Many Many Worlds

tegmarkMax Tegmark is a professor of physics at MIT, a major force in the development of modern cosmology, a lively expositor, and the force behind what he calls the Mathematical Universe Hypothesis — a vision of the Universe as a purely mathematical object. Readers of The Big Questions will be aware that this is a vision I wholeheartedly embrace.

Tegmark’s new book Our Mathematical Universe is really several books intertwined, including:

  1. A brisk tour of the Universe as it’s understood by mainstream cosmologists, touching on many of the major insights of the past 2000 years, beginning with how Aristarchos figured out the size of the moon, and emphasizing the extraordinary pace of recent progress. In just a few years, cosmologists have gone from arguing over whether the Universe is 10 billion or 20 billion years old to arguing over whether it’s 13.7 or 13.8.

    This part of the book concludes with an account of the theory of cosmic inflation — the process that triggered the Big Bang — which was spectacularly confirmed just a few weeks ago (and just a few months after this book was published) by a set of observations that Tegmark has called one of the most sensational scientific discoveries of all time. Inflation models tend to predict that our Universe is infinite (though of course we can never observe anything more than about 13.8 billion light years away). This will come up again later.

  2. A tour of the “landscape” — the multiverse in which, according to many versions of cosmic inflation, our infinite universe is a tiny bubble. (More precisely, though still less precisely than the explanation you’ll find in Tegmark’s book, our universe looks infinite from within but tiny to a hypothetical outside observer.) The landscape is a strange place indeed, filled with plasma and doubling in size an incredible 1038 times every second. Now and then bubbles form, and within these bubbles, the plasma cools into any of 10500 different varieties of space and time, giving rise to 10500 possible collections of laws of physics.

    The theory of the landscape is respectable but controversial among physicsists. Tegmark is a true believer, largely because the landscape explains the otherwise seemingly inexplicable hospitality of our Universe, where dozens of physical constants seem to have been fine tuned to make life possible. (For example, if gravity were a little stronger, the sun would have devoured the planets long ago; if it were a little weaker, the planets would have spiraled out into space.) If there are in fact 10500 possible states of physical law, then it’s not so surprising that one of them is hospitable to life, and then not surprising at all that that’s the one we evolved in.

    I have not attempted to read much of the scientific literature about the landscape (though it’s on my to-do list), and in any event I have little enough faith in my physical intuition that I’d hesitate to form my own judgment about its plausibility. But I think that the hospitality-to-life argument is quite undermined by Tegmark’s own Mathematical Universe Hypothesis (in which I do believe!). If the Universe is a mathematical object, then the great diversity of mathematical objects is enough to explain why there’s one where we can survive. We don’t need the additional diversity of the landscape.

    (Of course, if the Mathematical Universe Hypothesis is true and if the landscape is ultimately a mathematical object, then the landscape certainly exists and has physical reality — as does any mathematical object sufficiently complex to contain self-aware substructures. It does not follow that we live there.)

  3. An account of the many-worlds interpretation of quantum mechanics, — together with a very surprising (at least to me) tie-in to the earlier material. Namely: If our Universe really is infinite, then it contains an infinite number of balls 14 billion light years in diameter. But the laws of physics allow only finitely many possible histories for such a ball, so at least some of those histories must occur an infinite number of times. So we should expect that there are, far beyond the limits of our observation, balls with histories identical to the ones we live in, as well as others that differ from ours only in minor details. (I am reminded of Father Guido Sarducci‘s bit about a far-away planet identical to earth in every way except that they hold their corn on the cob vertically instead of horizontally.) Tegmark calls this collection of balls the “Level I Multiverse”, to distinguish them from the “Level Two Multiverse” arising from cosmic expansion. So the Level I Multiverse is a tiny bubble in the Level Two Multiverse, and it contains an infinite number of these balls where history plays out more or less as it does right here.

    Now, according to the many-worlds interpretation, the wave function of the Universe never collapses. It’s always seemed to me that the many-worlds interpretation should actually be called the one-world interpretation, since there’s just one wave function and hence just one history, but there’s a sense in which you can think of that history as constantly “branching”, so that the version of you who is reading this blog post and deciding whether or not to eat a cashew will soon branch into two of you, one of whom eats the cashew and one of whom doesn’t. Tegmark calls those branches the “Level III multiverse”.

    The big surprise is that in some sense, there’s no reason to distinguish between the Level I and Level III multiverses. Tegmark (with a pointer to an academic paper where he works through the details) shows that when you get down to the mathematics, there’s no important difference between the various versions of yourself who have branched off the quantum wave function and the various versions of yourself who inhabit far distant versions of the Universe.

    (Incidentally, Tegmark has the wonderful habit of including pointers to many of his — and others’ — academic papers, so readers can easily decide for themselves when they want to pursue topics in greater depth.)

  4. An account and a defense of the Mathematical Universe Hypothesis — the hypothesis that the Universe is, pure and simple, a mathematical object. I’ve reached a point where I can no longer remember why I ever thought this hypothesis even needs a defense. If the Universe isn’t made of mathematics, then what could it possibly be made of? It seems to me to be metaphysically reckless to postulate the reality of things (such as non-mathematical objects) that we have absolutely no reason to believe in.

    Look. To exist is to have properties. If statements about a thing are objectively either true or false, then that thing exists. Statements about (say) the natural numbers are objectively either true or false; for example, it is objectively true that every natural number is a sum of four squares. Therefore the natural numbers (and similarly other mathematical objects) exist. Mainstream physics accounts for all physical phenomena as mathematical phenomena (for example, an electron is a disturbance in a field, and a field is a mathematical function). These physical theories are surely only approximations to the truth, but as the approximations get better, they get no less mathematical. Therefore mathematical objects not only exist; they are the only things that we have any particular reason to believe in. To suggest that there’s anything else is to raise the apparently unanswerable question of why. In that sense, physical substance, apart from mathematical substance, is no different from God — a desperate pre-scientific attempt to postulate our way out of anything we don’t immediately understand.

    That, in essence, is my story, as told in The Big Questions. Tegmark’s got a good story too. He starts by presuming that there is such a thing as truth, independent of the way humans happen to think — and proceeds to demonstrate (at least to my satisfaction!) that any such truth must be purely mathematical. The basic point is that most of what we think of as reality — rocks, for example — appear to have individual identities only because of the way our brains work — we think of a certain collection of elementary particles as a “rock”, even though they fundamentally have nothing more in common with each other than with other elementary particles that we don’t consider part of the rock. Get rid of those artificial human distinctions, and all you’re left with are the particles. But what’s a particle? It’s a disturbance in a field, which our human brains find it convenient to conceptualize as a “particle”. Take away the human brain, and all you’re left with is a disturbance in a field. And what’s a field? It’s a bit of mathematics.

  5. A more detailed description of Tegmark’s version of the Mathematical Universe Hypothesis. Here he sometimes loses me. Obviously, he’s very smart and he’s thought very hard about this stuff, so perhaps the fault is mine. But Tegmark says a number of things that seem to me to make no sense. For example, he says that “to define the mathematical structure of [the ring of integers], we need to specify just the shortest computer program that can read in arbitrarily large numbers and add and multiply them.” But this is circular. How can you determine the length of a computer program without counting its lines? And how can you count its lines if you don’t already know a whole lot about the integers?

    Indeed, we know that the Peano axioms for arithmetic admit non-standard models — that is, structures that obey all the axioms of arithmetic, but are in fact very different, to the extent that their addition and multiplication rules cannot be implemented by any computer program of finite length. Suppose you managed to mistake one of these non-standard models for the standard integers. Then you could easily fool yourself into believing you’d written a finite computer program to implement your model’s arithmetic — but only because you’d be confused about what “finite” means. So we cannot rely on the existence of such computer programs to assure us that we’re talking about the standard natural numbers. For that and related reasons, the integers arguably form an extraordinarily complex structure.

