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.

10 comments:

Peg said...

Hey Doc, Lee STrobel has a book out called, A Case For Faith...he provides ideas as to why thinking about God can be illogical.
Interesting, he is writing from an interview he had with a man, can't remember his name off hand, that use to be as big if not better than Billy Graham and then he turned against God when he saw a picture one time in Time or Life magazine of an African woman who was crying over the death of her child due to the lack of rain that the country needed so badly. Just kind of an interesting read.

Doctor Logic said...

Hi Peg,

There are two ways to take Strobel's statement (or your paraphrasing of it).

I will assume he doesn't mean that it is illogical to contemplate God because God is meaningless. Otherwise, he would be advocating my position. :)

Unfortunately, I think he's saying something I consider rather evil. He wants people to take action based on contemplating God without being logical about it. That is, to obey religious edicts without trying to see if they make logical sense. This is morally and ethically wrong, in my opinion. This is exactly what I find most terrifying about religion - its lack of reason, its ability to make human life and human happiness secondary to something illogical.

As for the problem of evil, I think that the man who turned away from religion has a point. Nothing humans do can make any significant contribution to a world ruled by God. It would be like me expecting a 2 year-old to assist me with software engineering. Then to decide that some 2 year-olds should be tortured forever? This cannot be reconciled with my conception of justice. If there was such a God, he would deserve our contempt, not our love. And, if my family were condemned by God to be tortured forever because they couldn't believe in him, should I collaborate with that same God? I can only see collaboration as an option when the highest morality is saving one's own skin. Christianity just doesn't compute. And to act on something that doesn't compute is to throw ethics to the wind.

Now, if God brought everyone into heaven, that might be just... Or maybe bad people could be recycled through re-incarnation. You see what happens when you start to build a system of cosmic justice? You start architecting your own religion. Just like people have been doing for millenia. :)

I'm not saying that you (or everyone else) will see Christianity the way I do. You are probably Christian because the ethics of your denomination are agreeable to you. I think people choose their religion based on their morality, not the other way around. Some people choose Buddhism or a more liberal Christian doctrine because they want to subscribe to a belief system that is fair. I mean, who wants to believe in an evil God?

Peg said...

Hi Doc, what Strobel is saying is it is hard to understand and accept all phasses of God and make it logical, because on one hand God is a loving God, but then if God is a loving God why then would he allow certain things to take place, such as the death of the little African infant. It is hard to see God as a loving God and then as a God who would carry wrath or allow evil to flourish in this world.
But, again, our understanding of God is incomprehensible. We cannot possibly think they way God thinks or in the terms that God thinks and acts.
The job or "things" that humans do in this world is to do all things for the glory of God.
For instance if your physics teacher expects you to do all your work to obtain an A then you do all that work if you expect to obtain that A. But if you only do some of the work your grade will be calculated(justified) on the amount of work you do.
As for all people going to heaven, it is God's desire to see all go to heaven but many will not take that road. Okay so now you can say, well if God is all powerful and in control of everything, why does he not just allow everyone into heaven. Again I understand it as the example of the teacher.
One bad apple can ruin the entire bunch if allowed to remain in the bushel.

And unfortunately for those who need tangible evidence and contact, God would indeed be a questionable problem. For faith is the mainstay of a Jew or Christian. If you have no faith you are indeed walking blindly. Not to say that you will not waver, but for the most part you have faith.
And to me as much as I love philosophy, there are many holes to is design.
And yes, I could be considered an oxymoron with the exception though that given the two ideas I will back Christianity to the bitter end.
Leaving of course "religion" out of it, for I am not a religious person, but a Christian.
What say ye to that? ;-0}

leibniz said...

This post does a better job than previous ones of laying out your position, Dr. Logic, but unfortunately it fails to address my primary criticism of previous instantiations.

The key step in your argument occurs in the last paragraph, where you write:

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.

Suppose I grant for the sake of argument that statements about G are compatible with all possible "empirical axioms". You go on to argue as follows:

1. G+E is factorable into GxE.
2. (1) is equivalent to saying that propositions about God and reality are factorable into mathematical statements (G) and scientific reality (E).
3. Therefore, propositions about G (God and reality) are no more about the world than propositions about noncommutative algebra and reality.

