This is the interview with Donald Hoffman from episode 090 of the You Are Not So Smart Podcast.
In this episode of the podcast, we talk to Donald Hoffman, a cognitive psychologist at the University of California with a background in artificial intelligence, game theory, and evolutionary biology. Hoffman has developed a new theory of consciousness that, should it prove true, would rearrange our understanding of not only the mind and the brain, but physics itself.
David: So, do we all see the same color?
Donald: We actually know that even among people that are color normal, that there are wide variations in our color perceptions.
David: That’s neuroscientist Donald Hoffman.
Donald: I’m Don Hoffman, and I’m a professor at the University of California at Irvine. You can just call me Don.
David: And Don has come up with a…well it’s an idea, it’s a theory, it’s a hypothesis that is maybe even more challenging than the bicameral theory of consciousness, and we’re gonna talk all about that in a second, but here’s the rest of what he had to say about seeing colors.
Donald: For example, among color normal males, there are two alleles of the red cone photo pigment gene, and they differ by a single nucleotide. But that changes one amino acid in the pigment and that changes the receptive properties, the peak sensitivity of that. And so, when you actually test these men, you can actually…about two-thirds of the men have one allele, one-third have the other. You take them into the lab, do genetic testing, separate them into the tow groups and you find that they actually make different color matches. So, even among so-called color normal males and females, there are these wide differences. And then, of course, 7% of males are red green color blind, and about half a percent of females are red green color blind. And then there are some females that apparently have 4 color receptors instead of the normal 3, and they actually have an extra dimension of color perception that the rest…that no male has, and that most females don’t have. So their…if I ask you to image a specific color that you’ve never seen before…right? Nothing happens…I…smoke comes out of my ears, nothing happens. But these women, apparently, are living in a world of colors that the rest of us can’t even imagine concretely, so yes we all use words the same way when we describe colors, and it’s easy to assume that we’re seeing the same colors. But, we actually know that that’s false.
David: I’m wondering, and I’ve just been very eager to hear from someone who would know exactly the answer to this question or have great speculation on this. There’s a…what do you think of that whole bicameral theory of mind…the idea that we were monists for a very long time until the idea of dualism became common enough that we started to reframe our concept of consciousness in that way?
Donald: Well, the book of course was quite popular and had quite a following. Among scientists, it doesn’t have much cache at all. It’s interesting speculation, and it’s fine to speculate, but the evidence is pretty much against those kinds of claims. We’ve made a lot more strides into what neuroscience really tells us about the nature of our perceptions and also our thoughts about consciousness and do we really…I mean, he does claim that the Greeks perhaps heard the voices of gods because they externalized their own internal thoughts, things like that. Interesting speculation, but you won’t find any serious scientist who is taking that as the foundation of their work.
David: Well, that is a good segue into your stuff, because you have been…I wouldn’t say it’s an identical kind of challenge, but you certainly have introduced ideas into academia. And as you’ve demonstrated, you are a empirical scientific explorer of things, you’re a neuroscientist, you’re not one of these sort of, Rupert Sheldrake type…not to disparage Rupert Sheldrake, but you…despite all that, you have some very challenging ideas, and I’m going to run through this in a way, so we can ramp up to it. You say in your TED talk that we’ve made very little progress in understanding the relationship between brain activity and conscious experience, even though we’ve really worked hard on this problem it seems like. Why is it we’ve made such little progress and why are there so many of these ideas that can get really fringy and speculative, yet they’re all kind of considered in…they’re almost all regarded equally at this point even if there’s contention among neuroscientist. Why has there been such little progress on this specific problem?