    Tegmark prefers to believe that our Universe is not just any mathematical structure, but a computable mathematical structure. He goes on at some length about this. But I cannot figure out exactly what “computable” is supposed to mean in this context. I know what it means for a function to be computable, or for a set of axioms to be recursive, but what does it mean for a structure to be computable? My first guess would be that a structure is computable if the set of true statements about that structure (in some formal language) all follow from recursive set of axioms. But Tegmark seems to want to count the integers as computable, which rules out that interpretation. What other interpretation is there?

    It wouldn’t entirely surprise me to learn that Tegmark’s got good answers to these questions, but based on what’s in this book (and the bits of his papers that I’ve worked through) I don’t understand what those answers could be. In the scheme of things, though, who cares? The Tegmark vision of a mathematical universe is, I think, a final solution to the single greatest philosophical conundrum of all time, namely “Why is there something instead of nothing?”. If he’s got a few details wrong, that accomplishment is not significantly diminished.

  6. A scientific autobiography, vividly portraying the triumphs and frustrations of research, and the excitement of being on the scene just at the birth of “The Age of Precision Cosmology”. We get a lot of good anecdotes about Tegmark, the people he’s worked with, and his relations with the rest of the scientific community, including a remarkable email from a senior scientist warning him that if he keeps thinking so far outside the box he risks being labeled a crackpot. Tegmark’s enthusiasm, for science, for life in general, and above all for ignoring such warnings, is palpable and infectious.
  7. A bunch of self-indulgent asides of the sort to which every author of an excellent book (and this book is surely excellent) is well entitled, though we need not take them too seriously, except of course when we agree with them. So I’m happy to laugh off some of his ideas about what the most important issues should be in the next election, and happier still to see that he shares my equally inexpert (and hence equally self-indulgent) sense of the unlikelihood of intelligent life (see also here) elsewhere in the 14-billion light-year ball that constitutes our observable Universe.

Bottom line: A terrific book, really. The idea he really wants to sell is the mathematical universe idea, an idea that puts to rest (at least for me) the greatest philosophical problem of all time, namely “Why is there something instead of nothing?”. That is an astonishing accomplishment and one that, all by itself, makes Tegmark a towering figure in intellectual history. But even if you don’t buy that, or don’t care about it, I am not aware of any more lively or readable account of the history and current state of cosmology, including widely accepted ideas like cosmic expansion and more controversial ideas like the landscape. All lightened up with charming personal anecdotes and deepened with appropriate links to the academic literature.

I’m glad he wrote this book and glad I read it. You should read it too.

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54 Responses to “Many Many Worlds”


  1. 1 1 Keshav Srinivasan

    Steve, you say “A tour of the “landscape” — the multiverse in which, according to most versions of cosmic inflation, our infinite universe is a tiny bubble.” That makes it sound as if most versions of cosmic inflation accept the notion of the multiverse, but that’s not remotely true. Inflation is a commonly accepted phenomenon in cosmology, and the landscape is a controversial idea taken from string theory.

  2. 2 2 Steve Landsburg

    Keshav: For caution’s sake, I’ve changed “most” to “many” in the post — though the applicability of “most” can, I think, be defended depending on how one counts. (Though I am sure I don’t fully understand all the issues here.)

  3. 3 3 Erik

    The obersvable universe is much larger than 14 billion lightyears (approximately 90 billion lightyears in diameter), despite the fact that it’s only about 14 billion years old.

    http://en.wikipedia.org/wiki/Observable_universe#Misconceptions

  4. 4 4 nobody.really

    Steve, you say “A tour of the “landscape” — the multiverse in which, according to most versions of cosmic inflation, our infinite universe is a tiny bubble.” That makes it sound as if most versions of cosmic inflation accept the notion of the multiverse, but that’s not remotely true. Inflation is a commonly accepted phenomenon in cosmology, and the landscape is a controversial idea taken from string theory.

    For caution’s sake, I’ve changed “most” to “many” in the post — though the applicability of “most” can, I think, be defended depending on how one counts.

    Both statements are fair. These ideas are quite controversial from the perspective of those within our bubble. But when viewed from outside of it, these ideas gain much wider acceptance.

  5. 5 5 Roger

    Tegmark has papers on his MUH, but he is never mathematically precise, and no one knows what he means by computable mathematical structure. It is not clear that any current theories would qualify.

    Saying that the universe is math object certainly needs a defense. No physical object has ever been realized as a math object. Not even a single electron.

    It is not metaphysically reckless to believe in a rock.

    There are physicists who postulate some grand unified field theory or string theory that will somehow explain rocks and everything else. They have approximations, but they always admit that they do not have that final theory, in spite of decades of searching. When experts serach for decades and come up empty, you need to consider the possibility that it does not exist.

  6. 6 6 Keshav Srinivasan

    Steve, I’m not an expert but as far as I know, the only theory of inflation that invokes the landscape is brane inflation, which involves the Dirac-Born-Infeld action. That’s not the standard view of inflation (and the dynamics are completely different from the standard view), and I think the vast majority of astronomers don’t think that accepting inflation commits them to string theory.

  7. 7 7 Neil

    Stephen Hawking famously asked what breathes fire into the equations and makes a universe for them to describe. I think Tegmark (and Steve L) thinks there is no need for any stinking fire. I wonder whether it is the collapse of the wave function that breathes the fire. Before the collapse we have the Schrodinger equation, after the collapse we have a measured electron.

    I realize this is just replacing one mystery with another.

  8. 8 8 Dave B

    Steve,

    You say,

    ‘Inflation models tend to predict that our Universe is infinite (though of course we can never observe anything more than about 13.8 billion light years away).’

    I’ve seen the infinite extent of the universe stated or assumed in numerous popular science articles now and am puzzled as to how you would ever be able to experimentally distinguish between a universe that is actually infinite and one that is just very very big – say 10^10^10 light years across or bigger.

    Could there be any testable consequences that you know of?

  9. 9 9 Oona Houlihan

    Hi Dave B, as for the testable consequences – well first of all, a totally infinite universe poses the question if an infinite mass could have been “housed” in whatever exploded during that “Big Bang”. Is it realistic to assume that anything that ever unfolded into of what is thought to be the universe of which we can but see a part (or maybe not – and it is indeed the whole), is it possible (and can be modeled mathematically) that an infinite mass would not forever have been held together by its own gravity and never created a Big Band. (Of course if we think, according to the energy/matter equivalent) that this infinite mass was actually energy that later and partly condensed into mass, we might be onto something. But then there was no space before that as energy needs no space or does infinite energy and Planck’s quantum of action require a minimum space, even at the “very” beginning?

  10. 10 10 Roger

    The infinite universe theorists say that the big bang could have started with very small (not infinite) mass. There is no empirical evidence for an infinite universe, of course.

    Tegmark claims to no believe in physical infinities. I don’t know how he reconciles that with the various infinite multiverses he seems to believe in.

  11. 11 11 Wonks Anonymous

    “If statements about a thing are objectively either true or false, then that thing exists”
    What if the statement is “It does not exist”?

  12. 12 12 Steve Landsburg

    Wonks Anonymous: What if the statement is “It does not exist”? Bertrand Russell solved this one a long time ago. The statement that “The present king of France does not exist” is shorthand for “For every x, x is not the present king of France”. As such, it is a statement about you, me, and Barack Obama, but not a statement about the present king of France.

  13. 13 13 Mike H

    “we need to specify just the shortest computer program that can read in arbitrarily large numbers and add and multiply them.” But this is circular. How can you determine the length of a computer program without counting its lines?”

    Firstly, you don’t need the full power of the ring of integers to count things. Furthermore, we don’t need to count the lines of a program to determine that it’s shorter than another. We can just pair off the lines until one of the programs runs out of lines. This can be done by a finite state automaton.

