Does Math Reveal Reality?

nature is lawful as far as we can tell  the universe and all that it is made of   changes from moment to moment according to fixed  unchangeable immutable patterns and those patterns   as it tested to by a couple hundred years of  observation and calculation can be expressed   in the language of mathematics indeed  it is this mathematics right here   that describes the fundamental equations of  the world within this mathematics we have a   description of einstein's general theory of  relativity the standard model of particle   physics equations that describe how the force of  gravity works how the electromagnetic force works   how the strong and weak nuclear forces  work and all the particles of matter on which those forces act this takes us  tantalizingly close to the reductionist dream of a   theory of everything at least at the level of the  fundamental laws and the fundamental ingredients   now i can well imagine that if you're not familiar  with all of this you may still be wondering   what does that really mean i mean how do these  symbols describe these particles well the answer   is that we can use mathematics to delineate the  configuration of the particles we can use numbers   to describe where the particles are and how the  particles are moving or in the quantum mechanical   language the probabilities for those qualities and  we can use mathematics to describe the intrinsic   features of those particles right i mean  the the mass of a particle that's a number   the amount of electric charge that a particle has  that's a number the spin of a particle that is a   number two and here's the thing we can take all  of these numbers as input and use the mathematical   equations to produce as output another set  of numbers that delineates the configuration   and the properties of those particles at any  subsequent moment in time that is how mathematics   can encapsulate the patterns followed by all of  nature's ingredients the fundamental ingredients   of everything as they evolve from moment to moment  through time now this is a remarkable achievement   showing how mathematics can describe the universe  at a fundamental level but even long before these   modern developments mathematics had already  shown itself capable of giving deep insights   into the workings of the world i mean even even 20  centuries ago the greek mathematician eratosthenes   he was able to use mathematics to figure  out the size of the earth and the pattern   that eratosthenes used is a is a very simple one  he noted that the difference in the lengths of   shadows cast by two identical sticks at  a given separation from one another the   difference in the lengths of those shadows depends  sensitively on the size of the earth right i mean   earth gets bigger and bigger the less and less  the difference between those lengths become   and so by actually measuring the difference  in the lengths of the shadows cast by two   identical sticks actually calculating the angle  of those shadows but it's all the same in the end   eratosthenes was able to figure out the size of  planet earth that's a a remarkable demonstration   of the capacity of mathematics to reveal qualities  of the world that are seemingly beyond our ability   to measure now some centuries later isaac  newton took this idea of using math to describe   reality took it to an extraordinarily new level  through careful observation and thought about   how objects move newton was able to discern  patterns in the motion of those objects perhaps   most famously he found that if you if you push on  an object if you exert a force on it that object   it will speed up it will accelerate and that  acceleration is proportional to the force   and inversely proportional to the mass of the  object embodied in newton's famous second law   of motion now newton also found that if you  have two objects out there in space the pull   of one object on the other through gravity will  be proportional to the product of their masses   and inversely proportional to the square of their  separation that's newton's famous universal law   of gravity and these equations are central to  what came to be called classical physics and   what a what a stunning picture of the world  classical physics gives us if you provide me with   the positions and the velocities of all objects  all particles we can use the mathematics of   classical physics to predict the configuration the  motion and positions of those particles tomorrow   the next day the next year the next millennia  or a billion millennia into the future   newton kind of likened the universe to a grand  cosmic clockwork you you wind up the universe   and the mathematical laws then dictate how the  universe ticks forward in time whether it's a   turning cog or a spinning wheel or a tumbling  raindrop or a hurtling rock or the orbiting moon   newton's mathematics gave us an  astoundingly precise description of   the natural world and on occasion when newton's  mathematics seemed to be a little bit off   oftentimes it was actually just pointing toward  some hidden feature of reality back in the mid   19th century astronomers noticed that the orbit  of uranus had some irregularities that that could   not be accounted for using newton's universal  law of gravity but they also found that those   irregularities they had their own inner  pattern a pattern that could be explained   if one imagined that there was a hidden  hitherto unknown planet out there in space   whose gravity was tugging on the planet uranus  and indeed the astronomers were able to calculate   where that hidden planet should be and on a  september night 1846 when those astronomers   turned their telescopes to the predicted  position in the night sky they discovered   the planet neptune a remarkable example of  mathematics revealing qualities of the world but   even newton's ideas newton's mathematics  was heading toward its own reckoning   early years of the 20th century scientists begin  to examine with precision the microscopic realm   molecules atoms subatomic particles and in that  domain the newtonian ideas however successful   they had been in describing motion in the skies  and motion here on earth the newtonian ideas were   giving predictions that were completely at odds  with the data remarkably it just took a single   generation of scientists to overhaul newton's  classical ideas giving us the quantum mechanical   description of the world a description in which  the universe evolves according to a mathematically   precise game of chance with the mathematics  given by a new quantum equation schrodinger's   equation an equation that makes its own astounding  predictions about the micro world it tells us that   particles should be able to penetrate through  seemingly impermeable barriers it tells us that   two distant particles can somehow influence  each other instantaneously across any distance   the math even predicts the existence of  particles that had never been seen before   and all of those predictions have now been  borne out by a mountain of experimental data   the quantum mathematics is jaw-droppingly  precise we can even use it to describe certain   properties of particles such as the magnetic  properties of electrons and the mathematics makes   predictions to many decimal places and when we do  the experiment the measured values agree with the   theoretical prediction digit by digit by digit  by digit a clear demonstration of the power of   mathematics to describe reality even right here  right we can see the work of quantum mechanics   in the fact that in this domain right now  this is what this room actually looks like   but using the power of quantum mechanics which  has given rise to the powerful computational tools   of the digital revolution we can take  this room and make it look like this   now all these successes of the mathematical  description of the world raise a number of   difficult and thorny questions right  i mean is is mathematics invented   or is it discovered that is is is math a  product of the human imagination is matha   a language that we have developed over the  course of many centuries for the express   purpose of articulating the patterns  that we encounter in the natural world   in a sense then is our sense that the world  is mathematical nothing but the fact that   we have developed a particular tool a kind of  mathematical hammer that makes everything look   like a mathematical nail or to the contrary is  math discovered is math out there independent   of of our understanding independent of us is it  deeply woven into the very fibers of reality and   if that is the case is it still possible that  there's something beyond mathematics perhaps   there's something beyond math that can illuminate  the most ephemeral and precious qualities   of human experience like like consciousness  and the kinds of things that conscious beings   contemplate like value ethics morality purpose  and meaning these are the kinds of questions   that we're going to be talking about here today  they're difficult they're contentious there is no   consensus on the answers to these questions but  these questions will surely drive an energetic   vibrant conversation on the relationship between  mathematics and reality all right let's get to it so with that let me now bring in david albert  is the frederick e woodbridge professor of   philosophy at columbia university and a physicist  who explores the foundations of quantum mechanics   world-renowned for his insights into philosophical  questions about the nature of space time and   other problems of modern physics welcome david  albert next is sheldon goldstein a professor   of mathematics and physics and philosophy at  rutgers university he studies the foundations of   quantum theory probability theory and statistical  mechanics the notion of increasing entropy and   the arrow of time welcome shelley next is sylvia  jonas who is a professor of philosophy at munich   center for mathematical philosophy with a phd  in philosophy from humboldt university in berlin   she studies epistemology the philosophy of  mathematics