Thursday, September 23, 2010
Moving Blog
I'm moving the blog to TowardsRationalExplanationExistence.blogspot.com because I don't want to give the impression that I think I have all the answers.
Saturday, September 18, 2010
Maximality, uniqueness, and reliability
What is a definition of logical inevitability that is better than
(1) T® á$T
?
We want T to be reliable in the sense of Kaplan []. T is reliable if
(2) from kT infer káT
where k means “knowing”. The paradigmatic example is a speaker speaking “I am here now.” I don’t know if (2) is the best interpretation of reliable.
What can it mean for a logical system to “know” something? This is the structure the system is “aware” of. And this is the image of implication within the system, given all of T as axioms. Let kT be what T knows. Then (1) gives
(3) T kT á$T
There are arguments elsewhere that this relational existence is sufficient for actual existence. A big question is the one of the uniqueness of T. There had better be a maximal T, where
(4) T is maximal if for all theories S such that S® á$S, S Ì T.
If there is no maximal T, then there would be many things that have an equal right to exist. In this case a reinterpretation of (1) would be called for.
Maximality results usually proceed…
Friday, September 17, 2010
Time and Physical Probability
Probably the things most useful to identify in T are those which correspond to time and physical probability.
Thursday, September 16, 2010
Hawking and Stenger
Hawking and M... 's idea is that the gravitational potential energy exactly cancels out the positive energy of the gravitating masses. In terms of energy, it is the same as having nothingness. The problem with this kind of approach is that the notion of energy is given significance only by theories of physics. So we can always ask: why should such-and-such a theory be the correct one out of so many possibilities?
Also, to argue that zero total energy (or stress-energy or whatever) is equivalent to the energy of nothingness assumes a background physical theory. Once again we would want to know why that theory obtains.
Stenger has said nothingness is unstable. By "unstable" one means nothingness is likely to decay into a state of somethingness as time goes along. But this assumes both time and the background physical theory defining what it is that is unstable. So it suffers from the same problem.
Also, to argue that zero total energy (or stress-energy or whatever) is equivalent to the energy of nothingness assumes a background physical theory. Once again we would want to know why that theory obtains.
Stenger has said nothingness is unstable. By "unstable" one means nothingness is likely to decay into a state of somethingness as time goes along. But this assumes both time and the background physical theory defining what it is that is unstable. So it suffers from the same problem.
Tuesday, September 14, 2010
Hawking's Book
I read much of The Grand Design. It did not reveal how to get the existence of the universe.
Summary with Reliable T
I take a theory T in the sense of 1st-order logic to be such that T ® the existence of T is logically inevitable. This might take the form T ® ƒ$T. But this isn’t quite what we want, since it leaves the question of T’s existence undetermined.
Supposing existence is really just relational existence, what can be said about the relationship between T and ƒ$T?
Interpret necessarily true not in the sense that it is true in every possible world, but more in the sense that “I am here now” where “I” refers to the speaker.
The relationship in question is the one that happens between the theory T and what it proves, ƒ$T, or necessarily there exists T.
ƒ$T is to T like “I am here now” is to its speaker. In some sense, it is already true.
We know there is necessarily the possibility of such a T.
Hence, T exists, logically inevitability.
T is, and does not merely represent the physical universe.
If we found evidence T is our universe, then
we would have a rational explanation for existence.
The main problem I see, which not everybody will see as a problem, is that even if we grant all these suppositions, there is no way to account for qualia. By the hypothesis, physical reality is ultimately mathematical structure. But I do not see how one could account for the experience of greenness, for example, with just mathematical structure.
Friday, September 10, 2010
A Bunch of the Arguments Collected Together
Towards a Rational Explanation for Existence
Is it possible to give a rational explanation for existence? Below is a web of suppositions which, if borne out, would allow us to claim to have a rational explanation for existence. Of course one is free to reject the suppositions. Nevertheless it is interesting to see how far we can get with the project.
The basic idea is to have a mathematical theory assert its own existence. Then argue this theory exists relative to itself. Next conclude this relative existence is sufficient for absolute existence. Finally, postulate that our universe is that theory.
Here is an attempt to lay bare the argument:
1) 2 + 2 = 4 is in some sense true whether or not the universe or anything at all exists. The interrelatedness of the definitions of 2, +, 4, and = in some sense precedes the question of existence.
2) Now define a consistent theory T that contains the statement:
“The existence of this statement is logically inevitable.”
(How this might be implemented is discussed below.)
3) T is always possible, it is reliable relative to itself, and actual relative to itself.
4) In some important sense, existence does not recognize the difference between T’s self-apparent existence and our actual existence. They are equally valid. Perhaps our actual existence is a kind of self-apparentness.
