think that you have a proof that "2+2" denotes the same mathematical object as "4", when "same" is held to the rigorous definition of the law of identity, then produce it again, — Metaphysician Undercover

It would be pointless. It's a basic proof from PA that 2 + 2 = 4. At the time you indicated a terror of symbolic reasoning. Perhaps you've gotten over it.

and I'll show you how it fails, again. Maybe this time you'll pay respect to the law of identity. — Metaphysician Undercover

Our previous conversation is still up. If you cared, you could go back and check it out. And if you did, and you happened to find a post of yours that directly addressed the proof I gave, I would apologize for falsely claiming you didn't. That should be sufficient motivation. I see no need to type it in again. You are on record that 2 + 2 and 4 do not represent the same mathematical object. You are absolutely wrong about that. There's no need to rehash the conversation and no inclination on my part to do so.

Bill Clinton clarified the Law of Identity in his unique way: "It depends on what the meaning of the word ‘is’ is. If the—if he—if ‘is’ means is and never has been, that is not—that is one thing. If it means there is none, that was a completely true statement. … Now, if someone had asked me on that day, are you having any kind of sexual relations with Ms. Lewinsky, that is, asked me a question in the present tense, I would have said no. And it would have been completely true."

That should put an end to the discussion.

It's a basic proof that 2 + 2 = 4 from the Peano axioms. — fishfry

You know I do not dispute the fact that 2+2=4. That's why I'm fulling justified in ignoring your strawman proof.

You know I do not dispute the fact that 2+2=4. That's why I'm fulling justified in ignoring your strawman proof. — Metaphysician Undercover

For the record: You do deny that 2 + 2 and 4 represent the same mathematical object. Is that correct?

Why are you going on about this? We've had this pointless conversation, it was over long ago. After I gave up talking to you about it, many others have taken their shot and given up. For whatever reason, my recent offhand 2 + 2 = 4 remark seems to have triggered you. I don't know why. Do you?

Why are you going on about this? — fishfry

Me? I was happy to participate in this thread, and proceed in many other discussions with you, without ever mentioning this small disagreement we have. We probably agree on many other things. But it seems you have some need to keep bringing it up.

Metaphysician Undercover is on record stating that he does not believe that 2 + 2 and 4 denote the same mathematical object. He's wrong but confirmed in his belief. I did at one point present to him a clean proof from the Peano axioms in which I defined "2", "4", "+", and "=", and proved that 2 + 2 = 4. Of course the truth of any symbolic expression depends on the interpretation given to the symbols; but it is NOT in dispute that 2 + 2 and 4, with their standard interpretations, denote the same mathematical object. — fishfry

See, it's you who brought up the past. Obviously, for you it's not yet over. But that doesn't surprise me.

See, it's you who brought up the past. Obviously, for you it's not yet over. But that doesn't surprise me. — Metaphysician Undercover

You made a claim that you did acknowledge my proof three years ago. If that were true you could produce your post doing so. But you can't because you didn't.

I agree it was a cheap shot to bait you with 2 + 2 = 4. I'd say it was beneath me but of course it's not :-) I was surprised at your reaction since we already know where we both stand on the matter. It was not intended to provoke such a strong reaction and if I'd known it would I'd have skipped it.

As you yourself have repeatedly stated, in this thread, your proof is that 2+2=4. And, you've also stated that you recognize that I do not dispute the fact that 2+2=4.

So, I am acknowledging that your proof is irrelevant, as I've done before.

Why do you keep bringing up, and referring to, an irrelevant proof, as if it is something which is relevant to our disagreement?

Why do you keep bringing up, and referring to, an irrelevant proof, as if it is something which is relevant to our disagreement? — Metaphysician Undercover

Enough.

I'm over my head here but some wag once said, if you are the smartest person in the room, find another room. — James Riley

:point: :up: :clap:

As you yourself have repeatedly stated, in this thread, your proof is that 2+2=4. And, you've also stated that you recognize that I do not dispute the fact that 2+2=4. — Metaphysician Undercover

I was listening to a podcast on Frege, and the speaker mentioned his insight into sense and reference.

She (Patricia Blanchette) said that Frege noted that 2 + 2 and 4 refer to the same thing; but they have a different sense. 2 + 2 = 4 tells us something about 4 that merely saying 4 = 4 does not.

I believe this is the point you have been making, and in fact I might almost agree with you about this

However as I understand it, you are on record as saying that 2 +2 and 4 do not REFER to the same thing. And in this, you are wrong. You're wrong mathematically, and you're wrong according to Frege.

