ENSAIO MODELO - Nº13
Knowledge is true belief based on argument. — Plato, Theaetetus, 201 c-d
Is Justified True Belief Knowledge?” — Edmund Gettier, Analysis 23: 121–123
Throughout the many ages of human experience, knowledge was of fundamental importance. In fact, the idea of goodness or meaning for our existence is directly derived from a certain idea of truth, for, every enquiry has to firstly suppose that it is able to achieve some dimension of resolution – even that this means to achieve aporia or something alike. In mythology, we have gods and ancient creatures to assert what is, say, good or bad. In another words, we have the existence of these superior beings to justify morals. In ancient Greek philosophy, we have logos and cosmos; namely, we have the assumption that reality is somehow ruled by universal, unchanging rules and, further, that we, humans, are able to apprehend and understand these rules. In either case, what we have is the conception of justification, or even proof, of something that is actually true and we, or at least the ones of us that knows it, judge it to be so. Thereof arises the platonic notion of knowledge whose criteria are, in a more modern language,
S knows P iff (if and only if)
i) S judges P to be true, or believes that P is true;
ii) S is justified to believe that P is true;
iii) And P is true.
Although very appealing, this criteria is flawed. We can have those three conditions attended and yet S can not know P, as Gettier demonstrated. However, that’s not the topic of this essay. Instead, I will defend that the ‘classical’ idea of knowledge must to be reformulated, for, rigorously speaking, ii) and iii) can never be satisfactorily answered at the same time. Along my analysis, I will take for granted that, in despite of not sufficing for knowledge, those two conditions are yet necessary for it. In effect, this classical idea of knowledge cannot be dissociated from factuality and proof.
Case 1: when ii) is attended satisfactorily
Firstly, let us discuss the matter of entailment. What does it mean to say that A entails B? I think that the best way to address this question is through mathematics, for it is well-known that the mathematical approach is somewhat different from the scientific one; in such a way that some philosophers separated them as two distinct kinds of knowledge, for example John Locke. In fact, this great distinction between them arises from the certainty we put on the relations between mathematical objects, which often entails one and another through deduction and pure logic. In the natural sciences, on the other hand, the relations, or rules, are themselves topic of discussion. To clarify that, let’s consider two lines of thought:
1: The sun has arisen every single day before this one, thus it will arise today as well.
2: 1 + 1 = 2
Both statements are extremely simple and strongly acceptable, and, for a lot of people, they will be equally true. Nonetheless, if we consider this a little further, for philosophical accuracy, we should understand where did these assumptions came from. The first one arises from experience and induction, so the ‘nature’ of the objects has nothing to do with the relation between them. If we see the object A (the sun has arisen every single day before this one) entailing B (the sun will arise today) many, many times, we will assume that this relation (A, thus B) exists, even that we do not know anything about A and B – we only need to know how to identify their occurrences. Therefore, it is not necessary that this relation be true, but rather likely or very likely.
In the second statement, however, the relation does not arise from experience, but rather from the nature of the objects themselves. The axioms of arithmetic, namely the ones of Peano, determine that relation. The rules of operations are part of the definition of natural numbers. So we already know that 1 is the first natural number and that the operation of addition consists in moving along the natural line, the nth natural number plus one leads you to (n+1)th natural number, or, in mathematical language, An + 1 = An+1. Therefore, it is necessary that this relation be true.
Now, we can see that ‘mathematical knowledge’ attends ii), for the justification arises necessarily from what S is considering to be true; but does it attends iii)? To answer that question, we shall first ask ourselves what do we mean when we say truth. In fact, truth is usually understood as homorphological (Presents the same form) with reality. The statement ‘the sky is blue’ is only true if the sky is blue in our frame of reference, namely, reality. Here, we achieve a critical point, for ‘mathematical knowledge’ does not actually take this into account. Of course, the axioms are meant to be reasonable and similar to reality, but they are never really natural, because mathematical objects are not like natural objects. A good example of this is the case of Euclidian geometry, which presents us with a few axioms and postulates, and thereby proceeds to several theorems. The condition ii) is certainly attended throughout Euclid’s The Elements, every theorem logically follows its initial assumptions; however, the fifth postulate of it does not correspond to reality. It presents the planes in which the other geometric objects (points and lines) will lay upon as flat, whereas, in reality, the planes are curve due to gravitational attraction. One could claim, then, that it would suffice to correct those assumptions in order to achieve knowledge, but, in effect, mathematics does not care with the truthiness of its assumptions. Math is not concerned with ‘what is true’, but rather with ‘what is true if’ and that is fundamental, because the value of truth of your axioms, if they should intend to be true, would be based upon science, which is the actual one searching for ‘what is true’. Therefore we would be able to establish the following relation:
M: mathematics can reach knowledge
S: science can reach the truthiness of particular objects
M iff S
Let us consider for a moment that this relation is true in order to see if science is able to attend the iii). Later on this essay, we will return to it.
Case 2: when iii) is attended satisfactorily
The idea of science as the accumulation of truths as it was conceptualized by the positivism was mostly abandoned in contemporary epistemology. Since Karl Popper’s criterion of falseability, science was newly elaborated as a process that proceeds negatively; namely, by the refutation of false beliefs, instead of by proving the true ones. Anyhow, it does not mean that the idea of scientific truth was throw away. In effect, one of the conditions for science presented by Thomas Kuhn is the paradigmatic monism, that is, a certain agreement among the scientific community upon the truthiness of certain things. Modern science yields absolute truths in order to reach coherence, to the same extent mathematic ignores the values of truth of its assumption in order to do the same. Despite not achieving necessary truths, the enquiry of modern science goes indefinitely in the right direction, just like trying to catch a particular ball inside a box with infinitely many other balls by taking them one by one and throwing the wrong ones away.
Final considerations
If we go back to our previous relation, we will notice that it cannot make sense, because what justifies the conclusions of mathematics does not justifies its premises, once their premises come from science and not math. One last attempt could be made in order to look for axioms perfectly coherent, since we can assume that reality is coherent and that these axioms could be unique – although that would also need a proof. Unfortunately, as it was proved by Kurt Gödel’s incompleteness theorems, if a system of axioms can prove itself as correct or perfectly coherent, it is inconsistent. No matter which axiomatic system you choose, it can always present flaws.
Summarizing: science can dynamically (changing through paradigmatic shifts in face of discrepancies and irregularities) attend to iii), for it deals with real things and reach reasonable conclusions, yet not necessary ones, thus failing in ii), and mathematics can satisfactorily attend ii), but it does not deal with real objects and neither can achieve perfect axioms, thereby, failing in in iii).
SINTETIZANDO -
Autor da citação: Edmund Gettier (1927-) / Platão (428 a.C–348 a.C)
Posição em relação ao autor: Discordância em relação a Platão
Tese do ensaio: Nenhuma investigação filosófica pode ao mesmo tempo alcançar verdades, correspondentes a realidade, e ser inteiramente lógica, ou seja, basear em prova rigorosa.
Autores usados em suporte: Edmund Gettier (1927-), Kurt Gödel (1906-1978)
Ensaio escrito por Cauan Marques Negreiros