Video Transcript: Example Theories about 1 + 1 = 2
Welcome back. This is the one you've been waiting for. Okay, let's see it Clouser How are you going to show that theories are guided by and differ depending on the Divinity beliefs they presuppose? Or put it this way. The divinity beliefs presuppose by the person devising the theory. Are there theories in math anyway, one plus one equals two is not a theory. No, it's not. No, it's something we we learn by just observing the world around us. So that's the formula that we're going to worry about. Now, although that is not a theory, there are questions that naturally come to people's minds about that, which require a theory to answer. First of all, what are those marks stand for? The marks are numerals and plus and equal signs. But what do they stand for? And I guess the answer you feel like giving is well, they stand for numbers. Okay, what's a number? If you really want to spoil the mathematics department, Christmas party, all you have to do is lean over the Punch bowl, and say, what's a number and it could end in a food fight. There are very different, sharply conflicting theories about this. So let's start to look at it. Here's the question, what are those things? What's a number? Second question? How do we know that 1+1=2 everywhere? And all times? How do we know that 1-1=2 10 million years ago? How do we know it will make two a 100 years from now? How do we know that it makes two somewhere in the distant, far distant universe? Could it be different? Some thinkers say yes. And some say No, those are differences we're going to look at. So here's the first interpretation. And I'm going to call this the number world theory. Number World Theory. Here we go. What are numbers, according to the number of World Theory. And this sounds like Plato, because this is where it comes from. There's another dimension of reality that's got all the numbers in it, this one doesn't. In this, in this reality, we don't see the number two sitting on the floor, and number three, walk through the door. There are a number not numbers here, but things can be numbered, and they can be counted. And they can be measured by the use of numbers. But all the numbers exist in another realm, another dimension of reality, that includes this law, and all other laws that includes all numbers, all decimals, all fractions, it includes perfect lines and points and circles and triangles, all the apparatus of anything that's mathematical is in this other world, and this world is ruled by that one. What happens here, what comes into existence here? What passes away here is determined by the number world, the people who hold this. Okay? Plato, you already know, Pythagoras. And the great mathematician, Leibniz who invented the calculus, marvelous achievement, isn't it? You and I would like to be able to put on our resumes that we can do calculus, he puts on his that he invented it. But he's in the same group. This is how he sees numbers. Leibniz was asked one time by a student, how do we know that 1+1 always makes 2 and it always will? And the reply was this, one plus one equals two is an eternal, immutable truth. That would be so whether or not there were things to kill, or people to count them. This world might come and go, but the world of the numbers and
the laws that's eternal, it's uncreated, God did not create it. And it's changeless, immutable. Leibniz believed in God. Leibniz regarded himself as a Christian, he held that there God is the creator of the world, and that God used the number world to make this one. God, God did not create the number world. It is eternal, self existent and changeless. God looked to the number world to know what to do here in order to make the best creation he could manage. God's the great mathematician but God didn't create that. And that's in direct violation of Colossians 1 that says God did create it because it's not visible. It's either visible or it's not. Anything that's either visible or not was created by God. Leibniz says, no, it's not surprising that Pythagoras or Plato wouldn't say that they didn't know about God at all. But Leibniz should have known better in my view. That's the number World Theory. In case you think that's just an historical curiosity, let me tell you that this view is probably a plurality among mathematicians. It's not a majority, but it's the biggest single group people's, most mathematicians, from what I read are some sort of number of World theorist. So it's not just in the past. How about another theory, I'm going to put the name of a person here, because Bertrand Russell is famous for holding this view, he's not the only one that did. This is the view that says these numerals up here don't stand for objects in another realm. They don't stand for real things, somewhere, one plus one equals two is a shortcut way of doing logic. Doing logic, we can do logic without any reference to quantity at all says Russell. In fact, we can express 1+1=2, without any quantity. That sound startling, especially when you had mind. If you don't mind just may take a little bit but about this. There exists something x. There exists something y, and there exists a set, and we'll call the set S. And x is a member of the set S. And y is a member of the set S. And for anything, whatever, that's a member of the set S. Then z is identical with x, or Z is identical with y. Okay. So Russell's claim is that it's a shortcut way of writing this. And this doesn't refer to quantity at all. It's just all logic. Mathematics is whole reduces to logic. I think not, in the first place. The quantifiers here have to be read. There exists at least 1x or Y, or did you hear a number there exists at least one oops. Sure, there's quantity. And this sign, this symbol means a set is a member of the set is a member of the set does that sound to you? Like is one member of the set? Sure it does. So there's quantity in this all over the place? Much to Russell's embarrassment. It doesn't work. But it's a view of how to understand that. And that's the important point here. Why does Russell hold that? Why does he want to say things like that? Here's the one. Here's his comment. Philosophers have commonly held that the laws of logic which underlie mathematics, are laws of thought laws, regulating the operation of our minds. By this opinion, the true dignity of reason is very greatly lowered. It ceases to be an investigation into the very heart and immutable essence of all things actual impossible. logic isn't just in here, it doesn't just govern our thinking is in everything and making it what it is, is the immutable essence of all things,
