To do science the first thing to do is to dream. Good principles which explain everything we want to understand. Good theories which answer the big questions that arise. Intellectual virtues which enable us to discover the most beautiful truths. Good scholars who tell the truth reliably in their field of competence. Good teachers who teach the best knowledge to all who want it. Good knowledge about knowledge which gives us the means to recognize all good knowledge. A good community where all spirits in love with truth help each other to develop together this public good which is knowledge, this good for all, this good thanks to which all can be good...
An ideal of science serves as a criterion for recognizing good knowledge. Knowledge is good if it helps us to realize the ideal, provided the ideal is good. But how do we know that the ideal is good? And how to make the dream come true?
How can a dream come true? How can we achieve the goals we have imagined? How can we realize intentions? How does the will to reach an end have the strength to set in motion to reach this end? How can final causes be driving causes or forces?
A dynamical system is determined with a space S of states and a function f from S to S of change of state. It obeys the law of motion x(t+1)=f(x(t)) where x(t) is its state at time t. f is the mathematical representation of the forces because it takes into account everything that determines the movement at each instant. The law of motion determines the future of the system from any initial state x(0): x(n)=f^n(x(0)) for all natural numbers n, where f^n is defined by f^0(x)=x and f^(n+1)(x)=f(f^n(x)).
Reaching a fixed state x at a fixed time t is a simple goal. A set of goals determines two new goals, its disjunction and its conjunction. A disjunction of goals is reached if and only if at least one of the goals of the set is reached. A conjunction of goals is reached if and only if all the goals of the set are reached. For example a goal g can be defined by the disjunction of the elements of a subset G of SxN (N is the set of natural numbers. SxN is the Cartesian product of S and N, it is the set of all the couples (x,y) such that x is an element of S and y is an element of N). An element (x,n) of SxN represents a simple goal, reaching state x at time n. It is reached from the initial state x(0) if and only if f^n(x(0))=x. The goal g is reached from x(0) if and only if there exists n such that (f^n(x(0)),n) is an element of G. We can conclude that the capacity of a system to reach goals depends on its initial state and its law of motion. A rolling stone reaches the bottom of the slope.
A self-willed agent is more than a rolling stone because he chooses the goals he wants to achieve. By determining his will he determines his initial state. But the goals he can achieve do not depend only on him, they also depend on the initial state of his environment. How then can he select the goals he is going to achieve by determining only his intentions? Should we believe in magic? As if external reality mysteriously obeys our desires?
"You can't always get what you want But if you try sometime you find You get what you need." (Mick Jagger, Keith Richards)
Desires have no magic power, but we can still sometimes achieve them, provided we make the effort to adapt. We have not chosen reality but we can choose to adapt our intentions and actions to the reality we perceive.
An agent is a being capable of perceiving, wanting and acting. In the following we reason on a very simplistic model of an agent, but the reasoning would be the same with a more realistic model. Let E be the set of states of the environment, P that of the states of perception of the agent, W that of his will, A that of his actions. To model perception, we assume the existence of a function p from ExPxW into P which determines the state of perception as a function of the previous state of the environment and of the agent. To model action, we assume the existence of a function a from PxWxA into A which determines action as a function of the previous state of the agent. The law of motion of the environment is determined by a function e from ExA into E. To model the formation of will, one can assume the existence of a function w from PxW into W which determines the present will as a function of previous perception and will. This amounts to assuming that the agent is deterministic, that its destiny is completely determined by its initial state and that of its environment. The four functions e, p, w and a together define a state change function from ExPxWxA to ExPxWxA. The system constituted of the agent and its environment is therefore a dynamical system.
A deterministic agent is not free, but we can make a deterministic theory of free agents by reasoning about the set of their possible destinies. It is not the state of a free agent at a given instant which is determined by the law of motion, but only the set of his states accessible from an initial state.
The ability of an agent to achieve goals depends on his initial state, that of its environment, and their laws of motion. The laws of environment, perception, will and action make dreams come true or not.
