sunnuntai 3. huhtikuuta 2011

Methodology of science

Platonic school developed no fixed theory, although all Platonists shared some basic thoughts of e.g. a source of unity and oneness beyond the sense world. But it is not Speusippos or Xenocrates who we remember of Plato's successors, but Aristotle who founded his own school of philosophy. It is often discussed whether Aristotle was some sort of rebel who turned from the Platonic world of supernatural sources or ideas to nature or whether he still was close to Plato's own teachings. The question is difficult to decide, because we have no succession of Aristotelian writings starting from his Platonic days and ending in his own matured thinking, but only writings of different issues, often even collected from different eras of Aristotle's life and mixed sometimes in a rather confused manner – worst example being the so-called Metaphysics, which shows clearly its cut-and-paste origin. Because of this, we have to revert to the pre-Platonic manner of expressing the whole of Aristotle's philosophy as a whole with no genesis: although Aristotle's views must have progressed, it is usually impossible to determine this progression.

Although Aristotle's development from Platonic positions is usually difficult to demonstrate, it is still sometimes possible to show that Aristotle answered some questions that were discussed already by Plato in his dialogues: often Aristotle preferred considering actual examples, while Plato had tried to determine answer to them through discussion and arguments. Thus, Plato had considered whether there was some natural language or whether all language was merely based on custom: in effect, Plato was concerned about the possible divide between language and what was described by language. Aristotle noted that at least Greek language did not correspond perfectly to world, because one word might have had many different meanings: this phenomenon of homonymy is familiar in almost any language.

Although homonymic words made the relation between language and things more complex, they still did not completely break the connection between the two. Indeed, Aristotle took it for granted that words were used in describing things in some manner. Yet, he quickly noted that certain groups of words played different roles in these descriptions: these roles he investigated in his work Categories. One group of words did not so much describe anything, but named it: “Socrates” did not tell anything of a thing, but only named it. Such names were connected with individuals and answered the question “who or what is it”. What's that coming down the road? Oh, it is just Socrates. Indeed, these words were a sort of basis for all descriptions: we call something black, and when we are asked what is black, we can say it is our dog Fido.

Yet, the question “what is it” can be answered in another manner also: we can say of Socrates that he is a human being and of Fido that he is a dog and of both that they are animals. Yet, these answers are always secondary in comparison with the names of the individuals, because we can respond these answers with a new question “what dog” or “which human being is it”. These generic word “animal”, “human” and “dog” formed then hierarchies where e.g. humans and dogs were all animals.

Beyond names and generic words there were words of a completely different sort. Socrates is Socrates always and he is human being and animal also always, but he has also other descriptions that are not as necessary to him. Of these more arbitrary descriptions Aristotle discerned many different sorts. Thus, Socrates could have qualities like paleness. Such qualities had often opposites (paleness vs. darkness), but not always (what would be the opposite of redness?) and one could also be e.g. more or less pale. Furthermore, he could have quantitative properties like being 170 cm tall. Such quantities did not usually have opposites (although such indeterminate quantitative terms like large and small might have) and one could not be e.g. more or less 170 cm tall. Another interesting group formed of relative words: Socrates might have been described as a father, which then implicated that someone was his son.

In a similar manner Aristotle went through different contingent descriptions, and all of them answered to a different question (e.g. where is it, when was it, what did it do, what was done to it etc.). All of these words were then single words, but Aristotle noted in his work On Interpretation that we could also put such words together into combinations. Within these combinations the words might lose their usual meaning or gain new ones. The words and their combinations might also be modified in such a manner that these modifications are not independent words or expressions, but forms of the original word or expression: “Peter's” is not a word different from “Peter”, although it expresses the notion of Peter having something that is not implied in the original word. Furthermore, words and expressions might be combined with a negative: “not-man” or “not-blue”. These negative combinations are peculiar, because they do not refer to any object or property of an object, but to a lack of something.

An interesting differentiating characteristic is whether the expression in question has a reference to time. Expressions like “sheep”, “red bicycle” or “gas station near Alabama” do not refer to any particular moment of time. Then again, expressions like “is a vehicle”, “had a nap” and “will be going to have a terrible headache” all refer to some point in time, whether it be past, present or future. Although the examples of the previous paragraph were of expressions with no reference to time, the expressions referring to time can also be inflected and modified by negations: some examples are “wearing a ring” and “is not dark”.

