All numbers are represented by glyphs, so all numbers are graphics. The Roman numerals are supposed to be based on the image of a Roman architectural column.

The different glyphs are just different ways to combine images of a Greco/Roman architectural column.
So if numbers are just picture representations then suggesting they exist outside of spacetime, suggests that all human manifested symbols or imagined images exist outside of spacetime, which just brings us back to BS ways to try to find some significance in woo woo thoughts of 'outside this universe.'
I assume aliens will have an efficient way to describe how many planets are in the solar system they come from but that does not mean quantities have an objective reality.
I don't think quantities have an objective reality. I think they are subjective. Do you reference a football team or 11 football players or 22 arms or 110 fingers etc.
Do you see a pint of water or a combination of 2 half pints or so many water atoms?
How many branches make two trees?
Quantities are subjective. 1 star or 2 gases (hydrogen and helium)?

The correlated events or actions can be because of a common cause. — universeness

If locality is the case, then the common cause of the entangled particles' correlation is their initial preparation (see spontaneous parametric down-conversion).

I think the use of the term correlation for quantum entanglement is a wise use but really just indicates that the detailed nature of the relationship is not yet well understood. — universeness

:up:

So "tree" here is a reference to an individual. Is 'two" an individual in this way? — Banno

In set theory, I think we can say that the number two is an individual in this way.

Numbers are abstract objects. They do not actually exist. — Alkis Piskas

It might appear like a very acceptable approach, to say that abstract objects are objects, only a different kind of object from physical objects, but then we need acceptable principles to set the two apart, or else we'll have equivocation between two types of "objects" in logical proceedings. As Banno indicated, this is problematic, because it presents the issue of interaction between the two types.

What Plato showed, is that if there is a distinct class of objects which are abstract (intelligible objects), then we must place all the ideas, including moral ideas such as "just" and "virtue", and aesthetic ideas like "beauty" into this class of abstract objects. Then the subjectivity of the supposed 'abstract objects' becomes apparent.

What Plato described is that the objectivity of abstract objects is provided for by "the good", because the good is the "object", or "objective", in the sense of the goal. Abstract objects are "objective" in the sense that they are useful toward goals.

This sets up the solution to Banno's problem of interaction, the good is the means by which the two types of objects interact. Furthermore, we have the principles here to properly distinguish the two types of objects in an effort to avoid the logical fallacy of equivocation. The abstract, or intelligible objects are associated with intention, (goals), as "the good" which is desired, and the other type of objects, sensible objects, are associated with the material world as particular things, which Aristotle assigned the law of identity to. By the principles demonstrated in Plato's cave allegory, the intelligible objects are more 'real' to us because we understand them directly with the mind, rather than through the unreliable medium of sensation. That this is truly the case is supported by the fact that there is a separation between the identity which we assign to a particular sensible object, and the identity which the object has in itself.

If locality is the case, then the common cause of the entangled particles' correlation is their initial preparation (See spontaneous parametric down-conversion). — Andrew M

Phew! That link was to a physics level that is a bit high for me. I clicked on sub-links such as 'non-linear crystal,' 'etc to gain a better insight. But I found I had to click on further and further sub-links eg 'Schwinger limit' and then 'Birefringence,' to gain any clarity. I will go back to it, but you have moved past my current width and depth of physics understanding.

Does your mind create the concept of 2? Does the concept of 2 cease to exists when you stop thinking about it? — Art48

My mind creates my private concept of something in the world publicly named as "two".

Imagine at a particular place and time in the world there is something. The public name "two" is attached to this something by the authorities. From my observation of this something, in my mind I have the private concept two. Someone else observing the same thing will also have the private concept two. However, it may well be that my private concept two is different to their private concept two, but as we are both part of the same community, we will both name our private concepts as "two".

My mind has created my private concept two, someone else has created their private concept two. But as we are both part of the same community, the public concept "two" continues to exist even if I stop thinking about it.

