Structure and Randomness: Pages from Year One of a Mathematical Blog


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When the transistors are being updated? When the system clock ticks forward? When a higher being interprets the state of the simulation? When is the simulated world spawning into existence in that case? Whenever a monkey writes a one or a zero? In the mind of the monkey when it figures how to apply the rule? Does anything practical actually need to happen for the simulated world to even spawn into existence?

This argument seems to lead to the idea that the simulated world is a mathematical structure that just exists. Without assuming the Born probabilities, I believe we can say the following:. The Copenhagenist who says that quantum mechanics is simply a set of rules for making sense of our experiences and that there is no ontology being imposed, is making an implicit claim that this is the best we can do. We can at best say that the mathematical structure we write down is isomorphic to our experiences of the universe; which is a much weaker claim.

Thus the wavefunction is what the universe is and therefore is isomorphic to a mathematical structure, consistent with MUH.

The Entropy Decrement Method and the Erdos Discrepancy Problem

PS: I will be sad to see this discussion go offline. Update us if you reach some conclusions! It seems to me that the best you could ever do would be to transfer your brain onto a quantum computer, do an interference experiment on yourself, and see what it felt like! This plasticity of the brain seems to suggests that the same signal is being felt in a very similar way of course emotions often enter the picture and a symbol never exists in isolation. Tangentially, the point about superluminal signalling seems circular to me. We mainly care about superluminal signalling because we care about special relativity, but the topic at hand is how to get special relativity out of QM.

Is it harder to explain what a headache is than to explain what red is? You could try to tell what a headache is by pointing to a particular drink and telling that headache is what you feel during getting hung over drinking it. My question is, is there a difference between saying that only one of those people have a headache and saying that both of them have a headache only they experience it differently? Just google for these keywords.. Isomorphism needs a notion of permissible objects, and permissible maps between objects; an isomorphism is a pair of composable maps which compose in both directions to the identity in this case, you want to say that the encoding function and the decoding function are the maps.

And, in fact, the isomorphism given by your encoding procedure is an isomorphism between rocks-in-general or physical-objects-in-general, or whatever , and positive integers, not an isomorphism between a particular rock and a particular positive integer, whatever that would mean. To me, MUH seems to be a generalization of the the assertion that we can import this bit of math, the idea of isomorphism-invariance, into the real world, and that whatever physical existence is, it should be isomorphism-invariant: it suggests that once we find ourselves in a perspective where any structure isomorphic to reality can and should be identified with it, that is, where physical existence is granted to this restricted class of structures those isomorphic with physical reality , than it seems natural to extend this to a wider class of mathematical structures.

Suppose we profile a large portion of the population, and discover that the time it takes to recognize a color is normally distributed. About half of the population has a peak at red, and the other half has a peak at blue. Or does collapse permit interference experiments to succeed on beings which cause wave-function collapse?

Here is a social question that might depend on the validity of MUH: what is the moral status of duplicating, deduplicating, or shutting off computer simulations of conscious beings, and how does it compare to the moral status of instantaneously wiping all of humanity out of existence? I think this is more what you are saying it does not. So it seems to me that its not science. Glad the book is still interesting. My experiment- this comment has no unusual capitol letters, lets see if it still gets spam filtered.

Is the MUH not devoid of content i. If we take King Lear and translate it into French, German, and Spanish, the bits representing the play would be different in each language. To a person fluent in those languages, the play would be the same. There could be an infinite number of translations of Lear but any two or more would be recognizable as the same to anyone fluent in the languages.


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And yes, there might be aspects of the subjective, first-person experience of reading or watching King Lear that are not conveyed by just giving someone the text. On the other hand, presumably another person can generate the appropriate qualia in their own mind, after you do give them the text and they read it! It will be hard to encode something close to quantum field theory in these model theoretic terms. So the claim is that the world is describeable by some mathematical theory.

And, if you rellay insist, conversely that every theory describes some world. This kind of statement is commonly attributed to Galileo. After all, under powerful enough isomorphisms, everything is isomorphic to everything else! Even supposing we had a final theory of physics, there might be many different ways of carving things up…. Any formalized theory of physics is or will be a theory in this sense.

