tag:blogger.com,1999:blog-15005476.post2678234387917099648..comments2018-08-25T21:24:44.954-04:00Comments on Rationally Speaking: Rationally Speaking podcast: Max Tegmark and the Mathematical Universe HypothesisUnknownnoreply@blogger.comBlogger155125tag:blogger.com,1999:blog-15005476.post-7364818059177536262014-02-20T12:42:14.785-05:002014-02-20T12:42:14.785-05:00Pete: Does it matter if we know what protons are? ...Pete: Does it matter if we know what protons are? We have evidence of their existence, we can detect them, we can distinguish them from other particles, and as I said, we can manipulate them. Empirically, we know that a proton you detect today will have the exact properties as one I detected yesterday, and we have no reason to expect that will ever change. They are as physical as anything can be, and I say they exist, independent of our concept of "substance".<br /><br />I can also say that mathematical objects exist. But in mathematics, I don't think the word has the same meaning. I can't take the number 5 and put it in a particle accelerator and collide it with the number 7.<br /><br />DM, I agree, gliders exist in a real sense, from the point of view of inhabitants of the game of life. The game has fundamental particles (or the analogs of particles) which are the states of cells in the matrix. I'm not sure I agree about Luke Skywalker's chair, but that's because we can't establish that he and the chair inhabit a universe with a coherent set of rules governing its progress. What happens to Luke or the chair is what George Lucas tells us (or shows us), and narrative alone does not create physical worlds, no matter what some theologians may want to believe.<br />Richardhttps://www.blogger.com/profile/10042619745483254124noreply@blogger.comtag:blogger.com,1999:blog-15005476.post-82563070948050612342014-02-18T17:21:08.078-05:002014-02-18T17:21:08.078-05:00Hi Pete: I agree with you that it would be fascina...Hi Pete: I agree with you that it would be fascinating if we could devise a physical experiment whose outcome depends on whether something truly infinite exists. It would be even more intriguing if we could perform an experiment (performed in a finite volume of space during a finite time) whose outcome depended on a Gödel-undecidable statement. <br />If you can think of such an experiment, please let me know!Max Tegmarkhttps://www.blogger.com/profile/14025578388753974108noreply@blogger.comtag:blogger.com,1999:blog-15005476.post-66124140619288584882014-02-18T12:32:02.551-05:002014-02-18T12:32:02.551-05:00PS: Roger: It's not correct that Many-Worlds &...PS: Roger: It's not correct that Many-Worlds "ruins predictions"; there's a perfectly well-defined prescription for calculating probabilities for what we'll observer. In the same way, the the appearance of probabilities in the Copenhagen Interpretation interpretation doesn't "ruin predictions": it's just that with quantum mechanics, some of our predictions become statistical.Max Tegmarkhttps://www.blogger.com/profile/14025578388753974108noreply@blogger.comtag:blogger.com,1999:blog-15005476.post-63848326635126383442014-02-18T12:28:12.766-05:002014-02-18T12:28:12.766-05:00Hi Roger: We're currently stuck with the infla...Hi Roger: We're currently stuck with the inflationary measure problem (that we can't make predictions) whether we like it or not. It only occurs if space can be infinitely stretched, which is why I think it's a good idea to test that assumption rather than simply take for granted that it's true.<br /><br />We *don't* have good evidence that we need to get rid of infinity because of Gödel's incompleteness theorem, which I explore in detail in chapter 12 of the book (http://mathematicaluniverse.org).<br />Is the matematical universe hypothesis ruled out by Gödel's incompleteness theorem? No, not as far as we know. Given any sufficiently powerful formal system, Gödel showed that we cannot use it to prove its own consistency, but his doesn't mean that it is inconsistent or that we have a problem. Indeed, our cosmos doesn't show any signs of being inconsistent or ill-defined, despite showing hints that it may be a mathematical structure. Moreover, what were we hoping for? If a mathematical system could be used to prove its own consistency, we'd remain unconvinced that it actually was consistent, since an inconsistent system can prove anything. We'd only be somewhat convinced if a simpler system that we have better reason to trust the consistency of could prove the consistency of a more powerful system - unsurprisingly, that's impossible, as Gödel also proved. Of the many mathematicians with whom I'm friends, I've never heard anyone suggest that the mathematical structures that dominate modern physics (pseudo-Riemannian manifolds, Calabi-Yau manifolds, Hilbert spaces, etc.) are actually inconsistent or ill-defined. <br /><br />Please note that I'm using "mathematical system" to refer to "formal system", as distinct from "mathematical structure": the former can describe the latter and the latter can be a set-theoretic model of the former. <br />Max Tegmarkhttps://www.blogger.com/profile/14025578388753974108noreply@blogger.comtag:blogger.com,1999:blog-15005476.post-87750513052254062932014-02-16T20:55:23.961-05:002014-02-16T20:55:23.961-05:00Richard -