Because numbers have objective properties. — Art48

Some aspects are objective, others subjective

The something that I have observed in the world about which I have the concept two is objective in the sense that it exists independently of me. Because the public name "two" has been attached to this something, the concept "two" is objective in the sense that it exists independently of me within the community. As regards my private concept two, it is objective in the sense that it somehow exists within my physical brain, but it is also subjective in the sense that no one else can ever know my private concepts.

Numbers don't refer to individuals, they describe the parts of the individual

In the world is something that has been given the public name "two" about which I have the private concept two. Starting with two things in the world each of which exists, the question is, when brought together, does a new existence come into being, where this new existence has each thing as a part. Does the number two exist as a new whole in addition to the existence of the two parts that make it up ? Not according to Hume, Kant, Frege or Russell.

As the problem of numbers involves language, Bertrand Russell's On Denoting may shed light. For example, in the sentence "the author of Waverly was Scott", the phrase "the author of Waverly" is not a referring term, in that it doesn't refer to an individual having an independent existence. It is a quantificational expression, a definite description of a set of properties that makes up "the author of Waverly". Frege and Russell believed that existence was not the first-order of an individual, but the second-order of a concept.

Similarly, the phrase "the number two" is not a referring term, in that it doesn't refer to an individual having an independent existence, but rather is a quantificational expression, a definite description of the separate parts that makes up what is known as "the number two".

In language, the phrase "the number two" doesn't refer to an individual having a unique existence, but is a description of the separate parts that make up what is known as "the number two".

Why can't your 2 be greater than your 3? — Art48

It cannot

I observe something in the world that has the public name "one" and I have the private concept one. I observe something different in the world that has the public name "two" and I have the different private concept two. I observe "one" added to "two", and when observing this new something, I have the private concept three. As my concept of three has resulted from an addition to my concept of two, my concept of two cannot be "greater" than my concept of three.

The chief difficulty with Platonism is that while proposing a distinct type of reality of mathematical entities, it must then explain how this reality interacts with everyday things. — Banno

Good question What do you think of the following explanation for explaining interaction?

Regard a human being as having four-parts: body, emotions, intellect, and consciousness. Consciousness receives input from seven sources: the five bodily senses of sight, sound, smell, taste, and touch. It also receives input in the form of emotions and thoughts. (If an eighth sense, like ESP, exists, that won’t impact the argument.) So, when we “see” a tree, we actually see only light. (It could not be otherwise because we lack a specific tree-sensing sense). The actual input is patches of green and brown. Based on that input, the idea “tree” arises in our mind. If we “touch” the tree, we experience a rough surface, which gives us confidence that the tree is not an hallucination. If we see multiple patches of green and brown, the idea of number arises: we “see” two trees. So, it appears that “tree” and “two” are on equal footing: they are ideas which arise in our mind which help makes sense of the seven inputs.

This view, I think, is somewhat similar to Kant’s view that all we experience are phenomena. It differs in that it limits the phenomena we experience to sight, sound, smell, taste, touch, emotion and thought.

In this view, mathematical entities are not a distinct type of reality. They are ideas, just like “tree.”

It might appear like a very acceptable approach, to say that abstract objects are objects, only a different kind of object from physical objects, but then we need acceptable principles to set the two apart, or else we'll have equivocation between two types of "objects" in logical proceedings. As Banno indicated, this is problematic, because it presents the issue of interaction between the two types. — Metaphysician Undercover

You are right. The term "objects" I ised in saying "Numbers are abstract objects" might be confusing because it normally refers to something physical. I could use the word "things" --which is more general and can refer to anything-- but it's too commonplace and banal. So I prefer to use neither and say, "Numbers are abstract".

So, I believe we should use the word "object" only for material things, things we can perceive with our senses, things that actually exist. Thus, we can talk about objective things, which actally exist, indepedently of us, in contrast to subjective things, which are abstact and exist only for (each one of) us.

In this case, I think, there would be no equivocation, as you say, neither any kind of interaction of two types of objects.

