Band In One
Seeking Mathematical Platonism in the Stars
A potent but flawed idea
Perhaps one of the strangest yet historically most potent ideas in all of Western Philosophy was put forward by Plato in his World of Ideas or Forms. This was a world that existed separately and independently from our changing and often chaotic world. Only in such a world were things, the Forms or Ideas, whole and complete, changeless, and thus perfect. Entranced by this perfection, he held that they were the true realities, in contrast to everything we experience in the more immediate and transitory world, a world of things that are never complete, that are ever changing, that to his dismay "never really are". Over the centuries these essential views have inspired many thinkers in one or another form of idealism, even as they grappled with and ultimately rejected the idea of a separate World of Ideas. Plato's student, Aristotle, for example, snatched the Forms out of their independent realm and placed them firmly in things as essential substances, the defining characteristics of an object.
Plato's idea of persisting perfection has had a long run in mathematics, for mathematical objects such as numbers and geometrical forms seem to have a certain objectivity about them. Their properties can be determined through reasoning and established with "mathematical certainty". This kind of clarity epitomizes a type of rational perfection that is relatively resistant to change. Euclid's proof that there is no highest prime number stands just as valid and unassailable today as it did 2500 years ago. Concomitant with the elaboration and expansion of the properties and domains of mathematical objects, and particularly in the convergence of different threads of mathematical thought, there is a sense of wonder, a sense that we are glimpsing a world that is complete and whole in itself. This is most dramatically illustrated in Euler's identity, e ^(i π) +1 = 0 , which links five critical numbers, three of which, π , i, and e, have long, complex, and separate histories, and the other two, 1 and 0, are numerically foundational. The beauty, elegance, and power off this formula must amaze anyone with a mathematical brain, and it reinforces that sense of "other worldliness", an independent order or realm not subject to the contingencies of the natural world. This viewpoint, child of Plato's idealism, is called mathematical realism; real because the objects are held to exist independently of our understanding. They are admittedly abstract, but still abstractly existing, somehow, somewhere. But no one has ever provided a satisfactory account of how or where, nor have they explained how we might connect to them.
There are of course alternative mathematical philosophies. The most prominent, indeed, the currently dominant view is called mathematical formalism. Besides denying the separate existence of mathematical objects, formalist rely on a system of logical rules and principles to define and explore mathematical objects. What is essential are the axioms, the starting principles, and the definitions. The rest is, at least theoretically, formal deduction from these starting points, perhaps guided by intuition but that is sometimes admitted reluctantly. The original goal, articulated in the early part of the 20th century by the influential German mathematician David Hilbert, was to create one complete formal system for all of mathematics. But this program was undermined by the research of Gödel, Turing, and Church, which demonstrated that no formal mathematical system could be complete and consistent. Even with this "defeat", formalist theory, as a working method, still prevails. However, it is important to note that very few mathematical arguments are completely elaborated in a formalistic way. It is far too tedious and prone to typo error. Instead, the arguments are formally sketched and then described in a more informal but still technical way; there is a working assumption that a complete formal argument could be made.
Formalism allows for different geometries based on different starting points; for example, Euclidian geometry, where parallel lines do not meet, and spherical geometry where they do, each based on a slightly different set of axioms. I should note that many mathematicians who profess formalism are seen to be operating as Platonist, realists. It is hard to escape that sense of looking for and then discovering something that is there.
This is the briefest possible sketch of formalism, provided only for contrast. What I want to explore in this article is a different kind of mathematical realism, based on a highly speculative line of thought which must first conjecturally explore the probabilities of intelligent life on other planets. Say what? A little patience, please.
Note: what follows is a lengthy exploration of probable life on other worlds, on the evolution of life on this planet as a model, and on the nature of symbolic life, the rational and linguistic mind. If you would prefer to skip this excursion, you can use the "Go to Continuation" link to resume the exposition on Plato's theme.
And the heavens were full...
Modern astronomy has made many incredible discoveries, but there are two that are of particular importance to my purpose. First, is the immensity of the universe. This, of course, is an ongoing story since the universe has grown progressively and conceptually larger and larger over the last 200 years as our telescopes have grown more powerful and our understanding has deepened. The latest estimates, based on surveys, is that there are over 2 trillion galaxies.
[ 2 ] Assuming an average of 200 M stars per galaxy, a very conservative and minimum estimate, that comes to:
400,000,000,000,000,000,000 (4 x10^20) stars!
Minimum!
[ 2 ] Henry Fountain, Two Trillion Galaxies, at the Very Least, (New York Times; 10/17/2016)
But the second discovery is even more astonishing, and more recent: there are more planets than stars. Although we have found "only" 4000 or so exoplanets as of this writing, our detection methods are very limited. They can only see the small percentage whose orbital planes around their parent stars happen to intersect with earth's orbital plane. And the stars must be nearby neighbors of the sun; the furthest exoplanet that the Kepler mission found is less than 3000 light years away, near the limit of this methodology and a sliver of a smidgen of a drop in the great cosmic bucket as far as distances are concerned. However, from this small sample we can reasonably extrapolate to the whole cosmos since, by all the evidence that has accrued over 400 years of science, the laws of nature are universal. Stars are born in molecular clouds of gas and debris from the explosions of earlier eons of stars, and in the process, they are surrounded by disks of condensing gas and dust that progressively conglomerate into clumps and chunks, small and larger bodies, asteroids and comets, planetesimals, moons and planets. We have telescopic images of the process, of protoplanets sweeping out orbits around protostars, pictures from deeper in the Milky Way than the nearby exoplanets we have tracked. It would be the relatively rare star that isn't born with a small or large family of planets.
But there is more. There are hundreds of moons in our solar system, at least 20 of which are gravitationally large enough to maintain spherical shape. The Kepler data also has evidence of moons around two of the planets. Only a very large moon would show up in this method. We can reasonably expect that the average star has more moons than planets.