Voluntary agents can partly control the laws of their movements. They can give themselves rules to direct their action, their perception and their will. We don't choose the laws of Nature but we can choose the laws we obey. We can hope to realize our ideals if they are adapted to reality and if we give ourselves the rules which enable us to attain them. The power of our dreams depends on the right choice of the rules that we apply.
To find the truth, one needs good truth-producing devices. The law of the movement of the device must make it pass from its initial state to the desired truth.
An observation device is a truth-producing device. The observed reality is part of the initial state of the device. Its law of motion must lead him to a true observation statement.
In general, we observe localized events of the actual world. But we can extend the concept of observation. A law of Nature is an observation of the Universe, since it has been observed to obey the law. With imagination and theory one can observe beings that are not actual, only possible. Any truth is equivalent to the observation that it is a truth, since a statement is true if and only if it is true that it is true. If we understand the concept of observation with such generality, all truth is observation and a truth-producing device is always an observation device.
The being of a concept is to be attributed. Concept attribution devices make concepts what they are. Without attribution devices, there would be no concepts. To attribute a concept is to observe its presence. Concept attribution devices are observation devices, their proper functioning depends on the laws of their motion. No concepts without laws therefore.
Theories are used to make truth-producing devices. A theory is a system of axioms and definitions, the principles of the theory. Theorems are the logical consequences of principles. With the principles we recognize the true, the theorems, and the false, that of which the negation is a theorem. True principles infallibly guarantee the truth of all theorems, because logic always leads from truth to truth.
A theorem bears no visible marks that it is a theorem. Nothing in the appearance of a statement differentiates a theorem from other statements. One cannot therefore observe that it is a theorem simply by examining it. On the other hand, a proof bears the visible marks that it is a proof. It suffices to verify that all its premises are principles of the theory (or theorems already demonstrated, or temporary hypotheses) and that all the stages of the reasoning respect the rules of logic. Logical principles give us the means to make a universal device for observing proofs. It is universal because it enables to observe all the proofs of all theories.
In principle, a device for observing proofs is enough to make a device for observing theorems. Just look at all the proofs, in order of length, starting with the shortest. If the examined statement is a theorem, the device will eventually find a proof, since it examines all the proofs, and thus will provide the observation that the statement is a theorem. But even with a very powerful computer, this method of observing theorems does not generally provide results within a reasonable time. Even if we could wait billions of years, that wouldn't be enough for most theorems, except those with very short proofs.
Some truth-producing devices can function as oracles. We ask a question and they provide an answer, or they answer that the question is outside their area of expertise.
We acquire a lot of knowledge without even wanting to. Our natural faculties of observation and reflection, and the society in which we live, cause us to constantly produce or receive many truths, whether we want to or not. To acquire knowledge, it suffices to let the truth-producing devices that already exist, those given to us by Nature and culture, operate. But if we are satisfied with it, we do not always go very far. What already exists does not always say everything one would like to know, or says it in a very uncertain way. Moreover, truth-producing devices are mixed with falsity producers, and we do not always know how to tell the difference. To be truly learned, what one acquires without effort is not enough. To be truly competent, one has to want it. This is true for the production of knowledge as for all professions. "I want to know" is the "open sesame" for this cave of Ali-Baba which is the sum of all the knowledge that we can discover.
"- ... I'm willing to join you in examining virtue and inquiring into what it could be.
- But how will you inquire into this, Socrates, when you don't at all know what it is? For what sort of thing, from among those you don't know, will you put forward as the thing you're inquiring into? And even if you really encounter it, how will you know that this is the thing you didn't know?
- I understand the sort of thing you want to say, Meno. Do you see what an eristic argument you're introducing, that its not possible for someone to inquire either into that which he knows or into that which he doesn't know? For he wouldn't inquire into that which he knows (for he knows it, and there is no need for such a person to inquire); nor into that which he doesn't know (for he doesn't even know what he'll inquire into)." (Platon, Meno 80de, Fine 2014)
To want, one must know what one wants. How could one will if one has not determined one's will? To know what one wants is to know how to recognize if one has obtained it. How could we get what we want if we are not able to realize it?
