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
Of course the truth of any symbolic expression depends on the interpretation given to the symbols; — fishfry
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" — Metaphysician Undercover
Metaphysician Undercover is on record stating that he does not believe that 2 + 2 and 4 denote the same mathematical object. — fishfry
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
"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. — 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. — Metaphysician Undercover
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. — 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. — Metaphysician Undercover
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
and I'll show you how it fails, again. Maybe this time you'll pay respect to the law of identity. — Metaphysician Undercover
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. — Metaphysician Undercover
Why are you going on about this? — fishfry
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. — Metaphysician Undercover
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
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
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
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
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
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
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
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
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
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
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
They refer to properties, which are not things, by the law of identity. — Metaphysician Undercover
So, take my example, "blue and yellow" does not refer to the same thing as "green", because neither refers to a thing. — Metaphysician Undercover
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
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
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
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
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
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
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
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
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
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
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
* 2 + 2 = 4 is meaningless because neither 2 + 2 nor 4 refers to anything. — fishfry
* 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
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
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
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
The law of identity states that a thing is the same as itself. — Metaphysician Undercover
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
Do you recognize the difference between "the same as" and "equivalent to"? — Metaphysician Undercover
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
Of course you do not have to agree with the law of identity, many philosophers have argued against it. — Metaphysician Undercover
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
The intuitive response to "identity" is to think of the name of the thing as the thing's identity. — Metaphysician Undercover
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
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
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
So these uses of "=" have meaning, but the meaning is not "the same as", as dictated by the law of identity. — Metaphysician Undercover
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 might in common speaking, say that such things are true, but I would be using "true" in a less rigorous way. — Metaphysician Undercover
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
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
* 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
These I think I've already addressed. — Metaphysician Undercover
I've studied philosophy for many years. Have you not heard of platonic realism, and that some philosophers are opposed to it? — Metaphysician Undercover
Does 2 + 2 = green have meaning? — fishfry
If they don't refer to anything, how do you know they have meaning and what that meaning is? — fishfry
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
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
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
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
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
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
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
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
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
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