I guess that's one cool thing about math. If you want to make a statement, you prove that it's true. From the basic definitions of addition, multiplication, and so on, you can go on to prove entire branches of mathematics. It wasn't something like, say, world views or ways of life, where it was basically impossible to prove anything right or wrong.
You would think that math majors would be pretty good at proofs, but still, you can find math majors that aren't that good at proving things. Maybe a lot of people in general aren't good at understanding the things that are required for proving statements. (Of course, I can't prove that, so, maybe it's true. Maybe it isn't.) If you want to see if you understand some about proofs, here's a simple example I read from a book (I might add that whether or not you get the answer right or wrong, I don't think it proves anything about your ability to prove things):
There are cards that have a letter on one side and a number on the other side. Someone makes the statement, "Every card with a vowel has an odd number on the other side." You want to test if this statement is true. Say you have 4 cards on the table, and what you see face up on the table is: A, D, 2, and 5. Which cards are necessary to turn over to see if the statement is true for those 4 cards?
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| "Every card with a vowel has an odd number on the other side." Which cards do you have to turn over to test this statement? (It is guaranteed that there is a letter on one side of each card and a number on the other.) |
You probably got the A. If the A card has an even number on the other side, then the statement is false, so you have to turn over the A to see if the number on the other side is even or odd. You could also probably guess that you don't have to turn over the D. The statement says nothing about cards with consonants. You might have said that you needed to turn over the 5 card. If you did, you'd be wrong. What would you see on the other side? If it's a consonant, it doesn't disprove the statement as the statement says nothing about them. But if it's a vowel, well that doesn't disprove the statement either because 5 is an odd number. So it's not necessary to turn over the 5 card at all, because it won't give you any information. However, it is necessary to turn over the 2 card. What if there's a vowel on the other side? Then the statement is wrong.
The book also gave another example of the exact same problem, but in a different context, one in which people got the right answer more often. The statement is, "Everyone drinking wine is over 21." There are cards with people on one side and their drinks on the other. The 4 cards have an adult, a child, wine, and soda. Which cards do you have to turn over to verify the statement? Well that's easy, right? You have to check the child's card, to make sure he's not drinking wine, and you have to check the wine card, to make sure it's not a child on the other side. Well it is the same problem, with the wine being the vowel card, the soda being the consonant, the adult being the odd number and the child being the even number. In this case, checking the "odd" card would be the same as checking the adult's drink: you don't have to because the adult's already over 21 so it doesn't matter.
My Abstract Algebra professor told a story about how a failing student hoped to get extra credit to pass the class for his attempts to prove Goldbach's conjecture (which, as far as I know, is still unproven, and has been for a very long time). I think it's cool because it sounds really simple: "Every even integer greater than 2 can be expressed as the sum of 2 prime numbers." Even if you're not a math major, you surely know what an even number is, and perhaps you know what a prime number is as well. And you can say 4 is 2+2, 6=3+3, 8=3+5, 10=5+5 or 3+7... Apparently the student showed that it was true for even numbers up through the hundreds. Unfortunately, an approach like that will never work. I could just say, "What's the highest even number you proved it for? Okay, take 2 times that number, and you haven't even proved it for half the even numbers in the world because you haven't proven it for the higher half of the numbers from 2 to 2 times the highest you got to. And then take 2 times that and you haven't proven it for a quarter. Or for that matter, take 1 million, 1 billion, 1 googleplex times that. You haven't even proven it for .000000...0001% of all the even numbers, and, no matter how high you go, you'll never prove it for even .000000000...0001% of all the even numbers with that strategy."
It might be that people forget that when they look at the world. You want to make an absolute statement about the people in the world, or a specific group of people? Well you'll never be able to assert that it's true based on the fact that it's true for everyone you've ever known or met (unless of course, you've literally met everyone in the world which... isn't feasible). You can't even guarantee that it's true for most people, based only on the people you've met. The best you can do is say that it seems to be true. And even then, not everything that seems to be true actually is true.
Hey Daniel. I'm a new Applied Math PhD student at Mines and I came across your blog pretty recently when I created my own. I'm enjoying it so far. I have a few thoughts regarding this post that I'd like to share. I'd like defend the idea that you may be able to make certain true absolute statements about the 'life world' (e.g., about people or about morals). Your comments are welcome!
ReplyDeleteThe concept of a mathematical, or, more generally, a deductive proof, is pretty astounding. But there are some important things to note about the limitations of this concept.
1. Proofs by themselves do not necessarily uncover *truth*; rather, they provide a *justification* for a statement (the conclusion). There's a subtle but important difference. If any of the premises in the proof are false, then the conclusion is not guaranteed. One thing that is fantastic about math is that it works in increments, so the premises are always thought to be true. But, these premises of almost all mathematical arguments are the conclusions of other arguments; this goes back all the way to the axioms. Thus, all mathematical statements depend on the axioms for their truth. So, really, any assertion in math about the truth of a statement P is of the form "IF the axioms are true, THEN P is true."
