Phase 1 The philosophy of
4 PRESCRIPTIVE DESIGN METHODS
4.1 The Method of Analysis and Synthesis
The method of analysis and synthesis was crucial as a method for discovery in geometry in ancient times (Panza, 1997). As such, this relevant method has been successfully applied in other fields to support their methodological approaches. For example, it is known that Isaac Newton used the method of analysis in Physics and Bernard Riemann in Biology.
The link between the method of analysis and discovery has recently triggered the discussion about this method as a proto-theory of design (see Koskela and Kagioglou, 2006 and Koskela et al., 2008). These authors claim that the method of analysis comprehensively embeds the features presented in most contemporary candidate theories of design. In this respect, this section is developed on the assumption that the method of analysis is a proto-theory of design and as being first in order it is presented ahead of contemporary candidates.
4.1.1 Analysis and synthesis: Ancient Description
Detailed descriptions of the method of analysis and synthesis from ancient times do not exist. As such, the reconstitution of this method is based on the interpretations of material that is related to the method of analysis such as in the works of Aristotle
(Analytics, Nichomachean Ethics and Physics) and Euclides (The Elements). There are
several interpretations available such as those provided by Heath (1921) and Hintikka and Remes (1974) referring to the work of Pappus.
In this respect, the only available description of the method (as a method) is from a later period, around 300AD. This existing fragment of a description defines analysis and synthesis in the view of the Greek geometer Pappus as translated by Hintikka and Remes (1974).
“Now analysis is the way from what is sought - as if it were admitted - through its concomitants in order to something admitted in synthesis. For in analysis we suppose that which is sought to be already done, and we
until we on our backward way light upon something already known and being first in order. And we call such a method analysis, as being a solution backwards.
In synthesis, on the other hand, we suppose that which was reached last in analysis to be already done, and arranging in their natural order as consequents the former antecedents and linking them one with another, we in the end arrive at the construction of the thing sought. And this we call synthesis.
Now analysis is of two kinds. One seeks the truth, being called theoretical. The other serves to carry out what was desired to do, and this is called problematical. In the theoretical kind we suppose the thing sought as being and as being true, and then we pass through its concomitants in order, as though they were true and existent by hypothesis, to something admitted; then, if that which is admitted be true, the thing sought is true, too, and the proof will be the reverse of analysis. But if we come upon something false to admit, the thing sought will be false, too. In the problematical kind we suppose the desired thing to be known, and then we pass through its concomitants in order, as though they were true, up to something admitted. If the thing admitted is possible or can be done, that is, if it is what the mathematicians call given, the desired thing will also be possible. The proof will again be the reverse of analysis. But if we come upon something impossible to admit, the problem will also be impossible.” This description of the method of analysis does not provide much to understand what in reality the method is. In this respect, the only way to reconstitute the method is to rebuild it from explanations provided by more contemporary descriptions. For instance, mathematician Polya (2004) explains that “…whilst analysis constitutes the elaboration of a plan, synthesis is the execution of the plan.” In geometry, that refers respectively to the elaboration of a theorem and to the use of the theorem to solve a geometry problem. Thus, in the following section those features are presented through a compilation of interpretations given by those who attempted to use the method. The following description complements the description provided in Section 2.1.
4.1.2 Prior to Analysis and Synthesis
Before proceeding further, an important note about the problem-solving process is essential. According to Riemann (1866, as cited in Ritchey, 1991), three important foundations guide the process of gaining knowledge of any system:
• Previous knowledge: knowing the existing/available natural laws and principles
challenged. In reality, it is by challenging the current knowledge that most of the contributions to knowledge are made.
Riemann (1866) does not highlight the importance of making explicit the ontology and epistemology used to generate previous knowledge. As discussed in Section 2, a change in the ontological and epistemological approach can also generate the promotion of knowledge and discovery.
• System objective/function: the second foundation refers to knowing what the
system to be investigated performs, does or accomplishes. For Riemann this knowledge can be obtained by empirical observation or experiment of the system in different situations.