science metaphysics meta ethics   philosophy of religion and aesthetics welcome  sylvia finally last but absolutely not least   is max tegmark a physicist cosmologist and machine  learning researcher at the massachusetts institute   of technology and a co-founder of the future of  life institute welcome max so thanks to all of you   for joining us for this conversation  about these issues that really people have   thought about struggled with come to a whole wide  variety of perspectives on over the course of   of truly hundreds if not thousands of years in  one form or another and i thought i'd begin to   set the stage with giving some quotes from famous  accomplished renowned mathematicians in the past   who weighed in with their own views on some of the  questions we'll be talking about just to sort of   set a bit of a backdrop of how the field has  viewed these questions so let me start with   chronicler god made the integers all the rest  is the work of man famous mathematician g   h hardy i believe that mathematical reality  lies outside us that our function is to discover   or observe it andrew wiles more in our generation  to tell you the truth i don't think i know a   mathematician who doesn't think that math is  discovered and wouldn't be right to end this list   of quotes without bringing in albert einstein  himself who once said as far as the laws of   mathematics refer to reality they are not certain  and as far as they are certain they do not refer   to reality all right so those are a whole variety  of perspectives on the invention question the   discovery question whether it's out there whether  it is something mathematics emerges from the   machinations of the human mind but let's begin  at the beginning if you will by just having a   little bit of the history of mathematics i mean  oftentimes when we learn mathematics in the   classroom at least when i learned in the classroom  it was just presented as his body of knowledge   there is no sense that this knowledge even  emerged from the struggles of of humans thinking   about things but of course mathematics as we  understand it as we practice it as we use it is a   body of insight that emerged over a long course of  history so sylvia can you just give us a thumbnail   sketch of the history of mathematical development  sure yes there are four stages in the unfolding of   mathematical history the beginning was in ancient  mesopotamia and babylon a tool that was used for   example calculating lunar calendars or measuring  plots of lands in order to determine who owns what   in ancient greece mathematics became  something quite different the ancient greeks   associated the laws of mathematics with something  that they consider to be ultimate reality   plato for example thought that ultimate reality is  not physical reality which is what we think today   but a sort of heaven of eternal forms not only  mathematical entities but also things like   beauty justice truth all of these with capital  letters as it were there is this famous legend   that it said on top of his academy let no one  ignorant of geometry enter here he believed   that mathematical knowledge was the only kind of  perfect knowledge and in order to gain knowledge   about any other thing you would first have to  be proficient in mathematics so ultimate reality   was seen to be in this platonic heaven outside  and completely different from physical reality jumping ahead centuries during the time of the  scientific revolution things were completely   different all of a sudden scientists started  to discover that our physical reality follows   mathematical laws and that we can actually  explain empirical phenomena using mathematical   laws so in that sense ultimate reality was now  identified with physical reality and mathematical   and physical reality became two sides of  the same coin they were seen as interwoven   that picture changed again in the 19th century  with the introduction of mathematical concepts   that didn't have very straightforward relevance  for our physical world anymore non-euclidean   geometries for example so all of a sudden  mathematicians started to feel free to explore   mathematical entities that had no connection  to the empirical world anymore and that caused   a complete shift in the way mathematics was seen  no longer essential part of our physical reality   but as a tool that scientists can use but they  can also discard it their pleasure today things   have yet again changed we're at a point where we  can see that mathematical reality and physical   reality intersect but there are two separate  realities existing independently of one another   great so thank you for that history shelley you  know in that in that history that sylvia just   recounted and of course in your own experience  as a philosopher a mathematician who focuses on   questions about the physical universe do you  do you see math as this tool for describing the   external reality or do you see math as something  bigger than that in the language that sylvia was   saying touching upon a reality that perhaps goes  beyond the physical reality or is parallel to the   physical reality somehow separate from the  reality that's made up of tables and chairs   yes i i mean i certainly see it as a tool but  certainly see it as something beyond the tool   i guess for me however hard it is for even  me to understand even though it's my view   i think that mathematical reality  is the reality we know best   the reality that we can be most confident  about even if we have the sense that we   don't know where it is or how we could know it  given how we interact and gain with things and   gain knowledge there's so many mysterious things  about mathematics but at the end of the day i   do think it's the thing we know best and  it's the reality which we can have the most   confidence in and and so mathematical reality  it exists in in in human heads for sure right   so when we think about the ideas of mathematics  it amounts to something inside of a human brain   firing one way or another thinking about a square  root thinking about a factorial thinking about the   number 17 or whatever so so david are we fooling  ourselves in in thinking that math is somehow   out there touching some deeper reality when  in essence it's really just something that is   invented and lives the only place we're  certain that it lives in the human mind   i'm not sure that i have anything particularly  deep to say about this um and i should say to the   audience david always begins every answer that  way and he always has something deep but anyway i mean there's a famous 20th  century philosopher um named quine   one of the things he's famous for is a criterion  for what we should take to be real for what our   discourse commits us to thinking  actually exists basically the idea is   if there's something that we find we can't  do science without talking about as if it's   a real thing then we should count it as real and  otherwise we shouldn't quine famously thought that   that certain claims about numbers are the  kinds of things that you have to talk about   as if the numbers are real things other  people people who were very interested   in crime's criteria but thought he might have  made a misjudgment about this particular case   people famously like hartree field for example  have tried to show that there are ways of doing   science without acting as if numbers are  are real objects um out there in the world   i think it might be it might be useful to regard  this as a scientific question and you allow that   to guide your beliefs about what may exist and and  what may not exist and the guide is essentially   if you find that there's a certain kind of object  whose existence you need to commit yourself to   in order to get the scientific project done  then you ought to believe in that if not not   so so just every when you like are are  playing with the schrodinger equation from   quantum mechanics which is a subject that  you've spent a lot of time thinking about   uh if you're not thinking hard about the  question of invention versus discovery   do you have a sort of working methodology that's  in the back of your mind that views this as a mere   description of the world versus when you write  down schrodinger's equation you're like this is   the world there are lots of physicists who like  to say it's a great mystery this question of how   mathematics applies to the world why mathematics  applies to the world why we're so lucky   as for it to be the case that mathematics applies  to the world fraga famously says i don't see what   these people are talking about i can't imagine  a situation in which mathematics would fail   to apply to the world somebody says to me i  have two apples over here and i have another two   apples over here um isn't it a miracle that the  fact that mathematically abstractly two plus   two equals four applies to these apples i'm  tempted to turn the the question on its head   it's very difficult for me to imagine  what the world would have to be like   in order for that to fail to be the case so max  jumping off from that i know you have very strong   views on the reason why mathematics does work in  in the manner that we've been describing we're   going to come to that later on in the discussion  but just in a more cursory cursory way right now   math in your mind invented or discovered  discovered so it's out there yeah i think we   when people have these arguments is it invented or  discovered when people say it's invented i think   what they are often after is that the language  of mathematics is invented like we invent   the calligraphy by which we write the number five  because in china you can write it differently or   we invent the word for five you know  which in swedish is fem so obviously   that wasn't discovered we humans just kind  of made it up i think that's very analogous   to like if you look in the solar system and argue  about whether the whether the planet neptune was   invented or discovered obviously the name for  it was invented because in swedish we call it   neptunus there's a