I don't deny that two plus two equals four.

I only assert the obvious, that "2+2", which represents an operation of addition, does not represent the same thing as "4", — Metaphysician Undercover

If ““2+2”...does not represent the same thing as “4””, then in what sense are they equal?

.. or of anything/whatever, hence the utility of a calculator.
Say, a set of my left ear, that soccer match, the Moon, and the experience of vanilla taste I had the other day when eating icecream, comes to 4 in quantity; kind of trivial to count. — jorndoe

So you're agreeing with me, a quantity is necessarily a number of things, therefore a property of those things, hence a predication. You named that group of things as a thing itself, "a set", and you predicate "quantity" as a property of that named thing, the set..

You're now confusing quantity, predication, measurement, ...
Say, a set of you and I comes to a quantity of 2, |{you,I}|=2|{you,I}|=2; kind of trivial to count.
Say, where ϕϕ = is human (predicate), it so happens that ϕ(you)∧ϕ(I)ϕ(you)∧ϕ(I), I assume.
You're making a wicked mess of things. :confused: — jorndoe

I can't agree with your conclusion. To me, your use of strange symbols is what is making a "wicked mess". We can look at what is meant by "quantity", "predication", and "measurement" without leaving the English language. Quantity and measurement are both types of predication. You seem to agree with me, so why complain that I'm confused and making a mess of things? In your examples you predicate "quantity" as the property of a set. Do you not consider this to be predication?

If ““2+2”...does not represent the same thing as “4””, then in what sense are they equal? — Luke

In a mathematical sense, obviously. As I explained to fishfry, it's really the only sense of "equal" that there is. "Equal" is a mathematical term. To say that two things are equal is to render them in a form which allows us to proceed with mathematical operations such as counting them. We cannot count apples and oranges unless we say that an apple is equal to an orange. It is only by assigning equality to distinct things that we are able to count them One apple is equal to an orange, and then we count them, 1,2. So apprehending different things as having an equal value allows us to count them, 1,2,3,4....

Notice that it is necessary and fundamental to the operation of counting, that equal things are distinct things. If equal things were the same thing, we would count the same thing over and over again, and the count would clearly be invalid. Therefore, we can conclude that it is a fundamental and necessary axiom of arithmetic, that equal things are not the same thing. Otherwise all arithmetic would be invalid, like counting the same thing over and over again to claim a quantity of things, is an invalid count.

We cannot count apples and oranges unless we say that an apple is equal to an orange. It is only by assigning equality to distinct things that we are able to count them One apple is equal to an orange, and then we count them, 1,2. So apprehending different things as having an equal value allows us to count them, 1,2,3,4.... — Metaphysician Undercover

And why does "2 apples + 2 apples" "not represent the same thing" as "4 apples"?

You're now confusing quantity, predication, measurement, property, ...

To me, your use of [ standard notation that does away with confusion ] is what is making a "wicked mess". — Metaphysician Undercover

Confuzzlement has roughly gotten worse with each comment. :confused: Start over?

I am not sure I follow the argument from Fourier series to saying that "therefore the Planck scale is ontic." — fishfry

(A1) If something is random, it is either epistemically random or aleatorically random.
(A2) The position and momentum states of quantum particles are random.
(A3) The position and momentum states of quantum particles are either epistemically random or aleatorically random. (1,2, modus ponens)
(A4) If something is epistemically random, the uncertainty associated with that randomness can be arbitrarily reduced by sufficient sampling.
(A5) The uncertainty associated with the product of position and momentum (time and frequency) cannot be arbitrarily reduced by sufficient sampling. (If you localise in time, you disperse in frequency and vice versa)
(A6) The position and momentum states of quantum particles are not epistemically random (4,5, modus tollens).
(A7) The position and momentum states of quantum particles are aleatorically random. (1, 6, disjunctive syllogism).