You mention the existence of protons an...Richard -<br /><br />You mention the existence of protons and us being able to "do" things to them, but do we honestly even have a handle on what they are? I remember bringing up this same point on Massimo's post from 12/2013:<br /><br />"This problem is not one that is unique to the idea that the world is literally mathematics. Physicalism, a doctrine that virtually all scientists and a large portion of philosophers subscribe to, also runs into the same difficulties. I have always considered myself a physicalist as well, but after really contemplating what that means I start to revert to simply considering myself a naturalist, which contrary to many opinions can embrace abstract objects without a problem. When one really begins to delve into the nature of the "real" objects that are out there, their decidedly ephemeral nature is exposed. For one, objects in the universe are made almost entirely of empty space. The atoms that compose physical entities are something like 99.999% empty space, with a very tiny nucleaus and some extremely small electrons that occupy probability clouds around it. The nucleus itself is composed of even tinier quarks and gluons, the true nature of which (along with every other elementary particle in existence) is extremely hard to pin down. Maybe they're "vibrating strands of energy" as String Theory posits, or “knots” in the fabric of space-time as postulated by Loop Quantum Gravity. (I'll add to the original that the present description of fundamental particles is that they are in fact "point particles," which means they're 0-dimensional mathematical points).<br /><br />The point is, what we think about as being physical is really no such thing at all. At the bottom, it really seems to become mathematical equations and relations. And at that point, a la Massimo's invocation of no miracles and the idea of indispensability to our understanding of the world, you have a strong case for mathematical realism. The ironic part of the opposite position, that of nominalism with regard to mathematics, where something must be physically instantiated in space-time if it’s to be considered real (something which makes mathematics nonexistent), is its inherent assumption of physicality that is largely misunderstood and nonexistent in and of itself."<br />petehttps://www.blogger.com/profile/12969621709127674152noreply@blogger.comtag:blogger.com,1999:blog-15005476.post-70161529700291596332014-02-16T13:54:55.617-05:002014-02-16T13:54:55.617-05:00Hi Richard,
Again, I'd say there is no fact o...Hi Richard,<br /><br />Again, I'd say there is no fact of the matter on whether mathematical objects exist. It depends on your perspective. From the formalist's perspective, they don't. But this is compatible with the MUH if we assume the universe doesn't actually exist. This is a strange point of view to adopt which is why I prefer Platonism.<br /><br />>If "existence" does have semantics in mathematics, it doesn't seem to be the same semantics that the word has in physics.<<br /><br />This is compatible with my view that physical existence is a relative predicate. The chair I am sitting on physically exists from my point of view. The chair Luke Skywalker is sitting on is real from his point of view. Neither of our chairs exist from the point of view of the other.<br /><br />Let's take it back to math. I would note that if you need interaction to exist, we should consider only mathematical structures with something analogous to time, so the empty set is not the best example.<br /><br />A glider exists from the point of view of another in Conway's Game of Life, because it can do stuff to it. A character in a computer simulation exists from the point of view of another, but not to an observer outside the simulation.