Mathematics is a fundamentally human activity, and a semiotic one at that, which is to say, it is an activity of making and using signs in relation to the wider world of practices whereby humans relate to their worlds. While this might seem obvious, most working mathematicians self-identify as Platonists, which is to say, they take the position that they are working with realities which are “really there,” “mathematical objects” which are able to be discovered by means of techniques modelled on that of the discovery of physical objects in nature. Mathematical objects, which is to say, things like numbers and geometrical figures, are ideal entities whose contours are wrenched from the fabric of the ideal itself by means of the techniques of logico-mathematical proof. All of which is to say, even if there were no physical world, the truths of mathematics are “really there,” as if God given, hence the term Platonism, often worn with pride by mathematicians today. The locus classicus of this position is that of Leopold Kronkeker, in 1893 when he famously said that “God made the natural numbers, all the rest is the work of man.”
Nevertheless, Kroneker was responding to the foundations crisis which was beginning to shake the tree of mathematics. For example, logicists like Gottlob Frege had attempted to “found” mathematics upon the basis of its subsumption to the “rules of thought” articulated in his new logical calculi. The problem with this, however, is the at it hardly did what Frege intended, which is to say, to “ground” mathematics, and hence show its absolute necessity in “all possible worlds” (to use a term from Leibniz), but rather, to reveal just how ungrounded the seemingly incontrovertible world of mathematics actually was. When combined with the set theoretics developed by Georg Cantor, or the slippery attempts to ground number from linear continua as described by Richard Dedekind, it seemed as if just as mathematics had radically increased in power and rigor during the nineteenth century, it had also revealed in the process that perhaps despite or by means of this very power, it was all illusion, a slight of hand. Did the Emperor have clothes? Kroneker believed that blind faith was the answer, and so do many mathematicians today.
And of course, if one works far from the limits of the mathematical enterprise, which is to say, far from the applied aspects of mathematics which find themselve continually in dialgue with the physical world and its non-mathematical impingements upon the edifice of pure mathematics, then one is safe from these issues. Likewise, if one doesn’t stray too far into the realm of the pure, to the foundations of mathematics itself, one is also able to skirt around the issues of how precisely mathematics derives its authority or internal consistency. It is the in the “dirty middle” realm, from which both the shores of the physical world and the purely ideal world are both distant horizons, that the terrain of mathematics appears boundless. But at the shores, the issue becomes muddier indeed.
And this is what the foundations crisis that shook the mathematical world at the start of the century revealed. Some argued, with Hilbert, that mathematics was purely about signs, and was merely a game, and hence, should not be compared to the physical world. Any need for grounding was then moot, because mathematics grounded itself, circularly, and needed no justification beyond this. It’s own internal consistency made it a form of sophisticated play, and if it was useful in the world beyond mathematics, then so be it, but this was ultimately, accidental and not something worth the time of mathematicians. This “formalist” approach, however, simply ignored the fact that the engine of mathematical creativty had not only come from within, but without. The radical developments within mathematics during the early 19th century, for example, the great works of Leonard Euler, were often spurred by attempts to solve problems from mechanics, which is to say, very practical issues which engineering posed to the lofty realm of math, and to which it could not answer. Even analysis, the great discovery of Newton and Leibniz, had been wrested from the gods of mathematics by means of the push of the attempt to describe accurately the motion of heavenly bodies, not to mention the behavior of mundane physical objects. Whatever mathematics is, it is hardly pure.In contrast to this we see the Intuitionism proposed by L.E.J. Brouwer, who argued that mathematics should simply get rid of anything that couldn’t be grasped by the intuition of the mind as purely abstract nonsense. Brouwer attempted to “construct” mathematics on the intuitions of the mind, producing an analogy to the manner in which the mind intuits the objects of the physical world and the manner whereby it intuits the ideal realm of mathematical entities. A highly influential early twenteith century movement, one based to a large degree in Neo-Kantian ideas of scientific method and practice, Intuitionism largely fell out of favor, along with formalism, even as the limits of pure Platonist and applied approaches found their own limits in Goedel’s famous “incompleteness theorums” of 1929-31.
