Sun Xiaoli, ed.: *Philosophical Disputes in Modern Science*, 2nd ed., Beijing University Press, August 2003, (editor in charge: Su Xiangui~~, Gu Weiyu)
Du XunSun Xiaoli: New Features of Modern Mathematics
Pages 8–9
While mathematics was moving toward “specialization” and “complexification,” many ideas in the major disciplines were gradually becoming unified, and the boundaries among the major disciplines were becoming increasingly blurred. Therefore, although the content of mathematical science is immensely abundant and its scope enormous, the holistic conception of mathematics has reappeared, and workers in all kinds and branches of mathematical science have once again realized that they are engaged in a common undertaking.
////——This is good news.
Liu Xiaoli: Gödel’s Theorem and Its Philosophical Implications
Pages 39–40
What Gödel did was to state the axioms of Peano arithmetic, “the definition of the natural numbers + mathematical induction,” in the formal language of first-order predicate logic, and at the same time to denote the resulting formal arithmetic system as PA. In the paper published in 1931, “On Formally Undecidable Propositions of *Principia Mathematica* and Related Systems I,” he proved the following two important results:
Gödel’s First Incompleteness Theorem: If PA is consistent, then there exists a PA proposition p such that p is unprovable in PA; if PA is ω-consistent, then the negation of p, ¬p, is unprovable in PA (in 1939 Rosser proved that the condition “ω-consistent” can be replaced by “consistent”), that is, the system PA is incomplete. Such a p is called an undecidable proposition (that is, neither the proposition nor its negation is a theorem of the system).
Gödel’s Second Incompleteness Theorem: If the formal arithmetic system PA is consistent, then its consistency cannot be proved within the system PA itself.
Gödel’s two incompleteness theorems can be stated more generally as follows:
Gödel’s First Incompleteness Theorem: Any mathematical formal system sufficient for elementary number theory, if it is consistent, is not complete, that is, there must exist undecidable propositions in it.
Gödel’s Second Incompleteness Theorem: Any mathematical formal system sufficient for elementary number theory, if it is consistent, cannot prove its own consistency within the system. Another form of Gödel’s Second Incompleteness Theorem says that any sufficiently rich mathematical formal system, if it is consistent, cannot prove as a theorem the proposition expressing its own consistency.
////——One can feel, once one has only a slight understanding of the line of Gödel’s argument, how ingenious and marvelous it is!
Page 40
Between provable mathematical propositions and mathematical truth there always lies an infinite distance, and with finite methods alone there is not even any hope of approaching it. As Gödel himself said, “mathematics is not only incomplete, but also incapable of being completed,” and this is precisely the deepest implication of Gödel’s theorem.
////——Besides the impact on philosophy of artificial intelligence mentioned by the author later, the profound influence of Gödel’s theorem obviously extends to much broader fields. For example, in physics, Hawking has recently also begun to admit that there may perhaps be something analogous to Gödel’s theorem in physics, making the final discovery of a “theory of everything” impossible. Philosophically, the pursuit of “systematization” upheld since Aristotle and German classical philosophy has inevitably been struck by Gödel’s theorem. Many great philosophers throughout history have tirelessly tried to make philosophy scientific and science mathematical, but Gödel’s theorem ruthlessly shows that, let alone whether it is possible to make philosophy scientific or science mathematical, even once one reaches mathematics, which represents the most rigorous science, it is destined still not to be complete. A simple transformation of Gödel’s theorem becomes this: a philosophical system, if it is self-consistent, cannot prove that it is self-consistent. Of course, this only means that building a closed philosophical system by making philosophy scientific and science mathematical is impossible; it does not negate efforts to establish other kinds of philosophical systems by other means. In fact, Gödel’s theorem also precisely proves the value of metaphysics, because if the truth of a mathematical system cannot arise from within itself and must rely on something else, then the thing that guarantees the truth of science can only be something of faith or something metaphysical. Gödel’s theorem shows that even mathematics has its intrinsic limitations, but mathematics’ power lies in the fact that it can actually “prove” its own limitations. Truth not only can never be reached, it is not even hopeful to “approach.” We should also awaken to the inevitable limitations of human knowledge; the fantasy that human knowledge will ultimately fully grasp, or infinitely approach, all the knowledge of this world should be shattered.
