(September 13, 2018 at 8:02 pm)polymath257 Wrote: Let me put it this way. If everyone agrees what it is to be the number 4, what is it, precisely?

"Everyone agreeing" is not what is happening when we consider the concept of 4 objects. We discover the concept. The word we use is irrelevant.

(September 14, 2018 at 8:34 am)polymath257 Wrote:

(September 14, 2018 at 8:19 am)RoadRunner79 Wrote: Here you are arguing that they are ontologically objective.  The number of things did not change and was not effected by the subject or their opinions. We are describing something external and not internal to the subject.

No, I *am* saying the number is subjective: it does depend on the observer and their biases.

But, more importantly, the actual identity of the number 4 is not agreed upon. Is it the set {{{{}}}}? Or is it the set {0,1,2,3}, where 3={0,1,2}, 2={0,1}, 1={0}, and 0={}?

Or do we use the notion in the integers (equivalence classes of order pairs of natural numbers)? Or in the rational numbers (equivalence classes of order pairs of integers)? Or in the real numbers (Dedekind cuts? equivalence classes of Cauchy sequences?)? Or maybe in the complex numbers (ordered pairs of real number? Or the algebraic quotient field obtained from real polynomials after modding out by x^2 +1?)

Or do we simply mean SSSS0 in an inductive set with first element 0 and successor function S (in which case, there are many, many, many different specific notions)?

Or do we count in Spanish... you seem to be hung up on language again.

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(September 14, 2018 at 8:52 am)Jörmungandr Wrote:

(September 13, 2018 at 1:15 pm)RoadRunner79 Wrote: I’m not trying to guess where you are going. As I said, I don’t understand what you are saying.

A simple answer to your question here, I do believe that the whole exists and the parts of things exist, outside of human conception. I believe that hydrogen and oxygen exist, as well as the combination of them known as water, apart from a persons knowledge of it.

Okay, a simple thought experiment.  Let's say that you and I each have a lump of gold, me at my location, and you at yours.  Now we know that these two lumps of gold are not part of the same lump of gold, the atoms from each are separated by considerable distance.  Any two atoms in the respective lumps of gold are not part of the same lump of gold.  Now take a single lump of gold.  There will be two atoms in it that are adjacent to each other.  Both atoms are a part of the same lump of gold.  Between the atoms is empty space.  Now suppose we start gradually expanding the space between the atoms, first by 5%, then 10%, eventually by 100%, 200%, 3000% -- eventually the individual atoms will be as far apart as the lumps of gold in our respective pockets.  Keep expanding and eventually the two atoms are farther apart than the entire width of the universe.  The immediate question that comes to mind is, at what point do the two atoms stop being a part of the same lump of gold, but more importantly, why?  What is it about the distance between them that makes one pair of atoms, separated by an arbitrary amount of empty space, different from two other atoms, equally separated by an arbitrary amount of empty space?  This is similar in many ways to what is known as the Problem Of The Many, and as a glance at that page from the Stanford Encyclopedia of Philosophy will tell you, there are multiple incompatible answers to the paradox.  Some suggest that this is just an example of the sorites paradox or the paradox of the heap, that it is a problem in the vagueness of the boundary, that something changes, even though there is no clear line or point at which it changes.  Others disagree with this, and see the problem of the many differently.  One solution which occurs to me immediately is that when the atoms are close together, they interact with each other by way of forces such as gravity and the other four forces.  But this doesn't solve the problem, as no matter how far apart the atoms are, they still interact, albeit weakly, it is only a change in the strength of the interaction.  To argue that it has to do with interaction would then be to assert that there is an arbitrary strength of interaction which defines whether two atoms belong to the same object or not.

Now, I don't really expect to convince you that parts and wholes do not exist.  I suspect your intuitions about the matter, as well as perhaps your prior philosophical commitments make that unlikely.  But if you'll recall, the original complaint was that the position that number, which requires parts and whole distinctions, being subjective was not so far fetched that, as Neo put it, no rational person would believe that number, like parts and wholes, is subjective.  At minimum, I think this example, as well as the problem of the many, is not as far out as your intuitions, Steve's hyperbole, and Neo's contemptuous remarks made it out to be.  If not, then what is your answer to the lump of gold problem, and your answer to the problem of the many?  (And ultimately all objects are like the cloud in the problem of the many, a cloud of particles, interacting in various strengths through various forces.  What makes the cloud of atoms that is my desk a whole with parts, and a cloud composed of water droplets problematic?  There is the problem of the boundary in the case of the desk, as the surface of the desk trails off and it becomes difficult to say which atoms are a part of the desk, and which are not, and that indeed may be an example of the sorities paradox, yet the question remains, what makes this particular "desk-like ensemble of particles" a whole in the first place?)

I don't think that I understand how this has anything to do with whether the nature of numbers is objective or not.

