So in some possible world, Hesperus has no properties, and hence Hesperus does not exist in that world. It doesn't follow that Hesperus does not exist in some other possible world. — Banno

My next post will be about heat and the motion of molecules. One could easily become paranoid about being thrown off TPF for not sticking to the OP.

Regarding "possible worlds"
Kripke wrote: 1) What do I mean by ‘rigid designator’? I mean a term that designates the same object in all possible worlds. 2) All of this talk seems to me to have taken the metaphor of possible worlds much too seriously in some way 3) And if the phrase ‘possible worlds’ is what makes anyone think some such question applies, he should just drop this phrase and use some other expression, say ‘counterfactual situation,’ which might be less misleading.

IE, one can use the phrase "possible world", as long as one takes it metaphorically.

Existence
I proposed that if in this actual world, all the properties of Hesperus disappeared, then Hesperus would also disappear. You made the point that even if Hesperus didn't exist in one possible world, it may still exist in another possible world.

I agree that even though Hesperus no longer existed in this actual world, it could still exist in a possible world.

However, these are different kinds of existences. The first refers to something physically existing in the actual world and the second refers to a possible world existing in the mind.

And in your example, we need a similar a priori principle which states that one measurement of 12,103km is necessarily the same as another measurement of 12,103km. — Metaphysician Undercover

I know The Red Sox will win their next game, I know The Eiffel Tower is in Paris and I know that I am looking at the colour red. The word "know" is being metaphorically, in that it has degrees of certainty, because language is inherently metaphorical

Similarly, the word "same" is being used metaphorically having varying degrees of certainty.

I know a priori, before using language, that language is metaphorical. The a priori principle is that language is metaphorical. that "same" is being used metaphorically.

What do you make of this, pg 177-8: — Mww

Kripke asks on page 177: "Is everything that is necessary knowable a priori or known a priori?". He writes page on 178: "So we certainly do not know, a priori or even posteriori, that every even number is the sum of two primes”.

There is a difference between knowable a priori and known a priori.

Taking a simpler example of cardinal numbers, 1,2,3,4 etc. If numbers are invented, and only exist in the mind and not the world, it is certainly true that not every cardinal number is known a priori, because there are an infinite number of them. However, if numbers are invented, every cardinal number is certainly knowable a priori.

We may not know something that is necessary a priori, even though it is knowable a priori.

That is, what is it that the sentence quoted above is about? It seems that it is about Hesperus. If one asks what it is that you are suggesting we remove the properties from, the answer is "Hesperus", and this is so even if the properties are removed. That is, in Kripke's terms "Hesperus" is a rigid designator, while it's various properties may not be. — Banno

I agree that "Hesperus" will continue to exist in language as a rigid designator even if all the properties of Hesperus disappeared from the world.

I will use the nomenclature that "Hesperus" exists in language and Hesperus exists in the world.

"Hesperus" may exist in language even if it doesn't exist in the world
1) Hesperus as an object in the world has millions of properties, most of which are unknown, but includes properties such as being 12,103km in diameter, having a solar year of 117 Earth days, has a central iron core, has a rocky mantle and has an atmosphere 96% carbon dioxide, 3% nitrogen, etc.

2) Hesperus has been named "Hesperus". If all the properties of Hesperus disappeared from existence, Hesperus would no longer exist. There is no example of an object existing in the world that doesn't have any properties. However, "Hesperus" would still exist in language. For a word in language to have meaning, it must have a set of properties, such as "being 12,103km in diameter", "having a solar year of 117 Earth days", etc. No word in language has meaning if it has no properties, for example a word such as "xxyyxx".

Nixon may be named "Nixon".
Similarly, Richard Nixon as an object in the world has millions of properties, most of which are unknown, such as born in 1913, family home in California, attended Whittier College, had a spot on his lung, a good debater, enthusiastic, etc. There are different approaches to how Nixon is named "Nixon".

1) For Ruth Barcan Marcus, proper names are tags which refer to an object which is the bearer of the name. Tags are directly referential and without descriptive content. For example, in the morning Nixon is tagged "Nixon". The tag could be a blue cross or a sheet of paper with the word "Nixon" on it. In the evening, the person with the tag is by definition "Nixon", even though the person may in fact be George McGovern.

2) For Bertrand Russell, in the morning Nixon is described as "born in 1913", "attended Whittier College" and "a good debater", Such a description, such a cluster of properties, is judged sufficient to pick out an individual uniquely. In the evening, the person that can be described as "born in 1913", "attended Whittier College", "a good debater" is by definition "Nixon", even though in fact it could be George Elmer Outland.

3) For Kripke, from (1), page 163, x may be identical to y and x may have the property F. For "Nixon" to be a rigid designator, for "Nixon" to be "Nixon" in all possible worlds, "Nixon" must have essential properties, such as having a spot on his lung. As with the example of the lectern, a non-essential property could be being in a different room. Whether a property is essential or non-essential can only be determined by human judgement, and then codified by social institutions, either fixed in a dictionary or similar or by daily use. For "Nixon" to be "Nixon" in all possible worlds, "Nixon" must have essential properties, such as having a spot on his lungs, where the property having a spot on his lungs is one designator of "Nixon", and as fixed in all possible worlds, is a fixed designator.

"Unicorns" exist in language and may or may not exist in the world.
1) I can define a "standard weight" as having the property 12.102kg, even before ever knowing whether or not 12.102kg exists in the world. Having the property 12.102 kg is an essential property of a "standard weight", is true in all possible worlds, and is a rigid designation. If I subsequently discover 12.102kg in the world a posteriori, I know a priori that it is a "standard weight", in that having the property 12.102kg is a necessary property of a "standard weight".

2) I can define a "unicorn" as having the properties the body of a horse and a single horn in its forehead even before ever knowing that unicorns exist in the world. Having the properties the body of a horse and a single horn in its forehead are essential properties of a "unicorn", and is true in all possible worlds as a rigid designation. If I subsequently discover in the world a posteriori the body of a horse with a single horn in its forehead a posteriori, I know a priori that this is a "unicorn", as having the body of a horse with a single horn in its forehead are necessary properties of a "unicorn".

Kripke's proposition that "identity statements are necessary" is true
1) Objects are observed in the sky. By observation, as "Phosphorus" has a diameter of 12,103km, and as "Hesperus" has a diameter of 12,103km, "Phosphorus" is identical in diameter to "Hesperus". Therefore, the identity statement "Phosphorus is identical in diameter to Hesperus" is true.

2) The property being visible is a priori defined as non-essential, and the property of diameter is a priori defined as essential. As "Phosphorus" has a diameter of 12,103km, having a diameter of 12,103km is a necessary property of "Phosphorus". As "Hesperus" has a diameter of 12,103km, having a diameter of 12,103km is a necessary property of "Hesperus". As "Phosphorus" has of necessity a diameter of 12,103km, and as "Hesperus" has of necessity a diameter of 12,103km, "Phosphorus" is of necessity identical in diameter to "Hesperus". Therefore, the identity statement "Phosphorus is identical in diameter to Hesperus" is necessarily true.

