[Edit 02/11/2017: The final version of this essay can be found
here.]
Perception, including human perception,
has not always been a well-defined concept, but these days I believe
general agreement can be reached somewhere along the following lines.
Animals receive, through their nervous system, an assortment and
range of sensory experience from which is distilled an awareness of
the animal's environment, as well as a reaction back into that
environment. It is the distillation part of this process that
stands at the core of what we would typically call perception.
Perception is necessary because the entirety of sensory
information would be too much. Unfiltered and undifferentiated
sensory experience would lead only to a chaotic awareness of the
animal's environment and would make the enactment of targeted and
productive reaction problematic at best. Perception extracts signal
from sensory noise, perception distinguishes figure from sensory
ground. The foregrounded elements of sensory experience are
precisely those elements that an animal perceives.
As such, particular types of perception
can be described in large measure by highlighting the characteristics
of what tends to foreground within that perception and also
juxtaposing these against the characteristics of what tends to remain
ignored (unperceived). Applying this technique across the entire
animal kingdom is instructive, for it reveals a broadly consistent
and unifying theme. As any regular observer of television nature
shows could easily attest, the experiences and attentive focus of
untamed animals are both predictable and mostly unvaried across the
many species, and can be classified, almost entirely, under just a
small set of headings: food, water, danger, shelter, family, sexual
targets, sexual rivals, predators, prey, conspecifics. There is of
course nothing random or surprising in that list; each of its
constituents is an essential component in the drive for survival and
procreation, and this type of perception is one that efficiently
serves the biological process. Nonetheless, while noting these
characteristics of what tends to foreground within animal perception,
it is worthwhile also to consider those sensory elements that go
undiscerned. The wind rustling in the grass and leaves, wisps of
cloud overhead, an arrangement of bushes along the distant
horizon — unless such elements and characteristics happen to play a
direct role in an animal's quest for survival and procreation, they
will go almost entirely unnoticed. And this will be true for a very
large portion of an animal's sensory experience, it will simply fade
unobserved into the sensory background.
I would like to give a name to this
universal tendency to foreground primarily (perhaps exclusively)
those sensory features that are essential to survival and
procreation. I will call this tendency biological perception.
And on the other side of the coin, I would like also to give a
categorizing name to the sensory features themselves — that is, those
features that tend to foreground within biological perception (food,
water, danger, etc.). Since I am unaware of any such name in common
use I will invent one, darwinamatics, an awkward term to be
sure but one chosen because it corresponds nicely to its ready-made
counterpart, a counterpart we will consider shortly.
Human perception is intriguing because
it is both animal perception and it is not. Human perception adheres
to biological perception's rule of universality and yet it also
provides the only known exception to biological perception's rule of
exclusivity. That human perception is a form of biological
perception can be seen readily enough from two different
considerations. First, there is mankind's long anthropological
history, which reveals that for an extremely large portion of time
after the evolutionary split from the other apes, man's existence — and
along with it, his perception — must have remained as animal-like as
all the other beasts. From Australopithecus down through the
later genus Homo, there is little in the way of evidence to
suggest that mankind's foregrounded focus and endeavor ever deviated
far from the constraints of survival and procreation. Some might even
argue that this perceptual state remained constant until as recently
as fifty thousand years ago, but at whatever point one places the
timing of mankind's perceptual turn, it seems certain that our
species' perceptual characteristics must have comprised little more
than biological perception for an extremely long period of time.
The second consideration that
demonstrates human perception is a form of biological perception can
be observed directly today. For although modern human perception can
no longer be defined in terms of just biological perception alone,
modern human perception still retains the vast majority of its former
biological traits. When we observe what tends to foreground within
modern human awareness, we discover that food, sex, danger and all
the rest continue to play a prominent role — darwinamatics still
constitutes much of the locus of human attention and endeavor.
Indeed, a healthy dose of biological perception is considered to be
critical for both development and everyday functioning, with those
judged to be inadequately attuned to such things as family, rivalries
and conspecifics judged also to be the bearers of various
psychological or developmental disorders. Foregrounded elements of
survival and procreation no longer play the crucial role they once
did on the prehistoric savanna, and yet they still motivate and drive
much of the action in a modern human society.
Thus biological perception is not the
characteristic that distinguishes human perception from animal
perception, since that form of perception is still shared in common.