The problem with this argument (or one problem with it) is that you assume without argument that if a theory can be factored into disjoint sub-theories (or systems), then one of the systems must be mathematical, and the other empirical or physical. By why in the world would we think this, apart from a prejudice against metaphysics? Suppose I say that if G+E can be factored into GxE, then it can be factored into an empirical part and a non-empirical part, where the latter can be either mathematical (in which case it is not about the world) or metaphysical (in which case it is about the world, though perhaps not the physical world). For your argument to be successful, you need to show why this isn't a viable possibility.

Doctor Logic said...

leibniz,

What I am saying is that metaphysical systems are indistinguishable from mathematical ones. Is Euclidean Geometry about the world? But not the monster group?

Who is to say?

No one can reasonably say that a thing has a property without justification. I cannot say that there are 12 kinds of equilateral triangles when I have no way to distinguish the 12 categories.

I cannot say that there are 2 kinds of humans, Alphas and Betas, when I fail to say how to tell the difference.

Likewise, you fail to show that there are two kinds of mathematics, plain ol' mathematics and metaphysics.

I cannot accept that metaphysics has meaning if it cannot be distinguished from mathematics. There is so much emotional attachment with metaphysical systems, that confusion or self-delusion is a viable theory.

The burden of proof is on the metaphysician.

To begin with, one must at least provide criteria for deciding whether a system is metaphysical or not. I doubt that this is even possible because, as I have said before, there are no grounds for determining whether a metaphysical proposition is absolutely true.

leibniz said...

Likewise, you fail to show that there are two kinds of mathematics, plain ol' mathematics and metaphysics.

I fail to show it because I don't believe it. On my view, metaphysics isn't a kind of mathematics; rather, it can be thought of as a kind of tertium quid between mathematics and physical theory.

You say that the burden of proof is on the metaphysician to show that metaphysics can be distinguished from mathematics. But I have already met this burden. Metaphysical statements can be distinguished from mathematical ones on the basis of their content. Specifically, their content reveals that they are about the world. They include such claims as 'A necessary, uncaused being exists', 'Everything that exists is material', 'The universe came into existence a finite time ago', 'Space is a system of relations among things', 'Natural laws are relations among certain universals', 'It is wrong to torture merely for the sake of pleasure', 'Mathematical statements are not about the world'. All of these are claims about the world, but none of them can be falsified or verified empirically. (Incidentally, all of them are meaningful, as we can see from the fact that we all understand what is being claimed when someone makes such statements.)

Doctor Logic said...

leibniz,

You provided me with these propositions, I tried to find a context in which the proposition made sense. For each proposition, I was able to find some physical context in which I could regard it as meaningful by my logical positivist standards.

Let's take these one at a time:

A necessary, uncaused being exists

The proposition does not specify what the uncaused being is necessary for. And, to exist, a thing must have actual empirical attributes. If we could trace the history of a being back to the start of the universe, this could form the basis of a physical theory.

For example, if the universe could be considered a being, then a universe that starts at a singularity, e.g., the original Big Bang theory, satisfies this proposition. It is "necessary" as an axiom in a framework for the observed universe within the context of its physical model. It is uncaused, having a single initial event with no precursor events.

The Big Bang theory is falsifiable.

Everything that exists is material

Depending on your definition of material, this is just the definition of the verb "to exist." It is meaningless to say that something exists independent of its empirical attributes.

The universe came into existence a finite time ago

This is most definitely a proposition in a physical theory. It is a meaningful proposition within the Big Bang theory.

The Big Bang theory is falsifiable.

Space is a system of relations among things

If you mean Space as in the space between stars, then this could be a plausible definition of Space.

Natural laws are relations among certain universals

Natural laws are essentially conservation laws. Identifying conserved quantities in a subset of data enables us to predict future observations. Your statement is a mathematical precondition of all predictive physical theories.

Verifiable by showing that the mathematics of physical theories only permits constraints on predictions when the mathematics is isomorphic to the structure of physical laws.