So, the fact that the lectern is made of wood, and not made of ice, is supported by the empirical observations. But empirical observations do not make it necessary that the lectern is made of wood and not ice. The necessity, (that it is necessary that the lectern is wooden and not made of ice), is derived from the a priori law of identity, which states that a thing cannot be other than it is. — Metaphysician Undercover

Well, one consequence is that, that x=y may be discovered empirically - examples are given - but has necessary implications. While this may seem obvious now, it is contrary to both Kant and Quine, fir different reasons. The notion that an empirical fact implies a necessary truth is one of the novelties of this paper. — Banno

The next step, then, says that there is nothing contained in the conception of P that does not belong to the conception of H, therefore, P and H are the same thing, or, that P is H is a necessarily true statement. We don’t need the experience those conceptions represent, only that all of them are thought to co-exist equally in one object. — Mww

Kripke wrote: "To state finally what I think, as opposed to what seems to be the case, or what others think, I think that in both cases, the case of names and the case of the theoretical identifications, the identity statements are necessary and not contingent."

If "Hesperus" is "Phosphorus", then "Hesperus" is of necessity "Phosphorus", but Hesperus is not necessarily Phosphorus

I will use the practice that "Hesperus" is a name in language and either refers to or is described by its properties such as "bright", "visible", "ringless". Hesperus is an object in the world and is its set of properties bright, visible, ringless.

My belief is that Hesperus has no existence over and above its set of properties, in that, if all the properties were removed, then there would be no object, as argued by FH Bradley.

There are two identity statements to consider, "Hesperus is Phosphorus" and Hesperus is Phosphorus.

Analytic propositions
The statement "bachelors are unmarried" is an analytic proposition that is true solely by virtue of its meaning. As it is true by definition, its truth is a priori. As Kripke argues that this lectern made of wood is necessarily made of wood, a bachelor is necessarily unmarried. The meaning of words is determined by social institutions, and are codified either in dictionaries or similar or in daily use, as Wittgenstein proposed.

Empirical Observations
John and Mary observe an object first in position A and then later in position B. John believes the body moves smoothly from A to B. Mary believes the body moves in a series of jumps from A to B. It is empirically impossible to determine who is correct, as we can only infer what happens between A and B, from Hume's constant conjunction.

Axioms
If John is in the majority opinion within his society, the social institutions may codify the concept that well-behaved objects move smoothly between two points as an axiom, as the axiom of "spatio-temporal continuity". Axioms are regarded as being established, accepted or self-evidently true, as with Newton's Laws of Motion. However, it is in the nature of axioms that the axiom of "spatio-temporal continuity" may or may not be true, in the sense of corresponding with facts in the world.

In fact, if an object was observed to jump through space-time, by definition it wouldn't be a "well-behaved object". As the axiom of spatio-temporal continuity is true independent of any empirical observation, it is an analytic proposition, its truth is a priori, and well-behaved objects by definition necessarily pass smoothly through space-time.

"Phosphorus" is necessarily "Hesperus"
Phosphorus and Hesperus are objects. Phosphorus is observed as an object in the east, is named "Phosphorus". and Hesperus as an object in the west, is named "Hesperus". From the axiom of spatio-temporal continuity, and under the assumption that "Phosphorus" and "Hesperus" are well-behaved objects, moving smoothly from the east to the west, it may be concluded that "Phosphorus" and "Hesperus" is the same object, the same Planet. Note that "Phosphorus" exists in language, not in the world. For convenience this single object may be named "Venus"

If when observing the sky, what was thought to be "Hesperus" was observed not to be moving smoothly, then by definition it couldn't be "Hesperus" but must be another object.

Two possible identity statements
As the identity statement "Hesperus" is "Phosphorus is based on the assumption that both "Hesperus" and "Phosphorus" are well-behaved objects, and as well-behaved objects necessarily follow the axiom of spatio-temporal continuity, then the identity statement "Hesperus is Phosphorus" is necessarily true.

However, as it is impossible to empirically determine that when an object in the world has moved from one position to another that there have been no jumps, it cannot be proved that Hesperus is Phosphorus, meaning that the identity statement Hesperus is Phosphorus is not necessarily true.

I suppose I must stop now. All the very best to everyone in the New Year, whichever part of the world you are in. :smile:

I'd say necessity is implicit in Leibniz's law. He's just making it explicit because he's about to challenge the notion that apriori=necessary, and aposteriori=contingent. He's going to show that there can be a statement that is known aposteriori, but is necessarily true. — frank

Necessity is being used in two different ways, between objects and between an object and its property.

Necessity between objects - between a lectern and a rostrum
As regards (4), necessity is being used between objects. He writes: "For every x and y, if x equals y, then, it is necessary that x equals y."

(1), (2) and (4) make use of Leibniz's Law, where "if two objects have all the same properties, they are in fact one and the same".

It seems to me that the use of the word necessary is redundant between objects, in that what does "if two objects have all the same properties, they are in fact necessarily one and the same" add to "if two objects have all the same properties, they are in fact one and the same"

Necessity between an object and its properties - between a lectern and its property wood
As regards the lectern, necessity is being used between an object and its properties, where he writes "So we have to say that though we cannot know a priori whether this table was made of ice or not, given that it is not made of ice, it is necessarily not made of ice.

As he persuasively argues, this lectern, if made of wood, is necessarily made of wood, because if not made of wood it would have been a different object, and it wouldn't have been this lectern.

However, necessity between objects is irrelevant to the question of necessity between an object and its properties. Therefore, necessity may be removed from (1), (2) and (4) without affecting his argument about necessity between an object and its property.

You're asking what earth shattering consequences follow from Leibniz's law. Kripke is just setting the stage to show off a contradiction. That's all. Keep going. — frank

No, Leibniz's Law states that if two objects have all the same properties, they are in fact one and the same.

My question is, why does Kripke need to add the word necessary to Leibniz's Law. What does "if two objects have all the same properties, they are in fact necessarily one and the same" add to "if two objects have all the same properties, they are in fact one and the same"

If x and y are identical, that means x and y are two different names for the same object. Like say John's nickname is Tweezer. x is John and y is Tweezer. now plug that into the argument. — frank

Taking x and y as proper names, whereby x is John, y is Tweezer. From (4), if John equals Tweezer, then it is necessary that John equals Tweezer.

But this is a logical implication, which says nothing about the reality of what is being expressed. I could say that if I lived on Mars, then I would open a pizzeria, or if I was a nuclear scientist then I would work on small modular reactors.

The possibilities are almost infinite. If John equals Bill Gates, then it is necessary that John equals Bill Gates, or if John equals Alison, then it is necessary that John equals Alison or if John equals the President of France, then it is necessary that John is the President of France, etc, etc.

We learn nothing significant from these logical implications, other than if two things are equal then they are necessarily equal, which seems a redundancy. Why not just say that two things are equal.

I am trying to understand the relevance of (1) and (4) on page 163, which is central to the article.

Kripke writes for any objects x and y:
(1) If x is identical to y, then if x has a certain property, so does y
(2) Every object is necessarily self-identical
(4) For every x and y, if x equals y, then, it is necessary that x equals y

Example one - let x by the Moon, and y be the Eiffel Tower

(1) If the Moon is identical to the Eiffel Tower, then if the Moon has a certain property, such as having a diameter of 3,476 km, so does the Eiffel Tower.
(4) For every x and y, if the Moon equals the Eiffel Tower, then, it is necessary that the Moon equals the Eiffel Tower.

