![[personal profile]](https://www.dreamwidth.org/img/silk/identity/user.png){kind=small}
Ok, continuing what I was saying in part 3 regarding mathematical realism. Let me explain the way in which I currently view mathematics. As I say, this is what I consider a very difficult problem, so my views are still unfolding and may be refined in the future. But I think I've made a good amount of progress within the past few years so I'd like to summarize it.
Before getting to my views, let me mention that there is another book that I've read a portion of within the past couple years which has helped inform my opinions on these issues... The Emperor's New Mind by Roger Penrose. While Penrose is fairly well respected as a mathematical physicist, he has a rather unique view of mathematics, and is known for having very bizarre and non-traditional views both on that, the mind, and on quantum computing and quantum gravity. His view is that there are 3 different worlds: the physical, the mental, and the mathematical. The mental world for him serves as a bridge or a gateway between the physical world and the mathematical world. He sees mathematicians essentially as shamans, people who can reach out with their mind and contact another world, bringing back news from it to the rest of us who live mostly in the physical world. Needless to say, nearly everyone else in physics thinks he's crazy. That said, he's not even nearly as crazy as Godel was. Godel believed essentially the same thing about the mathematician's mind being in direct contact with the objects of mathematics, but he took it even further and also believed in God (he attempted to prove God's existence at one point) and other supernatural phenomena like psychics and contacting ghosts of past ancestors. He was also a hypochondriac and suffered from anxiety and paranoid delusions, especially throughout the latter portions of his life. Penrose, as far as I know, doesn't have any mental illnesses and his views are by comparison fairly sane.
My view is rather different from Penrose, Godel, and Plato. My view instead is most similar to the views of two other mathematical realists: David Deutsch and Max Tegmark (both outspoken advocates of the Many Worlds Interpretation). In fact, it's maybe worth my calling the former 3 Platonists and the latter 2 mathematical realists as there seems to be a big distinction here. I know a good bit about Deutsch's take on this and a fair bit about Tegmark's take, but I will nevertheless not try to speak for either of them and instead simply explain how I see the matter.
The main difference between the Platonists and I is that I'm thoroughly an empiricist. Godel and the others believed that mathematical truths could be justfied by "mathematical intuition" which results from a mystical connection between the mind and mathematics. None of that fits into my view at all. There is a longstanding tradition to treat mathematical truths as "a proiri" and "necessary" in a similar way that tautologies in logic are treated. I think this is a mistake. First of all, I don't believe there is a such thing as a priori truth. All truths must be justified empirically and so ultimately there is no truth which is not a posteriori in some sense. The so-called "truths" of logic I see as language conventions which are set up that way for convenience, not because they represent any actual truths about the world. But this does not make sense when you're dealing with actual truths about the mathematical objects themselves. The main thing that I think we need to do more of is to separate two historically conflated ideas from each other: Mathematics meaning a system of formal proof, and Mathematics meaning the structures which mathematicians study. But I don't think they are the same thing. Or if they are, it's not in an obvious way. My preference is for reserving the term Mathematics or "the mathematical world" or "mathematical objects" or "mathematical structures" for the things that mathematicians study, and to refer to the other thing as Formal Systems. There may be arguments for using other language here, but this is the convention I'll adhere to for the rest of the post.
Another difference between Penrose's view and mine is that I believe in one world whereas he believes in three. I acknowledge that you can approximately speak of a mathematical world, a physical world, and mental worlds (there are a lot of separate ones since there are a lot of different minds), but a part of my reductionist beliefs/suspicions is that each one reduces to the last. (This part I would thank Tegmark for.) Just as the mental worlds that constitute what people call "phenomenology" or "subjecthood" are illusions that ultimately reduce to the physical world, I suspect that our physical world is ultimately an illusion and reduces to a portion of the mathematical world. In other words, the property of physicality is an emergent one which certain mathematical structures acquire when they get complex enough. Once those physical objects combine and get even more complex, they start to acquire mental properties as well (they become conscious). Other types of worlds which can emerge once things get really complex is virtual worlds. For this you need a whole network of minds combining, for instance those connected through what we call the Internet, all participating in the same virtual world. In the future, I think virtual worlds will have the most practical importance, but the mathematical world is the root of everything, serving as the foundation of reality, supporting all of the other worlds on its back. The mathematical world is what I'd call the "real world"... the only one that is strictly speaking, not an illusion. The rest can all be deconstructed and shown to fall apart into nothingness, but mathematics is eternal and everlasting, and the whole of it will always remain a step beyond human comprehension. It's something we can't reach out and touch or feel or see, but it's always there behind our world, generating it and sustaining it.