  14. 14 14 Martin-2

    “though of course we can never observe anything more than about 13.8 billion light years away”

    Tegmark didn’t say this, did he?

    “If our Universe really is infinite, then it contains an infinite number of balls 14 billion light years in diameter”

    Now you’re mistaken on 2 levels.

  15. 15 15 Mike H

    @DaveB “Could there be any testable consequences that you know of?”

    Suppose we figure out that the chance of a universe being habitabe is 1 in a million. Yet, we observe a habitable universe. This is evidence that there exists O(10^6) or more universes.

    If the chances were 1 in 10^10^10^10, we could reject the idea that only 10^10^10 universes existed.

    I’m not sure how you’d ever deduce empirically that there were infinitely many universes, unless you could somehow show that the chance of life being possible was zero

  16. 16 16 Sam Roberts

    If statements about a thing are objectively either true or false, then that thing exists. Statements about (say) the natural numbers are objectively either true or false; for example, it is objectively true that every natural number is a sum of four squares. Therefore the natural numbers (and similarly other mathematical objects) exist.

    This is surely not the case. For example, define an oddeven number to be a natural number that is both odd and even. It is objectively true that every oddeven number is the sum of four squares since, a fortiori, every natural number is the sum of four squares. Nevertheless, no oddeven number exists.

  17. 17 17 Martin-2

    Sam Roberts (16): If you want to define an oddeven mathematical object then that’s your business, but you can’t just force it into the set of natural numbers. Then you’re talking about a different set.

  18. 18 18 srp

    “To exist is to have properties. If statements about a thing are objectively either true or false, then that thing exists.”

    Properties are necessary but not sufficient for physical existence. Captain Kirk was born in Iowa and had two eyes, but he is a fictional character who does not have physical existence. One of his properties is fictionality. And how about the luminiferous ether?

    It seems to me that you are begging the question by asserting that mathematical existence and physical existence are the same, or, equivalently, that the existence of a consistent description of something makes that thing real. I feel the ghosts of St. Anselm and Gaunilo (do they exist?) haunting this discussion.

  19. 19 19 Sam Roberts

    @Martin-2 I’m not at all “forcing” anything into the set of natural numbers – in fact I don’t know what that means. All I’ve done is to define an oddeven natural number as any natural number that is both odd and even.

    Of course those things are mutually exclusive – that’s clearly the point of my example – but the way I’ve made my definition is not in the least unusual, if that’s the content of your complaint. There are many definitions one could make that are not known at the time of definition to be satisfiable. Indeed, a friend of mine carried out an entire 3-year PhD on a set of numbers that was shown (crushingly to him) during his thesis examination to be empty.

    The fact remains that it is *objectively true* that any oddeven natural numbers are the sum of four squares, since any natural number is. Nevertheless, oddeven numbers do not exist.

  20. 20 20 Brian

    Steve,

    I am sympathetic to the claim that the universe is a mathematical object, though I can’t formally commit to such an idea because I’m not entirely sure exactly what you mean by that. But then you say

    “Look. To exist is to have properties. If statements about a thing are objectively either true or false, then that thing exists. Statements about (say) the natural numbers are objectively either true or false; for example, it is objectively true that every natural number is a sum of four squares.”

    Other posters have squawked over this and for good reason. Isn’t this claim susceptible to the same criticism often leveled at Anselm’s Ontological Proof for the existence of God? People have endlessly pointed out that having logically well-defined properties, even the property of existence, is not sufficient to guarantee that such an object ACTUALLY exists. And yet you appear to be making just such a claim.

    This claim seems to be obviously wrong. Sam Roberts’s counterargument presumably fails, of course, because the proposed object is self-contradictory (a number cannot be both even and odd), but there are lots of other, non-contradictory examples. Why can’t we derive all kinds of properties about fairies, for example, based on whatever we define them to be? But that doesn’t mean they exist.

    Ultimately, we see over and over again that structures in the universe are isomorphic to known mathematical structures. However unreasonable that correspondence might seem (ala Wigner), it likely indicates something profound. But does that mean that the universe is nothing but mathematics? I doubt it. Even more dubious is the claim that mathematically derived properties must exist. I am convincable on this point, but you’d have to lay out the argument rather carefully for me even to tak it seriously.

  21. 21 21 srp

    Brian: Thank you for noticing my point. Since it seems elementary I wonder why very smart people don’t seem to apprehend it which then makes me worry that I might be missing something obvious. Your concordance at least suggests that if this folly, I have company.

  22. 22 22 Steve Landsburg

    Brian:

    First, I am utterly baffled by your distinction between “existence” and “ACTUAL existence”. Does this heirarchy continue? Can a thing actually exist without actually actually existing? Et cetera.

    Next, as I said in an earlier comment, Bertrand Russell solved this problem long ago. The assertion that “the fairy who lives in my garden has wings” is shorthand for the assertion that “Something is a fairy and lives in my garden and has wings, and nothing else is a fairy and lives in my garden”. This (incidentally false) statement does not have as its subject “The fairy who lives in my garden”; it has as its subject “something”.

    By contrast, the assertion that “Every natural number is a sum of four squares” must be either false, or vacuously true (if there are no natural numbers) or non-vacuously true. I claim that this statement is neither false nor vacuous. If you grant that claim, then the natural numbers exist.

  23. 23 23 Keshav Srinivasan

    Steve, are you familiar with mathematical fictionalism? Under Russell’s theory of definite descriptions, both “Sherlock Holmes’ residence is on Baker Street” and “Sherlock Holmes’ residence is on Pennsylvania Avenue” are both false. Yet there’s a well-defined sense in which the first statement is right and the second statement is wrong.

    So how would you respond to the view that mathematical truths have the same status as truths about the fictional world of Sherlock Holmes?

  24. 24 24 Steve Landsburg

    Keshav Srinivisan:

    So how would you respond to the view that mathematical truths have the same status as truths about the fictional world of Sherlock Holmes?

    I would respond that they absolutely do not, insofar as the (allegedly) true statement “Sherlock Holmes’s residence is on Baker Street” is the result of a convention chosen by human beings, whereas the true statement “7 is a prime number” is independent of any human convention.

  25. 25 25 Brian

    Steve,

    The distinction is that one can imagine something existing, and the properties it would have, but that doesn’t mean those properties have been actualized. It doesn’t mean that the something, in fact, exists. I think you recognize this distinction by noting that something can be “vacuously true,” as opposed to being “non-vacuously true.”

    I’m not following the details of your second Bertrand Russell example, but for the first example you said

    “The statement that “The present king of France does not exist” is shorthand for “For every x, x is not the present king of France”. As such, it is a statement about you, me, and Barack Obama, but not a statement about the present king of France.”

    The statement “For every x, etc.” is indeed a statement about all of us. Another way to state it is that the position of King of France is not filled by any living person.

    But another way to interpret the original statement about the King of France is to say that such a POSITION no longer exists. That is, there is no legal or constitutional standing for anyone to claim such a position. This is a statement about the position, and not about any particular person.

    Let me note that you haven’t responded to my specific point (and srp’s oblique one) about the Ontological Proof for the existence of God. How exactly are you not falling into the same fallacy much criticized over the years by nearly everyone?

    Regarding the natural numbers, I agree that they exist AS A CONCEPT. That is, we conceptualize something we call the natural numbers. I also agree that many structures in the universe have properties isomorphic to the natural numbers. Beyond that, I’m not quite sure what it means to say that the natural numbers exist. The latter statement, if distinct from the preceeding possibilities, sounds a lot like some sort of Platonic view of the existence of pure idea. Is this your claim? In contrast, my language of potentiality and actuality come from a more Aristotelian point of view, if one were to translate these ideas into a more classical metaphysical language.

