Tuesday, September 27, 2005

Religiosity and Social Ills

What is the correlation between religiosity and social ills like abortion, teen pregnancy, sexually transmitted diseases, suicide and homicide?

The correlation is quite good. As The Times says:

According to the study, belief in and worship of God are not only unnecessary for a healthy society but may actually contribute to social problems.

The research paper that is the source for these claims has some wonderful charts on the topic. It's hard to pick a favorite, but I like this one:

(U=United States)

Eschew free thought, critical thinking and intellectualism, and these are the tragic results you get.

Quoting paragraph 18 of the paper:

No democracy is known to have combined strong religiosity and popular denial of evolution with high rates of societal health. Higher rates of non-theism and acceptance of human evolution usually correlate with lower rates of dysfunction, and the least theistic nations are usually the least dysfunctional. None of the strongly secularized, pro-evolution democracies is experiencing high levels of measurable dysfunction.

Once again, thanks to Pharyngula for the link.

Sunday, September 25, 2005

Magnificent Desolation

Magnificent Desolation is the new IMAX 3D film about the Apollo Moon landings. Seeing it yesterday, I had two thoughts.

First, if you've read about Apollo 15's mission to Hadley Rille and seen the photos, you'll appreciate how difficult it is to gauge scale and distance on the Moon's surface. It always bothered me that, though I was told I was looking at a 1,500-foot structure, I couldn't see it in the photos. In this film, you get to see what it's really like. What is it like? Well, it is almost as difficult to gauge distances on the Moon in 3D as it is in 2D. However, they do show the astronauts walk to the precipice overlooking Hadley Rille, and it's very impressive. It makes you realize just how dangerous the lunar environment can be, when you can't gauge how deep the adjacent Canyon is!

Second, though I would gladly accept a NASA Moon mission assignment, I can see the benefits of telepresence. Telepresence wouldn't have to get much better than an IMAX movie to be just like the real thing (minus the lunar gravity experience, of course). Now, how much bandwidth are we talking about? IMAX's 70mm frame is 10 times the size of 35mm film. If 35mm film is equivalent to around 14 Megapixels (just a ball park estimate), we're talking 140 Megapixels per frame, 280 MP for stereo. Multiply by 24 frames per second, and 4 bytes per pixel and you get about 26.8 Gigabytes per second, uncompressed. Okay, so live viewing might be tricky (especially given the transmission and processing delays), but it needn't be live. Not a lot happens on the Moon.

Why scientists dismiss 'intelligent design'

An excellent article found on MSNBC describing why ID has no scientific leg to stand on.

There's an upcoming court test of ID in public education in Dover, Pennsylvania. There are two reasons why this should be an easy victory for science. First, as my first link points out, there are no good scientific reasons to teach ID. The second is that the only remaining reason to teach ID in science class is religious. Indeed, it appears that the ID textbook proposed by Dover used the word Creation instead of Intelligent Design in its draft form, right up until printing time. Oops!

Tuesday, September 20, 2005

Existence Confusion

We sometimes paraphrase meaningful propositions in ways that are not strictly meaningful.

Take the proposition all Carbon 14 atoms will eventually decay. Literally, this proposition can never be tested. However, physicists would accept this as meaningful because they can translate it into a meaningful proposition, namely, the mean lifetime of Carbon 14 is 5,500 years. Technically, a Carbon 14 atom need never decay (though it would be extraordinarily improbable).

When you provide me with a philosophical proposition, I can try to squeeze it into a meaningful context, or paraphrase it so it does. I can almost always do this, as I demonstrated.

However, some propositions are designed to be resistant to this sort of reformulation. Suppose you give me just such a proposition:

A necessary, uncaused being exists.

If you accept my claim that a proposition must be part of a logical model and a choice of symbols to be meaningful, then we have to identify what that context is.

Here, you use the term "exists". You might mean this in a way that is different from when we say that "this cup of tea exists." We'll rewrite your verb as $exists until we know what it means.