In relation to evidence, it was discussed that its role is to sustain the status of knowledge to a proposition. For that, propositions are tested in relation to justification, truth and belief “levels” that the evidence promotes.
• System’s construction: the third foundation relates to understanding which are the
parts, components and elements of the system. This type of knowledge can be obtained by understanding the internal structure and processes of the system, especially the relationships between the parts and components.
In respect to evidence and knowledge, the components that give them “status” are justification, truth and belief and how together they escape the argument of illusion, i.e. Gettier’s problems. Despite the debate that these three components form an ill definition for evidence and knowledge, an alternative is not available. Therefore the search for another component or mechanism that brings them together is still not available.
The foundation for problem-solving is a point for reflection prior to when the application of the method of analysis starts. Although there is no known systematic rule for applying the method, the decision about which route to adopt for solving a problem should be considered first. According to Riemann (1866, as cited in Ritchey, 1991), two routes are possible:
to external input28. By doing so, what the system does (in parts and as a whole) is
identified by identifying its parts and how they operate. In other words effects are
identified from given causes29 and this is called the synthetic route.
• Alternatively, one can start by identifying what the system does and from there to
identifying which parts are necessary to accomplish what the system does. In
other words, that means to search for causes of given effects and this is called
the analytic route. Riemann in his studies used this route.
Riemann’s view is aligned with traditional thoughts regarding the method of analysis. However, there is a third possibility. Descartes disagreed that only these two routes were possible. According to Timmermans (1999), Descartes highlights (here in generic terms) that the solution of a problem depends on everything that precedes and follows the solution. For Timmermans (1999) the method that Descartes advocates does not start from causes to effects nor from effects to causes, but from the primary idea of the dependency or relationship between cause and effect (known and unknown), starting from what is easiest to solve and moving to what is most difficult. In Timmermans's (1999) words:
“Descartes might not go backward from the consequences to the principles any more than it divided a complex whole into simple parts, but that it revealed the correspondences, order, or relations between different terms, known and unknown, consequences and principles, wholes and parts”.
The process to identify relationships as the first step of analysis, as approached by Descartes, is not explicit. However, Timmermans (1999) highlights that to make such a correspondence between known and unknown, Descartes considered it important not to make any distinction between known and unknown, i.e. the unknown must be considered to be of the same nature as the known as the description described below extracted from Timmermans (1999, p.445):
28 The identification of parts requires the creation of taxonomies (identification and classification of parts). The elaboration of taxonomies and its relation to the method of analysis is examined further in this chapter.
29 It is extremely relevant to revisit the meaning of the word ‘cause’ in this context. Consider the example of a ball in movement. In this example ‘cause’ in the context of the method of analysis does not refer to searching for a fact “what has hit the ball so it causes it to move”. Rather, the analyst would be looking for the law of inertia or the three laws of motion as proposed by Newton.
“By this means Descartes managed to get round the logical problem of analysis in a different way than the logicians did. It was a matter no longer of drawing an uncertain consequence from a recognized principle or deducing truth from falsehood but of bringing to light the relations (operations) that connected the terms that one was taking into account, without supplying any prior hypothesis as to their order or rank, truth or falsity.”
Ritchey’s (1991) views of the works of Riemann’s (1866) and Timmermans (1999) in his analysis of Descartes work, do not make reference to the method of analysis as described by Pappus. In Pappus’s narrative “as if were admitted” the same approach as described by Descartes seems to be adopted by ancient geometers. In other words, the idea of starting from an assumption about the explanation of a conceptual relationship was already in the ancient method. Thus, the merit of Riemann and Descartes was to reincorporate this aspect of the method that, as explained by Koskela et al., (2008), was forgotten.
Finally, as highlighted by Holton (1998), Newton also contributed to the development of the method of analysis and synthesis. Conversely to Descartes, whose hypothesis was taken as assumptions discovered through intuition, Newton relied on experiments and observations to anchor the first principles in experience.
