What is Existence? - Existence as truth, relative to systems
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13-10-2016, 11:32 AM
What is Existence? - Existence as truth, relative to systems
One of the questions that comes up when talking about the existence of gods is the meaning of existence itself. How can a god "exist" outside space and time? What does "outside" even mean? And if it "exists" outside of the universe, which is "all that exists", does that even qualify as existence? What I aim to do is craft a well-defined meaning of existence, one that can answer these questions. The definition may not be rigorous, because I believe rigor is left to mathematics, which I'm not knowledgeable enough to use solely, but hopefully the way I define and use it will be clear. Also, I should note that I have no formal education in philosophy nor in rigorous, proof-based mathematics.

First, I'm going to begin with a reference to the Merriam-Webster definition of "system" because I think it allows for an intuitive understanding (but I do not continue to use it by this strict definition):

----"1: a regularly interacting or interdependent group of items forming a unified whole <a number system>"

The systems I'm going to use to illustrate my point are games: specifically, chess and Minesweeper. When creating a game, you can say, generally, that you start by defining some objects, the environment, and the rules about how they interact. If I create chess, I might say, "This game is played on an 8-by-8 board (environment), and you use 16 pieces (objects) to try and checkmate your opponent..." and go on explaining the rules of how to move each piece and what a checkmate is, etc. etc. What I want to argue first is that the objects and environment of the game are not independent of the rules, rather the objects and environment are defined by the rules.

For humans, we naturally create and treat things as objects with inherent properties, so we first think of things like "pieces" and "boards" before what makes those things what they are. For example, we think of a chair as a "chair" before thinking of it as a block of wood meticulously carved for the purpose of sitting. If you're not sure what I mean, ask yourself: "What is a 'piece' or a 'board'"? You can't define them, can you? Not without understanding the rules of the game already. The rules define what a 'piece' and 'board' is because the rules define all of the relationships and interactions of the system. It is only after the relationships and interactions have been formed by the rules that you can isolate specific patterns, such as "movable, able to be removed, can remove other identical patterns" and define this pattern as an object that can be called a "piece".

(For more depth, think about mathematics rather than games. Mathematics is constructed by axioms which describe fundamental relationships, rather than objects. Here is an example from Calculus, Volume 1: One-Variable Calculus, with An Introduction to Linear Algebra authored by Tom M. Apostol":

----"AXIOM 4. EXISTENCE OF IDENTITY ELEMENTS. There exist two distinct real numbers, which we denote by 0 and 1, such that for every real x we have x + 0 = x and 1 * x = x."

Using this axiom, we must first recognize the relationships "x + 0 = x" and "1 * x = x" and then from recognizing that relationship, we can form the idea of identity elements. Also, the denotation of 0 and 1, although mentioned before the equations, happens after the discovery of such elements, because they need to be communicated. Also, notice that the idea of "real numbers" is assumed in this axiom, but for completeness, one would have to recognize the axiomatic relationships that define real numbers as well.)

So now it is time to define existence. Curiously, if you take a look at the example used in the "further depth" section, you'll notice the description of the axiom uses the words "existence" and "exist". And I believe the way it used there is very accurate. To exist simply means to be a relationship in a system. That relationship can be a rule or axiom, which exists necessarily, for they are the fundamental, the defining relationships of systems, or the relationship can be derived from the rules. Things we turn into objects such as "pieces" and "boards" are very fundamental, defined directly by the rules. However, I can say "the position of a rook being next to a knight" exists just as well, because I could use the rules of the system to prove this relationship.

One thing you may notice is my usage of the term "prove". Proofs concern truths, so how did talk of existence bring up the topic of logic and truth? It is because "existence" is indistinguishable from "truth". The rules and axioms that define the relationships of systems are truth. And "using" the rules to find relationships means to apply logic to find more complex truths. This is precisely the way you can use logic to manipulate axioms to produce new results, how mathematics proves theorems. And whatever relationships you can't create by logically manipulating the rules, those relationships are false. They do not exist. . .or do they?