So as an empiricist, how do I know that mathematical structures exist? What experience have I had that gives me evidence they exist? Well, to start out very simply, consider the first mathematical objects which were discovered, the natural numbers. Note that these were discovered long before formal systems were discovered. Before anyone had any notion of proof or even logic, we knew about the natural numbers. The classic experiment you can perform involves 4 apples. Take two of them in your left hand, and two of them in your right hand, and now bring your hands together. Now count them... how many do you have? You've demonstrated empirically that 2+2=4. This was known before there was any such notion of proof. You can do the same physical experiment anywhere you like and any time you like, and you'll still see that it's true. Fast forward all the way to modern theoretical physics. We've discovered empirically that our microscopic world is very precisely described by an SU(3)xSU(2)xU(1) quantum gauge field theory in the spontaneously broken Higgs phase. This is a particular mathematical structure, and trust me... it's quite a bit more complex than the structure of arithmetic of natural numbers. Between them is a whole continuum of complexity building up, and centuries of history as we did more and more complicated experiments and uncovered more and more abstract mathematical structures in the world. The thing that gets really erie, which is something which was pointed out by Wigner at some point (google "Unreasonable Effectiveness of Mathematics in the Natural Sciences")... is that by the time we uncovered this particular SU(3)xSU(2)xU(1) structure (known as the Standard Model of particle physics) it had become commonplace for mathematicians to discover new mathematical structures on their own before we do the official experiments which "discover" them! What's going on here? Rationalists (or even weaker empiricists than myself) would say that the mathematicians used their own "a priori" methods to discover the truth before it was discovered empirically. One striking example of this is the discovery of non-Euclidean geometry before we realized that the physical spacetime we live in is actually described by non-Euclidean geometry. A second thing that happens is theoretical physicists take current mathematical structures we know describe the world, and extend them in consistent ways that are motivated by solving problems with the current theory... and they end up predicting new discoveries before they happen. An example of this is Dirac's prediction of antiparticles from pure mathematics and theory. He predicted that an antiparticle must exist (in order to avoid backwards causality and the kind of grandfather paradoxes associated with time travel) for every particle in the universe... just by figuring out what the only new mathematical ingredient could be in our world that would give just the right mathematical cancellation to ensure causality. About a year after his prediction, the first antiparticle was discovered in the laboratory. I take stories like this (and many others) as compelling empirical evidence that the methods we use in mathematics, whatever you may want to call them, allow us to gain real knowledge and insight about the world. It's not just random guessing.
So while some would say that the discovery Dirac made was justified "a priori" before it was later justified "a posteriori", I would disagree. If this were the first time anyone had thought of using math to try to predict the real world, then there would be absolutely no reason to think it would succeed... there would be no justification to think that it would work or to think that what was claimed was true. The reason we believe math works is because it is justified a posteriori in the same way that the existence of the natural numbers is justified a posteriori because it predicts the right number of apples when you put your hands together.
So here comes the really interesting part. . . the way in which mathematicians study mathematics is to use formal systems of axioms in order to "prove" statements about them. They set up the axioms and then it's just a matter of turning the crank. And if they've set up the axioms properly, it appears to work. By "work" I mean that if the axioms they use define a structure that we've already discovered directly through sensory experience with the physical world (such as the natural numbers) then they always predict the right future sensory experiences when dealing with that same structure. After working with these formal systems a lot, what it looks like is that these axioms are a very reliable tool for studying these mathematical structures. Again, note that the only reason we know we can rely on it is because it has been shown to work empirically. Not because some foolish rationalist has "proven" it will work. Ok, so we trust that certain mathematical structures exist, and we trust that the axioms are a tool to investigate these structures. What else do we know empirically? Well, we also know that if we take one of the axioms and modify it but keep the rest the same... there's a very good chance that we'll end up with another structure that exists somewhere in the physical world. Sometimes you get one that we already know exists, but the most interesting case is when you get one that is later discovered to exist. That's where we get empirical evidence that even something more exciting is true! From cases like this (such as the example of non-Euclidean geometry) I believe that what the empirical evidence is screaming at us is this: our physical world (the world we experience with the senses) is built out of mathematical structures, and those structures can be generated by building blocks themselves, and the building blocks are in some sense the axioms. By recombining the blocks we've already discovered to exist in one form, we can create a new form which also exists. We've seen this happen again and again. So we have reason to believe (or at least evidence which should make us suspect) that no matter how we recombine these blocks we will end up with other things which also exist.