Example two - let x be the object Hesperus, and y be an object that is not Hesperus

(1) If the object Hesperus is identical to an object that is not Hesperus, then if the object Hesperus has a certain property, then so does an object that is not Hesperus.
(4) For every x and y, if the object Hesperus equals the object that is not Hesperus, then it is necessary that the object Hesperus equals the object that is not Hesperus.

I may be misunderstanding, but I don't see any practical benefit to (1) and (4)

I hope this post isn't deleted as was my previous post for not being relevant to the OP.

“Hesperus is Phosphorus” is a pure analytic proposition, hence necessarily true. — Mww

I agree that analytic propositions are necessarily true, independent of any empirical knowledge. For example, "all bodies are extended", as the notion of extended is implicit in the notion of body.

If we are given two analytic propositions "Hesperus is Phosphorus" and "Hesperus is not Phosphorus", how do we know which is true, if the truth of an analytic proposition is independent of any empirical knowledge ?

It is therefore an analytical cognition, hence necessarily true, that Phosporus is Hesperus — Mww

I am surprised you say "hence necessarily true, that Phosphorus is Hesperus", as you also quoted Kant from the Critique of Pure Reason: "Secondly, an empirical judgement never exhibits strict and absolute, but only assumed and comparative universality (by induction); therefore, the most we can say is—so far as we have hitherto observed, there is no exception to this or that rule". (B4)

Running low on chilled Perrier over Christmas ? :smile:

I see something in the morning sky that is bright, visible, ringless and name it "Phosphorus". My knowledge that there is something in the sky is a posteriori. As I could have chosen any name, the connection between the name "Phosphorus " and something in the sky is contingent.

Henceforth using the convention that "Phosphorus" exists in language and Phosphorus exists in the world as a set of properties.

After looking at the sky on successive days, I infer that Phosphorus is Hesperus a posteriori. The connection between Phosphorus and Hesperus is contingent because it is an inference.

I can then say "I believe that Phosphorus is Hesperus". As I can only infer that Phosphorus is Hesperus a posteriori, the statement "I believe that Phosphorus is Hesperus" is synthetic

For convenience I rename both "Phosphorus" and "Hesperus" as "Venus". As I could have chosen any name, the connection between the name "Phosphorus" and "Venus" and between "Hesperus" and "Venus" are contingent. The statements "Phosphorus is Venus" and "Hesperus is Venus" are synthetic, as I can only know that Phosphorus is Hesperus a posteriori.

However, even though I can only know Hesperus and Phosphorus a posteriori, and as I can only infer that Phosphorus is Hesperus, then Phosphorus being Hesperus can only be contingent, the statement "if Phosphorus is Hesperus then Phosphorus is necessarily Hesperus" can still be true, as it is a logical implication.

Ruth Barcan Marcus argued that if x is y, then x is necessarily y. Although Barcan treats Hesperus as a proper name, a simple tag devoid of any further content, the truth of (if x is y then x is necessarily y) depends on whether a proper name such as Hesperus refers to an object Hesperus that exists in addition to its properties or refers to a set of properties that have been named.

If a proper name such as Hesperus refers to a set of properties that have been named, I can understand and agree that if x is y then x is necessarily y.

How can a necessary identity statement be derived from a contingent identity statement
Kripke wrote: "most philosophers have felt that the notion of a contingent identity statement ran into something like the following paradox."
(1) If x is identical to y, and if x has property F, then y has property F
(2) Every object is necessarily self-identical
(4) If x is identical to y, then x is necessarily identical to y

This idea was reinforced by Wiggins, who said: "Now there undoubtedly exist contingent identity-statements. Let a = b be one of them. From its simple truth and (5) [= (4) above] we can derive ‘☐ (a = b)’. But how then can there be any contingent identity-statements?"

Kripke argues that Phosphorus is Hesperus is a necessary identity statement
Kripke first writes that the common view is that Phosphorus is Hesperus is a contingent identity statement: "We may tag the planet Venus some fine evening with the proper name ‘Hesperus’. We may tag the same planet again someday before sun rise with the proper name ‘Phosphorus’.............When, at last, we discover that we have tagged the same planet twice, our discovery is empirical...........Surely no amount of a priori ratiocination on their part could conceivably have made it possible for them to deduce that Phosphorus is Hesperus."

However, Kripke later writes that he believes that Phosphorus is Hesperus is a necessary identity statement: "To state finally what I think, as opposed to what seems to be the case, or what others think, I think that in both cases, the case of names and the case of the theoretical identifications, the identity statements are necessary and not contingent."

Is (1) really a contingent identity statement ?
Kripke is saying that although (1) is a contingent identity statement, (4) can be derived from it, but (4) is a necessary identity statement, which seems a paradox.

However, is it really the case that (1) is a contingent identity statement ? (1) in being a logical implication, involving the terms if then, is, in Kripke's word "an a priori ratiocination", independent of empirical experience. However, by Hume's problem of constant conjunction, we can never know from empirical experience that Hesperus is identical to Phosphorus, we can only infer it.

For example, I observe Phosphorus at 9am and Hesperus at 9pm having a 180 degree separation. As I don't know what happened in the intervening period, I very weakly infer that Hesperus is Phosphorus. I observe Phosphorus at 9am and Hesperus at 9.01am having a 0.25 degree separation. As I don't know what happened in the intervening period, I very strongly infer that Hesperus is Phosphorus. No matter how close the period of time between my observations, I can never determine just from a posteriori empirical evidence that Hesperus is Phosphorus. The most I can do is infer through logical reasoning that Hesperus is Phosphorus. My logical reasoning is a priori in the sense that it is independent of empirical observation, although my logical reasoning is based on a posteriori empirical observation

We can never know from empirical evidence that Hesperus is identical to Phosphorus, the most we can do is make the judgement from logic and reasoning based on evidence that Hesperus is identical to Phosphorus. (1) is a statement about identity that is based on logical reasoning about empirical evidence, and therefore cannot be described as a contingent identity statement.

As both (1) and (4) are statements of logical necessity of empirical evidence, this doesn't support Kripke's statement that "This is an argument which has been stated many times in recent philosophy. Its conclusion, however, has often been regarded as highly paradoxical."

The suposition here is that an identity that we discover cannot be a necessary identity, and so there must be something amiss with the derivation (1-4). — Banno

I don't understand the logic of (1)

Kripke wrote: "for any objects x and y, if x is identical to y, then if x has a certain property F, so does y"

The sequence of (1) is:
A) starting with object x which has property F
B) knowing that object x is identical to object y
C) I then know that object y has the same property F as x

My supposition is that i) if there are no properties, then there is no object ii) if I cannot see any properties, then I cannot see any object.

However, this sequence seems more logical:
A) As object x has property F, I can know object x
B) I can only compare object x with object y if I know object y, and I can only know object y by knowing its property G. Therefore, I must know object y's property G before being able to compare object y to object x. If I didn't know object y's property, I wouldn't know that object y existed.
C) When comparing object x and its property F with object y and its property G, in discovering that property F is identical with property G, I then know that object x is identical to object y

IE, the problem with (1) is how can I know object x is identical to object y before I know object y's property?

Am I missing something.

The identity is within the object itself (as the law of identity states, it is the same as itself). The object's identity appears to any one of us as infinite possibilities because I can name it whatever I want. — Metaphysician Undercover

Does a single object as a thing in itself have infinite possibilities, or do we, as observers, see infinite possibilities in a single object ?

