Perception, including human perception, has not always been a consistently defined concept, but these days agreement can usually be reached somewhere along the following lines: animals receive, through their nervous system, a multitude of sensory impressions from which is distilled an awareness of the animal’s environment, as well as a reaction back into that environment, and it is the distillation part of this process that stands at the core of what is typically called perception. Perception is necessary because the entirety of sensory experience would be too overwhelming. Unfiltered and undifferentiated sensory experience would produce only a chaotic awareness of the animal’s environment, making enactment of targeted and productive reaction problematic at best. Perception extracts signal from sensory noise, perception distinguishes figure from sensory ground. It is the foregrounded elements of sensory experience that are precisely those elements that an animal perceives.
As such, particular types of perception can be defined in large measure by highlighting the characteristics of what tends to foreground within that type of perception, and by juxtaposing those characteristics against what remains ignored or 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 nature shows could easily attest, the attentive focus of wild animals is highly predictable and generally unvaried across the many species, and can be classified 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 components plays a critical role in the struggle for survival and procreation; this type of perception is universal precisely because it serves the biological process. Nonetheless, while noting these characteristics of what tends to foreground within animal perception, it is also instructive to consider those sensory features that go undiscerned. The wind rustling in the grass and leaves. Wisps of cloud drifting overhead. An arrangement of bushes along the distant horizon. Unless such features 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 of a very large portion of an animal’s sensory experience—it will simply fade unobserved into the sensory background.
It would be useful to give a name to this universal predisposition to foreground primarily, if not exclusively, those sensory features that are essential to survival and procreation. This predisposition can be labeled biological perception. And as a supplement, it would be useful also to give a categorizing name to the foregrounded sensory features themselves (that is to say, food, sexual rivals, predators, etc.). This category can be called darwinamatics—an awkward sounding name to be sure, but one chosen because it corresponds nicely to its ready-made counterpart, a counterpart that will be considered shortly.
Human perception is intriguing because it is both animal perception and it is not: like animal perception, human perception adheres to biological perception’s rule of universality, and yet unlike animal perception, human perception provides the only known counterexample to biological perception’s rule of exclusivity.
That human perception is a form of biological perception can be seen readily enough from two distinct considerations. First, there is anthropological history, which reveals that for an extremely long period of time after the evolutionary split from the other apes, human existence—and along with it, human perception—must have remained as animal-like as for all the other beasts. From Australopithecus down through the later species Homo, there is little in the way of evidence to suggest that mankind’s foregrounded focus and endeavor ever deviated far from the immediate constraints and concerns of survival and procreation. Some would argue that this perceptual state of affairs must have remained constant until as recently as around fifty thousand years ago, but at whatever moment one places the timing of mankind’s perceptual turn, it is certain that the human species possessed nothing more than a strictly biological perception for a very long period of time.
The second consideration demonstrating that human perception is a form of biological perception can be observed directly today. For although modern human perception can no longer be defined strictly in terms of biological perception alone, modern human perception still retains the vast majority of its former biological traits. In observing the features that tend to foreground within human awareness, one recognizes that food, sex, danger and all the rest continue to play an extremely prominent role—that is to say, darwinamatics still constitutes much of the focus of human attention and endeavor. Indeed a healthy dose of biological perception is considered to be crucial for both early 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. The foregrounded elements associated with survival and procreation may no longer play the critical role they once did on the prehistoric savanna, but 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—that characteristic is still shared in common. What distinguishes human perception from animal perception is that human perception, and human perception alone, has acquired a significant addendum. In the foreground of modern human awareness, in addition to the still influential components associated with survival and procreation, one finds also an entire host of sights, sounds and other sensory features that no wild animal would ever naturally discern. A list of such features would stretch to enormous length, and it would include not only the symbols of language, the architectural traits of buildings, the rhythms of music and the intoxications of perfume, it would include also the wind rustling in the grass and leaves, wisps of cloud drifting overhead, and an arrangement of bushes along the distant horizon. Modern man now foregrounds a vast range of sensory features not directly connected to the immediate concerns of survival and procreation, which is to say that modern man has acquired a second and entirely different type of perception.