5) The universe is T, and it admits a rational explanation for its existence, namely 1-4.
Some considerations
How to get an explanation for existence
Why does anything at all exist? This question already makes an assumption. It just assumes that nothingness is more warranted than the existence of something. But what justifies this assumption? Without making any assumptions all we can say is that it could be that something exists. Indeed it could turn out to be the case that the existence of some particular thing is logically inevitable, while nothingness is not.
Hypothesis: what ultimately exists is pure structure (mathematical objects)
I will not go in to this hypothesis except to say it is consistent with materialism and inconsistent with dualism.
By the hypothesis, we are after some mathematical structure that finds its own existence to be logically inevitable. It is sufficient to consider the case where the mathematical object is the web of sentences of some formal system T. Beyond that we want a theory that, in some sense, asserts it’s own inevitability. In other words, the mathematical theory T is such that
(1) T
But even supposing we could make sense of this and find such a T, this condition is not enough to guarantee actual existence. All we can assert is that there could be some mathematical structure, such as T, that derives its own logical inevitability. But all we really know is that the existence of T is possible. In the many-worlds semantics of possibility this leads to two cases, worlds W for which T exists, and the disjoint worlds V where T does not exist.
We want to motivate the existence of T. In the worlds W where T exists, there is only the issue of finding a (the) T that satisfies (1). In the possible worlds V however, it is not obvious how to motivate existence. In these worlds T does not exist, so it is not around to assert anything about itself. In this case there is no way to get existence off the ground, so to speak.
Happily, there are at least four possible ways out of this conclusion.
The first way out is to argue that the possible-worlds semantics of possibility is misleading here. In possible worlds semantics there are some worlds W in which T exists and some V where it does not. But this contains the trick of forcing an either-or decision upon us about T’s existence where it may be inappropriate. We do not want or else What we want is to consider the possible worlds in which it is possible that T exists. If we give the notion of possibility this new status, it might be enough, in view of condition (1), to allow the theory to bootstrap itself into existence.
Returning to the possible worlds V and W, a second way out is to argue that, for the possible worlds V, a theorem t may not exist , but t’s relations to other things might nevertheless exist. Thus there are second-order properties of t that claim existence in spite of t’s own non-existence.
We want to argue T’s self-apparent existence is sufficient because we can define T. We need the crucial supposition:
(2) definitions precede existence, and do not make an assumption about existence
Infinite numbers are definable (as in set theory), even though the physical universe may be only finite. Nevertheless we often “prove the existence” of this or that infinite number (such as a set). In this case we might be able to say that infinite numbers exist relative to the theory that defined it (e.g. set theory).
Then,
(3) if (1) is satisfied T (necessarily) exists relative to itself.
Now, it would be argued, this relative kind of existence is what we really mean by “existence”. This would have to apply to actual existence. And existence forms an equivalence class...
In this case, (3) is sufficient for actual existence.
What if things have “intrinsic” or “absolute” existence? In this case (3) would seem to be insufficient to generate actual existence…
But by the hypothesis, the universe is ultimately mathematical structure. Can’t parts of a mathematical structure exist relative to each other? Suppose we say
(4) the number 2 exists relative to the number 3
You might say: it may be true, but presupposes absolute existence. But how do you know? If absolute existence is, in fact, relative existence, we would not know the difference.
You might say: this is a false option, and (4) is neither true nor false. But it is either a theorem of the system that
(5)
or else it is not.
In the system T we want something like
(6)
which is not to be confused with
(7)
In (6) it says if there is an x then there also exists (relative to the formal system) the assertion that there is an x. This is plausible since x is itself an assertion of the formal system.
This requires some attention to the rule for - introduction. In addition to
(8)
a new rule is required, 2nd-order (nth-order) existence-introduction, or maybe some consistent implementation of (6).
To pick up where we left off with (9), we can say that relative to the definition, the theory necessarily exists. So relative to the definition, the definition itself exists, that is because the theory asserts it’s own logical inevitability. The interrelatedness of the definitions generates its own necessary existence, since the definitions are, in fact, so interrelated. Here is something (the interrelatedness of the definitions) from nothing (what the definitions are definitions of). The possibility of such a definition is all we need.
Then we would use the supposition
(9) absolute existence is relational existence
to conclude that absolute existence is given by the theory T.
( One might ask: where did the possibility for something from nothing come from? (It does not come from nothing, we do not assume nothing is more logically inevitable than the existence of some particular things. Instead of nothingness we take a stance of having no assumptions about the matter. I.e. why is the world so constructed as to allow for such a possibility? The short answer is that an answer to this question must lie outside of our capacity to reason (such as qualia do), since it is asking for a reasonable explanation of reason, i.e. the answer must involve reason, so the faculties to accommodate reason must already be accounted for.