What distinguishes human perception from animal perception is that
human perception, and apparently it alone, has acquired a significant
addendum. When we observe what foregrounds within modern human
awareness, in addition to those still influential components of
survival and procreation, we find also a host of distinguished
sights, sounds and other sensory features that no wild animal would
ever naturally perceive. An iterated list of such features would be
lengthy, and it would include not only the symbols of language, the
architectural traits of buildings, the rhythms of music and the
intoxications of perfume, but also the wind rustling in the grass and
leaves, wisps of cloud overhead, and the arrangement of bushes along
the distant horizon. Man now foregrounds a vast range of sensory
features not directly connected to the immediate urges of survival
and procreation; man has acquired a second type of perception.
In addition to their mostly
non-biological nature, the foregrounded elements in this second type
of perception can be seen, upon closer inspection, to carry a
consistent and unifying theme. At their core (perhaps tautologically)
these elements would appear to emerge in perception precisely because
they carry the properties that inherently defy chaos and sensory
background, and as was the case with biological perception, these
unifying properties can be listed under just a small set of headings:
symmetry, pattern, mapping, order, object, structure, form. When we
examine the attention-grabbing, ever-expanding innovations of the
modern human world we find everywhere an underlying cornucopia of
number, shape, order, rule. The distinctiveness of the present age is
a constructed distinctiveness, in fierce defiance of nature
and its biologically limiting constraints. The foregrounded elements
of man's second type of perception are characterized by the fact they
are drawn from just a small set of structural, mostly non-biological
features, ones that now emerge persistently and prominently from a
modern human's sensory background.
As was done with biological perception,
I would like to give a name to this exclusively human tendency to
foreground sensory elements that possess structural and mostly
non-biological characteristics. I will call this type of perception
logical perception. And again, as was done in the case of
darwinamatics, I would like to give a characterizing name to the
sensory features themselves — that is, those features that tend to
foreground within logical perception (symmetry, pattern, mapping,
etc.). This time, however, there is no need to invent a term, for
there is one already in widespread and common use. That term is
mathematics. The study of symmetry, pattern, mapping, etc. is
none other than the study of mathematics.
Thus we have theorized so far that
human perception consists fundamentally of two different types of
perception. The first type of perception foregrounds those sensory
elements directly connected with survival and procreation, and we
have labeled this type of perception biological perception and
have categorized the foregrounded sensory features themselves as
darwinamatics. We recognize that man shares this type of
perception with all the other animals, has inherited it from out of
man's animal past, and continues to experience its influence in the
modern age. The second type of perception, unique to humans and
acquired quite recently in man's long anthropological history,
foregrounds those sensory features that possess pattern, structure
and form, and we have labeled this second type of perception as
logical perception and have recognized its foregrounded
sensory elements as precisely those elements commonly studied under
the heading of mathematics.
Mathematics has always been something
of a philosophical puzzle. Intimately connected with space and time,
and the underpinning behind almost every facet of rational thought,
mathematics appears to be at the foundation of all non-biological
conception, and so it has been more than a little bit tantalizing to
determine the foundation of mathematics itself. The ancient Greeks
already were arguing the matter fiercely, including Plato and his
idealized forms, and in more recent times such esteemed thinkers as
Leibniz and Kant have made widely influential contributions. The
twentieth century saw the rise and battle of three competing schools
of thought — the logicist, formalist and intuitionist points of
view — and at various times and in various ways, the ultimate source
of mathematics has been attributed in turn to God, human intuition,
the objective world and the neuronal mechanisms inside the human
skull (the latter being the most mythical suggestion of them all).
And yet despite these many arguments and assorted proposals, the
philosophical puzzle remains as puzzling as ever.
Recognizing mathematics to be the
equivalent of the foregrounded elements in man's second type of
perception opens the door to a less mythical, more directly
observable explanation for the origin and foundation of mathematics.
The similarities between logical perception and biological perception
already begin to point the way, for there has not been the same
philosophical qualms about darwinamatics — such features have
typically been regarded as simply open to inspection. And the
placement of logical perception's rise within the time frame of man's
long anthropological history provides still more reason to take
mathematics as something other than mystical, more akin to a
biological/anthropological event. Of course if that is all there were
to it, one might reasonably complain that we have done little more
than change the aspect of the problem, have made out of logical
perception the same philosophical puzzle we have made of mathematics
itself. What is the source and foundation of logical perception? Is
it a gift from the gods? a synthetic, irreducible intuition? or an
evolutionary explosion of synaptic computational miracle? Here in the
early twenty-first century, we are slowly uncovering a patch of human
knowledge that makes it clear the source and foundation of logical
perception is in fact none of the above. Instead, logical perception
is directly attributable — and in a directly observable way — to the
presence and influence of an atypical group of people.