Aleatoric randomness is randomness that plays a causal role in a model - it is part of the modelled dynamics, epistemic randomness is randomness that does not. I don't think this gives an interpretation of how randomness works causally here, just establishes that it is a property of the entity modelled (the wavefunction's position and momentum observables). Whether it's appropriate to say that quantum shit really does behave like this because the model says so I think is a different question (realism vs anti-realism of scientific theories).

(A4) If something is epistemically random, the uncertainty associated with that randomness can be arbitrarily reduced by sufficient sampling. — fdrake

This is only assuming that all of the relevant data is being sampled. Assuming Bohm's interpretation, for example, you can never sample the value of the hidden variable, no matter how good your sample is. In the Everett interpretation, your sample does not include all branches of the wavefunction. Both interpretations are metaphysically deterministic, but both predict epistemically random measurement outcomes.

On the other hand, if you were sampling digits of pi, for example, then unless you already knew what you were sampling, you would never see that it is non-random from your sample, even if you were getting every digit with perfect accuracy. And if you knew what you were sampling, then the question would not arise.

And why does "2 apples + 2 apples" "not represent the same thing" as "4 apples"? — Luke

Quite obviously,"2 apples + 2 apples" signifies two distinct groups of two apples, and the "+" represents an operation of putting the two groups of two apples together into one group. There is no such two distinct groups of two, nor the operation of addition signified by "4 apples". Therefore it is very clear that the phrase "2 apples + 2 apples" represents something very different from "4 apples".

Do you understand the difference between saying "4 objects", and specifying a specific configuration of four objects? A specified configuration is not the same thing as the more general "4 objects". This is the type of difference I am talking about here. In both cases, "2 apples + 2 apples", and "4 apples", we are saying something about four apples, but "2 apples + 2 apples" says something more specific than the more general "4 apples". So we cannot say that the two phrases represent the same thing.

Start over? — jorndoe

Go ahead. You're the one quizzing me. It appears we have great difficulty understanding each other.

(A4) If something is epistemically random, the uncertainty associated with that randomness can be arbitrarily reduced by sufficient sampling. — fdrake

I don't think I agree with this (A4). If the appearance of randomness is caused by inadequate principles by which the samples are modeled, then no degree of sampling will reduce the appearance of randomness.

Quite obviously,"2 apples + 2 apples" signifies two distinct groups of two apples, and the "+" represents an operation of putting the two groups of two apples together into one group. — Metaphysician Undercover

So you agree that the group resulting from this operation of addition is “4 apples”? That is, you agree that “2 apples + 2 apples” = “4 apples”?

There is no such two distinct groups of two, nor the operation of addition signified by "4 apples". — Metaphysician Undercover

It is obviously signified by the equation “2 apples + 2 apples = 4 apples”. Both sides of the equation are equal in value or quantity. They “represent the same thing” in terms of value or quantity, which is the point of the mathematical equation. I’m not sure what point you are trying to make instead.

So you agree that the group resulting from this operation of addition is “4 apples”? That is, you agree that “2 apples + 2 apples” = “4 apples”? — Luke

Yes, I think I've stated about four times now, on this thread alone, that I agree that two plus two equals four.

It is obviously signified by the equation “2 apples + 2 apples = 4 apples”. Both sides of the equation are equal in value or quantity. They “represent the same thing” in terms of value or quantity, which is the point of the mathematical equation. I’m not sure what point you are trying to make instead. — Luke

I've been very clear in the point I'm making, so I don't understand why you're not clear about it. The point is that "2+2", and "4", do not represent the same thing, as "same" is defined by the law of identity, contrary to what has been claimed on this and other threads.

To say that "they represent the same thing in terms of value or quantity", is to qualify "same thing". It is to say that they have some quality which is the same, this quality being named as "quantity". Therefore it is does not say that they represent "the same thing" as determined by the law of identity. I can say "red roof" and "red car", and claim that these two expressions represent the same thing in terms of colour, just like you say "2+2", and "4" represent the same thing in terms of quantity. This does not mean that they represent the same thing in any strict definition of "same thing".

