In The Big Questions, I argued that math is all there is: The Universe we live in is a mathematical object and is no more or less “real” than any other mathematical object. Thus, for example, the Godel universe, where time moves in circles, so that everything eventually returns to the time and place where it started, is as real as our own — though far, far, less complicated, because it contains, for example, no sentient beings). (Though on the other hand, it’s entirely plausible that there exists a Godel-like Universe that does contain sentient beings, and the existence of such a Universe can, in principle, be settled by purely mathematical inquiry.)
Obviously, I can’t prove this, but I’ve tried to explain why it strikes me as far more plausible than any of the alternatives. It all comes down to Ockham’s Razor. I know these mathematical Universes exist (pick up any issue of any theoretical physics journal and chances are you’ll find a couple described in detail), and it seems ontologically extravagant to suggest that some enjoy a different kind of existence than others. In other words, the notion of “physical reality” is exactly the sort of unnecessary baggage that Ockham’s razor wants to cut away.
People do seem to want to believe that the Universe we inhabit is somehow “special”, which is why I believe they’ve invented the unnecessary concept of “physical reality” to distinguish it from all the others. But the history of science has not been kind to viewpoints that cast human habitats as special. People used to think that the earth occupied a special place in the Universe; Copernicus (crying “Give up your Ptolemy! Rise up and follow me!”) rejected that notion in what can be seen as a slick application of Ockham’s Razor. Nowadays, people are tempted to think that the Universe we occupy has a special status in the zoo of mathematical Universes; but as good Ockhamized Copernicans, we should resist that temptation.
The reason I bring all this up right now is that MIT Professor Scott Aaronson recently had a truly fabulous blog post where he applied the Ockham/Copernicus analogy not (as I have) to the mathematial Universe but to the many-worlds interpretation of quantum mechanics, according to which everything that can happen (according to the laws of quantum mechanics) does happen. A cat sits in a chamber where an atom might or might not emit a radioactive particle that triggers the release of a cat-killing poison. Does the cat live or die? According to the many-worlds interpretation (speaking non-technically and hence a bit imprecisely, but well within the spirit of things), the Universe splits in two, one with a live cat and one with a dead cat. And you, dear reader, split in two along with the Universe, one of you finding the cat dead and the other finding it alive.
(It’s important to recognize that in the view I’m espousing, the entire ensemble predicated by the many-worlds view is just one of the vast array of mathematical universes out there.)
What’s great about Scott’s post is that he makes an extremely persuasive case for accepting the many-worlds interpretation on Ockham/Copernicus grounds — and then criticizes and ultimately rejects his own argument. I find both his original argument and his criticism-and-rejection extremely compelling. But here’s the thing: When I copy his arguments over, replacing many-worlds with the mathematical Universe, it seems to me that his original argument survives, but his criticism-and-rejection no longer makes sense.
So I think that Scott’s blog post, while he intended no such thing, gives an excellent argument for throwing the notion of “physical reality” onto the ash heap of history.
Here, according to Scott, is one reason why the Copernicus analogy does not work well for many-worlds:
The second way I’d say the MWI/Copernicus analogy breaks down arises from a closer examination of one of the MWIers’ favorite notions: that of “parochial-ness.” Why, exactly, do people say that putting the earth at the center of creation is “parochial”—given that relativity assures us that we can put it there, if we want, with perfect mathematical consistency? I think the answer is: because once you understand the Copernican system, it’s obvious that the only thing that could possibly make it natural to place the earth at the center, is the accident of happening to live on the earth. If you could fly a spaceship far above the plane of the solar system, and watch the tiny earth circling the sun alongside Mercury, Venus, and the sun’s other tiny satellites, the geocentric theory would seem as arbitrary to you as holding Cheez-Its to be the sole aim and purpose of human civilization. Now, as a practical matter, you’ll probably never fly that spaceship beyond the solar system. But that’s irrelevant: firstly, because you can very easily imagine flying the spaceship, and secondly, because there’s no in-principle obstacle to your descendants doing it for real.
Now let’s compare to the situation with MWI. Consider the belief that “our” universe is more real than all the other MWI branches. If you want to describe that belief as “parochial,” then from which standpoint is it parochial? The standpoint of some hypothetical godlike being who sees the entire wavefunction of the universe? The problem is that, unlike with my solar system story, it’s not at all obvious that such an observer can even exist, or that the concept of such an observer makes sense. You can’t “look in on the multiverse from the outside” in the same way you can look in on the solar system from the outside, without violating the quantum-mechanical linearity on which the multiverse picture depends in the first place.