This plentitude of planets and moons is both intriguing and dismaying: intriguing because of the rich possibilities for life which, in my view, likely fills the universe in an unimaginable variety of forms, but dismaying because such wealth of forms seems so beyond our reach in terms of contact or visitation, and thus beyond our understanding.
There are at least two other equally relevant responses to this overwhelming number of planets. First, there is the Fermi paradox: "Where are they", or some such remark attributed to physicist Enrico Fermi. If life is ubiquitous, why do we not have clear evidence? This has always struck me as a little naïve, with a technician's touch of arrogance. Life might be common, but symbolic and technological life of a type that is capable of communicating its presence in some way may not be so common, or its chances of enduring may be far more fragile and effervescent than simple cellular life. Moreover, we have only just begun to search, with relatively primitive tools. It is ridiculously presumptuous to assume that we have done anything more that scratch the surface of a very thick and almost inexhaustible question.
Second, there is the whole issue of UFOs with a relatively small number of events that are impossible to explain. One of the major obstacles here is the limits imposed by our understanding of general relativity and the immense distances in time and space between the stars. We simply do not understand how traveling freely between the stars could be possible on the scale of lifetimes. How do you even have a conversation if the nearest star, …NEAREST!...is four light years away. If you do manage to travel near the speed of light, and go there…when you return, you'll find your family and friends are long dead. These are the time effects of relativity, effects that have been confirmed experimentally. Oh yes, we have our Star Wars and Aliens. But it is one thing to write in a film script, "Pop it into warp drive, Scotty". It is quite another to engineer it, and we haven't a clue. This is not to deny that it may be possible, or that our understanding of the rules of physics is incomplete, etc., but we are simply still too ignorant to evaluate the question. You can come up with all sorts of ideas based on analogies, but they are all empty balloons (though superficially entertaining) if they are not backed up with research and knowledge.
We understand the basic theoretical conditions of life, metabolism and replication, and the essential dynamics of evolution, but we only know these in relation to an earth-universal DNA and its familiar four bases, adenine, guanine, cytosine, and thymine. Although we have created and inserted a fifth base in the lab, we really know nothing of the possibilities of that occurring naturally or expanding to include additional and/or different bases, or even a code of life based on entirely different molecular structures. We expect life to require liquid water, the solvent and medium we are familiar with, but is that truly a necessary condition? In our own solar system, the large Saturn moon Titan has a complex climate and seasons like Earth, but it is based on hydrocarbons, not water, and it is very cold with an average surface temperature of -290 F. Can some form of life evolve under these alien and frigid conditions? This is actually a question we could answer by sending a probe to Titan stocked with the appropriate sensors and analytic tools. But the greater questions remain: what are the limits, what are the possibilities? Can there be a Horta (a la Star Trek), a silicon-based life form that eats rock? Authoritative answers to such questions may forever elude us. Tendentious opinions, little knowledge.
Would you care to try a sample?
Be that as it may, there is still much room here for useful thought. Whatever the limits for some kind of life arising under (for us) exotic conditions, with this wealth of planets we can be confident that there are, have been, and will be vast numbers of earth-like planets orbiting sun-like stars within what we consider a habitable zone, a distance that allows for liquid water. In the Milky Way alone, with roughly 200 - 400 billion suns, there are authoritative estimates of 17 billion earth-sized planets, although not all of these would be in the habitable zone. We can always imaginatively grab a small sample of, say, just a thousand of the habitable-zone planets, and ask interesting (though still speculative) questions, but at least it is speculation that is on familiar grounds.
So, we have a sample of a 1000 earth-like planets existing in the habitable zone, a region that allows for liquid water. And I must emphasize, this is a speculative sample, not a predictive tally; just something with which to pose interesting questions like, how many of these worlds would generate life? I am inclined to say virtually all. The essential elements needed for familiar life—carbon, hydrogen, oxygen, nitrogen, phosphorus, and sulfur— would likely be available on all earth-like systems. Many of the molecules that play an essential part in life on earth, like amino acids, have been found in meteorites, comets, and even interstellar space.
As yet we have no detailed description of the initial processes leading to life, but we have multiple and competing theories, all incomplete in terms of sequences of testable specifics, but each with a certain general plausibility. This wealth of theories may actually indicate a wealth of possible pathways that nature can follow; it does not have to commit itself to just one path to please a theoretician. We can also ask, if the planet persists in a relatively stable stellar environment, what factors would prevent life from emerging? Other than sterilizing and destructive radiation from a nearby supernova or black hole jet, I know of no other good anti-life candidates. Even the catastrophic effects of an asteroid collision, like the one that did in the dinosaurs, do eventually heal, and life resets. There is in fact evidence that the early earth suffered an extended period of over a billion years of celestial bombardment, even as life took hold. [ 3 ]
[ 3 ] Richard A Kerr, Asteroid Model Shows Early Life Suffered a Billion-Year Battering, (Science, 04/15/2011)
All the worlds in our sample are water worlds. We know that water is the "universal solvent", a liquid that can dissolve more substance than any other, and thus an apparently ideal environment in which large collections of diverse molecules can “collaborate” and "struggle" for stability in their interactions, with the ultimate prize being controlled replication, an essential and defining feature of life. The timescales for molecular interactions within a variety of different sub-environments are micro-microseconds, and the time available for experiment is billions of years; this means the probabilities would be overwhelmingly in favor of life. The longer you have to flip coins, the more likelihood you will get improbable sequential outcomes. I say that all the sampled planets take the first steps.
Assumably, it will be some bacterial-like or archaea-like form, which is to say some cellular form that has well-defined boundaries. Could it be non-cellular, non-bounded? This goes immediately beyond our knowledge. It is hard to see how the critical processes involving many species of complex molecules could be concentrated and localized without some sort of cell membrane of one kind or another which provides a protective and to some degree controllable environment. But let's leave room for that possibility and say that 5% of planets do something totally bizarre (to our prejudiced minds), taking a form and path that is beyond our experience and understanding. That leaves us with 950 planets with a cellular based life, and we will focus on these.