  26. 26 26 Steve Landsburg

    Brian:

    Let me note that you haven’t responded to my specific point (and srp’s oblique one) about the Ontological Proof for the existence of God. How exactly are you not falling into the same fallacy much criticized over the years by nearly everyone?

    Because existence is not a predicate.

    Beyond that, I’m not quite sure what it means to say that the natural numbers exist

    Are you quite sure what it means to say the planet Earth exists?

    To exist is to be within the range of a bound variable in a true sentence (or, less formally, to be the subject of a predicate, or still less formally, to have properties). To say that the number 13 is prime is to presuppose the existence of the number 13. To say that Earth is the third planet from the sun is to presuppose the existence of the Earth. The number 13, like the Earth, exists because it is the subject of at least one true sentence. (By contrast, despite superficial appearances, “God” is not the subject of the sentence “God exists”, because existence is not a predicate.)

    If you care to respond, I’d seriously like an answer to the question above: Are you quite sure that the planet Earth exists, and if so, why? I think you will have a hard time formulating an answer that doesn’t apply equally well to the natural numbers.

  27. 27 27 James

    Steve,

    Some problems:

    If you are correct to equate existence with having properties, you couldn’t know what you are referring to when you claim that something doesn’t exist.

    Also, membership in the set of items that Steve asserts not to exist is a property.

    How do you know that to exist is to have properties? As in: Is this an experimental result? Or is this a theorem you have proved from first principles? Or are you merely defining a term, in which case it would be nice if you would respect the fact that English speakers already use the same term to mean something a bit different.

  28. 28 28 Steve Landsburg

    James:

    If you are correct to equate existence with having properties, you couldn’t know what you are referring to when you claim that something doesn’t exist.

    Of course I do. When I say that fairies don’t exist, I mean that all things are non-fairies. I am referring to all things. If you’re going to object that I can’t possibly know what all things are, I agree, but I don’t see where that’s a problem.

    Also, membership in the set of items that Steve asserts not to exist is a property.

    Sure. And if anything has that property, then that is a thing regarding whose existence I am mistaken.

    How do you know that to exist is to have properties? As in: Is this an experimental result? Or is this a theorem you have proved from first principles? Or are you merely defining a term, in which case it would be nice if you would respect the fact that English speakers already use the same term to mean something a bit different.

    I am attempting to give a precise version of the everyday English meaning of the term.

  29. 29 29 srp

    Mathematical existence and physical existence can’t be the same thing because many mutually exclusive mathematical structures can be established that cannot all correspond to physical reality. Physical existence means corresponding to physical reality. False theories or models can easily “exist” mathematically.

    Physical existence is not about sentences at all–sentences may or may not accurately describe physical existence, and the sentences themselves exist physically as symbol strings in some medium, but what the sentences describe is not guaranteed physical existence.

    The Bertrand Russell argument appears irrelevant to the question of whether a given, self-consistent description refers to a physically existing object. There is no sense in which natural numbers are physical objects–they may or may not be descriptors of physical things, and they have their own internal logic in the same way that Sherlock Holmes’s or Captain Kirk’s fictional universe does. It so happens that people have spent a lot of time coming up with rules under which manipulating statements about the natural numbers are consistent, while they have not for the Sherlock-verse or the Kirk-verse, but in principle the latter could be axiomatized just as the former without conferring upon them physical existence.

  30. 30 30 Steve Landsburg

    srp:

    Mathematical existence and physical existence can’t be the same thing because many mutually exclusive mathematical structures can be established that cannot all correspond to physical reality. Physical existence means corresponding to physical reality. False theories or models can easily “exist” mathematically.

    1) I do not understand what you mean by “physical reality”. Do you mean “conforming to the laws of physics in the Universe we inhabit” or do you mean “conforming to the laws of physics in some Universe”?

    2) Of course mathematical existence and physical existence are not the same thing; I’m not sure why you need to point this out. The group with two elements is a mathematical structure, and surely exists, so I assume that it satisfies your criterion for “mathematical existence” but I’m equally sure that there is no reasonable sense in which it physically exists. I’ve argued in The Big Questions (have you read the book, by the way?) that a reasonable definition of “physical existence” is “sufficient complexity to contain self-aware substructures”. Some mathematical objects have that sort of complexity; most don’t.

    3) False theories or models can easily “exist” mathematically.

    I have no idea what you mean by a “false theory or model” that exists mathematically. A model (in the mathematical sense of the word) can no more be “true” or “false” than a rock can be “true” or “false”. Sentences (in which all bound variables have been interpreted) are the things that are either true or false.

    4) Physical existence is not about sentences at all–sentences may or may not accurately describe physical existence, and the sentences themselves exist physically as symbol strings in some medium, but what the sentences describe is not guaranteed physical existence.

    I honestly cannot make head nor tail of what your point is here. Nobody claims that physical existence is ‘about’ sentences. The claim is that things exist if sentences about them are objectively either true or false. Thus diamonds exist because it is objectively true that diamonds are made of carbon, and found in mines, and are frequently used to make jewelry. If these statements had no truth value, it would be because there is no such thing as diamonds. The reason they have truth values is that diamonds exist. So existence is not *about* sentences; instead, the objective truth of certain sentences is a criterion for *verifying* existence.

    But in any event, the objective-truth-of-sentences criterion is a criterion for *existence*, not for *physical existence*. So when you write “what the sentences describe is not guaranteed physical existence”, you are saying something obvious that nobody disagrees with.

    My claim is that if there are sentences about X that are *objectively* true, then X *exists*. *Physical* existence is another matter.

    (5) There is no sense in which natural numbers are physical objects

    Once again, you’re disputing something that nobody ever said. To say that some mathematical objects have physical existence is not to say that all mathematical objects have physical existence. An individual natural number is not the sort of structure that’s capable of having physical existence. A Riemannian manifold equipped with various fibre bundles and connections on those fiber bundles, on the other hand……well, the physics journals are full of papers that describe our Universe as just such a thing. Why shouldn’t we take them seriously?

  31. 31 31 Brian

    ” I’d seriously like an answer to the question above: Are you quite sure that the planet Earth exists, and if so, why?”

    Steve,

    Obviously, I am sure that Earth exists. Why? Let me put it this way. I observe that some part of my body is almost always in contact with a material we call the ground, or my body is in contact with something that is itself in contact with this “ground.” Whenever this contact ceases, it’s always very temporary. Everyone I know reports the same experience. We also notice that this “ground” appears to be continuous. I can shuffle my feet, always maintaining contact, and move from place to place without a gap in the ground. As a matter of convention, society has chosen to call the ground and all material in contact with it “Earth.” As long as this material is observed as being part of a continuous body, we say that Earth exists.

    I suppose I could call the test of existence the Missouri test. If someone claims the existence of something, I ask them to “show me.” If at least one person can show me the thing itself or a direct result of its action, the thing exists. Stomping one’s foot on the ground, for example, and pointing out the continuity of the stomp is sufficient to show that Earth exists.

    For the things whose objective existence I doubt, no one can pass the Missouri test. Fairies? No one can show me. Unicorns? No one can show me. Draw pictures? Yes. Talk about them? Yes. Make inferences about them? Yes. But actually SHOW me? No.

    And how about the natural numbers? How about the number 13 by itself? Can you SHOW it to me? Can you hold it in your hand or kick it? What actions result directly from it? Can I interact in any way with the number itself? I think the answer is no.

    Keep in mind that I fully understand that the Missouri test is rather primitive, and I am not proposing that it represents a necessary condition for something to exist. But I think it does the trick with things like Earth, which you specifically asked me about.

    In any case, my objection is to your claim that ” if there are sentences about X that are *objectively* true, then X *exists*.” This seems faulty, again, because we can make many objectively true statements about both real and imaginary things.