In contrast to this we see the Intuitionism proposed by L.E.J. Brouwer, who argued that mathematics should simply get rid of anything that couldn’t be grasped by the intuition of the mind as purely abstract nonsense. Brouwer attempted to “construct” mathematics on the intuitions of the mind, producing an analogy to the manner in which the mind intuits the objects of the physical world and the manner whereby it intuits the ideal realm of mathematical entities. A highly influential early twenteith century movement, one based to a large degree in Neo-Kantian ideas of scientific method and practice, Intuitionism largely fell out of favor, along with formalism, even as the limits of pure Platonist and applied approaches found their own limits in Goedel’s famous “incompleteness theorums” of 1929-31.
Goedel’s singular accomplishment was to put all four of these approaches to grounding math – Platonic idealism, Physical Realism, Intuitionist Neo-Kantian Subjectivism, and Objective Structuralist Formalism – to rest as aspects of the insoluability of the same problem. That is, math simply could not be grounded from within, nor could it be grounded from without without proving itself ultimately both grounded and ungrounded, and in fact, both and neither, from a mathematical point of view, in the process. What Goedel essentially did, then, was show that the very notion of “grounding,” at least as this notion was being framed by mathematicians of his time, was part of the problem. That is, mathematicians who wanted to ground mathematics from within, as the insurer of its own truth, would find only circularity, but no ability to say if this circularity applied to anything beyond math. Those who wanted to ground mathematics in something beyond math, such as the physical world or human intuition, or even the workings of ultimately meaningless signs, would find that all they could prove by means of the tools provided by math, from within at least, was that mathematics relied on something beyond it, but no way of showing the need of this, or the need of any relation to a particular grounding or another, by solely mathematical means. That is, to ground math with something beyond it (ie: human activity, human signs, human intuition, god), would require actually bringing something outside of math inside of math. And that would produce contradiction.
Circularity or contradiction, the result would be incompletion or incoherence, respectively. The final option, oscillation or inconsistency, was simply what most working mathematicians did, which is, to use whichever options made math “work best” in a given local instance, and leave “grounding” for some other time. What Goedel did was show that Hilbert’s famous dictum that math must prove itself “consistent, coherent, and complete” by its own means, which is to say, by means of mathematics, is simply not possible, and that this is not simply an accident, but part of the very structure of the way mathematics itself works. Goedel showed that math had its own limits.
Of course, some of this might seem like common sense. Math always counts (numbers) or draws (geometry) something which is not math itself. If I see a group of animals and count them, and label them “four dogs,” the dogs and the number “four” are fundamentally different things. Math is always both about and not about mathematics. When math tries to eliminate any aspect of it which is not math, or which is math, the result will always be, at least from wtihin math, paradox. So it is with any signifying practice. The same can be said about language. A dog is not the same as the word “dog,” and ultimately, one cannot “ground” the relation between the two, at least from wtihin language, without producing paradoxes. Just as Goedel showed this within mathematics, so Jacques Derrida famously demonstrated by means of linguistic deconstruction, and Ludwig Wittgenstein in his own way nearly thirty something years before. The fact that Goedel, Wittgenstein, and Derrida share what can be seen as variations of the same insight in different fields, one which resonates strongly with that of Heisenberg in physics, is likely perhaps not accident.
If mathematics cannot ground itself, perhaps it can ground itself in the fact that humans devised mathematics. That is, it is a signifying practice, like that of language, whereby humans describe aspects of their world so as to interact with them. Mathematics is a special type of language, but language nevertheless. Of course, most working mathematicians are likely not to like this, because it subordinates their activities to something, anything. But there is no unsubordinated position from which to view the world, we are always mediated in our relation to anything and everything, and likely, are nothing but mediations of mediations all the way down, fractally and holographically. That is, any notion that there is some ‘God’s Eye Perspective’ from which to survey the world seems singularly outdated as any other simplistic form of faith in the unbiased. All is perspective, and mathematics is simply one amongst others. It is useful, of course, but so is language, our bodies, our brains, etc. Each of these has been viewed by its partisans as the singular, privileged lens on the world, and each can be decentered by others. Why mathematics should be any different is beyond me.