The situation with mathematics is analogous to finding a toy house made out of legos. You take apart the house and build a toy car. Your friend says "that's nice, but you can only build a house or a car... you couldn't, for instance, build a boat!" And maybe after only two trials, you would believe him... until you tried it and were able to build a boat. Then you build a tractor, and a limosine, and a truck, and etc. etc. Every time your friend says "but we only have evidence that you can build A, B, C, and D... we have no evidence that you could build E!" But I would argue that this is not the case. I would argue that you have good evidence to believe you can put the blocks together however you want and you'll get a new structure which exists every time. This is my argument for how we know that other mathematical structures exist besides the ones which we observe directly in the physical world. And I hope I've explained clearly why I see the justification for that knowledge as entirely empirical... it's not something that follows from pure reason. And it's not something that requires you to believe in a priori synthetic truths, or any kind of a priori truth for that mater! I remain an empiricist, even though I believe we have evidence that there are other universes besides ours and that "all" mathematical structures exist (where the "all" is a bit fuzzy and I will try to clarify in part 5).
I still haven't gotten to Turing Machines. Turing Machines are the link between formal systems and mathematics... or one of the links, anyway. I think this provides further evidence that what I'm saying is true and we should take it seriously. Other things I need to talk about in part 5 are clarification on which sorts of mathematical structures exist, how to deal with such a huge ensemble, what implications this has for the possibility that we're living in a simulation, and connecting it back to part 2 where I mention the one way in which I could see Copenhagen maybe actually working. Whew! Thanks for reading, if you got this far. I've been saving all this stuff up for a couple years, wanting to write it down. A lot of it I would have to thank David Deutsch for, as well as Max Tegmark for other parts. Both of them influenced my thinking on these issues greatly.
Before getting to my views, let me mention that there is another book that I've read a portion of within the past couple years which has helped inform my opinions on these issues... The Emperor's New Mind by Roger Penrose. While Penrose is fairly well respected as a mathematical physicist, he has a rather unique view of mathematics, and is known for having very bizarre and non-traditional views both on that, the mind, and on quantum computing and quantum gravity. His view is that there are 3 different worlds: the physical, the mental, and the mathematical. The mental world for him serves as a bridge or a gateway between the physical world and the mathematical world. He sees mathematicians essentially as shamans, people who can reach out with their mind and contact another world, bringing back news from it to the rest of us who live mostly in the physical world. Needless to say, nearly everyone else in physics thinks he's crazy. That said, he's not even nearly as crazy as Godel was. Godel believed essentially the same thing about the mathematician's mind being in direct contact with the objects of mathematics, but he took it even further and also believed in God (he attempted to prove God's existence at one point) and other supernatural phenomena like psychics and contacting ghosts of past ancestors. He was also a hypochondriac and suffered from anxiety and paranoid delusions, especially throughout the latter portions of his life. Penrose, as far as I know, doesn't have any mental illnesses and his views are by comparison fairly sane.
My view is rather different from Penrose, Godel, and Plato. My view instead is most similar to the views of two other mathematical realists: David Deutsch and Max Tegmark (both outspoken advocates of the Many Worlds Interpretation). In fact, it's maybe worth my calling the former 3 Platonists and the latter 2 mathematical realists as there seems to be a big distinction here. I know a good bit about Deutsch's take on this and a fair bit about Tegmark's take, but I will nevertheless not try to speak for either of them and instead simply explain how I see the matter.
