Treatise on the Universal Function: Introduction

Introduction

Anyone who studies mathematics and has thought deep enough about it has eventually realized that the generality of mathematical thought seems like nothing else in our world. Nothing in mathematics depends on specific things in reality; nothing in mathematics depends on space or time. There is a universal, or general, character to it which does not seem to be empirical. Take the example:

Find x, where x is a real number:

x +54329 = 2544

We might not know which number x is, but we know it is some number. Are we thinking of any particular number when we talk about x? No, not necessarily. Yet, we know what a number is because we have dealt with several instances of it. When we are taught numbers, we are not taught the abstract concept of number, but the instances like 1, 2, 3... and only then we generalize to what a number is. Now take a linguistic example of you telling your friend the following sentence:

"I have seen the cat fall out the window."

The person does not know which cat you are talking about. Yet, she does understand the sentence because she has seen instances of cats and so she knows what a cat is. There is a general character to cat which is not any cat in particular - very much like mathematics. This is something which seems obvious, but is it really? How is it that we are capable of understanding in general terms when the only experience we have is of particular things? Well, that's easy, you say - we are capable of abstraction. Okay, to this abstraction I will call a universal and to every thing in reality which can be perceived by us in any way I will call an instance. The universal is then a general concept that captures whatever is similar between instances.

This text attempts to explain my hypothesis as to how we create and organize universals in different levels of abstraction; these levels of abstraction are bounded, that is, they are not infinite in their extension.

Main Argument

Let us begin with only two assumptions: 

1. Universals exist (i.e. we are capable of abstracting concepts).

2. One instance occupies a unique place in space-time as perceived by an observer. Another instance, indistinguishable from the first (except for the place it occupies), cannot occupy the same place as the first [If it could, then it would be indistinguishable from the first at all possible levels and it would not make sense to consider more than one instance].

Lowest Level of Abstraction (Lower bound)

Since universals capture whatever is common, or similar, between instances, we need more than one instance to form a universal. 

Instances as maps to Universals: To every set of more than one instances there corresponds at least one universal.

Let us imagine the simplest possible case - imagine a set of instances which are physically identical at every possible level. This might be, for example, a set of water bottles, balls or animals, such that they are physically indistinguishably from one another. Every aspect of similarity will be captured by the universal. Thus it makes sense to define the following:

Lowest level of abstraction: The universals which are mapped by instances that are physically indistinguishable from one another except for the place they occupy in space-time are grouped in what we call the lowest level of abstraction.

A set of water bottles might be physically the same at every trait level, but the elements still occupy different places in space and/or time. The location of the bottles or the time at which they are perceived by the observer are not necessarily the same. Thus we may define the following universal:

Space-time universal: A universal is called a space-time universal if it belongs to the lowest level of abstraction, i.e. if all the instances it represents differ only by their location in space-time and are identical in every other respect.

This is as low in abstraction of universals as we can go - every possible thing is physically identical to the other except for position in space-time; if the space-time location and physical characteristics were the same, then the identity would be the same, as stated in assumption 2. Of course, universals of instances which physically differ from one another by something more than only space-time location will occupy higher levels of abstraction. We thus conclude something very important:

Every universal is independent of space-time.

Thus we see that this independence of space-time arises naturally from our considerations. We will see how mathematics can be perceived in terms of universals.

Highest Level of Abstraction (Upper bound)

We can now think on what would be the other extreme - the highest abstraction possible. That would be a universal corresponding to a set of instances which have nothing in common - neither space-time nor physical characteristics; but then, how would the universal be a universal, since there is no similarity to be captured? Well, there is one thing that they have in common - instances all have the property that they can be perceived. In fact, if this is the case, there can only be one universal in the highest level of abstraction; if there were more, they would need to be distinguished by something, in particular by the instances that map them; but the instances are already distinguishable in all characteristics (except for one), so they must be mapped to the same universal, which leads to a contradiction that more than one universal exists in this level of abstraction. We then define:

Highest level of abstraction: The set of instances which are physically different in every possible way, except for the characteristic that they can be perceived, map to only one universal, called the supreme universal, which defines the highest level of abstraction.