Rather, we live in a world of networks, each of which supports the others, culture and nature, language and physics, human and animal, living and non-living, each an aspect of a wider whole which supersedes them all. Whether we call this whole experience, or the universe, these are also simply aspects of it, attempts to describe the whole. And as Goedel showed in his way, Derrida in his, and Heisenberg yet another, the attempt of any system to grasp the whole from within it is likely to founder in paradox. In mathematics, of course, this is the famous paradox of the barber, Bertrand Russell’s attempt to articulate the issue of the “set of all sets.”
That said, these problems all become less of an issue if we say something like mathematics is a human signifying practice, which is useful in its domain in relation to others. Of course, this begs the question of what use means, but since humans are those who determine that which is useful to them, and are also the originators of any math we have ever known, then we could perhaps say that mathematics reflects aspects of what humans value in the world. That is, mathematics has helped humans do the sorts of things they value, one form of which is mathematics. While some enjoy doing math for its own sake, the urge to do something like mathematics does seem to find its impetus in practical activity, which is to say, the attempt to describe the world so as to be able to do things in relation to it. There is no question that both pure and applied mathematics have given rise to new forms of mathematics, but as some of the more “naturalistic” philosophers of mathematics today have argued, math is always between the physical and the ideal, with one foot in each, dirty and impure to the core. That is, it is a form of media, just like language, or the body. A lens on the world of experience, if a particular one, like yet different from all other media in this way.
While naturalistic approaches to mathematics are in the minority among working mathematicians, and even philosophers of mathematics, it seems to be the only approach to mathematics which takes into account its foundations crisis mid-century. If it dethrones mathematics from its attempt to imagine itself as the queen of the sciences, well, let it join philosophy and every other dethroned discipline which aimed for such a role. For perhaps it is the very desire for centrality which is the problem rather than attempt to “find” a solution, for it seems to give rise to paradoxes, whereby the very fabric of, well, something, call it the world or otherwise, resists. Physics, linguistics, mathematics, the foundations crises of many disciplines of the twentieth century, they all seem to indicate that the center does not hold, and yet, centerlessly, they still do many things. Naturalism attempts to start from this, from activity in the world, and human activity at that, rather than ideal foundations, be they ideal in the classical sense, or the materialist inversions thereof.
Post-Structuralist Approaches to Mathematical Gaming
If mathematics is a human activity, then perhaps it may be possible to philosophize about it in regard to this perspective on it. Certainly mathematicians refer to specific “things,” which they describe with symbols which they manipulate. These signifieds of mathematics are represented by signifiers, which is to say, the graphs and equations scratched on paper, computer screen, and chalk-board as “representing” something generally called “mathematical.” If “mathematical objects” are signifieds, meanings, that which are described and represented by mathematical signs, considered as signifiers, then perhaps we can think of mathematics as a specialized type of language, and the practice of mathematics as a type of writing and speech. Certainly not one which is meaningless, as Hilbert famously argued. No, mathematics seems to be about the world as much as about itself, just as any language, and yet, it is a very particular sort of language at that.
As any language, mathematics can be considered, as Wittgenstein famously argued, a game. That is, it has rules, and people get quite heated if you break them, even if the rules of the game are always being changed from within as you go. Good moves in the game, in fact, change the very nature of the game itself, and in doing so, change what it means to play, the players, etc. In this manner, the rules of mathematical play are like that of linguistic play, which is to say, mathematics has a grammar, just like natural languages do, even if this grammar works differently than those of natural language. But this grammar is a grammar nevertheless. And so, a (post)structuralist analysis of mathematics is not only possible, but I would say, desirable.