At its root, autism is a condition
defined by perception. In fact in many ways, the distinction between
autism and non-autism — taken in their purest form — is the same
distinction as that between logical perception and biological
perception. What unifies autistic experience, classified today under
a broad assortment of behavioral, sensory and developmental
characteristics, is a diminished bias towards biological perception,
and in particular a diminished foregrounding of conspecifics.
Autistic individuals do not easily or naturally attune to the
particular features of the human world: they do not readily
foreground human voices, they do not focus energetically on human
faces, they do not enthrall to many of the most popular human
concerns. This diminished awareness towards mankind and its sensory
attachments is apparent from the earliest ages and remains extremely
consistent — to the point of being defining — across the entirety of
the autistic population. It is compensated for only slowly and with
great effort throughout an elongated developmental process, and it
continues to produce many subtle social anomalies well into advanced
age. The natural animal experience is to foreground first and
foremost those sensory features concerned with a species' survival
and procreation, but autistic individuals serve as the most obvious
counterexample to this nearly universal tendency. And thus autistic
individuals are the least animal-like of Earth's many biological
creatures, for they are the least determined by the constraints of
biological perception.
A diminished facility towards
biological perception means that autistic individuals are initially
hindered in gaining sensory footing. Little emerges as signal, there
is no figure against the sensory ground. If this condition were to
hold, autistic individuals would be in the most dire of straits, with
almost no sensory traction to aid in developmental progress or even
in the most essential requirements of survival itself.
Fortunately — both for autistic individuals and for the human world
at large — this condition does not hold. In the absence of a stronger
type of perception — that is, in the absence of biological
perception — an alternative, perhaps we should say a default, type of
perception swiftly assumes its place. It might be no more than
tautological to say that the sensory elements displaying symmetry,
pattern, mapping, etc. are the sensory elements that most naturally
foreground from sensory chaos, but be that as it may, these features
do naturally foreground, as is readily observable from the
inclinations and behaviors of the youngest autistic individuals.
Lining up toys, a fascination with spinning objects, flapping hands,
extreme repetition in video and song, precocious dexterity with
letters and numbers — these behaviors betray the deepest attention
and focus on those sensory features that have emerged the most
prominently. Instead of the common bias towards other humans and
their species-driven endeavors, autistic individuals are drawn first
and foremost to number, shape, order, rule. Instead of ease with the
material of darwinamatics, autistic individuals gravitate more
naturally to the material of mathematics.
(All this is observable. It is to the
great shame of modern science that in its insistence on medicalizing
autism and in its pursuit of so many mercenary distractions — including
an endless, self-serving touting of treatment, intervention and
cure — modern science has failed to make these simple observations
itself. It remains unclear to me when science can begin to make its
own perceptual turn, but for the moment I remain highly
pessimistic.)
The history of mathematics provides
still more evidence of a direct autistic connection. Although
biographical details are not always complete, and although nearly
every famed mathematician lived well before the recognition of
autism, even a glance at the lives of Archimedes, Gauss, Newton,
Euler, Riemann, Lagrange, Cantor, Fermat, Gödel,
Turing makes it clear autism must have been lingering somewhere near
at hand. There is not one social butterfly among these men, not one
glad-handing denizen of the weekly cocktail party, and we can assume
it must have been so even at the very beginning, when shape and
number were first espied. Mathematics is a lonely pursuit, a calling
more tantalizing to those unattached to the immediate concerns of
everyday society and more compelled by the patterned arrangements of
the external world. There is nothing coincidental about this. Those
who are biased towards biological perception tend to become salesmen,
managers and politicians; those who are biased towards logical
perception become mathematicians, physicists, programmers, engineers.
Everyone is drawn to the path he most clearly perceives.