there's nothing fundamental  but the existence of neptune i would very strongly   argue is not something we made up it's just out  there and in exactly the same way when you go and   look for mathematical objects out there like plato  as as you mentioned brian was really interested in   these regular shapes called platonic solids and  he could invent the name dodecahedron for the   one that's made of 12 pentagons he could have  called it schmodecahedron instead if he wanted   but he couldn't invent the sixth one because  it doesn't exist just like you can't just   invent another planet in the solar system in that  sense i feel that these mathematical structures   really are discovered even though we invent the  language for talking about them well let me ask   the group do any of you allow for the possibility  that one day we're having a conversation with some   intelligent alien civilization and they ask  us okay what have you guys been doing to try   to figure out reality and we show them our  latest greatest mathematical description   of the world and they turn to us and they  kind of roll their eyes or they sort of   pat us on the head and say we understand that  direction mathematics we try that for a while   too but then we found that it was a dead end  and the real way for describing reality is this   and they give us some other description of the  world or or way of thinking about the world that's   not math it's not even disguised math it's just  something else do you allow that as a possibility   i certainly allow it as a possibility although i  wish i could imagine what that would be like but   probably it's important to distinguish between  whether or not our world has to be described   in mathematical terms that mathematics is the  language to express the nature of physical reality   and whether even if physical reality could  best be described in other terms there would   still be mathematical reality the validity of  mathematical reality wouldn't be diminished it   wouldn't be as you could be an imperialist about  mathematics and say mathematics is everything   you could be the opposite and say mathematics is  nothing i think the reasonable view is mathematics   is something but not everything but one of the  arguments that certainly has been put forward over   a long period of time for why math should be  viewed as really out there is in some sense   you know the the unreasonableness of mathematics  to quote a famous paper in describing how things   actually work in the world so just to be a con  give a concrete example when isaac newton was   writing down the universal law of gravity he  had access to certain observations certain   data to a certain degree of accuracy a certain  precision and you could imagine he wrote down   this formula that was able to describe the things  at that level of accuracy but then as the years   and decades go by our ability to measure the  world gets better and better and shockingly   newton's ideas continue to work and that leads a  sense of okay this is some deep actual truth of   the world not just some human description of the  world but then the counter-argument is at some   point the data becomes so precise that newton does  fail and of course we have to bring in einstein's   description of gravity a different equation  different mathematics so then it leads you to   believe well maybe newton's was just a provisional  description made up by a brilliant human mind   in the late 1600s and it was later replaced by  another description by another brilliant human   mind namely albert einstein and that seems to pull  us back to the idea of description as opposed to   the math being out there sylvia does either  of those ways of looking at math pull you in   one direction or another more toward it being out  there versus invented or more toward the reverse   i um certainly would also think that  um the the success of mathematics in   application scientific applications points towards  mathematics being out there mathematics being   uh discovered and so sometimes so the examples  you were just referring to it it looked to   me like um what was happening was that we were  eventually revising our physical theories but if   we're talking about purely mathematical theorems  they are eternally true they don't once they're   approved they don't get revised anymore so i i  think it's important to draw that to draw that   distinction um and i think that applicability is  really one of the most forceful arguments in favor   of what philosophers call mathematical realism  the view that mathematics is out there um that   mathematical objects exist independently of human  minds but it's one thing for math to exist as both   you and shelly were referencing as something  that perhaps might be given the label real   versus mathematics being out there as the reality  that we can touch and feel and interact with as   opposed to something that we're able to cogitate  about in some abstract sense so i mean i'm going   to be straightforward about it i waffle on this  question i have changed my mind over the course   of my career between whether math is invented  or discovered i almost can tie it to how well   my research is going when the research is going  really well it kind of feels like all that i'm   doing is like discovering something that that's  out there it just is falling into place and you   feel it has to be out there because it's locking  together with such power and such economy and   and it's so gratifying to see the equations  just really doing what they're doing almost   independently of of the human who might be  scribbling them out on a piece of paper at   other times i've come to the reverse where it just  feels like we're just trying to trying to push   this mathematical description on the external  world and sometimes it resists because it's not   the the right language or the right approach  for actually describing physical arousal so i go   back and forth on this but but you you personally  do do do you come down on that issue with   one uh unassailable and immutable view or  or do you also find it going back and forth   i'm pretty i'm pretty certain that  mathematics is an objective um thing out there   so i've i've never felt very drawn  to fictionalist views of mathematics   um the notion of mathematical fictionalism  that really in some sense what we've done is   we've written this wonderful rubric called  mathematics which is has proven itself capable of   describing the world but we couldn't even describe  the external reality without already having that   rubric to begin with how would we even interpret  the motion of the moon if we didn't have the   mathematical language to talk about it how would  we talk about the motion of particles if we didn't   already have the vocabulary that mathematics  provides us so is that perhaps biasing our view it might be um though i think it to be fair  i think fictionalists about mathematics are   perfectly happy to say that mathematics is perhaps  an indispensable tool or a very important tool   that we need to uh to do empirical science um and  i think i just think that the the fictionalist   stance the fictionalist philosophical position  arises from an unwillingness to um posit the   existence of abstract objects out there so much  about much of the fiction in this position is a   reaction against mathematical uh realism or uh  platonism even and so i think um fictionalists   arguments have so far not succeeded in convincing  me that there is something wrong with mathematical   realism david earlier mentioned hartree  field he is famous for having managed to   nominalize newtonian gravitational  theory which means that he managed to   reformulate the theory in a way that doesn't refer  at all to mathematical objects or to mathematical   entities it doesn't use any mathematical language  anymore and that's an incredible um achievement   and so that's more than just translating  a mathematical sentence into an english   sentence like force equals g times m1 i mean  all that it's much more than that right than   that it's much more than that yeah there is a  like i said there is a sort of fairly precise   and and fairly compelling once once you follow  the arguments criterion that quine wrote down to diagnose whether this or that theory that  that you believe in is treating this or that   kind of object as existent okay and and field  in what he was doing was constrained by that   okay and wanted to show that newtonian mechanics  could be formulated um the newtonian mechanics   or the gravitational part of newtonian  mechanics could be formulated in such a way   as to satisfy the condition that according to  quine's criterion of existence it didn't count   numbers as existing objects and and so does did  that push you in a direction of thinking that   math is auxiliary that math is displaceable  to the extent that you believe something like   that and to the extent that you believe that it  could be extended to to all of natural science   i might say that hartreefield himself  um recently is more pessimistic about   uh the possibility of that project succeeding  than he used to be i was just bringing it up here   because these questions do numbers really exist  so on and so forth i it's what's precious in these   conversations is to find some way to get beyond  people scratching their chins and looking as if   they're having profound thoughts and actually find  some criterion that you could get your teeth into   and that you could do some work on and claim  to have discovered the beginnings of answers   to these kinds of questions maybe at the  end of the day this isn't going to pan out   this isn't going to be the right way to decide  which things exist and which don't and there   are going to be these intuitions of the kind that  shelley evoked that this is something mysteriously   that we feel we know with a level of certainty  that we don't experience in any other way and at   the same time it's not something that we can tell  a story as we can with neptune and with reference   to what max was saying i think there's a big  