(September 14, 2018 at 8:58 am)RoadRunner79 Wrote:

(September 14, 2018 at 8:34 am)polymath257 Wrote: No, I *am* saying the number is subjective: it does depend on the observer and their biases.

But, more importantly, the actual identity of the number 4 is not agreed upon. Is it the set {{{{}}}}? Or is it the set {0,1,2,3}, where 3={0,1,2}, 2={0,1}, 1={0}, and 0={}?

Or do we use the notion in the integers (equivalence classes of order pairs of natural numbers)? Or in the rational numbers (equivalence classes of order pairs of integers)? Or in the real numbers (Dedekind cuts? equivalence classes of Cauchy sequences?)? Or maybe in the complex numbers (ordered pairs of real number? Or the algebraic quotient field obtained from real polynomials after modding out by x^2 +1?)

Or do we simply mean SSSS0 in an inductive set with first element 0 and successor function S (in which case, there are many, many, many different specific notions)?

Or do we count in Spanish... you seem to be hung up on language again.

Ahh...but my examples are NOT just different languages. They are different fundamentally and are *all* definitions of the *number* 4 in different contexts.

I am hung up on language because mathematics *is* a language. And the number 4 is word in that language. It doesn't have an independent existence: it is a language construct. And, in math, different aspects of the language have wildly different *definitions* of the *number* 4.[/quote]

(September 14, 2018 at 8:58 am)RoadRunner79 Wrote:

(September 14, 2018 at 8:52 am)Jörmungandr Wrote: Okay, a simple thought experiment.  Let's say that you and I each have a lump of gold, me at my location, and you at yours.  Now we know that these two lumps of gold are not part of the same lump of gold, the atoms from each are separated by considerable distance.  Any two atoms in the respective lumps of gold are not part of the same lump of gold.  Now take a single lump of gold.  There will be two atoms in it that are adjacent to each other.  Both atoms are a part of the same lump of gold.  Between the atoms is empty space.  Now suppose we start gradually expanding the space between the atoms, first by 5%, then 10%, eventually by 100%, 200%, 3000% -- eventually the individual atoms will be as far apart as the lumps of gold in our respective pockets.  Keep expanding and eventually the two atoms are farther apart than the entire width of the universe.  The immediate question that comes to mind is, at what point do the two atoms stop being a part of the same lump of gold, but more importantly, why?  What is it about the distance between them that makes one pair of atoms, separated by an arbitrary amount of empty space, different from two other atoms, equally separated by an arbitrary amount of empty space?  This is similar in many ways to what is known as the Problem Of The Many, and as a glance at that page from the Stanford Encyclopedia of Philosophy will tell you, there are multiple incompatible answers to the paradox.  Some suggest that this is just an example of the sorites paradox or the paradox of the heap, that it is a problem in the vagueness of the boundary, that something changes, even though there is no clear line or point at which it changes.  Others disagree with this, and see the problem of the many differently.  One solution which occurs to me immediately is that when the atoms are close together, they interact with each other by way of forces such as gravity and the other four forces.  But this doesn't solve the problem, as no matter how far apart the atoms are, they still interact, albeit weakly, it is only a change in the strength of the interaction.  To argue that it has to do with interaction would then be to assert that there is an arbitrary strength of interaction which defines whether two atoms belong to the same object or not.

Now, I don't really expect to convince you that parts and wholes do not exist.  I suspect your intuitions about the matter, as well as perhaps your prior philosophical commitments make that unlikely.  But if you'll recall, the original complaint was that the position that number, which requires parts and whole distinctions, being subjective was not so far fetched that, as Neo put it, no rational person would believe that number, like parts and wholes, is subjective.  At minimum, I think this example, as well as the problem of the many, is not as far out as your intuitions, Steve's hyperbole, and Neo's contemptuous remarks made it out to be.  If not, then what is your answer to the lump of gold problem, and your answer to the problem of the many?  (And ultimately all objects are like the cloud in the problem of the many, a cloud of particles, interacting in various strengths through various forces.  What makes the cloud of atoms that is my desk a whole with parts, and a cloud composed of water droplets problematic?  There is the problem of the boundary in the case of the desk, as the surface of the desk trails off and it becomes difficult to say which atoms are a part of the desk, and which are not, and that indeed may be an example of the sorities paradox, yet the question remains, what makes this particular "desk-like ensemble of particles" a whole in the first place?)

I don't think that I understand how this has anything to do with whether the nature of numbers is objective or not.