Welcome to the Casino Earths…
How many will develop multi-cellular life? This was a big step on earth where life existed for roughly two billion years, or longer, before taking it. Some highly-informed researchers, such as biochemist Nick Lane, see major obstacles and improbabilities.
Looking at a vital ingredient for life - energy - suggests that simple life is common throughout the universe, but
it does not inevitably evolve into more complex forms such as animals. I might be wrong, but if I'm right, the
immense delay between life first appearing on Earth and the emergence of complex life points to another, very
different explanation for why we have yet to discover aliens.[ 4 ]
He goes on to describe the complexities and improbabilities evident in the integration and management of energy demands in multicellular life forms. Part of the process as it occurred on earth required a symbiotic relation between two cellular forms, one of which engulfed the other; the swallowed components eventually lost their independence and became the mitochondria and chloroplasts, the energy powerhouses of all eukaryotic (multi-cellular) life. But the details! All the vast and minute biochemical changes required for these delicate arrangements to evolve and function. However, we should quickly point out that, even though the evolving of such arrangements is highly intricate and chancy, the products are surprisingly stable, and serve as a reliable platform for subsequent evolutionary changes. If relative stability can be established in the outcome of the microevolutionary processes, would it not in principle be a factor in earlier evolutionary stages? In any case it is easy for our finite minds to be overwhelmed by these detailed and exacting processes. On the other side of the issue, these processes and the changes to them can take places over very small time scales. Bacterial replication is generally accomplished in hours, or even minutes, but mutational events and microbiological changes within a cell occur, again, in microseconds. Nature is always experimenting, tampering and testing, and the great Evolutionary Reaper governs; what works survives, replicates; what doesn't, dies out. If you consider the total (and immense) number of bacteria-like cells doing this experimentation at any one time, on very small time scales, and then recognize that this continually took place over 2 billion years, is it really so surprising that nature stumbled upon an answer, and multicellular life took off? In view of the number of microevents and total time, one could just as well ask regarding the "delay between life first appearing on Earth and the emergence of complex life”, what took you so long?
Alright, it's a crapshoot. My inclination is to hold that the vast majority of our 950 sample planets will take the multicellur step, but in deference to the skeptical biochemists who have a far greater, though still very incomplete, grasp of the details, I will compromise at 50 percent. Nothing less!
[ 4 ] Nick Lane, Life: is it inevitable or just a fluke?, (New Scientist, 6/20/2012). Lane explores the chanciness of multicellular life in detail in The Vital Question, (W. W. Norton and company, 2015).
Onward and upward
Now, with our 475 remaining but promising planets we have to ask, how many would evolve complex life forms, forms with specially adapted and configured cells and differential organs, means of locomotion, senses, brains. We must assume that early experiments of life all took place in water, in seas, lakes, ponds, and puddles. Assuming that no local cosmic events occur that would wipe out all life forms, I cannot imagine a principle by which nature would now stop experimenting, draw a morphological line in the sands of those seas and lakes, saying "I will go no further, this is sufficient." It must continue to experiment with the same determination, under the same discipline of variation and testing, replication and culling. Novel forms with more complex capabilities, locomotion instead of fixed in place, more complex sensation with dedicated organs instead of simple tactile contact, predation of larger forms instead of scrap filter feeding, etc.
Photosynthesis first developed in cyanobacteria, probably some of the first forms of life on earth. Once multicellularity hit the stage, was it not inevitable for cells to conglomerate in sheets of photosynthesizing cells, rechargeable batteries to support the energy demands of complexifying and differentiating organisms, the plants. Was it not inevitable for them to take to the land, basking more directly in the sun, and for them to be followed by their distant cousins, their mobile consuming cousins. There is some evidence for the inevitable nature of this expansion and diversification. For life on earth has often found the same or similar biological solutions in very different lineages. Thus, we find that insects were first to take to the air, to be followed much later by the reptilian pterosaurs, the first vertebrates to master powered flight. Still later came the dinosaurs with the evolution of birds from feathered theropods. And then the mammals discovered flight in the form of bats. There are many such episodes of parallel or convergent evolution in the story of life on earth, and it is reasonable to expect them to play out in an analogous fashion on other earth-like planets.
We must note: while the greater processes, general sequential stages, and general forms of life may have unfolded under strong deterministic pressures, there is still much room for contingency. It is difficult to separate the larger processes from the specific forms, because we learned about the former by studying the latter over long periods of time. However, whatever attachments that may generate, there is no certainty that specific kinds of animals must evolve or prevail at any stage on an earth-like planet. There could be planets without dinosaurs because of earlier evolution of a different kind and more dominating type of reptiles, or even more precocious mammal-like creatures. There could be planets where insect-like creatures overcome the size limitation of our local varieties by developing lungs and circulatory systems. Or perhaps a planet is predominantly a water world, with only small bodies of land intermittently available, a circumstance that would certainly limit the scope for large land-based animals.
There are too many contingent factors at the detailed level where life is most real to allow us to predict specific forms. We can only presume, in my estimate, that all the many sub-environments in our earth-like planets will be exploited by some complex forms of life. So, if we reluctantly but realistically estimate that around 5% of the remaining planets undergo some catastrophic cosmic event that demolishes their living worlds, we are left with roughly 450 planets in our sample. This probable destruction is, of course, not a happy thought, but it occurs at all levels and stages, from species and environments to planets and stars, to galaxies. Such things do happen, must happen. Nature honors causality, not sentiment, and we must try to align with its essential programs (laws) even as we try to deflect them towards more life-enhancing and humane outcomes.
