It is often said that math is somehow transcendant and objective, but I prefer to take the opposite view. It is a phenmonenon found within a clump of wet, throbbing tubes striving to continue its throbbing and indeed, enjoy the throbbing. Since the profession and discipline of mathematics has come to be within the total of our human body/mind, we can study mathematics from a biological perspective, as an aspect of an organism.
The first thing we have to realize is that every organism might understand math at lesat slightly differently, that is, to have a different math system. Yet, we, being different, can communicate in some way, and concede when we have been corrected in certain cases. Other times, one refers to a concept that comes up empty for another, or mismatched. Sometimes, one person thinks a theorem is obviously true, another person thinks it is "counterintuitive." The notion of partially corresponding mathematical systems itself seems to be mathematical, hence may seem less plausible to you than to me. If you agree to look at mathematics from a material stance, you'll find that reading even a single formula proves the partial correspondence. 1+3=3+1. Of course, I cannot convince you to step back altogether from another stance, if that's what you wish. [Platonism is a popular philosophical position in the community of pro math'ers]. It suffices that you entertain my perspective for the duration of this piece.
We can think of a living body as positioned "somewhere" with some trajectory. Insofar as any animal can think this way, it can understand itself as participating in its moving environment: Here will become somewhere through action. Animals discover which cavities they can fit into, what fits into their cavities, and necessarily find ways to infer such properties by visual recognition, touch, and other acts of perception. Risk and fear are other essential concepts for any animal, coloring places and shapes with qualities of painfulness, repulsion, panic, etc.
We might expect that certain geometric and topological concepts would come to be in almost any motive and sensitive organism, so long as its body provided opportunities for complex enough interactions with the environment. So, organismic mathematics can be distinguished as dealing with the concepts that arise uniquely from being a self-sustaining organism in nature. Concepts such as fuel and force derive unambiguously from our organismic existence.
When organisms communicate and cooperate, a certain degree of self-awareness is necessarily present. One must at least make the necessary motions to communicate, then navigate the world successfully while completing some collectively defined task. It requires some minimal abstract awareness of the body's position within the whole. We can thus distinguish "superorganismic" mathematics as that which would only tend to arise when organisms are aware and allied to one another in some way. Concepts such as rule sets, truth values, and hierarchies derive at least in part from the superorganismic aspects of our existence. Modern mathematical economics has a decidedly superorgansimic flavor. Genuine superorganismic "computations" can only be carried out by multiple people at once, as a discourse, interaction, or mutual observation of some kind, though theoretical work can be done alone. For instance, we can individually understand a law's stipulation that in certain trials, we are guaranteed a jury of 12 peers. But a verdict must then be carried out by no fewer than 13 people, including the judge.
Since humans are social animals, it easy to see why both the organismic and the superorganismic should be found throughout the history of math. It is far harder to imagine suborganismic mathematics or math which is non-organismic altogether. What would a self-regulating but non-aware part calculate? How can we think of an organ as perceiving and thinking? We must broaden the traditional definitions quite a bit.
"Sense" may be generalized so that any perturbation to the evolution of a (locally) closed subsystem counts, and "memory" may be generalized so that any structure maintained at the (local) expense of free energy counts.
A cell very obviously senses and remembers as it participates in the body, to be sure, but a cell is not a very good example - it is not so unambiguously partial as a heart, which is too complex for a basic example. Consider instead some inhabitants of the nucleus.
We find the genetic apparatus of a cell "sensing and remembering" epigenetic actions, which naturally depend upon the cell's interaction with its ambiguous environment. The science is somewhat new but an agreement seems to be that it is reflected on the effective topology of the chromosomes - parts of the DNA sequence are switched between "accessible" and "inaccessible" for the purposes of synthesis. Rather than "knowing" any mathematics behind the regulatory apparatus in the sense of a self-aware creature, we can think of the available epigenetic "states" [as it is put by the biochemists] as encoding possible "life plans" for a working cell (and we are indeed assuming the relevant nucleic acid apparatus is part of a living cell).
In actuality, the cell likely lives by continually modulating between these "life plans". *Human* mathematics of the cell could attempt to compute the effect of a sequence of life plans on the cell's constitution or behavior (given assumptions about its environs), but "nucleic acid mathematics" could not. It could "know" the maintenance of the cell's processes, but it doesn't know how it is maintaining the cell's organization, or even the cell in the first place. *We* know that. The protosymbolic structure of the nucleic acid apparatus ensures that a wide range of conditions/situations and mechanical traumas can be withstood without disrupting the total operation, since the apparatus has evolved in part to maintain and repair to cell's structures, and it is there we should look for anything that could qualify as mathematics. The notion of regulation is very close to that of computation, involving some idea of "reading" or "sensing" some structure. The analogy with human recognition alone is what seems to allow that mathematics may be generalized from human activities to those of living systems. In this way, every living (or partially living) system can be thought of as a (partially) self-computing "machine," whose "output" is its continued life in materia, its input being its raw fact of existence/configuration. In physics parlance, it is "prepared" in a state which will prepare further states to prepare further states to... Speaking this way, we presuppose subjectivity somewhere within nature, that is, we are interpreting nature. The apparent implication of teleology in the phrasing should not be taken too seriously.