It is wrong to torture merely for the sake of pleasure

This is an expression of preference by the speaker. That preference being an empirical measurement of that person's opinion. Wrongness is by definition a subjective reaction to some scenario. To say that something was absolutely wrong would be to misunderstand the meaning of the word "wrong."

It is falsifiable by measuring the speaker's emotional reaction (or, one day, neural reaction) to depictions of sadism.

Mathematical statements are not about the world

By my definition, a proposition "about the world" predicts or paraphrases some empirical measurement. Computation is an empirical procedure that usually arrives at the same answer given the same initial conditions and a recipe for the calculation. Mathematics is about the world only to the extent that, given the same axioms, we will always derive the same theorems if we do our sums correctly. I assume you are not trying to express this claim.

However, if you can show that mathematics predicts or paraphrases some empirical facts quite apart from computation, you will have falsified the proposition. (This reminds me of block transfer computation in Doctor Who!)

So it seems that the propositions you provided can be viewed as verifiable or falsifiable propositions of mathematics or physics (or as definitions).

If I have misinterpreted your propositions, please feel free to explain where I have misunderstood their meaning. My claim is that either a proposition is about the physical world, is about mathematics, or else the speaker is confused about the meaning of its constituent symbols.

Doctor Logic said...

Hi Peg,

Hi Doc, what Strobel is saying is it is hard to understand and accept all phasses of God and make it logical, because on one hand God is a loving God, but then if God is a loving God why then would he allow certain things to take place, such as the death of the little African infant. It is hard to see God as a loving God and then as a God who would carry wrath or allow evil to flourish in this world.

What you are saying is that the idea of God is self-contradictory.

But, again, our understanding of God is incomprehensible. We cannot possibly think they way God thinks or in the terms that God thinks and acts.

So why do you think you can know what he wants? Or that he is good?

For instance if your physics teacher expects you to do all your work to obtain an A then you do all that work if you expect to obtain that A. But if you only do some of the work your grade will be calculated(justified) on the amount of work you do.

I think this is exactly the line of thinking that the inventors of God had in mind. They created God by analogy with fathers, tribal leaders, teachers and so on.

Okay so now you can say, well if God is all powerful and in control of everything, why does he not just allow everyone into heaven. Again I understand it as the example of the teacher.
One bad apple can ruin the entire bunch if allowed to remain in the bushel.


I could understand this if God didn't have the power to ensure that we all behaved correctly, either in the first place or in the afterlife. He would have the power to make sure that the bad apple were fixed. He either can't do it, or doesn't want to. For me, this raises serious questions about his morality, especially when he doesn't just terminate the bad apples, but torture them for eternity.

And unfortunately for those who need tangible evidence and contact, God would indeed be a questionable problem.

But there can be no evidence. If a superalien shows up and claims to be God, how do we tell the difference? We cannot. This is because the concept of God is malformed unless he's a superalien subject to physical laws (even if not our physical laws). God is not properly defined, so I don't see how it would be fair to blame logically and linguistically correct people for not believing in him. If reason shows that the concept is meaningless, and yet God still punished people for being reasonable, then God would be evil.

And to me as much as I love philosophy, there are many holes to is design. And yes, I could be considered an oxymoron with the exception though that given the two ideas I will back Christianity to the bitter end.

I expect that your beliefs are not based solely on your intuitive sense that God is a meaningful concept. However, your belief that God is something meaningful (i.e., is a sensible, well defined concept) is a prerequisite for many of your other reasons for faith.

Imagine you attempted to prove that God was linguistically and logically reasonable. But in the process, you proved the opposite, i.e., that propositions about God were meaningless and nonsensical. Would that have any weight in your assessment of God?

I think that when I was a teen, I thought that God would want us to be honest with ourselves and that truth would be paramount. In searching for God (as I did), there were three possible outcomes. I could prove he existed, prove he did not exist, or have no proof of either. If I could prove he didn't exist, then God (at least, the only God I would consider worthy of worship) did not exist because he would not have let me prove he didn't exist.

I am curious to know whether a personal proof of non-existence (or of meaninglessness) would be (partially) persuasive to you, or not.

