You had me worried for a moment.

current principles of physics allow that the second body, the receiving body, is not necessary — Metaphysician Undercover

True, no second body is necessary for thermal radiation, in that the Sun has no "awareness" that the thermal radiation it emits will hit the Earth 8min 20sec later. The thermal radiation could continue into space without ever hitting a second body.

Radiation of heat from an object........heat can radiate — Metaphysician Undercover

Heat doesn't radiate. Heat is the transfer of thermal energy between two bodies.

There are three modes of heat transfer, conduction, convection and radiation. The transfer of heat by radiation needs no material carrier. Thermal radiation is electromagnetic radiation.

It is incorrect to speak of the heat in a body, because heat is restricted to energy being transferred.

Kirchhoff's law of thermal radiation is regarding a body at temperature T radiating electromagnetic energy. The body is not radiating heat, it is radiating electromagnetic energy.

The sun doesn't radiate heat, it radiates thermal radiation. If this thermal radiation doesn't hit a second body, as heat is the transfer of thermal energy between two bodies, no heat will be transferred.

When Theodore Parker said "Cities have always been the fireplaces of civilization, whence light and heat radiated out into the dark", he was using it as a poetic metaphor.

Well, no. Heat is measured in Joules. It is the flow of energy from place to place. I don't think we can finesse that away. — Banno

I wrote "heat is not energy". The consequence is that Kripke's statement “Heat is the motion of molecules.” is not true.

1) Heat is the transfer of thermal energy between two bodies, not the flow of thermal energy between two bodies.

Water when flowing along the Danube can exist independently of either Vienna or Budapest, however, heat cannot exist independently of the two bodies between which it is being transferred.

2) Are you saying that because heat is measured in joules and energy is measured in joules, then heat is necessarily energy ?

If so, then it would follow that because the height of the Eiffel Tower is measured in metres, and the height of the Empire States Building is measured in metres, then the Eiffel Tower is necessarily the Empire States Building.

3) Your argument is that heat, which is the transfer of energy, is energy.

From https://psiberg.com/thermal-energy-vs-heat
i) Thermal energy = It is due to the movement of particles in a system
ii) Heat = It is the transfer of thermal energy

Then it would follow that:

The banking system, which is about the transfer of money, is money.
Language, which is about the transfer of knowledge, is knowledge.
The football transfer system, which is about the transfer of football players, is the football players.

All these are true, but only metaphorically.

But given that a posteriori we know that heat = the movement of molecules, then it's the same in every possible world. Or at least, that seems to be Kripke's argument...........I don't see how molecules could exist but not their movement — Banno

Justfication three that "heat is the motion of molecules" cannot literally be true.

Relevant, as Kripke uses "heat is the motion of molecules" as evidence for necessity a posteriori.

I agree that space and time exist in the world, molecules exist in the world and the movement of molecules exists in the world.

I still don't agree that heat exists in the world in the same way that molecules and their movements exist in the world. As language transfers knowledge, heat transfers energy.

Heat
Heat can be transferred by conduction, convection or radiation. Conduction is a process in which heat is transported between parts of a continuum, through direct physical contact. Convection is the principle, wherein heat is transmitted by currents in a fluid, i.e. liquid or gas. Radiation is the heat transfer mechanism, in which the transition takes place through electromagnetic waves.

What these have in common is that heat is the process whereby energy is transferred from one body to another. Energy is due to the motion of molecules. Heat is the transfer of energy. Heat is not energy, heat is the transfer of energy. As energy is the motion of molecules, and as heat is not energy, heat is not the motion of molecules.

Heat is not a substance, it is a process
Mark Barton, PhD physicist with University of Glasgow, wrote: ""Heat" is a noun and is spoken of as a substance, even in technical language, but it's a misnomer. Strictly heat doesn't exist, it happens: it's the process of energy moving from one system to another via random microscopic interactions."

1) Heat is the transfer of energy.
2) Energy is due to the motion of molecules.
3) Heat is not energy
4) Therefore, heat is not the motion of molecules.

Heat is necessarily the motion of molecules. The sensation of heat is not......
But this does not undermine the broader case that sometimes if A=A, then ☐A=A. — Banno

The law of identity
I don't disagree that if A = A then A = ☐A. As you say "Hence, it is not that "Kripke seems to want to prove something like the law of identity.........I think he takes it as given." Yes, if heat is heat, then heat is necessarily heat. If the motion of molecules is the motion of molecules, then the motion of molecules is necessarily the motion of molecules. If the Eiffel Tower is the Eiffel Tower, then the Eiffel Tower is necessarily The Eiffel Tower.

However, the law of identity doesn't show one way or another that heat is necessarily the motion of molecules, rather than heat is the Eiffel Tower, for example.