The main difference between the Platonists and I is that I'm thoroughly an empiricist. Godel and the others believed that mathematical truths could be justfied by "mathematical intuition" which results from a mystical connection between the mind and mathematics. None of that fits into my view at all. There is a longstanding tradition to treat mathematical truths as "a proiri" and "necessary" in a similar way that tautologies in logic are treated. I think this is a mistake. First of all, I don't believe there is a such thing as a priori truth. All truths must be justified empirically and so ultimately there is no truth which is not a posteriori in some sense. The so-called "truths" of logic I see as language conventions which are set up that way for convenience, not because they represent any actual truths about the world. But this does not make sense when you're dealing with actual truths about the mathematical objects themselves. The main thing that I think we need to do more of is to separate two historically conflated ideas from each other: Mathematics meaning a system of formal proof, and Mathematics meaning the structures which mathematicians study. But I don't think they are the same thing. Or if they are, it's not in an obvious way. My preference is for reserving the term Mathematics or "the mathematical world" or "mathematical objects" or "mathematical structures" for the things that mathematicians study, and to refer to the other thing as Formal Systems. There may be arguments for using other language here, but this is the convention I'll adhere to for the rest of the post.
Another difference between Penrose's view and mine is that I believe in one world whereas he believes in three. I acknowledge that you can approximately speak of a mathematical world, a physical world, and mental worlds (there are a lot of separate ones since there are a lot of different minds), but a part of my reductionist beliefs/suspicions is that each one reduces to the last. (This part I would thank Tegmark for.) Just as the mental worlds that constitute what people call "phenomenology" or "subjecthood" are illusions that ultimately reduce to the physical world, I suspect that our physical world is ultimately an illusion and reduces to a portion of the mathematical world. In other words, the property of physicality is an emergent one which certain mathematical structures acquire when they get complex enough. Once those physical objects combine and get even more complex, they start to acquire mental properties as well (they become conscious). Other types of worlds which can emerge once things get really complex is virtual worlds. For this you need a whole network of minds combining, for instance those connected through what we call the Internet, all participating in the same virtual world. In the future, I think virtual worlds will have the most practical importance, but the mathematical world is the root of everything, serving as the foundation of reality, supporting all of the other worlds on its back. The mathematical world is what I'd call the "real world"... the only one that is strictly speaking, not an illusion. The rest can all be deconstructed and shown to fall apart into nothingness, but mathematics is eternal and everlasting, and the whole of it will always remain a step beyond human comprehension. It's something we can't reach out and touch or feel or see, but it's always there behind our world, generating it and sustaining it.
So as an empiricist, how do I know that mathematical structures exist? What experience have I had that gives me evidence they exist? Well, to start out very simply, consider the first mathematical objects which were discovered, the natural numbers. Note that these were discovered long before formal systems were discovered. Before anyone had any notion of proof or even logic, we knew about the natural numbers. The classic experiment you can perform involves 4 apples. Take two of them in your left hand, and two of them in your right hand, and now bring your hands together. Now count them... how many do you have? You've demonstrated empirically that 2+2=4. This was known before there was any such notion of proof. You can do the same physical experiment anywhere you like and any time you like, and you'll still see that it's true. Fast forward all the way to modern theoretical physics. We've discovered empirically that our microscopic world is very precisely described by an SU(3)xSU(2)xU(1) quantum gauge field theory in the spontaneously broken Higgs phase. This is a particular mathematical structure, and trust me... it's quite a bit more complex than the structure of arithmetic of natural numbers. Between them is a whole continuum of complexity building up, and centuries of history as we did more and more complicated experiments and uncovered more and more abstract mathematical structures in the world. The thing that gets really erie, which is something which was pointed out by Wigner at some point (google "Unreasonable Effectiveness of Mathematics in the Natural Sciences")... is that by the time we uncovered this particular SU(3)xSU(2)xU(1) structure (known as the Standard Model of particle physics) it had become commonplace for mathematicians to discover new mathematical structures on their own before we do the official experiments which "discover" them! What's going on here? Rationalists (or even weaker empiricists than myself) would say that the mathematicians used their own "a priori" methods to discover the truth before it was discovered empirically. One striking example of this is the discovery of non-Euclidean geometry before we realized that the physical spacetime we live in is actually described by non-Euclidean geometry. A second thing that happens is theoretical physicists take current mathematical structures we know describe the world, and extend them in consistent ways that are motivated by solving problems with the current theory... and they end up predicting new discoveries before they happen. An example of this is Dirac's prediction of antiparticles from pure mathematics and theory. He predicted that an antiparticle must exist (in order to avoid backwards causality and the kind of grandfather paradoxes associated with time travel) for every particle in the universe... just by figuring out what the only new mathematical ingredient could be in our world that would give just the right mathematical cancellation to ensure causality. About a year after his prediction, the first antiparticle was discovered in the laboratory. I take stories like this (and many others) as compelling empirical evidence that the methods we use in mathematics, whatever you may want to call them, allow us to gain real knowledge and insight about the world. It's not just random guessing.