Note: Of course, no confusion should be produced if the context, phrase or expression in which the word "object" appears, make it clear about its nature and meaning. E.g. my use of the expression "abstract objects" shows that the the word "object" is used as something non-physical, since something that is abstract can never be physical. Yet, as I said earlier, it is better to avoid the use of the word "object" altogether in these cases. Look what it has created! :grin:

Sorry, good try, but an appreciation of creativity and discovery comes with involvement, not philosophical chatter. — jgill

Sorry, I was just jotting down some preliminary ideas related to the OP, and to your notion of "Creative Step" and "Discovery". When you "decided to extend this idea to a more general realm" (specific-to-general) you were doing Inductive Reasoning, which is one kind of creative act in Philosophy, and in Mathematics. But, another approach is to break-down a broad general concept into more particular applications (general-to-specific) Deductive Reasoning. I suppose both can be creative, depending on their practical or theoretical implementation (involvement??).

It seems that the OP is an attempt to generalize from spacetime observations to something beyond spacetime : specifically Mathematics & Mental Images. The abstract concepts of Mathematics exist "beyond" space-time in the sense that Math objects are not affected by the physical laws that govern the behavior of material objects. I suppose that is trying to extend that spaceless & timeless aspect of Math, to the abstractions that we humans create to represent the Self (or Soul, if you prefer). That is not exactly a new idea, except for the connection to immaterial Mathematics & Logic, which some thinkers imagine existing eternally out-there beyond the limits of space-time (Ideality instead of Reality).

So, he seems to be expressing an ancient concept (we are souls) in more modern language : "we are math". Whether that's a creative step may depend on how he develops the basic notion into a philosophical position. Unfortunately, even the Ontological status of Mathematics is subject to philosophical debate. So, the notion of a soul-man is not a slam-dunk. :smile:

PS__"Philosophical Chatter", as you put it, seems to be how philosophers get involved in discovering new ways to look at old ideas. Are the mathematician's chalk-marks on the blackboard more involved than text-marks on a philosophical forum?


"More or less, although most math people give this question little thought. In my case, I was introduced to a notion years ago in my PhD studies. A little later on I decided to extend this idea to a more general realm - a sort of creative step. Once the basic ideas of the concept were set, then came the acts of discovery - finding what flows forth logically." ___jgill

"Which suggests that reality—that me, you, Earth, universe, etc.—is fundamentally some sort of abstract object existing outside spacetime. Hm." ___Art48

Imagine at a particular place and time in the world there is something. The public name "two" is attached to this something by the authorities. From my observation of this something, in my mind I have the private concept two. Someone else observing the same thing will also have the private concept two. However, it may well be that my private concept two is different to their private concept two, but as we are both part of the same community, we will both name our private concepts as "two". — RussellA

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. When I talk about two, I am talking about something utterly distinct from what you are talking about when you talk about two.

But suppose it so happens that your private concept and my private concept have some overlap, such that we agree publicly on certain aspects. Those public aspects are what make a difference, are what we use to count and do things. Any private, unshared aspects have no impact on what we do - if they did, they would thereby be public.

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.

This is part of the Private Language Argument. Meaning is not found in private concepts inside minds or brains, but is public and shared.

The remainder of your argument hangs on this. It needs rethinking.

If the meaning of "two" is a private concept in my mind, — Banno

In my view, it is not a private concept. It's a pre-existent idea which we encounter in the "mindscape," just as we encounter a pre-existent tree in the landscape. In my view, ideas are pre-existent.

What do you think of the following explanation for explaining interaction? — Art48

Excuse my being blunt, but it is wrong on multiple levels.

There are far more than five senses. Indeed it is problematic to treat the senses as discreet, as being seperate "inputs". Contemplate how difficult it is to seperate even smell and tase, for example.

Emotions and thoughts are more reactions rather than "inputs". I'll use scare quotes, because the analogy of an "input", derived from computer science, is misleading. It gives the impression of a one-way process. What we sense is very much dependent on our state of mind - on our emotions and thoughts. Sensing is better thought of as an interaction between our bodies and the world. Think of the way the eye changes it's physiology immediately that there is a change in the light, and of the various optical illusions you have seen. Similar things happen with each of the other sense.

We see the tree using the light reflected from it, and using our eyes and the light from the sun. We certainly do not see the light. If we did we would not have needed Young and Maxwell and others to set out the physics of light. What we see is the tree.