Jorndoe claimed that I confuse quantity with predication, but obviously quantity is a form of predication, and it is jorndoe who is confused, not I.

This is only assuming that all of the relevant data is being sampled. — SophistiCat

Yees. I am assuming the things accurately described as random are random. Do any of the interpretations you referenced remove the distribution from the theory?

On the other hand, if you were sampling digits of pi, for example, then unless you already knew what you were sampling, you would never see that it is non-random from your sample, even if you were getting every digit with perfect accuracy. And if you knew what you were sampling, then the question would not arise. — SophistiCat

I don't quite understand the relevance of this. Can you elaborate? Are you saying that the real world might have a hidden number that removes all the randomness associated with quantum variables?

Your point is simply that "2+2" and "4" are written differently or use different symbols. Or, as I said earlier, they are different expressions of the same value, or different ways of expressing the same value. Very profound :roll:

Yees. I am assuming the things accurately described as random are random. Do any of the interpretations you referenced remove the distribution from the theory? — fdrake

Well, I have a mostly pop-sci "knowledge" of QM, my college physics being too rusty to be of much use, but as far as I know the "pilot wave" of Bohmian mechanics would make measurements deterministic - except, of course, being hidden, it is not part of the measurement. And MWI says that the full wavefunction evolution is deterministic (as the Schrodinger equation shows), but we can only measure one of its eigenvalues at a time, since our subjective state in which the measurement outcome is recorded doesn't encompass the full quantum state. If you perform successive measurements on identically prepared systems, the branching wavefunction will leave a trail of random results in each individual branch, even though across all of the branches every set of measurement outcomes will be the same.

I quite don't understand the relevance of this. Can you elaborate? Are you saying that the real world might have a hidden number that removes all the randomness associated with quantum variables? — fdrake

I am saying that if it did, we wouldn't know it just from this one sampling. We might guess that it looks suspiciously like the digits of pi, for example (if we were lucky to sample from the already calculated range), but such numerology is perilous. For example, in the past there were a number of attempts to "derive" the empirically measured fine structure constant of particle physics from the ratios of integers and important transcendent numbers like e and pi. Nothing came out of it, as more accurate measurements successively falsified all such hypotheses. (Perhaps we shouldn't retrospectively dismiss these exercises as unscientific numerology, but instead look at them as failed heuristics that once in a blue moon do lead to discoveries.) But the moral is that we usually require more context to establish a causal mechanism behind a phenomenon than a single sampling, which may not reveal a regularity behind it, or conversely may trick us with an appearance of regularity that is not actually there (like in the case of the fine structure constant). Quantum mechanics is, of course, an example of a theory that was developed, first of all, on the shoulders of previous successful theories, and second, with the help of numerous independent lines of evidence, so we should feel pretty safe here.

Well, I have a mostly pop-sci "knowledge" of QM, my college physics being too rusty to be of much use, but as far as I know the "pilot wave" of Bohmian mechanics would make measurements deterministic - except, of course, being hidden, it is not part of the measurement. And MWI says that the full wavefunction evolution is deterministic (as the Schrodinger equation shows), but we can only measure one of its eigenvalues at a time, since our subjective state in which the measurement outcome is recorded doesn't encompass the full quantum state. If you perform successive measurements on identically prepared systems, the branching wavefunction will leave a trail of random results in each individual branch, even though across all of the branches every set of measurement outcomes will be the same. — SophistiCat

AFAIK the Schrodinger equation's time evolution is deterministic, but that doesn't make the states deterministic. The states are samples from probability distributions . It might be that someone can declare some aspect of the randomness "unphysical" and salvage a global determinism (if only we had (blah) we'd determine the output states!). I don't really know enough about it.