Donald: Well, we actually call it the hard problem of consciousness, partly because we’ve made no progress in actually proposing a specific scientific theory that has any beef to it at all. And, there is speculation about why we’ve made no progress. One speculation is that we just don’t have the conceptual systems that are necessary to solve this problem. We don’t expect monkeys to solve problems in quantum mechanics, and it might be that Homo sapiens simply is not equipped with the concepts needed to solve the problem of how is conscious related to our brain activity. That’s certainly possible, and I can’t rule that out, I mean I do take evolutionary theory quite seriously and it’s certainly quite possible that we don’t have the concepts. But I think that there’s a different reason, and that is that we’ve simply made some false assumptions. So that was how I began to look at this, because I like everybody else was trying to figure out how brain activity could cause our conscious experiences, and I couldn’t think of anything intelligent to say about that; much less frame a theory. But when you look at all the so-called theories out there, for example ‘integrated information theory’ or ‘orchestrated collapse of microtubule states’ and so forth…when you actually look at them, there’s no beef, there’s nothing…for example, there’s no specific account for any particular conscious experience, like say the taste of chocolate. What integrated information state of the brain is identical to chocolate? That’s never been put out there, I’ve asked [?] [27:19] in person just a few weeks ago. So, there’s not one concrete…they have no concrete case that they can point to. So, we now know that this causal integrated structure is chocolate and this other one is the taste of a lemon and here’s the principal reasons why. There’s nothing remotely like that and no idea about how to even get that. So, I think that the reason that we’ve not made any progress is that we’ve made a very very simple but false assumption…we’ve assumed that our perceptual systems give us some indication about the nature of reality as it is. That we evolved of course to stay alive, to be fit and to have offspring, and that in the process of evolving perceptual systems that kept us alive, we evolved perceptual systems that showed us reality as it is, not exhaustively, we don’t see all of the truth of course, no one believes that, but that our perceptions of a world in terms of space and time and physical objects that seem to have causal powers…like a white…a cue ball hits the eight ball and knocks it into the corner pocket, we think that this genuine causality of physical objects in space and time, because that’s the way we see the world…that assumption is the one that I’ve been looking at. The reason I’ve been looking at that is that we’ve assumed that certain objects in space and time, namely neurons have the causal power to create our conscious experiences, so that is the question that I went after. Is that assumption right? When we make that assumption, we can’t get anywhere, when we assume that neurons cause our conscious experiences and neural activity causes our behavior, we run into deep deep problems. In fact, we can’t make any theoretical progress so I said, “let’s step back and ask ourselves the question, does evolution by natural selection favor perceptions that are telling us about reality as it is? Does perception evolve to see the truth?”
David: So, this is your big challenge, because this is what I learned in school, the idea that the brain, yes the brain constructs reality and it’s sort of a virtual reality and that was communicated pretty well, but also that our ancestors who saw a more accurate picture of…had a more accurate perception of the objective reality that we don’t completely bring into our minds, but the ones that had a more accurate perception of it were more successful, so over time you become more fit by having a more accurate perceptions that better fit reality, and so on. So you challenge this completely, correct?
Donald: That’s right, you’ve I think nicely summarized the standard view in the field that those of our ancestors who saw more accurately had a competitive advantage over those who saw less accurately and therefore were more likely to pass on their genes that coded for the more accurate perceptual systems. So, we can be quite confident that we don’t see all of reality as it is, but we do see some of reality, namely the part of reality that we need, e see it accurately. And, that was the very question…the very assertion that I then questioned. We don’t have to wave our hands, evolution by natural selection is a mathematically precise theory, we have the tools of evolutionary game theory, evolutionary graph theory and genetic algorithms, and so we can pose a precise mathematical question and ask under what conditions would natural selection favor veridical…that’s the technical, the geek term that scientists use…veridical perceptions, namely perceptions that are accurate to reality…not exhaustively, but wherever we might need them. So, I posed that question with some of my graduate students, and we ran Monte Carlo simulations where we simulated hundreds of thousands of random worlds and put organisms in those worlds that could see all of the truth, part of the truth or none of the truth, and there were these things called fitness functions, which tell you how many fitness points you get for various actions you take, right? That’s the name of the game and evolution is the…in addition to reality, there’s a fitness function that governs your success in reproduction. And, what we found in our simulations was that organisms that saw reality as it is could never out-compete organisms that saw none of reality and were just tuned to fitness – as long as they were of equal complexity. So then I conjectured a theorem, and the theorem that I conjectured is that an organism that sees the truth can never be more fit than an organism of equal complexity that doesn’t see the truth and is just tuned to fitness. And I worked with a mathematician friend, Chetan Prakash and proved the theorem, so it’s under review right now. But it’s actually a theorem that if our perceptual systems evolved by natural selection, then the probability that we see reality as it is, in any way, is zero. Precisely zero.