The question without which there would be no questions
So now we come to the most difficult question. Our original sample of 1000 planets has been reduced to 450 with multicellular life, with complex and divergent life forms. How many of these will have symbolic life? I have chosen 'symbolic' as opposed to 'intelligent' for a reason: any animal that survives by knowing its world, at whatever level, is intelligent. I used "level" here without any intent to wade into the meaningless debate surrounding comparisons of intelligence, all of which are usually misleading, prejudiced, and of little value—alright, you're smarter than a chipmunk, but what are you doing with that intelligence as the earth roasts, the ice melts, the seas flood…Most importantly, from nature's standpoint the question is irrelevant: there is a certain "needs to know" principle operating in evolution's selection regime; nature provides by chance what is necessary, what is sufficient. In the final analysis no species is nature's pride and prize. With that said, we can still acknowledge that we humans are, now, on this planet, uniquely symbolic with highly developed languages.
As an aside, I do not doubt that Neanderthals and other predecessors in the hominin lineage also had some language, most likely in a more rudimentary form, and that would be so the further back you go; but still, most of them were somewhere on the symbolic track, a long slow ramp that climbs explosively when certain thresholds are reached. [That is another story.] I hold that language isn't something that just happened at once in us, but evolved slowly, incrementally in our predecessors. We happened to persist and survive, and thus to have developed and expanded this symbolic capacity to a high degree. And make no mistake, nothing of the human life that we are intimately familiar with would be possible without language, and in particular the written word. The vast range and development of tools and technologies, the many forms and structures of social organization and communication, the wealth of cultural variations in all these developments, the libraries, the laws, the arts, the sports, none of this is possible without written language. The symbolic mind, with its enhanced capacity for framing questions, imagining alternatives, and planning for their implementation, is intrinsically curious and creative, and must necessarily change itself and its circumstances, that is to say, evolve on a cultural level. And to return briefly to my opening theme for a moment, this whole elaborated domain of scientific and technological creativity would not be possible without mathematics.
So, how many of our planets will evolve symbolic life? At first glance we should not immediately impose classification limits based on our familiar categories; if we must look for analogs, it wouldn't necessarily have to be in primate-like or even mammalian-like species; it could be reptilian-like, amphibian…I cannot think of any intrinsic limitation here in what we would describe as the broader classification levels of animal life. You might be tempted to exclude the insects, but again, if they could have overcome their size limitations, why not?
However, given this broad morphological freedom, is it not rather curious that fully symbolic life evolved on earth in only one small lineage, the hominins, and if we measure from the first stone tools, it occurred over a period of "only" 2 to 3 million years—a small ramp of time considering the changes, but still a ramp with a gradual acceleration in those changes of form and behavior as time progressed. Dinosaurs were on earth for over 150 million years, including the (often smaller) bipedal predatory theropods which are estimated to be among the more intelligent of the class; there are no dinosaur cultural artifacts, no hint that any subgroup was symbolic. There was over 60 million years of mammalian evolution before the hominins started "chatting" in presumably the crudest possible ways and climbing their cultural ladders. Considering the number and kinds of animals that have evolved during these temporal timelines, it doesn't appear that symbolic aptitude is such a likely or high percentage pathway. Certainly not as common as, say, flight!
This is a bit of a mystery. The intrinsic and inherent powers of symbolic life are immense, as any glimpse of the modern technological world would testify. Yet at the same time there seems to be a fragility, particularly in the earliest stages. There is some DNA evidence of a bottleneck in human populations, that the total number of humans was rather small when "fully anatomically modern" humans emerged in Africa. [ 5 ] There is also the stark fact that all the other hominin forms, both contemporary to early Sapiens and prior to them, such as Neanderthals, Erectus, Denisovans, Naledi, etc., all of them, died out. Now, of course, one could say that they "evolved into" something else rather than "died out", and that we are their "fulfillment". There is an element of presumption in that. The general picture you get of hominin evolution over the course of 2 million years is of small pockets of separated types moving somewhat randomly over the changing landscapes and climates of Africa, Asia and Europe, occasionally interbreeding, but often left isolated, and then extinguished by the fickle hand of evolution. And we, the final torchbearers of symbolic life, just managed to carry on. So one of the reason that symbolism may not have happened in the dinosaurs or other earlier mammals is that it is most problematic and fragile in its nascent stages.
It is also possible that it requires, as an essential biological adaptation, the ability to manipulate the world in ultimately more subtle ways, an ability that was provided by the evolution of bipedality in the hominins which in turn freed their hands and minds. And the hands, as agents of the mind, are critical. They could now hold things before their eyes, and in our lineage that ultimately became a stimulus, a pathway for elemental curiosities to begin to pierce the fabric of the world, and to learn to manipulate it, to shape objects for their purposes beginning with simple sticks and stones, and moving on to animal bones, antlers, animal skins, shells, plant materials, a growing wealth of cultural artifacts. The therapods among the dinosaurs were also bipedal, but those harsh crude claws…? Could they ever have been the instruments of curiosity and refined intentionality, both a product of evolutionary pressures and selection, that the subtly tactile and grasping hands of primates are? Would they have even needed the potential assets of symbolic life, as a focus of evolutionary pressure, since they were robustly equipped with speed, teeth, and claws?
What about dolphins? They are demonstrably highly intelligent, but they don't have hands to manipulate the world. Instead, they developed remote sensing through sonar, and apparently communicate through both gesture and sound. The question of whether they have language is hotly debated, but even if we gave them the benefit of the doubt, animals with their anatomical characteristics, immersed in a water world, do not seem to me likely candidates for developing a technological civilization. Try doing rudimentary electromagnetic experiments under water. Of course you could write a science fiction novel of a water world with cetacean rulers and dolphin mad scientists, but no matter how sound the general science, you could not square this with the complex, detailed, and exacting causalities and probabilities by which evolution operates. Nice story.