This second type of perception is distinguished primarily by its gravitation towards non-biological targets and by its persistent foregrounding of structural characteristics, characteristics that can be classified under just a small set of headings—symmetry, pattern, mapping, order, form. The same structural principles that unify and organize the objective world now increasingly unify and organize human sensory experience. Furthermore, humans have been making prolific use of these newfound principles to reconstruct their surrounding environment, forging an increasingly receptive home for this new type of perception, nudging the distinctiveness of the present age towards a constructed distinctiveness, in fierce defiance of nature and its biologically limiting constraints. Humans have become as proficient with the artificial as they once were with the natural, and the human population is now foregrounding a veritable cornucopia of number, shape, order, rule. Thus quite unlike any other species on this planet—and quite unlike humans themselves of not that long ago—modern humans find themselves focusing on an entirely different type of sensory target, a target not strictly associated with survival and procreation, a target capable of producing radical influence and astounding effect.
As was done with the phrase biological perception, it would be useful to give a name to this exclusively human capacity to foreground those sensory features that possess structural and patterned characteristics. This capacity can be labeled logical perception. And again, as was done with the term darwinamatics, it would be useful to give a categorizing name to the foregrounded sensory features themselves (that is to say, symmetry, order, 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, order, mapping, etc. is none other than the study of mathematics.
Thus to summarize, modern human perception consists fundamentally of a blend of two very different types of perception. The first type of perception foregrounds those sensory features directly associated with survival and procreation. This type of perception has been labeled biological perception, and its foregrounded sensory features have been categorized as darwinamatics. It is recognized that humans share this type of perception with all the other animals, have inherited it from out of the species’ animal past, and continue to experience its influence within the modern age. The second type of perception, unique to humans and acquired quite recently within anthropological history, foregrounds those sensory features that make up the structural framework of the non-biological world and that possess the underlying characteristics of pattern, symmetry, structure and form. This second type of perception has been labeled logical perception, and it has been recognized that the foregrounded sensory characteristics of logical perception are precisely those characteristics commonly studied under the heading of mathematics.
Mathematics has always been something of a philosophical puzzle. Intimately connected to space and time and the underpinning behind almost every facet of rational thought, mathematics appears to stand at the core of all non-biological conception, and so it has been more than a little bit tantalizing to try to determine the foundations of mathematics itself. The ancient Greeks were already 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 and sometimes controversial 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 external world, and the neural mechanisms inside the human skull (the latter being perhaps the most mythical suggestion of them all). Yet despite all these competing proposals—or perhaps because of all these competing proposals—the philosophical puzzle has remained as puzzling as ever.
But here is a proposal: the above-stated recognition of mathematics as being the equivalent of the foregrounded elements within humanity’s second type of perception can open the door to a less mythical and 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 never been any philosophical qualms regarding the origin and foundation of darwinamatics—the targets of biological perception have always been approached as simply open to inspection, and there is no reason to think that the targets of logical perception cannot be approached in precisely the same way. Plus the placement of logical perception’s birth within the recent time frame of human history provides still more reason to conceive of mathematics as being something other than mystical—instead history suggests that mathematics can be more soberly described as simply the outcome of an anthropological event, as simply the outcome of the advent of logical perception.
Of course if that is all that were being offered, one might reasonably complain that this “sober” description of mathematics does little more than change the aspect of the problem, it makes out of logical perception the same philosophical puzzle that was previously being made of mathematics itself. What would be the origin and foundation of logical perception? Would it be a gift from the gods? how about an irreducible human intuition? or perhaps an evolutionary explosion of synaptic generation? As it turns out, in the early twenty-first century humans have been slowly uncovering a patch of knowledge that has the ability to make it clear that the origin and foundation of logical perception is in fact none of the above. Instead logical perception can be seen as directly attributable—and in a directly observable way—to the presence and influence of an atypical group of people.
