Reading: Mathematics and Logic

All right, it's time for me to put up or shut up, right? I've been saying things such as that. Even a simple thing like "Formula One plus one equals two" has ontological and epistemological assumptions to it. They, in turn, are driven by some religious belief or another. That is, whatever assumptions one takes on logically, epistemologically, are due in turn to what one regards as divine. That's a pretty wild claim. So, I promised to give you the reasons for it today and explain how it works.

First of all, let's say that there definitely is a level at which one plus one equals two for everybody. It doesn't matter whether you're a Buddhist, a Krishna, a materialist, or what. You add a column of figures the same way. But there are questions that can be asked about this formula, which are ontological and epistemological. And which mathematicians have long asked and tried to answer. First of all, what are these marks on this paper representing? What do those marks represent? What do they stand for? Let's call the marks on the paper numerals. And we'll say that these numerals stand for something else. And that "something else" they stand for are numbers. Okay, what's a number?

Now if you want to really spoil the mathematics department Christmas party, you lean over the punchbowl and ask, "What's a number?" That may end up in a food fight. But people disagree wildly on the right answer to that. That's an ontological question. What are the realities that these stand for? There have been people who take many different positions, including that there are no realities that they stand for. That's an ontological question. Here's an epistemological one: How do we know that the truth expressed by this formula is true everywhere and always? We treat it as though it is. We treat it as though it's been true no matter how far back in the past. We treat it as though it's true right now in the farthest distant reaches of the universe and that it will be true 10,000 years from now. How do we know that? I mean, on what grounds do we know that? That's a truth that can't fail no matter the time and place? It's always going to be true. So, we have an ontological question and we have an epistemological question. If you think that how you answer those questions makes no difference to how mathematics is done, then you have a rude awakening coming.

Let's take a look at how some people have answered these questions.

The first person I'm going to quote is Bertrand Russell, a mathematician who lived in the late 19th and first half of the 20th century. Russell thought about this kind of stuff. He was fascinated with mathematics, which he said was one of the great loves of his life. I have a quote here from Russell concerning this. He says, "Philosophers have commonly held that the laws of logic, which underlie mathematics, are the laws of thought, laws regulating the operations of our minds. By this opinion, the true dignity of reason is greatly lowered. It ceases to be an investigation into the very heart and immutable essence of all things, actual and possible, becoming instead an inquiry into something more or less human and subject to our limitations. But mathematics takes us from what is human into the region of absolute logical necessity to which not only the actual world but every possible world must conform." So, Russell posits that math is reducible to logic. And logic is the realm of absolute necessity. Logical laws are real and they govern all reality. Math is a shortcut way of doing logic for Russell.

This reminds me of what Leibniz said when asked a similar question. Leibniz and Newton invented calculus at about the same time, independently of one another. Leibniz was asked, "How do we know?" He replied, "One plus one equals two, like all other truths of mathematics, is an eternal and necessary truth that would not be affected even if the whole world were destroyed, with no one to count and no objects to be counted." So for both Russell and Leibniz, logical laws are real. Math is a shortcut way of doing logic. These logical laws govern all reality with absolute necessity and are the heart and immutable essence of all things.

That's very different from the answer given by another thinker, John Stuart Mill. Mill believed that all our knowledge is derived from perception. This means that we see often enough that when we have one thing and another thing, we have two things. We then generalize and treat it as a law. Do we really know that one plus one will always make two? No. It's just our generalization. Another thinker, Ernst Mach, believed that all mathematics was a game that we invented. We set the rules and the outcomes are always what we say because we made it up. It has no correspondence to reality.

John Dewey, a pragmatist, believed that if ideas and systems are instrumental to an active reorganization of the environment and they succeed, then they are true. They don't correspond to any reality. They are tools, and if they work, they work. That's it.

The view that people take on math affects how they use and develop it. Epictetus showed that pi, as far as we know, has no end, and this upset the Pythagoreans who believed everything was made of numbers. They threw him overboard. Leibniz, on the other hand, believed in a realm of numbers and mathematical truths. This led him to deny negative numbers. But negative numbers are real and effective in many calculations.

One of the greats was a man named Brower. According to his intuitionist view of mathematics, logic is part of language. It's not foundational to math. In Brower's perspective, math is more fundamental than logic. There can be logical conundrums that shouldn't concern us. This led him to reject certain proofs. For example, if a proof looks like this: "either p or q is true. If it's not p, then it must be q," Brower would argue that such logic isn't acceptable. He wouldn't entertain any argument of that form. His intuitionist stance also led him to reject the entire theory of transfinite numbers as developed by Cantor. He believed Cantor's theory wasn't even false—it didn't even qualify as being false. It was utterly meaningless. This perspective essentially discarded an entire branch of math.

These were significant disagreements among these scholars. To quote a historian of mathematics who wrote a book on the subject: "The current predicament in mathematics is that there's not just one, but many mathematical schools of thought. For various reasons, each fails to satisfy members of opposing schools. The once-held belief in a universally accepted, infallible body of reasoning—the grand mathematics of the 1800s, man's pride—is now an illusion. The present state of mathematics mocks the previously deep-rooted reputation for truth and perfection. Different ontological and epistemological bases are assumed, driven by what each thinker sees as the independent, self-existent reality. For [Ludwig] Wittgenstein and [Bertrand] Russell, it's logic. For intuitionists like Brower, it's mathematics, not logic."

For pragmatists like Dewey, the foundation lies more in a biological model. It's about ultimate usefulness in preserving life, underpinned by the belief that materialism combined with biological evolution provides all we need to understand humans, even influencing the bounds of mathematics. So, if I disagree with all of them, what's my position? I propose the "reformational" perspective—a term I've borrowed from Calvin's critique of the Scholastic view. Calvin argues that reasoning about the world's design is futile unless one submits to the "foolishness of the cross." In knowing Christ and the true God through Him, one realizes that even natural laws are a creation, not uncreated. Neither logic nor math is synonymous with the essence of God; they are His creations. Mathematics deals with the quantitative aspects of things, just as there are spatial, physical, biological, or logical aspects. There are real laws governing these properties—logical laws, mathematical laws—and they correlate with the real world God created. This perspective differs vastly from others and is critically important, yet hasn't received much attention. I aim to change that in our upcoming discussions. While everyone agrees that one plus one equals two, interpretations of this truth vary. Some see it as real, some as divine and uncreated, others as fictional. My stance is that we should view it as real but created, distinct from the Divine Creator.

In our next session, we'll do a general review. We've covered a lot in a short span. We discussed theories, their distinction from myths, and how they relate to philosophy, reality, and knowledge. We also delved into how religious beliefs influence theory-making, driving their foundational assumptions, which, in turn, shape the theory itself. It's a lot to digest, so our review will methodically re-examine these topics. Do some more reading, and this review will be very beneficial.

(Note: Some changes were made for clarity, conciseness, and to correct grammatical errors. Some names, like Wittgenstein and Russell, were inferred from the context to provide clarity.)