Here’s how I replied in comments:
1) Just as Ockham’s razor, properly interpreted, argues for dispensing with measurement as an unanalyzed primitive, it also (it seems to me) argues for dispensing with physical reality (as something distinct from mathematical reality) as an unanalyzed primitive. On this view, the many branches of MWI are just a tiny tip of the iceberg of reality. Mathematical structures exist; some mathematical strucures are sufficiently complex in the right sorts of ways that they contain self-conscious substructures; our Universe, with its many branches, is one of many such structures. To single out our universe from all the other mathematical structures, and to declare it uniquely “real”, strikes me as (to adapt your words) an unmotivated perversity that mangles the simplicity of mathematics for no better reason than a parochial urge to place our own experiences at the center of reality.
2) One might adopt your anti-MWI arguments for service here, asking, e.g. “Who is the observer who in principle can take a birds-eye view and see that our own many-branched universe occupies no particular privileged place among the many other mathematical universes that we know it’s possible to describe? The answer, of course, is the mathematical physicist, who has a lifetime of experience writing down different models of the universe and therefore sees something of the entire landscape. It’s true that all of the math-physicists’ models are far too primitive to include anything like self-conscious beings, but it’s also true that your theoretical Copernican observer flies too far above the earth to see the details. Our math physicist, like your Copernican observer, glimpses a highly undetailed picture of something that is nevertheless undoubtedly there. And there is absolutely nothing in the mathematics to suggest that, e.g. the Einstein-de-Sitter universe is any more or less “real” (whatever that means) than the Godel Universe or any of a hundred other universes that we see in the literature, many of which look like extremely rough sketches of the universe we happen to inhabit.
Bottom line: Ockham’s razor tells me not to add unnecessary primitives, especially (as you point out) when it’s easy to imagine an observer to whom those unnecessary primitives would seem entirely arbitrary. That’s an argument for the reality of the MWI branches, but it’s an even better argument for ditching the unnecessary primitive of physical reality, because in that case we know who the key observer is. And this has the nice side effect of relieving us from having to ask why the Universe exists in the first place. Mathematical objects exist; the Universe is a mathematical object; therefore the Universe exists. Of course this leaves the question of why mathematical objects exist, but I think I’ll stop here for now.
I’ll give Scott the last word; he replied to my comment here.
I like the term “mathematical everythingism”. Beyond that requires more thought.
I have this recurring thought which makes me reluctant to embrace mathematical Platonism (or any particular philosophy of mathematics). Though I can’t really fault Prof. Landsburg’s arguments, I am skeptical about their starting-point, the basic insight which drives them.
He comes to the broader question of why the universe exists and the question of the nature and status of maths as someone for whom mathematics just feels (correct me if I’m wrong) more real than anything else. His mind (like the minds of many mathematicians) is predisposed to see the world in a particular way, just as others are predisposed to a more empirical or physicalist way of looking at things. We are all biased in other words, and build our arguments on the (dubious) foundation of these ‘intuitions’.
My perspective may be deemed to be unnecessarily pessimistic but it is at least Copernican in spirit.
I have this recurring thought which makes me reluctant to embrace mathematical Platonism (or any particular philosophy of mathematics). Though I can’t really fault Prof. Landsburg’s arguments, I am skeptical about their starting-point, the basic insight which drives them.
He comes to the broader question of why the universe exists and the question of the nature and status of maths as someone for whom mathematics just feels (correct me if I’m wrong) more real than anything else. His mind (like the minds of many mathematicians) is predisposed to see the world in a particular way, just as others are predisposed to a more empirical or physicalist way of looking at things. We are all biased in other words, and build our arguments on the (dubious) foundation of these ‘intuitions’.
My perspective may be deemed to be unnecessarily pessimistic but it is at least Copernican in spirit.
I am struggling with the difference between your mathematical theorist and a quantum theorist in the MWI. If we forget entire universes for a moment, and think of a simple event; measuring a photon through a double slit. We can work out the wave-function and calculate the probability of the each slit being passed is 50%. Once we make the measurement, we find it actually passes through one of the slits. Is there anything special about the world we inhabit that experiences the photon passing through that slit? We can subscribe to the MW theory, and say we just happen to be in the world where this happened, and the other is equally real, or we can think the photon actually “did” pass through the one slit rather than the other. Scott cannot find the viewpoint from which we can observe the landscape and determine that “our” universe is not special (if I have it right, which I am not sure of).