Can you count without making a figure / ground distinction? If you have four, does not that require you to conceive of a "set" of objects which are parts of a whole? Number by its very nature is a relationship built on parts & wholes. If you only have one, undiffferentiated whole with no parts, how can you have number? Number only exists in so far as we can posit identities (another concept dependent on parts & wholes), sets (parts belonging to a whole), and relations (reflexive consideration of the set of objects both as parts [multiple members of the set] and as a whole [as composing the set as a whole]). Without part and whole relationships, we don't have sets, and without sets, we don't have number. So you have to ground what makes something a part and something a whole in order to talk about sets, and you have to be able to have sets to have number. The concepts are dependent on each other. To put it in concrete terms, I can't talk about four apples if I can't identify individual apples as being wholes (an apple) and as being parts belonging to a set (the collection of four apples), which are part of a whole (a universe that has parts that are not apples). So you can't have number without part / whole relationships. Until you ground the concept and application of part / whole distinctions as objective, you can't ground number as objective. So I'll ask you to reconsider my prior post in that light and answer the questions.

(September 14, 2018 at 9:27 am)polymath257 Wrote:

(September 14, 2018 at 8:58 am)RoadRunner79 Wrote: Or do we count in Spanish... you seem to be hung up on language again.

Ahh...but my examples are NOT just different languages. They are different fundamentally and are *all* definitions of the *number* 4 in different contexts.

I am hung up on language because mathematics *is* a language. And the number 4 is word in that language. It doesn't have an independent existence: it is a language construct. And, in math, different aspects of the language have wildly different *definitions* of the *number* 4.

Using a language construct doesn't mean that what you are describing is subjective in it's nature. We use language to convey things. Perhaps sometimes that language is not a precise as we would like, or a word can have multiple meanings which we have to discern. I'm not talking about the language (and I don't think the others are either).

(September 14, 2018 at 8:58 am)RoadRunner79 Wrote: I don't think that I understand how this has anything to do with whether the nature of numbers is objective or not.

Numbers can provide epistemic objectivity while being simultaneously ontologically subjective. In the least restrictive view of subjectivity, concepts are necessarrily subjective. In the most restrictive view, concepts are deeply biased by subjectivity.

Consider the position this puts us in. There is a range between the most and least restrictive sensible descriptions...but that range goes from necessarily subjective to incidentally but meaningfully subjective. It doesn't get us over the hill and into ontological objectivity. If, like me, you prefer the most restrictive view of subjectivity (to cut out any number of meaningless or trivial subjectivities, for example)...then the best we can say is that our deeply subjective concepts -seem- to have an objective referent. That doesn't make the concepts themselves any less subjective, and the concepts themselves are not the thing to which they refer. They are abstractions of varying accuracy and clarity. That makes them sufficient as a basis for epistemic claims, but it doesn't certify ontological claims.

@Roadrunner:

I added the following paragraph after you read and replied to my earlier post. I think it adds some context and explains where I'm coming from, so I'll repeat it for your benefit.

(September 14, 2018 at 8:52 am)Jörmungandr Wrote: Ultimately, as I said to Neo in the thread on delusion and religion, number, and the concepts it is dependent on, are a mystery. I can suggest that number, being an example of reasoning using parts and wholes, only exists in so far as we make arbitrary identity judgements, about what is a part of what, and what is a whole. But our intuitions tend to marshall against us, at the very least, and it's not entirely clear that number is subjective in its entirety, that it doesn't have an independent, objective substance of some sort. Yet when we go the other direction, and assert that number, and part/whole distinctions are objective, we run into problems in that direction as well, problems which seem equally intractable. So we're left with a mystery, I think, and to declare that number, or parts & wholes, is definitely objective, is, to my mind, to embrace an opinion that is not in any sense fully justified. At minimum, if you can't prove that number is objective, that leaves the door open, no matter how slightly, that number is subjective, as it must be one or the other, it can't be both. So, QED, as it were, I think I've shown that number and parts & wholes being a product of mind is not a view that is as far fetched as Neo and Steve made it sound. If you disagree, please explain why.

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(September 14, 2018 at 8:57 am)SteveII Wrote:

(September 13, 2018 at 8:02 pm)polymath257 Wrote: Let me put it this way. If everyone agrees what it is to be the number 4, what is it, precisely?

"Everyone agreeing" is not what is happening when we consider the concept of 4 objects. We discover the concept. The word we use is irrelevant.

Given your views on the subject, I'd like to solicit your response to the questions and arguments I've presented to Roadrunner.

Here and here.

Numbers are part of abstract systems we create, which can have whatever rules we want.

These systems may or may not be useful for modelling reality. The former is applied mathematics. But the numbers themselves only “exist” within the pure mathematical framework we are using*.

The thing is, people are so used to using simple maths to model reality, they sometimes forget they are doing it at all.

*barring some amazing coincidence where they somehow manifest somewhere

It's easy to see why we conflate the two.  Our model is our access to whatever reality lies beyond it...and it's good enough to catch some dinner and step over that snake...so....

(September 14, 2018 at 6:15 am)RoadRunner79 Wrote:

(September 14, 2018 at 5:27 am)Abaddon_ire Wrote: Does it? why does the "4" in 408 equal 32?

Different way of explaining or referencing the same thing.

So the "4" is simply a symbolic representation, right?