2. Axioms cannot be proven. They are statements that are assumed without proof. Depending on the sub-discipline, the axioms are either taken to be self evident (e.g., geometry) or as definitions (e.g., group theory). But the history of math shows us that what is at one time self evident is not always true. Non-Euclidean geometries, for example, are formed from rejecting Euclid's 5th axiom for another statement that is inconsistent with it. Mathematics itself cannot tell us which type of geometry is the "true" one (I'm using the word true here to mean "corresponding to reality"); that is of the business of science, which is almost certainly not a deductive enterprise. The statements proven in Euclidean geometry *cannot be proven* in Non-Euclidean geometry (and vise versa). All this to say that the truth of statements in a mathematical system are dependent on the truth of the axioms.
I think this has important implications for your claims about making statements about people (or about morality, empirical objects, etc.). There are plenty of moral philosophers who do, roughly, as mathematicians do: they start with certain claims that seem self evident, and then "derive" (here I'm using derive to mean either inductive or deductive logic; I do think deduction is possible though) what follows from that. For example, utilitarian philosophers (J.S. Mill, Peter Singer) start from some very basic claims like, "Suffering is bad," and then try to derive what our obligations are from these "axioms".
They certainly can use deduction too. For example, if I believe the statements (a) "If I hang my dog, then I will cause him to suffer," and (b) "I do not want to cause suffering," then I can deduce that "I should not hang my dog." Another example might be the following: I can know that "All all living human beings have a heart," since a heart pumping blood seems to necessary for calling something a human. Deduction works in "the real world", just like in mathematics. And just like in mathematics, if one wants to be sure of the truth of the conclusion to an argument, one just has to be sure of the premises (and axioms). Thus, I think there can be a fairly similar method in ethics to the method in mathematics.
ReplyDeleteFurther, although you point out that inductive logic (e.g., observing a property in sample set and then concluding that a larger, unobserved set has that property) will never guarantee truth, there are still extremely strong inductive arguments. For example, the claim, "The sun will 'rise' tomorrow," can only be justified inductively; it is only because it has happened so many times in the past, and because we know about regularities in the solar system that we can assert this claim. But, I'm pretty willing to bet my life savings that this claim is true; the inductive evidence is incredibly strong.
All of science is usually thought to be justified through induction. Observations are made in a small area, and then, through careful reasoning, one tries to generalize the observation to a larger area. If your qualms about induction cause you to doubt the ability to make true statements in morality or in human affairs, then it seems like you're also committed to doubting true statements in science.
What do you think?
I feel like I agree with most everything you wrote in your first comment. I had considered the idea of "moral axioms" and thought it was interesting but too difficult to use. Something like "suffering is bad" is simple enough, but what if there's a situation in which you have to choose between the suffering of 2 people or of 1 person? You choose the one, right? Well what if it's one adult or one child? Or what if it's between someone in your family and a stranger? Or what if someone's suffering while dying? Do you let them die to end their suffering? In the end you would have to make so many rules for all these cases (2 people suffering is worse than one, rules like that), and I think these would have to be axioms. And then you'll end up with one main axiom and thousands of little rules and exceptions for various situations. And maybe everyone agrees with the main idea, but not everyone agrees on the little rules.
DeleteTo some extent I do doubt the "true" statements in science. There's the formula for gravity, but it hasn't been verified at every point around in space, not even every point around the earth. I would say that it could be possible for there to be some point in space that doesn't behave as we would expect from the rules of gravity. However, I have no problem with taking gravity as an absolute truth, at least on this planet. I'm not sure anyone could find even the smallest rumor of gravity not acting as we believe it does in the stories of history or in the experiences of everyone who lives today. The same with the sun rising too. But, if you wanted to tell me that you're sure gravity works the same way a billion light years from here, I would have to ask you "How are you sure? How do you know?"
With people, I would say, well, what statement do you want to make? If it has to do with personality, can you do better than saying "most"? I mean, if you're saying that something is true for most people, you're essentially admitting that there are exceptions, and then the statement isn't absolute. And with personalities, how do you test things? It's easy to check if the sun rises the next day, but it's harder to test if the next person you meet is a selfish person, or something like that.
The only other thing is that sometimes people seem to believe too deeply in the type of inductive reasoning that is something like, "I have proven myself to be better than everyone I know at some skill, therefore, it's extremely unlikely that there are people better than me at the skill." But unless you have reason to believe that the people you've proven yourself against are the best in the world, the conclusion most likely isn't true.