Hume - Final Paper

David Hume wished to explore and understand how people know things. To do this, he divided knowledge up into its different kinds, with different levels of certainty for each one. One of the kinds of knowledge is “Matters of Fact,” which are only known from experience, and and so cannot be known to be true. They can only be probably true. Another of the kinds of knowledge is “Relations of Ideas,” which includes all mathematical reasoning. This second kind of knowledge Hume claims to be undeniably true. However, he does not explicitly explain both what he means by “mathematics,” and how he can claim that it is true. By examining what he does say, though, we will conclude that for Hume, "math" refers to an internally consistent system of reasoning, and when he says that math is "true," he is claiming that it is true because it logically consistent with itself, and not because it is necessarily consistent to our experience.

Our investigation begins in the very first paragraph of Section 4 of Hume's Enquiry, where he makes the bold claim that math is "intuitively or demonstratively certain." He backs this up with the example: the Pythagorean Theorem. "That the square of the hypothenuse is equal to the square of the two sides, is a proposition, which expresses a relation between these figures…Propositions of this kind are discoverable by the mere operation of thought, without dependence on what is any where existent in the universe." And a bit later, he specifically refers to Euclid's propositions as "truths." Given Hume's aversion to certain knowledge, calling Euclid true seems rather bold.

This seeming devotion to Euclid can be explained, oddly enough, by examining the work of mathematicians who disagree with Euclid. Fifty years after Hume's death, mathematicians began to wonder whether Euclid's geometry was actually correct. By changing some of Euclid's starting premises (called postulates), these mathematicians were able to deduce entirely different results, results that even contradicted Euclid's results [Marshall]. Now unfortunately, Hume died before seeing this non-Euclidean geometry come to life, but it is helpful for one to ask what Hume may have thought of it, had he been alive to see it. I believe in that case Hume would have stuck by his claim that Euclid stated truths, but also claimed that the non-Euclidean geometry was true. You see, both of these competing mathematics are examples of systems. A system is just a set of rules and premises, where every conclusion is drawn completely from these premises. So, if you start with Euclid's premises, you get one conclusion, and if you start with the other set of premises, you get a completely different conclusion. But neither of these conclusions can be judged true or false by some external standard. They are only true or false only in regard to their own system's rules. So, if a conclusion in a system is consistent with that system's premises, then it is considered "true" within that system. And so, Hume would say that the conclusions of Euclid are true within his system, even as the conclusions of the non-Euclideans are true within their system.

Hume seems to agree with this idea of separate systems. He states that "[t]hough there never were a circle or triangle in nature, the truths, demonstrated by Euclid, would forever retain their certainty and evidence." So, according to Hume, the truthfulness of Euclid's claims cannot rest on whether they match up with our experienced reality. Euclid's geometry is its own separate system, and its truth or falsehood is determined by how internally consistent and logical it is, i.e. how well the premises follow from the conclusions.

However, this idea we have just posited is not a complete picture of mathematics. If math were just in its own system, separate from our everyday life, then math would not have practical purpose to us. We couldn't know if the conclusions we drew from math were helpful in explaining real life. Yet obviously, many people use math every day to help them through their lives. So, Hume tackles this problem in paragraph 13 of Section 4. He says that we can determine certain laws of nature through experience (though these laws are only Matters of Fact, and so are not indubitably known), and then mathematics can act, based on these laws. "Every part of mixed mathematics proceeds upon the supposition, that certain laws are established by nature in her operations." For example, Newton's Second Law claims that the force on an object is equal to its mass times its acceleration. Hume would say that Newton discovered this law through experience, and then once he had the law, he was able to use mathematics to take advantage of it (e.g. to figure out the force on an object by multiplying its mass times its acceleration). So, Hume would say that math exists as its own system, but when a law of nature that exists in a separate system (e.g. our world) is found, math can be applied to that law, even though they exist in separate systems. Hume calls this mixture of systems: "mixed mathematics."

The real question is: if normal geometry is true, then can we claim that mixed mathematics (in this case, applying normal geometry to our real world) is also true? Or is it only probable, but not certain? This question can be answered by remembering that math is always based on rules. For example, Euclid's geometry is based on one set of rules. Since these rules exist on their own, apart from any real world experience, they are only judged true or false based on how well they follow from the premises. When we use mixed mathematics, however, we are adding new premises to the system. So when we use arithmetic to calculate the force on an object, we are accepting Newton's 2nd Law as one of the premises of arithmetic. Since the conclusion's truth is based on the truth of the premises, the answer to our problem about force is only as certain as Newton's 2nd Law is. If Newton's Law is not an accurate description of our reality, then the conclusion to our math problem will not be either. And vice versa, if Newton's 2nd Law is true, then the conclusion that the mixed math derives must also be true.

However, Hume makes it very clear that although we can have some probable knowledge of the laws of nature, we can't actually arrive at a 100% certain law. We can't know that Newton's 2nd Law is true; all we can know is that it is probably true. For example, Newton's Laws seem to accurately describe the world we experience. However, in some extreme locations like the interior of an atom, Newton's Laws cease to work [Simanek]. Since they do not accurately describe reality in these situations, if we tried to do our previous calculation about force in them, we would arrive at a false conclusion. So, since the result of a math problem is only as true as its premises, any mixed math based on the natural laws we have discovered is only probable at best.

In conclusion, Hume thinks that math is just a system with rules and premises that are logically dependent on these rules. Math is considered true only because it is internally consistent, not because it necessarily coincides with our experiences. When one applies math to our experiences, the conclusion may or may not be true. But when math stays in its own system, it is always true and irrefutable.