Autism, a distinctive set of biological, sensory and developmental characteristics, is usually labeled as a medical condition, but at its root, autism is more accurately and more insightfully described as a condition defined by perception. Indeed in many respects, what distinguishes autism and non-autism—each condition regarded in its purest form—is exactly the same distinction to be made between logical perception and biological perception. Because what determines and unifies autistic experience is its diminished bias towards biological perception, and in particular its diminished foregrounding of conspecifics. Most individuals, strongly under the influence of the species’ ongoing attachment to biological perception, easily foreground and attend to the other members of the human population. But autistic individuals, to varying degree detached from the influence of biological perception, find themselves with little perceptual affinity for the other humans around them. Autistic individuals do not readily foreground human voices, they do not focus energetically on human faces, they do not enthrall to the most common human concerns. This diminished awareness towards the other members of the species is observable from a very early age in autistic individuals and it remains extremely consistent—to the point of being defining—across the entirety of the autistic population. This diminished awareness towards the other members of the species 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. Whereas the common animal experience is to foreground first and foremost those sensory features connected with a species’ survival and procreation—including a strong proclivity towards the perceptual foregrounding of conspecifics—autistic individuals now serve as the only known counterexample to this otherwise universal tendency, and thus autistic individuals can be described as the least animal-like of Earth’s many biological creatures, because autistic individuals are the least determined by the characteristics and constraints of biological perception.
As a result of their diminished facility with biological perception autistic individuals are initially hindered in gaining their sensory footing—for them, little can emerge as sensory signal, there is almost no figure against the sensory ground. If this set of sensory circumstances were to continue to hold, then autistic individuals would find themselves in the most dire of straits, possessing almost no sensory traction to aid in their developmental progress or in their tackling of the essential requirements of survival itself. Fortunately, this set of sensory circumstances generally does not hold. In the absence of a strong type of perception—that is to say, in the absence of biological perception—autistic individuals find themselves with both the opportunity and the motivation to latch onto alternative types of perception, and the alternative type of perception that emerges the most prominently for autistic individuals is the type of perception described above as logical perception. It is those same sensory features that embody symmetry, pattern, mapping, structure and form that are the sensory features most capable of breaking a background sensory chaos, and thus autistic individuals, hungry for perceptual targets that can make order out of their otherwise chaotic sensory world, gravitate to these alternative perceptual targets with an almost fanatical intensity. Lining up toys, preoccupation with spinning objects, repetitive flapping of hands, extreme iteration over video and song, idiosyncratic dexterity with letters and numbers—these characteristic behaviors, observable from the earliest age in autistic children, betray a deep fascination with sensory targets composed out of pattern, structure and form. Instead of the common perceptual bias towards other humans and their species-driven endeavors, autistic individuals find themselves drawn to number, shape, order, rule. Instead of a natural ease with the material of darwinamatics, autistic individuals gravitate almost obsessively to the world 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 observations itself. It remains unclear when science will begin to make its own perceptual turn, but at the moment the prognosis remains highly pessimistic.)
The history of mathematics provides still more evidence of a direct connection between mathematical and autistic characteristics. Although the biographical details are not always complete and although nearly every famed mathematician lived well before the recognition of autism, even one glance at the lives of Archimedes, Gauss, Newton, Euler, Riemann, Lagrange, Cantor, Fermat, Gödel and Turing will make it obvious that 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 one can assume it must have been so even at the very beginning, when shape and number were first espied. Mathematics is a strange and 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. Those who are biased towards biological perception will tend to become merchants, managers and politicians; those who are biased towards logical perception will tend to become physicists, programmers and engineers; and those who are perceptually obsessed by the basic elements of pattern, symmetry, structure and form will become the prime candidates to serve as mathematicians. Everyone is drawn to the path he or she most clearly perceives.