natural impossible. And that's why he takes this interpretation of mathematics. It's logical laws that govern all of reality that make things what they are, they may be made out of matter, but only by the logical order of the universe. Russell certainly didn't include God in that. He was a very famous atheist in his own day. Let's get this off of here. So we'll have to be looking at this stuff. These aren't the only views. Another thinker John Stuart Mill, he wrote a three volume work that he called a work on logic. But, in fact, the last one deals the last volume deals with mathematics. And Mill disagrees with both these views. mathematics. The, the Formula 1+1=2 doesn't refer to numbers that are real, mysterious sorts of things that exist in another dimension of reality. Nor is it all reducible to logic, that doesn't work either. There's no way to state a logical formula that doesn't presuppose and include quantity. This is derived from our, our sensations, all we really know are our own sensations. We know sounds, taste, touches, smells, we see arrangements of colors. We feel something as smooth or hot. It's all the sensations combined into things. That's what we experiencd. And so one plus one equals two is just a summary of what we observe. We see that when we have one thing, and another thing, we have two things. So this isn't a law, it's a generalization over our sense perception over our past experience, and the past experience of other people who've written to tell us that they experienced the same thing. Does that mean then that long ago or far away or far into the distant future? One plus one may make something else? Mills answer is yes, it does. That would be odd, like finding a Black Swan, but not impossible. Somewhere right now in the distant universe. One plus one might equal five and seven eighths, we have no way of knowing that. This is Mill's position. Sound a little wacky to you? It's not a very popular position among philosophers. But Mill and a group of people who thought that way held that. Think about that, would you want to be an astronaut? If between the time you left for the moon and the time you came home, the math changed. Not I. Of course I wouldn't want to be an astronaut at all. But aside from the point, the point is here, that Mills got a different theory. It's just our observations. Whenever we've looked, this is the way things things seem. Could there be exceptions? Sure. But how likely is that we noticed that this follows pretty regularly. But that's not because there's such a thing as a law that makes it happen. It's only our observations. All we know are are our own sensations. Those aren't the only views. But we're going to take a look at one more. And here I will put two names. Ernest Nagel, and Richard Rorty. These fellows belong to a school of thought that's called pragmatism. It says that we believe what we do, because it works. It succeeds in a practical way it gets gets us where we want to go, it gives us what we want to get. Not because it's true. It's neither true nor false. It's just used for or not. So Nagel says that he says, Nobody ever forms a belief even in logic or mathematics without first testing it out. To see if it really gives us the conclusion has given us what we need, tells us what we want to hear. And Rorty lived closer to our time.
He died about five years ago, I think we're well into the 21st century. Nagel did most of his work on the previous I think all of his work in the previous century. But this view has been around for a while it originated in the early 1900s with people like William James and John Dewey, a pragmatic view of truth. Let's not worry about whether we can show that something isn't indubitable or certain. The attempts to try this in the past. This is how they're viewed those. The attempts, this is how their view goes, attempts to try this in the past had failed. Used to be we thought that a belief was certain if it can be proven or self evident. But that's not what we should be doing. Now we've learned better than that, right? For almost all the things, we just actually believe there is no proof. And it's not self evident. And they say that these truths aren't self evident because they themselves accept very rigid restrictions on what's allowed to count as self evident. Those restrictions are dubious in my mind, they are rather easily shown to be unjustified. But that's the way they go. This tradition of trying to give proof sort of see whether something's self self evident, that just hasn't paid off, says Rorty. Sometimes we have intuitions of self evidence, of course we do. But I'm saying we should try to stop having them. Because that's not the grounds from which to believe something. That's what he has to say, I'm quoting him. We believe something because we think that it will make us happier than we'd otherwise be, is a line from his book? Well, that's another way to look at mathematics. It's a tool. And this is what John Dewey said about it, too. It's a tool, we use it to do a certain job. The question is, whether it does the job or not? Is it true or false? You don't ask of a snow shovel or a rake? Is it true or false? You just want to know does it work? And that's the way 1+1=2 is, it doesn't it's neither true or false. It just either works or it doesn't. That's what they want to say. I don't think that I have to tell you that that's not very persuasive. We do mathematics now. In calculating how to get somebody say, to the moon, or Mars and back. And it's not just useful, we're counting on it to be the truth. It's going to correspond to reality. Or we're not going to get our feet in that rocket