Your "actual input" is a misleading notion. One's neural network, starting at one's retina, constantly and actively re-works the signal it receives in order to construct the sense of green and brown. The "idea of tree" is constructed much later in the neural net, perhaps involving the areas of the brain that handle language. Our resident Neuroscientist, @Isaac, might be able to explain with greater clarity.

(Excuse my invoking you yet again, Isaac)

@Art48, you seem to be working with a homunculus-like view of the self, as if you were sitting inside your head looking out, receiving raw inputs of information that you interpret using a priori scripts. That is a view often attributed to Kant, although there are Kantians who deny it. The homunculus is, for several reasons, to be rejected.

In conclusion, trees are not ideas. They are things found in the world.

But you seem to be claiming that the numbers are found in the world in the way trees are, and so are not individuals from some parallel wold of forms. I'd therefore refer you back to this argument:

Look again at 'The word "two" refers to the objectively real number 2, just as "tree" refers to an objectively real tree'. So here I take it that we are talking about, say, the tree in your yard? So "tree" here is a reference to an individual. Is 'two" an individual in this way? So are you saying that when I talk about your two feet and then the two dollars in my pocket, I'm talking about the very same, individual, two? That "two" is a proper name for an individual? The same individual, two, can't be both in your feet and in your pocket, so it must be outside of space and time, is that the thinking? — Banno

And I put it to you that there are alternatives that do not require such obtuse ontologies:

What if instead of "one mind creating the concept of 2", it is a construct of our communal capacity to use language - a way of talking about cases where we have two things? "There are two trees in my yard" as a way of talking about the trees, and so not a reference to some platonic form. — Banno

And not a reference to something else in the world, besides the two trees.

It's a pre-existent idea which we encounter in the "mindscape," just as we encounter a pre-existent tree in the landscape. In my view, ideas are pre-existent. — Art48

The homunculus, siting in a head looking out, using its Kantian a priori scripts to interpret what it sees.

Nuh. I don't think so.

Your "actual input" is a misleading notion. One's neural network, starting at one's retina, constantly and actively re-works the signal it receives in order to construct the sense of green and brown. The "idea of tree" is constructed much later in the neural net, perhaps involving the areas of the brain that handle language. Our resident Neuroscientist, Isaac, might be able to explain with greater clarity — Banno

Here’s an enactivist perspective:

…traditional neuroscience has tried to map brain organization onto a hierarchical, input-output processing model in which the sensory end is taken as the starting point. Perception is described as proceeding through a series of feedforward or bottom-up processing stages, and top-down influences are equated with back-projections or feedback from higher to lower areas. Freeman aptly describes this view as the "passivist-cognitivist view" of the brain.
From an enactive viewpoint, things look rather different. Brain processes are recursive, reentrant, and self-activating, and do not start or stop anywhere. Instead of treating perception as a later stage of sensation and taking the sensory receptors as the starting point for analysis, the enactive approach treats perception and emotion as dependent aspects of intentional action, and takes the brain's self-generated, endogenous activity as the starting point for neurobiological analysis. This
activity arises far from the sensors—in the frontal lobes, limbic system, or temporal and associative cortices—and reflects the organism's overall protentional set—its states of expectancy, preparation, affective tone, attention, and so on. These states are necessarily active at the same time as the sensory inflow (Engel, Fries, and Singer 2001; Varela et al. 2001).

“Whereas a passivist-cognitivist view would describe such states as acting in a top-down manner on sensory processing, from an enactive perspective top down and bottom up are heuristic terms for what in reality is a large-scale network that integrates incoming and endogenous activities on the basis of its own internally established reference points. Hence, from an enactive viewpoint,
we need to look to this large-scale dynamic network in order to understand how emotion and intentional action emerge through self-organizing neural activity.”(Evan Thompson, Mind in Life)

Cheers. Before we start to argue enactivism against process-oriented extended mind we might at least try to have @Art48 form the possibility of such alternatives.

you seem to be working with a homunculus-like view of the self, as if you were sitting inside your head looking out, receiving raw inputs of information that you interpret using a priori scripts. That is a view often attributed to Kant, although there are Kantians who deny it. The homunculus is, for several reasons, to be rejected. — Banno

What's wrong with the homunculus? That seems to almost exactly describe my conscious experience. It seems like I'm inside my head looking out, only not sitting, and I don't know if any of the "scripts" (they seem more like memories to me) are apriori or not.

