logical bob wrote:

JamesSS wrote:In the Conceptual Realm, each concept affects the others. A straight line affects what a square is. A curve affects what a circle is. And it is all merely a matter of convenience of thought. Without the Conceptual Realm, how does one speak of an ideal circle that can never physically exist?

Alas, this is nonsense.

A circle is the set of points in the plane that are equidistant from a specified centre. It can be shown that a circle with radius r centred at (a,b) is the set of points (x,y) that satisfy

(x-a)2 + (y-b)2 - r2 = 0

and as such circles are a particular example of algebraic curves, a curve being a set of points satisfying an arbitrary polynomial equation.

Which can never physically exist.

logical bob wrote:To say that a curve "affects what a circle is" is hopelessly vague and woolly.

It isn't something that I would normally bother to say. It merely maintains the consistency of the concept of a realm of existence.

logical bob wrote:Suppose there was a set bigger than the set of integers yet smaller than the set of real numbers [easy to do]. That would have some serious implications, so I suppose that (if we could straighten out your idea of how one mathematical object affects another to get something coherent) you'd want to say that if affected other things, and therefore existed [within that realm]. The problem is that it doesn't exist. Using the normal mathematical rules it's impossible to prove or disprove the existence of such a set.

You seem to be ignoring yourself a little.

You say that such a set doesn't "exist". But then you immediately say that such a set cannot be proven to exist or not.

Proof is in the eyes of the beholder, so I'm not sure what logic you forbid such as to disallow such a proof. Do you disallow cardinalities of infinity, or hyperreal numbers? Those are issues of logic, thus I can't disallow them. Math is supposed to just be logic applied to quantities, but some people prefer to see it as merely a set of authorized chosen rules disregarding any association with logic, thus there might be a few holes that disallow completeness and proofs.

logical bob wrote:
How are you going to account for the difference between mathematical objects whose existence has consequences and mathematical objects whose existence would have consequences without a better criterion for determining which ones exist?

Any defined concept that isn't an oxymoron "exists" as a concept. The concept might be unusable in your application, but it would still exist as a concept. So I don't know to what you are referring. Any change in logical association creates consequences. So I don't see how you can have any mathematical set that can be altered and yet not cause logical consequence to everything else. If you declare that 2+2=3, you change all mathematics, not merely that one equation.

Sendraks wrote:Your explanation was clear, as opposed to the word salad I'm used to seeing in the philosophy forum, I'm grateful for that.

Well, thank you, but actually today I feel like I am struggling for words and rambling a bit.

jamest wrote:Then you're missing the point of this thread, since I'm obviously talking about a coherent definition acceptable to philosophers.

Why bother? Do philosophers even exist?

JamesSS wrote:The concept might be unusable in your application, but it would still exist as a concept.

So, to sum up, anything exists (you know, as long as it's conceivable), and we can dispense with existence and move right along to application. Bring on the relata of a property exemplification nexus, and Yablo conceivability, FTW.

Now, a few words about objects and properties: some objects, like the rock you chose to discuss, exist somewhere, an idea you might have run across had you studied some philosophy. Other entities (shall we call them) don't exist somewhere. That is presumably why you chose conceivability, so you could wrap up all of ontology in a sentence or two, which is not merely lazy, but par for the course in internet philosophy. Misery loves company, don't it?

You and jamest, summing up existence in a sentence or two. One alternative is what I recommend, which is to say that working up a sweat on a technical definition of existence is like pushing on a locked door. In fact, you can work up a sweat, simply typing. That's what I do, but I don't pretend to be getting anywhere, because the doors I'm pushing on (those that open on the interior walls of serious ontologists) are locked from the inside.

Cito di Pense wrote:

JamesSS wrote:The concept might be unusable in your application, but it would still exist as a concept.

So, to sum up, anything exists (you know, as long as it's conceivable), and we can dispense with existence and move right along to application. Bring on the relata of a property exemplification nexus, and Yablo conceivability, FTW.

..as long as it is kept in mind as to which realm it belongs in. Today with science being promoted so greatly, the physical realm of existence with which science can deal is often promoted as the only realm.