The third way out of the conclusion above, slightly different from the second, is to first show that T exists in the sense of T. Then, the argument would go, it is sufficient that T exists in the sense of T, because this existing-in-the-sense-of is all we need for a kind of absolute existence. But it is not obvious what it means to exist in the sense of something that only possibly exists…
A fourth way out is to argue that all notions of existence are relative, and so there is no problem for the relative kind of existence in the third way out (above) because the existence predicate in is implicitly relative, and absolute existence has really just been relative existence the whole time, etc.
My own preference is for argument (1) combined with argument (3).
Now suppose we have a candidate theory T. T finds itself to be logically inevitable (and T does not find nothingness to be logically inevitable). According to the hypothesis above, T does not merely represent the universe, it is the universe. If we chose to say that T has physical existence, then it would be a physical circumstance that the universe is logically inevitable.
Finally, if we knew T is our actual universe, we would be justified in claiming to have a rational explanation for existence.
Finding T
Let t be any theorem of T. Make the definition
Now
(11)
We take (1) and (7) as axioms. To express these, the deductive system T must contain 1st-order predicate calculus, equality and the modal operators. We make no assumption about the model for T. For T to find its own existence logically inevitable, (1) must be true of it. Let T1 be the collection of resulting theorems. T1 is every theorem derivable from axioms (1) and (7) and 1st-order predicate calculus, equality, and the modal operators. Now construct a model, in the sense of model theory, M2, such that the universe of M2 is exactly the collection of theorems T1. Let T2 be the theory of M2. Construct a model M3 out of T2 in the same way, and continue this process indefinitely.
Conjecture: this process reaches a fixed-point theory TFP
Claim: TFP satisfies (1)
The idea is that TFP is our universe, and in being so asserts its own logical inevitability.
Somewhere in there it should be added that the reason we want an ultimate explanation is that existence cannot be contingent on anything—is that if the explanation uses any assumptions we can always ask why state of affairs described by that assumption obtains.
Paul Merriam
Thanks to Jesse Folsum and John Mazetier.
Is it possible to give a rational explanation for existence? Below is a web of suppositions which, if borne out, would allow us to claim to have a rational explanation for existence. Of course one is free to reject the suppositions. Nevertheless it is interesting to see how far we can get with the project.
The basic idea is to have a mathematical theory assert its own existence. Then argue this theory exists relative to itself. Next conclude this relative existence is sufficient for absolute existence. Finally, postulate that our universe is that theory.
Here is an attempt to lay bare the argument:
1) 2 + 2 = 4 is in some sense true whether or not the universe or anything at all exists. The interrelatedness of the definitions of 2, +, 4, and = in some sense precedes the question of existence.
2) Now define a consistent theory T that contains the statement:
“The existence of this statement is logically inevitable.”
(How this might be implemented is discussed below.)
3) T is always possible, it is reliable relative to itself, and actual relative to itself.
4) In some important sense, existence does not recognize the difference between T’s self-apparent existence and our actual existence. They are equally valid. Perhaps our actual existence is a kind of self-apparentness.
5) The universe is T, and it admits a rational explanation for its existence, namely 1-4.
Some considerations
How to get an explanation for existence
Why does anything at all exist? This question already makes an assumption. It just assumes that nothingness is more warranted than the existence of something. But what justifies this assumption? Without making any assumptions all we can say is that it could be that something exists. Indeed it could turn out to be the case that the existence of some particular thing is logically inevitable, while nothingness is not.
Hypothesis: what ultimately exists is pure structure (mathematical objects)
I will not go in to this hypothesis except to say it is consistent with materialism and inconsistent with dualism.
By the hypothesis, we are after some mathematical structure that finds its own existence to be logically inevitable. It is sufficient to consider the case where the mathematical object is the web of sentences of some formal system T. Beyond that we want a theory that, in some sense, asserts it’s own inevitability. In other words, the mathematical theory T is such that
(1) T
But even supposing we could make sense of this and find such a T, this condition is not enough to guarantee actual existence. All we can assert is that there could be some mathematical structure, such as T, that derives its own logical inevitability. But all we really know is that the existence of T is possible. In the many-worlds semantics of possibility this leads to two cases, worlds W for which T exists, and the disjoint worlds V where T does not exist.
We want to motivate the existence of T. In the worlds W where T exists, there is only the issue of finding a (the) T that satisfies (1). In the possible worlds V however, it is not obvious how to motivate existence. In these worlds T does not exist, so it is not around to assert anything about itself. In this case there is no way to get existence off the ground, so to speak.
Happily, there are at least four possible ways out of this conclusion.