In this case, I think, there would be no equivocation, as you say, neither any kind of interaction of two types of objects. — Alkis Piskas

Why assume no interaction? I see no need for this assumption, and the human use of mathematics and engineering in creating things in the physical world demonstrates that there definitely is interaction.

When you "decided to extend this idea to a more general realm" (specific-to-general) you were doing Inductive Reasoning, — Gnomon

I don't think inductive reasoning in its usual interpretation is appropriate in this comment. I am not arguing that since it works here and there in a number of different venues it probably works in general. I am going the other direction and investigating, using deductive logic, whether ideas hold in various more or less distinct areas.

PS__"Philosophical Chatter", as you put it, seems to be how philosophers get involved in discovering new ways to look at old ideas. Are the mathematician's chalk-marks on the blackboard more involved than text-marks on a philosophical forum? — Gnomon

Good point. Nice and appropriate comparison. The chatter I was referring to was efforts to abstract "creativity" in math without using actual examples.

Excuse my being blunt, but it is wrong on multiple levels. There are far more than five senses. — Banno

Being blunt (or frank) is virtue. We get a lot more accomplished that way. And it helps me clarify my own thinking.

You write “There are far more than five senses.” As I noted when I mentioned ESP, more senses aren’t a problem. Emotions and thoughts, like our physical sensations, are inputs to consciousness. If it’s not one-way, if our emotional or mental state impacts what we sense, that, too, is not a problem.

The basic picture is that we have transitory physical, emotional, and mental sensations. Sensations imply a sensor, an experiencer. A criticism I’ve seen of the homunculus idea is infinite regress: who sees what the homunculus senses? I don’t see the same problem with consciousness. In effect, the buck stops with consciousness.

I see consciousness fulfilling the role of experiencer. Of course, this assumes we have an enduring self. Hume just saw sensations, not the self. I think the answer was Hume’s self—in my view, his consciousness—was the self he couldn’t find, just as the eye can see everything but itself.

So, consciousness is one answer to the question of how personal identity persists through time. To me, it’s the best answer. Historical continuity, for example, gives a kind of personal identity, but it seems a somewhat superficial type of identity.

You write “actual input” is misleading and mention neural network processing. But I mean input to consciousness, after all processing has been done.

If trees are found in the world, who finds them? Can you explain the tree-sensing sense that allows us to sense trees? Is there such a sense? If so, what is the organ of our tree-sensing sense?

When I introspect, my consciousness experiences a stream on physical, emotional, and mental sensations. I feel my consciousness is “me” while the sensations, because they are transitory, are not me. So, my view is not merely mental. It describes more or less how I experience myself.

If locality is the case, then the common cause of the entangled particles' correlation is their initial preparation (See spontaneous parametric down-conversion).
— Andrew M

Phew! That link was to a physics level that is a bit high for me. I clicked on sub-links such as 'non-linear crystal,' 'etc to gain a better insight. But I found I had to click on further and further sub-links eg 'Schwinger limit' and then 'Birefringence,' to gain any clarity. I will go back to it, but you have moved past my current width and depth of physics understanding. — universeness

SPDC converts a photon into two entangled photons, each with half the energy of the original photon. For Type II SPDC, this prepares the singlet state (represented below by qubits):

$\ \ \ \ \ |\psi\rangle = \frac{|01\rangle - |10\rangle}{\sqrt{2}}$

So that anti-correlation is the consequence of a local physical process (i.e., the SPDC process).

Subsequently, the two photons (qubits) can be separated by a large distance. QFT, by construction, says that spacelike-separated events, including measurements, cannot be causally connected. So the measurement of a photon is a single localized event in spacetime which has no effect on the measurement probabilities for the other photon until a signal arrives there (not exceeding the speed of light).