Let's suppose God did not exist and there was proof of this. You could always argue that the only universe worth living in is one in which (a) your particular God exists, and (b) he allows proofs of his non-existence. In that case, you could at least motivationally justify your choice to act in a manner that is inconsistent with the reality that he doesn't exist.

You see, I don't have a great track record in converting people. I don't think that people are easily converted by logical proofs, especially not complex ones. However, in the future, human intelligence will be significantly higher, and mathematics and logic will be much simpler for everyone. Though few people have patience for my arguments now, I think people will prove them to their own selves in the future. They might still opt to ignore reason, but they will at least know that they're doing so. At least this is how I justify the time I spend debating this stuff. :)

leibniz said...

Suppose we number the statements in question.

1. A necessary, uncaused being exists.

The claim here is that there exists a being who is uncaused who exists necessarily. The idea isn't that its existence is necessary for something, but that its existence is necessary in the sense that the being couldn't have failed to exist. Now it may be that the universe is such a being, but the falsifiability of the big bang theory does not make (1) falsifiable. Suppose the BBT were false. That would not show that there is no necessary, uncaused being. So (1), understood in the usual way that any philosopher or well-educated person would understand it, has not been shown to be falsifiable or verifiable.

2. Everything that exists is material.

You seem to be claiming that by definition to exist is to be material, so that this claim is trivial and vacuous, like saying that everything that exists exists. But what you are proprosing as a definition of existence is false. It is not true that to exist is to be material. That is why this claim expresses something substantive, as most any educated person will readily admit. What you need is an argument for your claim that to exist is just to be material.

Incidentally, you say:

...to exist, a thing must have actual empirical attributes.
It is meaningless to say that something exists independent of its empirical attributes.

These too are highly tendentious claims, and therefore they need to be defended, not just asserted. I actually see them as expressions of an anti-metaphysical prejudice that has no adequate rational justification.

3. The universe came into existence a finite time ago.

The falsifiability of the BBT entails nothing about the falsifiability of this claim. Suppose the BBT is false. Does that rule out the possibility that the universe came into existence a finite time ago? Not at all. The inference from 'The BBT is false' to 'The universe has always existed' would be highly dubious.

4. Space is a system of relations among things.

If you mean Space as in the space between stars, then this could be a plausible definition of Space.

OK, but you've said nothing to show that (4) is verifiable or falsifiable.

5. Natural laws are relations among certain universals.

Your statement is a mathematical precondition of all predictive physical theories.

How so? It is a philosophical theory of the nature of natural laws that need not be true in order for us to make predictions of future observations. (5) could be false, as some philosophers think, and that would make no difference to our scientific predictions. Another way to put the point is to say that this philosophical theory of the nature of natural laws is completely independent of the contents of those laws, must as a theory of the nature of books would be independent of the content of those books (what they are about).

I think I'll stop here. Almost all normal, well-educated people at least think they understand these claims. And understood as they normally are, these claims are both about the world and not empirically verifiable or falsifiable. That alone is enough to meet the metaphysician's prima facie burden of proof. You, in making the argument against metaphysics, cannot simply ignore this third possibility (that there are statements that aren't empirical but are about the world).

Think of it this way: In giving your argument against metaphysics, your intended audience is either (i) people who agree with you that every theory is either mathematical (non-empirical, not about the world) or physical (empirical, about the world), or (ii) people who accept that there are metaphysical claims that are neither mathematical nor physical. If your argument targets only those in (i), then you are simply preaching to the choir, giving people an argument for something they already believe. But if you are also targeting the people in (ii), then your argument, in order to succeed, needs to take account of the possiblity that metaphysics is a tertium quid (non-empirical, about the world). Most people in (ii) are going to accept this view of metaphysics, so in arguing against them you can't simply rule that view out from the beginning.

Doctor Logic said...

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

Take the proposition all Carbon 14 atoms will eventually decay. Leterally, 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. If I understand you correctly, your example #1 is just such a proposition.

1. 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 comptational self-consistency. What makes such a proposition about the world?

P.S. I'm going to repost this as an official blog entry.