Out of these expressions more complex could be constructed, such as prayers (“please give me that horse”) or questions (“what is the quickest route to Paris”). Aristotle noticed that some combinations had a characteristic that single words did not have, namely they could be called either true or false: an expression was true, if what it expressed was so, and it was false, if what it expressed was not so. Simplest of such expressions or statements consisted of two parts, one of which did not refer to time, while the other referred: e.g. “young sheep is in the field”, which consists of expressions “white sheep” and “is in the field”. Although the simple statement itself could be called true or false, neither of its parts could: neither “white sheep” or “is in the field” is either true or false. Simple statements could then be combined into a more complex statement using words like “and”: e.g. “I went shopping and I met my mother”.

Aristotle goes on to classify various sorts of simple statements. In some cases a statement says that something is the case – these are affirmations – while other statements use negatives to deny something – these are called negations. Aristotle notes that if an affirmation and negation speak of the same thing, one affirming something that the other denies, the two statements cannot be true at the same time or in the same sense: if “I am in Paris” is a true statement, then “I am not in Paris” must be false. Such a pair of affirmation and negation is called a pair of contradictories. If affirmation and negation appear then to be true at the same time, Aristotle concludes, they cannot speak of the same thing or they are not contradictories.

Statements might designate the things they speak of by using words designating certain individuals or names – “Fido is a dog” – or they could use words referring to kinds of individuals – e.g. “dog was playing with a ball”. The latter sort of statements use generic words, but they still speak of individuals. Yet, unlike the statements using names of individuals, these statements do not speak of any determinate individual: thus, statements like “man is riding a bike” and “man is not riding a bike” might well be true at the same time, if they just don't speak of the same person.

Further classes of statements can be found by using instead of words their negatives: thus, we could say “not-man is white”, where we would speak not of a thing of a definite kind, but of an indefinite thing, which happens not to be a man; or we could say “man is not-white”, where we would not say anything definite of the man in case, but only pointed out his lack of whiteness – furthermore, both of these new statements might again be negated by saying either “not-man is not white” or “man is not not-white” or combined together into an affirmation “not-man is not-white” or negation “not-man is not not-white”.

Although generic terms can be used in referring to individuals, they can also be used in referring to all members or a kind or to a portion of that kind by using words like “every” and “some”. Hence, we could affirm “every sheep is white” or deny this by saying “not every sheep is white” or “some sheep are not white”. Similarly, we could affirm “some sheep are white” or deny this by saying “no sheep is white”. The two pairs of statements discussed in the previous sentences were contradictories: one denied what other had affirmed. Yet, in a sense the sentences “every sheep is white” and “no sheep is white” are also opposed: only one of them can be true. Still, there is a room for a third option, namely, that some sheep are white and some not. Thus, the two statements are not to be called contradictories, but contraries.

Two contradictory statements or affirmation and its negation cannot be true at the same time. Aristotle notes that one might demand a justification for this presupposition, but continues immediately that in this case no strict deduction cannot be given, because all arguments are usually based on this rule: indeed, the whole point about arguing about truth of some statements is the belief that all statements cannot be true. Still, Aristotle gives a sort of practical justification: if affirmation and its negation could be true at the same time, one might as well throw himself down the cliff, because after that it might also be true that he had not thrown himself down a cliff.

It seems then reasonable to suppose that affirmation and its negation cannot be true at the same time. In most cases it seems also that at least one of them must be true: either today is Monday or it is not or either I was here yesterday or then I wasn't. But when the affirmation and negation discuss about future events, it seems plausible to suppose that neither of them might yet be true: otherwise, it would already be determined that tomorrow it will rain or that tomorrow it will not rain, although it might be that it is up to chance whether tomorrow it will rain or not.

A further interesting modifications of statements occur when we describe some statements as necessary, possible, impossible, contingent or true. Here appear certain grammatical difficulties involving contradictories. If we think of a statement beginning with “it might be that” followed by some affirmation as involving possibility, then it might seem obvious that a contradictory of such statement would be stating the possibility of some negation. Yet, it can well be that e.g. I might be swimming and I might not be swimming: possibility of affirmation does not preclude the possibility of negation. Instead, the true contradictory would be the denial of possibility: I cannot swim. Similar considerations apply also to other terms like possibility.

