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What Explains the Laws of Logic and Mathematics?

Post Author: Bill Pratt

The laws of thought and mathematics are absolutely true.  The law of non-contradiction, the law of identity, and the law of the excluded middle – the three fundamental principles of thought, otherwise known as the laws of logic – are all undeniable.  To deny them is to assume they are true.

They are true regardless of time, place, or who is thinking about them.  There is no possible world where they could not be true.  Likewise with mathematics.  2+3=5, and this is true regardless of time, place, or who is thinking about it.  There is no possible world where 2+3 does not equal 5.

So, any worldview which claims to explain all of reality had better have a good explanation of how this could be true.

How does atheistic naturalism explain the laws of thought and mathematics?  Since everything, on naturalism, must be reduced to physical matter, an explanation for the laws of thought and mathematics will be hard to come by, for these laws are clearly not made out of matter.

Naturalists take a couple different routes.  First, some of them say that there is no explanation for these laws; they just exist and that’s it; they are brute facts of the universe.  But surely these laws that transcend time, space, and matter, that existed before humans ever came on the scene, and will still exist after humans are extinct, are uncomfortable bedfellows with electrons, skin cells, and hydrogen atoms – the things of physical science.

How will the scientific methods of physics, chemistry, and biology explain the laws of thought and mathematics when they are built on them and rely on them? It’s like trying to explain the cinder block foundation of a house by appealing to a second-story window.

A second explanation is the following: some naturalists deny that these laws actually transcend time, space, and matter.  They claim that these are merely human conventions, laws that human beings have simply invented.  But this claim seems incredible.  Are we to really accept that 2+3 does not equal 5 unless human beings say it does?

I feel quite confident that even Klingons would agree that 2+3=5.  Can you imagine there being any dispute between a human and a Klingon over math? Of course not. The idea is absurd. This explanation just won’t fly because these laws are absolute; it doesn’t matter one bit whether any one of us ever discovered these laws, as they would still be true.  You can’t imagine a time or place where these laws aren’t true.

How does Christian theism explain the laws of thought and mathematics?  Instead of denying that these laws are transcendent, Christian theism affirms our basic intuitions that they are.  Christians identify the source of the laws of thought and mathematics with God, who is timeless, spaceless, and has always existed.

These laws are a part of God’s eternally existent nature.  They are built into God, in a matter of speaking.  So Christian theism not only provides an explanation for these laws, it also provides an explanation that makes sense of the absolute and transcendent nature of these laws.  They have always existed because God has always existed.

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  • Andrew Ryan

    “To deny them is to assume they are true.”

    Indeed. I don’t understand, therefore, why they NEED explanation. Saying an explanation is required implies that we need to explain why they are this way and not another, whereas you point out yourself that they could NOT be any other way. For me it therefore follows that they would be that way with or without a God.

  • Ggodat

    Us and the Klingons agree but the Romulans are a different story…

  • Bill Pratt

    Epistemology vs. Ontology. This is a common theme that you don’t seem to understand. That we know something is true or exists does nothing at all to explain its source, or its place in reality. You seem to think that if we know something to be true, that’s just the end of the story and any further questions about the thing’s source are pointless.

    Moral laws are true, but they need no explanation. Logic and math propositions are true, but they need no explanation. The fact that truth even exists, for you, I’m guessing needs no explanation.

    All I can say is that the vast majority of philosophers and serious thinkers of the last 3,000 years disagree with you. Socrates, Plato, and Aristotle were obsessed with these kinds of questions. They all want to know how is it that these things exist. Where do they come from? What are they?

    As long as you don’t understand this line of questioning, you will be forever frustrated with philosophy.

  • Todd

    As you point out, the laws of nature exist because they are brute facts about reality. I would not however argue that all logic exists outside of time and space. For example, speed = distance/time. Without time, there is no concept of speed. In these cases, I think it’s safe to say are human conventions created by humans to explain our reality. If we step outside of our reality, say, to before the beginning of the universe it is quite plausible that many of these equations would not work.

    Of course, more to your point, there are concepts that hold true outside of time such as math. Math is transcendent of time and space, but justifies itself by being useful in predicting the nature of reality. But just because the concept of math is transcendent, does not mean the concept of god has any basis in reality. Nor, I think a Vulcan would say, is it logical to follow that math is part of gods nature. We can use the concept of math to predict things about the nature of reality. The concept of god has no such property. Before you can assert that the concept of god is transcendent, first I think you must prove that god as a concept has a basis in reality. If we accepted that god exists simply on the faith that his transcendence is real, we might as well accept that 2+3=6 for all the good it does to explain the truths of our world.

  • Andrew R

    ” You seem to think that if we know something to be true, that’s just the end of the story”

    That’s not what I said at all. I’m talking specifically about concepts like 2+2=4. If there’s no way possible it could be any other way, then it is transcendent. If you conjecture a ’cause’ for it, then you imply it could have been different, which I understand you believe not to be the case – ie you accept 2+2 could equal nothing but 4.

    “As long as you don’t understand this line of questioning, you will be forever frustrated with philosophy.”

    I don’t think you understood my point.

  • Bill Pratt

    Why do numbers exist at all? If no transcendent minds exist which can know numbers transcendently, then why would the physical universe produce them? Numbers and mathematics are useless without minds to think about them. What are numbers anyway? Why do mathematics apply to physical reality?

    Naturalism can’t even begin to answer any of these questions because all the methods of science presuppose numbers that are transcendent and that apply to the physical universe.

    Again, I am asking about the foundation of the house, and all naturalists can offer me is a discourse on the second-story window framing.

  • Bob Seidensticker

    “How does atheistic naturalism explain the laws of thought and mathematics?”

    For this to be an interesting question, we need to know that a universe without these laws is possible. Only in that case could we wonder why we got the universe we did. Maybe it’s just the case that these laws come along for the ride.

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