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Visit With Paul Davies on Block Time

  1. Apr 9, 2012 #1
    Please forgive another round on the block universe, but I had a brief visit with Paul Davies and wanted to give the folks on the forum a report along with some quotes from his book. Sorry for the lengthy post.

    I was at a high school reunion in Phoenix last weekend and was staying with my brother in Tempe, AZ. Paul Davies, author of several physics books (popularizations for the layman), is the Director of the Cosmology and Beyond Center in the physics department at Arizona State University in Tempe. I read again his book, “About Time” (again), during the flight out and was anxious to meet him. I caught him in the middle of a busy schedule, but he was very nice in taking a few minutes to visit and autograph my copy of his book.

    I mainly wanted to report for you a sense of his views on the block universe concept that we’ve worked over quite a bit here. It is clear from his book and comments to me that he embraces this concept and feels that most physicists do as well, although he seems to prefer the term, “block time.” I could not get him to buy into the idea that the 4th dimension, X4, is fundamentally of a spatial character, having the same basic spatial nature as X1, X2, and X3.

    I asked him, “Given the block universe concept, should we not regard X4 as having a spatial nature rather than thinking of it as a time dimension?” His comment was, “Well, you can think of it that way, but that’s not the world we live in.” I think that is pretty much in line with the thinking of most of our forum members here.

    Here are a few quotes from his book on the block time concept:


    He has a section in Chapter 2 titled, “Timescape.” Each section has a quote or notable saying under the title. This section has a quote by Einstein that many of you are familiar with, “The distinction between past, present and future is only an illusion, even if a stubborn one.”

    Page 71. Paul Davies writing, “…there is only one rational conclusion to draw from the relative nature of simultaneity: events in the past and future have to be every bit as real as events in the present.”

    “The idea that events in time are laid out ‘all at once’ motivated Einstein to write the words quoted at the start of this section.”

    Page 72. “If time can be spatialized, at least for the purposes of mathematical representation, then it must be treated as a fourth dimension…”

    Page 73, referring to the British scientist Charles Hinton’s 1880 essay “What is the Fourth Dimension” . “Hinton asserts that the now of our conscious awareness is merely a subjective phenomenon…”

    Paul Davies writing, “Einstein himself wasn’t too thrilled with the unified spacetime idea at first, dismissing Minkowski’s new four-dimensionality as ‘superflous’ pedantry, but he came around to the idea in due course.”

    Page 76. “Spatializing time like this may advance our understanding of physics, but a heavy price has been paid. Human life revolves around the division of time into past, present and future; people will not relinquish these categories just because physicists say they are discredited.”

    “This is perhaps what disturbs people most about block time. If the future is somehow ‘already there,’ then we can have no hand in shaping it.

    ”Weyl once wrote: ‘The world does not happen, it simply is.’ If you believe Weyl, Einstein did; hence the quote at the start of this section, penned in consolation to Besso’s widow following his death (just a few weeks before Einstein’s own.”

    “In their professional lives most physicists accept without question the concept of the timescape (block time), but away from work they act like everybody else, basing their thoughts and actions on the assumption of a moving present moment.”

    CHAPTER 12

    Page 253. “I have already explained how the theory of relativity leads to the notion of block time, and the picture of time as the fourth dimension simply ‘laid out all at once.’ Since Einstein, physicists have generally rejected the notion that events ‘happen,’ as opposed to merely exist in the four-dimensional spacetime continuum.”

    “...Oceans (rivers?) of ink have been spilled on the subject, yet still the flow of time is as mysterious as ever, So mysterious, in fact, that philosophers such as Smart have been forced to conclude that there is no river of time. It is, so to speak, all in the mind. ‘Certainly we feel that time flow,’ Smart concedes, but in his opinion, ‘this feeling arises out of metaphysical confusion.’ In fact, he believes it is merely ‘an illusion.’ ”

    Page 255. “David Park is a physicist and philosopher at Williams College in Massachusetts with a lifelong interest in a time which he too thinks doesn’t pass. For Park, the passage of time is not so much an illusion as a myth, ‘because it involves no deception of the senses… One cannot perform any experiment to tell unambiguously whether time passes or not.’ “

    “When it comes to the truly objective properties of the world, reference to the flow of time appears superfluous.”