The first way out is to argue that the possible-worlds semantics of possibility is misleading here. In possible worlds semantics there are some worlds W in which T exists and some V where it does not. But this contains the trick of forcing an either-or decision upon us about T’s existence where it may be inappropriate. We do not want or else What we want is to consider the possible worlds in which it is possible that T exists. If we give the notion of possibility this new status, it might be enough, in view of condition (1), to allow the theory to bootstrap itself into existence.
Returning to the possible worlds V and W, a second way out is to argue that, for the possible worlds V, a theorem t may not exist , but t’s relations to other things might nevertheless exist. Thus there are second-order properties of t that claim existence in spite of t’s own non-existence.
We want to argue T’s self-apparent existence is sufficient because we can define T. We need the crucial supposition:
(2) definitions precede existence, and do not make an assumption about existence
Infinite numbers are definable (as in set theory), even though the physical universe may be only finite. Nevertheless we often “prove the existence” of this or that infinite number (such as a set). In this case we might be able to say that infinite numbers exist relative to the theory that defined it (e.g. set theory).
Then,
(3) if (1) is satisfied T (necessarily) exists relative to itself.
Now, it would be argued, this relative kind of existence is what we really mean by “existence”. This would have to apply to actual existence. And existence forms an equivalence class...
In this case, (3) is sufficient for actual existence.
What if things have “intrinsic” or “absolute” existence? In this case (3) would seem to be insufficient to generate actual existence…
But by the hypothesis, the universe is ultimately mathematical structure. Can’t parts of a mathematical structure exist relative to each other? Suppose we say
(4) the number 2 exists relative to the number 3
You might say: it may be true, but presupposes absolute existence. But how do you know? If absolute existence is, in fact, relative existence, we would not know the difference.
You might say: this is a false option, and (4) is neither true nor false. But it is either a theorem of the system that
(5)
or else it is not.
In the system T we want something like
(6)
which is not to be confused with
(7)
In (6) it says if there is an x then there also exists (relative to the formal system) the assertion that there is an x. This is plausible since x is itself an assertion of the formal system.
This requires some attention to the rule for - introduction. In addition to
(8)
a new rule is required, 2nd-order (nth-order) existence-introduction, or maybe some consistent implementation of (6).
To pick up where we left off with (9), we can say that relative to the definition, the theory necessarily exists. So relative to the definition, the definition itself exists, that is because the theory asserts it’s own logical inevitability. The interrelatedness of the definitions generates its own necessary existence, since the definitions are, in fact, so interrelated. Here is something (the interrelatedness of the definitions) from nothing (what the definitions are definitions of). The possibility of such a definition is all we need.
Then we would use the supposition
(9) absolute existence is relational existence
to conclude that absolute existence is given by the theory T.
( One might ask: where did the possibility for something from nothing come from? (It does not come from nothing, we do not assume nothing is more logically inevitable than the existence of some particular things. Instead of nothingness we take a stance of having no assumptions about the matter. I.e. why is the world so constructed as to allow for such a possibility? The short answer is that an answer to this question must lie outside of our capacity to reason (such as qualia do), since it is asking for a reasonable explanation of reason, i.e. the answer must involve reason, so the faculties to accommodate reason must already be accounted for.
The third way out of the conclusion above, slightly different from the second, is to first show that T exists in the sense of T. Then, the argument would go, it is sufficient that T exists in the sense of T, because this existing-in-the-sense-of is all we need for a kind of absolute existence. But it is not obvious what it means to exist in the sense of something that only possibly exists…
A fourth way out is to argue that all notions of existence are relative, and so there is no problem for the relative kind of existence in the third way out (above) because the existence predicate in is implicitly relative, and absolute existence has really just been relative existence the whole time, etc.
My own preference is for argument (1) combined with argument (3).
Now suppose we have a candidate theory T. T finds itself to be logically inevitable (and T does not find nothingness to be logically inevitable). According to the hypothesis above, T does not merely represent the universe, it is the universe. If we chose to say that T has physical existence, then it would be a physical circumstance that the universe is logically inevitable.
Finally, if we knew T is our actual universe, we would be justified in claiming to have a rational explanation for existence.
Finding T
Let t be any theorem of T. Make the definition
Now
(11)
We take (1) and (7) as axioms. To express these, the deductive system T must contain 1st-order predicate calculus, equality and the modal operators. We make no assumption about the model for T. For T to find its own existence logically inevitable, (1) must be true of it. Let T1 be the collection of resulting theorems. T1 is every theorem derivable from axioms (1) and (7) and 1st-order predicate calculus, equality, and the modal operators. Now construct a model, in the sense of model theory, M2, such that the universe of M2 is exactly the collection of theorems T1. Let T2 be the theory of M2. Construct a model M3 out of T2 in the same way, and continue this process indefinitely.