We have reached the conclusion that the capacity for abstraction is bounded - below by similarity, above by every possible difference in instances.

Intermediate Levels of Abstraction

It is now trivial to see how something as x (in fact, any generalization) might be conjured: x is just a universal who was created by sets of instances which have nothing in common except they are real numbers. A cat universal was created from instances of cat, with some degree of similarities and (almost certainly) differences. These universals reside in the middle levels of abstractions:

Intermediate levels of abstraction: All the universals which are not the supreme or space-time universals define the intermediate levels of abstraction and are called tangible universals.

We now notice one peculiarity. Consider a set of vegetables, with some similarities and differences and the corresponding universal which captures every similarity of between them. If all the vegetables are green, then that is it; if not all vegetables are green, we can create a subset of all those which are green. This subset of vegetables have more similarities than the elements of the set from which it originates - thus the corresponding universal will be in a lower level of abstraction. If the universal is a space-time universal, then any subset of instances will have the same universal - because it cannot go lower in level of abstraction (lower bound). We then conclude the following:

Given any subset of a set of instances, the corresponding universal will be lower in level of abstraction than the universal of the original; except if the universal of the original set is a space-time universal, in which case any subset has the same associated space-time universal.

Conclusions

We have found a way to organize generalizations. On one hand, we consider that the capacity for generalization is a given, and if so, then it must be bounded below and above by differences and similarities. Of course, the highest and lower bound are only theoretically possible - it is virtually impossible to find a set of things which are exactly the same except for their position in space-time, or a set of thins which differ in all possible characteristic except in that they are capable of being perceived. If they are actually impossible, then we still find that our capacity for generalization is bounded. This hierarchy puts mathematics, concepts and linguistics as arising from instances of the real world and then gaining an existence of their own in different levels of abstraction.

Some analogies between life and mathematics

In math and physics, we often find equations which we cannot solve immediately because they are too general - we need what are called boundary conditions. These allow us to make the problem concrete and soluble - per example, if I want to find how the temperature is distributed in the room, I need to know the shape and size of the room.

In life, I might want to find the solution of how I should act. But this is too general - how you should act in what context? We have to first define what your space of action is, so that the boundary conditions are well-known. These boundary conditions can be your ethics, fears, troubles - such that they represent the limits beyond which you cannot act - and thus they define what your space of action actually is. Then a solution can be formulated - how should you act, given these conditions? Of course, in real life I would say that the boundary conditions are the real challenge to solving any problem. If I have a very clear sense of what my ethics, fears and values are, then I would have a perfect recipe to act in any given situation - unfortunately, the boundary conditions depend on the nature of the problem, so they are ever-changing.

Another example in math and physics is the concept of change of reference of frame, or let us just call it change of perspective. Some problems might not seem trivial - e.g. if we want to figure out the evolution in time of a spinning top as seen from an outside perspective. We can change our point of view, imagine that we are spinning with the top, study it from that perspective and then go back to our original outsider point of view with more information.

In life, a problem is often seen as unsolvable from the perspective of the individual. If taken from a different perspective, the problem might be trivial or even disappear at all - be it a psychological or a practical problem. At a minimum, a change of perspective often makes the problem more clear than it was before.

It is interesting to think that perhaps some strategies used in mathematics are general forms of what we learned in particular cases throughout life. I do not want to expand on this here, but just pin down a very abstract idea for the sake of later reflection.

The Ideal Tune of the Violin

The aspect of being was never easy to me. 

Ever since I remember remembering, I was in angst with existing in this world. It was like a disconnect - a noise produced by a poorly tuned violin. This destructive interference between my way of being and the tune of the world was the cause of my suffering and existencial crisis for about 20 years. How can a psychologically troubled kid like me express this pain in the most clear and concise manner? Even if I could, who would be capable of understanding my complexities, my own shadow? As far as I could tell, this shadow extended far beyond the capabilities of my verbal articulation - but it always accompanied me in my journey through life and was never ignored.