    Page 260. “Einstein scuttled the notion of a universal now, and pointed the way to ‘block time,’ in which all events—past, present and future—are equally real. To the physicist, human beings of the twenty-fifth century are ‘there’… They are there in the future.”

    Paul Davies's views would not be properly represented without his comment on page 283 (the last page):

    "In my opinion, the greatest outstanding riddle concerns the glaring mismatch between physical time and subjective, or psychological time. Experiments on human time perception are in their infancy; we have much to learn about the way the brain represents time, and how that relates to our sense of free will."

    "It is my personal belief that we are approaching a pivotal moment in history, when our knowledge of time is about to take another great leap forward. Einstein left us an important legacy. He showed us how time is part of the physical world, and gave us a magnificent theory that interweaves time with space and matter. Throughout the twentieth century, scientists have diligently explored the consequences of Einstein's time, both theoretically and experimentally. In doing so, they have unearthed some unnerving and bizarre possibilites, many of which have turned out to be true. Yet they have also encountered severe obstacles to a full understanding of time, hinting that Einstein's revolution remains unfinished. I believe its completion will prove a major outstanding challenge to twenty-first-century science."
    Last edited: Apr 9, 2012
  2. jcsd
  3. Apr 12, 2012 #2
    I thought I'd add a quick refresher for those who may not have followed some of the previous discussion on the block universe. The space-time sketch lower left summarizes the motivation for a block universe. Two observers moving relative to each other live in two different 3-D cross-sections of the 4-dimensional universe. They share the same event at the intersection of their two world lines, yet each of their instantaneous 3-D worlds includes two different 3-D instances out of the world line of a third observer. The brown guy has Red at event R2 in his world while the blue guy has Red's event R1 in his world. The third observer (Red) must be a 4-dimensional object for this to be possible.

    The sketch on the right depicts a thought experiment in which different observers record data at the various numbered events along their world lines. All observers get together at some later time to compare their data--which will provide experimental verification of a 4-dimensional universe occupied by 4-dimensional objects (including the bodies of the observers). We have to understand more about consciousness and time to complete our knowledge about the 4-dimensional universe.

    As Dr. Davies has commented, for physicists, unraveling the riddle of the glaring mismatch between the time of physics and our subjective time is one of the greatest challenges of the 21st century.

  4. Apr 12, 2012 #3
    Bob, thanks for this very nice and instructive summary!
    Yes indeed; some of those who embrace the block universe concept emphasize nevertheless that there is a physical difference between time and space which is also reflected in the mathematics.
    Well if it means what I think that he means with it (that RoS is proof for "eternalism") then that's obviously wrong, as coincidentally some of us are just now trying to explain under the topic "Relativity of Simultaneity" (and if it doesn't mean that, then I have no clue as to what it could mean!).
    I agree with that: "the flow of time" must be seen as just a metaphor, time is not a mystery but simply a comparative measure of the progress of physical processes.
    That brings back old memories of being riddled by the concept of "time". However, nowadays there is hardly any mismatch between the physical time that I measure and the subjective time in my head. Perhaps he meant with "physical time" the block universe interpretation of physical time?

    Last edited: Apr 12, 2012
  5. Apr 12, 2012 #4
    In Blue's reference frame event 5 is simultaneous with event R1.
    In Brown's reference frame event 5 is simultaneous with event R2.

    The Block Universe conclusion that events R1 and R2 are simultaneous is in direct contradiction to Relativity, because the the relativity of simultaneity predicts that if two events are simultaneous in one reference frame, then those same two events are not simultaneous in another reference frame that has relative motion. Therefore the Block Universe idea is not compatible with SR.

    Time is a difficult concept but one notion of time is that, time is what stops everything happening all at once. The block universe seems to take the view that everything does all happen at once.
    Last edited: Apr 12, 2012
  6. Apr 12, 2012 #5
    Nice to see you jumping in here, harrylin. Good comment. However, if you are referring to the imaginary sqrt(-1), I still think that relates only to the rotation of the X4 axis and is not related to time per se.

    Yes, that is exactly what he means.

    Yes, that's exactly what he is referring to--the block time interpretation of physical time.
  7. Apr 12, 2012 #6
    Hi, yuiop. I think we are having semantics difficulty. Davies has used language like "...all at once" to describe the 4-dimensional universe populated by 4-dimensional objects. What you have said about simultaneity is correct. However, the "all at once" phrase is trying to communicate the concept of 4-dimensional objects.