To want to know, one must know what knowledge one wants, since one must know what one wants in order to will. But if we already know the knowledge we want, we already have it, so we don't have to want it. This argument is a fallacy. If it were fair, there would be no sense in asking any questions. We can know what we want, a good answer to the question asked, without having obtained it. We can know what we're looking for before we find it.
As soon as we have an observation device, we are able to search while knowing what we are looking for, before finding it. We know what we are looking for if we are able to observe it. We find what we are looking for by observing it.
To pose a problem consists in giving oneself an end, a goal, an objective. We have solved the problem when we have reached the end we have set ourselves or when we know how to reach it. We know an end when we know how to observe if it is realized. Knowledge of a problem is not knowledge of its solution. We can know an end, so we know what we are looking for, before we have reached it, so we haven't found it yet.
To find an answer to a question we ask ourselves, we must know how to observe that it is a good answer. The will to know requires knowledge about knowledge. If we had no knowledge about knowledge, we would not be able to recognize it if we encounter it, and it would therefore be pointless to seek it.
A statement is a knowledge if and only if it can be produced by a good truth-producing device that has worked well in producing it.
The proper functioning clause is necessary because good truth-producing devices are not always infallible.
Intellectual virtue is the force to produce truth. The previous definition of knowledge is therefore equivalent to the following:
A statement is a knowledge if and only it can be produced by an act of intellectual virtue.
The reliability of truth-producing devices and the effectiveness of intellectual virtues depend on the laws that determine their movements toward truth. Without laws to guide us, there would be neither truth-producing device nor intellectual virtue. The existence of knowledge depends on the laws of Nature and the rules we apply. To find the truth, we need our natural faculties and rules that enable us to find it.
We know when we have acquired a capacity to produce the truth, when we ourselves are a truth-producing device. But there are many ways to produce the truth. We can know in a more or less reliable, more or less autonomous and more or less thoughtful way. Reliability is measured by the error rate in the field of competence. We are autonomous if we depend only on ourselves, not on others, to produce the truth. We are thoughtful if we know how to evaluate the reliability of the production of the truth, our own reliability or that of others if we depend on them.
Knowledge about knowledge makes the quality of knowledge, at the same time its reflexivity, its reliability and its autonomy. The better we know that we know, the more knowledge is thoughtful. The reliability of the use of a truth-producing device depends on knowledge about its field of competence, therefore on knowledge about knowledge. To be autonomous, one must be able to recognize for oneself that knowledge is knowledge, so one needs knowledge about knowledge.
When we depend on others to know, in a sense we know, because we have acquired knowledge, but in another sense we don't really know, because we have to be autonomous to really know. To really know what one knows, one must know that one knows it. But is it really possible? We can fear an infinite regress: to know, we must know that we know that we know that we know...
The knowledge about the knowledge about knowledge is not a knowledge that comes in addition to the knowledge about knowledge, because it is a part of the latter, because the knowledge about knowledge is knowledge. Knowledge about knowledge is a knowledge about all knowledge, a universal knowledge. Since it too is knowledge, it is necessarily knowledge about itself. We can always know that we know when we know, provided we have good knowledge about knowledge, because such knowledge can contain knowledge that it is good knowledge.
I don't know how to choose the wine. I want the wine that will be chosen by the guest who knows wines well. In a sense, I know what I want, the wine that will be chosen, but in another sense, I don't know what I want, since I don't know how to recognize it alone.
I want to know what scholars already know in their area of expertise. This is the first thing to do when one wants to become a scholar, to acquire the knowledge which is already known. It is not only a question of knowing truths, it is also and above all necessary to become a good producer of truth oneself by knowing how to use the right devices that produce truth. But if I'm a beginner, I don't know how to recognize by myself the good truth-producing devices, because I have to be already learned for that. If I encounter a good truth-producing device that I want, I am not able to realize that it is what I want. How then can I acquire the knowledge I want? A teacher can solve the problem, because she knows what I want. I don't know how to recognize what I want, but she knows it and she can show it to me.
