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.