Is it possible for you to clarify your thinking here? Are you saying that 2 + 2 and 4 don't refer to the same thing, in which case you're wrong; or rather that 2 + 2 and 4 tell you different things about 4, in which case Frege would say you're right, and I myself am still on the fence.

Frege published this work in 1882 and Peano's axioms date from 1889. In terms of Peano arithmetic, 4 is just another name for 2 + 2 once we break everything down to the successor function. Which is why I don't even think they have a different sense. But if we drop the Peano formalism, I can indeed see that 2 + 2 = 4 gives more information about 4 than 4 = 4 does. So I can go both ways on this.

But regardless, 2 + 2 and 4 do REFER to the same object, namely the number that we call 4. So if you are saying they refer differently, you're wrong about that. Can you please clarify your intent?

However as I understand it, you are on record as saying that 2 +2 and 4 do not REFER to the same thing. And in this, you are wrong. You're wrong mathematically, and you're wrong according to Frege. — fishfry

Look at what you say above, "2+2" says something about 4. What is 4? It is a quantity, or a value, it is not a thing. Remember, my argument is only applied to a strict definition of "thing", in which a thing has an identity according to the law of identity.

If for example, you assume that green is a thing, you might say yellow and blue make green, so that you might argue that "yellow and blue" refer to the same thing as "green". But this is what I deny. "Yellow and blue", might as you say above, say something about green, but neither "yellow and blue" nor "green" refer to a thing. Green is a property, something we attribute to a thing. Likewise 4 is a property, something we attribute to a group of things.

The intent of the law of identity is to distinguish true things, which have a real identity, from concepts, platonic ideas, which have no true identity and are therefore not things. So it makes no sense to say that "4" refers to a thing, or "an object" in any sense, as 4 has no true identity. It has an infinite number of representations, 2+2, 3+1, etc., and none can be said to be the true representation. If we affirm that 4 is the true representation, then the others must refer to other true objects, represented by 1 and by 2 and by 3.

Is it possible for you to clarify your thinking here? Are you saying that 2 + 2 and 4 don't refer to the same thing, in which case you're wrong; or rather that 2 + 2 and 4 tell you different things about 4, in which case Frege would say you're right, and I myself am still on the fence. — fishfry

Yes, this is exactly what I am saying, 2+2 does not refer to the same thing as 4 because neither refers to a thing. They refer to properties, which are not things, by the law of identity. So, take my example, "blue and yellow" does not refer to the same thing as "green", because neither refers to a thing. Notice that there can be many green things in the world, just like there are many groups of 4. But through the application of the law of identity we see that "green" itself does not refer to a thing, otherwise all the distinct instances of green would be the very same thing in violation of that law. Likewise "4" does not refer to a thing.

But regardless, 2 + 2 and 4 do REFER to the same object, namely the number that we call 4. So if you are saying they refer differently, you're wrong about that. Can you please clarify your intent? — fishfry

To make up an imaginary object, a platonic idea, called a number, and say that this is the object referred to by "4" does not resolve the problem, as I've explained to you already. Then, "2" must also refer to such an imagunary object. So, when you write "2+2", you denote the object, the number two twice. By what principle do you say that this object, the number two, when denoted twice actually refers to the number four? That's completely nonsensical, to say that two instances of occurrence of this object which we call the number two, magically becomes the object called the number four. You must respect the object actually referred to by "2", just like if you say that the colours are this type of platonic object you must respect the colours actually referred to by "blue" and "yellow", and not falsely conclude that "green" is being referred to when someone says "blue and yellow".

Look at what you say above, "2+2" says something about 4. What is 4? It is a quantity, or a value, it is not a thing. Remember, my argument is only applied to a strict definition of "thing", in which a thing has an identity according to the law of identity. — Metaphysician Undercover

For the first time I understand what you're saying. If you've been explaining this all along and I didn't understand, that's on me. But you haven't been explaining your position this clearly, well now you have. What you say is interesting. Still extremely wrong IMO but at least you have staked out a position that is coherent, and I would say very interesting.

Let me say this back to you. In the past I thought you were saying that 2 + 2 and 4 refer to different things. But what you are saying in fact is that 2 + 2 and 4 don't refer to anything at all. Because they are properties, or attributes. 4 is a property of a string quartet, for example. There is no number 4 by itself as an object. So it's wrong to say that 4 refers to anything at all.

Do I have that right? I disagree but I will put that aside so we can catch up to each other.

Your position: 4 and 2 + 2 are qualities or attributes or properties of things; but they do not refer to a specific thing. 4 is a property of a string quartet. But 4 by itself is a "potential property," if I may call it that, but it's not any particular thing and it does not refer to any particular thing except something that enjoys thinghood, like a string quartet. Have I got this about right?