For everyday, physical objects, existence is predicated on a specific definition of what empirical attributes the objects have. They are said to exist not when we can imagine them, but when we observe them.

For example, if you say that "a cup of tea exists on the table", and I look at the table and find that there is only a bowl of sugar on the table, then I have falsified your claim. Without these tests the verb to exist would be useless because elephants could exist on the table just as well.

So in your proposition, what are the specific, observable qualities of this being that would afford its existence if we observed them?

If you don't have any such qualities, then the verb you are using is just $exists not normal exists.

In the latter case, what does $exists mean? Perhaps, it is the fictional interpretation of exists, e.g., Chewbacca's third heart exists if we could only do an MRI on him in the Star Wars Universe. I doubt this because you have not shown what experiment, real or fictional would validate your claim.

Your claim is that this being (whatever a being is in this context) has a property of $existence which is as yet undefined. This is what I mean when I say it is meaningless.

The fact that confused people, some philosophers among them, feel they "understand" your proposition is not adequate to make your case. People think they understand a lot of things that they don't. In this specific case, people have an intuitive sense of what existence is. In their brains there is a cluster of neurons that fires when the concept of existence is triggered. This region forms as we grow up, using the verb to exist in the coffee cup sense. However, once we have a word for this existence property, it is tempting to trigger this region of the brain out of context. It is tempting to say that existence is just an attribute of a thing that makes it real.

This is confusing to us because we intuitively think things can exist independently of their properties. Next, we start saying X exists where X is something unobservable. It sounds grammatically correct, and it seems intuitively possible, but it is actually nonsense. It is a psychological illusion.

Now, your proposition may still form part of a logical structure, but unless you can define $exists in some rigorous way, you really aren't saying anything about the world.

Suppose you provide me some other interlocking propositions. Maybe "If a being A $exists, then there is a being B that does not $exist". (I have no idea what this means, but it would seem to logically relate to your proposition.) Even in that case, none of your interlocking propositions have any empirical consequences except for their own computational self-consistency. What makes such a proposition about the world? Nothing, I would say.

Friday, September 16, 2005

LogicSat 1

Gotta get me one of these handy space satellites.

10cm on a side, built and launched for around $50,000.

Thursday, September 15, 2005

Logical Positivism: Another Angle

This post combines some of my recent statements about metaphysics with the Principle of Verifiability.


Mathematical Systems
Mathematical systems are based on axioms, propositions that are assumed to be true without proof. From these axioms we can derive theorems, or truths which follow from those axioms. Pick different axioms, and you either get a different mathematical system, or you get an inconsistent system.

Physical Theories
A physical theory is a mathematical system that is augmented with axioms of empirical fact and with a correspondence function from mathematical propositions to empirical ones. The empirical axioms are assumed to be true because they have been observed. Physical theories make predictions by stating which additional empirical axioms would be compatible with the system of propositions. If we observe some empirical axiom that is not consistent with the theory, then the theory is "falsified." Physical theories are never intended to apply over all possible empirical observations. So, a falsifying experiment does not necessarily falsify the theory completely, it might just limit its domain of applicability. Newtonian mechanics has been falsified for speeds close to the speed of light, but this has only limited its domain of applicability. We still use Newtonian mechanics to build cars and bridges.

Fictional Worlds
A fictional world is a physical theory of a world in which the empirical axioms are not fixed by our experience, but fixed by our design. Such a system could make predictions about what other fictional empirical axioms could be added. For example, we know from fictional empirical facts that some life forms in the Star Wars universe are immune to Jedi mind tricks.

Products of Systems
I can take two independent mathematical systems and compose them into a new space of propositions. For example, I take two algebraic systems, R and F, and create a new system R×F ("the product of R and F"). The propositions of R×F are like (Ri, Fj).