Aristotle noted that these concepts have rather interesting relations to one another. For instance, if something is possible, it cannot be impossible and vice versa. A difficult problem is the relation between necessity and possibility, because the word “possibility” is actually used in an ambiguous manner. Sometimes we say “it may be so” and immediately conclude that “it may also not be so”: in this case what is possible cannot obviously be necessary. Then again we also say things like “fire might burn”, although we would not accept the conclusion “fire might also not burn”. In other words, we have two different ways to use words indicating possibility: either they indicate that something is merely possible, but not necessary or then they indicate that something is possible and perhaps also necessary. In any case, we seem to have three levels of possible matters. Firstly, things might be necessary, just like fire can hurt, because it will of necessity hurt when you touch it. Secondly, things might be possible and sometimes even true or actualised, just like I might be walking downtown, because I sometimes do. Finally, things might be possible, although they have never been and might never be actualised, just like a woman might conceivably be a pope, although no woman has ever been one.

Aristotle did not just analyse various sorts of statements, but noted that when some statements were known to be true, we could immediately conclude some other sentences to be true also: these relations between statements he considered in his Prior Analytics. In very simple cases we could assert a new statement just on basis of one statement by converting the order of the terms in the statement. Thus, if no wolves are lions, then clearly also no lions are wolves. Similarly, if some white animals are humans then some humans must be white animals. Then again, if all humans are animals then not all, but some animals must be humans.

A more complex conclusions involve then at least two statements called premises, from which a third statement is deduced to be true: such deductions or syllogisms involve then three terms, two of which are somehow connected in the conclusion. The third term can then be related to the other two in three manners: if the three terms are A, B and C, and we want to connect A and C, B might be feature of one, while the other is feature of B (e.g. A is B and B is C), B might be feature of both A and C, or finally, A and C might be both features of B. Thus, we get the so-called three figures of syllogism.

In syllogism of each type the premises could both have one of the many shapes a statement is known to have: they might be universal (“All As are Bs”) or particular (“Some As are Bs), affirmative (“As are Bs”) or negative (“As are not Bs”). Now, Aristotle went painstakingly through all the possible combinations of different types of premises in each figure and pointed out which combinations occasioned valid deductions: that is, from which type of premises we could without a doubt conclude something. In some cases the validity of deduction could be seen immediately: if all As are Bs and all Bs are Cs, then clearly all As are Cs. In some cases a syllogism needed to be proven through syllogisms known already to be true: for instance, if a) no C is B, but b) all As are Bs, then we know that no B is C and thus can conclude through previously known deductions that c) no As are Cs – so we can conclude statement c) from statements a) and b). In other cases such straightforward proof is not possible, but Aristotle shows that if some conclusion would not follow from certain statements, then some contradictions would occur. For instance, if a) all Bs are As and 2) some Bs are not Cs, then 3) some As are not C: otherwise, all As would be Cs, but because all Bs are As, all Bs would also be Cs, which was not the case.

If Aristotle would have just enumerated valid syllogisms, we would still be unsure whether the rest of the possible syllogisms would contain some valid syllogisms. Therefore Aristotle showed carefully that other combinations of premises did not lead to any conclusions. His method was to show two different examples where in both premises of certain types hold, but the possible conclusion was different. For instance, no swan or crow is horse (no As are Bs) and some horses are white (some Bs are Cs): still, no conclusion can be made, because in some cases some As are Cs (some swans are white), while in other cases no As are Cs (no crow is white).

The syllogisms Aristotle studied were thus classified according to three criteria: according to the relations of the terms, according to whether they were universal or particular and according to whether they were affirmative or negative. He also attempted to investigate what an addition of a fourth sort of classification of statements would do: that is, the classification of statements as possible, true or necessary. Here the complexity of the task apparently overwhelmed Aristotle. He accepted some dubious principles. For instance, he believed that a syllogism needed only one necessary premise for a necessary conclusion: hence, if I happen to have a triangular door, then because triangles have necessarily three angles, my door would necessarily have three angles – although a door could well have four angles also. Furthermore, Aristotle noted properly that possibility could have several meanings – “it is possibly so and so” could mean either “it is possible, but not necessary that something is case” or “it is possible and perhaps even necessary that so and so is case – and then just confused these two different interpretations whenever possible.