Whether every theory in the sense of mathematical logic should be called a theory of physics, as is the claim made here, is a different question. More interesting would seem to be characterization of those theories in formal logic which might qualify as theories of physics. One person who has worked on this kind of question throughout his life is William Lawvere, see here. Lawvere talks for instance about Toposes of laws of motion for certain infinitary theories that admit a formulation of equations of motion of the kind encountered in continuum mechanics.

They are not however just first-order theories as in classical model theory as in above. The concept of isomorphism here is clear and uncontroversial. The question is which piece of physics is being formalized. I predict that in simulated multiverses, Berkley PhD. This is what I mean with the MUH having a self-referential problem and so far no one has commented.

And anyway, why did it have to be a human? But wait: why would I even need to run the program? But wait: why would I even need to write the program? Why not just look at my brain, and consider the set of all possible programs that my brain had some nonzero quantum-mechanical probability of generating?

Anyway, as I said in my post, this is the slippery slope that Max simply rides all the way to the bottom. I would answer that you should ride that slippery slope all the way to the bottom. So where is the bottom? And if that is so, then this multiverse is absolutely empty of any content that matters in any way to us or anything in this universe. And at that point then physical matter itself is no longer necessary all hardware is itself software in nature. Pi and e are fixed numbers. What is probablistic about findiong patterns in fixed numbers? And what if we can decode your history from the digits, but only by applying a rather convoluted transformation e.

Does that count? Is that asking whether unique mathematical structures can all have an essence that is independent of any symbolic scheme used to write them? I said that for every finite T, I have some nonzero probability of living for T years—not that I have probability 1 of living forever!!!

The former is just standard QM, while the latter requires not only MWI, but a bizarre way of calculating probabilities over worlds where you get to postselect on remaining alive, no matter what you do. We know of at least one mathematical object that is definitely real in both MUH and non MUH theories: the human mind your own consciousness.

And there is one mathematical representation that is flexible and simple enough to both realize the mind and encode all mathematical objects known to men: the brain. So the brain is probably a good starting candidate for a universal math representation. Would you say there are two kinds of MWI?

The reasonable one would assume that, yes, a lot of world must exists, but none in which a Gladstone Gander can live forever by pure luck. The desirable one would assume that every measurement basis is valid Good news! Everyone willing to terminate some branchs can have the universe he desserves! Empirically, I observe that basically no one who says they believe MWI actually lives their life that way.

Is this really the accepted definition of computation? Because from my perspective computation has nothing at all to do with the underlying implementation, which I consider to be a fairly irrelevant detail. Computation, at its heart, is nothing more than rigorous logic. Further, it assumes all computation is imperative, with a predefined thread of execution.

This flies very much in the face of both declarative programming in its many forms, as well as distributed computing, which is increasingly looking more and more like high density matrix mathematics. My own provisional answer was there may well be no consistent measurement basis in which quantum suicide can works, e. You have pushed the argument one level deeper and asked why all these mathematical structures exist.

I think we all agree that Yes, we do exist; otherwise, who is doing the agreeing? We agree, Yes. It seems clear from the motion of the planets and from space probes. Also, I think we agree, Yes. Also, we can in principle, travel there and check. Also, I think we agree Yes, even though, there is no in-principle experiment we can do to go back in time and see if it did exist. We can only look at the present and posit the existence of the far past to parsimoniously explain what we see. We agree, Yes; even though sand is simply made of fundamental particles and you may say that only fundamental particles exist.

I think we all agree, Yes; it is a most singular experience. This is true even though, at best, we can point to patterns of human behavior and say that it is caused by Love. Or we could, if you like, point to a set of electrical signals in the brain and show it is correlated with certain behaviors associated to Love. I think we all agree, No.

Postulating Unicorns adds no explanatory power for our experiences. It may help in explaining why there are people who believe unicorns; though, we all agree that there are better explanations for belief in unicorns than unicorns. The right way to frame it is to ask whether admitting a certain hypothesis would improve the predictive power, simplicity compared to previous theories and explanatory power.

Some say Yes, some say No. Here the existence is also in principle directly uncheckable or at least it seems so. How is this different from the question of the existence of the far past? Also, people like David Deutsch claim that quantum computers are the evidence left by the many-branches. In the case of the existence of the far past, there is no other hypothesis that comes close in simplicity to explaining all that we see around us. But with quantum mechanics, one can hold other reasonable positions, such as the Copenhagen position.

Maybe yes. Here you see a user-dependent notion of explanatory power kicking in.

Has it increased our predictive power? I think that is a clear no.