Propositions in R do not contradict those in F, even if they are written using the same strings of symbols. For example, there might be a proposition (x = 7, x = 5) in R×F. This is perfectly consistent because the symbol x has a different context. Propositions in R are meaningless in F because they have no logical consequences in F. This is analogous to getting two consecutive algebra problems in a workbook, one with x = 5 and one with x = 7. No problem.

Factoring
Suppose that systems R and F share some axioms. I can combine R and F into a single system by creating a new system, R+F ("R union F"), which is founded on the axioms of both systems and contains every proposition of R and F. There is no guarantee that R+F will be consistent (e.g., if R contains x = 5 and F contains x = 7).

Suppose that R+F happens to be consistent. I can factor R+F into R×F by creating the two proper subsets of the axioms of R+F, in this case the set of axioms of R and the set of axioms of F, and rebuilding them independently as R and F.

Now suppose that R+F is a physical theory. If I factor R+F into R×F, and R contains no empirical axioms, then R is purely mathematical, and F is a physical theory. R contains mathematics which is extraneous to F. Any empirical axiom can be added to R without contradiction, but not every empirical axiom can be added to F without contradiction. That is, I can factor a physical theory with extraneous mathematics into pure mathematics plus a refined physical theory. This process of refinement aims to isolate just those axioms of a theory that have bearing on empirical axioms that might later be added to the system.

Suppose I have a system of Newtonian mechanics augmented with the mathematics of noncommutive algebra (NCA), I can factor this (N+NCA) into one system of Newtonian mechanics and another system of NCA (N×NCA). The NCA system doesn't have any correspondence with empirical facts, so it is extraneous to Newtonian mechanics. In other words, NCA doesn't do anything for my theory of mechanics. It just adds irrelevant propositions to it. I could have started from a system containing Newtonian mechanics and any random mathematical system that was without a correspondence function. The lack of a correspondence function linked to the axioms of this mathematical system allows me to factor out the extraneous mathematics.

Meaning of Strings of Symbols
Suppose you provide me with a string of symbols and claim it to be a meaningful proposition, P (e.g., P: E = mc2). I must ask in which system does P have meaning? Certainly, I could locate some mathematical system M and some choice of symbols for M where this proposition has some meaning. But, by choosing an alternate symbolic representation of M, P could have the opposite meaning. So, by itself, P has no meaning. It only has meaning in the context of a logical system where P can be related to other propositions in a symbol-independent way. P only has distinct meaning in a system when we designate what strings of symbols are consistent or inconsistent with it. That is, we must designate what propositions, if true, would be consistent with P and which would falsify it. If we cannot do this, then we cannot claim to know the meaning of P. In other words, the meaning of a proposition is its method of verification or falsification.

When I give you a string of symbols representing a proposition, your brain is working to find a context (a logical system and a symbolic representation of that system) in which the proposition has meaning. The goal of logical positivism is to make these meanings clear. Intuitively, we can almost always find a meaning for a string of symbols. Unfortunately, intuition is sometimes subjective and misleading.

If I provide you with a set of propositions like 'God is good' and 'God created the universe outside of time and out of nothing', you can only make sense of these propositions by confusing different systems. You create one logical system, G, in which God is a symbol, and in which propositions like 'God is greater than everything' are true. Then you use symbols like 'universe', 'create', 'nothing', 'time', and 'good' from the system of empirical science, E. So you try to make sense of the propositions in the context of G+E. Unfortunately, because propositions about G are compatible with every possible empirical axiom, G+E is factorable into G×E. In other words, propositions about God and reality are factorable into mathematical statements and scientific reality. The metaphysical propositions drop out from experience just as any extraneous mathematics would drop out of a physical theory. Propositions about God and reality are no more about the world than propositions about noncommutative algebra and reality.


Okay, it's a first draft.

I need to refine my definition of "product". I think I can say that

P+(R×F) = (P+R)×(P+F)

(R-F)+(F-R)+(R×F) = (R+F)×(R+F) = R+F

Where R-F is the set of axioms of R not in F.