As I say I will defer my objections to the end and for now content myself with merely understanding you after all this time.

If for example, you assume that green is a thing, you might say yellow and blue make green, so that you might argue that "yellow and blue" refer to the same thing as "green". But this is what I deny. "Yellow and blue", might as you say above, say something about green, but neither "yellow and blue" nor "green" refer to a thing. Green is a property, something we attribute to a thing. Likewise 4 is a property, something we attribute to a group of things. — Metaphysician Undercover

I don't like the color mixing example because color theory, additive and subtractive, is relevant to painters and physicists but not to philosophers or mathematicians. But I will take your point that green does not refer to anything except the "things" it does refer to, like houses and traffic lights. But there is no color green. There are green houses and green traffic lights but no green by itself. Forget the color mixing, that's not a good analogy IMO.

The intent of the law of identity is to distinguish true things, which have a real identity, from concepts, platonic ideas, which have no true identity and are therefore not things. — Metaphysician Undercover

Ok! I understand you. The law of identity x = x only applies to "true things," things that are deserving of thinghood; and not properties, which can apply to things, but are not themselves things. A green house is green, but green by itself is not a thing. Therefore green = green is meaningless. Is that your point? I wish you had said this earlier, and I suppose you'll say you did. But I understand you now. Green is not a thing and 4 is not a thing. 4 does not refer to anything at all. It's a property. 4 = 4 is therefore meaningless.

Do you assert that 4 = 4 is meaningless? Can you please explicitly confirm this and don't skip over it?

You assert that 4 = 4 is meaningless because 4 does not refer to a "true thing."

May I ask, who decides what is a true thing. But nevermind, I said I'd defer my objections to later.

So it makes no sense to say that "4" refers to a thing, or "an object" in any sense, as 4 has no true identity. It has an infinite number of representations, 2+2, 3+1, etc., and none can be said to be the true representation. If we affirm that 4 is the true representation, then the others must refer to other true objects, represented by 1 and by 2 and by 3. — Metaphysician Undercover

I would never say that 4 is the "true representation." All representations are equally NOT the thing they represent. 4, 2 + 2, 3 + 1, 3.999..., and 47 - 43 are all representations of the number (represented by) 4. I'm sorry I can't actually name the thing itself! Tricky, that. But YOU say that none of those representations represent ANYTHING. I bizarre belief but like I say, I'm happy I finally understand what your belief is.

Yes, this is exactly what I am saying, 2+2 does not refer to the same thing as 4 because neither refers to a thing. — Metaphysician Undercover

I really don't think you've ever been as explicit. I understand your point of view perfectly. 2 + 2 and 4 are properties that can apply to string quartets and, um, gallons of milk that contain 4 quarts of milk. But by themselves, they don't refer to anything ... and that therefor you claim that:

2 + 2 = 4 is MEANINGLESS.

Is that what you say? Please give me a clear yes or no on this. You assert that 2 + 2 = 4 is meaningless because neither expression refers to anything.

Are you SURE this is a position you want to defend?

They refer to properties, which are not things, by the law of identity. — Metaphysician Undercover

Can you point me to a discussion of the law of identity that explicitly confers thinghood on some symbols and not others? Do you call them symbols? There are "true things" and "false things," by what word do you call things in general? Entities perhaps? An entity can be a true thing or a property? What is your terminology?

So, take my example, "blue and yellow" does not refer to the same thing as "green", because neither refers to a thing. — Metaphysician Undercover

Color theory isn't a good example but I don't know, maybe it is. It's not resonating with me so I'm skipping over it.

Notice that there can be many green things in the world, just like there are many groups of 4. But through the application of the law of identity we see that "green" itself does not refer to a thing, otherwise all the distinct instances of green would be the very same thing in violation of that law. Likewise "4" does not refer to a thing. — Metaphysician Undercover

Yes I get that. So green = green is meaningless; and 4 = 4 is meaningless; and 2 + 2 = 4 is meaningless. Is this the proposition you're prepared to go forward defending?

To make up an imaginary object, a platonic idea, called a number, and say that this is the object referred to by "4" does not resolve the problem, as I've explained to you already. — Metaphysician Undercover

Let's not argue about whether you've explained it before. Let's just say I understand your point. When mathematicians say that 4 and 2 + 2 refer to the abstract thing called the number (that we call) 4, you are disagreeing. You say there IS NO number 4. That the symbol '4' does not refer to anything.