disanalogy between neptune and and claims about   the existence of the number five um or something  like that we know how to tell a story about how we   know about the existence of neptune it's a causal  story okay um it has to do with neptune's having   direct physical influences on a chain of  things that eventually ends up inside our heads   in the case of mathematics we don't have a  story like that a causal story that links   the existence of the number five to a certain  brain state of mind so there's a real dilemma   there there are real powerful intuitive forces  pulling you in two different um directions at the   same time and the thing that's and that's a that's  a tension in which it's imaginable just to remain   forever the thing that's refreshing  about an observation like kwines   is it is it might give you another  direction um another slightly different way   of raising these questions so max the number five  presumably it's never done anything to you you've   never felt the number five you've never reacted  the way you would if a black hole were next to you   what's that well there you go so it has affected  you in some metaphysical way but but does that   description that david just gave make you rethink  it all the analogy with neptune that you gave   i would put no i would push back actually  on this a bit i think the reason that um   the business with causality comes in and  so much in your in your arguments that   you so eloquently made there david is  because you know it's a causality only   makes sense even if once you have the concept of  time right first things are like this and then   that somehow causes things to be like that at  a later time the mathematics in our universe   is at a much deeper level that even transcends  time if you come back and ask questions about   the the things that mathematicians ultimately  really study in mathematical structures there   isn't necessarily even any time there like look  at the cube you know if you study the cube what   properties does it have it has it has six faces  it has 12 edges it has eight corners was the cube   created causally that did the cube once have only  five faces and then no the cube exists completely   outside of time and space as well for that matter  and when we when we look at how our physical   space is thought about in science right einstein  came along and said you know time we should think   of just as the fourth dimension of this  timeless thing called space-time which is   and uh if if the reality they were having  this discussion in right now were a movie then   space-time would be the entire dvd right so uh  what i'm saying here brian and the answer to your   question in response to your question is these  mathematical structures that mathematicians study   they don't exist in time or in space but space  and time exist inside of some of them like inside   of the four-dimensional space-time of einstein  which mathematicians refer to by the nerdy name of   minkowski space and five is just like that  mathematicians don't just study five they study   this one mathematical object called the integers  and five of them five is is is part of that where   the integers ever created there's no reference  to time in there at all and therefore there's no   element of causality in it it's in that sense  that i feel these mathematical structures just   exist and we're not obliged to explain what caused  them to pop into existence because they never did if i can jump in the the causal story wasn't  about questions about how they got there it came   up in connection with questions about how we know  about them we have a how information about them   if we're thinking of them as things outside of  our heads no less than tables and chairs are   outside of our heads okay there's a question  that arises about how they got from their   positions how information about them got from  outside of our heads to inside of our heads   okay in the case of tables and chairs we have  a story about that how that happens that story   involves causation i completely agree with you  that causation is already a higher level notion   it's based on mathematical ideas so on and  so forth nonetheless once we have this causal   language it gives us a satisfying story about how  we found out about these tables and shares there   may also be a causal story about how the tables  and chairs got there but that's not what we're   concerned with at the moment we're concerned with  how we found out about them i i think the cube   would still have six faces even if it never got  into our head you know if you rewind five billion   years ago and there were no humans if there  were some aliens for some who built this some   super computer that started classifying geometric  objects it it would still get the cube on the list   and the computer with calculator has it's got six  faces and it has eight corners and twelve edges   even if there were no heads at all in existence in  the in the universe at that time and it's in that   sense that i feel that that mathematical structure  it has those properties and data came along and we   found out about them but we didn't make it pop  into existence we just discovered its existence   something that i think and of course sylvia  knows much much more about this than i do   but something that i think has often puzzled  people about mathematical knowledge is that   that kind of an account of how this stuff got  into our heads doesn't seem to be available   in that case we cannot tell a story that is  convincing to us by our by today's standards of   how we have epistemic access to these mathematical  objects that are purportedly out there um i just   wanted to point out that um plato of course had a  story about it he thought that our souls before we   came into existence before we were born our  souls were communing with the eternal forms   and so during life where recollect it's you  know mathematical knowledge is a recollection   of that time but also mathematicians  like godel for example thought that well   we have our five senses with which we perceive  physical objects but we also have mathematical   intuition he believed that you know there  is this mathematical landscape out there   and we perceive it with an additional sense  perhaps we could call it our mathematical sense   and he thought there was nothing you know  particularly perplexing about telling such a story   i think it's just that nowadays we're used to  um you know asking for an empirical story an   empirical explanation for kinds of things  that we can or cannot know and that seems   very difficult in the case of mathematics the  reason we have trouble being satisfied with   how we come to know mathematics is precisely  because we know it so intimately and so well   this would require a lot of elaboration but  um i really do think that's what's going on   it's a bit like consciousness if we know something  directly and intimately maybe we shouldn't expect   to be able to give an account of how we know  it an account of the kind which we feel so   satisfied about about for example knowing that  neptune was there we do the analysis and we're so   satisfied it comes out right but that's because  we knew it indirectly not intimately and that's   true of so much of physics you know we've had  some conversation about math describing the   external world that you know we can touch and feel  and observe and measure and you just brought up   consciousness shelly which is the next section the  next chapter that i wanted to move into which is   the paradigm of equality of the world  that we can't grasp in the way that we can   more familiar physical objects it's something  that as you say we have deep knowledge of   we have intimate connection with and yet it's  something that most people i think intuitively   would think stands outside of mathematics not  something that can be described in that particular   language so is that enough to really understand  consciousness or is consciousness something that   is much more perplexing than any of the qualities  of the world that we've been talking about so far   here's what i tell my students i i basically  with regard to physics i'm basically a pretty   strong reductionist so at the end of the day  our simple like hope will be at the end a simple   t-shirt equation which captures all the physics we  don't have anything like that yet presumably but   the equations we do have are beautiful and they  explain most of what we say around it basically i   say that you know the physics fundamental physics  should explain everything but then i tell my   students once and then leave that out and forget  that for the rest of my of the course i tell them   yes but i lied to you consciousness is left out  physics does not explain everything so you really   think that consciousness is left out uh will not  absolutely i'm i'm absolutely convinced of that   why can you i would let me just say this is not a  challenge if you could convince me of that i would   be eternally grateful because my view is is much  more the first view the reductionist view that the   end of the day were bags of particles governed  by physical law and that's all there is to it   it's a kind of bleak but i think beautiful way  of looking at the world but tell me why i'm wrong   look i couldn't convince  look i had exactly that view   for many many years i think sometime after  i maybe after i finished graduate school   i i suddenly for me it was just an  eye-opening experience saying oh   i was wrong all these years i was so strongly  arguing against all these people who said no no   consciousness can't be accounted for by physics  or mathematics i couldn't understand why people   were saying that and suddenly you know the phase  transition can occurred in my thinking and do   you remember what induced that phase transition  yeah but i don't remember i know it was reading   vigner's remarks on the mind body problem but when  i went back to look at it later to see what was it   precisely in there that did the job for me i  couldn't see what it was so no i'm to me it's   not clear what