You see, if we're defining existence by a relationship in a system, then it seems that existence is relative to the system. For example, let's come back to our two games again, Minesweeper and chess. It is accurate to say that, given the rules, a "rook" exists in chess, because its relationship can be defined. However, a rook does not exist in Minesweeper. Similarly, mines do not exist in chess because there is no relationship in chess that defines the same properties a mine has in Minesweeper. So, do they exist or not? It seems like the obvious answer is that "rooks exist in chess, but not Minesweeper" and "mines exist in Minesweeper, but not chess". It doesn't seem like a problem, but consider this. To us, the humans that play these games, both the mines and rooks seem to exist, because the games themselves exist, which would imply everything in those games exist. You may see where this line of thinking will lead us, bet let's continue on to the "real" stuff now. Not games and math, the universe and god. And actually, it gets a lot easier.

Because the universe is a system, most people agree. There is the question of whether it's deterministic, how to assess the results of quantum mechanics, etc., but despite the seemingly probabilistic nature of the universe, we have still discovered, to the best that we can, mathematical formulas to describe it. So it seems like the universe as a system is a clear idea. And if it is the case that the universe is a system, then that means we have defined how things exist as relationships derived from the complex rules that govern the universe. Now that we have a proper definition of existence, let's have a look back at the questions we were pondering at the beginning.

"How can a god 'exist' outside space and time? What does "outside" even mean?"

Using our view of systems, "outside" can mean "not a relationship in a particular system". Let's regard the rooks and mines again. From our perspective, in the system of the universe, chess and Minesweeper both exist. However, relationships in one may not exist in the other, yet they still exist. So we could say mines in Minesweeper do exist in our system "the universe", but "outside" of the system of chess, because although they exist in our system, they do not exist in the system of chess.
So two things: One, we are fairly certain that space and time exist in or relate in our universe. And two, space and time, which seemed to be necessary for the definition of existence, is actually not at all a part of this definition. However, since space and time are necessary parts of our system, to exist but not be related to space and time would mean that a god simply doesn't exist in our system. It could exist in some other system, or it could exist in an encompassing system, like how our universe encompasses chess and Minesweeper. But the key implication is that things are not required to exist in our universe to exist. But hold on, that seems weird right? Well let's answer the next question:

"If a god "exists" outside of the universe, which is "all that exists", does that even qualify as existence?

First things first, as we define universe this way, the universe is not "all that exists", it is simply the system that we exist in. But even so, if this is all we exist in, and we can't possibly relate to any system outside of our own, like how chess can't relate to Minesweeper, then should we even care about things that exist outside our universe?

This is a far, far different question than anything I've tackled in this post. All I've tackled up until now is ontological, metaphysical. The current question is of what value should we assess to things beyond our reach. Should we bother making beliefs about that which we can't observe, and can we truly know what seems unknowable? Maybe you're still wondering how we can even conceive of such a reality, yet be unable to attain it. Of course, maybe I made a fatal flaw somewhere down the line and the whole idea will be dismissed--by most, I feel.

I question the finite nature of our so-called "physical" reality, in the face of the infinite nature of abstract, mathematical reality. In the face of such a question, I wonder, if it is so simple to create our own lower systems, why we would not be part of a greater system. I wonder why the fundamental nature of mathematics is undiscoverable, as proved by Godel, and, by extension, learning about the fundamental nature of this universe be impossible if it was self-contained.