    If you look at a waste baskect sitting over in the corner not moving--it's static as time passes, it is a trivial concept to you to observe that the whole basket is "all there at once." All points throughout the basket exist simultaneously in your perception of a 3-dimensional world. You would have to imagine some hyperdimensional beings observing our 4-dimensional universe, observing that it a static 4-dimensional object. In the block universe view, they would see all of the 4-dimensional objects occupying the static 4-dimensional universe as "all there at once" as time passes for those little green hyperdimensional beings.

    Special relativity is not at all incompatible with the block universe concept. It is actually implied from the relativity of simultaneity.
  8. Apr 12, 2012 #7
    I think that most people agree that however you put it (as sqrt(-1) or as simply a minus term) puts the "time" dimension in a different category as the three "spatial" dimensions. On that point I also agree with Davies.
    Then you may be interested to participate in the (now almost ended?) thread on Relativity of Simultaneity. :smile:

    Best regards,
  9. Apr 12, 2012 #8
    I find block time distasteful as it implies that everything that will ever happen has in effect already happened. It rules out free will.

    As I see it, block time is like a video recording. The ending of the film is predetermined, but we are only aware of the plot up to the point where the tape head is on the tape. Block time has it that some viewers are further ahead in the film but are not allowed to communicate what they know about the "future" to viewers in the "past". Block universe theory seeks to rule out notions of past and future, but does not explain how future events are isolated causally from past events. Lets say I am destined to win the Lottery tomorrow. Block time theory would have it that some fast moving observer is already aware of this fact, so what in block time theory prevents them communicating this fact to me now (along with the winning numbers so that I can be doubly sure the future unfolds correctly)? Block time just seems to be a way to make money from seminars and books without really coming up with any satisfying conclusions or practical applications.
  10. Apr 12, 2012 #9


    Staff: Mentor

    But SR, the actual *theory* (whatever "block time" might say) does *not* say this or imply it. More precisely, the theory of SR allows you to construct a *model* in which "block time" works this way, but it in no way ensures that such a model will give accurate predictions about actual reality.

    "Block time" talks about what is in the *simultaneous spaces* of various observers in relative motion. But nobody is "aware" instantaneously of what is going on in his simultaneous space. We are aware only of what is in our past light cones. So only portions of spacetime that have your winning of the lottery tomorrow in their past light cones can be said to be "aware" of it. And none of those portions exist "now", nor are any of them in the simultaneous spaces of anyone "now".

    And since our actual knowledge of the actual state of the universe is imperfect, we cannot know for *sure* that you will win the lottery tomorrow, or which observers will later observe that event. We can only extrapolate based on the data we currently have. And we can never have all the relevant data, since, again, we don't have data from our entire simultaneous space, only from our past light cone. And that does not even cover *any* entire simultaneous space; given any spacelike slice, "cut" from spacetime any way you like, our past light cone only intersects a finite portion of it.

    So the claim Davies appears to be making about "block time" is in fact false. More precisely, it is true only of a *model* in which we have perfect knowledge of all conditions on an entire spacelike slice, but not of actual reality, in which we do not have such knowledge.
  11. Apr 13, 2012 #10
    I certainly understand the logic leading to this conclusion but have to wonder if it is the only possible one.
    This would seem to be the case only if you attach real temporal significance to the relative clock readings and operational simultaneity of SR This seems to be antithetical to the basic premise of RoS. The only meaningful simultaneity is regarding events in close proximity.
  12. Apr 14, 2012 #11
    But, the block time model does give accurate predictions, because it directly represents geometry based specifically on Lorentz transformations, and taken at face value, SR does imply a block universe. It's just that the concept runs so contrary to our subjective experience.

    Paul Davies implied that those who rely on this argument to refute block time are in effect solipsists. The block time is not the only concept susceptible to such an argument. Essentially, if you take this position, you cannot validate any reality associated with your current “NOW” event in the universe, because you do not directly experience anything in your simultaneous space—you only experience events communicated from the past via signals just arriving from the past. However, we physicists never discount the reality of our simultaneous 3-D cross-section of the universe just because we do not directly experience it at the instant of our "NOW." We have ample opportunity to validate "NOWS" of the past later on. All experimental data is handled that way--data is acquired from signals arriving from the past, and then we process it later and evaluate the inferences. The logical positivists took things way too far down the path of requiring direct experiences--and they have been marginalized as a result.