    -Fairies are small humanlike creatures with wings.
    -Dogs have four legs and fur.
    -Unicorns are horselike creatures with a horn on their foreheads.
    -God is all-powerful. etc.

    But since, as you correctly said, existence is not a predicate, true statements about a given thing, statements in which the thing is the subject, do not imply that the thing exists. Such statements only show that the thing can be conceived and discussed as a mental construct; it does not mean that it exists independent of our conception.

    What apsect of your point am I missing?

  32. 32 32 Kavanna

    I’ve heard Tegmark speak a few times, and he’s very good. I just wish these guys would stop dragging out the multiverse at every drop of a hat.

    Of course, what you do in the privacy of your own bathroom is your own business.

  33. 33 33 srp

    I hesitate to respond because I think Steve has conceded the point (though with a surprising amount of semantic avoidance) that physical and mathematical existence are not the same thing and that having properties is a necessary but not sufficient condition for physical existence (in our universe, which is what I thought was clearly what I meant). That disposes of the MUH as a plausible proposition, although of course unimaginable future observations might change its status. (To answer your question, I have not read The Big Questions)

    Despite the apparent clarity of “of course I’m not saying that natural numbers physically exist,” etc., Steve continues to ask these odd rhetorical questions about whether the Earth exists or what I mean by physical existence. Brian gives a nice pragmatic answer for the Earth’s physical existence, and 18th century empiricist philosophers expended a lot of ink in trying to relate everything to the senses, but I think it is clear that even without these arguments the common sense and meaning of the idea of physical existence is almost a primitive in any sensible system of thought. Even if you believe you’re living in a simulation, or a simulation of a simulation, there is still a “real” substrate in which that simulation runs.

    Ditto the odd dispute about whether a theory can be false. If theories can’t be false, then science is dead. Does the luminiferous ether physically exist or not? If yes, then the theory of the ether is correct, or true, or whatever lingo you prefer. If no, the theory is false. I’m pretty sure everybody is familiar with the correspondence theory of truth, so it seems strange to have to say stuff like this out loud.

    I get a hint of where this is all going as I contemplate the mutual dependence of the view that all mathematical structures (of sufficient “complexity”) physically exist somewhere or somewhen and the desire (denied but apparently present in the rhetoric) to equate mathematical and physical existence. If every mathematical fantasy I cook up corresponds to a physical realm outside my observable horizon, then it becomes psychologically easy to elide the distinction between my description of a universe and the actual physical thing.

    Finally, not being a physicist or casmologist I do not have strong opinions about which speculative mathematical schemes that possibly represent actual, physical unobserved or unobservable universes ought to be taken seriously. But I am not willing to concede that a mere collection of speculations very far from observable foundations is ipso facto likely to describe physical reality, any more than I would theological speculations about the nature of transcendental realms.

  34. 34 34 Steve Landsburg

    Brian:

    Obviously, I am sure that Earth exists. Why? Let me put it this way. I observe that some part of my body is almost always in contact with a material we call the ground, or my body is in contact with something that is itself in contact with this “ground.” Whenever this contact ceases, it’s always very temporary. Everyone I know reports the same experience. We also notice that this “ground” appears to be continuous. I can shuffle my feet, always maintaining contact, and move from place to place without a gap in the ground

    No, actually observe no such thing. All you observe are your own direct perceptions. You have constructed a theory that says those perceptions are caused by something called the ground. You’ve also noticed that unless you adopt that theory, you’re pretty much unable to engage in any sort of useful discourse about the world. So have I. So we’re both pretty well committed to the idea that the ground exists, because that’s the only way we can make sense of our perceptions.

    We also have mathematical perceptions, like, oh, say, the commutativity of addition. The only way I can see to make sense of this perception is that commutativity is a property of the natural numbers. Just as with the ground, I find it quite impossible to converse meaningfully about these things without admitting that they’re real.

    And how about the natural numbers? How about the number 13 by itself? Can you SHOW it to me? Can you hold it in your hand or kick it?

    Can you hold the electromagnetic field in your hand? Can you kick it?

    Some, but not all, things, can be held in the hand. Some, but not all, things, can be kicked. Some, but not all, things, have spatiotemporal extent. The natural numbers happen not to. So of course they’re not the sort of thing you can hold in your hand.

    Your argument seems to go like this: “I can think of lots of things I can hold in my hand. I can’t hold the natural numbers in my hand. Therefore the natural numbers don’t exist.”

    I could equally well argue that “I can think of lots of things that have textures. An electron has no texture. Therefore an electron doesn’t exist.” Or “I can think of lots of people who have favorite movies. My wife has no favorite movie. Therefore my wife is not a person.”

    You can’t argue something out of existence just by noting that there are some sorts of properties it doesn’t have.

    Keep in mind that I fully understand that the Missouri test is rather primitive, and I am not proposing that it represents a necessary condition for something to exist. But I think it does the trick with things like Earth, which you specifically asked me about.

    Okay, fine. This answers much of what I just finished typing.

    But here comes the main point:

    -Fairies are small humanlike creatures with wings.
    -Dogs have four legs and fur.

    The first of these statements is not objectively true; it is true only because we’ve all agreed to pretend it’s true. The second is objectively true. It is true whether or not we happen to believe it. (Assuming that “dogs” have already been well defined by some other property, e.g. their genetic code.) That’s precisely because fairies don’t exist and dogs do exist, and it’s a good way to tell that.

    Now: The number 13 is prime, not because we’ve all agreed to pretend it is, but because it is, independent of whether or not we believe it. So the number 13, like dogs, but not like fairies, exist.

    statements in which the thing is the subject, do not imply that the thing exists.

    I disagree with this. In order for it to be objectively true that fairies have wings, fairies would have to exist.

    What apsect of your point am I missing?

    I’m not sure, but I’m hoping I’ve addressed it now.

  35. 35 35 Steve Landsburg

    srp: I’m not sure who bears what percentage of the blame, but it’s crystal clear from your latest post that you haven’t the vaguest idea what I’m actually saying. I hope to find time to come back to this and take one more shot at explaining how off base you are.

  36. 36 36 Steve Landsburg

    srp: Okay, I have a little more time now:

    I think Steve has conceded the point (though with a surprising amount of semantic avoidance) that physical and mathematical existence are not the same thing

    I have no idea why you’d call this a concession, since it’s a major part of what I’ve been saying all along.

    That disposes of the MUH as a plausible proposition

    I have no idea why you’d say this. The MUH says, in essence, that physical existence is a special kind of mathematical existence. Your argument seems to be perfectly analogous to “Steve has conceded that being a mammal is not the same thing as being an animal. That disposes of the hypothesis that all mammals are animals as a plausible proposition”.

    Steve continues to ask these odd rhetorical questions about whether the Earth exists or what I mean by physical existence

    I ask these “odd questions” because we’re throwing around words and phrases like “existence” and “physical existence” and it’s generally a good idea to make sure the person you’re talking to understands what you mean.

    Ditto the odd dispute about whether a theory can be false.

    Now you’re (probably inadvertently) just pulling a fast one. Before you were talking about mathematical theories; now you’re talking about physical theories. A mathematical theory cannot be false, being a mathematical theory has no semantic content. It is just a set of symbols and a set of rules for manipulating them. (All this is standard first-course-in-logic fare.) Of course physical theories can be false, but this has nothing to do with your earlier mistaken assertion that a mathematical theory can be false.

    the view that all mathematical structures (of sufficient “complexity”) physically exist somewhere or somewhen

    This phrase above all tells me that we are totally talking past each other. “Somewhere and somewhen” presumably means “at some place and time”. Places and times refer to points in the physical universe we happen to live in. Mathematical objects aren’t part of this universe, so they don’t exist “somewhere and somewhen”. That doesn’t mean they don’t exist.