David: Ok, so this is a lot to unwrap. The first thing that’s exciting this is that this is an idea that you would think would be trapped in philosophy forever, but here it is way outside, not even just in psychology, but it’s over here in computer modeling and evolutionary science and in neuroscience, so you’ve pulled it way over here and put it into a framework where it can actually be peer-reviewed and that’s pretty fantastic. So how does this differ from just the idea that…the umwelt idea, how does this differ from the concept that there’s not…of course we don’t see objective reality, we see subjective reality, we see a very small slice of it. And how does it…what elements of this theory that you’ve put together, expand on that foundation.
Donald: Well, so the umwelt idea that von Uexküll had, I think was very insightful for its time. He was pointing out that different organisms effectively have different perceptual worlds because they inhabit different niches, so what it’s like to be a bat is very different from what it’s like to be homo sapiens. So their perceptual worlds will be very different, and many neuroscientists and cognitive neuroscientists today like von Uexküll’s idea, but the way they take it is the notion of an umwelt means that different organisms see different parts of reality. The bat sees different parts of reality than we do, so their perceptual world might be very very different from ours, because reality itself is very very rich, and so different organisms can be tuned to different parts of that reality. This was sort of J.J. Gibson’s idea too, in his ecological optics point of view, that different organisms would see different…whatever they needed, the accordances they needed for their specific niche. So, I agree to a point that different organisms are effectively different perceptual worlds, but where I disagree is that what these worlds are, are seeing different parts of the truth. I don’t think that they’re seeing the truth at all, I think a much better metaphor is to say that we have different user interfaces. So, if you have a laptop computer and you’re writing an email, and the icon for that email that you’re writing to your friend is blue and rectangular and in the middle of your desktop…does that mean that the email itself inside your computer is blue and rectangular and in the middle of the computer? Well, of course not, that’s silly. Anybody who thought that is mistaking the point of the interface, it’s not there to show you the truth, it’s there to hide the truth. You don’t want to know about the diodes and resistors and voltages. If you had to know all that, you’d never get the email written, if you had to toggle voltages, good luck, you’d never finish the email. So the desktop interface is there to hide the truth, give you eye candy that lets you get the job done, whatever you need to do. And that’s what evolution does for us, our perceptual systems are desktops, they’re interfaces. Space and time is our 3D desktop…time is an extra dimension, so three dimensional space desktop with time. And then physical objects, like tables and chairs are the icons in our desktop, and we evolved this perceptual interface not for it to show us the truth, but to hide that truth. We don’t need to know the truth and in fact, knowing the truth would get in the way, just like toggling voltages would stop you from doing an email. So that’s where I differ from the umwelt point of view, at least the standard interpretation of that point of view by my colleges. They will say, “Yes, different organisms see the world in different ways; they effectively inhabit different perceptual worlds.” But what they’re really thinking is they’re inhabiting different parts of the truth. And I’m saying, “No, none of us, no species at any time has seen any aspect of the truth. We have user interfaces that are species specific that evolve for each different species to hide the truth from the species and give it little simple symbols that it can use to stay alive long enough to reproduce.”
David: And you’ve modeled this…this isn’t just…that’s what makes this so great, is this is not just some pipe smoking and some perusing some old textbooks and pipe smoking in front of the dorm room, or whatever…this is…you’ve modeled this in a very detailed way, and this has lead you into this territory and I’m hoping that up to this point, people listening have followed along and that this makes sense. But now we’re going into a place that’s going to be more difficult and this is the concept of ‘conscious realism’ where…and I think that…the way that this made…when I was looking through the different things you’ve said, the thing you said that made this make the most sense to me was when you used the concept of a corpus colostomy, a split-brain patient to help make sense of this. So if you could use…if you could start there, I think that we could maybe get into this.
Donald: Right, so the reason I started to thinking about consciousness as a fundamental nature of reality is that most of us, myself included, start off as physicalists. We assume that space and time and matter…this is the ultimate nature of reality, and the reason believe that is because that’s what we see. We look around and we see space and time and we see physical objects and we assume that we’ve evolved to see reality as it is and so we assume that space and time and matter is the ultimate nature of reality. And it’s a simple mistake, we’ve mistaken our interface for the truth. It’s like a person who thinks there really is a blue icon inside the computer, that’s what physicalism is…it’s that silly mistake.
