Sunday, August 29, 2010
The Funkiest Song in the History of History
The Funkiest song of all time is Maybe Your Baby by Stevie Wonder. Yup.
Thursday, August 26, 2010
Why do we exist? version 1b
Can we find a rational explanation for existence?
This note outlines a strategy to find such an explanation. We start with possible existence (of anything at all), move to a relative existence (things may exist relative to each other), and argue for a resulting necessary existence (of the universe). A prototype of an argument to find what the universe must necessarily be is then given.
Start with possible existence.
The existence of nothingness is a simpler hypothesis than the existence of something, so why does anything exist at all? But, really, how did we determine that nothingness is the better hypothesis? In fact it could turn out that the existence of some particular mathematical structure is more inevitable than the existence of nothingness. On the other hand, it is better not to assume the positive existence of some particular structure, either.
All we can say is that for any given mathematical structure, it might exist.
Part 2 is the “relative existence” move. What does it mean to exist? It is sufficient to consider the cases of relative existence. If X exists and it is evident to X that Y exists then Y exists.
Hypothesis 1: existence is mathematical existence (i.e. ultimately what exists are mathematical entities)
Let be “it is possible that” and be “it is necessary that”. The idea is that the nature of existence is such that
(1) if x xy then y
where xy means y exists in the sense of x.
Why should we believe (1)? There is the intuition that relative existence is sufficient for a kind of absolute existence. That, morally, if xy then x too. This latter term means that a state of existing relative to x is a state that exists itself. One could also argue that “exists” in x was (implicitly) relative also, in some way.
For y to exist in the sense of X when X is a formal system means that
(2) X y
What we want is a mathematical theory T that sees itself as inevitably existing, so that
(3) T t
for all propositions t in T.
Under these circumstances, if we could be sure T is our universe, then we would be justified in claiming we have a rational explanation for existence.
Is it possible to find out which theory T is? Call our first guess T1. Let T1 be standard first-order predicate calculus, enlarged to contain the two modal operators above. These are needed to express p. However, T1 does not contain any axioms, functions or constants. This would seem to be the choice with fewest assumptions. From T1 we can construct some model M2. The elements of the base space of M2 are exactly the propositions of T1, the relations in M2 codify implication in T1, etc. Then what is T2, the theory of M2? It would seem to at least contain T1 as well as the theorems about the Lindenbaum-Tarsky algebra of T1.
Repeat this procedure for T3, T4 …
Conjecture: this process reaches a fixed-point theory TFP.
Claim: the meaning of TFP is that what is logically inevitable is the existence of TFP
This note outlines a strategy to find such an explanation. We start with possible existence (of anything at all), move to a relative existence (things may exist relative to each other), and argue for a resulting necessary existence (of the universe). A prototype of an argument to find what the universe must necessarily be is then given.
Start with possible existence.
The existence of nothingness is a simpler hypothesis than the existence of something, so why does anything exist at all? But, really, how did we determine that nothingness is the better hypothesis? In fact it could turn out that the existence of some particular mathematical structure is more inevitable than the existence of nothingness. On the other hand, it is better not to assume the positive existence of some particular structure, either.
All we can say is that for any given mathematical structure, it might exist.
Part 2 is the “relative existence” move. What does it mean to exist? It is sufficient to consider the cases of relative existence. If X exists and it is evident to X that Y exists then Y exists.
Hypothesis 1: existence is mathematical existence (i.e. ultimately what exists are mathematical entities)
Let be “it is possible that” and be “it is necessary that”. The idea is that the nature of existence is such that
(1) if x xy then y
where xy means y exists in the sense of x.
Why should we believe (1)? There is the intuition that relative existence is sufficient for a kind of absolute existence. That, morally, if xy then x too. This latter term means that a state of existing relative to x is a state that exists itself. One could also argue that “exists” in x was (implicitly) relative also, in some way.
For y to exist in the sense of X when X is a formal system means that
(2) X y
What we want is a mathematical theory T that sees itself as inevitably existing, so that
(3) T t
for all propositions t in T.
Under these circumstances, if we could be sure T is our universe, then we would be justified in claiming we have a rational explanation for existence.