These things lead me to believe that we are not all there is to reality. That the universe is not equivalent to the cosmos. And I realize I've taken a departure from the topic at hand, simply defining existence, but the implications of this idea is incredible and drove me to open my world wider, even if I it is unknowable. I can no longer call myself an atheist, however, I can't also call myself a theist, not yet. Because while I believe in reality beyond our universe, I can not say I can believe that our universe was the product of an intelligence, certainly not sentient in the same way humans are. However, I do not close my mind to the idea. I'm willing to splash in the unknown and try to find the truths that can not be known until I'm satisfied with a proper worldview. As for what we can truly know, what is in our universe, what we can relate to and can discover, I'll leave that to the scientists. I'll listen to whatever new fundamental discoveries are made about our reality, and I will continue to wander in the dark abyss with only a lit match to light the way.
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13-10-2016, 12:06 PM
RE: What is Existence? - Existence as truth, relative to systems
I haven't read your entire post yet, but I have one quibble about your mathematical example. I am quite familiar with Apostol's book (I have been trying to work my way through it with no assistance since 1980! -- I have so far finished the introductory chapter and the first 10 numbered chapters, working every single problem along the way), and I was just reviewing the introductory chapter a few days ago. I also have some familiarity with axiomatic mathematics in general.

Now, Apostol takes pains to point out that no meaning should be attached to 0 and 1 other than what is specified by the axioms, and this is characteristic of axiomatic mathematics in general. It turns out that these are the familiar numbers 0 and 1 of our real number system, but that's not necessary. Once you study abstract algebra, you will find that it is possible to have identity elements (and entire structures) that are not the old familiar ones. I'm not sure how this affects your system, but I deny your claim that the identity elements have to be "recognized" before you can use them. I'm pretty sure Apostol explicitly states that this is not the case.

I may respond further once I have more time to read the rest of your post and think about it.
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13-10-2016, 01:13 PM
RE: What is Existence? - Existence as truth, relative to systems
(13-10-2016 11:32 AM)unknowndevil666 Wrote:  One of the questions that comes up when talking about the existence of gods is the meaning of existence itself. How can a god "exist" outside space and time? What does "outside" even mean? And if it "exists" outside of the universe, which is "all that exists", does that even qualify as existence? What I aim to do is craft a well-defined meaning of existence, one that can answer these questions. The definition may not be rigorous, because I believe rigor is left to mathematics, which I'm not knowledgeable enough to use solely, but hopefully the way I define and use it will be clear. Also, I should note that I have no formal education in philosophy nor in rigorous, proof-based mathematics.

First, I'm going to begin with a reference to the Merriam-Webster definition of "system" because I think it allows for an intuitive understanding (but I do not continue to use it by this strict definition):

----"1: a regularly interacting or interdependent group of items forming a unified whole <a number system>"

The systems I'm going to use to illustrate my point are games: specifically, chess and Minesweeper. When creating a game, you can say, generally, that you start by defining some objects, the environment, and the rules about how they interact. If I create chess, I might say, "This game is played on an 8-by-8 board (environment), and you use 16 pieces (objects) to try and checkmate your opponent..." and go on explaining the rules of how to move each piece and what a checkmate is, etc. etc. What I want to argue first is that the objects and environment of the game are not independent of the rules, rather the objects and environment are defined by the rules.

For humans, we naturally create and treat things as objects with inherent properties, so we first think of things like "pieces" and "boards" before what makes those things what they are. For example, we think of a chair as a "chair" before thinking of it as a block of wood meticulously carved for the purpose of sitting. If you're not sure what I mean, ask yourself: "What is a 'piece' or a 'board'"? You can't define them, can you? Not without understanding the rules of the game already. The rules define what a 'piece' and 'board' is because the rules define all of the relationships and interactions of the system. It is only after the relationships and interactions have been formed by the rules that you can isolate specific patterns, such as "movable, able to be removed, can remove other identical patterns" and define this pattern as an object that can be called a "piece".

(For more depth, think about mathematics rather than games. Mathematics is constructed by axioms which describe fundamental relationships, rather than objects. Here is an example from Calculus, Volume 1: One-Variable Calculus, with An Introduction to Linear Algebra authored by Tom M. Apostol":

----"AXIOM 4. EXISTENCE OF IDENTITY ELEMENTS. There exist two distinct real numbers, which we denote by 0 and 1, such that for every real x we have x + 0 = x and 1 * x = x."