Page 41
Once the theorem had been discovered, people had no choice but to readjust their ways of thinking. The famous mathematician Weyl once exclaimed in this regard: “God exists, since mathematics is undoubtedly consistent; the devil also exists, since we cannot prove this consistency.”
Page 43
After strictly distinguishing the functions of mind, brain, and computer, Gödel made a clear assertion: the brain’s function is no more than that of a computer, and the view that mind and brain are identical is a prejudice of our age; but the incompleteness theorem cannot be taken as direct evidence for the assertion that “the human mind surpasses computers.” To derive such a strong conclusion requires other assumptions as well.
Page 45
In 1961, the American philosopher Lucas first wrote an article in vehement terms, “Minds, Machines, and Gödel,” attempting to use Gödel’s theorem to directly prove the conclusion that “the human mind exceeds computers”: “In my opinion, Gödel’s theorem proves mechanism to be false, because no matter how complex a machine we build, so long as it is a machine, it will correspond to a formal system. Then one can find a formula provable within that system, and it will be struck down by the procedure of Gödel’s construction of an undecidable proposition. The machine cannot derive this formula as a theorem, but the human mind can see that it is true. Therefore this machine is not an adequate model of mind.” This is the famous Lucas argument.
////——Gödel’s theorem intervening in the debate on artificial intelligence is very interesting. But I feel that Gödel’s theorem cannot directly prove that “the human mind surpasses computers”; still, Gödel’s theorem should have an important influence on the development ideas of artificial intelligence.
Wang Yutian: Some Problems in the Philosophical Study of Systems Science
Page 78
For example, the pessimists cite Gödel’s theorem as evidence, whereas according to Gödel’s own view, this theorem proves not so much the limitations of machines as it proves the limitations of human cognition.
Hao Bailin: Is the World Necessary or Contingent? — The Revelations of Chaotic Phenomena
Page 102
Chaos is not disorder and confusion. Whenever order is mentioned, people often think of periodic arrangement or symmetrical shapes. Chaos is more like an order without periodicity. In an ideal model, it may contain infinite internal layers, with “self-similarity” or “not entirely similar” relations between the layers. When observation methods and resolution are not high, only the structure of a certain layer can be seen. After the resolution is improved, structures on a smaller scale will appear where they could not previously be recognized. *Yiqian Zhaodu* says, “Qi resembles substance; it is already complete but not yet separated—this is called chaos” (气似质具而未相离，谓之混沌), and it seems that the two characters hun and dun, “chaos,” reflect this state better than the English word chaos.
////——Personally, I feel that chaos is one of the most astonishing and admirable things to emerge in the new sciences of the twentieth century. Its marvelous philosophical implications should be far greater than those of relativity, and comparable to quantum mechanics; and chaos theory also seems easier to understand than quantum mechanics. Besides the question of necessity versus chance, chaos also has far-reaching effects on many other philosophical questions, such as the picture of our entire world, and on many fields such as economics, sociology, ecology, information science, and so on. It is well worth attention.
Zhao Kaihua: The End of the “Heat Death Theory”
Pages 137–138
V. The Big Bang and the End of the “Heat Death Theory”
The above account of the process by which the Big Bang cosmological model was established may perhaps seem somewhat “tediously repetitive” to us. My intention was to show through this that this model is by no means a product of pure speculation; it has a solid theoretical and observational basis. Once we accept that the universe was born in a Big Bang and has been expanding ever since, then its constituents will decouple from one another, evolving from a thermodynamic equilibrium state to an nonequilibrium state, from uniform temperature to temperature differences. This phenomenon cannot occur in a static cosmological model, and it was something neither Clausius nor his critics had thought of.