it was but certainly my thinking  just changed dramatically i mean i think david   chalmers does a great job in three or four hundred  pages of explaining why it is that he thinks   that consciousness transcends  physics and mathematics   but yeah you would like me to convince you of  it i mean i find this is the hardest thing in   the world to convince anybody yeah yeah yeah you  can't remember an hour or two hours three hours   weeks maybe months can do it of discussion  it's maybe it's useful just to list   a couple of reasons that people do give whether  any of these reasons capture what's going on   with shelley there are several different kinds  of reasons that people cite for being deeply   puzzled about how a reduction of consciousness  to physics would work somebody says to me   a bunch of particles arrayed like this and moving  in this way that's a table okay and if i go back   to them and i say why is that a table there's a  lot they can say well look solve the equations of   motion you'll find that particles assembled  in this way will hold up plates and glasses   and you can be convinced that the way we ourselves  recognize what a table is is by means of its   causal interactions with us and with the rest of  the world you can give a very convincing story   about why an assembly of particles  arranged in that way moving in that way   is a table if you try to play the same game with  say the sensation of green or the sensation of   red okay there's a feeling that somebody tells me  the sensation of green is a is a bunch of ions in   your head going this way and the sensation of red  is a bunch of ions in your head going this way   it's just hard to see why why there's something  greener about this kind of motion and redder   about this kind of motion it feels like it feels  like a case of comparing apples and oranges   you just don't know what to say you don't  see what they have to do with one another   you don't see what the connection is and this  is what what shelley talked about in terms of   the intimacy of our acquaintance with these things  is relevant here unlike identifying something as   a table or a chair or an orangutang or a  computer or anything like this plausibly   we pick out all of those things by means of their  causal effects on us the sensation of redness   doesn't seem like that we don't distinguish  red from green in terms of their different   causal effects on us we distinguish them from  one another somehow just by the state of being   in them and conscious phenomenal states  seem like the only things in the world   that have that kind of relationship to us so this  is one reason why people are puzzled um i don't   know if it's shelley's reason it's certainly  certainly there are other ones people cite   suppose i say of myself i'm just a machine i'm  just a collection of billiard balls bouncing   around like this and somebody says why do you  believe that you say i think i have good reasons   for believing that our scientific investigations  of the world suggest that something like that is   true and the guy says gee i'm puzzled you just  claimed that you're the kind of physical system   whose properties including your beliefs are  just determined by the physical laws okay um   but now you're acting as if there's something  else that goes into determining your   beliefs some kind of sensitivity to what's  reasonable but if your model of yourself is   is the correct one that model denies that there  is any kind of additional sensitivity to reasons   if somebody asked me why i believe that two plus  two is four i have the feeling that i can see   how it couldn't not be the case that two  plus two is for this is the kind of certainty   that shelley refers to but if i really take it  seriously that everything about me is just covered   by the initial conditions and the laws of physics  then there's only one explanation available   of why i believe that two plus two is four that's  because of the initial conditions of the world   and uh and the laws of physics okay and there's  nothing about the truth that enters into the   explanation of my belief that two plus two is  four so i would like to inject some optimism here   both to david who was worried that uh his  bouncing particles couldn't really be trusted   in their views about the outside world and to you  shelley who are worried that consciousness can   forever be on physics and reductionism and  also to you brian who was using the word   bleak to describe reductionism so if we start  with you david i think that the reason you   should actually put some trust into what your  bouncing quarks and electrons are having you   tell us about the state of the world is because as  brian said there is this additional story you can   add on top of it the super beans on on just  particle physics it just follows from it   and it's the story of darwinian evolution  we obviously there were probably a lot of   organisms who were really really lousy at taking  information from the environment and coming up   with correct predictions and they got eaten or  starved to death and those who were still around   have evolved to actually tend to have more  reasonable beliefs mostly about what's going on   just to i just want to get the shelley first also  so for the for the this notion that we're forever   doomed to never be able to explain consciousness i  think history actually gives a lot of hope here so   brian mentioned galileo before if if galileo threw  a grape and a hazelnut right he could predict   wonderfully how they would both move in a parabola  and when they would hit the ground but he had no   clue why the grape was green and soft and the  hazelnut was brown and hard and that really   seemed the most scientist at the time to be just  beyond physics how could he ever predict that it   was going to be green but then maxwell's equations  came along and showed that colors and light can be   described by mathematics the schroedinger equation  of quantum mechanics was discovered and helped   us calculate why the hazelnut is hard and and the  grape is soft and gradually physics went from this   situation where it could describe almost nothing  except motion of the phenomena in the world right   to where we are today where it can describe  almost everything and from the subatomic world   you know to our expanding universe and the  formation of black holes and interestingly   what's left the final biggest bastion of  ignorance that we have to try to conquer our   intelligence and consciousness which is exactly  where we we're going in now and um we will come   back and talk about this later but my guess is  that both intelligence and consciousness are   particular kinds of information processing that we  can ultimately also describe with math and in fact   that's the reason why ai has been so successful  recently and making some progress on on the   intelligence side and there are mathematical  theories by giulio tinoni and others also what   they try to do with consciences but i agree we  should be humble we're absolutely not there yet   but we should we should be optimistic that  there's hope and last brian uh you you said   it felt bleak to acknowledge that you're just  a blob of of quarks and electrons evolving   according to the standard model of particle  physics i object to you so i shouldn't say   spoken many people see it as bleak my own view  is it's spectacular that particles and bags and   collections of products do the things that we do  so my view is actually quite the opposite but i   don't know where you come down on it so uh richard  feynman who's inspired me enormously has this very   beautiful uh counter-argument to this you know he  was arguing with an artist who said that euphysis   just ruined everything by trying to describe it  reductionistically you hold up a flower and say   look how beautiful it is but you as a scientist  all take this all apart and it becomes dull thin   first of all i can't appreciate the beauty of a  flower at the same time i see much more about the   flower than he sees i could imagine the cells in  there the complicated actions inside which also   have a beauty i mean it's not just beauty at this  dimension of one centimeter there's also beauty in   the smaller dimensions the inner structure also  the processes the fact that the colors and the   flower evolved in order to attract insects to  pollinate it all kinds of interesting questions   which the science knowledge only adds to the  excitement the mystery and the awe of a flower it   only adds i don't understand how it subtracts the  feynman rose story is a beautiful one that i've   recounted many times too so i i totally agree with  that perspective sylvia i just want to turn to you   on this question of of the self-reflective inner  awareness that we human beings have within our   heads do you see that as part of a mathematical  description of the world or for instance would you   imagine that other languages or other perspectives  need to be brought in to have a full understanding   of that quality of the world i sometimes feel  that the puzzlement about consciousness and what   we can achieve with mathematical descriptions  of consciousness arise partly from different   meanings that the word description has so  um we can think of all kinds of descriptions   descriptions in different languages so if we think  of conscious conscious experience we could think   of trying to find a mathematical description  of that and that would help us understand um   for example what distinguishes uh conscious  beings from unconscious objects in the world   um and so what we the goal of such a description  would be to enlighten us on the physics underlying   conscious experience but we could have a totally  different goal for our description for example to   replicate or to induce a particular phenomenal  experience in another being that we're trying   to describe our conscious experience to and in  order to achieve that goal um a different kind   of description might be much more suitable so  let's say if you had never in your life tasted   vanilla ice cream and