Using this axiom, we must first recognize the relationships "x + 0 = x" and "1 * x = x" and then from recognizing that relationship, we can form the idea of identity elements. Also, the denotation of 0 and 1, although mentioned before the equations, happens after the discovery of such elements, because they need to be communicated. Also, notice that the idea of "real numbers" is assumed in this axiom, but for completeness, one would have to recognize the axiomatic relationships that define real numbers as well.)

So now it is time to define existence. Curiously, if you take a look at the example used in the "further depth" section, you'll notice the description of the axiom uses the words "existence" and "exist". And I believe the way it used there is very accurate. To exist simply means to be a relationship in a system. That relationship can be a rule or axiom, which exists necessarily, for they are the fundamental, the defining relationships of systems, or the relationship can be derived from the rules. Things we turn into objects such as "pieces" and "boards" are very fundamental, defined directly by the rules. However, I can say "the position of a rook being next to a knight" exists just as well, because I could use the rules of the system to prove this relationship.

One thing you may notice is my usage of the term "prove". Proofs concern truths, so how did talk of existence bring up the topic of logic and truth? It is because "existence" is indistinguishable from "truth". The rules and axioms that define the relationships of systems are truth. And "using" the rules to find relationships means to apply logic to find more complex truths. This is precisely the way you can use logic to manipulate axioms to produce new results, how mathematics proves theorems. And whatever relationships you can't create by logically manipulating the rules, those relationships are false. They do not exist. . .or do they?

You see, if we're defining existence by a relationship in a system, then it seems that existence is relative to the system. For example, let's come back to our two games again, Minesweeper and chess. It is accurate to say that, given the rules, a "rook" exists in chess, because its relationship can be defined. However, a rook does not exist in Minesweeper. Similarly, mines do not exist in chess because there is no relationship in chess that defines the same properties a mine has in Minesweeper. So, do they exist or not? It seems like the obvious answer is that "rooks exist in chess, but not Minesweeper" and "mines exist in Minesweeper, but not chess". It doesn't seem like a problem, but consider this. To us, the humans that play these games, both the mines and rooks seem to exist, because the games themselves exist, which would imply everything in those games exist. You may see where this line of thinking will lead us, bet let's continue on to the "real" stuff now. Not games and math, the universe and god. And actually, it gets a lot easier.

Because the universe is a system, most people agree. There is the question of whether it's deterministic, how to assess the results of quantum mechanics, etc., but despite the seemingly probabilistic nature of the universe, we have still discovered, to the best that we can, mathematical formulas to describe it. So it seems like the universe as a system is a clear idea. And if it is the case that the universe is a system, then that means we have defined how things exist as relationships derived from the complex rules that govern the universe. Now that we have a proper definition of existence, let's have a look back at the questions we were pondering at the beginning.

"How can a god 'exist' outside space and time? What does "outside" even mean?"

Using our view of systems, "outside" can mean "not a relationship in a particular system". Let's regard the rooks and mines again. From our perspective, in the system of the universe, chess and Minesweeper both exist. However, relationships in one may not exist in the other, yet they still exist. So we could say mines in Minesweeper do exist in our system "the universe", but "outside" of the system of chess, because although they exist in our system, they do not exist in the system of chess.
So two things: One, we are fairly certain that space and time exist in or relate in our universe. And two, space and time, which seemed to be necessary for the definition of existence, is actually not at all a part of this definition. However, since space and time are necessary parts of our system, to exist but not be related to space and time would mean that a god simply doesn't exist in our system. It could exist in some other system, or it could exist in an encompassing system, like how our universe encompasses chess and Minesweeper. But the key implication is that things are not required to exist in our universe to exist. But hold on, that seems weird right? Well let's answer the next question:

"If a god "exists" outside of the universe, which is "all that exists", does that even qualify as existence?

First things first, as we define universe this way, the universe is not "all that exists", it is simply the system that we exist in. But even so, if this is all we exist in, and we can't possibly relate to any system outside of our own, like how chess can't relate to Minesweeper, then should we even care about things that exist outside our universe?