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Maybe work on the measure problem holds promise. Is there some notion by which it is simpler? But notice that Tegmark demands only consistency and finiteness for existence [actually, he may demand other things; I really should read the book]. Thus all the universes are hopefully derivable from these conditions. Does it improve explanatory power? This is largely user-dependent as of yet. Maybe attempts to formalize can yield fruit. So as yet, the status of the existence of Tegmark universes is dubious. For example:. It also seems to me like your logic could be pushed even further.

In short: If unicorns do not exist then by definition there can be nothing this sentence refers to. Or in other words, if unicorns do not exist then there can be no direct evidence your statement refers to. I am never sure if this is a really deep puzzle or just trivial. But when we talk about the Born probabilities then Copenhagen is the simplest imho. It is at least simpler than Bohm-deBroglie and simpler than mwi by far. That is the big problem I have with quantum suicide arguments.

My life and my self condition can not be represented as a binary state. But it would present all kinds of new problems of course. So this particular malady would only affect you for finitely many of your infinitely many years. Scott: how about coffee next week? In short, the quantum suicide thought experiment assumes an optimistic future which there is no reason to suppose.

Perhaps I will live forever, but my personal state will devolve into ever greater levels of misery and pain. As I tried to explain, in an infinite time, tunneling gives a nonzero probability for essentially anything to happen to you, including your being miraculously cured, then injured again, then swallowed by a whale, etc.

So if you expect to live forever, then you should really only care about the equilibrium probabilities of being in one state rather than another—what happens to you in the next decades or century is completely irrelevant. I do believe in reincarnation though, fwiw. Is that correct? Copenhagen merely assumes the Born rule and the measurement postulates. At least de Broglie-Bohm dBB can derive the Born rule by assuming a simpler axiom, and no measurement postulates are needed.

Therefore at least dBB is more parsimonious than Copenhagen. Imagine the theory of groups had just one model…. In the 90s there was a wide-spread subconcious belief that string theory has only one, or just a handful, of models… the shattering of that belief is — by a curious course of history — the reason why some fundamental physicists these days turned to philosophy.

There is a lot of interesting things to be said in founding modern physics in formal logical theory. But classical model theory is unsuited, as far as I can see. From discussions like this here it almost feels as if there is suddenly a wide public interest in foundations of physics in formal logic and maybe classical or modern model theory. That used to be a topic that besides William Lawvere few people were working on.

On the other hand then it seems most of the interest here is in witty chat, not so much in mathematical or physical theory.

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I am wondering if there is a way that the occasion that model theory has been sneaked into the public debate through the backdoor could be used to increase scientific interest in it that goes beyond lay chat. Would he not be distracted by debating suicide of MWI proponents — maybe there is a psychologically interesting Freudian aspect here, but none of mathematical or physical interest. For instance I claim that within linear homotopy type theory one may say something interesting about this, which looks relevant to the question of whether the world has an origin in formal mathematical theory.

Ok, so Sid has insisted on dragging unicorns into this. Please bear in mind, that I am merely an enthusiastic amateur in regards to Mythology. We all know by virtue of platonism that indeed Unicorns do exist, and in fact they possess an attribute orthoganal to winged equines aka Pegasi that attribute is their velocity.

Remember: Unicorns are the fastest land equines. On land, they run so fast, their velocity is not even a number anymore. It makes C look like nothing. Well, when the afforementioned winged equines Pegasi happen to be in motion, they are the fastest flying equines. They travel so fast, that in the time that a unicorn moves one unit of length, they travel Aleph-null units of length WOW! How fast is that, you ask? And this repeats every tick. Urs : I fully agree with you that more rigorous work on these model-theoretic topics is interesting and worthwhile.

Yet, the devil is of course in the details. The empty set is not isomorphic to the integers, period. If, instead, you were to regard our causal patch as a closed system evolving unitarily, then the question becomes much murkier. For then, there will just be some wavefunction of the universe rotating eternally through Hilbert space. And that wavefunction will almost always have some nonzero component in which you remain alive. I was thinking this way. Say every possible string hypothetically occurs in e or pi.

Then say there is a universal string S which is the program. So this is deterministic correct? J Well, which are the strings H i from which you can recover the history for user i? Rather, it seems like the basic ideas of those theories can be represented formally in many different ways, and those ways will give rise to different structures. Even failed ideas can cause the conditions to lead to true insights by helping to map out the problem space! Can you recommend any other easier works on formal type theory?