    As I said in my reply to Brian, lots of things have textures. It does not follow that everything exists has a texture. Lots of people have favorite movies. It does not follow that everyone who exists has a favorite movie. Lots of things have spatiotemporal locations. It does not follow that everything that exists has a spatiotemporal location. Mathematical objects, in particular, do not have spatiotemporal locations.

    Nevertheless, mathematical objects have *properties*, and those propoerties are objectively true. For example, it is objectively true that the number 13 is prime. It seems to me that “having properties that are objectively true or false” (or more precisely, “being the value of a bound variable”) does a pretty good job of capturing the everyday meaning of the word “existence”. The reason I asked what *you* mean by existence is that you appeared to be rejecting this definition, in which case you seem to be using the word in something other than what I take it to be its standard sense.

    If every mathematical fantasy I cook up corresponds to a physical realm outside my observable horizon, then it becomes psychologically easy to elide the distinction between my description of a universe and the actual physical thing.

    Once again, this tells me we must be totally talking past each other. The WHOLE POINT is that mathematical objects exist BECAUSE THEY HAVE OBJECTIVE PROPERTIES, which is exactly the opposite of “because somebody cooked them up”. Nobody “cooked up” the natural numbers. They were there all along.

    As far as “the distinction between my description of a universe and the actual physical thing……”, my entire goal is to maintain that distinction. Our description of the universe — the one that comes to us through our senses — is, we know, very unlike the actual physical thing. We describe the universe in terms of solid objects, which are in fact mostly empty space. We describe the constituents of those objects as elementary particles, which are in fact disturbances in quantum fields. The goal is to strip away all the illusions (what Tegmark calls “baggage”) and see what’s left. It’s very hard to see what, other than mathematics, can be left after you throw away all that baggage away.

  37. 37 37 James

    “Of course I do. When I say that fairies don’t exist, I mean that all things are non-fairies. I am referring to all things.”

    That doesn’t work. You can’t claim anything is a non-X unless you know what it means to be a X.

    “Sure. And if anything has that property, then that is a thing regarding whose existence I am mistaken.”

    Do you not see the problem? If anything has the property of belonging to the set of items you assert not to exist, then it has a property even while it does not exist.

    “I am attempting to give a precise version of the everyday English meaning of the term.”

    A description of a language is an empirical matter, right? Consider that you are mistaken. A number of English speakers have shown up to disagree. How many more English speakers would need to tell you that you are wrong for you to suppose that you are mistaken about the meaning of the term?

  38. 38 38 Steve Landsburg

    James:

    “Of course I do. When I say that fairies don’t exist, I mean that all things are non-fairies. I am referring to all things.”

    That doesn’t work. You can’t claim anything is a non-X unless you know what it means to be a X.

    You are confusing meaning with naming. I know what it means to be a fairy, which is why I am able to assert that all things are non-fairies. Here the property of non-fairiness is predicate, not a noun.

    This is all quite well worked-over material in the philosophy literature. Before repeating well-known and elementary mistakes, you might want to do a little reading.

  39. 39 39 Brian

    Steve (#34),

    I won’t reply to the first part of your response because I don’t disagree with what you’ve written, other than to note that I did say

    ” If at least one person can show me the thing itself or a direct result of its action, the thing exists.”

    Even though my existence test is admittedly primitive, it does include either that the thing itself can be seen or its effects can be seen. For the natural numbers, what effect can you point to that indicates they exist independent of our conception of them?

    With regard to your later comments, you say

    “The first of these statements is not objectively true; it is true only because we’ve all agreed to pretend it’s true. The second is objectively true. It is true whether or not we happen to believe it. ”

    Part of what’s happening here is a difference in how we define objectively true. For me, objectivity can only be defined operationally. Objectivity to me means that large numbers of people acting independently and seriously following the evidence or the logic agree on something. If they all agree that its true, then its objectively true. So far as I can see, there’s no way to ascertain objective truth without this approach. But, if I am not misreading you, seem to take the definition of “objectively true” as including existence.

    Let me say it this way. The statement that “Fairies are small humanlike creatures with wings” is objectively true because everyone agrees that that’s what is meant by the term fairy. The statement “Fairies have fins and no wings” is objectively false, not because fairies don’t exist but because the statement violates the definition of what we mean by fairy. Every child could tell you “No, you mean mermaids!”

    Your definition of objectively true does not distinguish those two statements, but dismisses them both as being objectively false just because neither fairies nor mermaids exist. I don’t find this approach helpful.

    Let me express it yet another way. The statement “Fairies are small humanlike creatures with wings” is really a shorthand for saying “If fairies exist, they are small humanlike creatures with wings.” Everyone would agree that if the antecedent is true, then the consequent is true (because it properly expresses the definition of fairy). Since everyone agrees, the conditional statement is objectively true. You seem to want to argue that if A is false, then B is “not objectively true.” Of course, if A is false, nothing at all can be said about B by itself.

    Ultimately, I am insisting that both objectively true statements (in the operational sense) and objectively false ones can be made about pure conceptions, regardless of whether those conceptions exist independently of the conceiver. This is what happens with any proposed theory, whether it ends up being physically validated or not.

  40. 40 40 srp

    “The MUH says, in essence, that physical existence is a special kind of mathematical existence.”

    The controversial claim is not “If it physically exists then it mathematically exists.” (I would dispute this claim but on other grounds.) The thing people are exercised about is the claim “If a mathematical system has certain formal properties P then it physically exists.” It doesn’t make sense to go from the internal consistency implications of a set of axioms and definitions to the conclusion that the system physically exists. I believe that is the core proposition in dispute, not what Steve has stated above.

    “Your argument seems to be perfectly analogous to “Steve has conceded that being a mammal is not the same thing as being an animal. That disposes of the hypothesis that all mammals are animals as a plausible proposition”.”

    No. You are claiming that all animals are mammals. I do not dispute that any physically existing thing can be described mathematically and that the description mathematically exists. I do dispute that any mathematical system (of appropriate properties as stipulated by Tegmark) necessarily describes a physically existing object.

    “Now you’re (probably inadvertently) just pulling a fast one. Before you were talking about mathematical theories; now you’re talking about physical theories. A mathematical theory cannot be false, being a mathematical theory has no semantic content. It is just a set of symbols and a set of rules for manipulating them. (All this is standard first-course-in-logic fare.) Of course physical theories can be false, but this has nothing to do with your earlier mistaken assertion that a mathematical theory can be false.”

    At no time did I suggest that a mathematical theory can be false. I said and maintain that a mathematically consistent structure (e.g., the ether) may not correspond to any physical reality and so the physical theory “this structure describes something that physically exists” is incorrect. Not sure why this wasn’t clear. If you agree that a physical theory can “of course” be false then how can you support the idea that any consistent mathematical structure physically exists? Mathematical structures can be as fictional as fairy tales.

    “This phrase above all tells me that we are totally talking past each other. “Somewhere and somewhen” presumably means “at some place and time”. Places and times refer to points in the physical universe we happen to live in. Mathematical objects aren’t part of this universe, so they don’t exist “somewhere and somewhen”. That doesn’t mean they don’t exist.”

    Hey, I’m not the one pushing the multiverse; Tegmark is. You’re making the same error again of substituting mathematical existence for physical existence in my statements. I’m not engaging in a dispute about the foundations of math. Platonism is a respectable position. But that’s not Tegmark’s claim–his claim is about the physical existence somewhere and somewhen of anything a mathematician can cook up. It’s not impossible for that to be true but I don’t see any reason to believe it, any more than there is warrant for thinking other fictional worlds physically exist somewhere or somewhen.

    “As I said in my reply to Brian, lots of things have textures. It does not follow that everything exists has a texture. Lots of people have favorite movies. It does not follow that everyone who exists has a favorite movie. Lots of things have spatiotemporal locations. It does not follow that everything that exists has a spatiotemporal location. Mathematical objects, in particular, do not have spatiotemporal locations.