i was going to try to  describe to you what it what it tastes like   then perhaps a mathematical description would not  be the best one or let's think about a different   example ice cream is a little bit too too trivial  think about the emotion of fear um i could give   a mathematical explanation of what happens in a  subject's brain when that subject is in a state of   fear i could describe you know which neurons are  firing and what exactly has happened on the on the   particle level in that person's organism but in  order to make you understand assuming that you're   a being who's never had um who's never felt fear  in order to explain to you what fear is like   it might be more useful to use a completely  different kind of description using a different   kind of language let's say the language of  music um it's famously and this also relates   um to what we just heard about you know uh  disputes between physicists and artists famously   art can reveal and communicate let's say knowledge  can communicate things can reveal things to us   that cannot be revealed in a different in  a different way using a different language   so i think we always have to ask what exactly  is it that we want to achieve with a particular   description and if what we want to achieve is  to get a better sense in terms of the underlying   physics of consciousness then my intuition is more  on on max's side i think we can be hopeful that   empirical science will at some point uh be able to  figure out a mathematical model for consciousness   if it's about making another sentient  being understand what my inner life is like   i'm much more skeptical what about morality  just turning the gears a little bit further   human minds human brains are are able to  have a moral sensibility of right and wrong   and good and evil and all those qualities that  are so vital to a rich life here on planet earth   do does morality and those notions  do they stand in your view outside   of the mathematical description or the  mathematical universe or are they subsumed   within it in some deep way that might not be the  most efficient or effective way of describing   them but nevertheless they are accounted for in  a mathematical way of thinking about the world i think mathematics um gives us an interesting  paradigm or an interesting model um that can   help us to try and understand how morality fits  into our empirical world so we've discussed this   earlier in the program that you know people are  some people are drawn to the idea that there   exist mathematical objects other people object  to that idea and think that's completely crazy   because those objects would be so vastly  different from the objects we're familiar with   physical objects and pretty much the same thing  can be said about moral entities you know like um   moral values for example would  be what i call a moral object   and so there are some people out there let's call  them moral realists who believe that morality is   something much like mathematics it's it's eternal  there are eternal truths out there and what we're   trying to do when we think about morality is  to sort of um dock onto those truths like to   capture those truths and um represent them in  our beliefs but that of course entails that moral   objects exist and it's very confusing uh to some  people to to to imagine what that would look like i think that thinking about mathematics and  seeing that we have good reason to believe in   mathematical realism and we shouldn't be afraid  of positing those entities can actually help us   overcome our scruples also in the moral arena  so one one uh one of the main arguments that is   usually cited in favor of moral anti-realism in  favor of the view that morality is is something   invented something that we constructed for some  evolutionary purpose one of the main arguments is   always disagreement we vastly or human beings  vastly disagree on what are the moral truths   therefore there cannot be any substantial  truths out there now i think there are a   number of reasons why that argument is is flawed  but um the com the the comparison to mathematics   can lend credibility to the view of  moral realism and here is the comparison   if the reason to reject realism about morality is  fundamental disagreement we have the same also in   the mathematical arena it's true that mathematics  is often viewed as this sort of disagreement-free   cosmos where all that mathematicians are doing  is discover new mathematical theorems or prove   new mathematical theorems but in fact when when  we look at the foundations of mathematics of set   set theory for example we can see that there is  pretty vast disagreement on pretty fundamental   questions also in mathematics for example on the  question how many mathematical universes are there   and so i think that this this main argument  against moral and against moral realism   can be discarded once we turn to mathematics so  that's just one way in which the analogy between   mathematics and morality can be enlightening  well i'm not all that comfortable with the   plurality of mathematical truths in the sense  that genuinely conflicting mathematical truths i   that's something i have trouble understanding um  i do think the analogy between mathematics and   morality is a useful one just as justice sylvia  said and of course the big difference is that i   at least for most people and that's i think well  i don't know about most people for me in any case um mathematical truths force themselves on  me with a con compulsion which is at least   somewhat lacking when it comes to moral truths  i'm not a moral relative a moral realist but um   nonetheless uh the mathematical realism that  just seemed to that i that i thought i feel   much more strongly maybe a distinction however  is relevant not sure it's a sensible one or not   with a distinction which would apply both to  mathematicals mathematical reality and moral   reality and that the distinction is this what is  mathematical realism what is moral realism in both   cases you might say you might say the mathematical  realism you believe in the objectivity   of mathematical propositions or moral truths  you believe for example somehow that there there   really is it's not there are an infinite number  of primes and there always were and you did there   didn't have to be people around for that to be  true there simply are an infinite number of primes   that's certainly a view that a  mathematical realist would have   now for many mathematical realists if not most  and many would david might say how will you how   would you maybe you're forced to have this view  but i know many people would say but i do not   want to insist that there are mathematical  objects somehow i i don't believe there's a   separate realm of mathematical objects there's  a realm of mathematical truth i can certainly   say that somehow or other there's definitely an  infinite number of primes but i don't believe   prime numbers exist the mathematical objects  themselves don't exist i think that's a view   some people have david any uh any thoughts on the  morality uh math analogy there's an argument that   mathematics is indispensable mathematical some  kind of mathematical realism is indispensable to   the scientific project and i take it there's  no analogous argument in the moral case we   can teach people a whole course of theoretical  physics without mentioning the words good or bad   um but we can't do it without without mentioning  the word five right right can i just that yeah yes   judge our ontological commitments to  mathematical entities with reference to   the scientific project and that's because  the scientific project seems like um   itself like an intrinsically indispensable  project that science is our paradigm for   knowledge at the moment but here is an  alternative indispensability argument for   moral entities or a criterion as david called it  if we cannot not quantify over um moral entities   or normative entities in a project of our lives  that is intrinsic intrinsically indispensable   perhaps not science but something else then  that might also give us a reason to believe   in the existence of uh moral entities and so it  has been argued this is not my argument this is   david enoch's argument it has been argued that  quantifying or referring to normative entities   like reasons or values is indispensable  to the intrinsically indispensable project   of deliberation human beings cannot fail to  deliberate we deliberate all the time in order to   come up with uh well in order to decide  how to behave and how to interact with   the world and in this indispensable project of  deliberation we cannot fail to refer to normative   entities and so that's a sort of analogous  indispensability argument that i think has   definitely some force this question  is actually incredibly timely because   we're now putting rather mathematical based  objects artificial intelligence right in charge   of ever more decisions they're out there now  deciding who gets a loan who does not get a loan   they're having influence of who  gets probation who stays in jail   very controversially some people are even wanting  to build weapons that themselves decide who lives   and who dies right and and then this question of  sort of splits into two parts one is how can you   actually mathematically codify an ethical moral  system so that you can explain it to a machine   a self-driving car or or whatever there's and  this is a booming field now in machine learning   machine ethics and then the separate  question is well okay if you can quantify   mathematically all these different systems  of morality well which one is the best   uh this one clearly there's no consensus on but i  want to inject just a little bit of optimism here   too because frankly i don't think the reason our  world is in such dire straits right now is because   we can't find we haven't managed to answer  exactly that question almost everybody on this   planet agrees that hey it's better if humanity  does not go extinct than if we do go extinct   uh it's better to have