The universe is a self-gravitating system, and it is unstable both mechanically and thermodynamically. Mechanical instability causes a density-uniform universe to develop into clump structures. Self-gravitating systems have negative heat capacity, and systems with negative heat capacity are also thermodynamically unstable. They have no equilibrium state, and the ordinary second law of thermodynamics cannot be applied to them. If we are to speak of entropy here, we agree with Zeldovich’s view: for a gravitational system, a state of uniform density is not the most probable. In the process by which uniform matter in the universe condenses into clumps (galaxies, stars, etc.), gravitational potential energy is converted into kinetic energy. From uniformity to non-uniformity, the probability distribution in configuration space decreases, but the temperature rises, and the probability distribution in velocity space increases. Comparing the two, the total probability increases rather than decreases. This means that the formation of celestial bodies is a spontaneous process in a gravitational system, and its entropy increases. Since there is no equilibrium state, entropy has no maximum value, and its increase knows no end.
In short, what the expanding universe model presents before us is a scene entirely opposite to that attached to the “heat death theory”: in the early universe, things were basically a high-temperature, high-density “broth” in thermal equilibrium; from this monotonous state of chaos, ever more complex and diversified structures gradually developed step by step. Thus, on the microscopic level, atomic nuclei, atoms, and molecules were formed (from relatively simple inorganic molecules to advanced biological macromolecules); on the macroscopic level, under the action of universal gravitation, galaxy clusters, galaxies, stars, the solar system, and the Earth evolved. Then life evolved on Earth, until intelligent beings like human beings appeared, together with the increasingly advanced societies they formed. In the ancient Egyptian myth, the phoenix bird burns itself in fierce fire and is reborn youthful from its own ashes. This is truly a marvelous portrayal of the contemporary view of the universe! Not only will the universe not die (heat death), it will instead revive with vigor from an early state of “heat death” (thermal equilibrium). Of course, present-day cosmology still cannot predict the universe’s final end (if there is one), but the nightmare that has tormented the physics community for more than a hundred years—the heat death theory—can, as a page of history, safely be turned over. As for what the universe’s prospects are according to modern cosmology, that is another question, unrelated to this historical case.
////——As the author says, the author previously spent too much space discussing the Big Bang theory, but that is hardly surprising. In the late 1980s, when this essay was written, the Big Bang theory may not yet have become a universally accepted scientific theory in China. And once the Big Bang cosmological model became widely recognized, discussion in this area would naturally seem superfluous. The key question is: how can the Big Bang theory be made consistent with the second law of thermodynamics, that is, how can the universe develop highly ordered structures from a thermodynamic equilibrium state?
Questioning the physical views of the former head of Peking University’s physics department is obviously not something I am qualified to do, but I happened to have read some related popular science works earlier—apparently by Paul Davies, though I have not yet checked the source—and so I have acquired some understanding of the heat-death problem of my own. Let me say a little here according to my own understanding.
Zhao Kaihua points out an important key issue, which is also mentioned in the popular science books I have seen: when applying the laws of thermodynamics to a cosmological model, one must take the factor of “universal gravitation” into account. In a small closed system where universal gravitation is not considered, the most probable state, namely the thermodynamic equilibrium state, is indeed a state of uniform density. However, when the effect of universal gravitation is taken into account, the situation is reversed: matter spontaneously forming star clusters under gravitational attraction instead becomes a process of increasing entropy. But for a system under universal gravitation’s influence—leaving aside the expansion of the universe for the moment, and discussing first a sufficiently large spatially closed system—it is not the case, as Zhao Kaihua says, that there is no maximum of entropy. We can imagine further: forming galaxies is an entropy-increasing process compared with a uniformly distributed density state, and what will galaxies and clusters of galaxies eventually become? — Galaxies with sufficiently large mass will ultimately spontaneously form black holes, while low-mass galaxies and other free-floating matter, as they move randomly through space, will one day collide with some black hole and be swallowed by it; that is to say, everything being swallowed by black holes is the final destination of a stationary gravitational system. Black holes are the boundary of spacetime, and only black holes remaining in the universe means the end of the universe. Of course, according to Hawking’s theory, black holes will also radiate energy; the final fate of black holes is to disappear in an explosion, ultimately transforming all matter completely into energy. In fact, another key meaning expressed by the second law of thermodynamics is that matter can spontaneously transform into energy, but energy will not spontaneously condense into matter. By this point, one could say that a system of universal theory has reached its true endpoint.