“…the investigation of difficult things by the method of analysis ought ever to precede the method of composition. This analysis consists of making experiments and observations, and in drawing general conclusions from
them by induction30, and of admitting no objections against the
conclusions but such as are taken from experiments or other certain truths… By this way of analysis we may proceed from… effects to their causes… And the synthesis consists in assuming the causes discovered and established as principles, and by them explaining the phenomena proceeding from them, and proving the explanations (Newton, I., 2003).”
According to Holton (1998), Newton’s crucial addition to the process of analysis was to test propositions (that were assumptions in Descartes process) via observation and experimentation of the principles in practice: “from the phenomena of motion to investigate the forces of nature, and then from these forces (e.g. the postulated
universal gravitation) to demonstrate the other phenomena.” This debate is further
addressed in the next section.
In summary, three principles were presented that are key when investigating a phenomenon through the method of analysis and synthesis: to be aware of previous knowledge related to the phenomenon; to understand what is accomplished (effect); and to identify the existing parts and how they operate (causes). As described there are three routes that can be followed: the analytic route, i.e. seeking causes from given effects; the synthetic route, i.e. seeking effects from given causes; and the Descartes route (as it is called here), i.e. seeking relationships that link causes and effects, either through intuition (as by Descartes) or through experimentation (as per Newton). In the next sections the features related to the analytical and synthetic routes are further discussed.
4.1.3 Analysis
The debate about whether to start with analysis, synthesis or with the primary idea of the dependency between cause and effect is not addressed in this thesis. Here, an assumption is made that the three starting points are possible and we proceed by clarifying the features related to the method of analysis and synthesis.
From the previous description from Pappus as cited in Hintikka and Remes (1974) two types of analysis exist: theoretical and problematical. In theoretical analysis we attempt to establish the truth of theorems (for example, Pythagoras’ theorem a2 + b2 = c2) whereas in problematical analysis we aim at finding something that is unknown (Gaukroger, 1989) - for example, the length of a side of a right triangle. Thus, theoretical analysis deals with truth and falsehood, whereas problematical analysis deals with possible or impossible (Panza, 1997).
In this respect, the method of analysis has its origins in a rationalist science (geometry); that is to say that theoretical analysis is related to (the development of) knowledge a priori (Descartes as cited in Timmermans, 1999) whereas problematical analysis refers to knowledge a posteriori. In other words, problematic analysis generates ‘evidence’ for proposed theoretical truths (proof).
Theoretical analysis involves establishing a series of hypothetical 31queries (axioms) to enable the linking of a more complex theorem to a series of interconnected simpler
31 Hypothesis in the technical sense defined by Aristotle is an assumption of existence (Lee, H. D. P., (1935). Geometrical method and Aristotle's account of first principles. The Classical Quarterly 29(2): 11).
(previously known and easier to solve) theorems. Riemann (1866, as cited in Ritchey, 1991), argues that the hypothesis is used to explain what is accomplished. If the hypotheses are confirmed to be truth we are led to a good understanding of the problems that can be solved. On the other hand, if we find something false, the analysis ends and synthesis is not necessary (Panza, 1997; Timmermans, 1999).
On the other hand, in problematical analysis the ‘desired thing’ is hypothetically considered/assumed to be known (it is given – for instance, in geometry a problem such as “given the sides a and b of a triangle, calculate its hypotenuse” – the geometer will use a2 + b2 = h2) and we can reach either something that can be done (the calculation of the hypotenuse) where synthesis is necessary (i.e. to derive the theorem from the calculation of the hypotenuse) or we find something impossible to construct (for instance, if only the side “a” of the triangle was provided, then the calculation of the hypotenuse would be impossible). In other words, the geometrical figure to be constructed is considered as given and through analysis its parts and their relationships and the principles by means of which the figure can be constructed are found.
