Two books, William Byers’ How Mathematicians Think and John D. Barrow’s Impossibility, at a first glance seem to have nothing in common: the former deals with the process of creating mathematics, while the latter discusses the outer limits of science. But closer inspection quickly reveals that both works are eerily similar both in content and ideas, and while their authors seem to disagree on particular points of their respective exposés (such as the ordered relation between mathematics and logic), neither seems to present a particularly objective or dispassionate outlook on science or mathematics itself.
The purported goal of How Mathematicians Think is to acquaint the reader with fundamental creative processes of mathematics, in particular those based on ambiguity, contradiction, and paradox. But the increasingly technical approach of the book begs the question about its intended audience: mathematicians will find the content trivial and self-evident – they need not be reminded of rather basic mathematical concepts, nor be told how they do what they do – while non-mathematicians will find the text abstruse and forbidding, and so may never really learn what the author really tried to convey – that is, the notion of mathematics as a creative albeit formal and rigorous art.
Impossibility focuses more on science in general (including the social sciences); yet, unsurprisingly, mathematics still remains the focus of the author, John Barrow, a Cambridge mathematician himself. Here, too, the reader is faced with a plethora of scientific and mathematical curiosa. Many of them are identical to those used in Byers’ book, but, while unquestionably interesting, its wealth of graphs, pictures, theorems and equations still begs an answer to the classic question: “And the point is?”
Both authors tried for a book about mathematics, or about science, without imposing much linear structure on their presentation. After shortly presenting the effect of basic concepts of Platonism and constructivism on the philosophy of mathematics, Byers at long last discloses his real if somewhat anticlimactic purpose: it is his spiritual quest for the school of Zen, wherein he tries to employ introspective musings on the nature of mathematics to fulfill his transcendental yearnings for spiritual completeness. Barrow, after disposing of the relevance of religion in science and mathematics, addresses himself to another of the speculative arts: philosophy.
Having understood the objectives of both authors set forth on the first pages of their introductions, the reader is puzzled by pages upon pages of prolix elaborations, vividly and frequently interspersed by refreshing intermezzi of fun mathematical factoids that make the reading feel like strolling through a collection of curiously interesting oddities.
Sadly, science has long been misunderstood as one of the least creative fields of human activity – the popular image of a scientist portrays a person continuously engaged in trial and error, over and over, until the moment of random discovery. And yet, without creative thinking no discovery could become possible, no secret of nature can be unearthed. Encountering a book that deals not as much with science as it does with the philosophical aspects of science must be disconcerting for a reader unfamiliar with the subject matter. Still, such books provide a valuable glimpse into the functioning of a scientific mind, an introduction to the kind of big questions that researchers may eventually reach.
And yet the approaches of Byers and Barrow both seem to be missing something centrally important to science and mathematics: dispassionate objectivity, uninfluenced by religious, spiritual, ideological or philosophical ideas that have nothing to do with science or mathematics itself, but have everything to do with an individual need for soul-searching and for establishing a value structure. Whether taking refuge in the dicta of philosophers or in religious or spiritual traditions, a scientist does not contribute to the knowledge pool as he purports to do, but merely limits his own potential by allowing himself to be weakened by a quest for higher purpose and moral guidance that appears to enable him to “dare to contribute to Pandora’s box of human knowledge.” Experience shows, however, that ‘thinkers’ or ‘gurus’ are among the worst compasses that may guide a scientist, since they lack both scientific understanding and objectivity. And aren’t thinkers almost universally known to be more apt to devoting themselves to the creation of a faithful base of followers than to helping individuals succeed and potentially surpass their erstwhile teachers? Ideally and in the abstract, the primary objective of scientists is to further science (without, of course, entirely neglecting to make a name for themselves in the process). By contrast, the purpose of a guru or philosopher is to shape the world according to his individual convictions and beliefs (without, again, neglecting to make a name for himself). This inherent conflict of interests is all too readily apparent, and while it is only natural that a scientist’s human quality might feel an occasional need to seek spiritual or philosophical guidance, such crutches for the mind and soul should never be mistaken for scientific guidance, lest we obtain a generation of theology-bound scientists who either end up ostracized if not burned at the stake, personally or vicariously through their books, or ready to close their eyes to evidence not in line with the dominant, and therefore “orthodox” ideology. That was generally the seemingly eternal status quo of science before the Renaissance (but in some areas also later until the 19th century) under various ideological and/or religious regimes of “truth in certitudes.”
Those capable enough to push known outer limits of science usually have no time and resources to speculate and pontificate about “definitive” limits. This is best left to those with few ideas about science in the first place, such as philosophers and other intellectual speculators. The two books discussed here are an interesting exception to that common observation: both were written by practicing mathematicians who, for one reason or another, decided to stop on their path of discovery, look around, and speculate where it may eventually take them and society as a whole. Evidently, the authors saw valid reasons to take time off from what they are gifted enough to pursue as a day job for the sake of engaging in a much less demanding occupation. But whatever both authors’ personal needs, their results should by no means be confused with authoritative findings of actual scientific exploration. Newton, for one, was right and visionary in a great many areas – but his opinions on alchemy should be taken, politely, with a huge grain of salt. Even mathematicians expounding on an emerging “philosophy of mathematics” are, if truth be told, not necessarily much of an authority. Unless he is one of those rare giants of thought, say, Leibniz or Russell, someone writing about philosophy of science is frequently either not enough of a philosopher or not enough of a scientist to convince or at least intrigue a discerning reader. Attention-grabbing catchy and conclusory statements, like Emil Du Bois-Reymond’s famous “ignoramus et ignorabimus,” are much too often just plainly wrong.
The debate comes down to intellectual clarity and honesty about disclosing purpose and method, and about frequently reviewing one’s adherence to both. Epistemological concepts of truth, belief and understanding are critical to cognitive success but they yield only partial results for the analysis of limits to knowledge and discovery. Still, one might argue, much to the chagrin of theorists, that those limits will actually remain congruent with the limits of randomness and accident itself, so long as accidental discovery and recombination of existing knowledge resulting in new discoveries remain possible. To date, no meaningful reason has been presented in the abstract as to why limits other than temporary and technology-dependent ones should exist and why human capacity of cognition should be limited by anything other than by the capacity and processing power limitations of the human brain – and its man-made extensions.
Conceivably, theoretical limitations may exist in mathematics where the finite life span of the human body would require more time for acquiring the methodological mathematical tools and skills of their use requisite for pushing the limits of the common body of knowledge further out. It is easy to imagine that this might eventually apply even to very narrow and specific research topics. Yet, at the same time, it is overwhelmingly likely that the same state of cognitive evolution would not be limited to mathematics alone but also yield equally improved technological support for information processing and analysis. That would once more level the playing field, rendering again more or less constant and manageable the distance one generation may have to bridge from the legacy of knowledge handed down by their forefathers to the significant discoveries of their own acquired in a however extended lifetime.
Another scenario, albeit one whose potential implications for the human race cannot be adequately assessed beyond imaginative science fiction, involves delegating mathematical research tasks to artificial intelligence under some mechanism of joint control. This may, in fact, represent the one true limit of science because it is inconceivable that such a mechanism would not eventually fail as power to control the process is usurped once the genie leaves the bottle at a point of greater-than-human intelligence known as a technological singularity, never to return. Such a moment would constitute an intellectual event horizon beyond which events can no longer be predicted or understood.