Although Aristotle's dealing with possible and necessary syllogisms was not a success, at least his investigation with ordinary syllogisms was. Indeed, he did more than merely pointed out the valid syllogisms. For instance, he made several interesting observations on the different types of syllogisms. He noticed that through second figure – syllogism of the sort “A is or is not B, C is or is not B, thus A is or is not C” - one could only achieve negative results, while through third figure – syllogism of the sort “B is or is not A, B is or is not C, thus A is or is not C” - one could achieve only particular results. From this he could at once see that universal positive statements were particularly hard to deduce: in fact, there was only one valid sort of argument leading to universal positive conclusions, namely, of the type “all As are Bs, all Bs are Cs, thus, all As are Cs”. On the other hand, such statements were particularly easy to refute.

Aristotle also described how one could generally reduce syllogisms to other syllogisms, that is, how one could deduce these syllogisms through other syllogisms. Particularly he showed that syllogisms of other figures could be reduced to syllogisms of first figure, while all the syllogisms in this figure could be reduced to the only syllogism having a universal positive result – which in turn was not reducible to other syllogisms. This incontrovertibility of reduction underlines the importance of the syllogism for deducing universal positive statements: other syllogisms can be based on this statement, but not the other way around,

No Greek philosopher before Aristotle had discovered such a method for establishing true statements and only few had even considered the problematic. The only true predecessor of Aristotelian syllogistic was the method of division in the Platonic school, and Aristotle shows considerable effort in showing that his method beats Plato's. The method of division started from a known fact of the type “A is B”, e.g. humans are alive, and from another known fact of the type “B is either C or D”, e.g. living things are either plants or animals. From these premises one can only conclude that As are either Cs or Ds (humans are either plants or animals), but not yet that As would be, for instance, Cs: this can only be established by a proper syllogism, Aristotle says.

What is so revolutionary in Aristotelian syllogisms is that if they are applied to true premises, they invariably produce true conclusions. On the other hand, syllogisms might result in true conclusions even if they start from false premises. Thus, if one thinks falsely that all crows are horses and that all horses are black, one can still deduce the true conclusion that all crows are black: here the conclusion is correct, but it has been justified through incorrect reasons. Because sometimes false premises lead also to false conclusions, we can be certain of the truth of the conclusion only if we are certain of the truth of the premises.

Aristotle actually tries to show that all proper deductions or arguments producing truths from known truths are syllogisms or consist of syllogisms or at least use syllogisms without being syllogisms. Sometimes a deduction seems to be more complex than a syllogism, but then can be analysed into a series of syllogisms. At other times deduction or its part is based on showing that some absurdity arises from certain assumption and then discarding this assumption: yet, even here the deduction of this absurdity from the assumption requires some syllogisms. Aristotle even gives some instructions how to turn arguments into syllogistic form: the important thing is to find a term that repeats itself in various statements, because this is usually the middle term connecting other terms. This method of turning apparent arguments to syllogisms might even be used in evaluating whether an argument is valid or not, Aristotle surmised. Aristotle shares some hints as to how to pick out the terms for syllogisms. He also notes that although statements like “A is B” are the main area of application, other types of statement can also be used in syllogisms: for instance, if cows are animals and cows like to go into field every day, then some animals like to go into field every day.

Aristotle did not just satisfy himself with theoretical investigation of syllogisms, but he also gave practical advice on how to find syllogisms: something that often was forgotten after him. One wants to prove that A is B or that A is not B: what is one to do? Firstly, one should consider not contingent features of an individual A or B, but features of every A and B. Then one is to consider what one can deduce from something being A or B: whether As are Cs, whether Bs are not Ds etc. In addition, one is to consider which things are known to be As and Bs: e.g. whether Es are As etc. From all these considerations one should then try to discover some common elements that could be used in connecting A and B to one another: e.g. if all Fs are both As and Bs, one can instantly deduce that some As are Bs and vice versa.

Aristotle came from Platonic tradition where dialectical conversations and debates were thought to be the correct way of practising philosophy. It is therefore quite understandable that Aristotle wanted to show his syllogistic to be a good tool for such debates. Hence, Aristotle gave also instructions for construction of new syllogisms from known syllogisms: for instance, if we know how to justify some statement through some premises, we might be able to justify the premises through syllogisms, and if we know a possible justification of a conclusion, we can produce an appropriate syllogism for refuting that conclusion. Here it is clearly not important to find the true conclusion, but to beat the opponent in cleverness. This is even more apparent in Aristotle's instructions how to make syllogisms by using opposed terms, that is, how to deduce evidently false conclusions when someone accepts e.g. both that science is good and that science is bad; or in his instructions that we can justifiably object when someone tries to refute a statement with a syllogism ending up with a contradiction, when the statement itself has not been used in the syllogism as a premise.