Cito di Pense wrote:Now, a few words about objects and properties: some objects, like the rock you chose to discuss, exist somewhere, an idea you might have run across had you studied some philosophy. Other entities (shall we call them) don't exist somewhere. That is presumably why you chose conceivability, so you could wrap up all of ontology in a sentence or two, which is not merely lazy, but par for the course in internet philosophy. Misery loves company, don't it?

You and jamest, summing up existence in a sentence or two. One alternative is what I recommend, which is to say that working up a sweat on a technical definition of existence is like pushing on a locked door. In fact, you can work up a sweat, simply typing. That's what I do, but I don't pretend to be getting anywhere, because the doors I'm pushing on (those that lead to whatever you guys are thinking) are locked.

Some people attempt to fly and only "work up a sweat" while others invent the airplane. It kind of depends on attitude more than altitude.

JamesSS wrote:

Cito di Pense wrote:

JamesSS wrote:The concept might be unusable in your application, but it would still exist as a concept.

So, to sum up, anything exists (you know, as long as it's conceivable), and we can dispense with existence and move right along to application. Bring on the relata of a property exemplification nexus, and Yablo conceivability, FTW.

..as long as it is kept in mind as to which realm it belongs in. Today with science being promoted so greatly, the physical realm of existence with which science can deal is often promoted as the only realm.

Yeah, I had a sneaking suspicion there was another realm you wanted to tell us about.

I don't know what you're on about with all these references to 'promotion'. You make it seem like some kind of advertising campaign. Is that what it is? I never got that memo.

JamesSS wrote:
Some people attempt to fly and only "work up a sweat" while others invent the airplane. It kind of depends on attitude more than altitude.

Yeah, JamesSS. Stick with that realm, and you'll go far. As long as your airplane goes 'up' instead of 'out there'.

JamesSS wrote:
You say that such a set doesn't "exist". But then you immediately say that such a set cannot be proven to exist or not.

You seem to have your own idea of 'existence' and seek to impose it on mathematics. However, in maths, if you can't prove it exists, it doesn't exist. Such statements are known as 'unproven conjectures' or something like that. Statements aren't entities in the same sense as your rock example. You can make certain kinds of statements, but you'll just be flapping your gums or rattling your keyboard. The best you can hope for in some circumstances is that you'll give someone a laugh.

Proofs that there are no solutions to some equation can go on at some length.

Now, a few words about objects and properties: some objects, like the rock you chose to discuss, exist somewhere, an idea you might have run across had you studied some philosophy.

Or perhaps studied an upper ontology. This one says that objects are always physical, and thus always have a spacetime location. But there are also processes, which have a spacetime location but aren't objects.

The definitions here were settled on by various groups wanting to write internet services that responded to requests, data and everyday circumstances with a modicum of common sense about what the requests and data mean. The result was a mad proliferation of standards, with further standardisation destined to just make things worse.

Now what some researchers seriously want the internet services to do is intelligently figure out how to match two competing definitions of the same domain, or figure out how to diagnose such a mismatch when communicating with another service.

Fucking hell.

JamesSS wrote:Which can never physically exist.

True, but I thought we were discussing mathematical existence.

You say that such a set doesn't "exist". But then you immediately say that such a set cannot be proven to exist or not.

A mathematician may only invoke an object that exists, so if you can't prove it to exist you can't make use of it. Although the set in question can't be proven not to exist, it can't be proven to exist either, so use cannot be made of it, and it cannot "affect" anything.

Proof is in the eyes of the beholder...

It really isn't, you know.

... so I'm not sure what logic you forbid such as to disallow such a proof.

The question is called the Continuum Hypothesis. Look it up if you;re interested. The method that proves its inaccessibility to proof or disproof is a technique in set theory called forcing. I confess I don't understand the details myself.

Do you disallow cardinalities of infinity, or hyperreal numbers?

No.

Those are issues of logic, thus I can't disallow them. Math is supposed to just be logic applied to quantities, but some people prefer to see it as merely a set of authorized chosen rules disregarding any association with logic, thus there might be a few holes that disallow completeness and proofs.