Structuralism viewed languages as composed of utterances, often described as “parole,” in relation to structuring categories which were implicit yet made sense of utterances, or “langue.” There are, of course, several types of langue working in any given language. For example, in a natural language such as English, there is the langue of the semantics of the language, which is to say, the meanings of words, systematized in a dictionary, which a competent speaker of English would need to understand to “make sense” of a given utterance. Just as one couldn’t make sense of a sentence such as “The cat is on the mat” without knowing, for example, what it means to “sit,” likewise, one cannot make much sense of a mathematical sentence such as “x – 5 = 17” unless one knows the meaning of what “5” means, and how this mathematical “word” differs from that of “17.” While it is necessary to bring in forms of semiotics which deal in diagrams to describe how this could be applied to notions such as geometry, the semiotics of C.S. Peirce seems more than adequate to the task.
Below the level of semantics, or the meanings of words, are the deeper structures, whose which determine the ways in which these can be linked to each other. A word like “is” in “The cat is on the mat” presents a word which is really not merely a word, but a word which represents grammar, or syntax, which is the foundation out of which word meanings arrive, for it provides the fundamental and implicit categories which all the meanings of words to take form. And so, if one was to look up any word in a dictionary, one would find that the word “is” equivalent to this or that meaning. “Is” is both a word and a meta-word, so to speak, and this is what is meant by langue in relation to parole. Just as knowledge of the meaning of a term at a given level is necessary to understand an utterance, so is the meaning of the meanings which describe the meanings of these words, which is to say, the rules of the game as well as the particular move being made. And so, if one doesn’t understand what “is” is, then understanding the particular meaning of “The cat is on the mat” is likely impossible. The same with grammar markers in mathematics, such as “=,” which ultimately, is very similar to saying “is” in a natural language such as English.
As is likely apparent from the preceding, recurcsion is operative here, not only at each level, but at any level. That is, each and any utterance/parole is related to a langue, which itself is a parole in relation to at least one other langue, and this repeats fractally. This is the contribution to this sort of structural approach made by post-structuralism, namely, the attempt to show the paradox of any attempt to find an ultimate foundation at work in such an analysis. And so, rather than argue that a notion such as “is” represents a “deep structure” of the language of mathematics, and hence, in some way, the world itself, a post-structuralist approach uses relatively similar methods to show that the process can be carried on infinitely, with no ultimate ground in sight, or, if one wants, arbitrarily ended for convenience sake. But any attempt at ultimate ground will give rise to something like infinite regress, which is to say, incompletion, arbitrary end, or incoherence, or some mixture, which is to say, inconsistency. Post-structuralism and Goedel are on the same page on this one.
From a post-structuralist perspective, then, it becomes possible to say that mathematics has objects, which are meanings within its semantics. These objects are things such as numbers and shapes, or any of the other entities which mathematics attempts to “treat.” When mathematics deals with combinations as if they were “things,” which is the discipline of combinatorics, then we know we are in the realm of mathematical semantics. These things are then linked together to produce utterances, according to the rules of grammar implicit to the “game” of mathematics.
The sorts of utterances vary, however. Some are simple equations, such as “x – 5 = 17,” which are then transformable, by a known series of procedures, into “x=12,” such that it becomes possible to state that these two are themselve “equal.” There are procedures here, such as “solving” equations, which are the utterances. And these procedures produce the grammar whereby mathematical equations are transformed, one into the other.
But then there are meta-mathematical utterances, and these are proofs. A mathematical proof makes use of procedures within the language of mathematics to create utterances which attempt to alter the way the game is played. This is, for example, not all that different from the role of argument in philosophy. A philosopher might argue, for example, that we shouldn’t think of reason or god as this or that. Ultimately, the philosophers is using words to impact the way we use words, just as a proof indicates the ways in which a mathematician uses math to impact the way math is done. Of course, the goal is ultimately to impact the way people think about math, but seeing as mind-reading isn’t yet possible, the only way we’d know how people think is how they act, which is to say, how they “do” math, and a proof may aim at how people think, but ultimately, it only manifests its effects in how people “do” math. The same could be said of the role of argument in philosophy.