<br /><br />I therefore don't see any basis for confidence that there is any fundamental difference between abstract existence and physical existence.Disagreeable Mehttps://www.blogger.com/profile/15258557849869963650noreply@blogger.comtag:blogger.com,1999:blog-15005476.post-66736177070137307382014-02-15T23:42:42.519-05:002014-02-15T23:42:42.519-05:00To a formalist, "existence" is simply a ...To a formalist, "existence" is simply a predicate which has syntactic -- not semantic -- consequences for other objects (which may or may not "exist"). But few people would say that "existence" in the physical world is devoid of semantics. So formalism has its own consequences for MUH (except to those few), and those consequences are the opposite of Platonism. (I don't think either ism is more justified than the other.)<br /><br />If "existence" does have semantics in mathematics, it doesn't seem to be the same semantics that the word has in physics. The empty set exists, but you can't really do anything *to* it, although you can do things *with* it, like instantiate it in other sets. I don't even think that counts as doing something "with" it, because the containing set actually exists a priori, so all we can do is talk about the sets of a universe, not create them. On the other hand, a proton exists, and in the physical world, "existence" implies that you can do things *to* it (accelerate it through the LHC, for example).<br />Richardhttps://www.blogger.com/profile/10042619745483254124noreply@blogger.comtag:blogger.com,1999:blog-15005476.post-64786652162685994502014-02-14T19:51:59.545-05:002014-02-14T19:51:59.545-05:00Hi Richard,
You're probably right on Fibonacc...Hi Richard,<br /><br />You're probably right on Fibonacci.<br /><br />In a way I think you're right about Platonism. I've made a similar point about this on my blog. Basically, there is no fact of the matter about whether mathematical objects exist or not. It depends what you mean by exists. The concept of existence becomes ambiguous when referring to objects outside our universe. I prefer to say that mathematical objects exist because I think this is an intuitive and useful way to think about them.<br /><br />So you're right, it doesn't matter much either way. Until you come to the MUH at least. So what does no fact of the matter mean for the MUH? It just means that it doesn't make sense to ask whether this or any other universe exists. The concept of existence doesn't apply to universes any more than it does to mathematical objects. Either way, the existence of the universe is explained. Either its existence is an illusion or it does exist, but only in the way mathematical objects do.Disagreeable Mehttps://www.blogger.com/profile/15258557849869963650noreply@blogger.comtag:blogger.com,1999:blog-15005476.post-63178111240502228832014-02-14T18:07:44.133-05:002014-02-14T18:07:44.133-05:00I'm not sure if it is possible to define "...I'm not sure if it is possible to define "3+5=8" without the implicit knowledge that there are natural numbers in between those numbers. In fact, the Fibonacci set, taken outside the context of the natural numbers, loses much if not all of its semantics. "13" is no longer "the successor to 12" as it is in Peano arithmetic or set theory.<br /><br />In my case, by Platonism, I was referring to traditional Platonism, which might not be related to MUH at all, but mainly, I don't really think any of these isms are going to produce profound truths about mathematics. They are really just views about mathematics. And I'm just more inclined (nowadays) to be skeptical of ideas that I might have once accepted because of alignment with some particular ism.<br />Richardhttps://www.blogger.com/profile/10042619745483254124noreply@blogger.comtag:blogger.com,1999:blog-15005476.post-18112910986653538562014-02-14T17:32:49.696-05:002014-02-14T17:32:49.696-05:00Hi Disagreeable Me, it's been fun, but I don&#...Hi Disagreeable Me, it's been fun, but I don't think we are getting anywhere. Cheers!manyosohttps://www.blogger.com/profile/12384195364005229109noreply@blogger.comtag:blogger.com,1999:blog-15005476.post-35936792172342444202014-02-14T14:31:53.919-05:002014-02-14T14:31:53.919-05:00Hi Adam,
>I am afraid it is still too ambiguou...Hi Adam,<br /><br />>I am afraid it is still too ambiguous<<br /><br />Perhaps to do mathematics with, but I don't see why if we're having a metaphysical discussion. Can you challenge the definition with some border cases?<br /><br />>pick a formal system and language and define it vigorously<<br /><br />It is not possible to express my position in these terms because it assumes that there is a privileged formal system, which I don't believe. Instead, I could define a mathematical structure as anything that can be defined in any formal system.<br /><br />>However, they have to be defined by *at least one* formalism.<<br /><br />Mostly agreed, but that would probably exclude first order logic, as that is the most basic formal system and so not defined in any more basic system. I would count that as a mathematical structure too.<br /><br />>I think that is what you would need to defeat my argument. Do you see why?