You clearly have no idea what you're talking about.

Any defined concept that isn't an oxymoron "exists" as a concept.

What about "largest prime number?" Does that exist as a concept or is it a very subtle oxymoron?

Any change in logical association creates consequences. So I don't see how you can have any mathematical set that can be altered and yet not cause logical consequence to everything else. If you declare that 2+2=3, you change all mathematics, not merely that one equation.

So do you think mathematicians can distinguish between what, in a mathematical sense, exists and what doesn't?

Do imaginary numbers exist?

JamesSS wrote:

Sendraks wrote:

JamesSS wrote:
Rocks stop/block things. It has the property of inertia (or momentum if moving). If you take that inertia away, it stops or blocks nothing. Also take away its gravitational affect and it is left with no affect at all. It doesn't even have shape any more. It no longer exists.

You're simply talking about removing the rock. Which would have the same effect as removing its behaviour. The talk of removing the behaviour is redundant.

My point was that if you say that a rock is not equivalent to its behavior, then you should be able to conceptually remove its behavior and still have something left, yet you don't.

You might have something left. You might have the knowledge of that rock left. That knowledge might remain as an abstract constructor in someone's head or elsewhere. Not in another realm - what a quaint idea, because all abstract constructors have some physical reality and are most assuredly in this "realm".

logical bob wrote:

JamesSS wrote:Which can never physically exist.

True, but I thought we were discussing mathematical existence.

You say that such a set doesn't "exist". But then you immediately say that such a set cannot be proven to exist or not.

A mathematician may only invoke an object that exists, so if you can't prove it to exist you can't make use of it. Although the set in question can't be proven not to exist, it can't be proven to exist either, so use cannot be made of it, and it cannot "affect" anything.

Proof is in the eyes of the beholder...

It really isn't, you know.

... so I'm not sure what logic you forbid such as to disallow such a proof.

The question is called the Continuum Hypothesis. Look it up if you;re interested. The method that proves its inaccessibility to proof or disproof is a technique in set theory called forcing. I confess I don't understand the details myself.

Do you disallow cardinalities of infinity, or hyperreal numbers?

No.

Those are issues of logic, thus I can't disallow them. Math is supposed to just be logic applied to quantities, but some people prefer to see it as merely a set of authorized chosen rules disregarding any association with logic, thus there might be a few holes that disallow completeness and proofs.

You clearly have no idea what you're talking about.

Any defined concept that isn't an oxymoron "exists" as a concept.

What about "largest prime number?" Does that exist as a concept or is it a very subtle oxymoron?

Any change in logical association creates consequences. So I don't see how you can have any mathematical set that can be altered and yet not cause logical consequence to everything else. If you declare that 2+2=3, you change all mathematics, not merely that one equation.

So do you think mathematicians can distinguish between what, in a mathematical sense, exists and what doesn't?

Is JamesSS arguing in favour of some Mathematical Constructivism?

LucidFlight wrote:Do imaginary numbers exist?

If that's your bent.

Let's go back to talking about writing on the other side of the blackboard. (I'm not talking about those flippy ones that you can spin on an axis parallel to the floor.) It would be like exposing the soft underbelly of inscriptions.

Cito di Pense wrote:

LucidFlight wrote:Do imaginary numbers exist?

If that's your bent.

...

IF THAT'S YOUR BENT!? Is that the best you can do? How about "Imaginary numbers aren't really imaginary in the common, non-mathematical sense of the word."

logical bob wrote:A mathematician may only invoke an object that exists, so if you can't prove it to exist you can't make use of it. Although the set in question can't be proven not to exist, it can't be proven to exist either, so use cannot be made of it, and it cannot "affect" anything.

In that kind of case, what is it that you are calling existent and non-existent? To me, any concept that is defined exists as a concept. So what are you saying doesn't conceptually exist? Do infinite sets exist? What is your proof for them?

