Sunday, September 04, 2005

More Meaning

In my last post, I stated that for a proposition P to have meaning, we have to be able to show that there is some distinct proposition P1 that can be shown to be consistent with P, and some proposition P2 (distinct from ~P and ~P1) which can be shown to be inconsistent with P.

That is, if P has meaning, then we can find P1 and P2 such that

P1 => P or P => P1, while P1 is distinct from P

and

P2 => ~P or P => ~P2, while P2 is distinct from both ~P and ~P1.


Writing my previous post got me thinking about other ways to express my position that might more clearly show the fuzzy thinking that leads to metaphysics.

It all has to do with context.

Mathematical propositions are true only inasmuch as they are implied by assumed axioms. So, although we may be able to determine through calculation that x = 5, we were only able to do this by first assuming, say, x + y = 10 and y = 5. Thus mathematics is a collection of empirical facts associated with assumptions (axioms) about mathematical rules and symbols. The mathematical scientist tests axioms, and mechanistically derives theorems from them. There is no guarantee that resulting mathematical system will be consistent.

There are many mathematical systems that can be constructed, and they can all be said to "exist" simultaneously. However, the rules of logic cannot be applied between mathematical systems because there is no assumption or expectation of consistency between them. The axioms of one system may contradict the axioms of the other, or the theorems of one may contradict the theorems of the other.

Empirical propositions are those whose truth can be demonstrated experientially. 'The sky is blue' is empirical in the physical sense. A priori, there need not be any logical implication from one empirical proposition to another. Water oceans might be blue whether or not the sky were blue. Fortunately, we have reason to believe that there are rules of implication that constrain empirical facts. Experience shows that, under the same initial conditions, processes lead to the same results. Of course, such rules are not directly visible.

We can create mappings from mathematical systems to the empirical world, e.g., we can say that the force due to gravity varies as the inverse square of the distance between two masses. This kind of mapping almost always produces a scientific theory. This is because most axioms produce a large number of theorems which have implications for future measurements. It is only by using this kind of mathematical modeling can we show logical implication for any empirical proposition. A history of tides in New York Harbor is not a predictor of future tides without a corresponding scientific theory of tides.

As with a purely mathematical theory, a physical theory begins with axioms. We begin with mathematical axioms, and add a mapping to empirical propositions.

Suppose I construct two mathematical systems, M1 and M2, distinguished by their axioms. There is no requirement that propositions shared by M1 and M2 have the same truth values. Theorems that are true in one system need not be true in the other.

For example,

M1: x = 5

does not contradict

M2: x = 7

but does contradict

M1: x = 7

So, refining my definition: a proposition P in system M (M:P) has meaning in M when we can find 1) another distinct proposition M:P1 in M that is entailed by M:P, and 2) another proposition M:P2, distinct from M:~P and M:~P1, that entails M:~P.

I'll introduce a shorthand for this:

$ M:P | M:P1 | ~ M:P2

Think $ proposition | confirming | ~ falsifying. Like this:

$ M: x = 5 | M: x = 3 + 2 | ~ M: x = 3 + 3

[Note: What I meant in my initial definition that we could determine that P1 was consistent with P, so that was, indeed, entailment.]

There are two rules we can derive from this.

First, one cannot claim that M: (P and P1) is a confirming proposition unless you know that P1 is true, in which case you are really verifying P using M:P1. In my notation:

RULE 1:

$ M:P | M:(P and P1) | ~ M:P2

=>

$ M:P | M:P1 | ~ M:P2


Second, one cannot use propositions from other systems to confirm meaning in a given context:

RULE 2:

$ M:P | M1:P1 | ~ M2:P2 => M = M1 and M = M2


Otherwise, one could claim:

INVALID:
$ M:x @ 5 | M2: x @ 3 | ~ M: x @ 4


or

INVALID:
$ M:x @ 5 | M2: x @ 3 | ~ M2: x @ 5


where M2 is some other mathematical system with alternative axioms. In neither of these cases do we learn anything about the meaning of x @ 5.

However, meaning in the context of empirical theories is quite different. Physical theories are grounded in experience:

EN: conserved kinetic energy = 2Kg * Newton(100,000 m / s)

contradicts

EE: conserved kinetic energy = 2Kg * Einstein(100,000 m / s)

even though the mathematical systems that define Newtonian and relativistic physics have different axioms.

In that case, RULE 2 can be relaxed slightly:

RULE 2 - Relaxed:

$ M:P | M1:P1 | ~ M2:P2 =>
EITHER M = M1 and M = M2
OR M, M1 and M2 are physical theories that share empirical propositions


The Newtonian/relativistic physics example demonstrates the case in which two physical theories share empirical propositions.

Here's an example where meaning is obscured if physical theories don't have overlapping empirical propositions:

$ E1: levers amplify force
| E2: some rocks burn exothermically
| ~ E3: frogs have three legs


If E1 is inadequate to describe chemistry, and E2 is inadequate to describe basic mechanics the our proposition about levers cannot imply anything about burning coal or vice versea. If you can't explain why 'levers amplify force' implies 'some rocks burn exothermically', you can't be said to understand what 'levers amplify force' means. This wouldn't preclude us from obtaining a meaning for 'levers amplify force', but only if we produce an alternative meaning over a more restrictive range, e.g.,

$ E1: levers amplify force
| E1: force is measured at the endpoints of the lever
| ~ E1: the force at each end of the lever is equal


Let's try to translate a couple of the common arguments for God into this notation.

For example, those who use the design argument would claim that God is the designer of the universe, that is, he is the designer of all empirical propositions.

$ M: God exists
| X: The universe was designed
| ~ M: God has sub-maximal divinity


The meaning of the existence of God depends on the context X. I would manitain that engineering and design are physical processes, e.g.,

$ E1: Whittle designed the first Jet engine
| E1: Whittle created an accurate model of a physical system that predicted propulsion
| ~E1: Whittle was building modern art and accidentally constructed an engine


The only meaning we have for design is as part of physical theory E1.

We are left with only two choices to make God meaningful. We can either cook up a meaning for 'design' as part of M, or create interpretations for God and divinity within physical theories that describe design. If we choose to interpret 'design' as part of M, then it is reduced to a mathematical symbol with no physical meaning. If we try to phrase everything in E1, we get:

$ E1: God exists
| E1: The universe was designed
| ~ E1: God has sub-maximal divinity


In this case, there has to be some physical measure of divinity. If there isn't, then we don't have a meaning for God.


Now, consider the ontological argument:
1. God is the most perfect ('the greatest') being conceivable.
2. It is more perfect ('greater') to exist than not to exist.
3. Therefore, God must exist.

Okay, it's a pretty silly one, but some people seem to find it convincing.

God as defined by the ontological argument appears as:

$ M: God exists
| M: It is more perfect ('greater') to exist than not to exist.
| ~ M: God is not perfect


The problem here is that 'perfect' and 'greatest' only have meaning in the mathematical or empirical senses. The claimant hopes to import these meanings into a new context. We can say that the meaning of perfection is consistent with:

$ E1: X is perfect for Y
| E1: X can be used to accomplish Y
| ~ E1: Z can be used to accomplish Y in less time than X


Notice that the meaning of perfection depends on a notion of time, or space, or mass or some other measurable thing.

As with the argument from design, we have two choices to salvage meaning. We can come up with meanings for perfection and greatness in M, or else we have to make the existence of God empirical by claiming that it shares empirical propositions with the other two test propositions. We can only say God exists is a proposition about the world when we can predict, for example, that some state of physical affairs is categorically inconsistent with the perfection of God and can never be observed if God exists.

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