Monday, September 12, 2005

Hail Pharyngula!

Want to keep track of the creationist war on science? Look no further than P.Z. Myers' wonderful blog about evolutionary biology. It's called Pharyngula.

One of today's posts is about so-called irreducible complexity. Irreducible complexity is an invention of creationists. It is the claim that certain evolved structures, like the human eye, are not only complex, but would be useless if any of their component parts were missing. Creationists then conclude that the odds of evolution coming up with all of the pieces independently are too remote.

As usual, creationism gets to append this argument to its litany of failures. Myers' blog post explains just how seemingly complex structures can evolve. Creationists ignore the effects of mutations that are harmless and of mutations that are orthogonally beneficial. Evolution is quite capable of building systems that naively appear designed.

I shall add that I'm very impressed with Myers' ability to write great blog posts, and write them so fast. It seems to take me forever to write even the simplest post. (This post originally featured a poker analogy, but it was taking me way too long to write up.)

Sunday, September 11, 2005

Social Darwinism

Some creationists accuse evolutionary biologists of advocating Social Darwinism. They say that Darwin's great idea, that Nature used survival of the fittest to evolve man from simpler forms, would imply that survival of the fittest is the "natural" choice of social dynamic. This isn't so much an argument as a confused association.

Evolutionary science does not imply that Nature "wants" anything at all, it just explains the mechanics of evolutionary systems. Even if we were to ascribe intentionality to Nature, we shouldn't leap to the conclusion that Social Darwinism would be Nature's optimal choice for humanity, or even that we should do what Nature wants. Compassion bestowed an evolutionary advantage on early humans. Caring for the elderly and the frail improved disease resistance by maintaining genetic diversity, and preserved a pool of knowledge and experience that benefited young and inexperienced members of the society.

Ironically, the evolution-deniers who criticize evolutionary biology are often right-wing, conservatives - precisely the kind of folks who advocate Social Darwinist government policy.

The "conservative" view is that programs such as welfare, Social Security and Medicare are a curse on society because they breed dependency. Conservatives also argue that oversight agencies such as the FDA, the USDA and the OSHA constitute unreasonable government restriction on trade. It is true, there are people (and corporations) who have grown helpless and dependent on government programs. It's also true that regulatory burdens raise the cost of doing business, and if people paid attention to product reviews and acted more like ideal consumers, maybe we could do without government oversight. There's no doubt that consumer protection and the social safety net has its downside. The question is, what are we trading for, here?

Free markets work by selecting winners and losers. If one advocates free market rules for social services, one is also advocating that there should be losers in society. And, no, when I say 'loser,' I don't mean 'unlikable person.' People who lose in social free markets have failed lives. They die from curable diseases, work in dead-end jobs from which they will never free themselves, lose their livelihoods by consuming defective products, and pass on their lack of skills and resources to their children.

You can't fool free markets. If you're willing to bailout the losers, the whole system will fail. If you're not willing to let people die of curable ailments, or let them starve on the street, you're not serious about eliminating government's social safety net.

Besides, the savings one might hope to realize by cutting taxes for social and consumer protection programs are insignificant in comparison with the increased costs of fixing failed lives. Studies show that prevention of failed lives is far less expensive than fixing the resulting crime, disease and ignorance. There's really no good economic reason for Social Darwinism. No. Social Darwinism is a moral choice.

For Christians, the choice to back social programs should be obvious. Much of the New Testament describes Jesus caring for the less fortunate in society. Believers can hardly claim that the monetary cost of social programs is prohibitive. I doubt that Jesus would oppose a strong social safety net because the undeserving will also benefit from such programs. Christians are not supposed to accumulate wealth when there are poor people who can't fend for themselves.