Anything that might help a programmer by trade understand what is going on? Even classically, what should the elements be: the spacetime points? But then what about diffeomorphisms? The values of the fields at spacetime points? The connection? And should there be a relation between every two points? Different experts would probably formalize their knowledge of the fields in different ways some using Lagrangians, others using Hamiltonians, etc. And that is the universal structure to consider and try to formalize very graph-like obviously. And then try to formalize isomorphisms that apply between brain structures.

Actually, they are both measure zero subsets of the real numbers. Of course, you can define the trivial mapping to the terminal category where all objects are isomorphic. So I find myself working internally to homotopy type theory instead. This might be an empirical question, should we find a natural way to solve the mathematical measurement problem; if there are sufficiently few free parameters and one of them is which notion of isomorphism to use , then perhaps we ask which notion of isomorphism gives rise to the correct anthropic probabilities.

Alternatively, it might be the case that we will find that our hand is forced; if we have a way of interpreting various structures, this interpretation should be isomorphism invariant in the sense that for any structure which seems to be interpretable into our physical reality, any isomorphic structure should also be interpretable; some structures might not have enough information to obtain an interpretation.

Could this an example of a result that should strongly bias us towards something like MUH? Or is there some way to remove any such dependence in general? I think there is a fundamental characteristic for a candidate universal structure — it has to be discrete in nature. Any continuous structure has to be excluded.

Continuity would lead to zeno paradoxes, i. This I think also explains the struggle I had with the connection between consciousness and computation. Our mind is not just a series of snapshots of states but a connected structure that evolves both in time and space. This reduces to the key conservation and symmetry concepts like energy which are nothing more than limits on how fast and how wide things can grow.

That simple requirement captures any fundamental limitations on dynamics and complexity that we observe in our universe, and is a universal pattern — expansion of the universe since the big bang, formation of galaxies and solar systems, evolution of life, growth of individual organisms from a egg to a full adult , neurons forming connections in a learning brain, etc. The brain itself can be modeled with such a structure, and the brain can represent all human knowledge it is isomorphic to our universe by definition. Jason — The morphisms are measure-preserving maps, which are certain equivalence classes of functions.

And no, the empty set is not isomorphic to any mathematical structure with non-zero cardinality with this definition.

Terence Chi-Shen Tao

I look forward to our coffee, Scott! I take this as strong evidence that Q and N people indeed experience the same sensory input quite differently. How does a thing have no intrinsic properties and is yet somehow defined and categorised placed into an abstract set without humans? Who decides what set the thing goes into? Who decides the relationships?

Regarding properties emerging — you claim that there is a natural way to interpret the mathematical structures that are apparent in physics. How exactly does this relate to the MUH? Does the MUH claim that there is always a natural mathematical structure for anything we observe? Max Thanks for the article link! Greg — As I understand it, there are no functions from a non-empty set to the empty set, measure-preserving or otherwise. This would imply that there are no morphisms from the integers to the empty set, because all equivalence classes of such functions are empty.

Have I misunderstood something?

1 Introduction

Wolfgang — I am not arguing that there is no hard problem of qualia. In all cases, there are many equivalent ways of describing the same structure, and a particular methematical structure can be defined as an equivalence class of descriptions. Thus although any one description involves some degree of arbitrariness in notation, etc.

How do you pick which foundations of math or ambient logic to use? For example, what if two structures are equivalent only if you assume the axiom of choice or the continuum hypothesis, or if some structure is only definable if you assume the law of excluded middle, or if the finiteness of some structure is independent of the axioms of Peano arithmetic e.

Even so, there are many relativistic extensions of dBB and a few field theory formulations. Most of them include an unobservable preferred foliation, although no particular theory has gathered enough support to be considered canonical. What this shows is that John Bell was correct: the real problem of QM that needs serious investigation is non-locality. Waving it away by denying realism is just a cheat.

Jason — As I understand it, there are no functions from a non-empty set to the empty set, measure-preserving or otherwise. A morphism between two measurable sets is a partial function, defined everywhere except on a set of measure 0 and which is also a measurable function. Such a partial function is equivalent to what you get if you erase its values on a measure 0 subset. We have tried to make clear in these pages that we have taken our terms from mathematical set theory and applied them to the theory of computation.