    But, of course no one is really concerned about this philosophical stumbling block. All of our experimental observations in physics come from past events. Of course, those are events from within your past light cone. You just wait for whatever time is needed to receive that data (we had to wait a long time to get the data on the first super nova). Then, predictions are made about future events, which are then validated at a later time. I have already presented a space-time sketch illustrating a thought experiment making predictions based on the block time model. The predictions are accurate because the block time model represents the geometry associated with the Lorentz transformations.

    This has nothing to do with the validity of a model. When have physicists ever required knowledge of the universe outside of the light cone of the laboratory to validate a model?

    You have not made your case. Just stating that Davies’s claim about block time is false does not make it false. And your earlier comments have not proven your point. I don’t know why you insist the block time model must have data from outside the light cone of the lab, and data over an entire slice, when we never require that of models.

    Having said all that, I must say that I am, at a subjective level, quite sympathetic to the thoughts you've expressed here.
    Last edited: Apr 14, 2012
  13. Apr 14, 2012 #12
    Austin0, I have wondered exactly that same thing for quite a few years. You do have another option if you wish to consider the Lorentz Ether Theory (LET) of Lorentz and others. Other forum members have really been working this topic with intensity in another thread (I believe it is still active--I felt like they didn't really need my 2 cents on that one, but I'm finding it interesting). Lorentz's approach attempts to account for the relativity of simultaneity using an absolute fixed ether medium and the results of dynamical interactions which could account for apparent fixed value of c (light speed) for all observers, time dilation and length contraction.

    To be honest, although I have read much about the theory, I have not scrutinized it in sufficient detail to offer a valid critique. The theory never seemed to gain traction after Einstein's theory was widely accepted. Einstein seemed to feel like LET was too adhoc.

    But, if nature really works like LET, then there would be real temporal significance to the relative clock readings, etc.
    Last edited: Apr 14, 2012
  14. Apr 14, 2012 #13
  15. Apr 15, 2012 #14


    Staff: Mentor

    There is no such thing as "the block time model" as a separate generator of predictions, over and above standard SR; there is the *theory* of SR, with "block time" as one *interpretation* of what the theory is telling us. More precisely, there is the theory of SR combined with the fact that Minkowski spacetime gives a correct geometric model of that theory. That is all that is implied by SR "taken at face value". The "block time model" requires further assertions about the physical meaning of the geometric model, but those assertions are *not* required to give correct physical predictions.

    You are missing my point. I am not saying that the simultaneous 3-D cross-section of spacetime is not "real", or that it is "less real" than the past light cone. I am saying that, until we get light signals from events that are spacelike separated from us "now", we *do not know what happens at those events*. Therefore, we do not know that our *model* of our simultaneous space "now" is *correct*. The "block time" view assumes that it is; that assumption is not valid. Another way of putting this is that the "block time" view requires us to have perfect knowledge and powers of prediction; but we don't.

    Or to *refute* them. Sometimes our models are shown to be wrong. The "block time" model does not appear to allow for this possibility.

    Or *refuted* at a later time. Once again, you are assuming your predictions will always be right. But they can't always be right, because you don't have all the relevant data to predict from. See next comment.

    No, that's not enough by itself. For your predictions to be accurate, you must also have complete information about initial conditions, not just in your past light cone, but in the past light cones of *all* events in the "simultaneous space" that you are predicting. In your example, that would be complete information about all events in the past light cone of Red's worldline up to event R2.

    In your example, you just declare by fiat that you have all that information; but in the real world, that's impossible. There could be causal influences coming in to Red's worldline from way outside Brown's past light cone at event B, so Brown could not possibly know about them; and those influences could drastically change Red's worldline, so that he didn't even pass anywhere near event R2. Similar remarks hold for events all along Brown's "simultaneous space" at event B.

    The "block time model" claims that all the events along Red's worldline are "real"; but how can this claim possibly be justified, when it might turn out that Red's worldline never even passes through those events?

    When have physicists claimed that a model gave us accurate knowledge of what is happening "now" at someplace far away? For example, when have physicists claimed that they know for sure what is happening in the Andromeda galaxy "right now" based on extrapolations from light that reached us from there yesterday?