None of which is to say, of course, that I haven’t been doing precisely this in what I’ve already said. In fact, the preceding paragraphs are simply arguments, attempts to impact the way the game of philosophy of mathematics is done, from within it. And this sort of meta-gaming is part of how the game is played, even if the results of this are always uncertain, which is to say, incomplete, inconsistent, or incoherent, at least from within the game as it currently stands. But games evolve, and meta-gaming is how this happens, in math and language as much as any other sort of gaming. And so, if I decide all of a sudden that a bishop can now jump, but only over rooks, in chess, and this move catches on, and becomes part of the new rules amongst the “community” of chess gamers world wide, then I have made an utterance, not within the game, but also not beyond it. In a sense, I’ve made a meta-utterance or meta-move in regard to the game, thereby altering its grammar from within.
Mathematics does this all the time. In fact, that is precisely what Goedel did, and what others do when they “invent” new mathematics. Those meta-moves which catch on become movements, such as “category theory,” or meta-meta-movements, such as “post-foundationalist mathematics.” None of which is to say that the meta-games precede the games, since there was no such thing as “post-foundationalist mathematics,” or even the need for this, before the foundations crises. Sometimes the meta-games pre-exist, because they have already been called into existence (ie: people have been talking about the grammars of languages for quite a long time), while other times, they are produced, even giving rise to new layers in between existing layers.
In a similar manner, mathematicians can give rise to new objects. Certainly “category theory” has not only given rise to new mathematical grammars within and beyond it, but also, new mathematical objects, such as “functors” or “categories.” The field of abstract algebra, of which category theory is simply one form, in fact is the branch of mathematics which works to deal abstractly with various sorts of ways of relating objects and grammars to produce utterances and meta-utterances. From its start in set theory, modern algebra developed into group theory and beyond, and by means of Emmy Noether around the time of Goedel, because the meta-mathematical enterprise it is today. If Goedel destroyed the hope of a single meta-mathematics, Noether proved that the true name of foundations was “many,” even if meta- as a notion was only ever “one” in relation to a particular location. Noether showed that grammars and objects have plural ways of relating. And from such a perspective, it becomes possible to see that foundations are things, but verbs, processes of continually founding and refounding, which is to say, relating to levels of micro- and macro- scale within a given level of practice, semiotic or otherwise.
It is for this reason that many have turned to category theory as a possible inheritor to set-theory, as a possible “post-foundational” foundation for mathematics. What is so incredibly slippery about category theory is that it defines its objects, grammars, and moves relationally. That is, the very meaning of an object is what you can do with it, and these “moves” give rise to the very categories of objects in question, and vice-versa. That is, object, category, and move are interdependently defined, collapsing the distinction between utterance and meta-utterance, such that all utterance is meta-utterance and vice-versa. All of which is a way of saying that the constructedness and reconstructedness is not hidden behind a smokescreen of “this is the way things really are.” Category theory is a mathematics, not of being, such as set theory, but rather, a mathematics of relation.
There are, however, other potential post-foundational discourses. Fernando Zalamea has, in recent works such as “Synthetic Philosophy of Mathematics,” argued that sheaf theory can play this role in a way different from that of category theory. Sheaf theory, a form of mathematics which works to extract invariants from particular transits between mathematical objects in transformation, is fundamentally a mathematics of the in-between. It is a mathematics which extracts from particular motions particular symmetries, and like group theory, then works to put these to work themselves in transit between local and general. For example, sheaf theory may attempt to describe the ways in which particular figures can be sliced and re-glued to themselves in ways which maintain coherency even when that figure is transformed in a particular way, and to then learn from this possible insights which can be applied to different yet related types of slicing, re-gluing, and transformation. In many senses, sheaf theory is a meta-analytic formation, which is to say, it takes the sorts of tools of decomposition and recomposition, analysis and synthesis, seen in notions such as differentiation and integration, and generalizes them to ever wider terrain.
