[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.