You do know you're kind of out of step with pretty much everyone, right? Not that this is an argument against your idea. You could be right and everyone else wrong. But you do agree that virtually all philosophers and mathematicians believe in abstract numbers as "true things."

Tell me, do you believe that "4 is an even number," or "4 represents an even number," have truth values? If so, what would you say the truth values are?

Then, "2" must also refer to such an imagunary object. So, when you write "2+2", you denote the object, the number two twice. By what principle do you say that this object, the number two, when denoted twice actually refers to the number four? — Metaphysician Undercover

I could go nominalist and invoke the Peano axioms, or empiricist and note that 2 pebbles and 2 more pebbles is 4 pebbles. But why bother, i'm sure you've thought of this. For now I will defer direct argument with your ideas. I just want to confirm I've got your ideas right.

That's completely nonsensical, to say that two instances of occurrence of this object which we call the number two, magically becomes the object called the number four. — Metaphysician Undercover

I must say it seems rather commonplace to me. But let's defer that for the moment.

You must respect the object actually referred to by "2", just like if you say that the colours are this type of platonic object you must respect the colours actually referred to by "blue" and "yellow", and not falsely conclude that "green" is being referred to when someone says "blue and yellow". — Metaphysician Undercover

Ok. I am not going to write anymore in this post. I just want to make sure that I understand your position.

* 2 + 2 = 4 is meaningless because neither 2 + 2 nor 4 refers to anything. You would agree that 2 fish plus 2 fish is 4 fish, but you would NOT go so far as to agree that 2 + 2 = 4.

* 4 = 4 is meaningless because the law of identity only applies to "true things" and not to properties, and 4 is a property and not a "true thing."

* You would NOT agree that 4 represents an even number, because you don't think 4 represents anything at all.

* You do understand that you haven't got much if any agreement in the math or philosophy communities, yes?

* Who is the arbiter of "true thingness?"

Ok very interesting. I eagerly await your comments.

Can you tell me (I've asked this before) where you came up with these ideas? Are they written down somewhere? I've never heard it said that abstract numbers are not "true things" deserving of being equal to themselves. The Wiki page on the law of identity does not mention any distinction between "true things" and properties.

Let me say this back to you. In the past I thought you were saying that 2 + 2 and 4 refer to different things. But what you are saying in fact is that 2 + 2 and 4 don't refer to anything at all. Because they are properties, or attributes. 4 is a property of a string quartet, for example. There is no number 4 by itself as an object. So it's wrong to say that 4 refers to anything at all. — fishfry

Right, "2+2=4" has meaning without referring to anything, just like "green is a colour", and "the acceleration of gravity is 9.8 metres per second per second" have meaning without referring to anything. These are generalizations, abstract rules or laws, which have a broad application without referring to any particular thing. But even though they don't refer to any particular thing, they still have meaning.

Ok! I understand you. The law of identity x = x only applies to "true things," things that are deserving of thinghood; and not properties, which can apply to things, but are not themselves things. A green house is green, but green by itself is not a thing. Therefore green = green is meaningless. Is that your point? I wish you had said this earlier, and I suppose you'll say you did. But I understand you now. Green is not a thing and 4 is not a thing. 4 does not refer to anything at all. It's a property. 4 = 4 is therefore meaningless.

Do you assert that 4 = 4 is meaningless? Can you please explicitly confirm this and don't skip over it?

You assert that 4 = 4 is meaningless because 4 does not refer to a "true thing."

May I ask, who decides what is a true thing. But nevermind, I said I'd defer my objections to later. — fishfry

The law of identity states that a thing is the same as itself. So when it is expressed using = then "=" signifies is the same as. So if the law of identity is expressed as "A=A", then A=A is supposed to signify that an object is the same as itself. I wouldn't say "4=4" is necessarily meaningless though, because "=" in mathematics signifies "has equal value to". Do you recognize the difference between "the same as" and "equivalent to"?

As for your question of who decides what a true thing is, that's what the law of identity is the criteria for. If it has its own identity as unique particular, it is a thing. Of course you do not have to agree with the law of identity, many philosophers have argued against it.

Can you point me to a discussion of the law of identity that explicitly confers thinghood on some symbols and not others? Do you call them symbols? There are "true things" and "false things," by what word do you call things in general? Entities perhaps? An entity can be a true thing or a property? What is your terminology? — fishfry

The law of identity is not about symbols, it is about things. That's why it is quite difficult to grasp, and also why many argue against it. The intuitive response to "identity" is to think of the name of the thing as the thing's identity. But this is not what the law says, it says that the identity of the thing is the thing itself. This creates a separation between the identity which we assign to the thing, (it's name, description, or whatever we say about it to identify it), and its true identity, which is itself.