The first part of this essay explains why that's problematic. How do you respond to Kripke's point? — frank

Proper names refer to descriptions.

I wrote: "As the fact that John is in Paris is not part of the description of John's identity, John could equally well be in Rome. As the fact that John is doing some barbering work is not part of the description of John's identity, John could equally well be doing some plumbing work."

The Ancient Greeks saw in the evening something having the properties brightest natural object in sky, visible by naked eye during day, has no rings and tagged it Hesperus. They also in the morning something having the properties brightest natural object in sky, visible by naked eye during day, has no rings and tagged it Phosphorus. Pythagoras recognized that Hesperus and Phosphorus were in fact the same object, the planet Venus.

Kripke wrote: We may tag the planet Venus some fine evening with the proper name ‘Hesperus’. We may tag the same planet again someday before sun rise with the proper name Phosphorus’.” ....................“When, at last, we discover that we have tagged the same planet twice, our discovery is empirical, and not because the proper names were descriptions.”

Does the name Hesperus refer to Venus or describe its properties brightest natural object in sky, visible by naked eye during day, has no rings, appears in evening?

If Hesperus refers to Venus, does Venus refer to Venus or describe its properties brightest natural object in sky, visible by naked eye during day, has no rings, appears first in the morning and then in the evening?

Venus cannot exist independently of its properties, in that if Venus had no properties, Venus wouldn't exist. If I looked into the sky and saw no properties I would see no Venus. As Venus would not exist if it had no properties, Venus cannot refer to Venus but can only describe its properties. Venus cannot create itself by referring to itself.

IE, the proper names Hesperus and Phosphorus are descriptions, as in Russell's Descriptivism.

And this is the problem Kripke is addressing. If your identity is a description or definition, then it makes no sense to say you could have become a plumber. But we can say that. There's a possible world where you're a plumber, so it doesn't look like your identity can't be a description. So what is it? — frank

That John does some plumbing work is not part of his identity, in that neither is holidaying in Paris for ten days part of his identity.

Russell says that the name John is a description rather than a reference.

We can say "John is in Paris", "John is a barber", "John could running for the bus" or "John could be a plumber".

I am not defined as a person by where I live or what I do. The fact that John is in Paris, is doing some barbering work, running for the bus or doing some plumbing work is not part of his identity, and therefore not part of the name John, is not part of what the name John describes.

The sentence "John is a barber" illustrates the metaphorical aspect of language. John's identity is not that of being a barber, in that water is H2O, it means "John is doing some barbering work".

As the fact that John is in Paris is not part of the description of John's identity, John could equally well be in Rome. As the fact that John is doing some barbering work is not part of the description of John's identity, John could equally well be doing some plumbing work.

No more Kant. ↪Banno will take us to TPF court, and I can’t afford the fines. — Mww

Neither can I. :smile:

An object such as Phosphorus is a set of properties: brightest natural object in sky, visible by naked eye during day, has no rings, etc.

It depends whether the name Phosphorus is a reference or a description.

If Phosphorus refers to the planet Venus, through empirical observation, we can infer, along the lines of Hume's constant conjunction, that Phosphorus in the morning is the same body as Hesperus in the evening. From (1), x is Phosphorus, y is Hesperus, and as both x and y refer to the same body, x is identical to y.

If Phosphorus is a description, from Russell's Descriptivism, Phosphorus is a description of a set of properties, whereby Phosphorus has no existence over and above its properties. From (4), x is the set of properties bright, visible, no rings, y is the same set of properties bright, visible, no rings, and as both x and y refer to the same set of properties, x is necessarily y.

IE, as regards referring, two bodies having the same properties, but each body existing over and above its properties, are contingently the same a posteriori. As regards description, two sets of the same properties are necessarily the same a priori.

identity is within the thing itself, while logical necessity is within the human mind. Therefore identity will always present itself as infinite possibility — Metaphysician Undercover

How can an object such as an apple, having a self-identity, have infinite possibilities ?

Sorry, but if you read the paper, Kripke posits the logicality on the empirical finding that Hesperus is Phosphorus. — Shawn

Silly me, to think I posted a comment before reading the article.

Kripke starts off by writing that it is often taken for granted that contingent statements of identity are possible: “How are contingent identity statements possible?” This question is phrased by analogy with the way Kant phrased his question “How are synthetic a priori judgments possible?” In both cases, it has usually been taken for granted in the one case by Kant that synthetic a priori judgments were possible, and in the other case in contemporary philosophical literature that contingent statements of identity are possible."

He later writes that he believes that identity statements are necessary and not contingent: "To state finally what I think, as opposed to what seems to be the case, or what others think, I think that in both cases, the case of names and the case of the theoretical identifications, the identity statements are necessary and not contingent. That is to say, they are necessary if true; of course, false identity statements are not necessary. How can one possibly defend such a view? Perhaps I lack a complete answer to this question, even though I am convinced that the view is true."

Although Kripke writes "x has a certain property F", one questions how this fits in with Russell's Descriptivism where x is its set of properties.

So maybe Kant’s term isn’t a mere idiom after all. Which is neither here nor there with respect to the thread. — Mww

As Kripke mentions Kant's "synthetic a priori judgements" in the second sentence of his chapter, and as @Banno includes the same term in his OP, the meaning of "synthetic a priori judgements" cannot be irrelevant to the thread, otherwise, why mention it in the first place.

Kantian transcendental idealism, not needing any inverted commas — Mww

It deserves commas as it is a name, not a description. First, Kant was an empirical realist. Second, in edition B of the Critique of Reason, Kant inserted a refutation of idealism. Third, also in edition B, Kant said "Transcendental Idealism" was a poor choice of name. Fourth, there is debate as to how we can have transcendental knowledge, and whether what Kant calls transcendental knowledge is no more than knowledge by inference.

Kripke wants to unite the contingent with identity, which Kant deemed, if not impossible, then at least logically insufficient in regard to a brand new philosophy. — Mww

Kripke didn't want to unite contingent with identity, he wanted to unite necessity with identity. As he writes "According to this view, whenever, for example, someone makes a correct statement of identity between two names, such as, for example, that Cicero is Tully, his statement has to be necessary if it is true."

Sorry, but it's entirely legitimate to ascribe the predicate of existence of Mary in a possible world. Why is there so much confusion about counterpart theory or possible world semantics? — Shawn

The confusion is not about possible world semantics, the confusion is about the mixing up of metaphoric and literal meaning.

There is no confusion as to what "Mary exists in a possible world" means, as there is no confusion as to what "Mary has a heavy heart", "Mary is down in the dumps" or "Mary is as happy as Larry" mean.

There is nothing wrong with using poetic or metaphoric language, as such words are an integral part of language. The problem arises when poetic and metaphoric language becomes mixed up with language that is intended to be literal, after all, this is a philosophy forum where the meaning of words is important, not a poetry forum.

A possible world may or may not exist. If the possible world doesn't exist, Mary cannot exist in it, so "Mary does not exist in a world that does not exist" is true. If the possible world exists, then it is not a possible world, it is an actual world, so "Mary exists in a world that exists" is true. As a possible world is a modal world, if Mary exists within it, then Mary's existence should also be a modal existence. Therefore it would be better say "Mary may exist in a possible world", "Mary might exist in a possible world", "Mary can exist in a possible world" or "Mary could exist in a possible world".