<<br /><br />No. I think as long as it is not tied to any one specific formalism, your argument is defeated, because proving the consistency of a structure does not prove the consistency of any specific formalism - it only proves that there exists *some* formalism that can handle it. Proving the inconsistency of a formalism similarly does not prove the inconsistency of a structure defined with it - the structure may still be consistent if some other formalism can handle it.<br /><br />The Mandelbrot set as visualised by computers has to be consistent, because there are computer programs that can draw it out. Whatever they are doing, it is an operation defined by the algorithms they are running. It's not possible to implement an inconsistent or ambiguous (single-threaded) algorithm. Anything that is algorithmic has to be consistent.<br /><br />Let's say the Mandelbrot set is typically defined in ZFC. We know the Mandelbrot set (as visualised by computers) is consistent, but that in itself does not prove that ZFC is consistent. The analogy to our universe is that knowing that the laws of physics are consistent does not prove that ZFC is consistent.<br /><br />Now, suppose for a second that there is actually something inconsistent in the definition of the Mandelbrot set. To me, this means that the computer programs are not actually implementing the definition correctly, because this is impossible. So, we don't necessarily know that the Mandelbrot set is consistent, we only know that its visualisation algorithm is.<br /><br />Now let me bring this back to the universe. We know we are in a universe which is consistent, but perhaps we have not correctly identified the true laws of physics. The laws of physics we have found could be inconsistent, like the Mandelbrot set itself, but our universe is not actually following those laws, like the visualisation algorithm. So the existence of the universe does not prove that any particular mathematical structure is consistent, only that something like it is.<br /><br />So the existence of our universe can neither prove the consistency of any particular formalism or of any particular mathematical structure, therefore there is no problem from Godel.Disagreeable Mehttps://www.blogger.com/profile/15258557849869963650noreply@blogger.comtag:blogger.com,1999:blog-15005476.post-60974984074733685042014-02-14T14:11:33.093-05:002014-02-14T14:11:33.093-05:00Hi Richard,
>Certain operations -- but not the...Hi Richard,<br /><br />>Certain operations -- but not the operation that defines GoL, right?<<br /><br />My thinking was not to define Fibonacci in terms of ordinary addition but in terms of a restricted kind of addition, say "faddition" which is only permitted on adjacent Fibonacci numbers. It would be as illegal to "fadd" two non-adjacent Fibonacci numbers as it is to divide by zero. Possibly this way of thinking about it does not really make sense, but that was my intuition. In a similar way, the operation that defines the GoL is a restricted form of the more general operation that says "set the state of the matrix to M).<br /><br />I don't see why formalism is more true to the view that there is no one true mathematics than *plenitudinous* or *full-blooded* Platonism. That's basically the central idea.<br />Disagreeable Mehttps://www.blogger.com/profile/15258557849869963650noreply@blogger.comtag:blogger.com,1999:blog-15005476.post-76992983699986133272014-02-14T13:26:44.754-05:002014-02-14T13:26:44.754-05:00Disagreeable Me,
Does my definition of mathematic...Disagreeable Me,<br /><br /><i>Does my definition of mathematics make sense to you?</i><br /><br />I assume you meant your definition of mathematical structure... In which case, no I am afraid it is still too ambiguous. When I say I want you to define what you mean I think the best way to do this is in the language of mathematics... ie, pick a formal system and language and define it vigorously.<br /><br /><i>Do you now see what I mean by the independence of a mathematical structure from any particular formal system?</i><br /><br />No, not really. I get that you are saying that some of these structures you have in mind can be defined by more than one formalism and in this sense they are independent from any *one* formalism. However, they have to be defined by *at least one* formalism. No mathematical structure can be truly independent of *any* formalism. And I think that is what you would need to defeat my argument. Do you see why?<br /><br />Formal systems of mathematics are usually found in non disjoint sets. Some are weaker and some are stronger. I guess in truth each mathematical structure has at least one minimally weakest formalism that is still powerful enough to define it. In this sense, it will never be independent.