David: And you’re saying that goes all the way down to atoms and quarks and everything
Donald: That’s right. It goes down to anything that’s inside space and time, so even when physicists like Rutherford told us a century ago that, you know, a piece of metal that looks hard and solid is really mostly empty space, there’s all these atoms and subatomic particles whizzing around in empty space, and so it’s nothing like what we perceive. And that’s true and that’s perhaps surprising, but I’m saying something even weirder, I’m saying that is still in the desktop because those atoms are whizzing around in space and time. You haven’t gotten outside of the desktop until you let go of space and time itself. So I actually have a clear proposal that physicists are going to discover that space-time is doomed, it’s not part of the fundamental ontology that they can use in booting up a theory of physics. And already, they’re all over it, Nima Arkani-Hamed, a professor of physics at the Institute for Advanced Study at Princeton has lectures on exactly that point, that space-time is doomed, and they’re trying to figure out what will replace it. So, that was why we’ve always believed that physics…that physical stuff that’s non-conscious is fundamental. But now that we have to give that up, then the question is, “Well what ontology shall we have, what is the nature of the ultimate reality?” And, the first answer is, again homo sapiens may not be capable of understanding it, that’s one possibility, we just can’t know. But there’s another possibility, and that is, I may be wrong about everything I believe, and that’s certainly possible, that maybe everything I believe is fundamentally wrong, but if I’m right about anything, I believe that I have conscious experiences, I believe that I have experiences of pain, taste of chocolate, smell of garlic, feelings of love and so forth. If I…I mean I could be wrong about that…if I’m wrong about that, I’m wrong about everything, we might as well just drink beer, eat pizza and have a good time because science is impossible. So, I decided, I said, “Look, evolution by natural selection entails that our perceptions of space and time and matter are not an insight into reality; physicalism is false. Let’s try to boot up a different theory of reality, but make it rigorous, I want a mathematically precise theory of reality.” And so I said, “Let’s start with consciousness because if I’m wrong about having consciousness, I’m wrong about everything, so let’s just start there.” So I have a mathematically precise theory of consciousness, it’s actually a mathematical model of consciousness that you can write down on the back of a napkin, but it’s mathematically precise. And then a very strong claim that every aspect of conscious experiences and the dynamics of consciousness can be modeled without exception by this mathematical theory. So I’m probably wrong, I mean that’s the whole point of science – is to be absolutely precise down to the mathematical nitpicking. Be very very precise, make bold bold claims so that people can then…you put a target on yourself, on your theory, you put a target on your theory, so people can try to shoot it down, that’s the whole spirit of science. So be very very bold and precise so that we can shoot these theories down and then evolve them, make them…so I’ve got this theory of consciousness and then the bold claim that consciousness is fundamental, that the ultimate nature of reality is consciousness, what i call ‘conscious agents’ and if I’m right, then one prediction is that we’ll be able to use this theory of consciousness to, with mathematical precision, get back what we call quantum mechanics, general relativity and eventually even quantum gravity. In other words, get…we’ll solve the mind body problem, not by starting with physical stuff like neurons and trying to get consciousness to emerge from it, but going the other direction. Start with consciousness, mathematically precisely defined and modeled and show that we can get back quantum mechanics, general relativity and eventually quantum gravity from it, and solve the problem that way.
David: And so, if it’s not atoms and molecules, those are just representations on our desktop, um, what is it?
Donald: So, the idea would be that if it’s not some physical thing like space and time and matter, I’m proposing that the ultimate nature of reality is actually something that we feel somewhat intimately [?] [43:43] with, namely conscious experience – fears, emotions, smell of chocolate, taste of garlic and so forth. I’m not saying that we have deep deep insights ourselves into those conscious experiences, but we do have conscious experiences. By using that as our foundation and making a mathematical model of it, we can then start to get a science of consciousness that is a rigorous science, where we can actually get surprises. Right? We might find out about…things about our own consciousness that we would never have known if we hadn’t gotten a mathematically rigorous theory of it.
David: How do we…what about…this is gonna…please forgive my complete ignorance, but how do we…what about…uh…how do we escape the possibility that math is also part of this desktop? That it’s not…that math itself is a human construct and doesn’t help us model reality because it’s just part of the…it’s also trapped inside the desktop metaphor?