Heat, temperature and energy are concepts and don't ontologically exist in the world
Although heat may be transferred by conduction, convection and radiation, keeping with Kripke's description of heat as the motion of molecules. I accept (for the sake of argument) that moving molecules ontologically exist in the world. Consider a body having moving molecules:

Temperature is the measure of speed of these molecules, the higher the speed the higher the temperature. However, temperature as a concept, as a measure of speed, cannot exist independently of the moving molecules. It cannot have an ontological existence in the world over and above the moving molecules themselves. If there were no moving molecules, there would be no temperature.

Energy as a concept is a measure of the number of molecules and their speed. Similarly, as a measure it cannot exist independently of the moving molecules. It cannot have an ontological existence in the world over and above the moving molecules themselves. If there were no moving molecules there would be no energy.

Heat as a concept is a measure of the transfer of momentum from one molecule to another. If a fast moving molecule hits a slow moving molecule, the fast moving molecule slows down, and heat has said to have been transferred. Heat as a measure of the change in movement cannot exist independently of the change in movement. It cannot have an ontological existence in the world over and above the moving molecules themselves. If there were no moving molecules there would be no heat.

Heat is a concept
In Bertrand Russell's terms, the existence of heat is not the first-order of an individual but the second-order of a concept. If I tell someone that the next bus will be arriving in 10 minutes, their knowledge has increased, in that my knowledge has been transferred to them. This does not literally mean that knowledge ontologically exists in the world. Similarly, if someone says that heat has been transferred from one object to another, this does not literally mean that heat ontologically exists in the world. Knowledge and heat as concepts are figures of speech.

"Heat is necessarily the motion of molecules" cannot literally be true
As heat is a concept that exists only in the mind and not ontologically in the world, and as moving molecules do ontologically exist in the world, the statement "Heat is necessarily the motion of molecules." cannot literally be true as it is comparing two fundamentally different things, though still valid as a metaphor, however.

One of the things about the thermometer definition is it explicitly states how to pick out temperature without telling you anything about temperature. I think that's a feature..........In the case of counter-factuals, when we're talking about "heat is the motion of molecules" vs. "heat is a caloric substance that goes from one object to the other", then I think both must be picking out the same things in the case of the first part, but I'm not sure about the latter part still. — Moliere

The example of the thermometer may be a key into Kripke's necessary a posteriori.

We may know an effect without needing to know its cause
I observe the number on a thermometer change. I don't know what is causing the number to change, but I name whatever is causing the number to change as heat. The number isn't heat, but what is causing the number to change is heat.

Some hypothesise that what is causing the number to change is the motion of molecules, ie, heat is the motion of molecules. Others hypothesise that what is causing the number to change is caloric, ie, heat is caloric. If one of these hypothesise becomes with time self-evidently true, it becomes an axiom. Society then accepts as given the axiom that heat is the motion of molecules.

Both the motion of molecules and caloric are referring to the same thing, heat in the world, even if we never actually know what heat in the world is. In practice, we don't need to know what is causing the numbers to change, we don't need to know what heat in the world is, all we need to know is the effect of heat in the world, the numbers changing on the thermometer.

As long as we know the effect of heat, we don't need to know what heat is.

Kripke is using the word "heat" in two very different ways
We can have a particular sensation, which we name the sensation of heat. We name the cause of this particular sensation heat. The word heat is being used in two different ways, one as a name of an effect, the sensation of heat, and the other as the name of its cause, heat. The effect is very different from the cause, though they share the same name.

Kripke refers to the sensation of heat. Page 185: "There is a certain external phenomenon which we can sense by the sense of touch, and it produces a sensation which we call “the sensation of heat.”"

Kripke also refers to heat as the motion of molecules. Page 170: “Heat is the motion of molecules.”

Heat as the sensation of heat in the mind
Kripke discusses possible connections between the sensation of heat and its cause: i) "So, it might be thought, to imagine a situation in which heat would not have been the motion of molecules" ii) "the motion of molecules but in which such motion does not give us the sensation of heat" iii) "Martians, who do indeed get the very sensation that we call “the sensation of heat” when they feel some ice which has slow molecular motion, and who do not get a sensation of heat—in fact, maybe just the reverse".

Heat as the cause in the world of the sensation of heat in the mind
Kripke discusses possible causes of our sensation of heat: i) "First, imagine it inhabited by no creatures at all: then there is no one to feel any sensations of heat" ii) "the judgment that heat is the motion of molecules would have been false."

Rigid designators
The motion of molecules in the world is a rigid designator. As this lectern is made of wood , this lectern is necessarily made of wood, similarly, as these molecules are in motion they are necessarily in motion.

A sensation of heat in the mind is a rigid designator. As this particular sensation of heat in the mind is this particular sensation, it is necessarily this particular sensation, whatever it is named, in that it could have been named "heat", "cold", "apple" or "The Eiffel Tower".

