Evidence of Design in Mathematicshttp://reasonandscience.catsboard.com/t1360-evidence-of-design-in-mathematics#6485
IN 1913, THE DANISH physicist Niels Bohr formulated a new theory of the actions of electrons as they orbited the nucleus of the atom, a critical step towards the development of quantum mechanics in the 1920s, leading to his winning the Nobel prize in physics in 1922. He continued to be among the most influential figures among the physicists of the world for decades to come,
and is today commonly ranked among the greatest physicists of history. Like a number of other leading physicists of the twentieth century, Bohr also had a deep interest in the implications of the discoveries of twentieth-century physics for the overall understanding of the human situation in the world. He recognized that physics in its radically new portrayal of the workings of nature was also raising deep philosophical and religious questions. In his 1938 essay, “Natural Philosophy and Human Cultures,” an address delivered to the International Congress of Anthropological and Ethnological Sciences, he wrote that:
“it is impossible to distinguish sharply between natural philosophy” as developed by physics “and human culture. The physical sciences are, in fact, an integral part of our civilization, not only because our ever-increasing mastery of nature has so completely changed the material conditions of life, but also because the study of these sciences has contributed so much to clarify the background of our own existence” as human beings. It was thus of great significance that twentieth-century physics had required a “revision of the presuppositions underlying the unambiguous use of even our most elementary concepts such as space and time.” Indeed, “in the study of atomic phenomena we have [now] repeatedly been taught that questions which were believed to have received long ago their final answers had most unexpected surprises in store for us.” Of particular importance, it was no longer possible “to distinguish sharply between the behavior of [physical] objects and the means of [their] observation,” as historically was the view of physicists (and most other people). Since ultimately the means of observation are inseparable from the workings of the human sensory organs and human consciousness, this newly linked what had previously been two distinct fields of investigation, the physics of the natural world and the psychology of human consciousness.
Bohr would thus write that the scientific questions in the two areas resemble one another in that there is a “close analogy between the situation as regards the analysis of atomic phenomena . . . and characteristic features of the problem of observation in human psychology.”
It is impossible for us to describe, as Bohr comments, in an entirely objective manner our own “psychical experiences . . . such as ‘thoughts’ and ‘feelings’” because we combine inseparably in our human consciousness the roles of observer and observed. In atomic physics, it was surprisingly similar in that there is a “complementary relationship . . . regarding the behavior of atoms obtained under different experimental arrangements”
in which the observer becomes an integral part of what is actually observed.
The world of mathematics is “supernatural” in that it already exists outside nature, prior to and outside of any material realities in the natural world. Moreover, this supernatural mathematical order somehow—and of this we still have no clue as to how it works—serves to govern the full detailed workings of the entire “physical world”. A governing mathematical intelligence is thus a supernatural, superhuman entity which we call God.
If we then choose to define a god as such a supernatural entity having a fully developed mathematical intelligence, we can then say that at least one god very probably exists, a god of mathematical truth that rules the physical world.
At the age of twenty-one, a Russian mathematical prodigy, Edward Frenkel, still short of having earned his undergraduate degree in Moscow, was invited in 1989 to be a visiting professor of mathematics at Harvard University. Very much impressed with his abilities, Harvard invited Frenkel to remain at Harvard, and he obtained a PhD in mathematics there in one year,
and then joined the Harvard faculty. He moved to the University of California at Berkeley in 1997 where he today ranks among the leading mathematicians in the world.
Following others, Frenkel clearly and explicitly recognizes that this view of the universe can be traced back to Plato—that the Platonic mathematical world exists altogether “independent of physical reality.” One of the most important mathematical findings of the nineteenth century, for example, “Galois groups were discovered by the French prodigy [in the 1830s], not invented by him. Until he did so, this concept lived somewhere in the enchanted gardens of the ideal world of mathematics, waiting to be found” by some human being. If it had not been Galois, it would have been someone else later who would have made the same discovery because a mathematical truth is an objective fact, even as it has no physical reality. As Frenkel affirms, “mathematical entities are independent of our rational faculties” as human beings and thus “mathematical truths are inevitable.” The objective truth that two plus two equals four precedes any human existence and will be true for eternity, whatever happens to human beings. Frenkel is confident that in thinking this way he has a great deal of company among contemporary mathematicians and physicists. Indeed, he finds that “most math practitioners believe that mathematical formulas and ideas inhabit a separate world.” He notes that “Kurt Godel, whose work . . . revolutionized mathematical logic, was an unabashed proponent of this view. He wrote that mathematical concepts ‘form an objective reality of their own, which we cannot create or change, but only perceive and describe.’” In the nineteenth century, as Frenkel observes, the physicist “Heinrich Hertz, who proved the existence of magnetic waves, . . . expressed his awe this way: One cannot escape the feeling that these mathematical formulas have an independent existence and intelligence of their own, that they are wiser than we are, wiser than their discoverers.” They are, as such language suggests, similar to if not identical to a god.
