I have a theory, which has not let me down so far, that there is an inverse relationship between imagination and money. Because the more money and technology that is available to [create] a work, the less imagination there will be in it.

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David Hestenes and Garret Sobczyk: Clifford Algebra to Geometric Calculus: A Unified Language for Mathematics, p xii

Klein's seminal analysis of the structure and history of mathematics brings to light two major processes by which mathematics grows and becomes organized... The one emphasizes algebraic structure while the other emphasizes geometric interpretation. Klein's analysis shows one process alternately dominating the other in the historical development of mathematics. But there is no necessary reason that the two processes should operate in mutual exclusion. Indeed, each process is undoubtedly grounded in one of the two great capacities of the human mind: the capacity for language and the capacity for spatial perception. From the psychological point of view, then, the fusion of algebra with geometry is so fundamental that one could well say, **'Geometry without algebra is dumb! Algebra without geometry is blind!'**

Last night I went to a baby shower where a good number of the attendees were babies themselves. I kept thinking how ridiculous it is that people pour so much time and energy into supporting a single life, when there are so many others that need more support.

Interface matters to me more than anything else, and it always has. I just
never realized that. I've spent a lot of time over the years desperately
trying to think of a "thing" to change the world. I now know why the search
was fruitless -- things *don't* change the world. *People* change the world
by using things. The focus must be on the "using", not the "thing". Now
that I'm looking through the right end of the binoculars, I can see a lot
more clearly, and there are projects and possibilities that genuinely
interest me deeply.

Non-conformity is not the adoption of some pre-existing alternative subculture.

It seems like most people ask: "How can I throw my life away in the least unhappy way?"

I got this wild dream in my head about what would help mankind the most, to go off and do something dramatic, and I just happened to get a picture of how, if people started to learn to interact with computers, in collective ways of collaborating together, and this was way back in the early 50s, so it was a little bit premature. So anyways, I had some GI bill money left still so I could just go after that, and up and down quite a bit through the years, and I finally sort of gave up.

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Sir James Lighthill: discussion following The Recently Recognized Failure of Predictability in Newtonian Dynamics

*Q: Do you regard the chaos [within Newtonian mechanics] as immutable, forever remaining inexplicable; and that no new data, no more exact observations or no future theory will ever be able to explain it? I have in mind that the history of science has revealed time and time again a state of affairs where observed phenomena have been seen as irrational, inexplicable and 'chaotic' according to received theory and accepted laws of science but that subsequent refinement of the data and/or new hypotheses, by offering a new explanatory schema, have revealed that a new order lay unperceived within the older chaos....*

A: Perhaps I should make it clear that the results I described are not 'scientific theories'. They are *mathematical* results, based upon rigorous 'proof' in the mathematical sense. They are not capable of alteration therefore.

Admittedly the history of *science* confirms that our understanding of natural laws is constantly being further refined. Newtonian dynamics is itself an illustration of this because we have long recognized it as only an approximation to the true laws of mechanics...

My lecture, however, was about the mathematical properties of *systems assumed to obey exactly* the laws of Newtonian dynamics. The behaviour of such systems had long been thought to be completely predictable but is now known, for a certain proportion of such systems, to be 'chaotic' in a well defined sense.

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Hippolyte Taine: On Intelligence (quoted in Jacques Hadamard: The Psychology of Invention in the Mathematical Field)

You may compare the mind of a man to the stage of a theatre, very narrow at the footlights but constantly broadening as it goes back. At the footlights, there is hardly room for more than one actor. ... As one goes further and further away from the footlights, there are other figures less and less distinct as they are more distant from the lights. And beyond these groups, in the wings and altogether in the background, are innumerable obscure shapes that a sudden call may bring forward and even within direct range of the footlights. Undefined evolutions constantly take place throughout this seething mass of actors of all kinds, to furnish the chorus leaders who in turn, as in a magic lantern picture, pass before our eyes.

[re "cyberpunk"] Once's there's a label for it, it's all over.

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Marc Ettlinger: Quora: What are some English language rules that native speakers don't know, but still follow?

Almost everything we know about our native languages is what's called *implicit knowledge*. Stuff we don't know that we know, or stuff that we can't really describe, but we can do anyway. Like maybe riding a bike, or walking.

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Hermann Grassmann: Die Ausdehnungslehre, 1862 (4,I,II; 10), quoted in Crowe’s History of Vector Analysis, p 89

For I remain completely confident that the labor which I have expended on the science presented here and which has demanded a significant part of my life as well as the most strenuous application of my powers, will not be lost. It is true that I am aware that the form which I have given the science is imperfect and must be imperfect. But I know and feel obliged to state (though I run the risk of seeming arrogant) that even if this work should again remain unused for another seventeen years or even longer, without entering into the actual development of science, still that time will come when it will be brought forth from the dust of oblivion and when ideas now dormant will bring forth fruit. I know that if I also fail to gather around me in a position (which I have up to now desired in vain) a circle of scholars, whom I could fructify with these ideas, and whom I could stimulate to develop and enrich further these ideas, nevertheless there will come a time when these ideas, perhaps in a new form, will arise anew and will enter into living communication with contemporary developments. For truth is eternal and divine, and no phase in the development of truth, however small may be the region encompassed, can pass on without leaving a trace; truth remains, even though the garment in which poor mortals clothe it may fall to dust.