Most importantly, symbolism requires an unusually strong and mutually interactive social context to develop. Many animals, in all categories, are group animals, but the degree of social dependency, and the forms of cooperation that can be expressed in these societies are more limited than in the primate societies that underlie our evolution. (The cetaceans may be an exception.) The critical aspect is shared intentionality, an operative understanding at some level of the mental states of another animal. Animals in general communicate their emotional states through body-gesture and sound. For conspecifics this is most evident in breeding behaviors, and other resource protective behaviors; think of a strutting bellowing buck who is determined to defend his harem, or a predator that is prepared to defend his kill. This type of communicating is generally (not exclusively) extended in the primates to include things like grooming, sharing food resources, warnings for threats, etc. All of this is emotion driven as concrete expression of social bonds, but gradually in our lineage something else develops: a more refined and less emotion-limited information sharing. Not just what I feel matters, but what I know. And it becomes what we know as slowly the pieces of language develop, gestures and sounds take on reliable reference to salient objects, and over time means are developed to make subtle distinctions; a conventional sound is altered or another sound added to indicate the lion is dead. What a difference the right sound makes. We are now so glib and take for granted this marvelous aptitude which we have mastered, but it was built slowly, piecemeal over a very long period of time.
[ 5 ] Chris Springer, Lone Survivors: How We Came to Be the Only Humans on Earth, (Henry Holt and Company, New York, 2012), 179
Critical aspects of human evolution
The account of how we evolved is a complex and unfinished story, no doubt involving many causal factors. It is not my main purpose in this paper to provide an accounting, but I would like to put forward several key principles which I believe were among the critical governing factors. First: our lineage was not a top predator (at least initially) but a vulnerable prey animal, and one with limited biological resources to escape predation in a world filled with dangerous predators. We didn't have speed or bulk or horns or poisons. We also didn't breed in high numbers, and in fact became even slower developers with more extended childhoods. All we had was our wits and our hands, and a high level of group identity, social cohesion and cooperation. Second: as a product of our curiosity, and in search of safety and resources, we travelled, unlike our now closest primate relative, the chimpanzees who remained in their accustomed African forests and savannahs. We ultimately travelled all over of Africa, Europe, and Asia, confronting a wide variety of ecologies. Third: we did this travelling and evolving during a long period of climate instability: the last two millions years or so is called the Quaternary ice age, with extended periods of glaciation followed by interglacial periods of thaw. Many species went extinct under the rigors of this climate change regime, but we managed to survive because, fourth, the wondering bands of differentiating hominins would, with some regularity, interbreed with distant relatives. This process of semi-hybridization favored on average the propagation of the most adaptive genes and cultural practices.
All of these factors were stimulants for the fifth factor, the "needs to know" principle which simple means that an animal develops as part of its adaptive niche the specific kind of intelligence it needs to survive if it is to survive. The first four principles were the general challenges and evolutionary pressures under which the last principle was able to squeeze out a highly variable and adaptable animal with a symbolic life that was more and more a critical part of its existence. On such a tightrope we adapted and survived, and ultimately prospered.
So where does this leave us in the planet tally? This can only be wild guesswork. From our remaining 450 planets I am going to project that 100 develop some symbolic life, that is, simple cultures with some rudimentary language. And among this set, only 25 advance further to a level of civilization which would include more advanced languages, larger social units and more complex social organization, the written word and mathematics, and progressively elaborated technologies. All of this, of course, is sheer guesswork!
Back to the Math Question
I have reduced our sample of 1000 earth-like planets to 25 that would likely harbor civilizations with complex symbolic and technological life. This reductive procedure has no real authority—there are no authoritative answers to the questions I have raised—and I could have short-circuited the whole exercise by a simply asserting: with the wealth of planets, there must be (have been, will be) billions of civilizations among all those stars. But there is pleasure in such a provocative exercise, one that can elicit many other interesting speculations and reflections. And among these, (although it might not be on the top of everyone's list), as indicated in my opening paragraphs, I want to talk about the state of mathematics on these different planets with technological civilizations.
The simple direct question is: could their mathematics differ significantly in its fundamental concepts and methods from ours? Unwrapping a few parts of the question:
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Would they not all recognize the familiar number systems, integers, rational numbers, zero, and negative numbers? And in reflecting on these different systems, would they not discover properties of the numbers such as odd and even, triangular numbers, prime and composite numbers, etc.?
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Would they not have the same essential rules and algorithms for arithmetic: addition, subtraction, multiplication, and division. After learning to manipulate fractions, would they not see that certain mathematical procedures could not be satisfied by rational numbers alone, and thus define irrational numbers?
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Would they not discover the ratio of the diameter and circumference of the circle? Would their pi not evolve along a similar path of approximation as ours? Would they not discover the trigonometric ratios?
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If at some stage they are equivalently technological to us (or more advanced), would they not need the theorems of calculus? Could it be possible to build complex technologies and machine, computers, rockets, satellites, without calculus? This point is particularly important. While you can go a long way with simple experiments, to capture critical patterns in the results requires higher mathematics. It is essential for creating models which elicit and focus further experimentation. Without that, the mere-experimenters are working blind and hobbled, incapable of generalizing usefully their results.
In all of these and similar kinds of questions I hold that they must develop a mathematics along similar lines as ours and from the same foundations. Obviously we are talking about contents and not specific symbolic forms. Their mode of communication, the symbiology, will be different. But their essential contents will be the same. This is entirely analogous to the differences in the literate cultures in our world where we can translate common experiential observations, points of view, ideas. We must also recognize that there may be differences in emphasis, of what kinds of mathematical questions are found most interesting at different stages of their separate historical processes. But there must still be a fundamental overlapping, an identity of basic concepts, procedures, and logic. No technological civilization will have
pi = 5.8, or 21 as a prime number. If a civilization arrives at the concept of prime number, their prime numbers will be the same as ours.
Whatever this core of common mathematical concepts, procedures, and values is—and at this point I am not primarily interested in determining its boundaries—the fact that there is this core, that there must be this core, is a type of mathematical objectivity. What is its basis?