    See next comment.

    What is required of models depends on what the models are claiming. The more sweeping and comprehensive a model's claims are, the more we require of it. The "block time" model makes claims about the entire universe; it says that *all* of spacetime is "real all at once", so to speak. Therefore, it is quite proper to require such a model to justify such a sweeping claim. Which it can't, because it can't possibly have all the data required to do so.
  16. Apr 17, 2012 #15
    I'm not sure I understand the point you are making here. Let me take it one step at a time with a couple of questions about the sketch below. It is a space-time diagram of a muon created in the upper atmosphere of the earth, traveling at relativistic speed to the earth surface where it is detected in a physics laboratory. I know you are quite familiar with the experiment and understand that without relativistic factors one would not expect the muon to reach the earth's surface, given its short lifetime. The ability of the muon to actually make it to the earth's surface becomes clear when viewing the space-time diagram.

    Question 1) At the event of the muon creation, would you say that the muon exists in the simultaneous space of the physicist and test lab at t'A?

    Question 2) At the event of the muon creation, would you say that the physicist and lab at event t'B exist in the simultaneous space of the muon (at the event of the muon creation)?
    Last edited: Apr 17, 2012
  17. Apr 17, 2012 #16


    Staff: Mentor

    Based on the way you are using the word "exist", I would say no. The statement that is in your drawing, "the muon exists as a 4-dimensional object in 4-D spacetime", is, I would say, correct (subject to technical points about in what sense the complete worldline of the muon "exists" in a world of quantum indeterminacy--but that's a separate issue). But that in itself does not require the muon to "exist" anywhere except on its actual worldline. More precisely, from the "block time" perspective, the muon *is* its worldline--it *is* a one-dimensional curve in 4-dimensional spacetime. Saying that the muon "exists in" various simultaneous spaces sounds like saying the muon exists somwhere besides itself, which makes no sense.

    In other words, making claims about things existing in each other's simultaneous spaces is really a misstatement of what the "block time" model says. The "block time" model says that the entire 4-D spacetime exists, period. It does not say that individual objects within that spacetime "exist in each other's simultaneous spaces"; that phrase really has no meaning from the "block time" viewpoint. From the "block time" viewpoint, all you can really say is that a certain set of spacelike surfaces all pass through some particular event, such as the event of the muon being detected. But that says nothing about the muon "existing in" those spacelike surfaces; the muon exists as a one-dimensional worldline in 4-D spacetime, and the surfaces exist as 3-D surfaces in the same spacetime, and surfaces all intersect the line. That's it.
  18. Apr 18, 2012 #17
    Of course. The sketch indicates that the muon exists only on its worldline. I could have been more careful about referring to the existence of a 3-D cross-section of a 4-D object within a particular 3-D simultaneous space, but I assumed the context would make it clear.

    Actually, I should not have included the statement about the 4-dimensional existence of the muon, since that was to be a final conclusion to be arrived at after establishing the facts about the everyday understanding of objects sharing simultaneous 3-D spaces.

    Evidently there is no need for me to establish for you the existence of the 4-dimensional muon. That’s good. However, there should be no confusion over the concept pieces of an object existing as well as the entire object existing. For example do you find it confusing to consider the airplane you are looking at on the runway as existing—and then also consider that the left wing of the airplane exists?

    I directly observe 3-D objects that are in my current simultaneous 3-D space (hopefully we’ve all been exposed to enough special relativity to understand what we mean by simultaneous space). Those objects exist as 3-D cross-sections of the complete 4-dimensional object.

    No. On the contrary, the block time model is actually motivated out of an acknowledgement of multiple simultaneous 3-D spaces—a logical outcome of the relativity of simultaneity.

    We are in agreement on this statement--except for your "Period."

    Wrong. That dismisses the bedrock of the block time viewpoint. Again, the block time model depends on multiple simultaneous spaces and the analysis of which 3-D spaces are shared by which 3-D cross-sections of which objects.

    Why is it so difficult to think of the existence of a small part of a large object? Can I not think of my hand existing? Must I be restricted to referring only to my entire body as existing?