Heat in the world as the cause of a sensation of heat in the mind is a rigid designator. As this lectern is made of wood , this lectern is necessarily made of wood, similarly heat in the world as the cause of a sensation of heat in the mind is necessarily the cause of the sensation of heat in the mind, whatever it is named.

Kripke concludes that heat is necessarily the motion of molecules
He wrote: page 187: "To state the view succinctly: we use both the terms ‘heat’ and ‘the motion of molecules’ as rigid designators for a certain external phenomenon. Since heat is in fact the motion of molecules, and the designators are rigid, by the argument I have given here, it is going to be necessary that heat is the motion of molecules."

There are two possible meanings to heat is necessarily the motion of molecules
The motion of molecules in the world has one possible meaning, although heat has two possible meanings.

Meaning one: Heat in the world is necessarily the motion of molecules in the world. There may be heat in the world, and there may be molecules in motion in the world. Both the heat in the world and molecules in motion in the world are rigid designators, but it doesn't of necessity follow that there is a link between them. For example, both "Nixon" and "Caesar" are rigid designators in all possible worlds, but there is no necessary link between them.

Meaning two: Heat in the mind is necessarily the motion of molecules in the world. The sensation of heat in the mind is necessarily caused by heat in the world, but as there is no necessary link between heat in the world and the motion of molecules in the world, there is no necessary link between the sensation of heat in the mind and the motion of molecules in the world.

Conclusion
Heat is not necessarily the motion of molecules.

Exactly wrong. — Banno

True. My new understanding is:

From the SEP Rigid Designators, Kripke addresses the objection that we cannot talk about someone without first having some qualitative criterion of identity, an essence, and if we know of no such essence we cannot meaningfully talk about someone.

Kripke addresses the objection that we cannot meaningfully talk about you, with respect to another possible world, without first having some qualitative criterion of identity, some qualitatively distinguishing mark that allows us to pick you out from other objects in the world at issue, in order to assign your name to the right person, i.e., to you, as the individual that satisfies the qualitative criterion. This criterion would appeal to your essence (or be “an essence”: see Plantinga 1985, pp. 85–7; 1974, p. 98; recall, for this example, the minimal requirements of weak necessity), in the minimal respect that the criterion must be something that you and you alone have with respect to any given possible world. As an objection, the worry is that we know of no such qualitative criterion so we can not meaningfully discuss you, with respect to any merely possible world.

From Wikipedia Causal Theory of Reference, Kripke outlined a causal theory of names whereby you don't need to be able to describe what is being named, but after naming an individual in an "initial baptism" the name continues to refer through a causal chain. In fact, although the meaning of the name may change with time and use, the new meaning becomes the new "reality".

1) a name's referent is fixed by an original act of naming (also called a "dubbing" or, by Saul Kripke, an "initial baptism"), whereupon the name becomes a rigid designator of that object.
2) later uses of the name succeed in referring to the referent by being linked to that original act via a causal chain.

In lectures later published as Naming and Necessity, Kripke provided a rough outline of his causal theory of reference for names. Although he refused to explicitly endorse such a theory, he indicated that such an approach was far more promising than the then-popular descriptive theory of names introduced by Russell, according to which names are in fact disguised definite descriptions. Kripke argued that in order to use a name successfully to refer to something, you do not have to be acquainted with a uniquely identifying description of that thing. Rather, your use of the name need only be caused (in an appropriate way) by the naming of that thing.

Sure, you can use it any way you like, but if you want to maintain a distinction between knowing and believing, then I don't think loose or ambiguous usages are a good idea. — Janus

Totally agree, exactly my point.

as soon as the word 'if' appears, you are talking about a possible world, but then you doubly return us to 'this' and 'actual' world. But this actual world is necessarily the way it is, and not the way it would be if anything was different. — unenlightened

In this world, Hesperus exists. If Hesperus didn't exist in this world, it could exist in a possible world.

If Hesperus didn't exist in this world, in which world would Hesperus not exist in, this world or a possible world ?

I've suggested several times that possible worlds are a convenient way of dealing with counterfactuals. If you wish to call them a metaphor, go ahead. — Banno

Kripke wrote page 174: "All of this talk seems to me to have taken the metaphor of possible worlds much too seriously in some way."
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The salient piece for proponents of descriptions is that a proper name does not refer by making use of some description. — Banno

Within Bertrand Russell's Theory of Descriptions, a proper name refers to a set of true propositions that uniquely describe a referent.

Kripke and Donnellan rejected Descriptivism. Kripke described Descriptivism such that (1) To every name or designating expression 'X', there corresponds a cluster of properties, namely the family of those properties φ such that [speaker] A believes 'φX'

For example, to the name Aristotle there corresponds the properties Greek, a philosopher and a teacher. It is these properties that refer to the referent.