Next, we need to consider another factor raised by Zhao Kaihua that cannot be ignored, namely the expansion of the universe. And in fact, “expansion” may be precisely the secret by which the universe can take thermodynamic equilibrium as its starting point. Consider an expanding gravitational system: this system contains material particles as well as energy existing in the form of radiation. In equilibrium, their temperatures are the same; as the overall space expands, both will gradually “cool,” but the rates at which they cool are not the same (I have somewhat forgotten exactly how this works). Then this “temperature difference” brought about by expansion makes the original equilibrium state into a nonequilibrium state, making the flow of energy possible. And according to the popular science works I have seen, only when this expansion process is accelerating can such a nonequilibrium state be maintained indefinitely. Of course, this theory is probably still only a hypothesis.
From the above we obtain the result that if the expansion of the universe is to tend toward rest, then heat death appears as all matter first being swallowed by black holes and finally transforming into thermal radiation and dissipating. To avoid the nightmare of heat death, the universe can only be in constant expansion (and quite likely in ever-accelerating expansion). In that case, matter will inevitably keep endlessly receding from one another, and the final result, even if it is not heat death, will still be the universe’s final dissipation. Compared with heat death, this result is not all that different.
Of course, the development of modern cosmology has not completely negated the possibility that the universe will never perish (although this seems extremely remote). To study the end of the universe, one may need, besides relativistic physics, to rely also on quantum physics and even grand unified theories, and so on. But in any case, to say that the “heat death theory” can already be “safely turned over” seems a bit too simple…
He Zuoxiu: “The EPRParadox” and Related Philosophical Problems
Page 148
It is not difficult for us to draw the following conclusions:
(1) Quantum mechanics and the assumption of realism, that is, the doctrine that “the world is composed of objects that exist independently of human consciousness,” have no contradiction or incompatibility whatsoever.
(2) In quantum mechanics, deterministic realism has not received experimental support; what experiments support is a local, yet random or statistical, realism that “exists independently of people’s consciousness.”
////——The EPR paradox is a wondrous thing that I pay close attention to, as can be seen from the name of my blog. Of course, I cannot discuss such profound physical questions here. Still, He Zuoxiu’s article did not leave me very satisfied.
He Zuoxiu is right: quantum mechanics does not deny that the world exists independently of consciousness; what it actually reveals, however, is that whether consciousness participates in cognition, and in what way it does so, affects the results we come to know. Whether there are objects existing independently of consciousness—or, in other words, “things-in-themselves”—outside consciousness is indeed not denied by quantum mechanics. But what we can know, that is, what may be reflected in our consciousness, must certainly not be something independent of consciousness; it must certainly be something formed through the action of our consciousness.
Taking a step back, even if one says that the above conclusion has not been supported by quantum mechanics either, quantum mechanics has at least profoundly changed our conception of “reality.” Even if quantum mechanics can support some kind of “realism,” this “reality” will also differ enormously from what we traditionally mean by “reality.” As He Zuoxiu wrote in his second conclusion, this reality is astonishingly “random.” Can a “reality” that has lost determinacy still be called “reality”? In any case, quantum mechanics’ overturning of the traditional view of reality is immense.
Hong Dingguo: Quantum Mechanics and the Argument for Realism—Anti-Realism
Page 151
From the passive side, metaphysics is unavoidable; the question is how to take the correct attitude. Bohm put it well: “Many modern philosophers and scientists say that metaphysics should not be done, but in fact even those who swear not to do metaphysics have metaphysical ideas formed even in childhood. Thus they are all controlled by some kind of metaphysical thought. Therefore, it is worthwhile to explore one’s own metaphysical ideas, question them, and put forward new metaphysical ideas to replace them.”
From the active side, metaphysics is necessary, and this is something the Gödel theorem on the incompleteness of mathematical formal systems has enlightened us about. This theorem tells us: within any mathematical formal system, one can construct true propositions that are neither provable nor disprovable within that system. In other words, the truth of such propositions must be judged by means outside the formal system. There is no doubt that Gödel’s theorem can also be applied to formalized physical theories (it has only a logical-algorithmic significance). More specifically, a formalized physical theory must contain true propositions that it cannot itself prove, and cannot itself disprove. The truth of these propositions must be judged by means and ideas outside the theory. Therefore, in order to make the significance of a formalized physical theory complete, its philosophical expansion is necessary.