This is a far, far different question than anything I've tackled in this post. All I've tackled up until now is ontological, metaphysical. The current question is of what value should we assess to things beyond our reach. Should we bother making beliefs about that which we can't observe, and can we truly know what seems unknowable? Maybe you're still wondering how we can even conceive of such a reality, yet be unable to attain it. Of course, maybe I made a fatal flaw somewhere down the line and the whole idea will be dismissed--by most, I feel.

I question the finite nature of our so-called "physical" reality, in the face of the infinite nature of abstract, mathematical reality. In the face of such a question, I wonder, if it is so simple to create our own lower systems, why we would not be part of a greater system. I wonder why the fundamental nature of mathematics is undiscoverable, as proved by Godel, and, by extension, learning about the fundamental nature of this universe be impossible if it was self-contained.

These things lead me to believe that we are not all there is to reality. That the universe is not equivalent to the cosmos. And I realize I've taken a departure from the topic at hand, simply defining existence, but the implications of this idea is incredible and drove me to open my world wider, even if I it is unknowable. I can no longer call myself an atheist, however, I can't also call myself a theist, not yet. Because while I believe in reality beyond our universe, I can not say I can believe that our universe was the product of an intelligence, certainly not sentient in the same way humans are. However, I do not close my mind to the idea. I'm willing to splash in the unknown and try to find the truths that can not be known until I'm satisfied with a proper worldview. As for what we can truly know, what is in our universe, what we can relate to and can discover, I'll leave that to the scientists. I'll listen to whatever new fundamental discoveries are made about our reality, and I will continue to wander in the dark abyss with only a lit match to light the way.

Existence, being an axiomatic concept, can not be defined in terms of prior concepts. What would those concepts refer to except something which exists. So the concept existence can only be defined ostensively, by pointing to it. So existence is that which exists. It is tautological because at the level of the axioms, there is nothing else that we know other than that things exist. Your computer screen, the road outside your house, a pencil on your desk, these are all existence.

You are quite right to point out that positing something which exists outside the universe is nonsensical. It commits the fallacy known as the stolen concept. Theists seek to avoid this by defining the universe as all of creation or all physical matter or all matter, energy, space and time. But this means that they are defining the universe in terms of prior concepts and this causes one to ask what is their axiomatic starting point? It can't be God because God can not be defined ostensively but only in terms of antecedent concepts. The result is their true starting point remains hidden.

Do not lose your knowledge that man's proper estate is an upright posture, an intransigent mind and a step that travels unlimited roads. - Ayn Rand.

Don't sacrifice for me, live for yourself! - Me

The only alternative to Objectivism is some form of Subjectivism. - Dawson Bethrick
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13-10-2016, 02:08 PM
RE: What is Existence? - Existence as truth, relative to systems
I'll have to back up on my previous statement -- you weren't talking about "recognizing" 0 and 1 as real numbers, but about recognizing the relation or rule that defines them, and there is no logical problem with that. So basically, never mind my last post.

However, I question whether Goedel's theorems say anything at all about physical reality. They place limits on what can be accomplished within a formal logical system, but that's not reality. The whole idea of mathematical reality and physical reality being identical (or even congruent) seems wrong to me. Mathematics turns out to be very useful in modeling reality, but it is entirely possible to build mathematical structures that contradict each other while both remaining internally consistent (witness Euclidean and non-Euclidean geometries). At most one such system can match up with reality, and even then, it's an approximate model at best. And much abstract mathematics is more like a formal game, that doesn't necessarily match up with any kind of physical reality. It just has to follow its own internal rules. Bertrand Russell famously defined it like this: "Mathematics is the subject in which we never know what we are talking about, nor whether what we are saying is true." That's a bit tongue-in-cheek, but it also contains a lot of truth.
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13-10-2016, 02:23 PM
RE: What is Existence? - Existence as truth, relative to systems
(13-10-2016 12:06 PM)Grasshopper Wrote:  I haven't read your entire post yet, but I have one quibble about your mathematical example. I am quite familiar with Apostol's book (I have been trying to work my way through it with no assistance since 1980! -- I have so far finished the introductory chapter and the first 10 numbered chapters, working every single problem along the way), and I was just reviewing the introductory chapter a few days ago. I also have some familiarity with axiomatic mathematics in general.