Aristotle also considered a number of other , not as reliable forms of justifying statements that might also occur in a Platonic debate. Thus, if we know from a number of different types of objects that they have some feature (e.g. lions are mortal, humans are mortal, pigeons are mortal etc.) and we know that these types are all species of a certain genus (e.g. we have gone through all animals) then we can conclude that everything of this genus has this feature (all animals are mortal). Furthermore, if we know one example where thing of a certain kind has some feature, we can use this example as a sort of paradigm or exemplary case by which we can assume that other thing of the same kind probably has the same feature: if I know that Greece lost the football match against Serbia last year, I might assume that they lose it this year also. Finally, we can use features that are commonly associated with a thing or event of some sort as signs of thing or event being of this sort: that is, if we know that pregnancy causes female breasts to produce milk, we may reasonably conjecture that a woman with breasts producing milk is pregnant. All these modes of justification fall short of the trustworthiness of syllogism, even if they start from true premises, but they might come handy in Platonic debates.

The Platonic method of dialectical debates was obviously important for Aristotle: such dialectics practised intellectual capacities. Aristotle thus produced a whole treatise called Topics on how to gain dialectical skills. In a dialectical debate a proponent tries to make her opponent accept some conclusion. She can base such conclusions on previous propositions she can make her opponent accept. Therefore, the dialectician should know what propositions are accepted generally or at least by eminent persons who are held to be wise: the opponent is more likely to accept such propositions also. Furthermore, the debater should be aware of the meanings of words so that she cannot be deceived by e.g. using a name with different meanings: in best case, she may able to deceive her opponent. A knowledge of structural similarities is also helpful, if the debater wants to base her conclusion on analogies.

The Topics is full of rules of thumb for showing that an object has or does not have certain accidental characteristics – if thing has a capacity for one characteristic it must have a capacity for its contrary (if people can know things, they can also be ignorant), and if something has naturally a quality then they have this quality in a greater degree than something that doesn't have it naturally (berries taste sweeter than sweetened food) – that an object belongs or does not belong to a species – if a thing belongs to no subspecies of a species, it does not belong to the species either (if a creature does not belong to any species of animals, it is not an animal) – that a property either inclusively and necessarily characterises an object or not – if a lack of a property does not characterise a lack of an object in this manner, then the respective property does not characterise the respective object in this manner (deafness does not entail lack of sensation, thus, hearing is not the only sensation there is) – and that an object is or is not defined in a certain manner – if the supposed definition can be more intensive, while the defined thing isn't, the thing hasn't been defined correctly (because a desire to have sex with a person can become more intense, while the love towards that same person can remain at constant level, desire to have sex does not define love). Aristotle also notes that it is most difficult to show that something is the definition of an object, a little less difficult is to show that a property belongs to an object inclusively and necessarily, easier to show that an object belongs to a species and easiest to show that an object has an accidental characteristic: the difficulty levels of disproving these are opposite.

Aristotle, gave also a more normative instructions as to how a good dialectician asks his questions and how a good opponent answers them: both should be skilful, .but fair contestants. Thus, the answerer must accept those propositions that are widely accepted and that are not relevant to the question in case, but refuse to accept propositions that are not widely accepted or that would instantly lead to the questioner's victory: on the other hand, he must accept the seemingly absurd consequences that follow from the thesis he is upholding. Similarly, the questioner should approach his goal in a covert manner, but still avoid some obvious fallacious forms of reasoning. Aristotle even presents a number of such fallacies in another work, Sophistic refutations. These fallacies could be used in more aggressive debates, meant to embarrass the opponent, and one should learn both to use them and to answer them. Still, the Platonic debate should in its proper form not be used for such a contest, but for educational and research purposes.

But the true worth of syllogisms lies not in their use in such debates, according to Aristotle, but in their capacity to lead to true conclusions from true premises: this is what Aristotle's Posterior Analytics investigates. Thus, if we already know something without a doubt, we can use syllogisms to gain further knowledge. In this case syllogisms can be called demonstrations, which provide us with incontrovertible knowledge of facts and also some bases on which to justify and explain these facts. The presupposition of such demonstration is that we start from premises that are more certain than the demonstrated conclusions and that can be used to explain these conclusions.

