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- Local Features in Natural Images via Singularity Theory
- Local Features in Natural Images via Singularity Theory | James Damon | Springer
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- 1. Spacetime Singularities
- SIAM Journal on Mathematical Analysis
In spite of the fact that at this level there is no time, one can meaningfully speak about dynamics, albeit in a generalized sense. Space, time, and singularities appear only in the transition process to the macroscopic physics. This idea, explored here in more detail, clearly favors an atemporal understanding of creation. Volume 35 , Issue 3. The full text of this article hosted at iucr. If you do not receive an email within 10 minutes, your email address may not be registered, and you may need to create a new Wiley Online Library account.http://abutparker.com/pulpit-and-politics-competing-religious-ideologies-in-canadian.php
Local Features in Natural Images via Singularity Theory
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Local Features in Natural Images via Singularity Theory | James Damon | Springer
View Preview. Learn more Check out. Abstract One of the most important and most frequently discussed theological problems related to cosmology is the creation problem. Citing Literature. Volume 35 , Issue 3 September Pages If one wishes to think of Big Bang cosmology as involving, say, a sufficiently vast expanding ball of matter surrounded by empty space, the data supporting hot Big Bang cosmology provide no obstacle; of course, the Robertson—Walker metric will not apply in the exterior vacuum region.
A bit of historical perspective is useful here. It is not difficult to see how finitism could serve teleological world views by reducing the probabilistic resources for unguided evolutionary processes, or how its denial could serve nonteleological world views, but experiments have not resolved the issue and perhaps cannot. While a number of people noted above have regarded the Big Bang singularity as potent evidence for creation in time and hence for theism, perhaps the most visible sustained defenses have come from Craig Craig [ ]; Craig and Smith [ ]; Copan and Craig [ ] and from astrophysicist—apologist Hugh Ross Ross [ ], [ ].
The conditionalizing on GTR soon disappears, however, with a ringing endorsement of that theory: Today it can be said that no theory of physics has ever been tested in so many different contexts and so rigorously as general relativity. The fact that general relativity has withstood all these tests so remarkably well implies that no basis at all remains for doubting the conclusions of the space-time theorem Ross  , p.
By contrast, William Lane Craig argues more carefully and concludes more modestly.
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Therefore the universe has a cause of its existence Craig  , p. This argument is valid, at least if it is unproblematic to treat the universe as a thing, rather than a collection of things; I assume that there is no problem here. The truth of the second premise, or rather, the source of warrant for the second premise if it is true, is the key question. If one accepts, say, the Bible as divinely inspired, then one has already accepted theism and much else besides. The question therefore arises whether to take the singularity seriously as a feature of the real world, or to dismiss it as an artifact of incomplete physical understanding.
As one sees all the time in papers on quantum gravity, most people who work on quantum gravity take for granted that the Big Bang singularity is an artifact of incomplete physical understanding and expect or hope that uniting gravity with quantum mechanics in some kind of quantum gravity will resolve the singularity into some well-defined situation that admits extrapolation to still earlier times, ad infinitum. Jayant Narlikar has persuasively deployed this point as a critique of the argument from the singularity to theism Narlikar [ ].
Tolerating singularities and trying to learn from them, as Earman does, is an attitude that should commend itself only to GTR-exceptionalists, those who emphasize the differences between GTR and the other forces over the similarities between them. If one thinks that gravity as portrayed by GTR is importantly like other forces, then gravitational singularities are not appreciably more interesting than the singular electric field of a point charge, which simply needs to be resolved by a better theory, such as quantum electrodynamics. Then, however, one has strong technical reasons to doubt that singularities exist as part of space-time.
Thus there is no beginning required and premise 2 might be false, as far as physics can tell. While probably all who tolerate singularities are GTR-exceptionalists, many or most GTR-exceptionalists, such as those who work on canonical quantum gravity and loop quantum gravity, do not tolerate singularities. It appears, then, that whether one is tolerant or intolerant toward singularities, it turns out that there is no first moment unless one is installed by hand , because every moment is preceded by earlier moments. In order for the Big Bang singularity to provide a good theistic argument, the singularity must be well enough behaved to be a real and intelligible part of space-time, and badly enough behaved that it cannot have a past.