Metaphors do not provide good premises for logical proceedings. — Metaphysician Undercover

True. In the sentence "Mary exists in a possible world", "exists" means "could exist", so the sentence is incorrect. It should be "Mary could exist in a possible world".

However, if it was a deliberate intention to use "exists" as meaning "could exist", then this would have been a valid metaphorical use of language. Confusing, but valid.

There simply isn't any objects in logical possibilities (possible worlds), and nobody actually believes that there is, despite the fact that many people like busycutter, and Banno, argue that there is. — Metaphysician Undercover

In the expression "an individual exists in a possible world", the word "exist" is being used metaphorically, not literally, in the same way that it is being used metaphorically in the sentence "I existed on my desire for vengeance". The problem with a metaphorical language is that meaning depends on context and if the context is vague then the meaning is vague.

The problem is, that if we said "an individual exists in our actual world", are we still using "exists" metaphorically or literally ?

And then again, where does this "actual world" exist. I think it exists in the mind, though others would disagree. But even "the mind" is a metaphor.

IE, an individual exists in a possible world metaphorically, a possible world is a metaphor, exists in our actual world is being used either metaphorically or literally, and our actual world exists either metaphorically in our minds or literally as mind-independent.

The key topic of the paper is 'How are contingent identity statements possible?", or as Kant may have put it, "How are synthetic a priori judgements possible?"

Kant's synthetic a priori judgement is more in agreement with a Kripke necessary identity statement than a contingent identity statement, though Kant's synthetic a priori is more about knowledge by acquaintance than Kripke's knowledge by description.

Contingent identity statements versus Kant's a priori judgements
Kripke writes in the introduction: "“How are contingent identity statements possible?” This question is phrased by analogy with the way Kant phrased his question “How are synthetic a priori judgments possible?”. However, he later writes: "To state finally what I think, as opposed to what seems to be the case, or what others think, I think that in both cases, the case of names and the case of the theoretical identifications, the identity statements are necessary and not contingent"

As regards contingent identity statements, such as "Hesperus is Phosphorus", from Hume's principle of constant conjunction, we logically infer that Hesperus is Phosphorus, and therefore is logically contingent rather than logically necessary.

As regards Kant's synthetic a priori judgement, which is about a priori pure and empirical intuitions, it is not about the identity statement "the postbox is red", rather it is about the identity statement "this is red", and as a priori "knowledge", logically necessary.

Synthetic a priori judgements
I've always thought the phrase "synthetic a priori" was wrong, as it mixes two fundamentally different things. To my understanding, within language are synthetic and analytic propositions, some knowledge can be a priori and some a posteriori and within logic is the necessary and contingent. It is as if one said "anger is a heavy thing", not to be understood literally but metaphorically.

"Synthetic a priori" means no more than humans are born with certain innate abilities, such as the innate ability to be able to distinguish between a loud and quiet noise, something hot and something cold, etc. Children don't need to go to school to be able to distinguish between a sweet and sour taste, as this is instinctive.

The term a priori knowledge is not correct either, in that humans don't have a knowledge of the colour red before seeing it for the first time, but they do have the ability to see the colour red before ever seeing it for the first time. As an analogy, a wine glass passively shatters when the frequency of an opera singer's voice matches the natural resonant frequency of the wine glass, it is not the case that the wine glass is an active participant.

The term "synthetic a priori" should be understood as an idiomatic expression rather than as a literal guide to Kant's doctrine of "transcendental idealism".

Better copy of Identity and Necessity
There is a web site, but one needs to sign in through your library.
https://academic.oup.com/book/36436/chapter/320710138

Anyway, I have to go now to see what Santa Claus has left under the tree.

The answer I accept is that they exist outside of spacetime. In particular, mathematical objects exists outside space time — Art48

Where does Kripke's Identity and Necessity say that numbers exist

As regards Kripke's chapter on Identity and Necessity in his book Philosophical Troubles: Collected Papers, Volume 1, he writes:

"Independently of the empirical facts, we can give an arithmetical proof that the square root of 25 is in fact the number 5, and because we have proved this mathematically, what we have proved is necessary. If we think of numbers as entities at all, and let us suppose, at least for the purpose of this lecture, that we do, then the expression ‘the square root of 25’ necessarily designates a certain number, namely 5. Such an expression I call ‘a rigid designator’. Some philosophers think that anyone who even uses the notions of rigid or nonrigid designator has already shown that he has fallen into a certain confusion or has not paid attention to certain facts. What do I mean by ‘rigid designator’? I mean a term that designates the same object in all possible worlds."

On the one hand, he writes that numbers necessarily exist in all possible worlds, meaning that numbers ontologically exist in the world. However, he doesn't specify whether this world exists in the mind or is mind-independent. On the other hand, he writes that we are able to manipulate numbers independently of the empirical facts, meaning independently of any mind-independent world.

For Kripke's Identity and Necessity, as numbers ontologically exist in the world, and as we can manipulate numbers independently of any mind-independent world, the world he is referring to must be in the mind. IE, Kripke's Identity and Necessity infers that numbers exist in the world of the mind, not in a mind-independent world.

If numbers did exist outside our three dimensions of space and time, one wonders how a calculator physically existing in space-time when adding numbers is able to access numbers existing outside of space-time.

If numbers did ontologically exist mind-independently, as numbers exist as relations between individuals, one wonders how the ontological existence of relations in a mind-independent world can be justified.

But the lectern is identified via it's description - being wood - so in effect he is saying "the wooden lectern is necessarily made of wood". — Banno

Where do lecterns exist ?

Kripke gives the example of "here is a lectern" as a description of something made of wood, something that can only be known a posteriori and is an essential property.

However, what happens when we move from the demonstrative pronoun to the definite article.

There is no single property that lecterns have. Some are made of wood, some of metal, some have a flat base, some a legged base, some are grey in colour, some brown, etc. But as Wittgenstein pointed out, objects such as lecterns do have a family resemblance, such that a human observer can judge the difference between a lectern and a non-lectern.

As lecterns have no essential property, then lectern is more like a rigid designator than a description, as Mary as a name is a rigid designator, having no properties.

If lecterns exist only as a family resemblance between "this lectern" and "that lectern", and family resemblances is a human judgement, how can lecterns exist in the world, unless family resemblance is also something that exists in the world ?

In place of this I offer a picture of "two" as part of a family of activities that we engage in together — Banno

I agree that "two" is part of a family of activities that we engage in together, but where did "two" originate, allowing us to use it in our activities.

In answer to the question what are objects such as apples and what are numbers such as two, I can refer to the Standard Model, Kant's Critique of Pure Reason, Russell's On Denoting and Wittgenstein's Tractatus and Philosophical Investigations.

Within the Standard Model, in the world are fundamental particles, fundamental forces, time and space.

We are born with certain innate abilities, which have evolved over 3.5 billion years, elementary concepts such as the ability to distinguish between time and space, green or red, round or square, rough or smooth, tart or sweet, hot or cold, acrid or fragrant, loud or quiet, etc. In Kant's terms, from the Critique of Pure Reason, these are a priori pure and empirical intuitions. His term for the mind's ability to combine distinct parts into a unified whole is known as unity of apperception. Given innate elementary concepts, we can then discover correspondences between them and what we observe in the world.