A not unreasonable conjecture would be to say that logical perception first made its appearance on this planet starting around fifty thousand years ago, when autistic individuals first began to achieve significant presence and influence within the human population (rising to the one to two percent level of prevalence that can be measured today). Employing their structure-grounded proclivities to reconstruct various aspects of their environment—and thereby helping to introduce the features of language, art and number into the human surroundings—autistic individuals would have begun to pave the way to logical perception for the entire human population, leveraging the natural inclination of most humans to do what other humans do. In turn, the non-autistic population would have maintained the connection to the biological concerns and ambitions of species, helping to thrust both populations forward in an expansive and explosive conquest of survival and procreation. In today’s prodigiously human world, each individual now enjoys the benefit of this dual influence, with pure forms of either logical perception or biological perception, as well as the correspondingly pure forms of autism or non-autism, now exceptionally rare (and most often with challenging consequence). In the modern world, while continuing to display the outward behavioral signs of one’s more natural inclination, each individual learns to employ a blended form of both logical perception and biological perception.
Thus to summarize once more, it has been recognized that mathematics is the general term to be applied to the foregrounded sensory features that arise within logical perception, and it has also been recognized that the origin of logical perception can be traced to the atypical perceptual characteristics of the autistic population. This latter recognition places the subject of mathematics into a more natural context, it allows one to confidently forgo such unobservable notions of mathematics as being for instance the mind of God, the fruit of human intuition, the cold formality of the external world, or the magical product of neural modules inside the human head. Instead, mathematics can be described more directly and more openly as being the natural consequence of the presence and influence of autistic individuals within the human population, the natural consequence of their readily observable, albeit highly unusual form of perception. Thus mathematics can be grounded as an anthropological fact.
Recognizing mathematics to be an anthropological fact—a fact of perception—has consequences for the practice of mathematics. Throughout its historical development, mathematics has frequently become entangled in controversies of legitimacy, controversies spawned by questions not of calculation or deduction but questions over whether certain proposed concepts can be considered genuinely mathematical. Here too the ancient Greeks already were well engaged, wrestling with the status of irrational numbers and with the validity 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 easily argued but not so easily resolved, and thus opposing camps are apt to form and the debates to run on and on. The dilemma 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 taken to be God, intuition or perhaps a mass of neurons, the combatants are free to shift the foundation to fit their own particular case.
Thus the key to finding pragmatic means to help resolve mathematical legitimacy disputes is to develop some grounding for mathematics itself, and the key to developing some grounding for mathematics is to recognize the critical role being played by perception. The proposal being made here is that nearly every mathematical legitimacy concern comes down ultimately to a question of perception—and in particular, comes down to a question of foregrounding within perception. Always at the moment of dispute, always at the point of crossover from general agreement to widespread debate, there can be found 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 distinctiveness of this perception as surely as they trust their ability to construct the sequence’s numbers within the terrain of their physical environment. Furthermore, in addition to the constructed sequence itself, humans can 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 assessment by the senses, and thus no dispute or uncertainty ever arises about its nature.
But with an infinite sequence, something becomes different—perceptually different. The iterative sequential process remains entirely unobjectionable, because it remains essentially the same: each step in the process is still as prominent and surveyable as all the previous steps, with the fact that the steps now come to no end being seen as 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 doubtful whether finite phrases such as “actual infinity” or “infinite set”—or axioms attached to such phrases—can alleviate the uncertainty. Many humans are unsatisfied with symbols or axioms that serve as substitutes for perceptual foregrounding, especially when what those symbols or axioms represent remain hidden as noise within the perceptual field. The ancient Greeks, as well as more recent mathematicians such as Gauss, have readily 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, and the introduction of techniques such as the Dedekind cut were motivated precisely by the need to place the irrational numbers on much firmer ground. And yet when it comes to the firmer ground associated with logical human perception, much of the queasiness still remains. Dedekind cuts define the real numbers via unique divisions of the rational numbers into two order-based sets—for instance a Left set of rational numbers that are less than or equal to the given number and a Right set of rational numbers that are strictly greater than the given number. Proponents of the technique can then provide many examples that demonstrate how such cuts distinctly determine 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 these Left and Right sets, for anyone who has ever 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 a Dedekind cut gives the appearance, by and large, of a technique with strong perceptual grounds.