You can see why suborganismic mathematics is so difficult to conceive of: if it exists, it is not conceived of by any organismic mind, which is practically to say, it is non-conceptual. [For how could you speak to a body part? You cohere only in their plurality]. Perhaps insight can be achieved in so-called altered states of consciousness, rather to contract consciousess closer to unconsciousness in some way. Dreams and fraction-asleep states may hint at suborganismic organizational schemes. What do you think? Even more difficult is to conceive mathematics beyond life altogether, which might be used by some type of natural abstract actor, which persisted in a non-organismic way, that is, without the ability to organize itself. Perhaps they are just projections and phantoms of the autopoietic. I think it may be impossible to imagine. Perhaps digital intelligence would be a version of it. Can self-awareness be simulated ... to itself?
In the limit towards complete knowledge, mathematics would simply limit to actual mechanics, or "whatever actually is." Taking quantum mechanics into account, we cannot characterize nature as having well-defined configurations at "instants" in time, so human ideas of a complete sequence of pictures of nature wouldn't constitute absolute mechanical knowledge - the Truth wouldn't be visible. Nor could it be tangible in the bodily sense, since it would have to concern every possible body [including bodies that are part of bodies - a radically parallel existence in comparison to conscious life] The idea is that a god of whatever sort could understand the state and motion of nature as a whole in terms of its raw being, so that it could be written in terms of the absolute. As organisms are self-guiding *parts* of nature, we cannot hope to grasp mathematics of the absolute. Likewise, the "mechanics god" could not hope to grasp us our worlds, since it could not help but see those of our cells and every last bacterium that participates in our bodies superposed. Such a perspective would show every possible world at once - ours would be but one infinitesimal point in its would be perspective-space.
What if we were to deal with the world at "coarser" scales, rather than "finer" ones? Can we, as organsims, get an idea of super-superorganisms, super-super...-superorganisms, and the like? That is, when groups of people attempt to coordinate as distinct entities, or when many coalitions of tribes attempt to form a nation? I would argue that these are much more difficult than smaller social organisms for one person to coordinate. In reality, they are better coordinated at the correct scales: Superorganisms re-configure themselves to better fit a super-superorganism, rather than one organism attempting to discern how to best micromanage each superorganism in order to satisfy its desires as well as those of the larger overgroup. When would be collaborative political projects are attempted with a dictatorial approach, the subordinate parties tend to have their goals frustrated. Even without ill will or greed, peoples understand their interests better than do their neighbors.
So we see an interesting thing - as our notion of mathematics "scales" up and down in organizational capacity, we lose sight of what we are trying to imagine. Either there is so much detail to every small moment as to overwhelm our capacities, or it is too broad, too diffuse, evolves too slowly and over too many subjects to get a clear grasp of it from within the organism.
Shapes, motions, symmetries, harmonies, and symbolic quantities seem to have occupied mathematicians for most of history. The dawn of these concepts from animal experience pave the way for understanding nature in terms of navigation, aesthetics, economy. Human mathematics has wrapped itself comfortably between these, but there is no reason to believe that these concerns alone serve to represent the mathematical in the most general sense.
Anything organismic, that is, any aspect of experience itself, of the mind, may be 'noticed,' in such a way that it may be related to other things that have been noticed. There appear to be no limits or borders to these acts of analogy, aside from those which arise as a natural consequence of the bodily substrate. This affords any minds like ours incredible power to modify their own realities. It is in principle limitless, given a limitless resource flow to the organism (preferably in a way it is accustomed to).
One broad consequence of the second thermodynamics for life is that once a form finds the ability to produce a version of itself from its environment and self alone, life tends to spread. I mean this in the sense that so long as free energy is injected, life tends to turn it into more life. [Perhaps this is the true source of the notion of Freudian libido, a relentless energy, which manifests in modern economics as scarcity and the "tragedy of the commons."]
The corollary for organisms that must guide themselves, then, must be that whatever grasp of mathematics they find, it must grow with their form. In the same way that an ecological "safety cushion" can result for a population from a combination of solid ecological "strategy"/niche and low population relative to [expectations for] available resources, I argue that well-functioning and well-fed minds will have more room to experiment. The reasoning is more or less identical: When one need not work too hard to live, more organismic energy is available for "mistakes" of all sorts, whether they be exploring unknown regions, taking new routes, or trying new maneuvers. I have a feeling this would, over many millions of years, select for animals which had a propensity to get bored in times of surplus and safety. This is simply because the bored ones would take more risks when they had the luxury. Of course, "risks" must be interpreted more or less economically, as anything that is less certain than some "safe" strategy to produce good outcomes, whether they are expected or not.
If all this lines up, it wouldn't be surprising that animals could come to sophisticated ways of thinking about the world before they were ever ready to use them: the bats would never have survived with wings if they didn't know they could be used to fly. [Did not humans learn to fly by pipe-dreaming?] It must come to an idea of flying as a motion in the world at least in parallel to the development of wings. God, that's a mystery though, isn't it? Tempting as it is to blame the apparent shift from tree-dweller to sky-walker on some creator, I actually don't think it solves the problem. That answer doesn't remove the necessity of the evolution being carried out in materia, which still has to work out mechanically some how, whether or not it was all designed to take place. I think truly, the essence of the modern argument for a creator is that the story of evolution seems implausible, which I disagree with.