Humanists like myself will side with the Christians on this one. It is better to lift up the poor from death, disease and misery, even knowing that there will be freeloaders who abuse the system. Moreover, the humanist will have faith that, through science and reason, we can make the system more cost effective, more uplifting and more fair.

Wednesday, September 07, 2005

What is metaphysics?

In mathematical systems, axioms and proven implications of axioms (theorems) are tautological. On the other hand, unproven mathematical hypotheses may or may not have a known (or knowable) truth value, though they may have meaning. For example, the Goldbach Conjecture ("every even number greater than 2 can be written as the sum of two primes") is meaningful, though it may be unprovable. It is meaningful because we might find some even number that would violate the proposition.

Alone, empirical facts form a system of axioms and tautologies. If the mass, m, of a cylinder remains the same over some time period, we can restate these facts by saying that between time T1 and time T2 M(t) = m. This isn't a theory, just a tautological restatement of our observations.

A physical theory combines empirical facts (empirical axioms) with some set of mathematical axioms by creating a correspondence between the mathematics and the empirical axioms. Physical theories generate scientific hypotheses about which empirical facts can be added to the system with consistency. Experiments that confirm the theory are able to add the new empirical facts to the system as axioms of that system without breaking the consistency of the theory. Experiments that falsify the theory show that there are absolutely true empirical axioms that are inconsistent with the theory.

But where lies metaphysics? I see only two options:

1) Metaphysics might be purely mathematical, there being no special axioms of metaphysics except, perhaps, those axioms that we require for reasoning about the consistency of systems.

2) Metaphysics might somehow be grafted onto physical theories. If we add mathematics to a physical theory such that the mathematics has no implications for experiment, what have we done?

If case 1, then metaphysics would not be about the world, only a statement about possible mathematically consistent systems.

In case 2, we have more questions to ask.

Let's assume that the same metaphysical system is claimed to be true independent of the mathematical theory of the day. Yet, the mathematical theory is a function of empirical fact. As new empirical facts are discovered, the mathematical theory has to be modified. Empirical facts tell use what (infinite) set of possible mathematical theories are consistent with those facts ({E} constrains {T}).

We know that a metaphysical system does not imply anything about empirical facts ({M} does not constrain {E}). If it did, it would be empirically testable.

Therefore, to be about the world, metaphysical claims must have something to say about the allowed subset of physical theories we can use to explain empirical facts ({M} constrains {T}). The principle of causality might be such a metaphysical claim. Without causality, a mathematical theory might predict two outcomes for the same experiment. Causality limits the possible mathematical theories we can cook up.

However, if a metaphysical system does not restrict our physical theories, then that metaphysical system would be wholly detachable from physical theories, and we revert to case 1.

Propositions about God (e.g., 'God exists' and 'God is good') do not constrain empirical facts we might observe, i.e., they are not scientific. No surprise there. However, they also fail to constrain scientific theory-building. Therefore, it is totally separable from physical theories, so it must be pure mathematics, and not about the world.

Republican Kool-Aid

From the Washington Post:

It is a problem that now appears destined to follow Bush through the final years of his presidency -- a clear failure of his 2000 campaign promise to be a "uniter, not a divider."

A Washington Post-ABC News poll taken last Friday illustrates the point vividly. Just 17 percent of Democrats said they approved of the way Bush was handling the Katrina crisis while 74 percent of Republicans said they approved. About two in three Republicans rated the federal government's response as good or excellent, while two in three Democrats rated it not so good or poor.


Two in three Republicans rated the federal government's response as good or excellent? Apparently, Republicanism has been reduced to the Cult of Bush. Sick.

And what is it with the press? Can't they report the fact that federal response was utterly dismal without asking for partisan opinions? Go to the experts on disaster relief and ask them, not Bush's campaign manager.

Sunday, September 04, 2005

Leadership

I grow increasingly convinced that leadership is the key to making this country great.