    Perhaps our problem here is just one of semantics or word usage.
  19. Apr 18, 2012 #18


    Staff: Mentor

    NO YOU DO NOT. Sorry to shout, but you keep on saying this without stopping to think about what you are saying, and it's the crux of my argument against the "block time" view. Taking the above statement as it's given, you are saying that light travels at infinite speed. It does not. You do *not* directly observe 3-D objects in your current simultaneous space. You directly observe 3-D objects in your past light cone. The fact that, under ordinary circumstances, the difference is negligible (light travels about 1 foot per nanosecond, so ordinary light speed time delay is about eight orders of magnitude smaller than the threshold of conscious perception, and about six or seven smaller than the time scale of the underlying neural activity) does not change the principle.

    You *construct* your current simultaneous 3-D space by extrapolating observations taken from your past light cone. But that construction cannot possibly be complete or guaranteed to be correct. (Again, the fact that under ordinary circumstances the construction is more than good enough for your everyday activities does not change the principle.) The "block time" argument, taken literally, would require that construction to be complete and correct. So the "block time" argument fails.

    That's my position in a nutshell. Note that I nowhere said that objects do not have 3-D cross sections in multiple simultaneous spaces; of course they do. That does not change the fact that all of those multiple simultaneous spaces are constructions, not direct observations, and that the constructions are not complete nor guaranteed to be correct. Nor did I say anything about parts of objects somehow not existing even though all of the objects exist; obviously if a whole object exists, its parts exist as well. But again, that's irrelevant to my argument.

    If you have further arguments that directly address the above, I'd be glad to see them. But there's no point in repeating stuff that's irrelevant to the argument. If your position is simply that the construction of the multiple simultaneous spaces is sufficient ground for accepting the "block time" view, then we'll just have to agree to disagree, since I don't think it is.

    I think it's that you haven't been grasping the essence of my position. Hopefully the above makes it clearer.
  20. Apr 18, 2012 #19
    Regarding simultaneity and SR I think that a lot of the confusion in this regard is an unfortunate side effect of the conventions of Minkowski diagram graphing. I.e. The angled plane of the hypersurface of simultaneity. This is easily misconstrued as indication of a dichotomy , of two realities.
    In actuality this hyperplane is nothing more or less than the coordinate plane of the primed frame under an alias.
    In the real world it is superposed on the unprimed plane with complete correspondence. Not metrically but existentially.
    Every observer and every spatial point on both frames is absolutely simultaneous with its corresponding point by our only definition of actual simultaneity; Proximate occurence at a single instant.

    ALL observers agree on this. There are not two sets of events but rather a unique set of events with two interpretations.
    It is only when we take events at separate locations and try to establish a temporal relationship between them that the appearance of temporal disjunction and duality occurs. Of course this endeavor is futile from the onset if you actually accept the basic premise of RoS.And IMO misguided if you attach any significance to any apparent determination.

    In simple graphics: AN endless train on an endless track. Passing through time every point on the train is simultaneously occuring with a cojacent point on the track. They may disagree on the rate of coordinate time, spatial metric and clock synchronicity but they agree on this simultaneous local occurence throughout both frames and throughout time.

    FOr myself I see no compelling reason not to consider the frames globally simultaneous in occurence. ANd simply consider the disagreement to be a consequence of kinematics and clock convention not necessarily an intersection of actually disjunct time periods.

    I am not that familiar with block time but from what I know Peter has accurately described what might be called fundamentalist block time.
    You seem to be describing a particular derivation shared by many SR people.

    One interpretation of block time is that yes, all world lines are actually existent in their entirety in some hyperdimension but reality (or subjetive reality) still occurs as a globally simultaneous, dimensionless slice through those lines.
    One SR version is of course every particle having its own unique slice through the continuum.

    A little aside; I recently had occassion to read about Zeno where I found that I had made completely erroneous assumptions regarding his purpose and perspective on his own paradoxes.
    It appears that, not only was he seriously arguing the concept of the motionless arrow, but that it was in defence of his mentors, particularly radical, version of block time.
    In which, not only is all reality fixed and predetermined, but physics itself, motion, causality etc were actually mere appearences emerging from the passage through this preordered continuum.

    Approx. 2300 years and here we are again.
    Last edited: Apr 18, 2012
  21. Apr 18, 2012 #20


    Staff: Mentor

    :rofl: I hadn't quite thought of it this way, but you're right; I'm kind of arguing for a "fundamentalist" view of what we can be sure is "real" or "exists", based on the limitations of our ability to gather information.
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