less poverty than to have  more poverty so the reason we as a species are   still epically failing at avoiding you know  the risk of nuclear war and avoiding people   starving to death all the time isn't because of  this moral disagreement it's it's more about the   for some reason we just really haven't gotten our  act together and having and in the way we manage   our planet and people often push in a certain  direction because they think it's gonna be in a   way they think is morally good and and it isn't so  i think i think uh the optimistic part is i don't   think we're limited by our understanding of math  and morality and our opportunity to make a better   future so with that optimistic note i want to turn  to uh the final section max one of the ideas that   you are a proponent of in terms of discussion  we've had a little bit around a corner   where we've been discussing sort of the universe  and math either as a description of it or somehow   being part of it or somehow being you know deeply  connected with physical reality you go further and   suggest that perhaps we should turn that question  around and view math as the real reality which   every so often perhaps looks like the physical  reality that we experience but but it's the real   uber you know bottom line most basic reality  that exists i don't know if i'm describing that   accurately but if you could give us you know your  sense of of this view of mathematics and then we   can have some of the folks weigh in as to whether  that's a view that that resonates with them or not   that's right so if you look at the spectrum of  views you can have from it's all in our heads   you know all just invented to somehow it's in  some sense out there i'm as far as you can go i   think on the opposite extreme saying not only does  mathematics describe our universe but our universe   is actually mathematical in the sense that we  live in a gigantic mathematical object and when   you first hear that frankly it sounds ridiculous  it sounds absurd right if you all look around in   the room you're sitting right now if this is  all supposed to be mathematical well i mean   where is the math you know i i'm looking around  in my room outside my window now i see a tree here   you know if something is entirely mathematical  it means that all its properties are mathematical   properties it has only mathematical  properties but that tree there it's green   and brown and it's leafy you know that doesn't  sound like mathematical properties at all so what   am i even talking about well when i look at the  tree again though through my eyes as a physicist   i see a blob of quarks and electrons and what are  the properties of an electron minus one one half   one and of course we physicists have made up nerdy  names for those properties like electric charge   spin and lepton numbers but that was just  the language we made up to refer to those   numbers the properties are just numbers and  and you brian know as well as anybody that   the only difference between an electron and an up  quark is what numbers its properties are and then   what about so if all the stuff in our space has  only mathematical properties then what about the   space itself what properties does space have well  if it has the property three for example that's   the largest number of fingers i can put that are  perpendicular to each other again we made up a   fancy name for that we call it the dimensionality  of the space but that word is just the word we   made up like the word neptune the property of  space is three and then we heard earlier you   the non-euclidean geometry came up we now know  that space also has properties of curvature   and topology which you study in in graduate math  classes they're entirely mathematical properties   so if you accept the idea that all the stuff  in space as well as space itself actually   seems as far as we can tell as physicists have no  properties at all except mathematical properties   then it starts to sound a  little bit less insane that maybe our entire physical reality  is in fact a mathematical object max   is your view com is your view compatible with  the view that which you don't presumably don't   have but is it compatible with the view that  consciousness transcends mathematics and physics   i would guess that consciousness and intelligence  are certain kinds of information processing   uh not all kinds of information processing but  the one day someone will discover an equation and   maybe julia tinoni already has it such that when  the information processing obeys that there is   a subjective experience there like david albert so  eloquently um described um and information itself   of course as we learned from claude shannon is  shouldn't be described entirely mathematically   as well so by max along the lines of the you know  the the minus one and the half you know in terms   of electric charge and spin and so forth of course  i totally understand where you're coming from and   yes if you were to ask me what is an electron like  what really is it i would ultimately be forced to   give the description that you are rehearsing you  know i'd give you its charge as a number i'd give   you a spin and i wouldn't be able to talk about  electron-ness in any other more fundamental less   mathematical way but but that still could be a  human description of the electron as opposed to   the real intrinsic electron-ness of that particle  right it could still be us humans imposing a   description we developed this tool and it does a  good job a great job but it could still be that no   it's really fascinating i think  historically and i'm so grateful   sylvia for you giving us this  this historical perspective how   again and again uh if you took just traditional  materialism saying oh there's this stuff   as if we knew what that really was right how  all the non-mathematical properties we used to   talk about one by one have just sort of melted  away and we just seem to be left with only the   mathematical ones uh even the fact that there is  a particle there an electron we know now of course is also an oversimplification there is this thing  called a quantum field and how many electrons   are in there it just tells you something about  that state in the field it's like how much the   guitar string is vibrating how excited is some  mode of this um entirely mathematical right and   then came quantum mechanics and said well actually  you describe the whole thing with a wave function   that lives in this place called the hilbert  space completely mathematical things   and i i think i i find it very intriguing that  throughout the history of physics whenever we   have found that some mathematical description of  the world was inadequate it wasn't quite right   and we had a revolution and replaced it by  something else that something else was also   mathematical and the most ambitious attempt so far  to find a single set of equations you can put on a   t-shirt maybe in the future string theory as you  know better than anyone else on this here on this   this panel brian is entirely mathematical too i  really don't see any evidence so far from nature   that there's anything non-mathematical about  the world now if i understand you go much   further than what you've so far recounted you  would actually go so far as to say that all   of mathematics is as real or i should say any  piece of mathematics is as real as any other piece   of math you don't distinguish between math that's  relevant to reality and method isn't relevant is   that is that an accurate description that's right  they sometimes call that the level four multiverse   so when we talk about something existing if we say  that pink elephants don't exist what we secretly   tend to mean by that is why they don't exist  here on earth or anywhere where we've looked   um but maybe there's another planet really really  far away whether you actually have pink elephants   somewhere else right and um we've gone through  this progression in history where again and   again and again we've come to realize that the  totality of physical reality that exists is just   way bigger than we thought then everything we knew  about is just a small part of something grander   a planet you know solar system a galaxy a galaxy  cluster our observable universe it's probably much   bigger than the part that we can see that  light has reached us from etc and if what   i'm saying what i'm guessing is true then it's  bigger still so you have all of these different   mathematical structures that mathematicians like  shelley can study the cube a columbia manifold   you know minkowski space three plus one  dimensional manifold the hilbert space etc   if they're you know hilbert the mathematician who  was quoted earlier famously said that mathematical   existence is just a freedom from contradiction  there are a lot of different objects that exist   mathematically and my guess is that we inhabit one  of them and if there are other ones which are also   complicated enough that some of their inhabitants  are conscious and have experiences they're gonna   feel that their exists physically and just the  same as we feel that ours exists physically and   in just a sense the same sense that if you brian  were actually right now in a some future very   high tech computer game you know this governed  by mathematical laws you know you wouldn't feel   any different you would feel of course my  my little game world here exists and i can   punch the table it'll punch me back um so yes if  this is true i think this uh copernican revolution   needs to go one step farther and and there  is the ultimate reality is still much bigger   than we thought david any any thoughts on  whether we might be living in sort of the   the mathematical reality of all equations being  instantiated in some way shape or form i mean first of all they don't i all i'm going to be able to express about this is  a certain kind of bafflement um about what's going   on and how i'm supposed to understand it i mean  when um max when you say um our description of the   world has gotten more and more