The constraints on physical science
Before addressing that question, we can ask similar questions about physics. No civilization is going to arrive at our technological level (or beyond) believing for long in their equivalent of the philosopher's stone, the preoccupation of earthly alchemists, or even something like phlogiston, the substance that was (briefly) assumed to be cause of heat. At some point they must necessarily isolate the abstract and universal properties inherent in all physical processes, and they will arrive at the same laws of dynamics and thermodynamics, the same essential periodic table, the constants of nature, etc. Recognizing that our physics is far from complete, and that there is a strong interpretive element in all such conceptual systems—indeed, in all language—we must still acknowledge that there is a hard to isolate or define objective core: if there was no such objective core, we could not build the amazingly precise and reliable machines that are an intrinsic part of our modern technological way of life. This bears repeating and generalization: No advanced technological civilization can build such a culture without some understanding of the universal laws of physics, which requires a certain mastery of the necessary mathematical tools. We don't do heart transplants or neurosurgery without science; we don't build rocket ships and large particle accelerators without science; we couldn't even make cell phones and refrigerators without science. And we can't have science without some progressive understanding of the laws of nature expressed in mathematics.
I am calling this an objective core, a red flag in philosophy which for the most part has emphasized the subjective and personal in its scholarship over the last four hundred years. I will justify my objective core assertion by two broad examples. First, there are many extraordinary consequences that flow from Einstein's general theory of relativity (GR), and they have all been confirmed by subsequent experimentation. We are all familiar with the dramatic existence of black holes, which Einstein himself did not explicitly anticipate, but are in fact one of GR's major confirmed cosmological consequences. In 2015 scientists confirmed another predication of GR (almost exactly 100 years after its formulation) by detecting the diminishing remnants of gravity waves generated by the cataclysmic collision of two black holes far, far away. But the LIGO observatory results is not even my main story here. Another prediction of GR is that clocks run slower in gravitational fields (time dilation). In 2010 researchers at the National Institutes of Standards and Technology where able to confirm this with amazing precision. Using two atomic optical clocks and placing them one meter apart in height, they were able to detect time dilation; the lower clock ran slower because it was closer to the earth. When they then reversed the position of the two clocks, the different clock in the lower position now ran slower.[i] That the equations of GR could be used to calculate and predict such physical changes at this scale is absolutely extraordinary.
A second example: those of us who are space and astronomy enthusiasts could not help but be amazed and thrilled with the dramatic landings on Mars by the Curiosity and Perseverance rovers, both of which involved a sequence of separate deceleration and landing strategies, from heat shield to parachute deployment, and finally to the release of the sky crane which used retrorockets to maneuver over the Martian target area and lower the rover safely to that surface. The whole complex sequence had to be preprogrammed and executed locally by the space craft. A small but critical part of that coding was a value for the gravitational field of Mars which would affect the trajectory of the spacecraft. For the Curiosity project a generic Martian value was used, and this almost proved to be a costly error since the gravitational field varied slightly due to local mass concentrations. Thankfully it landed safely, but an analysis of the recorded telemetries determined an actual landing speed that was discrepant with what was planned and engineered. Moreover, subsequent measurements of the local gravity field confirmed that the difference in landing speed was due to the gravity coding error.[ii]
These two general examples could be multiplied a million-fold. What is illustrated in all such cases is a refined set of abstract concepts regarding physical forces and the properties and behavior of materials, all elaborated in the language of mathematics, and these hard-won tools allow us to predict, plan, and control the objects with startling precision. Again, we cannot do this if we are not "in touch" with nature, with natural law.
Even without the elaborated rational understanding and interpretation, any physical procedure with the same essential materials, if consistently followed, will yield consistent results because of the underlying and mostly unknown physics. Think about cooking and all the procedures we follow, most often without a clear and "scientific" understanding of their exact consequences; change them or omit something and you might have a culinary disaster.
Consider the early metallurgist in our history, who learned to recognize the ores that could be smelted, and with trial and error developed methods for the mixing and heating of materials, for working copper, tin, silver, gold, and iron. That trial and error was driven by the hope and expectation of gaining control over the material. They looked for alternatives, they looked for reasons. And that search, even when it is laden with superstition, is an essential expression of a critical component of rationality—there must be a reason. This is a very old instinct in the animal mind, shaped by evolution: there is a sound in the forest, so the animal looks to the sound; something made that sound—there is a reason.
It is, of course, a very long trek to distinguish, sort out, and catalogue those reasons, and our relations in the greater animal world do not do it symbolically. But we do. It is our now unique gift on this planet and underlies all the cultural and technological complexity that human life now manifests. In those worlds where a symbolic culture persists, the inhabitants will follow a scientific trajectory analogous to ours, and they will come up with similar principles and discoveries because they are part of the same natural order, governed by universal natural law. This real world imposes constraints and ultimately limits ideas. Highly symbolic creatures can escape some of these constraints for a time through superstition and fancy. They can build their mythologies, and live by them. But if they want to build a science, and manipulate nature technologically, they must come to terms with its laws, and with the objective world. To put it in Kantian terms, these constraints are the noumena making itself known in the phenomena—not easily separated or identified in language, and often obscured by it, but still there and operative.
[i] C. W. Chou, et. al. Optical Clocks and Relativity, (Science, 09/24/2010)
[ii] Mars Landings May Not Accurately Account for Local Gravity, (NASA, Lessons Learned System, Lesson 27901)
The constraints on mathematics
So what are the constraints, the noumena that are there for mathematics? There are, in fact, no prime numbers out there in the world. There may be seven donuts on the plate, but there are no prime numbers on that plate, and there's no "seven" either. You many resist this statement because you can clearly see seven particular objects; but you are ignoring your seeing, your counting, and your understanding. We may agree at a certain level there are “amounts” out there—an idea I will return to toward the end of this essay—but there no numbers. The numbers are constructs of the symbolic mind. In an analogous way, although there are circles and lines in the perceived world, there are no perfect circles or lines "out there". Where do these contrived and perfected things come from? I am going to say that the quasi-objective core for mathematics is based on the necessary and inherent rationality of sufficiently evolved mind interacting with the world. There's a loaded statement, but let's break it down.