In your model, the observer is the mathematical physicist, who can see something of the whole landscape. But in the MWI, the quantum physicist has just as good a view of the landscape of the two choices, and can equally declare neither to be special.
It would still seem that the mathematical universe is preferred under Occam’s razor, but if Scotts point is valid, does it not remain so for the mathematical universe?
I feel like the problem with this hypothesis is that one has to be aware of it in order for it to be true. Otherwise, we’re talking about aleph^aleph universes here. It can’t simply be true that “anything that can happen does happen,” it must necessarily be true that “anything that WILL EVER BE A POSSIBILITY must ALREADY EXIST.”
Why? Because I can think of a virtually infinite number of things I could do after writing this blog comment. Each of those things is merely the initiation of a sequence of events, and each step in each of those sequences is another node at which another infinite number of universes is created.
It literally is an infinite set of infinite sets. Aleph^aleph.
Not only does Shroedinger’s Cat live AND die, it also exists in a multitude of universes in which it was never placed in a box, and placed in the wrong box, and ran away before the box was procured in the first place, and, and, and…
I can’t disprove this theory but it strikes me as being so absurd that it’s difficult to believe. But don’t worry – in some other universe I subscribe to it whole-heartedly.
I struggle with the premise of MWI that each time an action occurs, a branch is created in the universe. Does this imply that if a “cat” dies or does not die on some planet in a far away galaxy, that my existence branches? That makes no sense to me, given that any information passed from the death of the cat to me would take thousands or millions of years to reach me. As a next logical step, it would seem reasonable to think that each branch propagates at the speed of light, but ultimately dissipates as it spreads across the universe. And yet this implies that MWI is somehow local, which is equally counter-intuitive.
Thus, it seems to me more reasonable to imagine that there are simply an infinite number of universes, one where every possibility exists, but there is no branching occurring, no connection between the universes.
Interesting hypothesis Steve. I can conceive that the line of enquiry might be justified; given that the universe is only a relatively tiny mathematical object in comparison to the scope of entirety of mathematical objects as a whole, there is reasonable justification in accepting the many-worlds theory that all possible alternative histories and futures have some kind of existence parallel to our own. If mathematical objects can be said to exist, then the infinite alternative possible realities can be said to exist within that much larger mathematical object, and not all of them will be physical. So, physical-only = parochialism. That’s one theory.
But it seems we might be taking an epistemological bridge too far, because I don’t see that just because mathematics contains infinitely more potential configurations than our own universe’s configurations, we can then say that that ‘mathematics’ can create universes (physical or not). Numbers exist outside of our physical universe, but what does it mean to be a number, and how can numbers or mathematics create, or be responsible for, physical universes? As far as human being are concerned, we may only know mathematics because we have minds that can conceive of formal tokens (symbols) that are ideal for expressions of the reality we perceive. I don’t know what it even means to talk of mathematics being able to create or facilitate or design or bootstrap a universe, and I don’t see how anyone else can explain this either. Any suggestions?
Steve, let me preface this by saying that I’m a Platonist (albeit a weaker one than you), but I have a few issues with your reasoning:
1. Why stop at mathematical universes? Can’t we equally well say that ALL conceivable (and perhaps inconceivable) universes, including the Lord of the Ring universe, exist? Just as the mathematical physicist can survey all the solutions to the Einstein equations, can’t any human survey all the universes he has thought of? What is the special ontological status of mathematical objects that distinguishes them from such things? As I commented in one of uour previous posts, I don’t think Godel’s completeness theorem suffices as an answer, because it just reduces the question of whether models exist for consistent theories to the question of whether sets exist, so the existence of mathematical objects is left unresolved.
2. You believe that the physical universe is a mathematical object because its behavior seems to be describable by laws that can be written in the language of mathematics. As such, that is no more noteworthy than saying that the Universe can be described by statements in the English language; presumably the reason you think it is important is that you think mathematics is indispensable to understanding the physical world. If so, are you familiar with Hartry Field’s work on mathematical fictionalism? He’s tried to express all the known laws of physics using no mathematics whatsoever, not even real or natural numbers! You can read his book Science without Numbers. He seeks to show that mathematics is just a useful, but unnecessary, fiction, and he believes that mathematical truth has the same status as other fictional truth; just like “Sherlock Holmes lived on Baker street” is a meaningful in the fictional works of Arthur Conan Doyle, Field wants to say that the Pythagorean theorem is nothing more than a meaningful statement in the fiction of mathematics.