You might say then, that to have "thinghood", is to have independent existence, separate from whatever we might say about the thing. This is to have an identity, to be something.

Yes I get that. So green = green is meaningless; and 4 = 4 is meaningless; and 2 + 2 = 4 is meaningless. Is this the proposition you're prepared to go forward defending? — fishfry

No, this is not what I am saying at all. What I say, is that "=", when used in mathematics, does not mean "the same as", as dictated by the law of identity. So these uses of "=" have meaning, but the meaning is not "the same as", as dictated by the law of identity.

You do know you're kind of out of step with pretty much everyone, right? Not that this is an argument against your idea. You could be right and everyone else wrong. But you do agree that virtually all philosophers and mathematicians believe in abstract numbers as "true things." — fishfry

I don't agree with this. Many philosophers argue against platonic realism. Saying that "4" refers to an object which is a number is nothing other than platonic realism.

Tell me, do you believe that "4 is an even number," or "4 represents an even number," have truth values? If so, what would you say the truth values are? — fishfry

No, philosophically I believe in a correspondence type of truth, so strictly speaking these statements are valid within a logical system, but it doesn't make sense to talk about truth here. I might in common speaking, say that such things are true, but I would be using "true" in a less rigorous way.

* 2 + 2 = 4 is meaningless because neither 2 + 2 nor 4 refers to anything. — fishfry

No, "=" as it is used in mathematics means to have the same value. It does not mean "the same as" as dictated by the law of identity. This has always been the heart of our disagreement. So "2+2=4" has meaning, it just does not mean that "2+2" refers to the same thing as "4", it means that they have equal value.

* 4 = 4 is meaningless because the law of identity only applies to "true things" and not to properties, and 4 is a property and not a "true thing."

* You would NOT agree that 4 represents an even number, because you don't think 4 represents anything at all.

* You do understand that you haven't got much if any agreement in the math or philosophy communities, yes?

* Who is the arbiter of "true thingness?"

Ok very interesting. I eagerly await your comments. — fishfry

These I think I've already addressed.

Can you tell me (I've asked this before) where you came up with these ideas? Are they written down somewhere? I've never heard it said that abstract numbers are not "true things" deserving of being equal to themselves. The Wiki page on the law of identity does not mention any distinction between "true things" and properties. — fishfry

I've studied philosophy for many years. Have you not heard of platonic realism, and that some philosophers are opposed to it?

Right, "2+2=4" has meaning without referring to anything, just like "green is a colour", and "the acceleration of gravity is 9.8 metres per second per second" have meaning without referring to anything. — Metaphysician Undercover

I have a quibble about that. You say 2 + 2 = 4 has meaning even though you assert that neither 2 + 2 nor 4 refer to anything. And you've said that green doesn't refer to anything. Questions so that I can better understand:

* Does 2 + 2 = green have meaning? If no, then why does 2 + 2 = 4 have meaning? None of 2 + 2, 4, and green refer to anything by themselves. So why can we combine 2 + 2 and 4 on either side of an equal sign, but not 2 +2 and green?

* Does 2 + 2 = 5 have meaning? What does it mean? If none of 2 + 2, 4, or 5 have meaning; then how can you distinguish 2 + 2 = 4 from 2 + 2 = 5?

These are generalizations, abstract rules or laws, which have a broad application without referring to any particular thing. But even though they don't refer to any particular thing, they still have meaning. — Metaphysician Undercover

If they don't refer to anything, how do you know they have meaning and what that meaning is? Does dfslkjf have meaning? Does it refer to anything? What gives 2 + 2 meaning; and what exactly does it mean, if it doesn't refer to anything?

The law of identity states that a thing is the same as itself. — Metaphysician Undercover

Yes. And you assert that it only applies to "things." So that Joe Biden = Joe Biden is an instance of the law of identity; but 4 = 4 is not.

And I asked you, how do we determine thinghood?

So when it is expressed using = then "=" signifies is the same as. So if the law of identity is expressed as "A=A", then A=A is supposed to signify that an object is the same as itself. I wouldn't say "4=4" is necessarily meaningless though, because "=" in mathematics signifies "has equal value to". — Metaphysician Undercover

Well now you have introduced a brand new term, "value." What does it mean? If 4 does not refer to anything, how do I know what it's value is? If fglfdkjgldj does not refer to anything, how can I determine what its value is?

Do you recognize the difference between "the same as" and "equivalent to"? — Metaphysician Undercover

Of course. I wonder if you do. Equality is a special case of equivalence. But there are equivalences that aren't equalities. For example 4 = 4 is an equality. $4 \equiv 6 \pmod 2$ is an equivalence that is not an equality. Is that what you mean? That's the mathematical sense of the word but perhaps you have something else in mind.