Sheaf theory is then a mathematics of transits, of the temporary reification of an aspect of a transit, only to reapply this to another. In this sense, the objects, categories, and grammars are also relationally interdetermining, and the relation between utterance and meta-utterance are constructed and reconstructed continually, as with category theory. One difference, however, is that category theory is ultimately a logical enterprise, and doesn’t get into the specifics of particular figures and their equations, but rather, attempts to describe the logical grammar “beneath” the mathematical language used to describe and manipulate these. In a sense, then, one could say that category theory and sheaf theory are both instantiations of post-foundational foundations of mathematical practice, but in regard to differing aspects of the mathematical enterprise. Category theory is a meta-gamic approach to logic, while sheaf theory plays this role in regard to transformation within a particular field which already has instantiated categories of objects, categories, and grammars (ie: figures, types, and rules to regulate transformations).
None of this is to say that category theory is “more” foundational, but rather, it is more abstract, and this is different. Abstraction here indicates that this is a further move away from the physical world, and closer to the ideal, which is simply the realm stripped of specifics. Between a relatively “concrete” aspect of the world, such as a stone, there is the abstract representation of this, such as the word “stone,” or the number “1” which can be used to count this stone. Category theory leans to the latter side, and sheaf theory the other, but depending on the particular pole one is using to base one’s practice at a given moment, be this the concrete or the abstract pole, or any other for that matter, then the “foundational” orientation of one’s meta-gamic practice will proceed differently. Foundations are always foundationing, and as such, one always creates and recreates them by means of meta-gamic moves. The whole point of a post-foundational foundationalism is that it aims to produce the potential for foundations everywhere, rather than prohibit them, less this simply become foundationalism in reverse. The hope isn’t to proscribe foundations, but to liberate them, and in the process, practices, from the straight-jacket of both foundationalism and its evil twin, anti-foundationalism. Post-foundationalism, on the contrary, embraces creativity, which is to say in relation to mathematics, the potential production of new mathematical games and meta-games to give rise to new ways of descriging our potential relations to the world.
Foundations as Foundationings: Or, Mathematico-(Meta)gamic Ethics
If abstract and concrete are two poles which can help us orient in this process, these two should be seen as merely categories produced by meta-gamic moves, and hardly necessary, but produced and reproduced at each and any moment in which they are operative. Just as Zalamea finds this set of polarities useful, I also find those of the human body helpful. That is, if all mathematics can be constructed as the product of human activity, then perhaps, as with natural language, it makes use of categories which can be seen as potentially deriving from the form of our embodiment. Natural languages, for example, have nouns, adjectives, linking words (ie: prepositions), and verbs, four primary parts of speech, and some theorists, including Gilles Deleuze, have argued that these can be seen as the result of the ways in which human embodiment in the world makes use of things, qualities/categories, forms of relation, and actions, which is to say, nouns, adjectives, ‘linking words’ (ie: of, on, in, is, therefore), and verbs. The grammar of human language, then, can be seen as a way of representing some of the primary categories which humans have, by means of the media of their bodies, extracted from their worlds of experience. None of which is to say that these categories are necessary, but rather, that they are produced and reproduced, continually, by gaming and meta-gaming in the worlds of our experience, with language being one of the effects thereof.
From such a perspective, it might not be far-fetched to argue something similar about mathematics. That is, there are mathematical objects, which function like nouns, and categories of these, which are similar to adjectives, giving rise to semantics from the relation between these. From here, utterances and meta-utterances formed by means of these objects and their meta-linkages in categories, utterances, and meta-utterances give rise to modes of relation, which are represented alongside the objects and categories within utterances and meta-utterances by means of “linking words” which represent within these grammartical structure. But all of these are ultimately the result, sedimentation, and reification, if only partial, of the processes which give rise to these. Mathematics and meta-mathematics, two sides of the same, ultimately, like a language and its grammar, are processes of gaming and meta-gaming.