Conjecture: this process reaches a fixed-point theory TFP
Claim: TFP satisfies (1)
The idea is that TFP is our universe, and in being so asserts its own logical inevitability.
Somewhere in there it should be added that the reason we want an ultimate explanation is that existence cannot be contingent on anything—is that if the explanation uses any assumptions we can always ask why state of affairs described by that assumption obtains.
Paul Merriam
Thanks to Jesse Folsum and John Mazetier.
Thursday, September 9, 2010
how do you get mathematical symbols on blogspot.com???
I give up: how do you get mathematical symbols on blogspot.com???
Infinite numbers are definable (as in set theory), even though the actual universe may be only finite. Nevertheless we often “prove the existence” of this or that infinite number (set). In this case we might be able to say that infinite number exists relative to the theory that defined it (such as set theory).
Then,
(2) if (1) is satisfied T (necessarily) exists relative to itself.
Now, it would be argued, this relative kind of existence is what we really mean by “existence”. This applies to actual existence: everything that actually exists, does so relative to everything else in actual existence… (Existence forms an equivalence class...)
In this case, (2) is sufficient for actual existence.
What if things either do or do not have “intrinsic” or “absolute” existence? In this case (2) would seem to be insufficient to generate actual existence…
But by the hypothesis, the universe is ultimately mathematical structure. Can parts of a mathematical structure exist relative to each other? Suppose we say
(3) the number 2 exists relative to the number 3
You might say: it may be true, but presupposes absolute existence. But how do you know? If absolute existence is, in fact, relative existence, we would not know the difference.
You might say: this is a false option, and (3) is neither true nor false. But it is either a theorem of the system that
(4)
or else it is not. In the system T we want something like
(5)
which is not to be confused with
(6)
Then,
(2) if (1) is satisfied T (necessarily) exists relative to itself.
Now, it would be argued, this relative kind of existence is what we really mean by “existence”. This applies to actual existence: everything that actually exists, does so relative to everything else in actual existence… (Existence forms an equivalence class...)
In this case, (2) is sufficient for actual existence.
What if things either do or do not have “intrinsic” or “absolute” existence? In this case (2) would seem to be insufficient to generate actual existence…
But by the hypothesis, the universe is ultimately mathematical structure. Can parts of a mathematical structure exist relative to each other? Suppose we say
(3) the number 2 exists relative to the number 3
You might say: it may be true, but presupposes absolute existence. But how do you know? If absolute existence is, in fact, relative existence, we would not know the difference.
You might say: this is a false option, and (3) is neither true nor false. But it is either a theorem of the system that
(4)
or else it is not. In the system T we want something like
(5)
which is not to be confused with
(6)
Wednesday, September 8, 2010
About the Ontological Argument
About the ontological argument:
Can you really imagine a red sports car? If we imagined red things but there were no red objects in the universe, would we still say red exists?
Can you really imagine a red sports car? If we imagined red things but there were no red objects in the universe, would we still say red exists?
Yet Another Attempt
Here is yet another attempt to lay bare the argument:
1) 2 + 2 = 4 is in some sense true whether or not the universe or anything at all exists. The interrelatedness of the definitions of 2, +, 4, and = in some sense precedes the question of existence.
2) Now define a minimal theory T that contains the statement:
“The existence of this statement is logically inevitable.”
Perhaps this could be given the expression T t for all t
3) T is always possible, it is reliable relative to itself, and actual relative to itself.
4) In some important sense, existence does not recognize the difference between T’s self-apparent existence and our actual existence. They are equally valid.
5) The universe is T, and it admits a rational explanation for its existence, namely 1-4.
1) 2 + 2 = 4 is in some sense true whether or not the universe or anything at all exists. The interrelatedness of the definitions of 2, +, 4, and = in some sense precedes the question of existence.
2) Now define a minimal theory T that contains the statement:
“The existence of this statement is logically inevitable.”
Perhaps this could be given the expression T t for all t
3) T is always possible, it is reliable relative to itself, and actual relative to itself.
4) In some important sense, existence does not recognize the difference between T’s self-apparent existence and our actual existence. They are equally valid.
5) The universe is T, and it admits a rational explanation for its existence, namely 1-4.
summarizing the argument
Here is another attempt to lay bare the argument:
1) 2 + 2 = 4 is in some sense true whether or not the universe or anything at all exists. The interrelatedness of the definitions of 2, +, 4, and = in some sense precedes the question of existence.
2) Now define a minimal theory T that contains the statement:
“The existence of this statement is logically inevitable.”
Perhaps this could be given the expression T t for all t [how to get math symbols to appear in the blog?]