Note that the singlet state is perfectly anti-correlated in every basis (X, Y, Z, etc.). That is, if each qubit is measured in the same basis, then one will be measured as 0 while the other will be measured as 1.

Compare QFT to the case of Bertlmann’s socks, which John Bell recounts:

The philosopher in the street, who has not suffered a course in quantum mechanics, is quite unimpressed by Einstein-Podolsky-Rosen correlations. He can point to many examples of similar correlations in everyday life. The case of Bertlmann’s socks is often cited. Dr. Bertlmann likes to wear two socks of different colours. Which colour he will have on a given foot on a given day is quite unpredictable. But when you see (Fig. 1) that the first sock is pink you can be already sure that the second sock will not be pink. Observation of the first, and experience of Bertlmann, gives immediate information about the second. There is no accounting for tastes, but apart from that there is no mystery here. And is not the EPR business just the same? — Bertlmann's socks and the nature of reality - John Bell, 1981

Observing one sock to be pink doesn't cause the other sock to not be pink. Instead the common cause of the sock anti-correlation is Bertlmann's initial choice. That's analogous to QFT, where the SPDC process is the common cause of the photon anti-correlation. In both cases, locality is maintained.

The key difference is that the observation of Bertlmann's socks is explained by the pre-measurement colors of the socks. Whereas the observation of the entangled qubits can't be explained by pre-measurement values, as shown by Bell's Theorem. Which is why quantum interpretations become relevant (note the Local dynamics column).

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".

I'm just talking about the word "object" and it's use. The interaction I mentioned was between the two uses and meanings of the word. Mathematics and engineering have nothing to do here.

Anyway, nevermind. Too much has been already said about the subject. Let's move on.

The key difference is that the observation of Bertlmann's socks is explained by the pre-measurement colors of the socks. Whereas the observation of the entangled qubits can't be explained by pre-measurement values, as shown by Bell's Theorem. Which is why quantum interpretations become relevant (note the Local dynamics column). — Andrew M

Ok, It seems to me that your the sock example you site is similar to the glove example.
So if both gloves are individually boxed and are sent a great distance away from each other, then the moment I open my box, I know instantly, the contents of the other box, based on one left and one right handed glove, regardless of the distance between the two boxes. I understand that in this sense, the two gloves are set at their creation. Is SPDC simply asserting the same for these two entangled photons?
So, when you locally open one of the boxes, that act, does not affect the state of the gloves.
But a qubit can have more that two states due to superposition states. A qubit does not just resolve to 1 or 0, it can be in a superposition of 1 and 0.
The nature of the gloves as left or right handed is there from the beginning, just like in QFT, a coordinate in spacetime can manifest any of the known particle states/field excitations (almost like an interdimensional vibrating superstring). Entanglement may 'correlate' (as a field excitation 'travels') the states of two spacetime coordinates, regardless of the distance between them.
What's wrong with the imagery I am invoking, if it's incorrect.

There's a need to sort out two seperate discussions here. You were replying to my critique of the Platonic view of there being a distinction between the world of the senses and the world of forms. In your OP you propose that the world of the sense is the world of forms.

What you are proposing in the OP is not a dualism such as I was addressing, a physical world and a world of mathematical forms, but that the physical world is exactly the world of forms. Some form of idealism.

So you set consciousness as the pivot on which everything depends. Hence, you can talk without apparent irony of an

input to consciousness, after all processing has been done. — Art48

I'd submit that consciousness is the very processing you dismiss. Again, you are not sitting inside your head looking at the results of the processing, but rather you are the processing.

The obvious difficulty, as for all forms of idealism, is how you can avoid solipsism. If all you have is your own consciousness, then, by that very fact, you are on your own.

...those private aspects of the concept two make no difference". — RussellA

Almost there.

Now we can go back to this:

Imagine at a particular place and time in the world there is something. The public name "two" is attached to this something by the authorities. From my observation of this something, in my mind I have the private concept two. Someone else observing the same thing will also have the private concept two. However, it may well be that my private concept two is different to their private concept two, but as we are both part of the same community, we will both name our private concepts as "two". — RussellA

You go to the grocer and hand over a piece of paper with "Two apples" written on it. The grocer pulls out the bin with a picture of apples in it. She then looks at a chart showing the numbers with tally marks next to them, scans down the chart until she find "two", and pulls out one apple for each tally mark.