Navigational schemes must develop in parallel with bodily forms. A tardigrade navigates its world very differently than a worm; just watch them. [Youtube.]
So we should expect a cornocopia of mathematical forms corresponding to the beauty of the biosphere, and my primary question is how mutually intelligible they could possibly be. Mathematics across the same species is pretty easy. "Give me four of those. I'll give you three of these." "But I gave you four in advance last time, remember?" "Fine! Here's six of these then." Across humans and other apes species, it is a little more difficult, but still possible for those who think most like us. Across from humans to other intelligent species, like crows and dolphins, communication is even more difficult, but it is absolutely evident that we use some concepts which are mutually intelligible. All three groups of organisms live in different ecological domains, yet have come to similar understandings of abstract organizational principles. Rats, raccoons, octopuses, and other birds and mammals of diverse types display at least some ability to navigate rectangular human worlds and operate some of the apparatus of human organization.
Certain geometric concepts appear unavoidable. If not points, there are at least places, or things in places, which definitely differ from one another so that they can be assessed, selected, and navigated. If not lines, then at least curves or paths; thinking things make plans to walk in a direction. If not planes, then surfaces - the surface of the land, of a tree, of a prey or predator, and of course, of the body. The only essential thing for a foraging organism to understand is that is interfacing with the environs. There must a notion of bordering the environment, of the skin as a boundary of the self. The body can be thought of as going right up the border, as filling a volume, but the self-on-the-boundary is the site of interface with tangible reality.
If one makes complex enough use of a body, it has to be mapped in a topological way, that is, we must have some way of knowing what places are near which, ideally to have an idea of the total nervous structure of the body. From this, we find the "surface" of our bodies. If you trace a path along the surface with a finger, what you are feeling is this constructed topology: you feel everything near its neighboring points on the skin. The degree of access between two points on skin is known merely by feeling distinct sensations there and there.
If a stranger slides their hand slowly up your leg on public transport, you may find it disturbing, or perhaps frightening or more. When they pull away after you glare at them, and say "I didn't do shit," you will know they were lying whether or not you *saw* them do it.
Probably this topological method is used extensively by Terran animals, but I am for more interested in the subtle differences that may emerge as we attempt to study the problem. That is a large project, enough for a life's worth of research, so I will leave it there. For a more complete understanding of ecosystems self-organize, we would ideally understand how every inhabitant maps the surface of its body [if at all, obviously].We should remember that our modern mathematical understandings of surfaces have been flavored irreducibly by the development of the rest of culture through history, and we should suspect any population would be similarly influenced in their understandings of mathematics by the psychological, emotional and social realities a population must cope with. Aside from our cognitive differences to begin with, this would inevitably cause obstacles to communication and understanding.
Anything which eats special forms must find a way of recognizing them, as well as motivating their meaning. Mammalian brains, at least, tend to associate in any possible way, so that a shape once linked to a taste can evoke the taste on sight in the future. Thus symbolism is another basic animal act, constructing from the raw geometric picture a directly accessible world which consists of potential symbols which gradually resolve to genuine symbol-copies if the perceptual process happens to compute that way. The symbols are felt by whatever it is they might be thought to "do" or feel like.
Complex-bodied animals, whose environments are persistent relative to the times scales of their actions, will benefit from remembering where everything "was." One then must construct a topological map of the home as well, with easily bodily-accessible world-places corresponding to mental places that feel close. Each "point" on the map can be made to correspond to a hypothetical places are felt by what was there, either in terms of geometric forms or symbolic ones.
Since one inevitably comes to a practical theory of motion in/around places (i.e., how to navigate shapes one finds in the world), its plausible to treat places-maps as containing some sort of information about possible ways to navigate a place. The smallest or shortest paths, or motions, through a place are its accessibilities, or "infinitesimal open sets," in classical-modern parlance, which can be thought of as generating a topology upon any place in a world. Such motions are purely from a bodily perspective, and may take into account gravity and other "forces" of the environment. Most maps will also attach abstract "motions" to world symbols, when materials are encountered and symbolized, such as food, water, types of shelter, plants, animals, rocks, landscapes and planetary/ecological forms of all sorts.
Regardless of how these maps have been learned, they contain easily accessible abstractions as the organism navigates the cosmos.
This is an incredibly large and fascinating topic, but I'm stopping here. This small piece is meant only to introduce the notion that mathematics can be understood as part of a biological process, which plays an immensely important role in the mechanics of evolution and ecology. Moving beyond our very mode of existence, we can think of life, hence mathematics, perhaps even more abstractly as a component of a physical process. That is to say, biological forms necessarily "compute" their future selves, and math emerges with awareness of self-guidance. I encourage you to explore and critique what I have suggested and asserted here, particularly in the final paragraphs, so we can begin study of the general organismic "mapping" process, and explore what it suggests about the ecosystems we participate in, and about our relationships with our own mapping processes.