If I can be honest about my usage, I have only read some some of the introduction when I tried out the complementary MIT OCW Course. But, for some reason, reading about your experience with the book makes me feel like going through it again, perhaps with a little more depth. Although. . .

Quote:Now, Apostol takes pains to point out that no meaning should be attached to 0 and 1 other than what is specified by the axioms, and this is characteristic of axiomatic mathematics in general. It turns out that these are the familiar numbers 0 and 1 of our real number system, but that's not necessary. Once you study abstract algebra, you will find that it is possible to have identity elements (and entire structures) that are not the old familiar ones. I'm not sure how this affects your system, but I deny your claim that the identity elements have to be "recognized" before you can use them. I'm pretty sure Apostol explicitly states that this is not the case.

. . .I don't think we disagree! I should probably just clarify what I mean a little more because I think there's just a minor misunderstanding. The first thing I say after the example is:

----"Using this axiom, we must first recognize the relationships "x + 0 = x" and "1 * x = x" and then from recognizing that relationship, we can form the idea of identity elements."

I didn't say we have to "recognize" (which I suppose means "take as true") the identity elements first. I said we have to understand the definitions/relationships ("x + 0 = x" and "1 * x = x) first because they define the concept of (these particular) identity elements. And I did try to be careful to mention how the 1 and 0 in the equations aren't what they seem. Also, just to be clear, when I mentioned "the idea of 'real numbers' is assumed in this axiom", I pointed that out not in reference to 1 and 0, but to "x", which represents a real number. I'll be sure to read a bit more of at least the introduction so that I can have a more concrete idea of this stuff in my head Thumbsup
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13-10-2016, 02:35 PM
RE: What is Existence? - Existence as truth, relative to systems
(13-10-2016 02:23 PM)unknowndevil666 Wrote:  
(13-10-2016 12:06 PM)Grasshopper Wrote:  I haven't read your entire post yet, but I have one quibble about your mathematical example. I am quite familiar with Apostol's book (I have been trying to work my way through it with no assistance since 1980! -- I have so far finished the introductory chapter and the first 10 numbered chapters, working every single problem along the way), and I was just reviewing the introductory chapter a few days ago. I also have some familiarity with axiomatic mathematics in general.

If I can be honest about my usage, I have only read some some of the introduction when I tried out the complementary MIT OCW Course. But, for some reason, reading about your experience with the book makes me feel like going through it again, perhaps with a little more depth. Although. . .

Quote:Now, Apostol takes pains to point out that no meaning should be attached to 0 and 1 other than what is specified by the axioms, and this is characteristic of axiomatic mathematics in general. It turns out that these are the familiar numbers 0 and 1 of our real number system, but that's not necessary. Once you study abstract algebra, you will find that it is possible to have identity elements (and entire structures) that are not the old familiar ones. I'm not sure how this affects your system, but I deny your claim that the identity elements have to be "recognized" before you can use them. I'm pretty sure Apostol explicitly states that this is not the case.

. . .I don't think we disagree! I should probably just clarify what I mean a little more because I think there's just a minor misunderstanding. The first thing I say after the example is:

----"Using this axiom, we must first recognize the relationships "x + 0 = x" and "1 * x = x" and then from recognizing that relationship, we can form the idea of identity elements."

I didn't say we have to "recognize" (which I suppose means "take as true") the identity elements first. I said we have to understand the definitions/relationships ("x + 0 = x" and "1 * x = x) first because they define the concept of (these particular) identity elements. And I did try to be careful to mention how the 1 and 0 in the equations aren't what they seem. Also, just to be clear, when I mentioned "the idea of 'real numbers' is assumed in this axiom", I pointed that out not in reference to 1 and 0, but to "x", which represents a real number. I'll be sure to read a bit more of at least the introduction so that I can have a more concrete idea of this stuff in my head Thumbsup

No problem. I realized later that I was misunderstanding you, so my second post basically cancels the first one.