Having a different tune than the world can be hard as hell - but it was by no means the whole story. There was more - there was this ideal. The ideal tune; the pure tune. I could feebly hear it - far beyond in the distant future, deep into the abyss of my own consciousness and in the background of the metaphysical landscape. Every time I paid attention, it was like seeing this picture which captured the true dichotomy of what I was. This picture was two-fold. In the first scenario, I was in chains - weak, coward, oversensitive, destructive; the dissonance was the background music which I knew so well. In the second scenario, I was unchained - strong, brave, confident, balanced, flying; the background music was a finely tuned violin song.

I never knew why this ideal was with me - I just remember always having it, like my shadow. I tried desperately to make sense of this painful way of being to someone - to my mother (in her own words, I was an "(...) extremely complicated kid"), to my grandfather and to characters in books. No one could hear me, no one could tell me. Even the other children did not seem to have this dissonance - they were not smiling all the time, but they seemed happy with the world in general. This only added to my resentment. Having the notion of an ideal was painful, since my self-awareness was right in telling me that I was as far as I could from it - perhaps that it was even unattainable. The years of self-hate that followed were a predictable reflection of this distance that existed between what I was, what the world was and what the ideal was.

It was clear to me that I had two choices if I wanted to keep on living. Either I would bend the knee to the dissonance of the world and make myself one with it - at the risk of losing my identity and ceasing to hear the weak but beautiful melody of the ideal violin - or I would struggle to remove the distance between me and the ideal - at the risk of suffering so much from the fight that I would die.

The reason why I chose the latter, in retrospective, is not so obvious. As far as I could tell, the world was filled with people whose knees were bent to suffering. Not only did they accept suffering, they did not think about it - their shadow was the true sculptor of their way of being. Beings sculptured by shadows are resentful because they hurt inside - because they hurt, they hurt others, intentionally or not - because others are hurt, suffering takes over as they let their shadows tame them - and the so goes the cycle. Human beings can only stand a certain amount of suffering without giving up their construction to their shadows - that is because it is far easier, when you are tired of fighting, to give up responsibility than to take it up. I did not want to contribute to this cycle of existential suffering - so I made a choice.

My mind was made up on taking responsibility for my own suffering; on closing in distance to the ideal tune. The question is clear, the answer is not: how to do such a thing? It took me years of action and thought and destruction and reconstruction of myself to figure out the answer and a clear way of articulating it.

I look at it in this way. My shadow represents an amalgam of all my fears, insecurities and troubles. It represents the chains that hold me in my place; the distance that separates me from my ideal; the boundaries of my comfort zone. Beyond this zone lies what I do not want to face - darkness. At the same time, that is precisely what I have to face in order to get hear the sound of the violin louder and louder and get constructive interference between what I am and the world I live in. So I take a deep breath, get closer to the boundary - I am shaking - and take a step into the darkness. It is cold, it hurts but I can hear the violin. My eyes adjust to the darkness, my body gets used to the cold and when I look back my comfort zone is a little wider. I did not die from taking a step forward - in fact, I expanded my zone of action by being brave and thus shedding light on the surrounding darkness. I can now hear the violin a little louder and my being is more resonant with the world I live in. The answer was clear:

Whatever things you fear and do not face become your limits.

The things you fear are nothing but little monsters which, if you do not tame, will consume you. If you face them, the fear will eventually go away - you become their master, you are now in control. You take the chains that bound you and use them to tame your own shadow so that it does not define your limits. This requires bravery and suffering and determination - but as far as I am concerned, it is the proper way of living. I started facing my fears; speaking my mind even in face of fear; being okay with failing over and over again. Lack of confidence was tamed and it turned into confidence; over-sensitivity became balanced and I dominated the fear of being hurt - the list goes on. The meaning of freedom becomes clear:

I am as free as I can be, precisely because I refused to be bound by fear.