Just to help us get at what you mean, and give a point of reference for discussion, can you give an example of one of the simplest real universes you can think of (or at least a very simple universe)? Talk of Godel universes is all very well, but I can’t really intuitively grasp it.
Keshav,
Very interesting points and I tend to agree. To add, I guess what I’m struggling with is that even though what we would describe as our physical universe may indeed, at least someday, be perfectly mathematically described, I don’t see how it necessarily follows that everything that is describable by mathematics has the same ontological status.
Because we don’t see or understand why our physical reality is ‘privileged’ doesn’t mean that it isn’t. In fact doesn’t it kind of have to be different in some fundamental way because we interact with this universe quite differently that all the ‘others’ regardless if we understand the distinction or not.
What’s the furthest we can see in all directions? Is it different in one direction than another?
I will leave it others more knowledgable than I to answer Aaronson directly, but I have read a fair amount of MWI and have never encountered the notion that “our branch” of the universe is any more real than any other (except to us). Indeed, the main point is that the other branches are equally real, just not the one a particular observer is following, so I do not understand his criticism that MWI is parochial.
Copernicus certainly did not believe that every model of the solar system had its own reality.
You can certainly believe in imaginary universes if you wish, but such a belief is not simpler and does not have fewer hypotheses. If you really want to apply Occam’s Razor and reduce your assumptions, then you could stop assuming that the universe is a mathematical object.
Roger:
If you really want to apply Occam’s Razor and reduce your assumptions, then you could stop assuming that the universe is a mathematical object.
You seem to have glossed over (or ignored or misunderstood) the main point, which is (in brief, and hence in caricature) that if the Universe is not a mathematical object, it must be some other sort of entity. Thus if we deny that it’s a mathematical object, we must posit the existence of non-mathematical objects, in violation of Ockham’s Razor.
When I think of mathematics, I think in terms of relationships. The Pythagorean Theorem is a relationship of proportionality, for example, and it is always and everywhere true. But in what way is it an object?
The universe as a set of mathematical relationships – or at least, relationships that can be described reliably and accurately using mathematics – is something that I find true and uncontroversial.
Maybe the phrasing is what throws people off. What is a mathematical object?
In MWI, if everything happens, that implies a uniform distribution
of possible events, does it not? Then how does it deal with real,
non-uniform distributions?
Steve: One could assume that the universe is, or is not, a mathematical object. Ockham does not tell us which assumption is better.
Paul T: MWI does not deal with probabilities very well, and that is a bit drawback to MWI. The unlikeliest events are real in some universes, according to MWI.
Roger: One could assume that the universe is, or is not, a mathematical object. Ockham does not tell us which assumption is better.
It seems to me that he does. Of course, one could reasonably argue that he’s wrong, but I think it’s at least pretty clear what he’s telling us.
To distinguish between universes that are “physically real” from those that aren’t, we must admit the notion of “physical reality” into our metaphysics. Admitting unnecessary metaphysical concepts is exactly what Ockham tells us not to do.
There seems to be jump made from being physically real and being special. Not sure I understand that.
I think it was Stephen Hawking who asked what breathes fire into a particular set of equations and makes a universe which we experience and describe. I guess that is the same question I’d ask of those who think our universe is a mathematical object no different than other mathematical objects.
Ockham’s razor does not tell us what is true or not true. It is just a preference for simplicity. I really don’t see how it gets to be a reason to believe in all sorts of bizarre universes that no one can ever observe.
If the notion of “physical reality” is not in your metaphysics, will you be posting about the Easter Bunny and the Tooth Fairy?
The recurring nature of the mental object to which I alluded in comment #2, seems to have manifested itself in the latter’s unwelcome (but thankfully single) iteration as comment #3.
Steve, you say “I know these mathematical Universes exist (pick up any issue of any theoretical physics journal and chances are you’ll find a couple described in detail)”. Is it your position that anything which can be described in detail must exist? Is that your argument for the existence of mathematical objects? Or is it Godel’s completeness theorem?