The word "rational" in this context is a leaden word that must sink into the muck of controversy. We have little choice but to wipe it off as best we can since there really are no better alternatives. I would exclude for my immediate purpose all applications that would apply to behavior for which the word "reasonable" is usually a viable alternative, and "emotional" a common antipode, although emotions obviously have their own--usually biological--'reasons'. Such rational/reasonable behavior, whether in values, beliefs, courses of action, etc., usually will encompass alternatives; under the same circumstances there is usually more than one reasonable response. But in analogy with the nature/physics interface, the rationality that I am aiming at has strict constraints. The constraint necessarily includes a sufficiently evolved symbolic mind (culture) that must eventually interpret certain relations and characteristics of these objects in certain ways.
Let's return to numbers: the world of objects is manifestly a multiplicity, and if a symbolic culture gets to a level of counting beyond one, two, and many, then it will necessarily build a number system (in whatever base, 2, 10, 16, 60…) that is equivalent to and translatable into our familiar decimal system. It does not really matter, of course, how it is symbolized, although the symbolization may have some consequences in terms of how easily the system is algorithmically manipulable—think of the awkward math of Roman numerals. But whether you say 14, fourteen, quatorze, vierzehn, chaudah (चौदह), Shísì (十四), etc., it will still be followed by 15, fifteen, quinze, fünfzehn , pandrah (पंद्रह), Shíwǔ (十五) etc. This is a function of the rational mind's inherent way of dealing with the numericity of objects. There are no alternatives of equivalent completeness; that is, if you can count to one hundred, you can only do this in one essential way. Any extraterrestrial system of counting to 100 and beyond will necessarily align, one-to-one, with our counting systems. Anything else would be error, incomplete, irrational.
This ultimate lack of choice will presumably manifest itself through a historical process where multiple individuals will make partial and perhaps conflicting contributions before a dominant consensus is established. Dominant consensus? There are virtually no philosophical positions for which there is 100% agreement. In general (beyond mathematics) this is partly attributable to the ambiguities of language. But it also carries over to different "mindsets" which are frequently irreconcilable. For example, the conflict between classical mathematical logic (Hilbert) and intuitionist mathematical logic (Brouwer) over the law of excluded middle. This arena of philosophical conflict may seem to negate the efficacy or universality of the presumed rationality that I am championing. However, we are too easily drawn to and overemphasize conflict. It is equally important, perhaps more important, that we acknowledge there are significant areas of agreement between all protagonist in these cases; in the example above, both Hilbert and Brouwer counted the same, they would have recognized the same trigonometric functions, the unit circle, etc. They would have been in agreement on a vast number of things. There is a core!
There is obviously creativity in the discoveries (definitions) and establishment (demonstrations and proofs) of core mathematical objects, principles, and techniques, and sometimes controversy in their promulgation (often involving mathematical politics and egotism—e.g. Leibniz vs. Newton). But in the end, after the history has played out, that core will be established and accepted by the majority without mathematical doubt. A perfect example is the imaginary numbers which are involved in the square roots of negative numbers. For a long time such numbers, when briefly encountered or entertained, were considered absurd, logically impossible…until Italian Renaissance mathematicians found them to be inherent elements in a formula they created to derive roots of cubic equations (a more complex formula analogous to the quadratic formula we all learned in high school). It took time for their insights to be accepted, for techniques to be developed, and the properties to be established, but no one who does mathematics now doubts the validity and importance of imaginary and complex numbers.
The role of logic
The discovery of these mathematical objects and their historical refinements will require an exercise in logic, which is essentially concerned with the formal validity of arguments, of how one proposition (or set of symbols) is derived from another, and the principles of inference that are entailed. Logic is not directly concerned with content: you can form perfectly valid and logical arguments about nonsense, about fantastical fictions. As part of its evolution, mathematics will necessarily refine its logical principles, but it will remain very concerned with content, with the definitions, properties, and relations of mathematical objects.
Since Cantor, Frege, and Hilbert in the late 19th and early 20th century, the broad field of mathematical logic has gained some prominence with the purpose of providing a formal and structural foundation for all of mathematics. This includes subfields such as set theory, model theory, proof theory, etc., with set theory in fact providing the now common language for mathematical presentation. However, as previously mentioned Hilbert's formal and foundational program foundered on issues of consistency and completeness discovered by Gödel, Turing, and Church. As a consequence it is more or less universally acknowledged that neither first-order nor second-order logic are adequate for establishing a consistent totality. The details of this rational "disability" are not critical to my theme. From my point of view it is far more important to recognize that over 2500 years of good and great mathematics was done, establishing fields, principles, theorems, and values, all without the benefit of the more recent and rigorous formalities. In other words, the functional core was still defined, whatever the more prim and proper logicians would later think about it. The calculus that Newton and Leibniz pioneered in the late 17th century had logical flaws; for example, it relied on a poorly defined concept of infinitesimals which Bishop Berkeley mocked as "ghosts of departed quantities". Calculus was not logically cleaned up until the mid-19th century. But despite those historical flaws, its theorems and methods proved to be powerful and unprecedented tools for the understanding and manipulation of nature.
From the current perspective of the philosophy of mathematics, the logical cleanup is essential, and I am not disputing that. We have different and more exacting standards now for mathematics. But it is noteworthy on this issue that physics is often pushing the envelope as indicated in this observation from the Princeton Companion to Mathematics:
...mathematicians are far more interested in finding rigorous proofs, whereas physicists, who use mathematics as
a tool, are usually happy with convincing argument for the truth of a mathematical statement, even if that argument is not actually a proof. The result is that physicists, operating under less stringent constraints, often discover
fascinating mathematical phenomena long before mathematicians do.