Satisfying both conditions seems difficult and unlikely to be achieved. Moreover, there are various reasons, some quite good, for not tolerating singularities, which it will be worthwhile to explore. One might think that Leibniz has provided the prototype for a good critique of the singularity argument for theism. In the Leibniz—Clarke correspondence Alexander  , pp.
Whereas one sometimes encounters biological dysteleology arguments, this is a physical dysteleology argument. Whether Newton and Clarke deserved this criticism need not concern us. If the physical world is like a watch, then it ought to be able to run forever without breaking down. There is no obvious analogy to the repair of the watch. Thus the singularity theorems arguably show that GTR demonstrates its own inadequacy, one might conclude.
Because God would not build the world so incompetently, it follows that GTR is not the correct theory for describing gravitational collapse; the true theory would not yield singularities.
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But GTR is time-reversal invariant, and the Big Bang singularity is simply the time reversal of a specific model of gravitational collapse of a star, which uses a Robertson—Walker metric for the stellar interior Misner et al. Thus the Big Bang singularity would be eliminated along with the singularities of gravitational collapse, or so the argument might go. This sort of argument continues to be used today, often without the explicit theism, by theoretical physicists. Thus Abhay Ashtekar, one of the dominant figures in contemporary work on quantum gravity, opened a recent review of the field with the following motivation: Big-Bang and other singularities: It is widely believed that the prediction of a singularity, such as the big-bang of classical GTR, is primarily a signal that the theory has been pushed beyond the domain of its validity.
A key question to any quantum gravity theory, then, is: What replaces the big-bang? He lived in a still Christian age and took himself to know many things by divine revelation; many contemporary physicists cannot say the same. Thus Elliott Sober has recently warned against this sort of a priori theological claim about what God would or ought to do Sober [ ], [ ]. The Leibnizian intuition that the world would not break down after a finite time has a certain appeal for physicists, but reflection suggests that it might be difficult for the argument to get traction with those not already disposed to accept it, except perhaps for theistic rationalists such as Leibniz.
That this Leibnizian intuition perhaps stripped of its theistic justification in fact appeals to physicists may reflect the fact, discussed in theories of science by Thomas Kuhn, Imre Lakatos, and others Laudan et al. The Leibnizian intuition might well be truth-conducive, but it is difficult to argue for that conclusion. This phenomenon bears some resemblance to the sort of stalemate that can result in the scientific realism—antirealism debate Kukla [ ]. The story of blackbody radiation and the development of quantum theory is complicated Kuhn [ ] , more so than the view given in modern physics textbooks, but the following selection should suffice.
A century ago, there were good classical theoretical arguments for the Rayleigh—Jeans law for blackbody radiation, according to which the energy density for radiation at a given frequency increased with frequency. The Rayleigh—Jeans law was also demonstrably false empirically, but in some ways that is an irrelevant accident for present purposes.
Lakatos has already extolled the virtues of rationally reconstructed history. Ordinary objects, especially black ones, approximate blackbodies, so they would radiate away their energy immediately in a blinding flash, contrary to experience. The answer of Planck and others to the threatening inference of ultraviolet catastrophe, history shows, helped to lead to quantum mechanics. New theoretical foundations were brought in that yielded an exponentially decaying factor to counteract the Rayleigh—Jeans power law growth, and thus give a convergent integral up to infinite frequency.
The exponentially decaying factor preceded widespread worries about the ultraviolet catastrophe, but previously it was motivated on more empirical grounds. From a logical point of view, the ultraviolet catastrophe was a reductio ad absurdum of the classical physics underlying the Rayleigh—Jeans radiation law. The solution was new physics of a quantum kind, which averted the catastrophic infinity; no miracle was required.
Using this case and others, one might argue inductively that just as quantum mechanics resolved these problems, so it will resolve the singularity problems of GTR. A theory that predicts disaster here and now clearly needs to be changed, if disaster is not observed. However, GTR fits the data rather well. What of its ultimate mathematical breakdown in singularities?
It would seem that the supposed inductive argument relies on a metatheoretic criterion besides empirical adequacy to determine that GTR breaks down, one that strongly resembles the Leibnizian intuition previously discussed. So this inductive argument will not persuade those who are not persuaded by the Leibnizian intuition.