I am saying that if it did, we wouldn't know it just from this one sampling. We might guess that it looks suspiciously like the digits of pi, for example (if we were lucky to sample from the already calculated range), but such numerology is perilous — SophistiCat

I'm reading this as a claim that there's some source that determines the observed quantum states deterministically, it's simply that we don't (or cannot) know the behaviour of the source? Analogously, Pi's digits pass tests for statistical randomness, but they're determined given a way to arbitrarily accurately evaluate Pi.

Your point is simply that "2+2" and "4" are written differently or use different symbols. Or, as I said earlier, they are different expressions of the same value, or different ways of expressing the same value. Very profound :roll: — Luke

If you were looking for something profound, you've come to the wrong person. I was just pointing out the mistake of those who say that "2+2" and "4" refer to the same thing. It's really quite trivial, but some people seem to act like I'm attacking their God. Perhaps it was the behaviour of others which made you think I might be saying something profound.

By the way, in case I didn't make this clear last time, I consider "different expressions of the same value" to be ambiguous nonsense, and "different ways of expressing the same value" does very little to clarify what you could possibly mean. Do you even know what "value" means? It refers to the desirability of a thing, or what a thing is worth. How do you apprehend "2+2", or "4", as an expression of what a thing is worth?

By the way, in case I didn't make this clear last time, I consider "different expressions of the same value" to be ambiguous nonsense, and "different ways of expressing the same value" does very little to clarify what you could possibly mean. Do you even know what "value" means? It refers to the desirability of a thing, or what a thing is worth. How do you apprehend "2+2", or "4", as an expression of what a thing is worth? — Metaphysician Undercover

Perhaps you are unaware that a word can have more than one meaning or use. You seemed to have little difficulty understanding what I was talking about when I spoke of “value or quantity”. I also assume you were not talking about value as “the desirability of a thing” when you said:

It is only by assigning equality to distinct things that we are able to count them One apple is equal to an orange, and then we count them, 1,2. So apprehending different things as having an equal value allows us to count them, 1,2,3,4.... — Metaphysician Undercover

That's because we qualified value with "quantitative" value. But your assertion "different expressions of the same value" does not show that qualification. Hence the ambiguity. And I know the way you argue through ambiguity, I've been exposed to it too many times. So I would not accept an ambiguous proposition from you.

Do we agree that "value", being inherently subjective, is not a thing?

, in the interest of avoiding equivocation and ambiguity, can you differentiate the uses of the word "value" in these two sentences? :)

"We calculated the value of the national carbon footprint for last year to so-and-so."

"I greatly value a cold beer on a hot summer night."

I'm guessing the use here is like the former. Contextual reading matters.

That's because we qualified value with "quantitative" value. — Metaphysician Undercover

Yes, because our discussion was in the context of mathematics. Or do you think that mathematics is all about monetary value (i.e. desirability/worth)? Don’t be daft.

AFAIK the Schrodinger equation's time evolution is deterministic, but that doesn't make the states deterministic. The states are samples from probability distributions (generalisations of probability distributions I guess? I vaguely recall that they break a few rules). It might be that someone can declare some aspect of the randomness "unphysical" and salvage a global determinism (if only we had (blah) we'd determine the output states!). I don't really know enough about it. — fdrake

The more traditional interpretations treat the equation as only one component that is needed to determine the actual physical state - position, momentum. Everett just reads it literally as the equation of state, sacrificing some of our traditional notions of what a physical state is.

I'm reading this as a claim that there's some source that determines the observed quantum states deterministically, it's simply that we don't (or cannot) know the behaviour of the source? Analogously, Pi's digits pass tests for statistical randomness, but they're determined given a way to arbitrarily accurately evaluate Pi. — fdrake

With some effort we could interpret, for example, the spigot algorithm for calculating the digits of pi as some exotic physical process. Looking at it from the other end, if we were performing physical measurements, would we be able to figure out the underlying mechanism? Not without more context; taken on their own, measurements would appear quite random. Or take a chaotic Newtonian system: an observation that is limited to setting up the system, letting it run and then performing a measurement would lead us to conclude that it is aleatory. My point is that we need to probe nature not just many times, but in many different ways, in order to establish the causal mechanism with reasonable confidence (and always with the assumption that nature is not much trickier than we think it is, but that assumption is part of what goes into "reasonable confidence").

