The first of the logical arguments for God is the Cosmological argument. This is one that is commonly misunderstood by both believers and unbelievers. Since the standard presentation (which I believe leads to the standard misunderstandings) can be found elsewhere, I am going to take a different approach in how I explain this argument.
Before I begin, I want to clarify some terms. I will be using, throughout these posts, the term atheist in a particular way. Atheism can be defined as an affirmative belief in no God, as a strong opinion that there is no God, or as a belief in a religious philosophy that has no God. I am going to be very particular in how I present the term, though, to encompass exactly the meaning that I am going for. When I speak of atheism, I am not speaking of a particular type of atheist, but rather the world that would exist if atheism was correct. The atheism worldview is the way the world would have to work if there was no God. It is more closely akin to reductive materialism (that everything in the universe is nothing but the matter and energy that make it up, and that with a sufficiently good computer and algorithms we could have predicted everything that would ever have happened thereafter). This type of atheism is far more common among scientists than among philosophers -- a scientist becomes so enamored with science that they begin to believe that science, cause, and effect are all that exists.
Atheism requires a cause for every effect. Science can explain why the billiard ball, when struck just so, rolls into the corner pocket. It can explain why the cue had to be hit in a certain way. It can explain the force used by the player. It can explain the birth of that player, and the origin of the materials that created the table. Science is truly amazing in that regard. It can continue to track backwards on and on -- each effect related to the cause that preceded it (presumably -- I disagree but that is not relevant to this discussion). The effect of the planet's formation arises from the cause of a nearby supernova some time ago. This effect was caused by the stellar formation, which was caused by two other stellar formations before it. Atheism can trace it all the way back to the Big Bang bang.
That's where things get murky, but we are going to ignore that murkiness. Instead, we will push beyond that arbitrary boundary and state that somehow science could explain the effect of the Big Bang by its preceding cause. And that effect by another cause. And so on and so forth. Do you see where this is going? Even assuming science could answer every question -- there were no 'gaps' in knowledge, so to speak, there would still need to be an infinite number of causes to result in any effect. Blame the Big Bang on the many worlds and branes bumping into each other just leads to the question of where the branes came from, and where their cause came from, and so forth. To put it more bluntly, why is there something rather than nothing? An infinite past of infinite causes is logically inconsistent with atheism. Atheism, at its heart, cannot answer the question of why there is something rather than nothing. They must take the existence of something on faith.
Compare that with the theist (or, more particularly, the LDS) view. We believe that there has always existed certain things. Matter, spirit, intelligence. These things are eternal, and because they are eternal they are uncaused. Nothing need cause something that always existed -- it just always existed. This seems very counter intuitive to us, because we live in a materialistic world that follows a pattern of cause and effect. But the doctrine as revealed to Joseph Smith clearly indicates that it is true. The wider Christian world also believes in an eternal, uncaused God (they just differ on our natures). Note that we are not talking about multiplying Deity here, as it is true for us and for Him and for every uncaused, eternal being.
Look at the result of that. Atheism makes sense to us in a world of cause and effect, but we can run it against a simple experiment. If atheism is true, and there cannot be an infinite regress of cause and effect (because how else would the first cause occur), then there should be nothing. If there is nothing, then atheism is right. If there is something, then atheism is wrong. This is what is known as a self-refuting hypothesis. The presentation of the belief is contradicted such that it cannot be true on its own terms.
But wait, says the atheist, that is no different that what the theist argues. It is all word games and logical tricks. Not so. The theist says, 'we don't understand it, because God hasn't explained it, but we know from God that certain things are eternal and uncaused.' This is a worldview consistent with the evidence that there is something rather than nothing.
To present it as a logical argument:
1) In a reality that is controlled solely by cause and effect, there could never be a first cause because nothing would exist to cause the first cause;
2) Therefore, in a reality solely controlled by cause and effect, nothing would exist rather than something existing;
3) Something does exist, therefore reality is not controlled solely by cause and effect; and at least one uncaused cause must exist from which all other effects and causes were derived;
4) An uncaused cause is impermissible and contrary to atheism, while an uncaused cause is permissible and mandatory to theism;
5) Therefore atheism cannot be right on its own terms and theism may be right on its own terms.
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