In general, two things are equivalent when they live in the same equivalence class of some equivalence relation. For example if we say two people are equivalent if they live in the same state, then the equivalence classes are the states, and I am equivalent to Gavin Newsom. That's equivalence.

Equality is a special case of equivalence in which each equivalence class contains exactly one element.

I fail to see how equivalence has anything to do with what we're talking about, though.

As for your question of who decides what a true thing is, that's what the law of identity is the criteria for. If it has its own identity as unique particular, it is a thing. — Metaphysician Undercover

Isn't that circular? The law of identity only applies to things. And how do we know if an entity (if I may use that word to mean something that might or might not have thinghood) has thinghood? It does if the law of identity applies to it. That's circular!

Given an entity, like Joe Biden or 4, how do I know that the former has thinghood, so that Joe Biden = Joe Biden is an instance of the law of identity; and 4 DOESN'T have thinghood, and 4 = 4 is NOT an instance of the law of identity?

Yet in the latter case you assert that although 4 = 4 is not an instance of the law of identity, it does nevertheless have meaning. And what is its meaning, exactly? Does 4 = 5 have meaning? If so, what is its meaning?

@Meta you are tying yourself into knots trying to make sense of your claim that 4 does not refer to anything.

Of course you do not have to agree with the law of identity, many philosophers have argued against it. — Metaphysician Undercover

I'm not disagreeing with the law of identity, you are. You claim that 4 = 4 is not an instance of the law of identity. That is a radical claim that requires much more evidentiary support than you've so far provided.

And you THEN claim that even though 4 = 4 is not an instance of the law of identity, it nevertheless has "meaning," by virtue of the "value" of 4. But you haven't defined these terms and clearly seem to be using them in a nonstandard way. If 4 has a value but doesn't refer to anything, you have some 'splainin' to do.

The law of identity is not about symbols, it is about things. That's why it is quite difficult to grasp, and also why many argue against it. — Metaphysician Undercover

But I'm not arguing against it. I'm asking you to give me a coherent account of your idea that 4 = 4 is not an instance of the law of identity, yet 4 = 4 has "meaning" by virtue of the "value" of 4. If 4 doesn't refer to anything, how can it have a value?

By the way philosophers of mathematics have long settled on perfectly sensible answers to this question. We can define 4 as the class of all sets of cardinality 4 (which was the original definition, but suffers from the drawback that this class is not a set); or we can use the modern definition, which isthat we choose a canonical representative of 4-ness, namely 4 = {0, 1, 2, 3}. This was von Neumann's clever idea..

The intuitive response to "identity" is to think of the name of the thing as the thing's identity. — Metaphysician Undercover

I am not making that error, this is a strawman argument.

But this is not what the law says, it says that the identity of the thing is the thing itself. This creates a separation between the identity which we assign to the thing, (it's name, description, or whatever we say about it to identify it), and its true identity, which is itself. — Metaphysician Undercover

More strawmen. You claim that 4 = 4 is not an instance of the law of identity; yet it has "meaning" by virtue of the "value" of 4. I am asking for a coherent account of that claim.

You might say then, that to have "thinghood", is to have independent existence, separate from whatever we might say about the thing. This is to have an identity, to be something. — Metaphysician Undercover

How do I know that Joe Biden has thinghood and that 4, 2 + 2, and green, don't?

And if 4, 2 + 2, and green don't have thinghood, how can you claim that 2 + 2 = 4 has "meaning," but 2 + 2 = green doesn't?

No, this is not what I am saying at all. What I say, is that "=", when used in mathematics, does not mean "the same as", as dictated by the law of identity. — Metaphysician Undercover

You're just flat out wrong about that. In math, = means "the same as" as dictated by the law of identity. As I've been telling you for three years.

But the problem is that having denied that, you can't give a coherent account of "meaning" and "value" by which 2+ 2 = 4 has meaning and 2 + 2 = green doesn't.

So it's not that you're not allowed the opposite opinion. It's that your account of the opposite opinion is not coherent. I'm not just disagreeing with your position. I'm asking you to make sense of it.

So these uses of "=" have meaning, but the meaning is not "the same as", as dictated by the law of identity. — Metaphysician Undercover

As I say. Give me an account of your idea so that I too may see why 2 + 2 = 4 has meaning but 2 + 2= green does not, even though none of the terms have thinghood and neither equality (according to you) is an instance of the law of identity.