"We calculated the value of the national carbon footprint for last year to so-and-so."

"I greatly value a cold beer on a hot summer night." — jorndoe

The use of "value" in the first statement is extremely ambiguous because it is not related (grounded) to anything. The supposed "value of the carbon footprint" needs to be substantiated by a scale of some sort in order to have meaning. Without that scale the supposed "value" is meaningless. In the second case, "I" substantiates the value, with personal beliefs and a personal hierarchy. So the meaning of "value" is revealed by your use of "I".

Yes, because our discussion was in the context of mathematics. Or do you think that mathematics is all about monetary value (i.e. desirability/worth)? Don’t be daft. — Luke

So, back to my point then. If it is true that "4" expresses a value, then "2+2" does not express a value. The latter expresses a mathematical operation which is a different type of expression than an expression of value. Do you see the difference? Do you see that if "4" is an example of an expression of value, then in "2+2" there are two distinct values expressed, "2" and "2" whereas only one value "4" is expressed with "4"?

If it is true that "4" expresses a value, then "2+2" does not express a value. — Metaphysician Undercover

2+2=4. You said that you don't deny this equation. How can "2+2" and "4" be equal if "2+2" does not express a value (i.e. a quantity, number, amount)?

Do you see that if "4" is an example of an expression of value, then in "2+2" there are two distinct values expressed, "2" and "2" whereas only one value "4" is expressed with "4"? — Metaphysician Undercover

What does '+' do?

The use of "value" in the first statement is extremely ambiguous because it is not related (grounded) to anything. — Metaphysician Undercover

Really? And yet you understood it fine? And well enough that you could, say, go look up annual carbon footprints and such...? (I could start listing examples ... maybe another day)

2+2=4. You said that you don't deny this equation. How can "2+2" and "4" be equal if "2+2" does not express a value (i.e. a quantity, number, amount)? — Luke

I don't see how this can be so difficult for you.
Let's assume "4" represents an amount, number, or quantity. And we can also say that "2" represents an amount, number, or quantity. Doesn't "2+2" represent two distinct amounts, numbers, or quantities, related to each other with "+"? If "2" represents an amount, number, or quantity, how can you not see that there are two such amounts, numbers or quantities represented by "2+2"?

So, "2+2" does not represent a value, it represents two distinct values related with "+", and we say that this is equal to the value of "4". Here's an example, go to the store, and pick out three items. Now you have item number one, item number two, item number three, each has a distinct value. Place a plus sign between them and you have the cost of item number one plus cost of item number two plus the cost of item number three. Clearly what is represented here is three distinct values being added together. And we say that this is "equal" to one value, which is the sum of the three.

Therefore, a phrase such as "2+2", which does not express a value, but expresses a multitude of values in a specific relation, can and does, equal "a value".

Really? And yet you understood it fine? And well enough that you could, say, go look up annual carbon footprints and such...? (I could start listing examples ... maybe another day) — jorndoe

I really can't say I understood it at all. I have absolutely no idea what "the value of a national carbon footprint" is. Examples would not help. As I said, you need a scale of some sort.

So, "2+2" does not represent a value, it represents two distinct values related with "+", and we say that this is equal to the value of "4". — Metaphysician Undercover

“2+2” is equal to a value of 4. I don’t see how this can be so difficult for you.

You are no longer arguing about identity. You are now arguing against the mathematical equation which you formerly said you did not deny. I have no interest in trying to teach or convince you of basic mathematics that most children can master.

