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.