    Nevertheless, mathematical objects have *properties*, and those propoerties are objectively true. For example, it is objectively true that the number 13 is prime. It seems to me that “having properties that are objectively true or false” (or more precisely, “being the value of a bound variable”) does a pretty good job of capturing the everyday meaning of the word “existence”. The reason I asked what *you* mean by existence is that you appeared to be rejecting this definition, in which case you seem to be using the word in something other than what I take it to be its standard sense.”

    I am baffled. I made it clear that I was disputing the physical, not the mathematical, existence of structures like the ether. It turns out that you respect this distinction between physical and mathematical existence, with the former in your view being a subset of the latter. So what’s the problem? If something physically exists then it has a spatiotemporal location, unless I’m missing some tricky exception. So when Tegmark says that these mathematical structures must physically exist in some multiverse precinct, that is not the modest Platonist claim about the foundations of math that you are now belaboring. It is instead a much more far-out idea that if I can describe X consistently (mathematically) then X must physically exist.

    “The WHOLE POINT is that mathematical objects exist BECAUSE THEY HAVE OBJECTIVE PROPERTIES, which is exactly the opposite of “because somebody cooked them up”. Nobody “cooked up” the natural numbers. They were there all along.

    As far as “the distinction between my description of a universe and the actual physical thing……”, my entire goal is to maintain that distinction. Our description of the universe — the one that comes to us through our senses — is, we know, very unlike the actual physical thing. We describe the universe in terms of solid objects, which are in fact mostly empty space. We describe the constituents of those objects as elementary particles, which are in fact disturbances in quantum fields. The goal is to strip away all the illusions (what Tegmark calls “baggage”) and see what’s left. It’s very hard to see what, other than mathematics, can be left after you throw away all that baggage away.”

    By description in that context, I meant mathematical description, not common language description, of a purported physical entity. Tegmark says, I believe, that if we find a mathematical structure then a)it uniquely maps to a particular set of observations and b) those observations would actually be seen somewhere and somewhen. (If you don’t like non-Platonist foundations of math, fine, substitute “find” above for “cooked up.” It changes nothing.) I think you are wrong about a), but I’ll stipulate it for argument’s sake. What I’ve been decrying from the beginning is b).

    I ask you plainly, do you believe that out there in the multiverse there physically exists a realm where the luminiferous ether is the correct description of the behavior of light? If not, isn’t that a rejection of Tegmark’s claim?

  41. 41 41 Steve Landsburg

    srp: It will take me a while to work through your long post but I can’t resist singling this out for special mention:

    But that’s not Tegmark’s claim–his claim is about the physical existence somewhere and somewhen of anything a mathematician can cook up.

    This is entirely entirely wrong. If you believe this, then you don’t understand what the whole discussion is about.

  42. 42 42 Steve Landsburg

    srp: Having read your post a bit more carefully now, it’s crystal clear to me that you really don’t understand the position you believe yourself to be criticizing. Since much of what you’re refuting has almost nothing to do with anything that either I or (as far as I know) Max Tegmark has ever said, it’s too far off topic to merit a response.

    There is evidence of your confusion throughout your post, but I’ll confine myself to one statement that pretty much sums up the depth of your misunderstandings:

    But that’s not Tegmark’s claim–his claim is about the physical existence somewhere and somewhen of anything a mathematician can cook up.

    There are at least three major misapprehensions here.

    First, the MUH is emphatically NOT about things that human beings can cook up. It’s about mathematical objects that exist in their own right, without any human intervention. That’s a critically important piece of the story, and if you’ve missed it, then you’ve necessarily missed the whole story.

    Next, nobody — certainly not I and certainly not Max Tegmark — has attributed physical existence to arbitrary mathematical objects. My own preference (I’m not sure whether Tegmark would agree) is to *define* a structure to “physically exist” if it contains self-aware substructures. That excludes almost every mathematical object that we know about.

    And next, it is *absolutely* contrary to the spirit and the meaning of the MUH — (and contrary as well, in my opinion, to anything any sane person could possibly believe) to say that mathematical objects exist “somewhere and somewhen”. The Riemann Zeta function does not exist anywhere or anywhen. Neither does any other mathematical object that I know of. Spatiotemporal locations are not properties of mathematical objects. I don’t know of anyone who believes otherwise, so it seems a little bizarre for you to spend so much space contesting a position that nobody has ever taken.

    Similar remarks apply to the rest of your post. You might ultimately have something to say, but because you are totally oblivious to what the rest of us are saying, you’re putting a lot of energy into the unnecessary task of talking us out of things we never believed in the first place.

  43. 43 43 Steve Landsburg

    srp: I’ll at least answer your final question:

    I ask you plainly, do you believe that out there in the multiverse there physically exists a realm where the luminiferous ether is the correct description of the behavior of light?

    I have no idea what you mean here by “THE multiverse”, so I’ll skip over that part and edit your question down to “Do you believe that there physically exists a universe where the luminiferous ether is the correct description of the behavior of light?”. The answer is of course: How the hell should I know? This sounds like a really hard math problem, one that could be answered in principle but is probably intractably hard in practice.

    Here’s a far far simpler question: Do you believe there’s a finite group that is not the Galois group of any finite extension of the rational numbers? (Or, if you prefer, do you believe there’s an even number that is not the sum of two primes?) That’s perfectly analogous to your question, in that a) it asks whether or not there is a structure with certain properties, b) the question is far too hard for anyone to answer right now, and c) the right answer is in principle discoverable. Math is full of questions like that. Asking the questions is easy. Asking people what they expect the answers to be, absent any actual argument, is, well, stupid.

  44. 44 44 srp

    Wow. Have you noticed the pattern in this exchange? I ask about physical existence and you come back with a response about mathematical existence, even though you agree that these are not the same thing.

    “I have no idea what you mean here by “THE multiverse”, so I’ll skip over that part and edit your question down to “Do you believe that there physically exists a universe where the luminiferous ether is the correct description of the behavior of light?”. The answer is of course: How the hell should I know? This sounds like a really hard math problem, one that could be answered in principle but is probably intractably hard in practice.”

    Here’s what’s odd about your response. This is not an open mathematical question. The classical physicists worked out an internally consistent theory of the ether and its purported properties. So we know it exists mathematically. They were pretty sure back then that it also described our physical universe, even though some of the properties it needed to have seemed a bit far-out. The question of the physical existence of the ether in OUR universe is not a mathematical question–it is an empirical one that was (provisionally, if you are a strong agnostic) settled by a series of experiments that appeared to disprove it. It was an unwittingly fictional though internally consistent story about how the universe works. It fit many of the things we could observe, but not others and so was ultimately rejected as a physical description of what is. But there was no mathematical flaw in it–you could generate axioms within which it was perfectly self-consistent and so existed mathematically. Those axioms turned out to be not good enough as a model of the physical world but the structure derived from them is perfectly fine as math.

    I don’t wish to be exasperatingly obtuse or waste your time, so if you still feel that I’m being stupid despite the clarity of your writing please feel free to ignore me or just put in a terminating last word. But I feel fairly certain that anyone reading this thread all the way down, especially one who has some idea of what Tegmark has been saying in interviews and blog comments, is going to be more confused by your remarks than by mine.

  45. 45 45 Steve Landsburg

    srp: I don’t wish to be exasperatingly obtuse or waste your time

    I admit I was starting to feel a little exasperated, but your latest post convinces me that I was being a bit unfair. So I’m happy to spend more time on this.

    This is not an open mathematical question. The classical physicists worked out an internally consistent theory of the ether and its purported properties. So we know it exists mathematically.