To me, the set of all integers, "n's", added to the set of all "n+0.5"'s forms a set of numbers that is greater than the set of all integers but less than the set of all reals. But you are saying that some part of that cannot be proven to exist. So what do you mean by "exist"?

logical bob wrote:

Proof is in the eyes of the beholder...

It really isn't, you know.

Then there is no such thing as proof. To prove is to convince beyond doubt. You can say that you have logically or mathematically proven something, but if no one knew how to read your proof, to whom have you proven it? If no one accepts your axioms, you haven't proven anything even if your axioms were true.

... so I'm not sure what logic you forbid such as to disallow such a proof.

The question is called the Continuum Hypothesis. Look it up if you;re interested. The method that proves its inaccessibility to proof or disproof is a technique in set theory called forcing. I confess I don't understand the details myself.

The CH is an issue with mathematicians involved with ZFC theory, and expresses one of the reasons why mathematicians and physicists should stay away from metaphysics - what exists or doesn't exist. ZFC is an effort to avoid all paradoxes. The problem is that paradoxes don't actually exist in the first place. A paradox is merely a mind game that gives the first appearance of an impossibility. When the logic is worked out properly, the paradox disappears.

In their case, Russel's paradox concerning sets that don't include themselves apparently had them fooled (easy to do with mathematicians and physicists). The problem is that Russel's paradox is merely another case of an incoherent definition ("oxymoron"), which disqualifies it as a definition of any "thing", such as a "square circle" or in Russel's case, "a set of all sets that do not include themselves". It is easy to formulate a self contradicting declaration, eg. "this sentence doesn't exist", "X is not X". These are nonsense, oxymoron statements, not actual paradoxes.

And the CH specifically states that there is no CARDINALITY between the set of all integers and the set of all reals. It isn't that there are no sets between the two. It is a matter of the definition of "cardinality". And in this case, the issue resolves as one of defining "infinite".

Cardinality designates the number of elements in a set. But when it comes to an infinite set, there is a problem: infinity comes in varied sizes yet all are called "infinite". If you do not specify which size of infinite you are talking about, all kinds of illogic can arise (as you can find in many wiki nonsense proofs).

If I specify an infinite set of apples and another infinite set of oranges, together I have an infinite set that is twice the size of either original infinite set. That is a set that has a cardinality between the set of all integers and that of all reals. And the reason that works is that by stating a set of all of a physical item, I have specified a specific infinity:

InfiniteA == [A,A,A,...A] numbering a quantity equal to [1+1+1+...+1]
InfiniteB == [B,B,B,...B] also numbering a quantity equal to [1+1+1+...+1]

And then the sum of them has twice the number/quantity of elements:
[A,B+A,B+A,B+...A,B] numbering a quantity equal to;
[1+1+1+...+1] + [1+1+1+...+1] = [2+2+2+...+2] - as a total quantity of elements thus a cardinality.

But the set of all reals is infinitely larger than that set:
[infiniteA+infiniteA+infiniteA+...+infiniteA]

So the whole CH concern is a bit silly (as are all proposed actual paradoxes).

logical bob wrote:You clearly have no idea what you're talking about.

Quite the contrary it appears.

logical bob wrote:

Any defined concept that isn't an oxymoron "exists" as a concept.

What about "largest prime number?" Does that exist as a concept or is it a very subtle oxymoron?

The concept of infinity would make it appear as though the "largest number" is a definite oxymoron, because infinite MEANS no largest number. But I don't know that I could personally ever prove that there isn't a largest "prime number", even though it seems intuitive that there wouldn't be. Although I seriously doubt it, perhaps there is some phenomenon that happens concerning extreme prime numbers.

So in the interim, I would feel safe calling it an oxymoron.

logical bob wrote:
So do you think mathematicians can distinguish between what, in a mathematical sense, exists and what doesn't?

As explained above .. NO!.

Russel was a logician. Logicians toy with mathematicians and physicists.

LucidFlight wrote:Do imaginary numbers exist?

Do they affect anything? If you took them out of the set of all possible numerical concepts mathematics would look a little different, but would it actually say anything different than it had before?