From my point of view, the impact of mathematics on technological culture is most profound in the play and elaboration of core concepts. These are the necessary and critical ideas, the intuitive heart and soul of mathematics, though not its accountant mind, and are thus more important. With them you can create a technological world; without them you have logical nothing.
So, the core of mathematics is a rational response of a symbolic mind which terminates after a historical process in a kind of mathematical certainty. Mathematics is absolutely unique in this respect. No other science or body of knowledge can remotely approach this level of rational certainty, and for this reason: all other sciences must still reference the real world of changing objects, the arena of intransigent hardnosed facts, the black box of surprises. Mathematics escapes the onerous demands of this burdensome world by defining and perfecting its objects independently of any real world embodiments. Those source objects may have instigated it imaginary rational flights, but its primary loyalty is to the concepts and structures this has engendered. This is much as Plato wished and anticipated. But where he wanted to believe this entailed the contemplation of a preexisting and independent order, mathematics was in fact discovering what is implicitly necessary in the interactions between rational symbolic mind and nature. Without that mind, there is no mathematics.
By embedding mathematics in the interactions between symbolic brain and the world, mind in matter, we escape the connection problem inherent in Plato's separate world of ideas. There was never any viable answer to the question of how we could connect to such an independent world, and now we don’t need one. Mathematical ideas, like all concepts of a language, are abstractions, a necessary product of the symbolic mind. All concepts can be applied to many objects (things, actions, events). This generality of symbolic concepts has its roots in the non-symbolic animal brain, for all animals perceive and catalogue their world into operational categories that are important to their functioning and survival. The categories, the concepts are general, but the things they reference are individual. You may have a “spouse”, but you are not related to the concept spouse, but to a particular and unique individual. Certain behaviors may be expected based on the concept, but one of the great tensions in language and human experience is that things (and people) do not behave consistently with their category. (And, of course, we all have our own individual understanding of all words, concepts, categories…)
The objects of mathematical thought are not individual, although each is unique. The uniqueness is governed by the definitions and deductions of their properties, But the instantiation is immaterial. The number π has been applied to countless equations, but it is the same π in each and every one. (I am ignoring approximations and human error.) The unit circle, a workhorse in mathematics, is a “pure concept” inasmuch as all unit circles, the one in this textbook and the one on that blackboard, are essentially the same; their individualities, this print ink on this paper or that chalk on that slate, are irrelevant. Mathematics is not concerned with individuality, at least not the individuality of its embodied concepts. This "indifference to incarnation", again, makes mathematics somewhat unique from other intellectual disciplines.
However, the "purity" of mathematical concepts does not mean that they have different metaphysical roots from other concepts. They are still the expression of the brain of a living symbolic and rational organism (to state it as broadly as I can). The other-worldly aspect of mathematics ultimately is informed with a particularly intractable divide of thought, namely, the nature of consciousness: one large block of thinkers assumes it is all reducible to neural algorithms or the like, while another finds such a viewpoint as totally inadequate to the character and nature of immediate conscious experience. I have mainly used the expression "symbolic mind", and acknowledge that it is rooted in the brain; this language points in both directions. I would like, however, to add this observation: The drive towards formality initiated by Frege, Hilbert, et al. was in part a reflection of the positivist spirit in scientific circles of the time, a particularly aggressive form of materialism. Their formal mathematics emphasized the external symbol and deemphasized the "meaning", a questionable ambiguous aspect to be treated like an embarrassing relative. One could perhaps ask if the formalist in their hygienic efforts of cleansing mathematics and the world from superstitions did not throw away its creative soul, in both senses of the word.
Math in Nature
In describing above the constraints on mathematical objects I said that on a certain level they were inherent products of a symbolic mind which must view certain aspects of the world in a certain way. Yet at the same time we must acknowledge that there is something deeper, and more objective than this in the world. This touches upon one of the great mysteries in physical science; why is it that the laws of physics can be expressed in mathematics? I do not expect to answer this question fully, but an answer would seem to involve at least two principles.
First, amounts matter in nature, and thus in physics. Having more or less of something (of matter) matters, makes a difference in behaviors and consequences, and the differences scale in consistent ways. This, of course, will involve the whole complex industry of measurement, but the upshot is we can meaningfully (and usually without ambiguity) talk about agreed objective amounts, and there are patterns in how behaviors relate to those amounts. The dynamics might scale simply: twice as much of this gives twice as much of that. Or it might scale more complexly as in the square of the distance laws that apply to gravity and electromagnetism. This scaling is not arbitrary, at least in the paradigm cases. Consider again the laws of gravity that were operative in the examples I gave above.
Second: natured is quantified in many respects. That is, at a certain level only certain values are possible. You can't have 2 1/2 Hydrogen atoms in a water molecule. The elementary particles that the standard model of physics describes involve such quanta as rest masses, charges, spin values, etc. On this level, involving the intrinsic mathematical consilience of nature, we would have to agree with Aristotle: the forms are in things. But there is another level; and not all mathematics is so solidly exemplified. For every impressive equation that works broadly and brilliantly in describing events in nature, such as GR, there are a thousand that don't; all, of course, equally valid mathematically, but simply not applicable. Statistics, a critical mathematical discipline for all of science, has no simple embodiments. There are no "means" or "modes" moving about in the world, no "standard deviations" walking around, no "distributions of random variables" cluttering the marketplace. And there is no probability doing its chancy thing. Probability is the mind's necessary heuristic for dealing with the unknown. But Nature knows no probability; it only honors causality.
The two levels, the one inherent to nature, the other intrinsic to the interactions of symbolic mind with nature, do not necessarily and perfectly align. A simplifying mind, even a rational one, might expect it to be so, but nature is under no obligation to honor the expectations of a simplifying mind. Even a rational one.