Thinking in these terms also allows for mathematical gamings to be linked up to philosophical notions as well. Set-theory, then, is clearly, as Alain Badiou has argued, the mathematics of being, and as such, has fundamental resonance with the philosophical notion of ontology, even as category theory seems to be something like a mathematics of relation. Sheaf theory seems an attempt to describe something like a mathematics of becoming, of the transits between the figural and the numerical, and within each. If, as Brian Rotman has argued, geometry is the language of space, and algebra the language of counting, it becomes possible to see these as so many layers of semantics and syntax in relation to each other, and yet, also permeated by that often forgotten stepchild of linguistics, which is to say, pragmatics. There is no vocabulary or grammar without a context which makes these relevant and meaningful. Without something like human bodies in something like a world of experience which has particular structures of space and time such as we know them, it is unlikely that anything like mathematical grammar or vocabulary, let alone those of natural languages, would make any sense. Likewise with what we tend to think of as constituting proof or argument.
By framing languages as composed of semantic vocabularies of objects, linked in syntactic categories which produce series of potential relations between these, which are grasped in meta-syntatic categories which are the grammars of that language which ultimately articulate the pragmatic linkage of that language to its contexts beyond itself, then what gives rise to all of this? Creation, it would seem, but also recreation. Math emerged from our world, and continues to reemerge from it, as does language. Semantics, syntactics, and pragmatics are simply aspects of this continual process of recreation. And if this essay attempts to do anything, it is to deconstruct attempts to hide the manner in which recreation, which is to say, emergence, is the ever present potential within any and all, here and now.
Mathematics and language, just as with any other sedimentation of our actions into representations thereof, are simply ossifications which stand out from our practices, which we then treat as if “real,” which is to say, as if necessary. And yet, they are only ever produced and reproduced by our actions, even as these are only ever produced and reproduced by that of our contexts in turn. We are networked, linked nodes of praxis within others, at potentially infinite levels of scale. And yet, within our particular zones of this, there remains the potential to give rise to effects which ripple widely beyond, leading, under the right contextual conditions, to the potential for cascades which give rise to a sea-change in how things are done.
There will always be those who will try to control the way things are done, who will try to close off our sense that, to quote the revolutionaries of the past, “beneath the stones, the beach,” or, in more contemporary idiom, that our world is only ever of our own making and remaking. No one entity can ever shift the whole, and yet, any one entity can sit at the fulcrum of others and provide the critical shift in a grain of sand that creates a massive change. Or participate in structuring such a set of conditions so that something else can provide that final push.
There are times, of course, in which it is necessary to restrict the manner in which things recreate themselves, for in fact, too much change can dissolve and destroy. But our world has almost never been in danger of such things, and when it has, it is the chaotic dissolution has nearly always served to reproduce the deeper aims of control. But creativity, radical creativity, doesn’t care if it is the center of the world, but only that it resonates with the creativity within it in its own way, amplifying the power of liberation and emergence, being emergence itself, even if towards an end beyond it. The reason for this, of course, is that emergence knows no fruit, it is its own reward. As anyone who has ever created anything knows, creation is powerful stuff.
Mathematics is a form of creation. If god created the natural numbers, and humans created god or are god, or even tap into some aspect of the godlike nature of creativity within fabric of the world as such, that which has never yet ceased to surprise us with its novelty, then by creating new mathematics, we allow the world to recreate itself through us. And in doing this, we learn something deep about ourselves, and about the manner in which we emerge from the world in and through things like mathematics. Human languages aren’t thinks, they are emergences and reemergences, as are we, as is, it seems, the universe from some primordial singularity.
While the end of this essay may seem far from questions of mathematics, the reason for going to such metaphysical issues is to make the point that they are omnipresent, and that reification obscures this, with control and petrification as the result and aim. Liberation and emergence can also be result and aim, and both are self-potentiating tendencies, as paradoxical as anything described by Godel, but dynamic beyond such at attempt to grasp it in snapshots. None of which is to say that we need dispense with reification, but rather, that we simply need not be seduced by its productions as being anything ultimate. Rather, there is always a play of natura and naturans, to use Spinozist terminology, or to cite Schelling, producer and produced. Forgetting that the produced is only a product and not the producer is the classic means of taking the present and its products as necessary. Those in control of our world have always relied on the stability of appearances to maintain the notion that the world has to be as it is.