Aristotle was convinced that demonstrations were truly possible. Indeed, he could always note that the mathematicians of his time had discovered some real demonstrations: for instance, they had shown that the sum of all angles of a triangle was always equal to sum of two right angles Now, if all statements required some justification from statements known to be true, demonstrating couldn't really begin anywhere: hence, Aristotle was committed to the idea that demonstration was not the only source of knowledge. Aristotle tried even to show directly that infinite chains of justification would be impossible. A chain of demonstrations must begin at least from some final thing of which we can say something, but which cannot be feature of anything, that is, from individuals and their immediate classes. Similarly, a chain of demonstration must end at least with some features that cannot be described any further, that is, the ultimate classes, like substances and qualities. The only other option beyond infinite and finite chains of demonstration would have been to endorse the possibility of circular demonstrations where conclusions are first based on premises and premises then on the original conclusions. Such a circularity would destroy the essential difference between basic and applied knowledge and would also lead to a mere moving in familiar circles without any novelty being reached, therefore, it was not to be accepted. True knowledge should thus be based on some indubitable premises from which all the rest of the truths should be deduced through demonstrations.

The final premises of such a string of demonstrations should confirm to some criteria. These premises should describe properties of all members of some class of objects. Furthermore, these properties should be either essential to these objects – although all men would have crooked noses, this would not still be essential to them being men – or they would have to form an essential division of the class of objects – just like all numbers are either odd or even. Finally, the premises should deal only with the highest possible classes having certain properties: that is, a statement “right-angled triangle has three sides and three angles” would not be a final premise, because we could demonstrate this statement from further premises like “triangle has three sides and angles” and “right-angled triangle is a triangle”. This final demand for the premises is actually applicable also to demonstrated knowledge: it is better to demonstrate that the sum of angles in all triangles equals two right angles than to demonstrate this just for right-angled triangles. Because the premises of a proper demonstration should be necessary and essential and the demonstrations should also produce necessary and essential knowledge, we could never demonstrate anything inessential, Aristotle says: for instance, we could not truly demonstrate that a person has a crooked nose, because he might as well have a straight nose.

Proper demonstrations should then begin from premises describing essential characteristics of certain kinds of entities. Therefore, Aristotle suggests, we should be able to divide demonstrations into different sciences which all concern some peculiar kind of entities. Thus, we get a sort of hierarchy of sciences where a science higher in hierarchy turns into a science of lower level, when some new premises are added to it. Aristotle goes even so far as to suggest that results of one science cannot be used in another science, if neither investigates subspecies of the issue of the other science: after all, they investigate completely separate kinds of objects. In cases where such application apparently happens – like when we use arithmetical calculations in geometry – it is actually a case of us using results of higher science which contains the two sciences as its possible applications. Indeed, all sciences use some common principles, like the fact that same thing cannot at the same time and in the same sense be and not be characterised by the same feature. Still, every science has its own proper problems, and problems of one science can be answered only through the premises and assumptions of this science: thus, we cannot expect a mathematician to have an answer to a medical problem. Still, one might be able to use higher science explain some facts that are taken as mere given facts by a lower science: e.g. mathematics might explain some optical phenomenon.

The special status of the first figure of syllogism is even strengthened when we are talking of demonstrations. First figure was the only means by which all sorts of statements could be justified, and particularly it is the only figure through which universal affirmative statements could be justified. Because statements concerning essential characteristics of some sort of objects are obviously universal and affirmative, Aristotle concluded that first figure was of the utmost importance in scientific considerations. In addition to being the correct source of true statements, in general, demonstrations of universal statements are more important than demonstrations of particular statements, because universal statements potentially contain particular statements: if I know that all triangles have three angles then I know this of any individual triangle I happen to come in contact with. Similarly affirmative statements are more important than negative, because all syllogisms require some affirmative premises, and therefore it is also better to prove statements straightforwardly and not through showing their contradictories to be false.

Aristotle considers also how erroneous statements or generally ignorance could arise. Some errors arise through syllogisms using false premises, and some errors arise because we do not have the necessary means by which to find the correct premises, for instance, because our sense perceptions are faulty (if we cannot see colours correctly, our beliefs about colours might be erroneous).Still, mere individual perceptions cannot make a science: we cannot know necessarily that all triangles have angles sum of which equals two right angles just by looking at individual triangles. At most sense perception will lead us to have uncertain opinions, which we might assume to be true, but which we think might also be otherwise (e.g. I might see that Rufus has a beard, but I still cannot infallibly demonstrate that he has a beard). We can, undoubtedly, use such opinions as premises in syllogisms, but these syllogisms won't be true demonstrations, because the opinions do not describe the essence of any object.