We find, however, that infinite self similarity in any finite time interval promises self organized, and therefore self limiting, maps of short local intervals to long global intervals. What about your friend Sean Carroll? Admittedly, he does not claim deriving the Born rule. Here is slide 34 from his presentation. So anyway, I have been pondering the implications of MUH for a long while, and attempting to poke holes in it. A single universe in a Level 4 Multiverse consists not just of a set of laws, but also an initial state.

In fact, in MUH, every possible combination of initial state state and set of laws would define a different universe.

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The odds of us finding ourselves in universe with a highly complex initial state would have to be considerably higher than if our beginning were simple. So why is our initial state so simple or why does it seem to be? Where are the really oddball laws in our universe? These functions are just as valid in the mathematical definition of a universe, and would seem to be at least as numerous again with the measure problem as the simpler functions.

Why are all our laws so seemingly simple? But yes, he and I do have somewhat different perspectives about MWI. If Alice and Bob have a joint state, then the state of Alice alone is for all quantum purposes described by a density matrix. In conjunction with other reasonable physics, you do expect to see decoherence, thermalization, approximate Maxwell-Gibbs distributions, and ultimately physical objects which have classical behavior expressed in density matrices. We only know it as a partial-trace entity; the density aspect is purely formal.

After all, if almost all of the wave function of an electron is near an atomic nucleus, then even without any quantitative Born rule, you would informally say that the electron is near the nucleus. Not to mention that much of the calculus of density matrices is mathematically identical to classical probability, even if those density matrices happen to come from partial trace and physical locality rather than from the Born rule.

It is dubious to reject that coincidence as philosophically meaningless. The universal wavefunction of mwi follows objectively a deterministic evolution. Therefore probabilities have to be subjective — and this is indeed what you use in your derivation: Alice does not know the state of the environment. But once you go the subjective route, why not use Copenhagen, which basically uses the fact that you know the state of your mind by definition — therefore you reduce the wavefunction every time it affects your state of mind i.

The point is that she may have tested subjective Copenhagen with millions of repeated trials; it is then endorsed by the law of large numbers. So, if you are anti-Copenhagen, which of these considerations should be deprecated? Is the probing is part of the state? Is there a computation analogy?

Turing machine capturing its current own state as data? This is not necessary. They may contain a lot of fat and there may exists simpler theories that perform exactly the same service. It only makes the much weaker assumption that high concentration of quantum state is perceived. It follows as a corollary that the exact Born rule is perceived as well.

This is a very confusing position because, as I want to argue, you can practically derive subjective Copenhagen from unitary QM. Okay, I looked at the definition in Appendix A. If you want to be rigorous about all of this, then I think you appeal to a model of isomorphism in a subcategory of the category of sets.

The problem is that it is not at all clear that a subcategory of Set is a very good model for our universe. In general, a lot of categories are most naturally constructed as subquotient categories of Set rather than as subcategories, or even by more general constructions than subquotients. For instance, the homotopy category of topological spaces is naturally a quotient of Top, which is a subcategory of Set. Now, yes, there is an elementary result in category theory that every category embeds as a category of Set. But not very naturally, and not uniquely either.

If you emphasize mathematical isomorphism, and if you also emphasize in your Appendix A morphisms in the category Set, then presumably the specific implementation of the witnessed universe in Set should matter! Is an atom aware of what it experiences? Your Web is as generalised for opportunity. Some searches of WorldCat will now receive sure. The hopeful brothers or rules of your shelling ebook ASP. NET 3. Please download noncompatible e-mail seasons. Contact Us Close panel.

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Structure and Randomness: Pages from Year One of a Mathematical Blog Structure and Randomness: Pages from Year One of a Mathematical Blog
Structure and Randomness: Pages from Year One of a Mathematical Blog Structure and Randomness: Pages from Year One of a Mathematical Blog
Structure and Randomness: Pages from Year One of a Mathematical Blog Structure and Randomness: Pages from Year One of a Mathematical Blog
Structure and Randomness: Pages from Year One of a Mathematical Blog Structure and Randomness: Pages from Year One of a Mathematical Blog
Structure and Randomness: Pages from Year One of a Mathematical Blog Structure and Randomness: Pages from Year One of a Mathematical Blog
Structure and Randomness: Pages from Year One of a Mathematical Blog Structure and Randomness: Pages from Year One of a Mathematical Blog

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