At no stage does she refer to a private conception, in her mind or in yours. The process is entirely public.

Can you now see that in using "two", say in asking for two apples, what counts is that you receive an agreed quantity of apples, which is a public activity? That this satisfies your private conception of "two" is of no consequence to the action involved.

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

What you are proposing in the OP is . . . the physical world is exactly the world of forms. Some form of idealism. — Banno

I’m not sure idealism applies. I’d say our consciousness directly experiences its physical, emotional, and mental sensations, and so we can be certain the sensations exist. (Much like “I think therefore I am” although I’d replace “think” with “experience”.) What causes the sensations? Are we a brain in a vat? Or are we experiencing the world more or less as it is? Or is what we experience Platonic forms? Or is there a monist entity responsible for what we experience? I can make some intelligent hypotheses, but I just don’t know.

I'd submit that consciousness is the very processing you dismiss. Again, you are not sitting inside your head looking at the results of the processing, but rather you are the processing. — Banno

I think of consciousness as what is aware of the sensations. I think some philosophers view consciousness in the same way. Thus, the “hard problem of consciousness.” And, thus, the concept of philosophical zombies, which have all the sensations but no consciousness.

Neuroscience can explain (to a certain degree) our bodily, emotional and mental sensations but I don’t think it can yet explain consciousness (except for the view that it’s “what the brain does,”, i.e., consciousness has a physical basis).

I understand that in this sense, the two gloves are set at their creation. Is SPDC simply asserting the same for these two entangled photons? — universeness

If locality is the case (per QFT) then the correlation is a consequence of the SPDC. However, unlike with the gloves, the final measurement values are not predefined (see counterfactual-definiteness).

So, when you locally open one of the boxes, that act, does not affect the state of the gloves.
But a qubit can have more that two states due to superposition states. A qubit does not just resolve to 1 or 0, it can be in a superposition of 1 and 0. — universeness

Yes. Or in the case of the singlet state, the qubit pair is in a superposition of 01 and 10.

The nature of the gloves as left or right handed is there from the beginning, just like in QFT, a coordinate in spacetime can manifest any of the known particle states/field excitations (almost like an interdimensional vibrating superstring). Entanglement may 'correlated' (as a field excitation 'travels') the states of two spacetime coordinates, regardless of the distance between them.
What's wrong with the imagery I am invoking, if it's incorrect. — universeness

I'm not sure I follow the imagery. There isn't an image or picture that explains the correlation, unless one goes with an interpretation (such as Many Worlds or Superdeterminism).

Very interesting stuff! I will continue to try to understand all of the aspects of it, better than I do at the moment.

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.

Outstanding.

I think it's important (whether it's new to you or you've forgotten) to familiarize yourself with the Frege-Russell definition of number. Here's a link to an abstract of an article by R. L. Goodstein that provides a nice description. Note that he uses (1, 1) to stand for the more common "one-to-one".

https://www.jstor.org/stable/3609188#:~:text=Russell%20definition%20of%20number%20is%20the%20identification%20of,1%29%20related%20to%20some%20class%20of%20n%20members.

Re the OP :

For instance, the number “2” exists outside spacetime. — Art48

Others have commented on this already, but here's my take : most materialists and idealists (recognizing that there are various flavors of each) must disagree with this premise, since both camps consider mind-stuff to be part of the world (i.e., "spacetime"). Only dualists would agree that thoughts are divorced from the "world out there".

With apologies, the better question is, "Do numbers exist independent of minds?" * And of course, debate over mathematical Platonism has raged for many decades, if not centuries, and will not be answered on TPF any time soon. Still, it's fun to plant your flag and defend it. My own view, following from the Frege-Russell definition, is that numbers do NOT exist independent of minds because the act of placing objects in a one-to-one relationship is a cognitive one.

*Assuming most idealists assert that ALL is mind, their answer to the question of mind-independence must be no, correct?

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