I don't agree with this. Many philosophers argue against platonic realism. Saying that "4" refers to an object which is a number is nothing other than platonic realism. — Metaphysician Undercover

I'd be willing to accept that 4 doesn't refer to anything, since my claim is that it refers to the "abstract number 4" which has much less claim on thinghood than a rock or Joe Biden. Mathematicians get around that problem either by defining 4 to be the proper class of all sets of cardinality 4; or else they choose a canonical representative of that class, {0, 1, 2, 3}, to stand in for the number 4. Although as Benacerraf points out, 4 "can not possibly be" {0,1,2,3} and of course he is right about that.

So I'm willing, for sake of discussion, to grant you that 4 has a rather tenuous grasp on thinghood.

But then 4 = 4 is not an instance of the law of identity, yet you say it has MEANING by virtue of the VALUE of 4. That's where your thesis is in need of support.

It's easier to just take 4 as {0,1,2,3} so that we can have a sensible theory by which 4 = 4 IS an instance of the law of identity. That's what I do and that's what most philosophers of math do.

What you have done is propose an alternative but you are stuck on "value" and "meaning." If you can explain these, please do.




No, philosophically I believe in a correspondence type of truth, so strictly speaking these statements are valid within a logical system, but it doesn't make sense to talk about truth here. [/quote]

This was in response to my question of whether you think "4 is an even number" is true. I agree that if one is a strict formalist then it's not a true fact about the world. But that's an argument against strict formalism, and not an actual stance one should take if one wishes to be sensible.

I might in common speaking, say that such things are true, but I would be using "true" in a less rigorous way. — Metaphysician Undercover

As I say, I take this as an argument against formalism, and not as a sensible position to take. Even on my formalist days I am stuck on "5 is prime." That seems to be a true fact of the world no matter how we fight against Platonism.

No, "=" as it is used in mathematics means to have the same value. It does not mean "the same as" as dictated by the law of identity. — Metaphysician Undercover

As I've noted, you're wrong about that. But if you're not, then you need to give a coherent account of "meaning" and "value."

This has always been the heart of our disagreement. So "2+2=4" has meaning, it just does not mean that "2+2" refers to the same thing as "4", it means that they have equal value. — Metaphysician Undercover

Once you explain to me what meaning and value are, I'll be in better position to understand your viewpoint. If 4 and 2 + 2 don't refer to anything, how can they have meaning or value? And how can I know that 4 has meaning and value but dflsklsjlslds doesn't?

* 4 = 4 is meaningless because the law of identity only applies to "true things" and not to properties, and 4 is a property and not a "true thing." — Metaphysician Undercover

You've been saying it has meaning, even though it's not an instance of the law of identity because 4 lacks thinghood. Now you say 4 = 4 is meaningless.

In fact in your very first sentence of the post I'm replying to you said: "Right, "2+2=4" has meaning without referring to anything, ..."

And now yuou say 4 = 4 is MEANINGLESS. Can you see that you contradicted your own position? And how could you not? You are claiming that 4 doesn't refer to anything, and that 4 = 4 is therefore not an instance of the law of identity, and now you are hard pressed to give an account of 4 = 4. You say it has meaning and then you say it's meaningless. And then you wave your hands and say that I don't understand the law of identity, as if that would make your position coherent.

These I think I've already addressed. — Metaphysician Undercover

Not to my satisfaction. But I'll settle for a coherent account of 4 = 4. Does it have meaning? Does 4 have value? Does kfkdfjkdjdkd have meaning or value?

I've studied philosophy for many years. Have you not heard of platonic realism, and that some philosophers are opposed to it? — Metaphysician Undercover

I take your point, but even a Platonist has to come to terms with the truth of "5 is prime" and "4 is even." And "4 = 4".

Hi Enrique

Sorry - I missed your post on 9th April, due to all the other comments.

The reason why I don't think that an electromagnetic field generated by the equipment would explain the effect is because the original experiment conducted by Thomas Young in the early 1800s used candles, not lasers.

Does 2 + 2 = green have meaning? — fishfry

Yes, "2+2=green" has meaning. But I would reject this statement as inconsistent with the principles that I already understand and accept.

If they don't refer to anything, how do you know they have meaning and what that meaning is? — fishfry

I don't know, the answer to these questions. "Meaning" is not an easy topic. That's why there's so many philosophical debates about it. I really do not know what meaning is, or how I know that something has meaning. Those are questions yet to be answered.