Keshav Srinivasan: Godels theorem tells me that theories have models, i.e. those models exist. But the models that are presented in physics journals are generally constructed explicitly, so there’s no need to invoke Godel. (In fact, there’s also no opportunity to invoke Godel, because, as a general rule, no theory is explicitly stated, so there’s nothing to apply Godel to.)
RPLong (5): “Because I can think of a virtually infinite number of things I could do after writing this blog comment”
I’m not sure that all of those infinite things you’re thinking of are possible. You could certainly do any of them if you wanted to, but considering you’re going to choose one eventually I don’t see why there need be another universe where you choose differently.
^ And I mean all of that given Steve’s analysis.
Steve, it’s true that the universes presented in physics journals are explicitly constructed, but they are explicitly constructed out of sets. What is your argument for the existence of sets (and other mathematical objects for that matter)?
I think the real kicking question hasn’t been answered; What does it mean to talk of mathematics being able to create or facilitate a universe? We know that the fundamentals of mathematics is numbers, but how can numbers be responsible for physical universes?
Keshav Srinivasan:
What is your argument for the existence of sets (and other mathematical objects for that matter)?
What is your argument for the existence of the physical universe? I expect that your question and mine have the same answer: We perceive it directly. I can’t see any argument for the existence of the physical universe which is not also an argument for the existence of mathematical objects.
I’m reminded of the proof of God that goes like this: God, by definition, is the embodiment of everything perfect; existence is an attribute of perfection; therefore, God must exist. The problem is, I think, that the conclusion is implied in the premise. Similarly, purely mathematical universes must exist, because it would violate Occam’s razor to deny existence to a mathematical universe.
What does it mean to exist?
Serious question, by the way. I know what it means to exist in an everyday context, but I can’t write down a definition of existence in terms of other things, that we would all agree on.
Steve, responding to your comment:
1. Here’s an argument that distinguishes the physical universe from mathematical objects. Our perceptions of the physical universe are contingent – they could have been something other than what they actually were. For instance, suppose you have an apple in your hand, and when you open your eyes you see that it is red. Well, you could have just as well opened your eyes and seen that it is green. So all the sensory information we have about the physical world could have been different, and hence we infer that there is something that is “out there” that has certain properties, but could well have had other properties, and thus it is an object of study which we can learn about by perception.
2. On the other hand, even if you concede that it’s possible to perceive the natural numbers directly, our perceptions are not contingent. If we find that there are infinitely many prime numbers, then it’s impossible for us to have had a different perception, of there only being finitely many primes. Since there is nothing we can learn by perception of numbers that contradicts what we conclude about them without using perception, we can infer that there is nothing really “out there” that we are learning about through our perception, and thus we are perceiving nothing more than figments of our imagination.
3. As a practical matter, I don’t think most people believe that they perceive mathematical objects at all; they can imagine numbers, just like they can imagine letters, and they can imagine dragons, but they are not aware that they have a PERception, as opposed to a CONception, or numbers. And yet you maintain that everyone has this “sixth sense”, so how would you show people that they do in fact have it?
Nice post.
“If all you have is a hammer, everything looks like a nail.”
This quote seems appropriate. If you build houses, you’ll come to see all of reality as just being a big house, if you’re a computer scientist, reality is just a bitstring, and if your a mathematician, it’s a “mathematical object.”
“Make things as simple as possible, but not simpler.” –Einstein.
This quote also comes to mind. I think things are being made too simple here.
Landsburg: “Admitting unnecessary metaphysical concepts is exactly what Ockham tells us not to do.”
But Ockham doesn’t tell us whether “physical reality” (the metaphysical concept) is unnecessary or not. no?
If you can do away with notions of “physical reality” without reducing explanatory power, do it, but whether you can do it is not determined by [if you can do away with notions of “physical reality” without reducing explanatory power, do it].
It seems to me Landsburg is just asserting “everything is a mathematical object”, and this might be true but there is no evidence here.
I think Ockham’s razor just means trying to find the Kolmogorov complexity of things, not evidence for any kind of Truth out there.
Some David Berlinski if you’re interested.
http://www.c-spanvideo.org/program/204696-1
Keshav Srinivasan:
For instance, suppose you have an apple in your hand, and when you open your eyes you see that it is red. Well, you could have just as well opened your eyes and seen that it is green.
The only way I can parse this is: “In another Universe, very similar to ours, that apple is green”.
Likewise, in the ring of rational integers, the number 3 is prime. In other number rings, rather similar to the rational integers, the number 3 is not prime.