Apostol's Calculus is the best mathematics text I've ever encountered, but it's also one of the most difficult. If you want to tackle it, be prepared to work hard. Some of the problems are real killers (although you could just skip those -- you don't have to do every problem like I've been doing).
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13-10-2016, 02:40 PM
RE: What is Existence? - Existence as truth, relative to systems
(13-10-2016 02:08 PM)Grasshopper Wrote:  I'll have to back up on my previous statement -- you weren't talking about "recognizing" 0 and 1 as real numbers, but about recognizing the relation or rule that defines them, and there is no logical problem with that. So basically, never mind my last post.

Oops, a little lateTongue
Quote:However, I question whether Goedel's theorems say anything at all about physical reality. They place limits on what can be accomplished within a formal logical system, but that's not reality. The whole idea of mathematical reality and physical reality being identical (or even congruent) seems wrong to me. Mathematics turns out to be very useful in modeling reality, but it is entirely possible to build mathematical structures that contradict each other while both remaining internally consistent (witness Euclidean and non-Euclidean geometries). At most one such system can match up with reality, and even then, it's an approximate model at best. And much abstract mathematics is more like a formal game, that doesn't necessarily match up with any kind of physical reality. It just has to follow its own internal rules. Bertrand Russell famously defined it like this: "Mathematics is the subject in which we never know what we are talking about, nor whether what we are saying is true." That's a bit tongue-in-cheek, but it also contains a lot of truth.

Yes, I did kind of add my own interpretation on Goedel's theorem by applying it to reality, which is a bit of a jump on my part. Furthermore, although I do still claim to have some understanding of what the theorems imply, I certainly don't have a fully comprehensive understanding, so I'm open to being shown the light on that front.
It is an interesting point that different mathematical structures can contradict each other while still being consistent. Interestingly, that seems to go hand in hand with the idea of things existing relative to systems, although I suppose it might more accurate to call the "system" a "mathematical structure" in this case.
I'm personally not convinced in the idea that mathematical reality is separate from physical reality, but a difficult part of it is there is a lot of semantics involved with the issue. For example, see if this makes sense: Although I am more convinced that the realities are one in the same, I also agree that "at most one such system can match up with reality, and even then, it's an approximate model at best". Seems like a contradiction, right? But the distinction I use is that even if reality is mathematical, from within the system, we can't truly discover the actual structure completely. Which, from my possibly flawed understanding, I think is at least similar to the conclusions of the Incompleteness Theorems. Although these ideas in particular are quite abstract so I can't be certain of my own beliefs.
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13-10-2016, 02:49 PM
RE: What is Existence? - Existence as truth, relative to systems
(13-10-2016 01:13 PM)true scotsman Wrote:  Existence, being an axiomatic concept, can not be defined in terms of prior concepts. What would those concepts refer to except something which exists. So the concept existence can only be defined ostensively, by pointing to it. So existence is that which exists. It is tautological because at the level of the axioms, there is nothing else that we know other than that things exist. Your computer screen, the road outside your house, a pencil on your desk, these are all existence.

This all seems a bit fishy to me, but I did a quick search of the "stolen concept" fallacy and it's interesting to say the least. I'll have to look into it a bit more to see how valid it is in regards to my claims and arguments. I guess the one thing I mainly wonder is how you are so certain that existence is an axiomatic concept. What other axiomatic concepts are there? Consciousness?
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13-10-2016, 02:59 PM
RE: What is Existence? - Existence as truth, relative to systems
(13-10-2016 02:40 PM)unknowndevil666 Wrote:  
(13-10-2016 02:08 PM)Grasshopper Wrote:  I'll have to back up on my previous statement -- you weren't talking about "recognizing" 0 and 1 as real numbers, but about recognizing the relation or rule that defines them, and there is no logical problem with that. So basically, never mind my last post.