From Russell's On Denoting, these innate elementary concepts may be combined by the mind into compound concepts. For example, the elementary concepts circular, sweet and red/green may be combined into the compound concept of apple.

From the Picture Theory of Wittgenstein's Tractatus, it may be discovered that these elementary and compound concepts in the mind correspond with what can be discovered in the world, and once a correspondence has been discovered, that concept can be named. For example, in discovered that our elementary concept of red corresponds with pictures of red in the world, we can name this concept "red".

From Wittgenstein's Philosophical Investigations, these named elementary and compound concepts can then become part of a coherent language. For example, in the statement "an apple has the properties circular, sweet and red/green in colour".

Using the above, an object, such as an apple, is a set of related properties, such as circular, sweet and red/green. But as relations don't ontologically exist in the world, apples can only exist in the mind. Similarly, a number, such as two, is a relation between two individuals. But as relations don't ontologically exist in the world, the number two can only exist in the mind. Therefore, objects such as apples and numbers such as two exist only the mind as compound concepts.

In answer to the question posed in the OP, We Are Math?, the answer is yes, we are math.

I think Wittgenstein's approach can wholly replace Russell's — Banno

Russell and Wittgenstein fundamentally differ in that Russell's logical atomism requires both knowledge by acquaintance and description, whereas for Wittgenstein's meaning as use, knowledge by description is sufficient.

The question is, is it possible that Wittgenstein's approach includes knowledge by acquaintance.

I don't think it does. As he wrote in On Certainty, the proposition "here is a hand" is more about how the proposition is used rather than making an empirical claim about hands in the world. It may be objected that Wittgenstein's language games are circular, in that the meaning of the word comes from the game. As there is no external link, there is one problem of how to choose between different games, and another problem that there is no allowance for discourse between different games. For example, an atheist using one language game may not be able to criticise a religious believer using a different language game. A particular language game within a particular society may well be coherent, but such a language may not correspond with the world that the society lives within.

IE, Wittgenstein's language game of knowledge by description includes no link to knowledge by acquaintance.
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The capacity to differentiate colour is there, but it is trained by our interaction with others. It follows that what is to count as an "elementary colour" is not entirely innate, but learned by interaction with the world. Similarly, what counts as an elementary concept, a simple, is dependent on one's interactions with the world, including other people, and language. — Banno

Russell distinguished between two ways of thinking about things. He made the contrast between knowledge by acquaintance and knowledge by description, those things we think about directly and those things we think about indirectly. Knowledge by acquaintance includes sense data, universals, relations and oneself. As regards universals, he wrote "Not only are we aware of a particular yellows, but if we have seen a sufficient number of yellows and have sufficient intelligence, we are aware of the universal yellow"

The question is, are Russell's universals in fact not innate but learned by interaction with the world. If so, then Russell's knowledge by acquaintance becomes part of Wittgenstein's knowledge by description

I don't think they are. Consider those elementary concepts such as green or red, round or square, rough or smooth, tart or sweet, hot or cold, acrid or fragrant, loud or quiet, etc. I may have learnt many things over the past few years, but my perception of green, for example, one of these elementary concepts, has remained constant throughout my life. I certainly may have learnt more about the occurrences of green within the world, grass is green, traffic lights become green etc, but my innate ability to see green has not changed since the day I was born.

I agree that even in the absence of green I have the potential to see green, but this potential hasn't been taught, it was something I was born with. It is true, however, that I had to be taught that the name of my elementary concept of green is "green". It is also true that even though I have the potential to see green, I have to interact with the world, otherwise there would be no green for me to see. .

My ability to see green is innate, though I can learn by interactions with the world its occurrences in the world and can learn by interactions with other people its name.

IE, Elementary concepts such the innate ability to see the colour green cannot be learnt by description within a language game.
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One of the major differences between the Tractatus and the Investigations was Wittgenstein's realisation that what is to count as a simple is dependent on the task at hand. The meaning of "simple" varies with use. — Banno

We can call our perception of the colour green a "simples". Is Russell correct in treating such a simple as independent of context and as knowledge by acquaintance or is Wittgenstein correct in treating such a simple as being dependent on context and as such knowledge by description.

In fact, Russell and Wittgenstein are talking about different things. Russell's simples are within the philosophy of the mind and epistemology, where such simples have neither meaning nor can be true or false, Wittgenstein's simples are within language, can have meaning and can be either true or false. As noted by the SEP article on Wittgenstein's Logical Atomism "The so-called “colour-exclusion problem” is a difficulty that arises for the Tractatus’s view that it is metaphysically possible for each elementary proposition to be true or false regardless of the truth or falsity of the others (4.211)."

Wittgenstein's simples as being within language cannot be independent of the context they are within, as Wittgenstein explains, whilst for Russell, simples in existing independently of meaning, truth and falsity can be independent of any context they are in.

IE, Wittgenstein's approach of knowledge by description within language cannot include Russell's knowledge by acquaintance outside of language.

In summary, Wittgenstein's approach cannot wholly replace Russell's, as Wittgenstein's approach doesn't include knowledge by acquaintance, which Russell's does.

All that presupposes “I think” has some irreducible meaning. Whether we actually do think or not, is irrelevant, insofar as the very seeming of it requires an account. — Mww

I know that I can think of an apple and I know the concept of an apple, therefore thoughts and concepts must exist.

Kant Critique of Pure Reason A108 - "Just this transcendental unity of apperception, however, makes out of all possible appearances that can ever come together in one experience a connection of all of these representations in accordance with laws. For this unity of consciousness would be impossible if in the cognition of the manifold the mind could not become conscious of the identity of the function by means of which this manifold is synthetically combined into one cognition."

Consciousness, the unity of apperception in the mind is mysterious.

It seems that when the mind perceives a whole, which may be a set of parts, the mind is able to concurrently perceive each possible combination of parts as a unity, where each unity is distinct and irreducible. For example, the mind when perceiving a set of parts such as circular, sweet and red/green is able to perceive these parts as a distinct unified whole, an apple, and having a unity, irreducible. It will also be the case that when the mind perceives each possible combination of parts making up the whole, such as circular and sweet, the mind will also treat that combination as a distinct unified whole, and having a unity, irreducible

Similarly, each thought, such as the thought of an apple, is a distinct unified whole and as a unified whole is not only irreducible but has meaning.

Ok, but I take exception to compound conceptions. I know what is meant by it, but I think it a misunderstanding. Some thing, with a set of properties in the form of conceptions subsumed under it, is still represented only by its own conception. — Mww

I can have the concept of a single thing such as the colour yellow, or I can have the concept of an apple, which is a set of things, round, sweet and red/green.

When just looking at something round, my concept will be of something round, when just tasting something sweet, my concept will be of something sweet, when just looking at something red/green, my concept will be of something red/green.

However, what happens when I experience all of these things at the same time, something round, sweet and red/green, ie, an apple ?

Either i) I experience a single concept made up from a set of concepts, a unity of apperception, or ii) I will experience a set of concepts, discrete and separate ?

By "compound concept" I mean compound in sense i) rather than sense ii).

However, there may be a more technical term than elementary concept and compound concept.