A reasonable conjecture would say that
logical perception first began to make its appearance on this planet
around fifty thousand years or so ago, when autistic individuals
would have first begun to achieve significant presence and influence
within the human population (rising to the one to two percent
prevalence we can measure today). Employing their structure-grounded
perception to reconstruct aspects of their environment — and thereby
introducing language, art and number into the human
surroundings — autistic individuals would have paved the way to
logical perception for all, since of course most humans are naturally
inclined to do what other humans do. In turn, the non-autistic
population would have maintained the connection to the biological
concerns of species, helping bring both populations forward in an
expansive, explosive conquest of survival and procreation. In today's
prodigiously human world, each individual enjoys the benefits of the
dual effect, with pure forms of either biological perception or
logical perception, as well as the correspondingly pure forms of
autism and non-autism, now exceptionally rare (and most often with
challenging consequence). Each individual learns to employ a blended
form of both logical perception and biological perception, with each
individual continuing to display the outward behavioral signs of his
more natural inclination.
To this point, we have recognized
mathematics as the general term for the foregrounded sensory features
arising from logical perception, and now we have traced the origin of
logical perception itself to the atypical perceptual characteristics
of the autistic population. This discovery casts the subject of
mathematics into clearer, more natural light, for we can say with
more confidence that mathematics is not the mind of God, it is not
the fruit of human intuition, it is not a characteristic of the
objective world (and it is most certainly not a neural module inside
the human head). Mathematics is simply the natural consequence of the
presence and influence of autistic individuals within the human
population, the natural consequence of their readily observable,
albeit unusual, form of perception. We have thus grounded mathematics
as a biological/anthropological fact.
Recognizing mathematics as a
biological/anthropological fact — a fact of perception — has
consequences for the practice of mathematics. Throughout its
development, mathematics has frequently become entangled in
controversies of legitimacy, spawned by questions not of calculation
or deduction but concerns of whether certain offered concepts are
genuinely mathematical. Here too the ancient Greeks already were well
engaged, wrestling with the status of irrational numbers and the
allowability of actual (completed) infinities. In more recent years,
disputes have arisen regarding infinitesimals, the cardinal number of
sets, and existence proofs that rely upon the law of excluded middle.
These matters are not easily resolved: opposing camps form, debates
run on and on. The trouble here is that if mathematics itself is not
well grounded, then there are no practical means for settling
questions of legitimacy; when the ultimate arbiter is God, intuition
or a magical neuron, anyone is free to shift the foundation to fit
his case.
But if mathematics itself can be
grounded, there arises pragmatic means for assessing legitimacy. It
is my contention that nearly every mathematical legitimacy concern
comes down ultimately to a question of perception, and in particular
a question of foregrounding within perception. At precisely
the moment of dispute, at precisely the point of crossover from
general agreement to widespread debate, we find ourselves
face-to-face with a mathematical concept struggling to achieve its
perceptual grounds.
Take the case of an actual (completed)
infinity. By and large, modern humans have little difficulty or
disagreement about foregrounding a finite sequence (one, two, three,
four); they sense the distinctness of this perception just as surely
as they trust their ability to construct the numbers within their
physical environment. Furthermore, in addition to the constructed
sequence itself, humans foreground quite easily each step of the
iterative sequential process (take something, add one to get its
successor, take the successor, add one to get the successor of the
successor, and so on). This recipe is sharply defined and open to the
senses, and no dispute or uncertainty ever arises about its nature.
But with an infinite sequence,
something becomes different — perceptually different. The iterative
sequential process remains fine, each step still as prominent and
surveyable as all the previous steps, with the fact that the steps
have no end inconsequential to their perceptual foregrounding. But
the completed sequence is another matter. A fully realized
infinite set is precisely the thing that does not foreground within
human perception, and it remains dubious whether finite words such as
“actual infinity” or “infinite set” — or axioms attached to
such words — are adequate to alleviate the uncertainty. Many humans
are not satisfied that the symbol or axiom itself perceptually
foregrounds, not when what that symbol or axiom represents
remains hidden as noise within the perceptual field. The ancient
Greeks, as well as more recent mathematicians such as Gauss, have
dismissed the notion of an actual infinity, while many other
mathematicians have firmly disagreed.
As another example, the irrational
numbers have long produced a sense of queasiness among
mathematicians, with the technique of the Dedekind cut
introduced to place the irrationals' mathematical existence on much
firmer ground. And yet when it comes to the firmer ground based upon
the notion of logical human perception, the queasiness remains.