mathematical more  and more of the non-mathematical things have been   eliminated i'm not sure i understand what that  means you talk about you know plato and aristotle   sitting around or cavemen sitting around the  the nobody would have doubted that the tree that   they're looking at has some volume okay and you  can assign a number to that okay um what number   you design is gonna is a sign is gonna depend of  course on your choice of units and that's no less   true for the electron spin um um than it is for  the people sitting around looking at a tree but   do we think they would have denied that the tree  has a volume do we think they would have denied   that the tree has a mathematical description  it's just that you seem to want to collapse   um any distinction between the mathematical  description of these things and the things being   mathematically described and there i just really  get confused it's as if you know if somebody comes   up to me and tells me all of these perplexities  that you're having are going to evaporate   if you can just get it into your head that a  cucumber is really a ferrari okay and and if you   can just get your head around that everything's  going to be fine and i sit there saying cucumber   ferrari cucumber all i have to do is figure out  how to see that a cucumber is really a ferrari   i can't do it i don't know what i'm being asked  i don't know what i'm being asked to imagine   and i don't know what i'm being asked  to imagine when somebody tells me   no it's not just that the tree has a volume that  can be expressed by a number it's that the tree   is the number um like i say i'm just being  thick-headed here uh i'm just being stupid   i don't understand what it is that i'm asked to  being asked to imagine so it's a it's a it's a   fair question but i think it's one that has a  good answer there is something that's really   fundamentally changed in our understanding of  trees in the past thousands of years in the   beginning when archimedes was studying trees and  galileo was studying trees there were some very   limited aspects of the of the properties of the  tree that they could describe mathematically they   could describe how long it would take for a branch  to fall down if it fell off and stuff like that   but they had no idea why the wood was so rigid  and stiff and why the bark looked brown and so on   there were measurements you could make of the  tree that you just had no mathematical way of   calculating the answer to that's just really  changed today we really there are still there   are there's no evidence i would say that there  is that you couldn't if you were to sufficiently   if you were to that you couldn't at least in  principle derive all properties of a tree from   the standard model of particle physics we know  in math just because you have the fundamental   equations does not mean you can always calculate  things accurately you know we can't even calculate   the mass of the proton relative to the neutron  accurately even though we really think we know   gladys qcd right um we're in computing exactly  how the tree is gonna grow up just from knowing   its dna you know right now it's beyond our  ability to compute but i would say there's   the evidence that there's something  more to the tree than a quark blob has   sort of melted away that that's not true of all  of physics uh consciousness and intelligence   especially consciousness i think we have to  be very open-minded that maybe we're missing   something big but certainly with with trees i  think uh there's every reason to believe that   we've got the fundamental description  of it right and it's mathematical and   now we're struggling with how we can  link that to the sort of higher level   uh processes that are going on of a growing living  max if i can just push back a little bit look   everybody would have agreed thousands of years  ago that trees have tiny little pieces okay and   that trees are in some sense the sum of their tiny  little pieces okay um um and probably they would   have agreed that the pieces are in some sense  simpler than than the whole tree itself still   i don't understand how that takes us one inch  in the direction of saying something like the   tree just is numbers um we we're getting better  and better at not only describing the tree using   numbers but predicting how the tree is going to  behave in a way that that um that in a limitably   makes use of numbers okay but if somebody  says um here you got two apples here um   so i don't know what the next move is to be is  supposed to be so really all you have here is   the number two if you really think of it i i it it  really feels like cucumbers and ferraris i don't   want to talk smack about cavemen or physicists  from 2000 years ago but they did not understand   that trees are made of air primarily that  they actually take carbon dioxide out of this   out of this the air and sunlight they take energy  to bind it in and and and so on uh these are new   things they were just missing they didn't have a  complete understanding of of the makeup of trees   that we now do right and whereas now i think we're at the point where there  really is no evidence that trees are made of a new   kind of quark that hasn't yet been discovered  or that you have to overthrow string theory   to describe trees the second point that you  raise which is also very important is i think   the distinction between saying that something  is described by math and saying that it is   mathematical and that might be related to your  cucumber ferrari issue so right in mathematics   there's a very beautiful uh body at work on  on equivalence there are many different ways   in which you can mathematically describe  for example the integers piano wrote down   five axioms and the certain descriptions you  can you can start in many different places   you can use different symbols different notation  and when we talk about a mathematical structure   what we actually mean isn't any one description  of it but we mean the equivalence class of that   thing which is described by all the equivalent  descriptions that's the thing that exists so   there isn't in the in in plato's mathematical  realm to different cubes that there is just one   mathematical structure of the cube right and in  the same way if it turns out our physical world if you if you think of a mathematical structure  just a bunch of abstract elements with relations   between them if that corresponds one-to-one with  the abstract element with with elements in that   we call physical things like super strings or  quarks or whatever and relations between them   and you just have basically two different  descriptions of the same structure   then they are one and the same thing and it's  that in that sense that i say that our physical i   think our physical world really is a mathematical  structure you know you could even write a computer   program that starts to just classify one by one  different structures that exist in mathematics   my guess is that there is a mathematical  structure which is in fact the one that we live in   and since we live in it we have come  up with also the human words to refer   to the entities we talk about we call  them electrons and quarks or whatever   but they are just the names that we humans  have come up with for referring to abstract   mathematical entities well i've certainly enjoyed  our conversation with all of you mathematical   objects if you allow me to use that language  clearly this is an argument that could keep on   going for some time and it's a sharper version  of the very question we began with in terms of   is math a description is math a human invention  is math actually out there or in max's vision is   math the be-all and end-all of reality wherever  we come down on questions like that i think all   of us here and hopefully everyone who's watching  this can get a sense of how beautiful mathematics   is how powerful it is at describing the world  perhaps it's even more than that i began with   some quotes at the beginning of our conversation  i want to end with a few quotes that capture that   spirit first from bertrand russell mathematics  rightly viewed possesses not only truth but   supreme beauty a beauty cold and austere like that  of a sculpture without appeal to any part of our   weaker nature without the gorgeous trappings  of painting or music yet sublimely pure and   capable of a stern perfection such as only the  greatest art can show mariam murzakani said you   have to spend some energy and effort but if  you do you can see the beauty of mathematics   from g.h hardy the mathematician's patterns  like the painters or the poets must be beautiful   the ideas like the colors or the words must  fit together in a harmonious way beauty is   the first test there is no permanent place in the  world for ugly mathematics and finally paul erdos   why are numbers beautiful it's like asking  why is beethoven's ninth symphony beautiful   if you don't see why someone can't tell you i know  numbers are beautiful if they aren't beautiful   nothing is so with that let me just  thank all of you for this wide-ranging   conversation on math and reality thanks so much  for joining us enjoy the conversation and maybe   we'll have the pleasure of picking up these  controversies sometime in the future thank you so you

Channel: World Science Festival

Views: 1,177,471

Keywords: Brian Greene, Nikola Tesla, Does Math Reveal Reality?, Mathematical Universe, Mathematical mysteries, math, mathematics, Quantum Physics, search of answers, prime numbers, nature of infinity, geometry of hyperspace, mathematical truths, infinity, boundaries of math, masterminds, 2021, breakthroughs, Hypothesis, David Z. Albert, Sheldon Goldstein, Silvia Jonas, Max Tegmark, Orman Quine, David Chalmers, Kurt Godel, Hartry Field, Richard Feynman, History of Math

Id: VN19VOMHxkk

Length: 96min 2sec (5762 seconds)

Published: Thu Aug 19 2021

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[u/Ravishnu83]

...the stars began to connect, at which point I had an epiphany... I am a NUMBER... 😉 (Episode 70)