By way of illustration, Frenkel comments that “the discovery of quarks” in the 1960s “is a good example of the paramount role played by mathematics in science.” Quarks “were predicted not on the basis of physical data, but on the basis of mathematical symmetry patterns.” Physicists then made an educated guess that certain physical phenomena would later be found to correspond to these mathematical patterns. It turned out in this case, as in many other great scientific discoveries, to be true. As Frenkel explains, “this was a purely theoretical prediction, made within the framework of a sophisticated mathematical theory of representation of the group SU.”
In developing the physics of quarks, “a seemingly esoteric mathematical theory empowered us to get to the heart of the building blocks of nature.” As Frenkel puts it, “How can we not be enthralled by the magic harmony of these tiny blobs of matter, not marvel at the capacity of mathematics to reveal the inner workings of the universe?” Truth be told, there is no other way to see it other than as the manifestation of something supernatural.
As Wigner pointed out and Frenkel now again affirms, there are thus two miracles to be confronted, first the magical existence of a world of mathematical truths independent of human existence and physical reality, and second the magical correspondence of at least some of these mathematical truths to the “physical” workings of the natural world (as we perceive them in human consciousness). No one has any scientific explanation for any of this. As Frenkel comments about the total mystery, although physicists and other “scientists have been exploiting this ‘effectiveness’ [of mathematics] for centuries, its roots are still poorly understood”—indeed, are not really understood at all, as Frenkel would likely concede, if pressed on the matter.
There is no explicit mention of a god or explicit discussion of theology in Love & Math. But the book is nevertheless an important work of theology for our time. Contemporary naturalists take for granted that matter must be fundamental,
since physical science—for them the one all-powerful method of understanding the world—is all about explaining the workings of the material world. Frenkel is not the first to do so but he offers a strong up-to-date statement from a world-class contemporary mathematician that mathematics is more fundamental than matter
. This is all based on a rational analysis as he develops it, employing the skills in reasoning that are found at the highest levels among mathematicians. Indeed, while Frenkel may not realize it himself, he is laying out a rational argument for the very probable existence of a god.Many mathematicians and physicists believe that the mathematical world has an objective reality outside any physical domain of time and space.Max Tegmark:
“we don’t invent mathematical structures—we discover them, and invent only the notation for describing them.”
They simply exist in a realm of abstract ideas and have always existed there. Mathematicians (or physicists doing mathematics) can visit this realm through their own heroic rational efforts within human consciousness. The role of physicists is then to search for exact correlations between such mathematical ideas and the behaviour of the physical world—again as it is necessarily perceived in their consciousness.
Seemingly in a miraculous fashion, even though all this necessarily takes place within the consciousness of each individual physicist, in practice the community of all the physicists of the world can often reach a consensus, establishing a worldwide common agreement on these laws—something altogether impossible outside the physical sciences. It is yet another piece of evidence that a supernatural god must lie somewhere behind all this—what else could be the explanation?
At the deepest level of understanding of reality (the goal of Tegmark’s inquiry), therefore, with respect to the physical world there is seemingly nothing but the mathematics itself, plus the human “baggage” of words—the “poetry” of physics—that are added by physicists to aid heuristically in their own thinking and in communicating their results to others. But if we want to discover the deepest reality that transcends any human linguistic additions, it will be necessary to limit our claims to an understanding of the mathematics itself. Hence, as Tegmark concludes, if we are to seek a physical reality that is independent of any human influence, it follows that such an “external reality is a mathematical structure.” In other words, “something that has a complete baggage-free description” outside human language “is precisely a mathematical structure” in and of itself alone.
Since the mathematical world and its rational truths have always existed, even before the arrival of human beings on earth, the mathematics itself becomes the ultimate bedrock of the reality of the universe. Tegmark, like Wigner, Penrose, and other physicists today, admittedly has no way of explaining why our perceptions of a “physical” world in our human consciousness (as often aided by measuring devices) have turned out to correlate so exactly with one or another eternal mathematical truth that human beings have rationally discovered and explored. It may be all part of the giant mystery of the universe that seemingly only a god could reveal.The nonphysical workings of human consciousness remain well outside our physical scientific understanding at present.
Human consciousness as yet another “internal reality” (besides the physical and mathematical realities) that consists of “the way you subjectively perceive the external reality from the internal vantage point of your mind.” This third internal reality of human consciousness— Tegmark acknowledges some similarities to the three-world views of fellow physicist Roger Penrose —“exists only internally to you; your mind feels as though it’s looking at the outside world, while its only looking at a reality model inside your head.”
1. God? Very Probably: Five Rational Ways to Think about the Question of a God