On the origins of mathematics
> They seem rather to form a second plane or level of reality, which confronts us just as objectively and independently of our thinking as nature.
I’m pleased that. . . you advocate a cautiously Platonistic point of view. To me a Platonism of this kind (also with respect to mathematical concepts) seems to be obvious and its rejection to border on feeble-mindedness.
Platonism properly understood is a question: how can this be possible? For we find impossibility here. The old line is better turned: any sufficiently advanced product of evolution is indistinguishable from magic. This is merely to accept that it is what it does, that the principles of the world are indeed this way. We must accept that the mathematical realm is improbably true. There is a sense in which aligns with the subtle laws of this world, that it seems unified and invisible in a way unlike any other sphere of practical knowledge. Mathematics seems to take place of a piece, entirely together, and this in a way outside the world—and more: that it does not enter the world. It is perhaps this final feature which is most surprising: that mathematics, despite its importance, despite its preeminence in Western civilisation, only ever enters the world indirectly. This is precisely what makes mathematics so perfect, for in this we find it is timeless, even universal. The question then is the precise ontology of this structure: what does it have to do with the world, with us?
To answer this we must know where it comes from, as all things are best known. This is a notion formalised by Spinoza: essential knowledge of an object comes best from its practical description.1 The circle, for instance, is best defined by a compass. Something similar is so more generally: a thing is best known through its origin. The fact that this is so rarely asked in the history of philosophy is for the thing being unthinkable then. The notion of evolution, of our being produced in such a fashion, is of recent vintage—and yet it is not really, for we find much the same shapes in even the earliest mythologies. All of these tell us about the faculties by way of their origins. The whole is situated in a story. Some shame these stories, “just-so” tales, and they are that, but it is with humility that we must accept them as about the best we have.
These stories are our best option if we intend to really know anything. The other way is merely to know locations; that this is how consciousness is described, for instance: the frontal cortex, the parietal—or they may likewise tell us when it is: 170 milliseconds, 230. So much of science is the identification and arrangement of fragments, an aesthetic which analytic philosophy apes. I do not advise the excess of continental philosophy as its alternative, not the mere telling of stories, but I do accept this influence. I accept these both as elements of the proper image: evidence, yes, but also characters and plot. This is the key also to understanding mathematics.
Who are the characters in this play? They are the living tradition of mathematics, of course. It is found first in them, only then the world. They find it in the world only when they go looking. It is in the world also, of that there is no doubt, but it is found first by them. It is thus a thing discovered, but it is not akin to any object. If anything, it has the image of an ancient civilisation. We feel ourselves to have encountered something sublime, seemingly eternal. There is something in mathematics which still resembles the old perfection of Platonism—indeed, this has always been the truth of Platonism, for Socrates in the Meno grounds his anamnesis in geometry.
Socrates draws upon the ground, there the boy follows him. That the boy can follow him in the territory of mathematics is taken as proof that he remembers a previous place, that we once were of a world wherein these truths were certain and not obscure. The notion that all truth is recollection. This narrative of the spirit coinciding with epistemology, if such a word can be used to describe anything prior to Kant; it can, that is our advantage. We have the advantage also of evolution, but even Socrates explains by the genesis of this capacity, it is known in terms of its origin.2
The slave boy was remembering in a sense, if we accept a much broader store of human memory. This is true. There is a human head beyond ours, a whole of which we are part. This is true in Hegel, although only half understood for his standing on the other side. We can see both sides because we are over here, because we have followed Marx. The idea is not the cause of the world except in the sense of the child; in the sense of Man, the world is the cause of the idea. The idea has been negotiated with the world by man, an innate ritualism has brought it into being.3 Every word is an egregore summoned in this way, every institution, every human activity and object.
Winnicott said that there was no infant without a mother; that is, the infant is co-constituted by maternal care as an ongoing act. What there is without a mother, even if it be a man, even if it be a wolf—well, that he does not say. Soon it will be a corpse, we know this much at least. The child must eat, but it is unable to provide for itself. Say, I heard once of a child that starved when its parents were away because it could not find a spoon. There was plenty of food, but they did not eat because they had no proper implement. This is not stupidity or even an absurd arrogance, rather they simply did not know there was such a thing as food without a spoon.
By the spoon we enter the world of mankind, that our needs are the thread by which we are pulled into this world. This in the story of Enkidu also, that he is brought into civilisation by desire. This may be how man is civilised, but the child is made human through his stomach. Hunger is the way into the world, the whole built upon hunger. This is its basic prerogative. Kropotkin knew this, saw further how it held us here, a people will never be free so long as they cannot feed themselves. This is the way we are dragged into their system, just as each is from the first.
The spoon would not exist without man just as mathematics would not. This does not mean that spoons cannot be used to eat, nor does it mean there is not such a thing as steel, nor that there was not the possibility of work and heat and the calories necessary and all that preceded this in its production. The spoon is produced in a factory, alright, but it is more importantly produced in the activity of its use. This latter might be considered reproduction, though really these are one and the same.4
The spoon is produced through its activity; it is not real until it is realised. This follows the form of the world, fits the structures of embodiment and homeostasis. The spoon is conditioned by the hand, by the requirement of caloric intake, by the sort of food we eat, hence the geography, the wider trade situation, etc. The child is conditioned by their environment, in this way learns the meaning of the spoon, in this way learn its truth, come to understand how it can be used; only thus is it made real.