Oops, a little lateTongue
Quote:However, I question whether Goedel's theorems say anything at all about physical reality. They place limits on what can be accomplished within a formal logical system, but that's not reality. The whole idea of mathematical reality and physical reality being identical (or even congruent) seems wrong to me. Mathematics turns out to be very useful in modeling reality, but it is entirely possible to build mathematical structures that contradict each other while both remaining internally consistent (witness Euclidean and non-Euclidean geometries). At most one such system can match up with reality, and even then, it's an approximate model at best. And much abstract mathematics is more like a formal game, that doesn't necessarily match up with any kind of physical reality. It just has to follow its own internal rules. Bertrand Russell famously defined it like this: "Mathematics is the subject in which we never know what we are talking about, nor whether what we are saying is true." That's a bit tongue-in-cheek, but it also contains a lot of truth.

Yes, I did kind of add my own interpretation on Goedel's theorem by applying it to reality, which is a bit of a jump on my part. Furthermore, although I do still claim to have some understanding of what the theorems imply, I certainly don't have a fully comprehensive understanding, so I'm open to being shown the light on that front.
It is an interesting point that different mathematical structures can contradict each other while still being consistent. Interestingly, that seems to go hand in hand with the idea of things existing relative to systems, although I suppose it might more accurate to call the "system" a "mathematical structure" in this case.
I'm personally not convinced in the idea that mathematical reality is separate from physical reality, but a difficult part of it is there is a lot of semantics involved with the issue. For example, see if this makes sense: Although I am more convinced that the realities are one in the same, I also agree that "at most one such system can match up with reality, and even then, it's an approximate model at best". Seems like a contradiction, right? But the distinction I use is that even if reality is mathematical, from within the system, we can't truly discover the actual structure completely. Which, from my possibly flawed understanding, I think is at least similar to the conclusions of the Incompleteness Theorems. Although these ideas in particular are quite abstract so I can't be certain of my own beliefs.

I am not an expert on Goedel either -- the technical details of his theorems are fearsome, and I'm just not interested enough to work that hard. But my understanding is that they say more about the limitations of logic than they do about either mathematics or physical reality (although I will note that Bertrand Russell thought -- and tried to prove -- that mathematics and logic were one and the same. Ironically, it was Goedel who put a stop to that project). If Chas happens to wander into this thread, I'm sure he could tell you more about Goedel -- he has formally studied logic.

As far as I know, we actually have a very good and very complete understanding of the structure of both Euclidean and non-Euclidean geometries. That's not the problem -- the problem is determining the structure of reality. For over 2000 years, we thought that Euclidean geometry was a perfect match. Eventually, we had accurate enough instruments to discover that it wasn't. Space (or more accurately, space-time) is curved. We think we understand how it's curved, but I don't trust that to be the last word either. Reality is always surprising us, and I'm skeptical that any mathematical structure exactly corresponds to physical reality. If there is any professional mathematician or physicist who thinks that it does, I'd love to hear about him/her.

Incidentally, the fact that reality is always surprising us is the reason why I'm an agnostic atheist rather than a gnostic one. We don't know what we don't know.
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13-10-2016, 04:05 PM
RE: What is Existence? - Existence as truth, relative to systems
(13-10-2016 02:59 PM)Grasshopper Wrote:  Reality is always surprising us, and I'm skeptical that any mathematical structure exactly corresponds to physical reality. If there is any professional mathematician or physicist who thinks that it does, I'd love to hear about him/her.

I agree, we always seem to be a step (or many, too too many steps) behind.
Max Tegmark is one such astrophysicist/cosmologist that believes that the universe can be described as purely mathematical. I haven't read to extensively into him (I've only watched some of his talks) but he is a physics professor at MIT and he has authored the book "Our Mathematical Universe". That would probably be of great interest.
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