When my learning is supervised, in a sense I know what I want, the knowledge that scholars can give, but in another sense I don't know what I want, since without them I am not able to recognize it if I meet it. I know how to recognize what I want only at the end of the apprenticeship, when I have become competent enough to recognize competence on my own. Usually we know how to recognize what we want before we get it, but if the learning is supervised, we know how to recognize what we want only after we got it.
Good supervised learning makes us autonomous. The teacher teaches how to do without her, how to find in oneself the strength to produce truth.
Supervised learning is heteronomous. Another gives us the rules we apply. But the knowledge that we teach has not always been so, it first had to be found. If learning is supervised, the truth is taken from the cistern, but for there to be cisterns, there must be sources. How do we take the truth at the source?
One must be learned to observe knowledge. If one is not learned, one can learn with the help of a teacher, but in this way one acquires only the knowledge that already exists. What do we do if we want to find new knowledge that no one has ever taught?
If we have knowledge about knowledge, we don't need others to acquire knowledge, we can find it on our own. We can learn autonomously because with knowledge about knowledge, we can know what we want when we want to know and know if we have obtained it. Knowledge about knowledge gives critical thinking. To be critical is to observe whether a claim to knowledge is really knowledge or not. We have to be critical, and above all self-critical, to know autonomously.
We learn every day, even without wanting to and even if no one teaches us anything. To learn, it suffices to use the truth-producing devices that we already have. We can also discover new truth-producing devices. Knowledge about knowledge makes one capable of finding new ways of producing knowledge, because it makes one capable of recognizing them. It makes one capable of finding by oneself the rules that must be applied in order to produce knowledge. The more one has knowledge, the more one is able to acquire more of it, because truth-producing devices provide the means to produce both knowledge and new ways of producing knowledge.
Knowledge about knowledge makes one capable of learning by revealing that one is capable of learning. Sometimes we have to know that we are capable in order to become capable, because if we don't know it, we don't even try, we become incapable because we think we are incapable, we lose our abilities because we don't exercise them. Knowledge about knowledge reveals to us the immensity of our capacities to discover truths. It activates our potential by showing us how to fully develop it.
If I want knowledge that neither I nor anyone can observe, it seems that this desire is futile. Even if I meet what I want, I wouldn't realize it, so I have no chance of finding it.
We know what we are looking for when we are able to observe if we have found it. But one is not always able to observe intellectual virtue, reason or wisdom. To recognize wisdom, one must already be wise. How to seek wisdom if one does not know how to observe it? Even if we stumbled upon it by chance, we would not know that we met it.
When we look for reason, we don't always know what we're looking for, because we don't have the means to observe all forms of reason.
I seek reason when I want to learn all the good knowledge I can learn if I want to learn it. In a sense I know what I want, the good knowledge that I can acquire if I want to, but in another sense, I don't know what I want, since I could encounter it without realizing it. I still have the ability to recognize what I want, but it is not immediate. To know if I can learn, I have to try. If I succeed then I know I have found good knowledge that I can learn if I want to. If I don't succeed, I don't know, maybe it's bad luck, but maybe it's also that there's no good knowledge to be found. Anyway, I can observe my ability to learn, so I'm able to recognize if I'm getting what I want, so I know what I want, even though in the beginning I might encounter it without realizing it.
I can acquire knowledge that no one knows how to observe, because I can autonomously learn to observe it. Knowledge about knowledge makes one capable of discovering new ways of observing knowledge. It is not necessary to know in advance what we are looking for to look for it, we can learn it along the way. We can learn to observe and we don't know in advance what we will be able to observe. We don't know ourselves very well. We can search without knowing what we are looking for because we are looking for what we can become.
If one seeks knowledge, one does not always know how to observe at the beginning whether one has obtained what one wants, because one does not have a knowledge of all knowledge which gives the means of observing them all. The knowledge about knowledge that we have at the start is always of limited range and reliability. We are generally able to observe knowledge that is not very different from what we already have, but knowledge that is too different runs the risk of not being recognized.