My pride in America was demolished in 2004 because Bush did actually win the election. He used opposition to gay rights as a wedge, and absurdly smeared Kerry during the campaign, but Americans fell for it. As Janeane Garofolo once said, "being Republican is a character flaw," and 50+% of Americans suffer from it.

So, while I'm politically active, and I work to strengthen democracy in the country, I'm forced to wonder what the average American would do with the vote if they actually exercised it. Probably not a lot of good. Americans are too lazy to pay attention to what's going on in the world, and half of those who do pay attention get it wrong.

As with any financial market, you need the right constraints and oversight to keep a democracy healthy. I think this concept is well beyond the comprehension of the average voter, let alone the average citizen.

Elect the right Democrat, and you'll build a strong public education system, create effective government institutions, and once again make government a trustworthy ally for consumers. This won't happen just by getting more people to vote, or by fixing tax systems or anything else. We simply have to elect great leaders.

I'm not really saying anything deep here. It's just that I always used to think of our democratic institutions as being self-correcting. Now, I'm not so sure. It seems as though our country depends on its leadership for its health and safety, and that no amount of bureaucracy or paper pushing is a substitute for a good chief executive.


Our congressional representative is a Republican. He's a slick character and considers himself to be "independent." He's actually pretty right wing. To the average Joe, his voting record looks fairly benign. The reality is that Congress isn't being presented with any legislation worth voting for. That's where leadership comes in. Without a president who will propose legislation for the people of the United States, and veto perverse, corrupt, pork-barrel legislation, our representatives are going to be presented with precious little legislation worth voting for.

Tweaked Meaning

I wish to update my definition of meaning again. Here goes.

If P has meaning, then we can find P1 and P2 such that

P => P1, where P1 is distinct from P

and

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

I was shown by my colleage, leibniz, that my previous definition was plagued by the possibility that P1 was a function of P, and that did very bad things.

This revised definition says that for P to have meaning, P must be verifiable and falsifiable. My previous arguments show that, if P is meaningful, then P must be either mathematical or empirical.

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.

Saturday, September 03, 2005

On Meaning

I've been debating philosophy on other blogs lately. Occasionally, I come across a debating colleague of good temperament, who challenges me to think carefully and refine my ideas. The following is an edited version of one of my recent posts about meaning.

leibniz: What reason do we have to think that 'the meaning of a proposition is its method of verification or falsification' can be empirically verified or falsified?

Clearly, we cannot answer this question without some definition of meaning.

My claim is that the definition of meaning is such that, for a meaningful proposition, P, one can determine one or more distinct propositions that are consistent with P and one or more propositions that are not consistent with P (excluding ~P). This determination need not be complete. For example, I may determine that test proposition P1 is consistent with P, and that test proposition P2 is inconsistent with P, but not know whether test proposition P3 is consistent with P. In other words, P must have logical consequences to be meaningful.

Indeed, this is the principle of verifiability, i.e., the claim that any meaningful proposition is ultimately equivalent to some set of empirical test propositions or else it is part of some self-consistent mathematical system (i.e., it is equivalent to mathematical test propositions). Again, in my view, mathematics is empirical. Let's refer to propositions that meet my definition of meaning as verifiable propositions.

Now, you can challenge my claim with the counterclaim that my definition of meaning is wrong or incomplete.

Let's suppose that meaning is deeper than I have suggested and that not all meaningful propositions are verifiable propositions. In that case, we might be able to construct two propositions Q1 and Q2 that have different meanings, but which are compatible with exactly the same set of logical or empirical test propositions. This hardly seems reasonable, since we could not then distinguish the meanings of the two propositions except by looking at the symbols used to express the two propositions. For example, we would have to admit the possibility that given that the two propositions, "the cat is in the hat" and "the tac is in the hat", can be substituted for one another within any set of logical and empirical test propositions, they don't mean the same thing. In other words, we would have to claim that raw symbol substitution changes meaning. This isn't supported by the evidence that meaning can be preserved under language translation or changes in typeface.