3) T is actual relative to itself (and there is necessarily the possibility that T)
4) If the universe were T, we would not know the difference between existing in the sense of T (i.e. wherever there is a use of ““, and existence in some outside, preconceived sense.
5) The universe is T, and it admits a rational explanation for its existence, namely 1-4.
1) 2 + 2 = 4 is in some sense true whether or not the universe or anything at all exists. The interrelatedness of the definitions of 2, +, 4, and = in some sense precedes the question of existence.
2) Now define a minimal theory T that contains the statement:
“The existence of this statement is logically inevitable.”
Perhaps this could be given the expression T t for all t [how to get math symbols to appear in the blog?]
3) T is actual relative to itself (and there is necessarily the possibility that T)
4) If the universe were T, we would not know the difference between existing in the sense of T (i.e. wherever there is a use of ““, and existence in some outside, preconceived sense.
5) The universe is T, and it admits a rational explanation for its existence, namely 1-4.
synopsis
So, the interrelatedness of (potential?) definitions precedes existence. Then, by the way this particular definition is set up, the definition asserts that it exists logically inevitably. Now, it is actual relative to itself. If the universe were the definition (or the structure so defined), we would not know the difference between existing in the sense of T (in the sense of the definition) and existence in some outside, preconceived sense. This case of relative existence is just as valid as "absolute" existence.
That is a synopsis of the reasoning.
What do you think about it?
That is a synopsis of the reasoning.
What do you think about it?
Tuesday, September 7, 2010
Hawking's Grand Design
I haven't read it yet. But Hawking has a new book "The Grand Design" coming out. From Wikipedia: "Hawking wrote... that "Because there is a law such as gravity, the universe can and will create itself from nothing. Spontaneous creation is the reason there is something rather than nothing, why the universe exists, why we exist. It is not necessary to invoke God to light the blue touch paper and set the universe going."[57][58]"
I don't see how this gets us past the "and then a miracle occurs" stage of explanation. Of course, it is good he is breaking the taboo about speculating on the origin of existence. They say he is a smart guy, and maybe there is more in the book...
I don't see how this gets us past the "and then a miracle occurs" stage of explanation. Of course, it is good he is breaking the taboo about speculating on the origin of existence. They say he is a smart guy, and maybe there is more in the book...
Monday, September 6, 2010
possibility can be different from possible-worlds semantics
A predicate calculus that asserts its own existence would appear to exist to itself, and that notion of existence would be indistinguishable from our current notion of actual existence.
Predicate calculus P
P exists in the sense of P, i.e. P asserts qua P that there is something that exists which is P i.e. PP.
This notion of existence is the same notion as actual existence, i.e. if our universe is P we could not tell the difference between existence-in-the-sense-of-P and the kind of existence that is actual.
Because of this, it is plausible that all we need is the possible existence of P.
I will give an example from quantum mechanics where the possible existence of something is different from a possible-worlds interpretation. In the many-worlds semantics, one says it could be the case that it does not exist and it could be the case that it exists. We want to say instead that it is possible that P exists, perhaps using a kind of intentional logic/model. An example of this phenomenon comes from quantum mechanics: we can write y-spin up when any value of x-spin is possible,
(1)
but we cannot decompose this situation into one where there is a set of possible worlds where y-spin is up and x-spin is up, together with a set of possible words where y-spin is up and x-spin is down:
(2)
If we can find such a P, and we can show that our universe is P (by making predictions from P about physical phenomena), then we would have a rational explanation for existence.
(It is not germane that we can also define a set of possible worlds in which the electron is in a superposition.)
Predicate calculus P
P exists in the sense of P, i.e. P asserts qua P that there is something that exists which is P i.e. PP.
This notion of existence is the same notion as actual existence, i.e. if our universe is P we could not tell the difference between existence-in-the-sense-of-P and the kind of existence that is actual.
Because of this, it is plausible that all we need is the possible existence of P.
I will give an example from quantum mechanics where the possible existence of something is different from a possible-worlds interpretation. In the many-worlds semantics, one says it could be the case that it does not exist and it could be the case that it exists. We want to say instead that it is possible that P exists, perhaps using a kind of intentional logic/model. An example of this phenomenon comes from quantum mechanics: we can write y-spin up when any value of x-spin is possible,
(1)
but we cannot decompose this situation into one where there is a set of possible worlds where y-spin is up and x-spin is up, together with a set of possible words where y-spin is up and x-spin is down:
(2)
If we can find such a P, and we can show that our universe is P (by making predictions from P about physical phenomena), then we would have a rational explanation for existence.
(It is not germane that we can also define a set of possible worlds in which the electron is in a superposition.)
Sunday, September 5, 2010
indistinguishable notions of existence
A predicate calculus that asserts its own existence would appear to exist to itself, and that notion of existence would be indistinguishable from our current notion of actual existence.