Is it possible to find out which theory T is? Call our first guess T1. Let T1 be standard first-order predicate calculus, enlarged to contain the two modal operators above. These are needed to express p. However, T1 does not contain any axioms, functions or constants. This would seem to be the choice with fewest assumptions. From T1 we can construct some model M2. The elements of the base space of M2 are exactly the propositions of T1, the relations in M2 codify implication in T1, etc. Then what is T2, the theory of M2? It would seem to at least contain T1 as well as the theorems about the Lindenbaum-Tarsky algebra of T1.
Repeat this procedure for T3, T4 …
Conjecture: this process reaches a fixed-point theory TFP.
Claim: the meaning of TFP is that what is logically inevitable is the existence of TFP
Tuesday, August 24, 2010
Why do we exist?
The purpose of this blog is solicit help in solving the deepest question: that of a rational explanation for existence. The philosophy relating to existence needs to be clarified and a particular mathematical theory needs to be identified.
What we are after is a rational explanation for existence. But which explanation is it?
Is there a theory T that appears to itself to exist? By “appears to itself” I mean T as a model satisfies T as a theory. If so, we would have a rational explanation for existence at hand.
We can use T’s apparent existence because all that is required of T is that it appear to inevitably exist relative to the elements of the base space. The appearance is ontologically real in the same sense that it has mathematical existence—it cannot tell the difference between existing in the sense of T or not existing at all. In some sense T would be more inevitable than non-existence.
For a first guess at what T is take predicate calculus. Call this theory T1. It is a deductive system that has no axioms, functions or constants. What is the theory of T1, call it T2, that T1 is a model of? T2 is the theory of the Lindenbaum-Tarski algebra of T1. I don’t know if T3 is different from T2. Anyway, continue this process until it reaches some fixed point, call it Tfp. Then Tfp appears to itself to exist as itself, and intrinsically contains the explanation for its own existence.
Then the project would be to look for evidence of Tfp in our universe.
But, would Tfp be a theory of absolutely everything? Actually, I don’t think so. The problem, as I see it, is those entities well-known in the philosophy of mind, qualia. It is likely that an explanation for them is itself an experience (as opposed to a mere concept about something), and thus not a mathematical structure. But if you have a better idea lets hear it!
Appendix
Okay, so what’s T3? It is the Lindenbaum-Tarski albegra of T2. But does every proposition in T2 count as an element in the model’s base space or do we mod out by something for some reason? In the first case, it is clear the naked predicate calculus T1 forms a different base space than that of its LT algebra as a base space. Therefore T3 is different from T2.
What we are after is a rational explanation for existence. But which explanation is it?
Is there a theory T that appears to itself to exist? By “appears to itself” I mean T as a model satisfies T as a theory. If so, we would have a rational explanation for existence at hand.
We can use T’s apparent existence because all that is required of T is that it appear to inevitably exist relative to the elements of the base space. The appearance is ontologically real in the same sense that it has mathematical existence—it cannot tell the difference between existing in the sense of T or not existing at all. In some sense T would be more inevitable than non-existence.
For a first guess at what T is take predicate calculus. Call this theory T1. It is a deductive system that has no axioms, functions or constants. What is the theory of T1, call it T2, that T1 is a model of? T2 is the theory of the Lindenbaum-Tarski algebra of T1. I don’t know if T3 is different from T2. Anyway, continue this process until it reaches some fixed point, call it Tfp. Then Tfp appears to itself to exist as itself, and intrinsically contains the explanation for its own existence.
Then the project would be to look for evidence of Tfp in our universe.
But, would Tfp be a theory of absolutely everything? Actually, I don’t think so. The problem, as I see it, is those entities well-known in the philosophy of mind, qualia. It is likely that an explanation for them is itself an experience (as opposed to a mere concept about something), and thus not a mathematical structure. But if you have a better idea lets hear it!
Appendix
Okay, so what’s T3? It is the Lindenbaum-Tarski albegra of T2. But does every proposition in T2 count as an element in the model’s base space or do we mod out by something for some reason? In the first case, it is clear the naked predicate calculus T1 forms a different base space than that of its LT algebra as a base space. Therefore T3 is different from T2.
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