Science according to Aristotle is not described merely by its method, but also by the problems it tries to solve and questions it tries to ask. One sort of question science considers is of the form “is this such and such”, e.g. “do triangles have angles sum of which equals to two right angles”. These questions are answered by finding a demonstration where the questioned fact is proven or disproven from known facts. A bit different, though clearly related form of question says “is there such and such”, e.g. “are there triangles”. These questions of existence should also be answered by finding a demonstration where the existence of such objects is either proven or disproven from known facts.

Now, the two questions could also be turned over. Instead of asking whether triangles exist or whether they have such and such angles, we may ask why there are triangles and why they have such and such angles. Aristotle notes that such questions can be understood in four different senses. Firstly, they may be thought as asking for conditions of something: what does it require to make a triangle or an angle of a triangle? Secondly, they might be understood as asking for the thing or person accountable for the existence of this particular object: who or what has drawn this triangle with its angles? Thirdly, we might think these questions as asking for the purpose of something: what use could a triangle and its angles be? Aristotle suggests that the most important question is the fourth of the essence of things: what triangles and their angles are or what is their essence. The fourth question in a sense contains all the others within it: by knowing in what conditions triangles exist and in what conditions they have certain properties, who makes them and for what purposes, we know what it means to be a triangle.

Facts can be known through demonstrations, but essences of sorts of objects are defined: triangle is such and such an object. Now, Aristotle tries to show that definition is a method differing from demonstration. Indeed, demonstrations can reveal that something is not the case, while definitions always say that something is the case: “triangle is not square” does not define anything. Furthermore, definitions should provide the indemonstrable starting points for the demonstrations. In fact, demonstrations and definitions have completely separate tasks: definitions reveal what it means to be something, while demonstrations presuppose such meanings and then show what other properties such an object must have. Still, demonstrations can in a sense reveal the essence of something: when we know some fact, like that an earth quake is occurring, we can explain this fact through a demonstration based on a definition of earth quake (earth certainly quaked today, because the events corresponded with the essential features of an earth quake).

How does one then make definitions? Aristotle suggests the method of collecting properties that the definable species shares with some other species of same genus, because some set of such properties should characterise the species and nothing but the species: thus, number three can be defined by being an odd prime number (seven is also) which is not a sum of numbers larger than one (like two). Aristotle also suggests that the Platonic method of division helps to constitute definitions. True, divisions do not prove definitions – divisions do not prove anything, as we have seen – but they do help to classify all essences in a systematic manner: through a properly effected division we can define a whole classification of kinds of objects. Aristotle advises to begin definitions from the lowest species of objects (such as triangles) and working one's way slowly upwards to genera containing that species (like figure), because mistakes are made more often in defining higher genera.

Aristotelian science contains then a series of demonstrations based on definitions and other principles or axioms. We have seen how to find definitions, but what leads us to axioms? Aristotle suggests the following scenario. Human beings begin their quest for knowledge from sensations they receive from objects around them: this characteristic they share with all animals. In humans, and in all higher animals, Aristotle would probably admit, individual perceptions leave traces within human being, so that she can later recognise perceptions similar to those she has had before. This capacity of recognition or memory enables human beings then to make generalisations: e.g. when we see that triangles always have three sides, we at once understand that all triangles have three sides. The capacity of generalisation or “understanding” should be the property solely of rational beings, just like the capacity of systematic demonstration. Unlike demonstration, understanding should not be based on any proofs: we “instantaneously” or without any mediation see some general truth instantiated in particular events. On this immediately certain ground is based the edifice of the Aristotelian science.

Aristotelian view of science betrays its Platonic origins: indeed, we might say that Aristotle has merely purged Plato's methodology from its mythological elements. A person is brought to the firm starting point by showing her how general structures are embodied in particular instances: this is undoubtedly where Aristotle imagines the aforementioned dialectics is practised. The outcome of this practice is intuitive clarity on the essential structures of all types of things, from which then certain conclusions can be deduced. This picture of science is based not only on Plato, but also on contemporary mathematics, which indeed was the science most respected by the Greek and in which particular truths were seemingly based on intuitively clear and certain axioms and definitions: we shall see later how this mathematical ideal of science limited the understanding of other, more empirical sciences.

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