Friday, September 3, 2010
Indistinguishable Kinds of Existence
Consider
Hypothesis 1: the universe is ultimately pure mathematical structure
If the hypothesis is right, what kind of universe results from the mathematical structure
(1) the existence of this mathematical theory is logically inevitable
?
The resulting universe exists, relative to itself.
And this relative existence is a kind of absolute existence.
If we were it, we would expect our existence to be logically inevitable, and that notion of existence would be indistinguishable from our current notion of actual existence.
Now if the actual universe could be said to be that mathematical structure, we could be said to have a rational explanation for existence.
A predicate calculus that asserts its own existence would appear to exist to itself, and that notion of existence would be indistinguishable from our current notion of actual existence.
Hypothesis 1: the universe is ultimately pure mathematical structure
If the hypothesis is right, what kind of universe results from the mathematical structure
(1) the existence of this mathematical theory is logically inevitable
?
The resulting universe exists, relative to itself.
And this relative existence is a kind of absolute existence.
If we were it, we would expect our existence to be logically inevitable, and that notion of existence would be indistinguishable from our current notion of actual existence.
Now if the actual universe could be said to be that mathematical structure, we could be said to have a rational explanation for existence.
A predicate calculus that asserts its own existence would appear to exist to itself, and that notion of existence would be indistinguishable from our current notion of actual existence.
Wednesday, September 1, 2010
Can we give a rational explanation for existence?
Can we give a rational explanation for existence?
How to get an explanation for existence
Why does anything at all exist? This question already makes an assumption. It just assumes that nothingness is more warranted than the existence of something. But what justifies this assumption? Without any assumptions what we can say is that it could be that something exists. Indeed it could turn out to be the case that the existence of some particular thing is logically inevitable, while nothingness is not.
Hypothesis: what ultimately exists is pure structure (mathematical objects)
I will not go in to this hypothesis except to say it is consistent with materialism.
By the hypothesis, we are after some mathematical structure that finds its own existence to be logically inevitable. It is sufficient to consider the case where the mathematical object is the web of theorems of some formal system T. Then we want a theory that, in some sense, asserts it’s own inevitability. (How a theory that talks about itself might be implemented is discussed below.) In other words, the mathematical theory T is such that
(1) T
But even supposing we could find such a T, this condition is not enough to guarantee actual existence. All we can assert is that there could be some mathematical structure, such as T, that derives its own logical inevitability. But all we really know is that the existence of T is possible. In the many-worlds semantics of possibility this leads to two cases, worlds W for which T exists, and the disjoint worlds V where T does not exist.
We want to motivate the existence of T. In the worlds W where T exists, there is only the issue of finding a (the) T that satisfies (1). In the possible worlds V however, it is not obvious how to motivate existence. In these worlds T does not exist, so it is not around to assert anything about itself. In this case there is no way to get existence off the ground.
Happily, there are at least four promising ways out of this conclusion.
The first way out is to argue that the possible-worlds semantics of possibility is misleading here. In possible worlds semantics there are some worlds W in which T exists and some V where it does not. But this contains the trick of forcing an either-or decision upon us where it may be inappropriate. We do not want or else What we want is to consider the case that it is possible that T exists. If we give the notion of possibly existing this new status, it might be enough, in view of condition (1), to allow the theory to bootstrap itself into existence.
Returning to the possible worlds V and W, a second way out is to argue that, for the possible worlds V a theorem t may not exist , but t’s relations to other things might nevertheless exist. Thus there are second-order properties of t that claim existence in spite of t’s own non-existence. One example of this might be to say that if t is some assertion that is true, then that truth exists even if t does not (or only possibly exists).
A third way out, slightly different from the second, is to first show that T exists in the sense of T. Then, the argument goes, it is sufficient that T exists in the sense of T, because this existing-in-the-sense-of is all we need for a kind of absolute existence. It is not clear what it means to exist in the sense of something that does not (or only possibly) itself exist. What would it mean for a purple cat to exist in the sense of a flying pink elephant? Well, for all the flying pink elephant knows, the purple cat is real...
A fourth way out is to argue that all notions of existence are relative, and so there is no problem for the relative kind of existence in the third way out (above) because the initial existence predicate in is implicitly relative, and absolute existence is a priori relative existence anyway, etc.
My own preference is for argument (1) combined with argument (3).
Anyway, suppose we have a candidate theory T. T finds itself to be logically inevitable (and T does not find nothingness to be logically inevitable). According to the hypothesis, T does not merely represent the universe, it is the universe. If T is physical, then it would be a physical circumstance that the universe is logically inevitable.
Then, if we had good reason to believe T is our actual universe, we would be justified in claiming to have a rational explanation for existence.