But there are more compelling reasons for rejecting the argument from the Big Bang to theism. One might wonder why theological significance should be ascribed to the Big Bang singularity, but not to other physical singularities. This worry takes its most acute form when one considers the similarity of the Big Bang cosmology to the time reverse of the gravitational collapse of a star to a black hole with a central singularity. Assuming homogeneous matter distributions, both Big Bang cosmology and stellar gravitational collapse use the Robertson—Walker space-time metric in the matter-filled region Misner et al.
If one chooses a Big Bang model with a bounded matter distribution, as one certainly may Bondi [ ]; Layzer [ ]; McCrea [ ]; Callan et al. The main differences are the direction of time and the distance scale. The distance scale does not seem important for present purposes. If the Big Bang strongly indicates that there exists a God who created the universe, do formally similar time-reversed events such as the gravitational collapse of stars to form black holes with singularities imply that there exists a God who supernaturally destroys annihilates, ceases to uphold the interiors of stars?
That God is the Destroyer of collapsed stars is a rather surprising conclusion. While Christianity portrays God as creating, sustaining, and redeeming the world, the idea of God's absolutely annihilating either the physical world as a whole or a bit here and there from time to time appears to be a novelty or certainly a rarity in Christian theology.
Of course rocks, buildings, and animals perish from time to time, but their physical remains persist, which is precisely what does not happen in the stellar case. Given the divine policy of upholding the universe after creating it, for God to stop upholding, say, a star that collapses to a black hole, would be a miracle. Annihilating an object ceasing to uphold it in existence might not be the same kind of miracle as creating one ex nihilo , but it is a miracle nonetheless. I would not be appreciably less astonished by witnessing the sudden annihilation of my car than by seeing the abrupt appearance ex nihilo of a horse.
Thus, whatever asymmetry of time might exist in relation to causation, it does not help to avoid the conclusion that stellar gravitational collapse terminates in a miracle. The proponent of the Big Bang argument therefore needs to explain why the termination of stellar gravitational collapse is not a miracle, or why such a miracle is not pointless and hence absurd. It is somewhat comforting that God, on the view in question, would only annihilate regions of high density, perhaps typically surrounded by an event horizon.
Thus there is plenty of warning so that we may avoid these episodes of annihilation, and the farmer who stores wheat in a silo need not fear the annihilation of his wheat. But the theoretical oddness remains. The theological distinction between ordinary and special providence, though probably never absent historically else the distinction between miracles and ordinary events would be meaningless or hopelessly vague , received a boost in the twelfth century in the west Grant  , pp. Given this robust though qualified affirmation of the general integrity of natural processes, which likely was shared outside the Reformed tradition that produced the Westminster standards, the claim that God annihilates collapsed stars would have seemed odd even to pre-Enlightenment orthodox believers if we may imagine them informed about twentieth century physics , for whom miracles were no embarrassment.
If GTR were the only possible theory to describe the divine governance of the world in matters gravitational, perhaps one could manage to accept the conclusion that God miraculously destroys collapsed stars. But given the underdetermination of theories by data and consequent avoidability of that conclusion, that inference appears to be a reductio ad absurdum. There is a misleadingly persuasive move made by proponents of the Big Bang argument, stacking the deck in favor of GTR and thus of Big Bang cosmology against nonsingular rivals. The move is generally not made explicitly and in detail, so what follows is a reconstruction of the reasoning process that would underlie any good argument in the vicinity.
It is often suggested that potentially nonsingular rivals to GTR are speculative, whereas GTR is well confirmed, so GTR and its retrodiction of the Big Bang singularity ought to be accepted as the default view that challengers need to overcome with better empirical results. There is a grain of truth in this claim: some or perhaps many of the theories or models actually proposed as rivals to the Big Bang in the more speculative literature do not form part of a well-tested theory that is known to reproduce the empirical successes of GTR.
Thus some of these challengers might be refuted by data already in hand, should someone think to do the necessary calculations and apply the relevant empirical data to the challengers. Thus one cannot simply collect dozens of papers that discuss nonsingular cosmological models and thereby conclude that there are that many live challengers to Big Bang cosmology at the moment.