With his limited knowledge of knowledge, a beginner is able to solve beginner's problems, recognize a beginner's knowledge and mistakes. He is not able to observe the good knowledge that he ultimately wants to acquire, but his beginner's knowledge is enough to get started, it's a first step on the way to learning. Each step makes one more capable of observing and acquiring knowledge. The ability to observe knowledge progresses along with the acquisition of knowledge. This enables us to learn how to solve increasingly difficult problems. This is how one becomes truly knowledgeable. We have to want to move forward, take one step at a time, and always be willing to learn what comes along the way. In the end, if one has become really competent, one has become able to observe the good knowledge that one wanted in the beginning.
We can reach a summit even in the fog, even if we don't know where it is. Just go up. In the same way, one can seek and find a good knowledge even if at the beginning one does not know how to observe it. It is enough to know how to observe if we progress when we learn. A beginner does not know how to observe the good knowledge he wants, but he sometimes knows how to observe if he has progressed, in a way that is not always reliable but which can be enough to learn autonomously. To learn by doing, by trial, error and success, it is not necessary to know where we are going, it is enough to want to progress. Dreaming of reason makes us want to learn and can help us progress, even if we don't know what we're looking for. We can be carried and guided by ideas without knowing where they lead us.
If we know what we are looking for, we can acquire knowledge by passing from certainty to certainty, knowing at each step that we have obtained the good knowledge that we wanted. But the paths of knowledge are not always so secure, they are rather paths of doubt than paths of certainty, because we don't really know what we are looking for. This initial ignorance invites modesty and imposes a duty of hospitality. If we don't know what we are looking for, we must let it come and welcome it so that it reveals reason to us.
A problem is well defined when the statement of the problem is sufficient to completely determine all of its solutions. We know what we are looking for when the problem is well defined, because knowing the statement of the problem is enough to observe the solutions. For a question that we ask ourselves to be a well-defined problem, we must have in advance the observation device or the theory that determines the right answers. Very often the questions we ask ourselves are not well-defined problems. We don't have the observation devices or the right principles that enable us to answer, we must first find them (Aristotle, Topics).
When we look for good principles to answer a question, we replace one question with another: what are the good principles with which we can answer the question? But this new problem is not better defined than the previous one, because we do not have in advance a device for observing all the good principles, because we do not have a knowledge about knowledge which is enough to observe all knowledge.
The great theoretical problems (what is Reason? Virtue? Wisdom? ...) are not and cannot be well-defined problems. For them to be well-defined problems, it would be necessary for us to know in advance all the principles which determine all the forms of reason, virtue and wisdom. We would have almost nothing more to learn. All that remains is to verify knowledge known in advance in each particular case. But for us the reason is not to know everything in advance, it is rather the opposite. We know very little in advance. We have to always have an open mind and welcome whatever comes our way if we want to find reason.
How to find the good principles? - Good principles are recognized by their fruits. - How do we recognize the fruits? - Reason bears fruit when it helps us to think well and live well. But we do not have in advance a device for observing all the forms of good thinking and good living. One must already be learned to recognize the fruits of reason. It is not always easier to recognize the fruits than to recognize the good principles. And good principles are themselves part of the fruits.
Reason bears fruit when it helps us to think well, to do well, to live well. But it is easy to delude ourselves. It is very easy to believe that we think or act well for very bad reasons. Reason does not always provide clear-cut answers because the difference between real fruits and illusions, between the wheat and the chaff, is not always clear and clearly marked.
When we seek good principles, good knowledge, reason, virtue or wisdom, we seek without knowing what we are looking for, because we do not have sufficient knowledge in advance to recognize all the good principles, all good knowledge or all the fruits of reason. We must be willing to welcome whatever reason can teach us.
A touchstone is a rough stone on which a sample of precious metal is rubbed to test its purity. The assayer identifies the metal from the trace it leaves on the stone. We are both touchstones and assayers for reason. We experience reason on ourselves and assess it from its traces on our minds. We are for ourselves a testing ground for reason.