The square root of a negative number is an oxymoron because it requires that it be able to be multiplied by itself and render a negative value. That is the same as having a square-circle. It is a self-contradicting definition. So no, they do not "exist".

John Platko wrote:You might have something left. You might have the knowledge of that rock left.

The knowledge of something is not a property of that thing, but a property of the knower.

John Platko wrote:That knowledge might remain as an abstract constructor in someone's head or elsewhere. Not in another realm - what a quaint idea, because all abstract constructors have some physical reality and are most assuredly in this "realm".

Again, it is an ontological choice of convenience to refer to a realm specified as a set of all "too complicated to keep having to explain the physical existence of" items. A realm is merely a specified set or category. Concepts CAN BE represented in someone's brain. But does that mean that a circle never existed until someone thought of it? What about the "laws of physics"? Did they not exist until Newton? You can claim that if you want to, but it won't change anything.

An ontology is a choice of language. It is a "true" ontology as long as it is:

1) consistent/coherent
2) comprehensive
3) relevant

Many varied ontologies can be true as long as you don't mix them (such as physics speaking of angels or gods).

JamesSS wrote:

LucidFlight wrote:Do imaginary numbers exist?

Do they affect anything? If you took them out of the set of all possible numerical concepts mathematics would look a little different, but would it actually say anything different than it had before?

The square root of a negative number is an oxymoron because it requires that it be able to be multiplied by itself and render a negative value. That is the same as having a square-circle. It is a self-contradicting definition. So no, they do not "exist".

Protip, JameSS:
Demonstrating ignorance of a subject is not a very good way to get people to take you seriously when you're talking about that subject. Imaginary numbers are essential to mathematics. If you don't understand why, well, I'll be in the peanut gallery, pointing and laughing.

Here's a video. I didn't know there was such a handy video before I did a quick search, just now. Because we're on the internet right now and that means there's no excuse for ignorance.

ScholasticSpastic wrote:
Protip, JameSS:
Demonstrating ignorance of a subject is not a very good way to get people to take you seriously when you're talking about that subject. Imaginary numbers are essential to mathematics. If you don't understand why, well, I'll be in the peanut gallery, pointing and laughing.

Here's a video. I didn't know there was such a handy video before I did a quick search, just now. Because we're on the internet right now and that means there's no excuse for ignorance.

You might want to take your own advice regarding that. Your video has two problems:
1) It doesn't show except on a Youtube website
2) It doesn't address the topic of existence.

So I would suggest that before you accuse of ignorance, you learn something of the topic of discussion.

JamesSS wrote:
You might want to take your own advice regarding that. Your video has two problems:
1) It doesn't show except on a Youtube website
2) It doesn't address the topic of existence.

So I would suggest that before you accuse of ignorance, you learn something of the topic of discussion.

1) Why is showing on Youtube a problem?
2) It directly addresses a statement which you made that demonstrates a profound ignorance of mathematics. Ignorance of sufficient depth that we needn't concern ourselves with resources more advanced than Youtube until you've managed to make some improvements in your understanding of the subject matter, or until you learn not to speak about subjects about which you know so little.

If this is the extent of the rigor with which you intend to argue your points, I commend you to the philosophy subforum, for it is, indeed, a worthy home for you. If I see you in the science subfora, well..... I probably won't be the first shark to take the chum.

ScholasticSpastic wrote: It directly addresses a statement which you made that demonstrates a profound ignorance of mathematics.

Certainly not true. But you obviously get a sociopathic turn on by making such accusations.

So in other words, you know so little, that you can only proclaim that others know too little?

Is that your trip? Because you are obviously not adding content to this discussion.

And the CH specifically states that there is no CARDINALITY between the set of all integers and the set of all reals. It isn't that there are no sets between the two. It is a matter of the definition of "cardinality". And in this case, the issue resolves as one of defining "infinite".

That shows you, logical bob. Did you not realise the CH is about CARDINALITY, not Russell's paradox! And CARDINALITY is about INFINITY, which is a problem, like IMAGINARY NUMBERS. And there may or may not be a largest prime because that's about number theory.