No, I'm not arguing against the notion that 2+2=4. And, the fact that I've explicitly stated this numerous times, and explained that what I am arguing is the difference between being the same and being equal is a clear indication of this. It appears like you have not a good capacity to read English, because you intentionally interpret ambiguous words in a way which is inconsistent with what I've explicitly stated I am arguing, such that they mean something inconsistent with what I am arguing, when it is possible to interpret them in a consistent way.. This is why I did not want to follow you down this road of ambiguity, into discussing the meaning of "expressing a value", because I am familiar with this mode of argumentation of yours. You intentional misinterpret another person's writing, just for the sake of saying "see you've contradicted yourself". But the apparent contradiction is just intentional misinterpretation for the straw man purpose, which is a symptom of bad interpretation. .

If you cannot see the difference between representing the value "one dollar", (which is represented as $!), and representing something equal to a dollar ("four quarters", or "ten dimes"), then I think you've got a problem. Or, as fishfry seems to do, you do recognize the difference but deny that it's a difference, as if it's a difference which does not make a difference, or something like that..

If you cannot see the difference between representing the value "one dollar", (which is represented as $!), and representing something equal to a dollar ("four quarters", or "ten dimes"), then I think you've got a problem. — Metaphysician Undercover

Feel free to explain the difference between "representing the value one dollar" and "representing something equal to a dollar" to anyone who cares to listen.

I really can't say I understood it at all. I have absolutely no idea what "the value of a national carbon footprint" is. Examples would not help. As I said, you need a scale of some sort. — Metaphysician Undercover

You can't be serious.

"Based on those samples we calculated an average value of so-and-so."

My young nephew and niece understand what's meant in the English language. If you can't, then you're missing something.

Feel free to explain the difference between "representing the value one dollar" and "representing something equal to a dollar" to anyone who cares to listen. — Luke

I just did. This is how I represent "one dollar", like that or like this, $1. Something equal to a dollar is "ten dimes", or "four quarters". I really do not believe that you can't see the difference, I think you're in denial.

I explained the difference, in reference to "2+2+4" and I will explain it again. "One dollar", just like "4", says something very general. It allows for "four quarters", "ten dimes", or whatever, just like "4" allows for "2+2", "3+1", 6-2", whatever. However, "ten dimes" refers to something specific. It cannot be represented as "four quarters", or anything else, because the particular form, "ten dimes" is specified, and ten dimes is not four quarters, though they both equal a dollar. Likewise, "2+2" represents something specific, and though it is equal to "6-2", we cannot represent "2+2" as "6-2". They have distinct meaning, just like four quarters has a meaning distinct from ten dimes.

Do you understand that "four quarters" represents something different from "ten dimes"?

Last time I explained this to you, just above, you completely ignored it and made no indication whether you understood what I said or not, arguing that in terms of value, they say the same thing. Sure, that's why they are equal, but 'the whole point is that say they are equal in terms of value does not mean that they represent the same thing.

Do you understand the difference between saying "4 objects", and specifying a specific configuration of four objects? A specified configuration is not the same thing as the more general "4 objects". This is the type of difference I am talking about here. In both cases, "2 apples + 2 apples", and "4 apples", we are saying something about four apples, but "2 apples + 2 apples" says something more specific than the more general "4 apples". So we cannot say that the two phrases represent the same thing. — Metaphysician Undercover

So please, before you proceed, give me some indication that you've understood what I have said here.

You can't be serious.

"Based on those samples we calculated an average value of so-and-so."

My young nephew and niece understand what's meant in the English language. If you can't, then you're missing something. — jorndoe

Sure, I'm missing something, I do not know what a national carbon footprint is. There's more than samples involved here, there's principles and a scale. Of which I have absolutely no understanding. Do you know how they take some samples and produce "the value of a national carbon footprint" from them? You might explain it to me, but I don't see how it's relevant.

Something equal to a dollar is "ten dimes", or "four quarters" — Metaphysician Undercover

Redacted. No point in talking with Meta.