"Mathematics, far from being stymied by this situation, finds enormous value in it. The fecundity of ‘randomness’ is astounding; it is an inexhaustible source of scientific riches. Could ‘randomness’ be such a rich notion because of the inner contradiction that it contains, not despite it? The depth we sense in ‘randomness’ comes from something that lies behind any specific mathematical definition." (William Byers, "How Mathematicians Think", 2007)
"Mathematics is about truth: discovering the truth, knowing the truth, and communicating the truth to others. It would be a great mistake to discuss mathematics without talking about its relation to the truth, for truth is the essence of mathematics. In its search for the purity of truth, mathematics has developed its own language and methodologies - its own way of paring down reality to an inner essence and capturing that essence in subtle patterns of thought. Mathematics is a way of using the mind with the goal of knowing the truth, that is, of obtaining certainty." (William Byers, "How Mathematicians Think", 2007)
"Mathematics provides a good part of the cultural context for the worlds of science and technology. Much of that context lies not only in the explicit mathematics that is used, but also in the assumptions and worldview that mathematics brings along with it." (William Byers, "How Mathematicians Think", 2007)
"The concept of zero is so familiar that it takes a great deal of effort to recapture how mysterious, subtle, and contradictory the idea really is." (William Byers, "How Mathematicians Think", 2007)
"The immediate evidence from the natural world may seem to be chaotic and without any inner regularity, but mathematics reveals that under the surface the world of nature has an unexpected simplicity - an extraordinary beauty and order." (William Byers, "How Mathematicians Think", 2007)
"The infinite more than anything else is what characterizes mathematics and defines its essence. […] To grapple with infinity is one of the bravest and extraordinary endeavors that human beings have ever undertaken." (William Byers, "How Mathematicians Think", 2007)
"A conceptual system is an integrated system of concepts that supports a coherent vision of some aspect of the world. A conceptual system is personal; it is a 'way of seeing', that is, a 'way of knowing'. [...] You cannot do mathematics or science without a conceptual system but such systems are not objective and permanent. They are subject to change and development. Therefore we cannot claim that the reality that we experience and work with in science is independent of the mind of the scientist." (William Byers, "Deep Thinking: What Mathematics Can Teach Us About the Mind", 2015)
"A conceptual system is inevitably associated with a particular way of thinking. Mathematics and science involve different modes of thinking of which deep thinking is the most difficult, radical, and important." (William Byers, "Deep Thinking: What Mathematics Can Teach Us About the Mind", 2015)
"Abstraction is indeed an essential element of concept formation and mathematics is the discipline that has investigated the process of abstraction in greater depth than anywhere else. [...] Mathematics is simultaneously the most abstract and the most concrete of disciplines." (William Byers, "Deep Thinking: What Mathematics Can Teach Us About the Mind", 2015)
"All cultures organize themselves around a story, which tells them how the world came into being - a creation myth." (William Byers, "Deep Thinking: What Mathematics Can Teach Us About the Mind", 2015)
"All scientific theories are incomplete and approximate. They may function well at the centre of the domain that they describe but tend to break down at the boundaries. Paradigms are usually deep insights into some aspect of reality but they never capture reality definitively." (William Byers, "Deep Thinking: What Mathematics Can Teach Us About the Mind", 2015)
"An act of creativity is the result of an insight that arises discontinuously. Of course the insight must be preceded by something that is deeply problematic; it is so deeply problematic that a resolution may well seem impossible. This is why the resolution does not arise through systematic means but only occurs when all systematic approaches have been exhausted to no effect, that is, if you want to be creative you must sometimes be prepared to fly blind. This is not easy to do. Creativity involves living for protracted periods with the kind of tension that arises in situations of cognitive dissonance." (William Byers, "Deep Thinking: What Mathematics Can Teach Us About the Mind", 2015)
"Any simulation, no matter how brilliant in conception, is qualitatively different from what it simulates. Human intelligence and creativity are primary phenomena that are real and immediate, whereas simulations are not real in the same way - they are secondary phenomena. Simulations arise from applying deep thinking to a real situation that exists outside of and prior to the model." (William Byers, "Deep Thinking: What Mathematics Can Teach Us About the Mind", 2015)
"Change is real; the unchanging is an illusion. Classical science focused on systems that were in equilibrium whereas modern science also looks at states that may be far from equilibrium. Equilibrium situations can give you the feeling that things are unchanging but this is always only a temporary condition. Equilibriums beak down and when they do, when the system is far from equilibrium, then the dynamism of the system is most visible. In such situations one is studying change directly." (William Byers, "Deep Thinking: What Mathematics Can Teach Us About the Mind", 2015)
"Creativity does not merely involve the production of novelty. It does not arise from applying the same procedure to a series of different situations. Creativity involves coming to see some situation or phenomenon in a substantially different way - it is not the phenomenon that has changed but rather the manner in which one views the phenomenon." (William Byers, "Deep Thinking: What Mathematics Can Teach Us About the Mind", 2015)
"Facts and concepts only acquire real meaning and significance when viewed through the lens of a conceptual system. [...] Facts do not exist independently of knowledge and understanding for without some conceptual basis one would not know what data to even consider. The very act of choosing implies some knowledge. One could say that data, knowledge, and understanding are different ways of describing the same situation depending on the type of human involvement implied - 'data' means a de-emphasis on the human dimension whereas 'understanding' highlights it." (William Byers, "Deep Thinking: What Mathematics Can Teach Us About the Mind", 2015)
"For every conceptual system there comes a time when the system is overwhelmed by a critical mass of new phenomena and problematic elements that do not fit within the old paradigm." (William Byers, "Deep Thinking: What Mathematics Can Teach Us About the Mind", 2015)