Finding T
Let t be any theorem of T. Define t as
x such that x t
To express (1) and (2), T must contain 1st-order predicate calculus, equality and the modal operators. For T to find its own existence logically inevitable, (1) must be true of it. Since T is going to talk about itself, (1) is true in it. Therefore assume (1) as an axiom. Let T1 be the collection of resulting theorems. Now construct a model, in the sense of model theory, M2, such that the domain space of M2 is exactly the collection of theorems T1. Let T2 be the theory of M2. Construct a model M3 out of T2 in the same way, and continue this process indefinitely.
Conjecture: this process reaches a fixed point theory TFP
Claim: TFP satisfies (1).
TFP is our universe, and in being so asserts its own logical inevitability.
Paul
How to get an explanation for existence
Why does anything at all exist? This question already makes an assumption. It just assumes that nothingness is more warranted than the existence of something. But what justifies this assumption? Without any assumptions what we can say is that it could be that something exists. Indeed it could turn out to be the case that the existence of some particular thing is logically inevitable, while nothingness is not.
Hypothesis: what ultimately exists is pure structure (mathematical objects)
I will not go in to this hypothesis except to say it is consistent with materialism.
By the hypothesis, we are after some mathematical structure that finds its own existence to be logically inevitable. It is sufficient to consider the case where the mathematical object is the web of theorems of some formal system T. Then we want a theory that, in some sense, asserts it’s own inevitability. (How a theory that talks about itself might be implemented is discussed below.) In other words, the mathematical theory T is such that
(1) T
But even supposing we could find such a T, this condition is not enough to guarantee actual existence. All we can assert is that there could be some mathematical structure, such as T, that derives its own logical inevitability. But all we really know is that the existence of T is possible. In the many-worlds semantics of possibility this leads to two cases, worlds W for which T exists, and the disjoint worlds V where T does not exist.
We want to motivate the existence of T. In the worlds W where T exists, there is only the issue of finding a (the) T that satisfies (1). In the possible worlds V however, it is not obvious how to motivate existence. In these worlds T does not exist, so it is not around to assert anything about itself. In this case there is no way to get existence off the ground.
Happily, there are at least four promising ways out of this conclusion.
The first way out is to argue that the possible-worlds semantics of possibility is misleading here. In possible worlds semantics there are some worlds W in which T exists and some V where it does not. But this contains the trick of forcing an either-or decision upon us where it may be inappropriate. We do not want or else What we want is to consider the case that it is possible that T exists. If we give the notion of possibly existing this new status, it might be enough, in view of condition (1), to allow the theory to bootstrap itself into existence.
Returning to the possible worlds V and W, a second way out is to argue that, for the possible worlds V a theorem t may not exist , but t’s relations to other things might nevertheless exist. Thus there are second-order properties of t that claim existence in spite of t’s own non-existence. One example of this might be to say that if t is some assertion that is true, then that truth exists even if t does not (or only possibly exists).
A third way out, slightly different from the second, is to first show that T exists in the sense of T. Then, the argument goes, it is sufficient that T exists in the sense of T, because this existing-in-the-sense-of is all we need for a kind of absolute existence. It is not clear what it means to exist in the sense of something that does not (or only possibly) itself exist. What would it mean for a purple cat to exist in the sense of a flying pink elephant? Well, for all the flying pink elephant knows, the purple cat is real...
A fourth way out is to argue that all notions of existence are relative, and so there is no problem for the relative kind of existence in the third way out (above) because the initial existence predicate in is implicitly relative, and absolute existence is a priori relative existence anyway, etc.
My own preference is for argument (1) combined with argument (3).
Anyway, suppose we have a candidate theory T. T finds itself to be logically inevitable (and T does not find nothingness to be logically inevitable). According to the hypothesis, T does not merely represent the universe, it is the universe. If T is physical, then it would be a physical circumstance that the universe is logically inevitable.
Then, if we had good reason to believe T is our actual universe, we would be justified in claiming to have a rational explanation for existence.
Finding T
Let t be any theorem of T. Define t as
x such that x t
To express (1) and (2), T must contain 1st-order predicate calculus, equality and the modal operators. For T to find its own existence logically inevitable, (1) must be true of it. Since T is going to talk about itself, (1) is true in it. Therefore assume (1) as an axiom. Let T1 be the collection of resulting theorems. Now construct a model, in the sense of model theory, M2, such that the domain space of M2 is exactly the collection of theorems T1. Let T2 be the theory of M2. Construct a model M3 out of T2 in the same way, and continue this process indefinitely.
Conjecture: this process reaches a fixed point theory TFP
Claim: TFP satisfies (1).
TFP is our universe, and in being so asserts its own logical inevitability.
Paul
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