<br /><br />And please see above where I take issue with your point #4. I think this is the crucial question and we'll never be able to resolve it unless/until you can define "mathematical structure" in a rigorous way I think or give an example of one of these structures which can be proven consistent and still can be defined by an inconsistent formalism. I don't think you'll find any such examples, but feel free to look.<br /><br />Cheers,<br />Adammanyosohttps://www.blogger.com/profile/12384195364005229109noreply@blogger.comtag:blogger.com,1999:blog-15005476.post-59643140441055817402014-02-14T12:10:05.069-05:002014-02-14T12:10:05.069-05:00"I understand what you are saying but I do th..."I understand what you are saying but I do think that I could make a case for the GoL not being closed under certain operations." Certain operations -- but not the operation that defines GoL, right? Otherwise I'm not sure what that case would look like.<br /><br />I share the view that any consistent set of axioms is as valid as any other -- at least, as far as mathematics is concerned. Nothing about Woodin's approach undermines Platonism as a whole, but studying it made me revisit the idea more thoroughly. Woodin is pushing (as a Platonist) for Omega consistency (rather than consistency) as the standard for validity. It is a stronger standard that would resolve CH in the negative, but it is unnecessary in my opinion. I would think formalism is less constrained over standards than Platonism, and holds more strictly to the view that "there is no one true mathematics". But I don't know if I can really associate myself with any of these "isms". Or maybe for me it is just "pragmatism". I'm not going to look for empirical reasons to accept or reject AC or CH, for instance, because I don't think they will be found, and I don't think they will ever have physical consequences. It is enough to leave them undecided.<br />Richardhttps://www.blogger.com/profile/10042619745483254124noreply@blogger.comtag:blogger.com,1999:blog-15005476.post-45806902685036676312014-02-14T05:31:05.229-05:002014-02-14T05:31:05.229-05:00This comment has been removed by the author.Disagreeable Mehttps://www.blogger.com/profile/15258557849869963650noreply@blogger.comtag:blogger.com,1999:blog-15005476.post-34711667762303141122014-02-14T05:30:26.251-05:002014-02-14T05:30:26.251-05:00Hi Adam,
Hope you'll get back to me on this. ...Hi Adam,<br /><br />Hope you'll get back to me on this. Does my definition of mathematics make sense to you? Do you now see what I mean by the independence of a mathematical structure from any particular formal system?<br /><br />DMDisagreeable Mehttps://www.blogger.com/profile/15258557849869963650noreply@blogger.comtag:blogger.com,1999:blog-15005476.post-86036170474658502532014-02-14T05:25:21.333-05:002014-02-14T05:25:21.333-05:00Hi Richard,
I understand what you are saying but ...Hi Richard,<br /><br />I understand what you are saying but I do think that I could make a case for the GoL not being closed under certain operations. But anyway, it doesn't matter at all so we can drop it.<br /><br />I find it curious how one particular proposal could undermine platonism as a whole. Perhaps you were not a full-blooded/plenitudinous Platonist? This view of Platonism, my own, holds that all consistent axiomatic systems are equally valid, so there is no one true mathematics. On this view, if any one axiomatic system is flawed (i.e. inconsistent), then the others are still fine.Disagreeable Mehttps://www.blogger.com/profile/15258557849869963650noreply@blogger.comtag:blogger.com,1999:blog-15005476.post-9292278259688507062014-02-14T05:16:24.187-05:002014-02-14T05:16:24.187-05:00>Under MUH mathematically distinct digital demi...>Under MUH mathematically distinct digital demigods for each and every possible state of the multiverse are uncountably more probable than the entire multiverse and the same goes for each miraculous variation on it.<<br /><br />I don't see that, at all.<br /><br />>You seem to be thinking of the MU as a sort of physical realm.<<br />Why do you say that?Disagreeable Mehttps://www.blogger.com/profile/15258557849869963650noreply@blogger.comtag:blogger.com,1999:blog-15005476.post-13662278373668764042014-02-14T00:34:06.326-05:002014-02-14T00:34:06.326-05:00Richard -
