On Similarity (2000-)

"Complexity is the characteristic property of complicated systems we don’t understand immediately. It is the amount of difficulties we face while trying to understand it. In this sense, complexity resides largely in the eye of the beholder - someone who is familiar with s.th. often sees less complexity than someone who is less familiar with it. [...] A complex system is created by evolutionary processes. There are multiple pathways by which a system can evolve. Many complex systems are similar, but each instance of a system is unique." (Jochen Fromm, The Emergence of Complexity, 2004)

"Equifinality is the principle which states that morphology alone cannot be used to reconstruct the mode of origin of a landform on the grounds that identical landforms can be produced by a number of alternative processes, process assemblages or process histories. Different processes may lead to an apparent similarity in the forms produced. For example, sea-level change, tectonic uplift, climatic change, change in source of sediment or water or change in storage may all lead to river incision and a convergence of form." (Olav Slaymaker, "Equifinality", 2004)

"It makes no sense to seek a single best way to represent knowledge - because each particular form of expression also brings its particular limitations. For example, logic-based systems are very precise, but they make it hard to do reasoning with analogies. Similarly, statistical systems are useful for making predictions, but do not serve well to represent the reasons why those predictions are sometimes correct." (Marvin Minsky, "The Emotion Machine: Commonsense Thinking, Artificial Intelligence, and the Future of the Human Mind", 2006)

"Metaphorizing is a manner of thinking, not a property of thinking. It is a capacity of thought, not its quality. It represents a mental operation by which a previously existing entity is described in the characteristics of another one on the basis of some similarity or by reasoning. When we say that something is (like) something else, we have already performed a mental operation. This operation includes elements such as comparison, paralleling and shaping of the new image by ignoring its less satisfactory traits in order that this image obtains an aesthetic value. By this process, for an instant we invent a device, which serves as the pole vault for the comparison’s jump. Once the jump is made the pole vault is removed. This device could be a lightning-speed logical syllogism, or a momentary created term, which successfully merges the traits of the compared objects." (Ivan Mladenov, "Conceptualizing Metaphors: On Charles Peirce’s marginalia", 2006)

"Mathematical ideas like number can only be really 'seen' with the 'eyes of the mind' because that is how one 'sees' ideas. Think of a sheet of music which is important and useful but it is nowhere near as interesting, beautiful or powerful as the music it represents. One can appreciate music without reading the sheet of music. Similarly, mathematical notation and symbols on a blackboard are just like the sheet of music; they are important and useful but they are nowhere near as interesting, beautiful or powerful as the actual mathematics (ideas) they represent." (Fiacre 0 Cairbre, "The Importance of Being Beautiful in Mathematics", IMTA Newsletter 109, 2009)

"The reasoning of the mathematician and that of the scientist are similar to a point. Both make conjectures often prompted by particular observations. Both advance tentative generalizations and look for supporting evidence of their validity. Both consider specific implications of their generalizations and put those implications to the test. Both attempt to understand their generalizations in the sense of finding explanations for them in terms of concepts with which they are already familiar. Both notice fragmentary regularities and - through a process that may include false starts and blind alleys - attempt to put the scattered details together into what appears to be a meaningful whole. At some point, however, the mathematician’s quest and that of the scientist diverge. For scientists, observation is the highest authority, whereas what mathematicians seek ultimately for their conjectures is deductive proof." (Raymond S Nickerson, "Mathematical Reasoning: Patterns, Problems, Conjectures and Proofs", 2009)

"In the telephone system a century ago, messages dispersed across the network in a pattern that mathematicians associate with randomness. But in the last decade, the flow of bits has become statistically more similar to the patterns found in self-organized systems. For one thing, the global network exhibits self-similarity, also known as a fractal pattern. We see this kind of fractal pattern in the way the jagged outline of tree branches look similar no matter whether we look at them up close or far away. Today messages disperse through the global telecommunications system in the fractal pattern of self-organization." (Kevin Kelly, "What Technology Wants", 2010)

"A bell cannot tell time, but it can be moved in just such a way as to say twelve o’clock - similarly, a man cannot calculate infinite numbers, but he can be moved in just such a way as to say pi." (Daniel Tammet, "Thinking in Numbers: How Maths Illuminates Our Lives", 2012)

"The exploding interest in network science during the first decade of the 21st century is rooted in the discovery that despite the obvious diversity of complex systems, the structure and the evolution of the networks behind each system is driven by a common set of fundamental laws and principles. Therefore, notwithstanding the amazing differences in form, size, nature, age, and scope of real networks, most networks are driven by common organizing principles. Once we disregard the nature of the components and the precise nature of the interactions between them, the obtained networks are more similar than different from each other." (Albert-László Barabási, "Network Science", 2016)

"One of the roles of mathematics is to explain phenomena in the world around us, especially phenomena that crop up in many different places. If a similar idea relates to many different situations, mathematics swoops in and tries to find an overarching theory that unifies those situations and enables us to better understand the things they have in common." (Eugenia Cheng, "Beyond Infinity: An Expedition to the Outer Limits of Mathematics", 2017)