"For knowledge to be useful it must be understood. Whereas knowledge is primarily social, understanding is individual - it is tied to a particular individual." (William Byers, "Deep Thinking: What Mathematics Can Teach Us About the Mind", 2015)
"Functions are ambiguous creatures - they come with multiple representations. [...] These representations break down into two categories: those that see the function as a static object - a graph or list - versus those that see it as a process - the calculator button, the input-output machine." (William Byers, "Deep Thinking: What Mathematics Can Teach Us About the Mind", 2015)
"Learning is a dynamic event and so the belief that learning is primarily about the acquisition of facts is fundamentally flawed - the acquisition and manipulation of data is at best a prerequisite to learning. Real learning involves acquiring knowledge and understanding." (William Byers, "Deep Thinking: What Mathematics Can Teach Us About the Mind", 2015)
"Machines live in a universe of data; but the existence of knowledge necessitates a human presence." (William Byers, "Deep Thinking: What Mathematics Can Teach Us About the Mind", 2015)
"Mathematics courses are hierarchical but every new course begins with the assumption that the student is at the level of conceptual development that would be implied by an optimal understanding of the previous course. Unfortunately many mathematical ideas are so subtle and logically complex that it may take students many years to develop an adequate conceptual understanding. As a result, in practice there is a lot of 'faking it' going on and not merely on the part of the students." (William Byers, "Deep Thinking: What Mathematics Can Teach Us About the Mind", 2015)
"Paradigm change necessarily involves a discontinuous jump. Reality is singular and each paradigm evokes its own reality. This is the reason that scientific paradigms are not changed without a great deal of conflict; the reason why deep thinking is so difficult and involves overcoming so much resistance both in the individual and in the larger culture. In fact it has been said that a scientist never really gives up the paradigm within which she has been trained." (William Byers, "Deep Thinking: What Mathematics Can Teach Us About the Mind", 2015)
"Reality is necessarily viewed through a conceptual system and is inseparable from the system through which it is viewed. But reality is by definition singular - there is only one reality; there cannot be two or three. Something is either real or it is not. The notion that reality is relative or that there can be two competing and inconsistent realities is disorienting and produces untenable cognitive dissonance." (William Byers, "Deep Thinking: What Mathematics Can Teach Us About the Mind", 2015)
"Systems always contain problematic elements in a sense that is usually not clear until the implications of the new system are sufficiently explored. Often problems can be stated in the language of the initial system but can only be resolved by creating a new system. [...] Problems that can be stated in the language of one system often cannot be solved within that system because the solution depends upon the development of a new system." (William Byers, "Deep Thinking: What Mathematics Can Teach Us About the Mind", 2015)
"Teaching mathematics does not involve writing down a set of axioms and deducing their logical consequences. It involves introducing students to a new system of concepts. These concepts are often extremely subtle and deep." (William Byers, "Deep Thinking: What Mathematics Can Teach Us About the Mind", 2015)
"Technological change is discontinuous and difficult. It is a radical change in that it forces people to deal with the world in a different way, that is, it changes the world of experience."
"The best way to think about mathematics is to include not only the content dimension of algorithms, procedures, theorems, and proofs but also the cognitive dimensions of learning, understanding, and creating mathematics." (William Byers, "Deep Thinking: What Mathematics Can Teach Us About the Mind", 2015)
"The introduction of a new concept in mathematics conventionally begins with a definition, which is followed by examples, and then the logical consequences of the definition—properties of the defined object and connections with other mathematical objects or processes. In this sequence the mathematical object [...] is carefully and precisely defined because we regard the concept as identical with its definition." (William Byers, "Deep Thinking: What Mathematics Can Teach Us About the Mind", 2015)
"The moment of insight is the moment in which one grasps the concept or makes the creative leap from one conceptual system to another." (William Byers, "Deep Thinking: What Mathematics Can Teach Us About the Mind", 2015)
"The problem of teaching is the problem of introducing concepts and conceptual systems. In this crucial task the procedures of formal mathematical argument are of little value. The way we reason in formal mathematics is itself a conceptual system - deductive logic - but it is a huge mistake to identify this with mathematics. [...] Mathematics lives in its concepts and conceptual systems, which need to be explicitly addressed in the teaching of mathematics." (William Byers, "Deep Thinking: What Mathematics Can Teach Us About the Mind", 2015)
"The problem with artificial intelligence and information technology is that they promise a methodology that would lead to a way of solving all problems - a self-generating technology that would apply to all situations without the need for new human insights and leaps of creativity." (William Byers, "Deep Thinking: What Mathematics Can Teach Us About the Mind", 2015)
"The true foundations of mathematics do not lie in axioms, definitions, and logical inference, which are the foundational elements of formal mathematics. The true foundations of mathematics lie in the minds of mathematicians as they interact with and try to make sense of their world - in their ideas, their intuitions, and their aesthetic sensibility." (William Byers, "Deep Thinking: What Mathematics Can Teach Us About the Mind", 2015)
"The words 'imaginary' and 'complex' again demonstrate how difficult it is to make a major change in conceptual systems - a difficulty that we already encountered with negative numbers, fractions, zero, and irrational numbers. The word 'imaginary' tells us that these numbers are unreal from the perspective of someone grounded in the real number system." (William Byers, "Deep Thinking: What Mathematics Can Teach Us About the Mind", 2015)
"What is this reality? It is our internal concepts, our mental models, which are real objects with real properties, (italics in original) and which are congruent to each other, which fit together and match." (William Byers, "Deep Thinking: What Mathematics Can Teach Us About the Mind", 2015)
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