Donald: Great…that’s a very very important question, and it does distinguish my point of view from certain religious points of view that have tried to use evolution to take down science in a very different way. Alvin Plantinga, for example has used evolutionary arguments to say that all of our cognitive abilities are not reliable, and so therefore evolution sort of shoots itself in the foot. And my argument is very different, my argument only applies to one cognitive capacity, namely our perceptual capacities. It doesn’t apply to our logic and math, and in fact when you look at logic and math, a different analysis from evolution seems to apply. There are selection pressures to do certain elementary operations in logic and mathematics correctly. Two apples give me roughly twice the fitness payoff of one apple. Right? It’s about fitness, so that’s the key thing. Evolution is all about fitness, not about truth, but when you’re arguing and when you’re doing reasoning about fitness, you do need to get that right. And so even though you’re no arguing, or reasoning about the truth, you do have to do the mathematics and logic correctly to get the fitness correct. And so although evolution by natural selection uniformly pushes or shapes our perceptions not to be true, it does not uniformly push our math and logic not to be true. There are small selection pressures…now by the way, on the other hand, I’m not saying that there are selection pressures to make us mathematical geniuses, far from it. What there are are these little enclaves of ability, so two apples are better than one in terms of fitness, or in terms of…as a society, as a social group we’re hunter gatherers, we cooperated…at the end of the day, I would share with you if you didn’t have enough, and the next day I would expect that you would share with me. But there’s a logic of reciprocation that then evolves, I mean I gave you some yesterday, the day before, in fact I’ve been giving some to you for the last month and gee, you’ve never given anything back to me. Now wait a minute, so there’s a little bit of logic that comes into play here, just a little bit that says, “If I do this, then I expect this from you and vice versa.” And so we get these little, little bastions of logic and mathematics that evolution gives us, that are not there…designed to show us the truth just to make us fit. But then every once in a while the genes come together and you get a von Neumann or some other brilliant mathematician who really is…a Kurt Gödel…who really is, from our point of view and outright genius in these areas. So evolution by natural selection shapes our perceptions uniformly away from the truth, but it does not shape math and logic uniformly away from the truth. It gives us a little foot hold here and there, and every once in a while, the genes come together and you get a real genius in these areas.
David: Let’s see if we can help people understand this idea of conscious agents. You mentioned that a split-brained patient basically has…a person has one consciousness and then a split-brained patient…all those great experiments by Gazzaniga, where they have…do all these amazing…we’ve talked about it on the show before, so there’s a…in essence, you can frame them as being a person with two consciousness that are housed within one organism. And you can…I’ve seen this in some of your work, you can extend this to…why not divide it into four and then sixteen and then forty-five. And this sort of reminds me of the Minsky stuff until you finally get down to little binary agents.
Donald: Yes, yep.
David: So how does that plug into your concept of this conscious realism?
Donald: That’s a great example, so we all feel as though we’re a single consciousness, right? That’s the way it feels subjectively…sometimes we have inner conflicts, but it feels like we’re pretty much one single consciousness. But it turns out that the way your brain is organized, there are two hemispheres, a left and a right hemisphere that are connected by 200 to 225,000,000 million fibers, axons. So it’s like a…think of it as a cable, a computer cable between two massive computers. And in certain bad cases of epilepsy where you have a part of the brain that’s bad, say on the right hemisphere and it sends random signals throughout the whole brain that make you go unconscious, they couldn’t in some cases fix it with drugs. So, Joe Bogen…who I had the privilege of knowing while he was still alive…was a famous surgeon who would actually cure these people pretty much by taking off the cranium, taking a scalpel and cutting the corpus callosum and just splitting the two hemispheres, cutting off. And what it did was, it confined the bad electrical activity to just one hemisphere. So one hemisphere might go down, but the other one wouldn’t, and what they found was that the two, in careful experiments, the two hemispheres actually…from my interpretation of it is…they have two separate kinds of consciousnesses in the following sense: you can ask a question of the right hemisphere and ask the same question of the left hemisphere and you get different answers. For example, what do you want to be when you graduate? One fellow was asked this; the left hemisphere wanted to be a draftsman and the right hemisphere wanted to be a race cart driver. It couldn’t be more different, they have different personalities, different likes and dislikes. They can play twenty questions with each other, you could literally set up a situation where the left hemisphere and the right hemisphere play twenty questions, and it’s a real game because the two hemispheres have different contents of consciousness. So it feels to us like we’re a single unified consciousness, and we might be. But just a simple knife, a scalpel can cut your consciousness in two. And there are two separate spheres of consciousness, and so if two, then why not more? How far down does…might it go? Minsky had the notion of the society of mind, and I have a precise mathematical notion of what I call ‘conscious agents’ that can be divided further and further until you get down to agents that are literally as simple as is possible. One bit of perception and one bit of action. Literally a zero or a one for perception or a zero or a one for action. And then the mathematics allows me to build up conscious agents as complex as humans.