    Ah, but the theory they worked out was never intended to be a complete model of the universe we live in (it did not include, for example, my great-great-great-great-grandfather’s hypothalamus). It was (like all physical theories written down by humans so far) a *toy* model intended to have important features in common with this universe.

    Is there in fact a mathematical structure that *both* incorporates something we’d want to call a luminiferous ether *and* contains self-aware creatures? I have no idea, and neither did the classical physicists.

    I remind you that my *definition* of physical existence is “containing self-aware subobjects”. (Tegmark might not approve of this usage; I’m not sure.) Nobody has ever written down a model detailed enough to include the ether and to physically exist by that definition. So I am necessarily agnostic as to whether such a model exists.

    The question of the physical existence of the ether in OUR universe is not a mathematical question–it is an empirical one that was (provisionally, if you are a strong agnostic) settled by a series of experiments that appeared to disprove it.

    Of course I totally agree with this.

    But there was no mathematical flaw in it–you could generate axioms within which it was perfectly self-consistent and so existed mathematically.

    Yes. So I am quite happy to say that there is a universe with a luminiferous ether. Whether there is a universe with a luminiferous ether *and* physical
    existence (by my definition) is, however, still an open question.

  46. 46 46 Ken B

    Steve,
    Does the universe contain all mathematical objects?
    What about inconsistent theories, or incompatible theories?

  47. 47 47 Steve Landsburg

    Ken B:

    Does the universe contain all mathematical objects?

    Surely not. “All mathematical objects” do not even form a set. (Also, I’m not sure what you mean by “contain”. Do you mean as elements or do you mean as subobjects?)

    What about inconsistent theories, or incompatible theories?

    a) I don’t understand what the question is.

    b) What does “What about” mean in this context?

    c) What is an “incompatible theory”?

  48. 48 48 srp

    Given that most neuroscientists believe that the macro-phenomena of the brain are classical rather than quantum in their foundation, I don’t see any reason why a classical universe wouldn’t support self-awareness, even though I can see room to doubt this current consensus. But we are obviously in the murk here because of our difficulties with understanding self-awareness, the mind-body problem, intentionality, and the rest.

    This state of affairs leaves me with a zero comfort level in using the existence of self-aware beings as any kind of criterion for assessing fundamental physics theories. If nothing else, I would need a proof or persuasive argument that self-awareness is not a higher-level property capable of existing across many, most, or all self-consistent reductionist theories. (Given your orientation, I presume you’re familiar with Greg Egan’s Permutation City and the Dust Theory, which dramatically addresses some of these issues.) More fundamentally I’d need to know just what the heck the formal boundaries of a definition of self-awareness are. Some would argue that I’m not self-aware (and I guess there are days where that might be a plausible claim) or that our perception of self-awareness is all an illusion, etc.

    It seems to me that Tegmark has other criteria in mind for his MUH–he talks about computability in some blog comments I’ve seen–but isn’t his exact set of criteria in the book you reviewed?

    I’m guessing that what YOU are saying is that any mathematical structure that “contains” or “is consistent with” self-aware beings must physically exist because self-awareness is a purely informational or mathematical thing, so that a description of self-awareness, which is itself information, is the same thing as the physical existence of self-awareness. In other words, self-awareness can’t be simulated because any attempted simulation of it would be physically self-aware itself.

    I’m skeptical of this view because I’d bet that self-awareness involves substantial irreversible energetic processes, not just information processing (and not just the minimal energy needed to do information processing). In other words, self-awareness is likely to look like metabolism or reproduction, inextricable from material substrates and therefore capable of being simulated by formal descriptions that are not themselves self-aware. Hence, mathematical descriptions of universes with self-aware beings need not physically exist.

  49. 49 49 Steve Landsburg

    srp:

    I’m guessing that what YOU are saying is that any mathematical structure that “contains” or “is consistent with” self-aware beings must physically exist because self-awareness is a purely informational or mathematical thing, so that a description of self-awareness, which is itself information, is the same thing as the physical existence of self-awareness.

    No, I am not saying that.

    I am saying that:

    a) Most people who throw around the phrase “physical existence” never bother to define it, and therefore most of what they say is meaningless.

    b) We are free to define physical existence any way we want, but it’s a good idea to define it in a way that conforms to our intuitions.

    c) I believe that “containing self-aware beings” is pretty darn close to our intuitive notion of physical existence. The group with three elements contains no self-aware beings, and nobody would claim it physically exists. The universe we inhabit contains self-aware beings, and we all agree that it physically exists, etc.

    d) Therefore I adopt the existence of self-aware beings as a *definition* of physical existence.

    e) Because all I’ve done here is to adopt a definition, I have not made any claim with any content. When I say that “I define a mathematical object to have the property of physical existence if and only if it contains self-aware beings”, it’s like saying “I define a number to be perfect if it is equal to the sum of its divisors”. That’s not a statement you can argue with. You can, if you like, argue that this is a *poor* definition because it’s misleading or because some other word would work better, but you can’t argue that it’s a *wrong* definition, because definitions, by their nature, cannot be wrong.

    f) Your paraphrase of (your understanding of) my position seems to have actual content. My position (on this limited matter of how I’m going to use the phrase “physical existence”) has no content whatsoever. Therefore your paraphrase cannot be accurate.

  50. 50 50 srp

    Steve, thanks for the clarification. I thought you were going into the whole Dust Theory zone, but you’re not.

    It seems like your definitions of physical existence and whether particular theoretical structures physically exist requires a kind of holistic assessment. If I want to know if a particular theoretical entity–say, the ether–exists, then I would normally first try to test that through experiment without worrying if I had a complete theory about how the brain works.

    I don’t see why a theoretical structure X could not be a) logically consistent with supporting self-aware beings while b) actually, in our universe as determined by observation and experiment, not physically existing. In that case, X’s consistency with self-aware beings would be a necessary but not sufficient condition for X’s physical existence. And then it would NOT follow that every consistent mathematical structure X describing a world that supports self-aware beings would also physically exist.

  51. 51 51 Steve Landsburg

    srp:

    The only reason (as far as I can see) that we’re getting different conclusions is that we’re using different definitions. I define physical existence by the presence of self-aware subobjects. You seem to use the phrase “physical existence” only for substructures of the particular universe we happen to inhabit.

    By my definition, it is an open question whether there is a physically existing universe with something we’d recognize as a luminiferous ether. By your definition, the luminiferous ether does not exist. I am quite sure we’re both right.

  52. 52 52 Ken B

    Steve
    I am wondering how you avoid Russell like paradoxes for a start. I didn’t assert it was a set. But does the universe contain every set?
    I am using contain in whatever sense you use it. The universe surely contains things, either as elements or some other way. So you must have a mathematical correlate mustn’t you? In that sense, whatever it may be.
    I might be asking if your notion of contains is well defined.

    I should have been clearer. Mutually incompatible theories. ZFC, ZFM where M is an axiom inconsistent with C.

  53. 53 53 Steve Landsburg

    Ken B: Either I don’t understand what you’re asking, or I don’t understand why you’re asking it.

    Does the group of invertible three by three matrices contain every set? Does the euclidean plane contain every set? Does the Fermat hypersurface of degree 7 contain every set? Does the empty set contain every set?

    All of these questions seem to me to be exactly as well-motivated as the question “Does the universe contain every set?”. If someone asked me one of these questions, I’d be torn between saying “Ummm….obviously not” and “Your question makes so little sense that I can’t believe I’m understanding it correctly.” Ditto for your question.

  54. 54 54 Ken B

    Steve,
    I thought you were arguing the universe was a mathematical object, or a collection of mathematical objects. Is that not so? I am not sure what “the universe is made out of math” could mean otherwise. So rather than chasing what may prove to be a wild goose, let me seek clarification on that point. Is the universe a mathematical object or a collection of mathematical objects?

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