There is footage of a deaf-blind child learning to use a spoon. They sit on a stool at a table with a bowl before them. They wear a little bib and there is a woman behind them. There is food in the bowl, the woman has a spoon. She leans over, places the spoon in the hand of the child. She wraps the child’s hand in her own, holds the spoon within this. Here Ilyenkov notes the child’s response in stages: at first they resist the imposition, then their hand goes loose and is carried through the activity, at last they enter into the activity themselves. This the process by which we are born.5
This is the way in which mathematics enters the world also. The child is brought into the activity which constructs mathematics. This is an activity wrought fundamentally of signs. The early history of mathematics follows from geometry, that the first efforts were there. The Nile would flood yearly but farmers were taxed regularly according to their land. When the floods washed away the stone markers that had been placed, then it was necessary to measure and assign an equivalent area.6
Piaget describes the learning of geometry in children: that their intuition here is first a practical affair, is initially dependent on virtual actions. They construct the truth in much the same way as that slave boy from the Meno, by the creative recombination of embodied practices. The defining feature here is that this activity involves a plane, it occurs by way of inscription. This is the case even when we surpass geometry, when all is abstracted into the pure signs of algebra. The same structure underlies each of these, both alike deal with signs but the plane fades ever more into the background.
The same movement is apparent in the development of sign languages. They begin with an early iconism and then tend increasingly towards abstraction, that likely due to the degrees of freedom this allows, effected by creative recombinations of prior aspects. Meaning is scaffolded by its own structure, thus grows upwards as a plant.
This art, however abstract, remains a matter of inscription; in other words, a variant of written language. The question here is how language contains any truth, with mathematics seen only as a last redoubt, a particularly universal phenomenon, one which cannot so easily fall victim to relativism. Though we may use a different base system than them, say—still these languages are perfectly and precisely translatable. That much can be said for hardly any other; this because of its extreme abstraction.
The perfection of mathematics is its distance from material affairs: that it encounters our world only indirectly, that it floats behind reality and portrays a vast connective structure. There are principles within this that simply work. The same is true of a boat as much as a spoon: the boat works, displacement—what is this natural law? It is simple: it is a regularity in the world, one which language can comprehend but never grasp. This is easy enough to say, but why this way and not some other?
I come this far but no further. There is no answer here, only necessity. We find these things within ourselves, within mankind, but they are not our own.7 Life has taught us to dance, and that is something more—
One must admit that perception and what depends upon it is inexplicable on mechanical principles, that is, by figures and motions. In imagining that there is a machine whose construction would enable it to think, to sense, and to have perception, one could conceive it enlarged while retaining the same proportions, so that one could enter into it, just like into a windmill. Supposing this, one should, when visiting within it, find only parts pushing one another, and never anything by which to explain a perception.
You might as well read Spinoza, the seed of all this is there, see here my entire argument—and this much clear than my efforts—as adumbrated in his Treatise on the Emendation of the Intellect:
Now that we know what kind of knowledge is necessary for us, we must indicate the way and the method whereby we may gain the said knowledge concerning the things needful to be known. In order to accomplish this, we must first take care not to commit ourselves to a search, going back to infinity—that is, in order to discover the best method of finding truth, there is no need of another method to discover such method; nor of a third method for discovering the second, and so on to infinity. By such proceedings, we should never arrive at the knowledge of the truth, or, indeed, at any knowledge at all. The matter stands on the same footing as the making of material tools, which might be argued about in a similar way. For, in order to work iron, a hammer is needed, and the hammer cannot be forthcoming unless it has been made; but, in order to make it, there was need of another hammer and other tools, and so on to infinity. We might thus vainly endeavor to prove that men have no power of working iron.
But as men at first made use of the instruments supplied by nature to accomplish very easy pieces of workmanship, laboriously and imperfectly, and then, when these were finished, wrought other things more difficult with less labour and greater perfection; and so gradually mounted from the simplest operations to the making of tools, and from the making of tools to the making of more complex tools, and fresh feats of workmanship, till they arrived at making, complicated mechanisms which they now possess. So, in like manner, the intellect, by its native strength, makes for itself intellectual instruments, whereby it acquires strength for performing other intellectual operations, and from these operations again fresh instruments, or the power of pushing its investigations further, and thus gradually proceeds till it reaches the summit of wisdom.
This is likewise the way with most mythology, perhaps all; that it proceeds in the order of creation, either by generations (as in the Greek) or by an order of acts (as in the Hebrew).
The idea is our self-consciousness of a summed process, whether capitalism or a cat.
This is the level at which Marxists should be dealing with the world, instead they have economic myopia. Walter Benjamin comes the closest in his The Work of Art in the Age of Mechanical Reproduction. It was through him that I first came to a self-consciousness of this idea, although he does not quite seem to have realised its full depth and generality.
These are the stages identified also by Kierkegaard: aesthetic, ethical, religious. These categories are broader than he knew. The three proceed is particular to universal and then, by moving further through this from the point of the particular, into the particular again. This is the process of life in its fullness, and it is known more or less to each of us. This is the same process described by Kuhn, that we enter into a paradigm and it is this which structures the activity of a sphere; yet there is always the possibility of exceeding this sphere, rather: of enlarging it. This due to the incompleteness of knowledge: that it is open.
The construction of any area out of right-angled triangles: Hahn suggests that this underlies Thales’ metaphysics, that there might be a single thing from which everything else was made. The same was seen, however, in Akhenaten’s monotheism and might as well have come from there. The Jews and the Indians surely knew about this also. The Eastern races also, in their own way (by negation).
This is Lenin’s argument for logic, that the forms have been engrained in us by thousands of years of practice. Note further that the structure of Aristotelian logic corresponds to the logic of physical containers, that this is the basis from which it is abstracted and then bent back upon itself thus giving rise to the strange and varied permutations that are its progeny.