David: That is so…I’ve got a book here…I’ve got a really old…I’ll never find it…oh yeah, this reminds me so much of this old idea of the recursive universe from…this stuff…the Conway’s Game of Life stuff.
Donald: Oh right, right. Yes.
David: And it feels so…I really adore how there are…many of these ideas seem to be converging within your theory. And instead of being sort of lost out there in Deepak Chopra land, they seem to be in real academia. So, I’m wondering how has this been received by your peers and maybe by other people in academic silos. I mean, I think at first glance, this feels like woo woo, this feels like Deepak Chopra, the feels like you know, people talking about quantum consciousness and things that have generally been very poorly regarded in neuroscience and in psychology and in other social sciences. How has this been received, and how do you feel about that?
Donald: Well, the initial reaction of my colleagues in academia is generally surprise and shock, but there are…I’m doing the normal academic thing, I’ve published papers with mathematical models and I actually go to conferences and various universities and give colloquia where I present the mathematical models, and so it raises the bar. It would one thing to say, “I think consciousness is fundamental.” That would mean that there’s no beef there. But to say, “Here is the mathematical model of evolution by natural selection and what it does to our perceptual systems.” And that clearly shows…and here’s my theorem now, I’ve got a theorem and a proof that says the probability that our perceptions of space and time and physical objects and physical causality…the probability that that’s an insight into the nature of reality is zero. Now, so I put it out there, anybody that wants to take a pot shot at me can. If they think there’s some assumption in the theorem, or some misstatement in the proof, fair enough, go for it. No one’s found one and this has been out for a long time, I published a paper with my…a couple of my colleagues over a year ago in one of the most prestigious journals, and ten professors from around the world had a chance to write commentaries and take their best pot shots and then we replied to them. And we came out just fine, so we’re doing the normal academic thing of, here’s the precise mathematical theory, here’s the target, take your shots at it. And if you hit, great then we’ll try to evolve the theory. If it’s a devastating hit, then maybe we need to abandon it. So we’re doing the normal scientific thing of trying to be precise and evolving a theory. So far, no one’s been able to take it down.
David: This is great, and I cannot…I’m going to rip into this myself. I want to know…if people who are listening to this are like, “Ok, I want to take a look at this, I want to understand it.” And we have plenty of listeners who are in different academic fields…how can they find you? How can they find your stuff? How can they start looking through it and how can they get…how can they get into what you’re into?
Donald: So, probably the easiest way…all of my papers and interviews and videos are in fact listed chronologically on my vita. So if you just do a google search for Donald Hoffman, and then you’ll see on my home page there’s a link to my vita. It just says vita, if you click on that then you’ll have everything I’ve done with hot links to free sources. So all of my papers are there. I published a paper, you’ll see it on my vita, in 2010 in the Journal of Theoretical Biology, it’s the premier journal in the field, it’s where all the really major results in evolutionary biology have been published…theoretical results. And we there have our Monte Carlo simulations where we show that evolution by natural selection never favors seeing reality as it is when you’re competing against equal complex perceptual systems. So, it’s right there, it’s in the premier journal in the field and you can go and read the mathematics for yourself. Then the…there’s…Psychonomic Bulletin & Review published my target article last year, with the ten commentaries and replies; it’s called the Interface Theory of Perception, that’s the name of the article and so you can download the article and read it. And then there’s also my mathematical model of consciousness called Conscious Realism…no, the paper’s called Objects of Consciousness, it came out just a year or two ago and it’s online for free. And it’s gotten, for a geek paper, it’s pretty good, it’s got 52,000 hits…not bad for really geeky mathematics, the thing is math from start to finish, so it’s not an easy read, but it’s a precise theory of consciousness with precise roadmap for how we could make connections with physics. So the ideas are all out there, the math is clear and people can take shots at it, because I’m not trying to wave my hands. I’m saying, “Here it is, here’s the mathematical details. Any questions?”
David: I think that what you’ve done is unbelievably fascinating and challenging and I hope that people seek you out and try to understand it deeply before they either absorb it without question or dismiss it without question. So, I really appreciate you giving us so much of your time. Thank you.
Donald: It’s been my pleasure, thank you very much David.