So we have here two differing approaches to the nature of the apples being purchased at our grocer. On the one hand we have Russell's view that the apple consists in a concatenation of "constituents with which we are acquainted", something like "Green or red and round and waxy and smooth and tart or sweet". On the other hand we might set out the nature of an apple by setting out the roles it might play as we go about our daily activities: The thing we pick, sell, bite, stew, bake in a pie and so on. — Banno

I agree that Russell's work on denoting is not without criticism, and Wittgenstein's meaning as use, the language game and family resemblances are important aspects. But perhaps both are needed to arrive at an understanding of the process of buying two apples.

The mind and the language it uses need both Russell's elementary concepts and Wittgenstein's compound concepts

The elementary concepts of "logical atomism" and the compound concepts of "meaning as use"
At the moment , it seems to me that apple as a thought in the mind and "apple" as a word in language may be understood as a combination of the elementary concepts of Russell's logical atomism and the compound concepts of Wittgenstein's meaning as use, in that neither is sufficient by itself, but each provides an essential part of the whole.

Elementary concepts
Following Russell, there are things with which we are directly acquainted: green or red, round or square, rough or smooth, tart or sweet, hot or cold, acrid or fragrant, loud or quiet, etc, and the mind can judge the difference between these binary opposites.

In Kant's terms, trying to add a chilled Perrier moment, the human ability to judge between such binary opposites is an a priori intuition, an epistemic condition, an innate ability we are born with. It is the product of 3.5 billion years of life evolving in synergy with the world within which it finds itself, an Enactivist understanding whereby a person's understanding of the reality they observe in the world has been determined by the evolution of life within the world before they were born. Sentient life is a physical expression of the world it finds itself within. IE, the function of schools is not to teach children how to distinguish between green or red, round or square, etc as these abilities are innate, but without these innate abilities, being taught more complex concepts would be impossible.

Compound concepts
Given these simple concepts we can then combine them in various ways into compound concepts. Any combination is possible, but some combinations are more useful than others. For example, I have discovered that the combination round, sweet and red/green is of particular use, in that I have discovered that the apple is beneficial to my existence in the world. For convenience, rather than keep saying "pass me the thing that is round, sweet and red/green", I could name it "apple" and say "pass me the apple". I could equally as well have named it "camel", and said "pass me the camel", with the intended meaning pass me the apple, but as it has turned out, in the English language, something round, sweet and red/green has been named "apple".

But any possible combination of elementary concepts can be named, regardless of whether the particular combination is useful or not. For example I could name the combination green, square and smooth as "grasquim", not something that I have ever discovered to be useful to me.

The "apple", as a compound concept, exists as a relationship between the elementary concepts round, sweet and red/green. "Grasquim", as a compound concept, exists as a relationship between the elementary concepts green, square and smooth. As Russell in On Denoting showed, neither "apple" nor "grasquim" refer to an individual having its own existence, but describe the parts, the properties, that make it up. As both "grasquims" and "apples" have the same existence as a set of properties, if we said that "grasquims don't exist", then we would have to say that "apples don't exist", and if we said that "apples exist", then we would have to say that "grasquims exist". But Russell's On Denoting overcomes this problem in that neither "grasquims" nor "apple" are subjects that are predicated as either existing or not existing, rather, they are descriptions of a set of properties, not individuals being referred to.

It may well be that the "apple" plays an important role in our daily activities, and the "grasquim" plays absolutely no role in our daily activities, but both "apple" and "grasquim" have a meaning, in that "apple" means round, sweet and red/green and "grasquim" means green, square and smooth.

When Wittgenstein says "meaning as use", " meaning" can be interpreted in more than one way. In one sense of meaning, the "grasquim" has meaning even though it has no use. In another sense of meaning, the "grasquim" has no meaning because it has no use, in the same way that someone could say " travelling to Mars doesn't mean anything to me", knowing that they will never travel to Mars. Perhaps Wittgenstein's "meaning as use" refers to the second interpretation.

Kripke criticised Russell's Descriptivist Theory using a modal and epistemic argument
As regards the epistemic argument, Kripke pointed out the flaws in Russell's treatment of compound concepts as being able to be known a priori, inferring that compound concepts such as "government" can be known a priori, which is certainly not the case. Kant is different, in that Kant treats elementary concepts as being a priori, not compound concepts, which is certainly the case, in that humans are born with the innate ability to distinguish green from yellow, for example.

As regards the modal argument, Kripke said names should be rigid designators, true in all possible world. This requires that the elementary concepts building up a compound concept must be necessary rather than contingent, in that "apple" is true in all possible worlds, providing the elementary concepts building it up are round, sweet and red/green and not round, sweet, red/green and on the table.

Both Russell's "logical atomism" and Wittgenstein's "meaning as use" are needed
In summary, humans for survival and development within the world need both compound concepts and the elementary concepts they are built from. Some compound concepts mean more to us than others because of the use we can make of them, in that the "apple" means more to us than "grasquim", ie, Wittgenstein's "meaning as use"

Yet, we wouldn't have compound concepts without the elementary concepts they are built from, the constituents with which we are acquainted, as it were those fundamental indivisible atoms on which the rest of matter is made, where such atoms have been discovered through logical reasoning rather than intuitive feeling, ie, Bertrand Russell's "logical atomism".

I tried to include a reference to Kant's philosophy of mathematics and a priori intuitions, but I know Banno isn't a fan.

Hence any private mental stuff is irrelevant to the meaning of "two". — Banno

I'm sorry about the length of reply.

Buying two apples needs both private concepts and public names

Does meaning is use have implications for the status of numbers
I agree that the meaning of "two" is how the word "two" is used. But what is the implication for the status of numbers ?

Objects are publicly named in performative acts
Prior to the interaction between me and the shopkeeper, it is necessary that we both have the same chart. Alongside the picture of one apple the name "one", alongside a picture of two apples the name "two", alongside the picture of an apple the name "apple", etc.

However, it could well have been that alongside a picture of one thing was the name "red"
and alongside the picture of two things was the name "yellow", but we can assume that in some prior performative act by someone in authority, a picture of one thing had been named "one" and a picture of two things had been named "two", thereby establishing a public language.

I wake up hungry and have the image of two apples in my mind. I compare the image in my mind to the pictures on the chart, and see the name "two". I go into the shop, tell the shopkeeper "two apples", who looks at the chart, and by comparing the picture on the chart to the image of what is in the bin, is able to give me two apples.

The number "two" is redundant in this transaction
In fact, the number "two" is redundant in this transaction. I could just have shown the shopkeeper the picture of two apples. Numbers may be convenient, in that the number "two hundred" is more convenient than a picture of 200 things, but fundamentally, within this transaction, what can be done in numbers could equally well have been done in pictures.

Perhaps this is the point of Hartry Field's nominalism, an opponent of the Quine-Putnam Indispensability Argument for mathematical Platonism. Field rejects the claim that mathematical objects are indispensable to science, arguing that it is possible to reformulate scientific theories in such a way that mathematical objects are replaced by relationships.

The transaction couldn't happen without private concepts
What is fundamental in using numbers is our ability to compare two images, either a memory of an image with a picture on a chart, or a picture on a chart with an image of something in the bin. Yet it is inevitable that the image of two apples in my mind, the picture of two apples on the chart and the image of two apples in the bin will be different. A judgement will need to be made that two things having some differences and some similarities both fall under the same concept, in that we have the concept apple even though no two apples are the same. It is an inherent human ability to be able to look at several different things and discover a commonality within them, and discover that they fall under the same concept.