Along the way, I noticed that my ideal was way closer to me than I originally thought. My ideal was precisely the realization of the way for me to properly live - I felt in resonance for the first time. An ever-growing light circle; a dissonance that grows weaker and weaker; a gameplay between looking at my vast comfort zone but always wanting to step into the darkness, seeking to hear the violin louder and louder. Beside me, all my past fears that are now my friends - they are the part of my shadow that was tamed. Ahead of me, the remaining darkness.

Curious, excited - I plunge.

Os Frutos não nascem podres. [texto de 2012].

As Batalhas não decidem a Guerra, e os Frutos não nascem podres.

É como tentar remar contra a corrente. Quando se toma a decisão de lutar, está-se inevitavelmente destinado a encontrar obstáculos. Não há evolução sem desafios, é como tudo. Está-se destinado não só a sofrer, como provavelmente a perder batalhas atrás de batalhas. Dizem que batalhas perdidas não decidem o desfecho final da guerra... Mas quantas batalhas temos de perder até ficarmos aptos o suficiente para ganhar? Quanto sangue temos de derramar, nosso e alheio? Quanto cansaço acumula? Quanta esperança morre? Tanta esperança morre... Morre e renasce, demora a renascer, demora a morrer, tudo ao mesmo tempo. Existem momentos de glória, pequenas batalhas são ganhas, batalhas que, apesar do seu tamanho, nos enchem de triunfo e esperança, mais do que poderíamos esperar duma batalha tão pequena. O sol vai e vem, alterna com a lua, como se não houvesse espaço para os dois lá em cima; sempre foi assim. As tempestades vão e voltam... os fortes alimentam-se dos fracos. Os fortes e fracos lutam com fracos e fortes, e as guerras continuam, tal como o sol, a lua, e as tempestades, e o sangue, e a vida e a morte.

O mundo está para além do bem ou do mal. Somos um misto, um misto de bom e mau, de quente e acolhedor, de frio e inóspito. Somos um misto dentro de um misto. Bem, se uma fruta está metade podre, chamamos-lhe “fruta podre”. “Está podre”, dizemos com toda a certeza. Bem capaz. O que eu costumo fazer com essa fruta é cortar a parte podre antes que se alastre a corpo inteiro, antes que contamine as irmãs, as vizinhas, as sementes. A podridão alastra-se, e quem não vê, consegue certamente cheirar, esse cheiro horrendo a morte. Isto falando de frutas é claro. Falando de pessoas, bem... Como se corta o podre? As pessoas são feitas de carnes e osso, de tecido e nervos, de corpo e alma. Não existe uma faca que corte alma, mas existem armas que cortam pessoas. Existem armas que acabam com o corpo das pessoas, e a alma e o podre desaparecem por não ter onde se agarrar. Já que não se pode cortar só o podre, acaba-se com a pessoa por inteiro. Vai tudo à vida. Oh, mas sejamos sinceros, existem armas que cortam especificamente o que interessa ou não interessa, sim. Armas que podem fazer bem ou mal, mas são demasiado difíceis de manusear, que são mal usadas, ou usadas em pessoas erradas, ou por pessoas erradas, e que acabam por alastrar o podre. O que se faz? Tenta-se extrair o podre usando as armas certas, ou entra-se em guerra e destrói-se tudo por completo. Destrói-se tudo por completo, a outra opção é demasiado arriscada, é demasiado difícil, demasiado.

Mas porquê temer a podridão alheia quando já temos suficiente da nossa? E de armas na mão, as guerras começam, e o cheiro a morte e a podre alastra. E o sangue derrama, e a esperança morre, e renasce, e demora a renascer e a morrer. Enchemo-nos de glória por pequenas vitórias ganhas, e suores frios escorrem pelo corpo cada vez que uma batalha ameaça ser perdida, e o cansaço acumula. O sol brilha num momento, e depois não, e as tempestades vão e vêm.

E a árvore nunca apodrece, e os frutos nunca nascem podres.