That's an excellent point. I shoul...Richard - <br /><br />That's an excellent point. I should have further elaborated on one of the main reason's why this is an important result for the philosophy of mathematics. There are many mathematicians, including examples such as Lakatos (who definitely subscribed to a strand of Platonism), that believe elements of empiricism should be incorporated into mathematics to discover new truths. Even though proof theory has been ingrained into mathematical practice (and very much deservedly so), these individuals believe that empirical evidence has an extremely important place in uncovering new mathematical truths and practicing mathematics. In fact, empiricism came well before mathematics in the development and evolution of the discipline, preceding the formal axiomatic systems by many centuries.<br /><br />If these patterns are discovered in nature, and they hold for a large number of cases (which grows over time), then that should be taken as strong evidence in favor of it being a mathematical truth. If this result can then be proven only through large cardinal axioms, this would by extension lead one to believe that large cardinal are themselves real.<br /><br />The philosopher James Robert Brown has an excellent book about the philosophy of mathematics (he strongly leans towards Platonism) that develops just this type of argument for empiricism. I would definitely encourage anyone interested in the subject to read it:<br /><br />http://www.goodreads.com/book/show/469677.Philosophy_of_Mathematicspetehttps://www.blogger.com/profile/12969621709127674152noreply@blogger.comtag:blogger.com,1999:blog-15005476.post-34436813011868446172014-02-13T19:42:14.850-05:002014-02-13T19:42:14.850-05:00DM,
"To me, this is not obviously different ...DM,<br /><br />"To me, this is not obviously different from the fact that there are certain ways for the matrix of Conway's Game of Life to evolve which are forbidden by the rules."<br /><br />I see a significant difference. Within the context of the natural numbers, the operation of addition has an unrestricted domain. But unrestricted addition in the domain of the Fibonacci numbers generates numbers outside of that "universe". In GoL, there is an operator that defines how the matrix evolves, so: state[i+1] = G( state[i] ). For any state of the matrix at stage i of the game, the operation will generate a valid state for stage i+1. The GoL universe is closed under this operation, with no restriction of the operator's domain within the universe of GoL. That certain paths of evolution are forbidden is therefore beside the point, since the universe consists of cells and states. It does not consist of an unrestricted set of "paths".<br /><br />As to your question, there wasn't really any specific counterargument that changed my mind. Rather, it was disenchantment with mathematical platonism in general, triggered in part by Woodin's Omega logic proposal, which seemed to me (to the limited extent that I could interpret it) to be overreaching.<br />Richardhttps://www.blogger.com/profile/10042619745483254124noreply@blogger.comtag:blogger.com,1999:blog-15005476.post-4542768444175602282014-02-13T17:03:45.479-05:002014-02-13T17:03:45.479-05:00Under MUH mathematically distinct digital demigods...Under MUH mathematically distinct digital demigods for each and every possible state of the multiverse are uncountably more probable than the entire multiverse and the same goes for each miraculous variation on it. You seem to be thinking of the MU as a sort of physical realm.Robinhttps://www.blogger.com/profile/16015911138886238144noreply@blogger.comtag:blogger.com,1999:blog-15005476.post-43550351923801948412014-02-13T14:09:36.670-05:002014-02-13T14:09:36.670-05:00Hi Richard,
I understood what you meant. I know t...Hi Richard,<br /><br />I understood what you meant. I know that the Fibonacci numbers are not closed under arithmetic. I'm just not sure that closure under arithmetic or any other operation is a good criterion for ruling out certain structures as universes. In the context of the Fibonacci numbers there are certain operations you can't perform while still restricting yourself to the Fibonacci numbers. To me, this is not obviously different from the fact that there are certain ways for the matrix of Conway's Game of Life to evolve which are forbidden by the rules. But perhaps it is different.<br /><br />Might I ask what lead you from being a firm believer to being a skeptic? I'm obviously a firm believer myself, but I want to test my conviction to see if it holds up to various counter-arguments. Were there any in particular that impressed you?Disagreeable Mehttps://www.blogger.com/profile/15258557849869963650noreply@blogger.comtag:blogger.com,1999:blog-15005476.post-2650133304491364822014-02-13T13:54:24.426-05:002014-02-13T13:54:24.426-05:00DM,