So I don’t see what distinction you’re drawing. In some Universes, the apple is red; in others it’s green. In some number rings, 3 is prime; in others it’s not. These seem equally “contingent” (or non-contingent).
I’m not a math guy, so I probably should refrain from commenting on a post like this, but since nobody knows me, I guess it can’t hurt to embarass myself.
From what I understood by reading the Big Questions, the universe is a particular INSTANCE of a mathematical object. Just like a particular triangle written down on a piece of paper is an INSTANCE of the mathematical object known as a triangle, the universe is an instance of the mathematical object that describes the universe. If this interpretation is correct, then it needs to be explained why this instance exists. After all, not every abstract triangle that can exist,DOES exist as a particular instance.
Analogies are tough for this, but here goes…
So cats exist. We have big cats, small cats, pictures of cats, movies of cats, and drawings of cats. Does it seem ontologically extravagant to suggest that some of these cats enjoy a different kind of existence than others? No.
Vic: I would not express things as you have. A triangle written down on a piece of paper is not an instance of the mathematical triangle; it is more like a representation of it.
The instances of the mathematical triangle are all as abstract as the triangle itself; e.g. the triangle with vertices (0,0), (0,1) and (1,0) in the euclidean plane is an instance of a triangle — but no *picture* of a triangle in a *picture* of the plane is an instance of a triangle (though we do, of course, call these pictures “triangles” in casual conversation).
Steve, I’m not talking about whether the properties of the universe could have been different. I’m talking about whether what we discover about the universe by means of perception could have been different. The distinction I’m trying to articulate is that sensory perception of the physical universe can “surprise” us – it can reveal things we could not have anticipated just using reasoning without such perception. In contrast, even if we say that we do perceive mathematical objects, such perception does not have the ability to “surprise” in this sense. Think about it this way: no matter how many measurements we do, we we’ll never know with certainty that Heisenberg’s uncertainty principle holds in our universe. But we have no uncertainty about the how many prime numbers there are. How is it that we can prove a theorem concerning one structure and not the other, if our perceptions of both are on even footing?
Also, as I asked before, how would you convince someone that they perceive mathematical objects directly, when they don’t think that they do? What makes our thoughts about numbers, as opposed to our thoughts about unicorns, qualify as perception?
Steve >> What is your argument for the existence of the physical universe? I expect that your question and mine have the same answer: We perceive it directly. I can’t see any argument for the existence of the physical universe which is not also an argument for the existence of mathematical objects.<<
Asking what does exist is tricky because, as this thread has shown, even the qualification for 'existence' is debatable. Lots of people concieve ideas and impression of God, and they distil purpose from those ideas and impression. So in at least one sense God can be said to exist in conceptual form.
I think the crux of this is what Ken Arromdee said – what does “exist” mean. It is quite frequent for words to mean different things in different contexts. We had an example of the word “legitimnate” in a recent post.
In one context, exist means “has a physical reality”. In another, as in mathematical jargon “there exists a solution”, it has a different meaning. Steve seems to be saying that we can just do away with the distinction, and say the two “exists” are the same. As a way to think about this, would it be possible to use a different word in in one context? Is the apparent application of Occams Razor an illusion due to a semantic coincidence? How about other languages – do they alway suse the same word?
Typo alert, obviously we did not have the example of “legitimnate” recently.
Steve >>In some Universes, the apple is red; in others it’s green. In some number rings, 3 is prime; in others it’s not. <<
This doesn't make sense to me; a prime number is a number that can only be divided by 1 or itself. 3 would be a prime number in any conceivable universe, because 3 can only ever be divided by 1 or itself.
James Knight: 3 = ( 1 + i Sqrt[2]) (1 – i Sqrt[2]) .
Keshav Srinivasan:
The distinction I’m trying to articulate is that sensory perception of the physical universe can “surprise” us – it can reveal things we could not have anticipated just using reasoning without such perception. In contrast, even if we say that we do perceive mathematical objects, such perception does not have the ability to “surprise” in this sense.
Mathematical facts surprise me all the time, just as physical facts do.
The reason we can’t prove theorems about our Universe is partly that it’s a much more complicated structure than mathematicians usually study, but primarily because we don’t know exactly what it’s structure is; that’s what the theoretical physicists are trying to approximate.