The transaction would not have been possible if either me or the shopkeeper had no concept of either an apple or the number two, in that without concepts we would be still sitting in the corner of the room motionless. It may well be that my private concept of "two" is actually three, and the shopkeeper's concept of "two" is actually four, but we will never know, and is in a sense irrelevant. What is essential is consistency of concept, in that yesterday when I saw "two" my concept was of three, today when I see "two" my concept is of three, and tomorrow when I see "two" my concept will still be of three.

It is true that for the transaction to proceed, no reference is ever made to our private concepts, in that the shopkeeper does not need to know my private concepts of either apple or two, but it is equally true that the transaction could never have happened if either of us had no private concept of either apple or two. The process of buying two apples needs both a private aspect and a public aspect. As regards the private aspect, each participant must have a private concept of both two and apples, and as regards the public aspect, there must have been a priori performative act by someone in authority linking a picture of an apple to the name "apple" and linking a picture of two objects to the name "two".

Concepts don't exist in a mind-independent world
As regards the public aspect, two objects are linked to the name "two". What exactly is this link? It is the same problem Achilles had with the tortoise. When the tortoise started to move his castle diagonally, Achilles said that that move wasn't in the rules. The tortoise replied "where is the rule that I have to follow the rules". Similarly, there is the public rule that what is pictured is given the name it is linked to, such that when an apple is linked to "apple", then "apple" means apple, and when two objects are linked to "two", then "two" means two objects. But as the tortoise would say "where is the rule that a name means what it is linked to"

These linkages are relations, and as relations don't ontologically exist in a mind-independent world, then neither do these linkages. But as we do perceive linkages in the world, and as these linkages don't exist in the world, they can only exist in our minds, meaning that things like apples and two can only exist in our minds.

How is the our concept of apple related to our word "apple", and our concept of two related to our word "two". In On Denoting, Russell argued that words such as "apple" and "two" are not referring terms, in that they are not referring to an individual having its own existence, but is in fact describing those properties or parts that it is composed of. As Russell wrote "Every proposition which we can understand must be composed wholly of constituents with which we are acquainted", where those things we can think about directly are sense data, universals, relations and oneself. Similarly concepts such as apple and two are not referring terms, referring to an individual having its own existence, but in fact describe the properties or parts that it is composed of and with which we are directly acquainted. Therefore, our concepts and words do the same job in describing the properties they are composed of and which we are directly acquainted.

The process of buying two apples needs both a private and public aspect
In summary, the process of buying two apples could not happen without both a private and public aspect. If either me or the shopkeeper had no private concept of either apple or two, we would remain motionless in the corner, unable to act. If either apple or two had not been publicly named in a performative act "apple" and "two", I wouldn't be able to communicate with the shopkeeper.

Regardless of what X is (in this case metaphor) I can't see this as anything other than bullshit. — khaled

Bye.

I would say a mind requires variation to exist and not the other way around, for there must be variation before a mind capable of distinguishing variation exists (specially when such a mind collects information by detecting changes in the environment through the senses of its body). If there is variation and there is "stuff" then there are patterns automatically. — Daniel

I agree that within a mind-independent world variations exist, and it is these variations that a sentient being observes as patterns. However, "pattern" is a word that exists in language, and the question is, what exactly does this word correspond to in the world it is describing.

My starting position is my belief that elementary particles and elementary forces do exist in a mind-independent world of time and space. These elementary particles and forces combine to form what we know as patterns, rocks, water, etc.

What we see in the world as a whole is a set of parts. In a sense, a "part" is a metaphorical rather than real entity, in that parts have parts which have parts, etc until we arrive at the elementary particles.

In treating a pattern, rock, water as a whole made up as a set of parts, in accepting that the parts exist mind-independently, the question is, does the whole also exist mind-independently, or only in the mind of an observer.

I agree that a water molecule surrounded by water molecules will behave differently to a water molecule surrounded by rock molecules, ultimately because the behaviour of an elementary particle is affected by the elementary forces acting upon it, and it is this difference in behaviour that eventually accounts for what we observe as patterns, water, rocks, etc.

One doubt I have that patterns, etc exist mind-independently (though I have another) stems from the problem of naming. For example, a sentient being can judge when a rock is worn away and becomes a pebble, in that a sentient being can judge the difference between a rock and a pebble. But if rocks exist mind-independently, and pebbles exist mind-independently, when a rock is slowly worn away to become a pebble, at what stage does the set of molecules change from existing as a rock to existing as a pebble. I agree that as outside observers we could judge, but what is there in a mind-independent world to make that same judgement

Similarly for patterns, as a pattern slowly becomes a non-pattern. Midway between a set of molecules existing first as a pattern and then as a non-pattern, what in a mind-independent world can determine that the set of molecules has changed from existing as a pattern to existing as a non-pattern. My belief is that if there is nothing in a mind-independent world that can determine when a pattern becomes a non-pattern, then neither can there be anything to determine when something exists as a pattern rather than a non-pattern.

Basically, "patterns", "rocks" and "water" exist as names within language, and as Bertrand Russell pointed out in On Denoting, names don't refer to an individual having an independent existence, but are definite descriptions, quantificational expressions, of the parts that make them up.

And what are these "laws of nature" exactly in your view? Given that you do not believe patterns exist ontologically. — khaled

The phrase "laws of nature" is a metaphor.

The problem of trying to describe literal truths in the world using language is that language is inherently metaphorical. For example, just taking the sentence "Patterns we see in nature are inevitable if things move and the laws of nature are constant", the following words are metaphors - patterns, we, see, in, nature, are, inevitable, if, things, move, and, the, laws, of, constant.

If the meaning of "two" is a private concept in my mind, and is different to a private concept in your mind, then you and I literally do not share the same concept of two. — Banno

It could well be, as in theory no one other than me knows what's in my mind. However, in practice, as we share more than 99.9% of our DNA, and we both have the same ancestor, "mitochondrial Eve", I would infer that our private concepts are very similar.

So those private aspects of the concept two make no difference, and it is only the public aspects that have a place in our affairs. — Banno

I agree. Elaborating, given something in the world that has been given the communal name "two", it may well be that my private concept of this something is actually three and your private concept of the same thing is four.

However, when I see something named "two" and have the private concept three, I will interact with the world in a particular way. Consistency is important, in that the next time I see something named "two", even though my private concept is still three, I will interact with the world is the same way as before. As you say, as regards my interactions with the world, " those private aspects of the concept two make no difference".

Do the rings of a tree give us information or just facts and from there it takes an intelligence to make the facts meaningful information. — Athena

I would say that the rings of a tree give us facts and from there it takes an intelligence to make the facts meaningful information.

As Wittgenstein wrote in Tractatus: i) “The facts in logical space are the world” and ii) "A logical picture of facts is a thought"

not only are patterns instantiated in nature, but nature is also receptive to patterns — Pantagruel

Patterns we see in nature are inevitable if things move and the laws of nature are constant.

If a particular event ends as it began, for example, the earth travelling around the sun, given the constancy in the laws of nature, and all things being equal, the same event will happen a second time, and a third, etc, and this is a pattern.

Nature is receptive to patterns in that at one time the metronome didn't exist in nature, but once a part of the natural world exhibits a pattern in its behaviour.