I was referring to this definition of closure...DM,<br /><br />I was referring to this definition of closure: http://en.wikipedia.org/wiki/Closure_(mathematics)<br />Ordering of the elements does not come into play in that definition. So, the Fibonacci numbers are most definitely not closed under addition.<br /><br />Regarding your last paragraph, I don't reject MUH outright. I might have been a firm believer once upon a time, but now I approach it with skepticism.<br />Richardhttps://www.blogger.com/profile/10042619745483254124noreply@blogger.comtag:blogger.com,1999:blog-15005476.post-17676598259399186802014-02-13T13:26:42.139-05:002014-02-13T13:26:42.139-05:00Pete: "The idea of concrete incompleteness is...Pete: "The idea of concrete incompleteness is something mathematician Harvey Freidman has worked on extensively. It is the idea that certain patterns inherent in numbers (that can be easily observed) actually depend on certain large cardinal axioms."<br /><br />We're getting a little off-topic, but that would be very unexpected if Friedman had shown such a thing. But what the linked article says is a bit different: "In particular, the existence of large cardinals is the condition needed to tame Friedman's unprovable theorems. If their existence is assumed as an additional axiom, then it can indeed be proven that his numerical patterns must always appear when they should. But without large cardinals, no such proof is possible."<br /><br />To show that the patterns depend on LCAs, you need to show that assuming the LCA implies the pattern exists, and assuming the negation of the LCA implies the pattern does not exist. But this is actually "assuming the negation of the LCA implies that there can be no proof that the pattern exists." And this leaves the possibility that the pattern exists independent of the LCA. So, not really anything new...<br /><br />I suspect that what Friedman has done is shown how a finite proof of such a statement can be written if we assume some particular LCA (still a remarkable accomplishment). Then the statement is (finitely) provable in (ZFC+LCA), but not in ZFC (assuming ZFC is consistent). That means the statement is true, provided that (ZFC+LCA) is consistent. But in the natural numbers, either a pattern holds out to infinity, or does not (decidable in a countable number of steps), so if (ZFC+LCA) is consistent, then the statement is true (but not provable) in ZFC, something we already expect to find, thanks to Godel.<br />Richardhttps://www.blogger.com/profile/10042619745483254124noreply@blogger.comtag:blogger.com,1999:blog-15005476.post-44314082182217534482014-02-13T11:24:59.922-05:002014-02-13T11:24:59.922-05:00Hi Adam,
Thanks for bearing with me a while longe...Hi Adam,<br /><br />Thanks for bearing with me a while longer. I will attempt to clarify what I'm saying further.<br /><br />>A set of axioms expressed in a formal language sufficient to form a model of the natural numbers using a system of logic. These axioms are taken as a priori as a starting point to reason from.<<br /><br />OK, so I am not claiming that the universe is a formal system then. I am claiming that it is a mathematical structure.<br /><br />>You do not define "mathematical structure" anywhere that I have seen.<<br /><br />OK, here is my attempt to define what I mean:<br /><br />Definition: a mathematical structure is an abstract object which can be defined entirely unambiguously.<br /><br />Explanation: If an abstract structure is defined unambiguously, then there can be no disagreement about its properties or what they entail. I think mathematics is the study of such objects. The lack of ambiguity makes it possible to e.g. prove theorems. To propose such a definition may in some cases require a formal system to be established first, but not necessarily as long as everyone can understand clearly what is meant. Formal systems themselves are mathematical structures on this definition, as is first order logic, which as the most basic formal system is not itself grounded in any deeper formal system.<br /><br />>This seems nonsensical to me. Read the sentence again and tell me if you can see where I have a problem with it.<<br /><br />That's why I emphasised the word "by". The idea "My car is grey" is not defined *by* English, it is defined *with* English. By this I mean that the existence of the English language does not entail that my car is grey or vice versa. I can also express the same idea with other languages. That is what I mean when I say the Mandelbrot Set is not defined *by* ZFC but *with* ZFC (or other formal systems).<br /><br />Finally I would agree with you that the MUH is neither a scientific nor a mathematical idea. It is a metaphysical idea, but one which I think must necessarily be true if we accept:<br />1) naturalism<br />2) mathematical platonism<br />3) the computational theory of mind<br /><br />I explain why on my blog.<br /><br />Regards,<br />DMDisagreeable Mehttps://www.blogger.com/profile/15258557849869963650noreply@blogger.com