Suppose the challenge to my definition of meaning is dropped, but instead the claim is made that we can still form a mutually consistent set of metaphysical statements that is meaningful. Let's look at what distinguishes a mathematical system from a metaphysical one.

We can create a large number of independent mathematical systems by devising axioms and deriving theorems within each system using the rules of that system. Each mathematical system is independent of the others when its axioms differ. If I start with an algebraic system, A1, and add the axioms x = 5, x + y = 10, I am not precluded from creating a new system, A2, in which x = 7, x + y = 8. The two systems are never in contradiction because their contexts are different. The axioms of A1 may contradict those of A2, but this is unimportant because the systems exist in isolation. Every proposition of A1 coexists with A2 and vice versa.

Similarly, all of these mathematical systems are independent of the physical world. However, we may be able to construct a new system that incorporates an association between a chosen mathematical system and physical phenomena (i.e., a scientific theory).

For example, noncommutative algebras (NCAs) may form a consistent mathematical system, independent of whether or not free neutrons decay in 15 minutes. However, we could form a scientific theory about neutrons that relied on NCA, and which incorporates connected empirical and mathematical propositions from NCA. However, if the theory were falsified by some experiment, the mathematics of NCA would not be falsified, because the mathematics is subject only to its own axioms. Only the new system incorporating the association between the mathematics and the empirical propositions would be falsified.

Now, to metaphysics.

Metaphysical systems also consist of a set of propositions {M} = M1, M2,...,MN which are inter-related by logical operations. For example, we can say that God is supreme and therefore Satan is not supreme and so on. In that sense, {M} is as meaningful as any mathematical system because, for any Mi with can find a distinct Mj that is consistent with Mi or a distinct Mk that is inconsistent with Mi.

However, metaphysical systems need more than just internal mathematical relations. Metaphysical systems have to mean something about the world. That means that we must be able to locate a set of empirical propositions about the world that contains propositions consistent with and inconsistent with {M}. If we fail to do this, and every proposition in {M} is consistent with every empirical proposition, then how can we distinguish {M} from a mathematical system? We cannot. Metaphysics just pushes symbols around on paper, and, at best, creates some mathematically consistent system. It never has anything to say about the world because it never makes a falsifiable prediction.

In summary, I have tried to show that

1) meaning is defined as a logical property in the context of a set of propositions.

2) that if meaning were not so defined, then one would have to conclude that meaning is dependent on the specific symbols used to express it. This is not supported by empirical evidence of meaning.

3) under my definition of meaning, mathematical systems are meaningful in their own context.

4) that scientific theories associate mathematical systems and physical empirical propositions in a falsifiable way.

5) that metaphysical systems are indistinguishable from isolated mathematical systems, and have no meaning relative to the world, only to themselves.

Imagine taking a test in algebra class, and solving problem #1 that begins "x = 5, ...". Then imagine being profoundly confused when problem #2 on your exam begins "x = 3,...". How can x be both 3 and 5?

Metaphysicians suffer from this very same delusion. They confuse truths within one isolated mathematical system with truths within other, independent systems.

Thursday, September 01, 2005

National Disgrace!

Though it has been three days since hurricane Katrina devastated New Orleans, we still don’t have the enough food, medical supplies, shelter, security or transportation for the victims.

Why do we not see convoys of relief vehicles carrying victims to safety and ensuring that humanitarian needs are met? Why are there not enough National Guard troops on the ground to ensure security?

President Bush needs to be held accountable for this outrage. Not only has the President failed to maintain a viable national disaster response program, he has failed to show even the minimum level of concern and leadership that we would expect of any responsible citizen in his place.


If you feel the same way, please write to your local newpaper.

You think to yourself, "surely, the government must have evacuation plans for major cities in the event of disaster or terror attack, right?" Now, you know. They don't. George W. Bush should be impeached for this debacle.