A property, such as being Greek, is also a description. As a proper name corresponds to a cluster of properties, and as properties are also descriptions, then surely doesn't Descriptivism make use of descriptions ?
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I agree the notion of removing all of an objects properties is problematic. I don't think that is mentioned anywhere in the article - is it? — Banno

Kripke wrote: "We can talk about this very object, and whether it could have had certain properties which it does not in fact have. For example, it could have been in another room from the room it in fact is in, even at this very time, but it could not have been made from the very beginning from water frozen into ice."

Kripke also wrote:"In fact, it would seem that both the terms, ‘my pain’ and ‘my being in such and such a brain state’ are, first of all, both rigid designators. That is, whenever anything is such and such a pain, it is essentially that very object, namely, such and such a pain, and wherever anything is such and such a brain state, it is essentially that very object, namely, such and such a brain state."

I agree that the possibility of removing all the properties from an object is not specifically mentioned in the article, but the problem of which properties may be removed from a rigid designator without affecting its status as a rigid designator must surely be important in understanding the article.

Objects have properties. Possible properties of this lectern are made of wood, made of ice, in another room, etc. A name is a rigid designator by virtue of having certain essential properties, whereby non-essential properties may be removed without affecting its status as a rigid designator. Kripke doesn't address the problem of how is it determined which properties are essential and which non-essential.

For example, some may believe that whether this lectern to be in this room or outside is clearly not an essential property, yet others may believe that the location of this lectern is an essential part of its identity, in that a stand made of wood outside a lecture theatre is not functioning as a lectern.

The article can only make sense to me if I can understand how is it decided which properties can be removed from a rigid designator before it no longer is a rigid designator.
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I'd also draw your attention to the difference between picking out an individual using a name and picking it out using a demonstrative. — Banno

Even Donnellan admits of description within proper names.

He wrote: "Nevertheless, so long as the user of a name can fall back on such a
description as 'the person referred to by Aristotle', the principle of identifying descriptions may be salvaged even if at expense of having to elevate one type of description to special status."

In the first case "know" is used incorrectly; you cannot know that The Red Sox will win the next game — Janus

If I am using "know" metaphorically, ironically, wryly, jokingly, humorously or sarcastically, it is not being used incorrectly.

You only have this issue if you do not distinguish between the subject and the predicate, or object and property. — Metaphysician Undercover

Yes, much of language is like that, ambiguous, in that rarely in practice if someone says "object A is object B" do they say in what sense they are using "is".

an object with no properties is beyond imagination — frank

I agree. Objects such as lecterns cannot exist in the world independently of their properties, as objects in the world are no more than the set of their properties.

I would say, that your example shows the existence of ambiguity rather than metaphor — Metaphysician Undercover

I more or less agree, but my long-term project is to show that language is fundamentally metaphorical. "Time is a thief" is a metaphor in that time is not the same as a thief. "Object A is object B" is a metaphor in that object A may be similar to object B, but object A can never be the same as object B. "Hesperus is Phosphorus" is a metaphor in that Hesperus may be similar to Phosphorus, but Hesperus can never be the same as Phosphorus.

And, Kripke supposes, this goes for any equivalence between heat and the motion of molecules. If "heat" is a rigid designator for that sensation, and "the motion of molecules" is a rigid designator for molecules in motion, then if heat is the motion of molecules, it is necessarily so. — Banno

Heat and the motion of molecules independent of any observer

Kripke wrote: "First, imagine it inhabited by no creatures at all: then there is no one to feel any sensations of heat. But we would not say that under such circumstances it would necessarily be the case that heat did not exist; we would say that heat might have existed, for example, if there were fires that heated up the air."

Beginning with this lectern, this lectern is made of wood. If it had not been made of wood it would have been a different object, so this lectern is necessarily made of wood. This sounds reasonable.

This raises the question as to whether objects such as this lectern exist over and above their properties, in that if all the properties of the lectern were removed, would this lectern remain. Would Hesperus remain if all its properties were removed. Would heat remain if there were no molecules in motion. As no example of an object existing having no properties can be found in the world, it must be concluded that an object cannot exist in the world independently of its properties.

Kripke would say heat has essential properties. These essential properties exist in all possible worlds, and as such heat is a rigid designator. Objects have properties, and for Kripke, some properties are essential and some non-essential. This can only be a human judgement. Human judgement cannot exist independently of any observer. In a mind-independent world the motion of molecules cannot be judged to be essential or non-essential, meaning that heat cannot be a rigid designator, as the world has no means of judging which properties are essential and which non-essential.

IE, heat cannot be rigid designator in a mind-independent world.

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.