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 been developed to foreground particular types of 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 convinced only by the evidence firmly linked to their own perception, a Dedekind cut arrives as little more than the proverbial white elephant. In those cases of irrational numbers that can already be perceptually foregrounded through an alternative technique, a Dedekind cut comes across as ostentatiously superfluous; and in those cases of irrational numbers that possess no conceivable foregrounding technique, the Dedekind cut proves to be little better than useless. Nonetheless, there are many mathematicians who would strenuously argue otherwise.
Finally, one might consider the circumstances surrounding the concept of negation and the arguments reductio ad absurdum based upon negation. A sense of the controversy can be highlighted with just a rough sketch:
In this image, there is a square region that foregrounds perceptually, and within that square region there are two clearly demarcated sub-regions (A and B). Outside the square region there is an unbounded region that has been labeled C, and this unbounded region is intended to depict whatever has not been otherwise described—a phrase intended to be precisely vague. Perceptually speaking, negation within the context of the square is unproblematic, because every region and sub-region foregrounds quite easily. For instance, within the context of the square, the negation of A is the region B and the negation of B is the region A, and neither of these negations is perceptually troubling. But negation in the context of the entire picture is perceptually more ambiguous. For instance, the negation of the square region itself (that is, the negation of A union B) comes across much differently than in the former case: the square region still foregrounds quite easily, but the negation of the square region does not—in fact, the best that might be said of the region C is that it does an admirable job of forming the background chaos. When mathematicians treat these two instances of negation as similar or equivalent, it is not surprising that disputes can quickly follow. And in a similar manner, lurking behind almost every instance of an argument over an existence proof relying upon the law of excluded middle, one can find a similar instance of perceptual ambiguity, a piece of the mathematical landscape struggling to be clearly seen.
It is not the intention of this essay to adjudicate these matters. The purpose behind these examples is to demonstrate that mathematical legitimacy disputes are still quite common and go generally unresolved, and this because mathematics itself has remained largely ungrounded. But with what has been said regarding perception—biological, logical and autistic—and with an understanding that mathematics has grown organically out of human circumstances, there is an opportunity to examine mathematics with a more anthropological eye. Armed with an awareness of the history behind logical and autistic perception, and recognizing that issues of perceptual foregrounding lurk behind nearly every known mathematical dispute, one can begin to approach these disputes from an entirely different direction, one more on par with efforts taken towards biological perception and darwinamatics. Thus it is no longer appropriate to make mathematical appeals to such concepts as divinity, intuition or specialized neurons. Humans will be much better served by grounding their mathematics in history, in anthropology, in the details—the perceptual details—of their everyday lives. Humans will begin to make progress on questions of mathematical legitimacy when they begin to place their mathematical concepts on the same perceptual footing as they would a sexual encounter, a live birth, or a tasty meal.
This short essay has offered a fresh journey through the world of mathematics. The journey began with the topic of perception and with the recognition that in addition to the animal-inherited characteristics of biological perception, humanity has recently in its anthropological history acquired a second type of perception, logical perception, in which the foregrounded sensory features are precisely those features that belong to the category of mathematics. Next, the behaviors and inclinations of autistic individuals have been explored with an unprejudiced eye, giving particular focus to the atypical characteristics of autistic sensory attention, leading ultimately to the conclusion 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, it has been ventured that the establishment of mathematics as an anthropological fact has provided means for reassessing many mathematical disputes, means that are more practical and more informative than resorting to myth, intuition or unexplained neural magic.
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