Steve, I didn’t mean to say that the theorems of mathematics can’t be surprising. I am saying that the extra-sensory perception you claim to have of mathematical objects cannot “surprise” you, in the sense that it cannot reveal to you new information that mathematical reasoning cannot. In contrast, the five senses provide all sorts of facts that we presumably could not have come up with just by reasoning.
Concerning the ability to prove theorems, you claim to perceive the natural numbers directly. But how you be do sure that they are the natural numbers? Why doesn’t the same exact kind of uncertainty accompany this perception?
Steve >>James Knight: 3 = ( 1 + i Sqrt[2]) (1 – i Sqrt[2]) .<<
Are you saying that you think there are universes in which 3 is not a prime number? I don't find this possible unless we have a universe in which prime numbers mean something different, in which case, the subject becomes incoherent.
Here's an example: take three masses coupled with two springs, no friction etc. If even just one the springs' elasticity is an irrational number, then no matter what the configuration is now (neglecting the one where both springs are completely relaxed) it will not in any universes by definition repeat in the future. This is a consequence of irrational numbers. If we change what we mean then things no longer hold. If we keep our definition of prime numbers then surely 3 must be a prime number in any conceivable universe.
James Knight:
Are you saying that you think there are universes in which 3 is not a prime number? I don’t find this possible unless we have a universe in which prime numbers mean something different, in which case, the subject becomes incoherent.
I have no idea what you mean by a “universe” in this context. There are certainly number rings in which 3 is not prime.
Ok, Steve, I meant universes with different number theories, but I suppose we could just say different number theories, because number theories don’t need a universe to exist.
This was looking at the standard divisibility in ℤ. I think what you’re saying is that what is prime or composite can change depending on the setting, as this relates to the existence of non-unit divisors. So in Gaussian integers there are more possible factorizations, to fewer real integers are primes there. So 5 is no longer a prime, being (2 + i) (2 – i) = 2² – i² = 4 + 1 = 5. But 3 remains a prime in that setting, as do other integer primes that are one less than a multiple of four.
If I understand you aright, you are saying that 3 = ( 1 + i Sqrt[2]) (1 – i Sqrt[2]) is a different number theory in which the number 3 no longer is prime. I hope I’ve understood you aright – I’m not all that familiar with alt number theories.
PS – Thanks for a great blog, Steve, and for your insighful mind in your books – I really enjoyed reading them.
Sorry, typo – Thanks for a great blog, Steve, and for your insightful mind in your books – I really enjoyed reading them.
Steve (44): “3 = ( 1 + i Sqrt[2]) (1 – i Sqrt[2])”
Or more succinctly
3 = sqrt(3)^2
3 = 3/2 * 2
3 = 1 * (-1)^2 * 3
If you allow for irrational factors then there’s no need to invoke complex numbers.
Keshav Srininvasan and Steve. I think the discusion of the abilty of the universe to surprise us is another way to discuss whether the universe is deterministic.
Maybe Max Tegmark is wrong when he says “all structures that exist mathematically exist also physically”. Perhaps it’s the case that the physical universes make up only a very tiny proportion of mathematical objects.
That way, “all structures that exist physically also exist mathematically” is true, and “all structures that exist mathematically exist also physically” is false.
James Knight: The whole point is that to say “not all mathematical universes exist physically”, you have to introduce the concept of “exist physically”. In Tegmark’s picture, physical existence is not a separate primitive concept; it’s just a new name for something that already exists. So if we’re averse to introducing unnecessary metaphysical concepts, then we should choose the view that does not require us to introduce a separate notion of physical existence.
Steve,
In #38 you already introduced the notion of “exist physically” when you distinguished a “representation of” a triangle from an “instance of” a triangle.
This universe may, in your language, be a “representation of” an instance of a class of mathematical systems.
Occam’s razor, contrary to your original assumption, would say that we should not assume the existence of other such representations. We know there is one. We need not assume anything more complex.
Tom:
In #38 you already introduced the notion of “exist physically” when you distinguished a “representation of” a triangle from an “instance of” a triangle.
Not at all! Consider the triangle with vertices (0,0), (0,1), and (1,0) in the euclidean plane. This is an instance (or ‘representation’) of the more abstract ‘triangle’, but it doesn’t follow that I have to attribute some separate sort of existence to it.
T%he language “instance (or ‘representation’)” sounds like you are saying that the two terms mean the same thing. But comment 38 implies that they are not the same thing.