In this regards, Timmermans (1999) and Koskela et al. (2009) highlight the qualitative difference of the start and end points of analysis. According to Koskela et al. (2009), the starting point in both theoretical and problematical analysis consists of something unknown as if it was known – a hypothesis or the geometrical figure (even if the geometrical figure is unquestionably given, its constructability is unknown). In both cases, the end point is something known (i.e. truth) or can be done (i.e. possible). In this respect, Wallace (1993), in his investigation about Galileo’s scientific method, argues that for Galileo, the starting point could be something known (in Enlightenment terms - a supposition32) or something “almost” known – a hypothesis.
The series/system of hypotheses here has a relationship with Plato’s paradox of analysis (Meno’s paradox): “Either we know what something is, or we do not. If we do, then there is no point searching for it. If we do not, then we will not know what to search for (Plato (80d). Meno.)” To escape the paradox, Socrates adopts the position that knowledge cannot be sought, but recollected. That means that when admitting something to be known, the known thing will need confirmation.
32 Wallace (1983) uses the term ‘suppositione’ (for supposition) which (even if not stated by Wallace) fulfills largely the JTB criteria of knowledge than hypothesis. The meanings attributed to both terms are
Here, we argue that such an approach escapes Plato’s paradox of knowledge by rejecting the status “known” and “unknown” and adopting the status of “uncertain”. By doing this, the method of analysis becomes a method of validation (or invalidation) of the hypothesis. Thus, it is only after analysis is concluded that the status “known” is granted. In this regard, analysis serves to determine how the unknown quantities of a problem depend on the given (‘known’) ones (Mäenpää, 1997).
Furthermore, the method of analysis has different modes of reasoning. Beaney (2003a) studying the concepts 33 of analysis distinguishes three main modes:
regressive (or directional), decompositional (or configurational) and transformative
(or interpretive) analysis. These modes are further described in the following.
The regressive mode of analysis is described by Beaney (2007) as “the process of
identifying the principles, premises, causes, etc., by means of which something can
be derived or explained”. Working back to principles (axioms) and previously proved
theorems suggests that the regressive conception of analysis reflected in Pappus's account is central (Beaney, M., 2003b). In other words, regressive analysis is the seeking for causes of a given effect and cause of the cause and so on.
Mäenpää (1997) argues that the regressive mode of analysis is related to reduction (as in mathematical reduction)34 and deductive reasoning. According to Mäenpää the method of reduction (apagoge) is a precursor to the method of analysis, and Hintikka and Remes (1974) only consider reduction in their translation. However, for Mäenpää examples of Greek mathematics shows that analysis also considers deduction. As stated by him, “Pappus’s description of synthesis as complementing analysis would be pointless if he regarded analysis as purely reductive, because this would make synthesis trivial and superfluous”.
33 By concepts Beaney means the different meanings given to analysis whereas by modes he means the different approaches.
34 “In mathematics, reduction refers to the rewriting of an expression into a simpler form. For example,
the process of rewriting a fraction into one with the smallest whole-number denominator possible (while keeping the numerator an integer) is called ‘reducing a fraction’ (Wikipedia, 2012 – entry: Reduction (mathematics))”
It follows that Galileo and his peers expanded this concept of regressive analysis. The concept of regressive analysis, as generated in mathematics, could not be seamlessly applied in “scientiae mediae” such as astronomy (Wallace, 1983). The key difference, as highlighted by Wallace was that suppositions could also be based on fictitious suppositions. Wallace also argues that there was confusion (even in some of Galileo’s text) in relation to the ancient and present (at that time) meaning of the words resolution and composition (as referring to analysis and synthesis). For Wallace, “in mathematics a resolution (analysis) is made when one first supposes that a theorem is true or a problem solved, and then on the basis of this supposition deduces consequence after consequence until he arrives at a manifest truth or falsity. If the original theorem is true, he will eventually come to a manifest truth, on the basis that only truth can come from truth if the matter and form of the reasoning