However, the grain of truth in this objection sometimes conceals the grain of falsehood that it also contains. Though not entirely trivial, it is possible to construct theories that reproduce the empirical successes of GTR in all tested regimes to date, but which differ in the ultra-strong field regime relevant to Big Bang cosmology. Granting the success of weak and medium field tests of GTR involving light bending, gravitational redshifting, time delay, and the like, why think that GTR, rather than one of its perhaps not yet unproposed competitors that fits the data currently in hand, is the right extrapolation?
However, positive energy has yet to be proven and might be false for massive variants of GTR Boulware and Deser [ ]; Pitts and Schieve [ ] , so Big Bang singularity apologists can still hope that the massive theories are vicious.
1. Spacetime Singularities
Apologists for or rather, from the singularity need to hope that a great many epistemically possible theories are vicious, in fact, a hope that most likely will disappoint eventually, if it has not already. The need to reconcile gravity and quantum mechanics all but proves that there exists a consistent theory of gravity that matches GTR in some classical limit, but which differs from it in regimes when dimensional arguments suggest that quantum effects should be large.
There might well be many such theories of quantum gravity. The works of Abhay Ashtekar and collaborators, such as Ashtekar et al. Loop quantum gravity is a part of the modern nonperturbative canonical quantum gravity project Pullin [ ] , which began in the s when Abhay Ashtekar proposed new variables that helped to resolve long-standing problems faced when using the older metric variables. In such a context, curvatures comparable to the inverse square of the Planck length arise, so neglected quantum terms should be large and the classical theory becomes a bad approximation.
In addition to the modern canonical quantization program, one should also keep an eye on string theory as tending to resolve singularities Gasperini and Veneziano [ ] , not to mention various classical proposals that alter the dynamics in the strong field regime. Clearly, they do in the present context. Kyle Stanford argues that the problem of unconceived alternatives is an even more serious problem for scientific realism than are more commonly discussed worries Stanford [ ], [ ].
Once the rights of unborn theories are respected, the default status allegedly held by GTR and hence of Big Bang cosmology as described by GTR near the singularity disappears. Is one truly rationally compelled, or even rationally encouraged, to accept an infinite extrapolation from a curve that fits the data in some finite region?
Surely not. This is a severe curve-fitting problem. But there is no reason to restrict the competitors of GTR to theories that someone on Earth has already proposed. The relevant set of competitors for GTR includes the set of theories that agree with GTR on all experiments to date, whether already entertained on Earth or not. This set might be infinite, might well be large, likely contains several members, and almost certainly has at least one member, a quantum theory of gravity.
The set most likely has at least one member that resolves the singularities of GTR. Thus it is not at all clear why one should take GTR seriously in the strong-field regime near the supposed Big Bang singularity. Certainly, Einstein did not Earman and Eisenstaedt [ ]. Making an induction over this history, one is supposed to learn the lesson not to appeal to special divine action in new cases, lest one make religion look foolish yet again when the gaps close Saunders [ ].
Del Ratzsch notes, for example, that the argument form is valid. These worries might also be overstated historically. Are they part of the same complex of distortion as the Huxley—Draper—White thesis that the characteristic mode of interaction between science and religion has been warfare White [ ]? This claim has been refuted by modern historians of science Lindberg and Numbers [ ]; Brooke [ ]; Olson [ ]. The warfare thesis, a large-scale generalization, keeps company with some specific claims that are simply false, such as that the medievals believed in a flat Earth refuted in Russell [ ]; Grant [ ].
Given how many flaws have been diagnosed in the Huxley—Draper—White story by recent historians, one might wonder whether the definitive history of God-of-the-gaps arguments also has yet to be written. Even if one concedes that some gaps arguments for theism might not be bad arguments, it remains clear that the Big Bang singularity argument is a bad argument from gaps to God.
That is clear from the Destroyer reductio : there is no nonconventional relevant difference between the Big Bang singularity and the stellar gravitational collapse, and the latter surely has no theological significance and will likely disappear due to improved physics. Here I have assumed an intolerant attitude toward singularities. A related problem with arguing from the Big Bang to creation in time and hence theism is that such arguments depend crucially on various highly technical premises which most people cannot even entertain, much less evaluate.