Yes. And you assert that it only applies to "things." So that Joe Biden = Joe Biden is an instance of the law of identity; but 4 = 4 is not. — fishfry

By asking this question you indicate that you paid not attention to my explanation of what "=" signifies, or means in common mathematical usage, and what "is the same as" signifies or means in the law of identity. Since you still cannot see the difference here, after I've explained it countless times to you, it seems pointless to explain it again. It's actually a very simple difference, and very easy to understand.

Well now you have introduced a brand new term, "value." What does it mean? If 4 does not refer to anything, how do I know what it's value is? If fglfdkjgldj does not refer to anything, how can I determine what its value is? — fishfry

We learn values in school. If you still haven't learned the value of 4 yet, you could talk to a primary school teacher, or look it up on the web. You'll find there's a lot of educational stuff like that if you google it .

Isn't that circular? The law of identity only applies to things. And how do we know if an entity (if I may use that word to mean something that might or might not have thinghood) has thinghood? It does if the law of identity applies to it. That's circular! — fishfry

Yes, that's a very real problem. We often do not know whether a word signifies something with thinghood or not. That is the case with quantum physics and wave/particle duality. Physicists cannot determine the "thinghood" of the described phenomena.

But I'm not arguing against it. I'm asking you to give me a coherent account of your idea that 4 = 4 is not an instance of the law of identity, yet 4 = 4 has "meaning" by virtue of the "value" of 4. If 4 doesn't refer to anything, how can it have a value? — fishfry

Since you seem to have no idea of what "value" means I suggest you do some research into this topic, and come back to me when you get above the primary school level.

You're just flat out wrong about that. In math, = means "the same as" as dictated by the law of identity. As I've been telling you for three years. — fishfry

Obviously you never looked into this, and haven't looked beyond your own nose to see what others say about what "=" signifies. The following is the opening paragraph from the Wikipedia entry

The equals sign (British English, Unicode Consortium[1]) or equal sign (American English), formerly known as the equality sign, is the mathematical symbol =, which is used to indicate equality in some well-defined sense.[2][3] In an equation, it is placed between two expressions that have the same value, or for which one studies the conditions under which they have the same value. — Wikipedia: equals sign

Notice the mention of "the same value".

But then 4 = 4 is not an instance of the law of identity, yet you say it has MEANING by virtue of the VALUE of 4. That's where your thesis is in need of support. — fishfry

Do you accept that there is a difference between "is the same as" and "has the same value as"? The former phrase is the phrase used by the law of identity. The latter phrase is what is signified by "=", as the Wikipedia entry indicates.

Until you accept that there is a difference here, further discussion is pointless.

The reason why I don't think that an electromagnetic field generated by the equipment would explain the effect is because the original experiment conducted by Thomas Young in the early 1800s used candles, not lasers. — Gary Enfield

I address this in the OP, might as well quote it for clarification purposes.

It is easy to imagine a stream of wavicles interfering as they diffract through the slits to produce an array of light and dark bands on the florescent screen corresponding to in phase and out of phase waves. This would resemble the classic experiment performed in the 19th century (nonelectronic context), where a beam of light was diffracted by a single aperture to then pass through double slits as a spreading field which apparently interfered with itself and produced a similar result.

Interference patterns from one at a time particle emission (the modern electronic emitter context) are a thornier outcome to account for. The typical explanation is that the wavicle passes through both slits to interfere with itself, spreading out in the double-slit chamber and then spontaneously collapsing in some way upon contact with the absorber surface to give a particulate signature. This “wave function collapse” mechanism is quite the brain teaser: does the wavicle spread out invisibly in the chamber as it diffracts and then somewhat mystically end up at a very localized endpoint? Why would many localized end points with no likeness to waves at all look like in phase and out of phase waves as they accumulate on the absorber screen? What exactly is going on? — Enrique

A single wavicle experiment has never been performed nonelectronically.

I believe the wave spreads out and where it converges and crosses the sensor, a single atom is excited, the wave carries on but some energy has exchanged. Wave with no collapse. Its just my idea, probably wrong but solves the riddle. If you can believe a wave of light can lose some part of its energy but carry on anyway.

Hi Enrique

True - single particle experiments can only be done with modern equipment, but I don't see why there is a need to imagine a different mechanism at work. All we are trying to do is find out about the original effect and why it is occurring. Unless you can show a reason why a different mechanism might be kicking-in, there is no reason to suppose that it's a different factor.

There may be different factors in some of the strange effects recorded in certain experiments - eg. the disappearance of the interference pattern when monitoring equipment was put near the slots, but that is not what you seemed to be discussing here.

Again - instead of imagining that particles miraculaously turn into waves, and then back again to form a dot on a detector, why not accept the simpler option of a hidden pool of stuff that is causing the wave which the particles ride?