Dedekind cuts define all real numbers via unique divisions of the
rationals into two order-based sets, for instance a Left set of
rationals that are less than or equal to the given number and a Right
set of rationals that are greater than the given number. Adherents to
this technique will then provide many examples showing how this cut
distinctly determines particular irrational numbers — the square root
of 2, the arctangent of 3, the natural logarithm of 5. Although
doubts may linger about the use of completed infinities to form the
two sets, for anyone who has followed the mechanics of an actual
Dedekind cut, it is hard not to be impressed by the vividness of the
technique. In the examples typically offered, the process of the
Dedekind cut would appear, by and large, to perceptually foreground.
Unfortunately, perceptually speaking,
the examples typically offered are not the instances most in
question. Long before a Dedekind cut was ever considered, various
mathematical techniques had already been developed to foreground
particular irrational numbers — including for instance, the square
root of 2, the arctangent of 3, and the natural logarithm of 5.
Indeed in many cases it is precisely the existence of such techniques
that makes an actualized Dedekind cut conceivable in the usual sense.
And so for those humans who are are convinced only by the evidence of
their own perception, the Dedekind cut arrives as something of a
white elephant: in the cases of irrational numbers that can already
be perceptually foregrounded through an alternative technique, the
Dedekind cut appears to be ostentatiously superfluous, and in the
instances of irrational numbers that would possess no conceivable
foregrounding technique, the Dedekind cut comes across as little
better than useless. Of course there are many mathematicians who
would argue otherwise.
Finally, we might consider the
circumstances surrounding the concept of negation and the arguments
reductio ad absurdum based upon negation. The potential
controversy can be outlined with just a rough sketch:
In this image, there is a square region
that foregrounds perceptually and two clearly demarcated regions
within that square (A and B). Outside the square is an unbounded
region labeled C that is meant to depict everything else (and I do
mean everything else, whatever that happens to mean). Negation
within the context of the square is unproblematic, because everything
foregrounds. For instance, within the context of the square, the
negation of A is the region B and neither A nor B is perceptually
troubling. But note that negation in the wider picture is
perceptually more ambiguous. For instance, the negation of the square
itself (the negation of A union B) comes across differently than the
former case: the square itself still foregrounds quite easily, but
the negation of the square does not — in fact, the region C might not
be anything more than the background chaos. When mathematicians treat
these two instances of negation as similar or equivalent, disputes
quickly follow, and I believe that behind almost every instance of an
argument over an existence proof relying upon the law of excluded
middle, one can find a similar region of perceptual ambiguity, a
piece of mathematical landscape struggling to be clearly seen.
It is not exactly my intention to
adjudicate these matters. The purpose behind these examples is to
demonstrate that mathematical legitimacy disputes are still common
and go generally unresolved, and this is because mathematics itself
has remained ungrounded in any observable anthropological fact. Armed
however with an understanding of the history of logical/autistic
perception, and recognizing that issues of foregrounding lurk behind
nearly every known dispute, we can begin to approach these matters
from an entirely different direction, one more on par with our
approaches to biological perception and darwinamatics. Some may have
noticed that the insights suggested by reference to logical
perception are similar in many respects to those principles held by
the intuitionist school of thought. But one must also notice the
significant distinction. The intuitionists' banishment of many of the
techniques of classical mathematics is a banishment that is itself
not entirely well grounded, other than that is (as their moniker
would suggest) an appeal to intuition. It is past time for
mathematical appeals to divinity and intuition. We are better served
by grounding mathematics in our biology, our anthropology, our
history. We can begin to resolve the questions of mathematical
legitimacy when we place our mathematical concepts on the same
perceptual footing as a sexual encounter, a live birth or a tasty
meal.
In summary, we have taken a fresh
journey through the world of mathematics. It began with perception,
and with the discovery that in addition to the animal-inherited
characteristics of biological perception, man has recently in his
anthropological history acquired a second type of perception — logical
perception — in which the foregrounded sensory elements are precisely
those elements recognized as belonging to mathematics. We then
employed the observable behaviors and inclinations of autistic
individuals to conclude that logical perception must have arisen
directly from autistic perception, and that it has been the presence
and influence of the autistic population that has served as the
catalyst for bringing logical perception and mathematics into the
human world. Finally, we ventured that the establishment of
mathematics as a biological/anthropological fact provides means for
reassessing many mathematical disputes, means that are much more
practical than either myth, intuition or unexplained neural magic.