and go anywhere. Besides that, mathematics has led to discoveries. How can it just be a tool since the quarter that's neither true to force, force, nor false. And so we get it out and use it to see see what it works for. But what's happened in recent years, the mathematics of physics has suggested the date for the origination the big bang of the universe has suggested other dimensions to things than those that we've experienced that has suggested dark energy and dark matter. These things are taken then to correspond to the truth. When we get these results, we say Oh, well, that's the way reality is, not just oh, well, this may work for something. But right now, the idea of dark matter or dark energy doesn't work for anything. But then there would be no way to discover anything using that. And that's exactly what we find happens. It gives us discoveries, it predicts what we'll see about the world, the same way, the atomic, the charge of relative atomic weights and so on that does the same thing. When people work
that chart out, they notice several several holes in the chart. There ought to be such and such kind of elements here, and they looked around and found it. The chart predicted it.If the pragmatists were right, that should be impossible. Well, there are a lot more views than just these too. And I want you to hear what one historian of mathematics has to say about that. There are schools of thought in mathematics that accept different axioms, and differ both in their methods and
the results. For example, there are disagreements between logicists, like Russell, trying to reduce math to logic. Intuitionist, such as Brower empiricists, like Mill, and formalists, such as Hilbert. These differences are very sharp, as has been recognized by Brower, who holds the intuitionist view. Brower says of the schools of thought. Either school operates with mathematical entities, not recognized by the other schools of thought. There are intuitionist structures which cannot be fitted into any classical logical frame, and there are classical arguments that don't apply to any introspective image. Likewise, in the theories mentioned, mathematical entities, recognized by both parties on each side are found to satisfy theorems which for the other schools are false, senseless, or even self contradictory. There is a great divide over how to under what those marks stand for. And whether we can know that the truth Express is true. anytime, anyplace. In fact, the divisions, differences go even deeper than that. What what's approved for one of these schools of thought isn't for another. Einstein praised Georg Cantor, who came up with a way of handling infinities in mathematics, called theory of transfinite numbers. Einstein called that the greatest advance in mathematics in a 100 years, the intuitionist say, it doesn't even rise to the dignity of being false. It's meaningless. Just in the same class with Twas brillig, and the slithy toves Did gyre and gimble or the wave, it's nonsense talk. That's all doesn't mean a thing. So, the historian of mathematics, Maurice Klein has written this in his book, The current predicament of mathematics is that there is not one but many mathematics. And that for numerous reasons, each one sale fails to satisfy the members of the opposing schools, is now apparent that the concept of a universally accepted, infallible body of reasoning, the majestic mathematics of 1800, is the grand illusion. The present state of mathematics is a mockery of the hitherto deep rooted and widely reputed truth and logical perception, perfection of mathematics. From a book he wrote, called mathematics, the loss of certainty. I think there's a Christian view of this, one that doesn't try to reduce math to our intuitions, or to logic, or to observations of sense perception and just generalizations over perceptions. Nor is it truths that have to do with another realm where there are real things called numbers and logical laws, none of those would be acceptable from a Christian point of view is my point. Mathematics has to do with the quantitative properties of things in this world. And we discover those properties, we assign measures to them, such as the natural number theories, we invent the numerals, but they stand for quantity that we find in the world. And then we
see how this quantities are related. And we discover laws. And it goes from there. It's a very realistic view of math, but one that doesn't deify it, which is our, which is one of our two great concerns. Not deifying what we're studying, and not trying to reduce what we're studying, to any other piece of creation, as though it's divine. I hope this clears up what I mean by a religious belief, directing guiding a certain point of view. Here, it's the numbers themselves, and the relations, the law relations between them that are accorded divine status. They are self existent, uncreated, and the advocates of this theory add they are also changeless. For Russell, it's math. That's a very essence, the immutable heart of all things. And therefore of math. For Mill, Everything is sense perception, guided perhaps by logic as that exists in the human mind. But everything is either perception or logic combination of them. They're just generalizations, we don't know that they've always been true. They don't need to be. And for Nagel, and Rorty who have a rather, biological view. These truths are affected by how we just happen to have evolved. So we see it this way. That's got nothing to do with whether it's true or false because after all, mainly, what we want to do is increase our comfort and increase our chances for survival. So it's just whether it works, whether it contributes to our enjoyment and our survival. That's it. So that math is kind of biological tool. I hope that's clear. I'm going to come back to it and go over it a bit in our next session. But then I want to concentrate more on what theory should look like if we put God as the controlling divinity belief. And no part is creation, not the quantity, not the space not the logic, not the matter, not the life. Only God is self existant and is produced all the rest of those.