Page 155
In BBB theory (the abbreviation of the surnames de Broglie, Bohm, and Bell), the “observer” is no longer in a foundational position in the theory, and observable quantities are replaced by existable quantities. Their existence does not depend on “observation”; on the contrary, observing instruments and observing acts, and even the observer, arise from the quantum existable quantities. In BBB theory, the cross-correlation between the states of global existable quantities and the behavior of local existable quantities is determined by the dynamical equations of quantum mechanics. This radically new relationship between the whole and the local on the ontological level runs through all quantum-mechanical problems, whether measurement or non-measurement, and is an essential feature that distinguishes quantum ontology from classical ontology. This feature is called the EPR nonlocal correlation of quantum systems, or the inseparability of quantum systems, or the wholeness of quantum reality. However, when the influence of the global existable quantities of quantum mechanics on the local existable quantities becomes far weaker than the influence of classical interaction, the quantum system disintegrates, and quantum reality degenerates into classical reality that can be described by classical physics. Therefore, in this extended model of quantum mechanics, a continuous transition between quantum reality and classical reality can be achieved, with no unbridgeable gap between the two.
Hu Xin and Luo Jiachang: Relational Realism—A Realist View of Quantum Mechanics
Page 162
The most complete and powerful discussion of relational realism comes from Jammer’s celebrated works The Conceptual Development of Quantum Mechanics and The Philosophy of Quantum Mechanics. In the “Conclusion” of the former, he points out that the reality of relations has been “fully confirmed in modern physics”; relativity and quantum mechanics respectively reveal that primary qualities are relative to the reference frame and to the means of observation. In the latter, he specifically analyzes the “relational concept of the quantum state,” pointing out that the description of a state is not confined to the particle to be observed, but represents the relation between the particle and the entire instrument. Therefore, one cannot separately ascribe to the former the various attributes of physical reality.
You Guangjian: On the Unity of Physics
Page 185
The process of unifying physics has greatly expanded the field of research in physics. On this, Einstein once said: “What we call physics includes a class of natural sciences whose concepts are based on measurement, and whose concepts and propositions can be expressed in mathematical formulas. Therefore, within all our knowledge, that part which can be expressed in mathematical language is classified as the domain of physics. With the progress of science, the domain of physics has expanded to such an extent that it seems to be limited only by the boundaries of this method itself.” (Collected Works of Einstein, vol. 1, Commercial Press, 1976, p. 384)
Yu Mouchang: Discussion on Catastrophism
Page 266
(Xu Jinghua) asks Wendao: “Is there a struggle for existence in the world? Is it the key to evolution? When great calamities come, is there enough time? Are survivors always the fittest? Or, to put it plainly, are they the luckiest!” (Disasters from the Sky, pp. 19–20)
Gong Yuzhi: Humanistic Reflections on the Development of Science and Technology in the New Century—Also on the So-called Anti-Scientism
////——In this article, the old gentleman begins by pointing out that when it comes to “humanistic reflections on the development of science and technology,” Marx and Engels are the “old ancestors.” I quite agree with this point; Marx’s relevant views remain quite forceful even today. And the subsequent reflections the old gentleman makes on the currently “popular” “anti-scientism” are also quite reasonable; this article is worth reading carefully. Of course, I cannot agree with many of the old gentleman’s arguments. Present-day China, on the one hand, lacks scientific spirit and scientific method in many places, and this is an important problem; but on the other hand, the overinflation of “science”—“experimental science” as an instrumentality of reason, rather than the “scientific spirit” inherited from ancient Greece—is also quite an important problem. “Scientism” still needs to be “opposed.” We hope to leave room for things outside science, such as religion, but we will always insist on using scientific methods and the scientific spirit (rather than religious or mystical speculation, or any irrational method) to reveal the limits of science itself, and to refute the claim that science is omnipotent through scientific modes of thinking. —The “science” in the sentence above refers to the rational spirit inherited from ancient Greece.
January 27, 2006
Translated from the Chinese original with AI assistance. The original text is authoritative.
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