It follows that the vast majority of people, even educated ones, simply are not entitled to beliefs on the matter, apart from relying on the testimony of experts. But most people, even most educated people, cannot even reliably identify relevant experts. Most astronomers and physicists are not relevant experts, though they might well write popular books and make statements to the media on such issues. Supposing that one manages to identify relevant experts, the problem remains that their expert opinions will or should vary rather rapidly with the winds and waves of research fortune.
Many once-credible energy conditions have fallen by the wayside or are seriously threatened Barcelo and Visser [ ]. It is now known that quantum field theory violates the local classical energy conditions, such as the Weak Energy Condition, though apparently quantum field theory still satisfies certain averaged energy conditions: energy density can be negative here and there, but not for very long and only with greater compensation of positive energy nearby Ford and Roman [ ]; Barcelo and Visser [ ]; Ford [ ].
Nonminimally coupled classical scalar fields violate energy conditions, as does massive gravity Visser [ ]. Are journalists, sociologists, homemakers, and truck drivers supposed to accept an argument whose premises are so technical that they cannot understand them, and so unstable that they could prove false in the next issue of Physical Review D?
It is not clear why.
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Martin Bojowald recently wrote Bojowald [ ] : Because of genuinely quantum geometrical effects the classical singularity is absent in those models in the sense that the evolution does not break down there, contrary to the classical situation where space time is inextendible. This effect is generic and does not depend on matter violating energy conditions, but it does depend on the factor ordering of the Hamiltonian constraint. Most people have no idea what that means, and thus no idea what sort of plausibility to assign a particular factor ordering of the Hamiltonian constraint.
Even if the singularity argument could not be decisively refuted, could it establish theism to a significant degree for anyone besides the few dozens of people expert in factor ordering in quantum gravity? It is unclear how. Probably not. But perhaps the choice of factor ordering is not so important after all Date and Hossain [ ] ; the fluctuations based on detailed technical premises continue.
Perhaps neither the singularity argument nor any other argument is the basis for religious faith. Craig holds to something like Reformed epistemology private correspondence and Cowan et al. Thus the religious believer does not need to read and understand every relevant paper in Physical Review D to maintain theistic belief rationally.
Maybe this is the correct way to understand the warrant for theistic belief. But if the point of making the singularity argument is to provide an argument that rationally ought to persuade some people of theism, then Reformed epistemology is simply irrelevant to the task at hand. The fluctuating for the experts or inaccessible for nonexperts character of such warrant if any as an argument from the singularity to theism can provide, is another reason that theistic apologists ought to abandon this strategy. If the Big Bang singularity does not provide a good theistic argument, one might still consider whether Big Bang cosmology is somehow especially congenial to the eye of faith.
Many hold that theism benefited from the victory of Big Bang cosmology over steady state cosmology. As a sociological—historical claim, doubtless theism did so benefit, but is there any real philosophical advantage? But since ideal points are not points of spacetime, the sense in which God can be said to cause or bring about the universe by operating at these points is very remote from the usual causal notions of science and everyday life that are concerned with connections between events in space and time.
This is not to say that theistic talk about God creating the universe is illegitimate.
SIAM Journal on Mathematical Analysis
But it is to say that such talk finds no special purchase in the big bang. Neither Big Bang cosmology nor some eternal-world cosmology has a beginning in the relevant sense of a first moment. Those who are willing to insist on a real first moment of space time, perhaps in accord with the Fourth Lateran Council, have the option of chopping off the front edge of the space-time model somewhere, whether in Big Bang cosmology or in some eternal-world cosmology, in an exercise of selective scientific antirealism.
Such a front edge presumably would be a Cauchy surface, everywhere space-like, but perhaps nearly null in places if one wishes. The question of just where to chop is vexing, however. Along with C. Clarke Clarke  , pp. If one adopts the image of spacetime as being generated or built up as time passes then the dynamical version of the principle of sufficient reason would ask why the Creative Force would stop building if it is possible to continue Earman  , p.
On the other hand, Leibniz himself held anachronism aside that space-time is maximal toward the future, but it is not maximal toward the past Alexander  , p.