So while some would say that the discovery Dirac made was justified "a priori" before it was later justified "a posteriori", I would disagree. If this were the first time anyone had thought of using math to try to predict the real world, then there would be absolutely no reason to think it would succeed... there would be no justification to think that it would work or to think that what was claimed was true. The reason we believe math works is because it is justified a posteriori in the same way that the existence of the natural numbers is justified a posteriori because it predicts the right number of apples when you put your hands together.
So here comes the really interesting part. . . the way in which mathematicians study mathematics is to use formal systems of axioms in order to "prove" statements about them. They set up the axioms and then it's just a matter of turning the crank. And if they've set up the axioms properly, it appears to work. By "work" I mean that if the axioms they use define a structure that we've already discovered directly through sensory experience with the physical world (such as the natural numbers) then they always predict the right future sensory experiences when dealing with that same structure. After working with these formal systems a lot, what it looks like is that these axioms are a very reliable tool for studying these mathematical structures. Again, note that the only reason we know we can rely on it is because it has been shown to work empirically. Not because some foolish rationalist has "proven" it will work. Ok, so we trust that certain mathematical structures exist, and we trust that the axioms are a tool to investigate these structures. What else do we know empirically? Well, we also know that if we take one of the axioms and modify it but keep the rest the same... there's a very good chance that we'll end up with another structure that exists somewhere in the physical world. Sometimes you get one that we already know exists, but the most interesting case is when you get one that is later discovered to exist. That's where we get empirical evidence that even something more exciting is true! From cases like this (such as the example of non-Euclidean geometry) I believe that what the empirical evidence is screaming at us is this: our physical world (the world we experience with the senses) is built out of mathematical structures, and those structures can be generated by building blocks themselves, and the building blocks are in some sense the axioms. By recombining the blocks we've already discovered to exist in one form, we can create a new form which also exists. We've seen this happen again and again. So we have reason to believe (or at least evidence which should make us suspect) that no matter how we recombine these blocks we will end up with other things which also exist.
The situation with mathematics is analogous to finding a toy house made out of legos. You take apart the house and build a toy car. Your friend says "that's nice, but you can only build a house or a car... you couldn't, for instance, build a boat!" And maybe after only two trials, you would believe him... until you tried it and were able to build a boat. Then you build a tractor, and a limosine, and a truck, and etc. etc. Every time your friend says "but we only have evidence that you can build A, B, C, and D... we have no evidence that you could build E!" But I would argue that this is not the case. I would argue that you have good evidence to believe you can put the blocks together however you want and you'll get a new structure which exists every time. This is my argument for how we know that other mathematical structures exist besides the ones which we observe directly in the physical world. And I hope I've explained clearly why I see the justification for that knowledge as entirely empirical... it's not something that follows from pure reason. And it's not something that requires you to believe in a priori synthetic truths, or any kind of a priori truth for that mater! I remain an empiricist, even though I believe we have evidence that there are other universes besides ours and that "all" mathematical structures exist (where the "all" is a bit fuzzy and I will try to clarify in part 5).
I still haven't gotten to Turing Machines. Turing Machines are the link between formal systems and mathematics... or one of the links, anyway. I think this provides further evidence that what I'm saying is true and we should take it seriously. Other things I need to talk about in part 5 are clarification on which sorts of mathematical structures exist, how to deal with such a huge ensemble, what implications this has for the possibility that we're living in a simulation, and connecting it back to part 2 where I mention the one way in which I could see Copenhagen maybe actually working. Whew! Thanks for reading, if you got this far. I've been saving all this stuff up for a couple years, wanting to write it down. A lot of it I would have to thank David Deutsch for, as well as Max Tegmark for other parts. Both of them influenced my thinking on these issues greatly.