I am the source, the middle and the end of reason, the source because reason is born from my thoughts, the middle because it develops in me when I seek it, the end because it is accomplishing itself when I am accomplishing myself.
I am for myself a fundamental criterion of recognition of good knowledge, since I recognize it by recognizing my competence. But I don't know in advance what is good for me, I have to learn it by experience.
True knowledge can always be shared. It makes me competent because it can make all minds competent. If I acquire knowledge without knowing how to explain it, and how to give proof acceptable to all minds, it is because I did not understand it well. To master knowledge, one must be able to teach it clearly to all those who want to acquire it.
Each mind is for himself as for all the others a criterion of recognition of reason, because reason is necessarily what is good for all minds.
Students teach teachers how to teach. We need experience to recognize if a teaching is good. If a teaching makes the students really competent, we learn that it is good. Otherwise, we don't know, maybe it's good but it's not adapted, it could be good for more advanced, more motivated or more hard-working students, maybe it is just not good.
We justify our knowledge by proofs based on principles. But the principles must themselves be justified. They have to prove themselves by helping us develop good knowledge. Everyone can use his own experience to put principles to the test and learn to recognize their value. But one must not limit oneself to one's own experience. When one takes a principle as the basis of a reasoning, one implicitly asserts that it has a universal value, that it can serve all those who want to reason. A principle must therefore be put to the test of all the experiences of all human beings. A principle proves itself by helping all minds develop good knowledge.
A beginner is not always able to recognize the fruits of reason and the good principles, because he is not yet a very good assayer of reason, he must learn it, but he is nevertheless a beginner assayer, able to recognize the fruits and the good principles accessible to the beginner. He becomes aware of good principles when they make him progress, when they make him more competent. Good principles must make people competent. If they do not make them competent, they are not good principles. Reason must be good for everyone, otherwise she would not be reason.
The development of reason is above all a collective work (Leibniz 1688-1690, Goldman 1999), in which each human being can participate as soon as he wants, that he knows that he is capable of it and that he voluntarily submist to its discipline: justification and critical evaluation.
In order to evaluate our proofs we must voluntarily submit them to the criticism of all human being. Objections and attempts at refutation may lead us to modify our reasoning, and sometimes even to abandon it, if refutation is decisive. We develop knowledge by preserving the principles and the proofs which resist the critical tests and renouncing the others.
All the development of knowledge can be conceived as the resolution of a single and vast problem. The objective is a knowledge which satisfies our desire to think well and live well. We explore the space of possibilities whenever we examine knowledge in order to evaluate it. Critical tests are designed to select promising opportunities. Criticism is therefore a heuristic, a way to find a way, that helps us to solve the problem of the development of reason (Goodman 1955, Rawls 1971, Depaul 2006). But we seek without knowing what we are looking for, because we do not always know in advance how to recognize reason.
A good critical community is a universal knowledge production device. It can recognize without ceasing to be critical that it is a good critical community, which produces good knowledge even when it declares that it produces good knowledge. It's circular reasoning, but it's not vicious, if the community really is a good critical community. This solves the problem of infinite regress in the recognition of knowledge. There is no infinite regress because knowledge about knowledge can recognize itself.
Reason makes us capable, but of what? What can we achieve with the skills we develop rationally? What can we hope?
We do not know in advance the scope of our ability to solve problems. We discover it through exercise. By solving problems, we become more aware of our abilities. The better we know them, the more we can extend their field of application. We thus discover ourselves as rational beings, that is, capable of developing reason. We discover that we are able to discover reason.
If the list of problems that we can solve rationally was known in advance, we would know what to hope. But precisely, it is not known in advance. We do not know the range of skills that reason can give us.
Since we do not know what reason makes us capable of, we can place our hopes very high, that the ephemeral present be the splendor of eternal truth, or very low, reason will never be more than a poor consolation in a valley of tears.
The development of reason is the story of a perpetually renewed astonishment. The sciences have exceeded our expectations. Nature has revealed many more secrets to us than we could dream of.
To know what reason makes us capable of, the best and only way is to try. If we don't try we have no chance of realizing what we can do.