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M Mitchell Waldrop: The Dream Machine: J.C.R. Licklider & the Revolution That Made Computing Personal, p 12

Considering all that happened later, Lick's youthful passion for psychology might seem like an aberration, a sideline, a twenty-five-year-long diversion from his ultimate career in computers. But in fact, his grounding in psychology would prove central to his very conception of computers. Virtually all the other computer pioneers of his generation would come to the field in the 1940s and 1950s with backgrounds in mathematics, physics, or electrical engineering, technological orientations that led them to focus on gadgetry -- on making the machines bigger, faster, and more reliable. Lick was unique in bringing to the field a deep appreciation for human beings: our capacity to perceive, to adapt, to make choices, and to devise completely new ways of tackling apparently intractable problems. As an experimental psychologist, he found these abilities every bit as subtle and as worthy of respect as a computer's ability to execute an algorithm. And that was why to him, the real challenge would always lie in adapting computers to the humans who used them, thereby exploiting the strengths of each.

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Hal Abelson and Andrea diSessa: Turtle Geometry: The Computer as a Medium for Exploring Mathematics

We encourage you not to lose sight of the most important reason for a combined look at turtles and vectors: Turtle geometry and vector geometry are two different representations for geometric phenomena, and whenever we have two different representations of the same thing we can learn a great deal by comparing representations and translating descriptions from one representation into the other. Shifting descriptions back and forth between representations can often lead to insights that are not inherent in either of the representations alone.

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George Lakoff and Rafael Núñez: Where Mathematics Come From: How The Embodied Mind Brings Mathematics Into Being

In recent years, there have been revolutionary advances in cognitive science... Perhaps the most profound of these new insights are the following:

1. *The embodiment of mind.* The detailed nature of our bodies, our brains, and our everyday functioning in the world structures human concepts and human reason. This includes mathematical concepts and mathematical reason.

2. *The cognitive unconscious.* Most thought is unconscious -- not repressed in the Freudian sense but simply inaccessible to direct conscious introspection. We cannot look directly at our conceptual systems and at our low-level thought processes. This includes most mathematical thought.

3. *Metaphorical thought.* For the most part, human beings conceptualize abstract concepts in concrete terms, using ideas and modes of reasoning grounded in the sensory-motor system. The mechanism by which the abstract is comprehended in terms of the concrete is called *conceptual metaphor*.

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George Lakoff and Rafael Núñez: Where Mathematics Come From: How The Embodied Mind Brings Mathematics Into Being

Symbolic logic is not the basis of all rationality, and it is not absolutely true. It is a beautiful metaphorical system, which has some rather bizarre metaphors. It is useful for certain purposes but quite inadequate for characterizing anything like the full range of the mechanisms of human reason...

Mathematics as we know it is *human mathematics*, a product of the human mind, [using] the basic conceptual mechanisms of the embodied human mind as it has evolved in the real world. Mathematics is a product of the neural capacities of our brains, the nature of our bodies, our evolution, our environment, and our long social and cultural history.

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Vincent Toups: Duckspeak Vs Smalltalk: The Decline of the Xerox PARC Philosophy at Apple Computers

HyperCard was, by comparison, much closer to the Dynabook ethos [than the iPad]. In a sense, the iPad *is* the failed "HyperCard Player" brought to corporate fruition, able to run applications but completely unsuited for developing them, both in its basic design (which prioritizes pointing and clicking as the mechanism of interaction), in the conceptual design of its software, and in the social and legal organization of its software distribution system.

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Carver Mead: The Universe and Us: An Integrated Theory of Electromagnetics and Gravitation (18:44)

We had started out [via de Broglie and Schrödinger] having a physical picture of the electron as a wave propagating around the proton... it all made perfect sense intuitively. But then you got some fancy mathematics that made it unnecessary to have the physical picture. And then Bohr argued that "We're above all that now. We don't need physical pictures. We don't need to use intuition." ...

Now don't get me wrong there's nothing wrong with mathematics. But what got propagated was the notion that mathematics had become the guide to physical theory. ...

One of the things that I've developed through my life is an enormous respect for the power of mathematics. I know a lot of mathematicians, they're very bright, and of the things I've come to realize is that you can, if you're good enough, develop a mathematics for any physical theory -- whether it's what nature does or not. So in fact if you say that mathematics is going to guide what physics does, all you're saying is that you've let go of the fact that *what the real world does* should be guiding what your physics is. Because the mathematics can express anything.

That's where we are today. Mathematics took over, and we now have essentially all our physics taught with increasingly sophisticated mathematics and less and less physical insight.