While creation for its own sake can dissolve all that’s good in the world if taken to its limits, our world has always erred on the side of reification for its own sake, minimal necessary creativity to avoid ossification. This of course makes sense in world in which survival is that of the fittest. But in a world in which we can now feed everyone, in which the worst predators humans face are other humans and their products, the very evolutionary situation has changed, and so, our ability to survive depends on us unlearning the very lessons that evolution evolved into us to survive the brutal period of biological evolution. Surviving cultural evolution is the next step, and this requires walking backwards the paranoia we needed to develop out of the primordial oceans.
All the games we play, from mathematics to baseball, politics to economics, these are all human products, and products of the world beyond this in turn. Those creations and recreations we give rise to reveal what we value. And so often, we tend to forget that we have choices. Of course, if we are to truly make choices, we have to deal with the meta-gamic question of what we value, and why. To avoid precisely such meta-gamic questioning, it is often easier to simply pretend that our world is not even partially of our own making, and that our agency is small indeed. But while our agency is distributed and relational, it is hardly small.
The ethics of this essay is that of emergence, neither creation for its own sake, thereby giving rise to chaos, nor consolidation and control for its own sake, giving rise to stasis. Because our world overvalues the second, course correction would seem to value a shift to the other side, to novelty, but again, this should hardly be seen as necessary, only situational.
Mathematics has in fact engaged in a radical move towards creativity during the twentieth century, and put the seduction of reification behind it after the foundations crisis of the early part of the century, itself potentially a reflection of so many other crises of foundations early in the century, and their often violent repercussions. Foundations are always foundationings. They are statements of value and valuation, but if one looks for the values underpinning one’s values, there is only ever refraction to the whole, because the whole notion of value itself is simply the manner in which one is able to relate a particular move within a particular game to the meta-game played at the level of a given whole. The ultimate meta-meta-game, however, something like the world or experience, is precisely what the question of value addresses, and the answer is only ever inconsistent, incoherent, or incomplete. And yet, the creations and recreations we give rise to in the way we game and meta-game in relation to this is what makes everything worthwhile.
And if there is something that gives value to meta-gaming as such, it seems to be the emergence of (meta)gaming as such, and its continued emergence, the robust complexity of the games playable within the larger gamespacetime. Everything in our world worth valuing, after all, seems to only exist contextually within such situations. Life, for example, or the love which it can give rise to, these depend upon the complexity, and robust sustainable complexity, of the systems which give rise to them. Look at the best within the game, and use it as a guide to establish new guidelines for future gaming and meta-gaming, and continually readjust.
For games without rules, such as life or love, it’s always about values, and values are always about the relation of moves to the context of the whole, which is always in the process of emergence. But is that emergence getting better, and in a manner which is seems to lead to that in the future? Perhaps that is the best we can ask. If there is an ethics to (meta)games such as mathematics, perhaps it is in praxes like these. And while many may balk at the notion that math could ever have an ethics, there is no action in the world which does express values and valuation, and hence, have an ethical component. Humans value things in the world which allow them to continue to live and grow, which is to say, to recreate their ability to create and recreate. We eat food, build buildings for shelter, and create things like mathematics, and in doing so, we expend our energy and time, things we value, in the process. We value mathematics, and as such, it is an expression of our value, valuation, and values, even humans themselves are expressions of the value, valuation, and values of the contexts which produced us in turn.
There is an ethics to every move in every game, including language and mathematics. I’d like to think that emergence is, ultimately, the only way to imagine winning, which is to say, to emerge more robustly in relation to any and all in one’s context. An ethics of emergence, with ramifications for all gaming and (meta)gaming, including the creation and recreation of mathematical worlds.