A remarkable aspect of art of the present century is the range of concepts and ideologies which it embodies. It is almost tempting to see a pattern emerging within the art field — or alternatively under an a posteriori umbrella — similar to that which exists under the umbrella of science where the general term covers a whole range of separate, though interconnecting, activities. Any parallelism is, however, in this instance at least — misleading. A scientific discipline develops systematically once its bare tenets have been established, named and categorized as conventions. Many of the concepts of modern art, by contrast, have resulted from the almost accidental meetings of groups of talented individuals at certain times and certain places. The ideas generated by these chance meetings had twofold consequences. Firstly, a corpus of work would be produced which, in great part, remains as a concrete record of the events. Secondly, the ideas would themselves be disseminated through many different channels of communication — seeds that often bore fruit in contexts far removed from their generation. Not all movements were exclusively concerned with innovation. Surrealism, for instance, claimed to embody a kind of insight which can be present in the art of any period. This claim has been generally accepted so that a sixteenth century painting by Spranger or a mysterious photograph by Atget can legitimately be discussed in surrealist terms. Briefly, then, the concepts of modern art are of many different (often fundamentally different) kinds and resulted from the exposures of painters, sculptors and thinkers to the more complex phenomena of the twentieth century, including our ever increasing knowledge of the thought and products of earlier centuries. Different groups of artists would collaborate in trying to make sense of a rapidly changing world of visual and spiritual experience. We should hardly be surprised if no one group succeeded completely, but achievements, though relative, have been considerable. Landmarks have been established — concrete statements of position which give a pattern to a situation which could easily have degenerated into total chaos. Beyond this, new language tools have been created for those who follow — semantic systems which can provide a springboard for further explorations.
The codifying of art is often criticized. Certainly one can understand that artists are wary of being pigeon-holed since they are apt to think of themselves as individuals — sometimes with good reason. The notion of self-expression, however, no longer carries quite the weight it once did; objectivity has its defenders. There is good reason to accept the ideas codified by artists and critics, over the past sixty years or so, as having attained the status of independent existence — an independence which is not without its own value. The time factor is important here. As an art movement slips into temporal perspective, it ceases to be a living organism — becoming, rather, a fossil. This is not to say that it becomes useless or uninteresting. Just as a scientist can reconstruct the life of a prehistoric environment from the messages codified into the structure of a fossil, so can an artist decipher whole webs of intellectual and creative possibility from the recorded structure of a “dead” art movement. The artist can match the creative patterns crystallized into this structure against the potentials and possibilities of his own time. As T.S. Eliot observed, no one starts anything from scratch; however consciously you may try to live in the present, you are still involved with a nexus of behaviour patterns bequeathed from the past. The original and creative person is not someone who ignores these patterns, but someone who is able to translate and develop them so that they conform more exactly to his — and our — present needs.


When people who are talking don’t share the same culture, knowledge, values, and assumptions, mutual understanding can be especially difficult. Such understanding is possible through the negotiation of meaning. To negotiate meaning with someone, you have to become aware of and respect both the differences in your backgrounds and when these differences are important. You need enough diversity of cultural and personal experience to be aware that divergent world views exist and what they might be like. You also need the flexibility in world view, and a generous tolerance for mistakes, as well as a talent for finding the right metaphor to communicate the relevant parts of unshared experiences or to highlight the shared experiences while demphasizing the others. Metaphorical imagination is a crucial skill in creating rapport and in communicating the nature of unshared experience. This skill consists, in large measure, of the ability to bend your world view and adjust the way you categorize your experiences. Problems of mutual understanding are not exotic; they arise in all extended conversations where understanding is important.
When it really counts, meaning is almost never communicated according to the CONDUIT metaphor, that is, where one person transmits a fixed, clear proposition to another by means of expressions in a common language, where both parties have all the relevant common knowledge, assumptions, values, etc. When the chips are down, meaning is negotiated: you slowly figure out what you have in common, what it is safe to talk about, how you can communicate unshared experience or create a shared vision. With enough flexibility in bending your world view and with luck and charity, you may achieve some mutual understanding.
Communication theories based on the CONDUIT metaphor turn from the pathetic to the evil when they are applied indiscriminately on a large scale, say, in government surveillance or computerized files. There, what is most crucial for real understanding is almost never included, and it is assumed that the words in the file have meaning in themselves—disembodied, objective, understandable meaning. When a society lives by the CONDUITmetaphor on a large scale, misunderstanding, persecution, and much worse are the likely products.
Later, I realized that reviewing the history of nuclear physics served another purpose as well: It gave the lie to the naive belief that the physicists could have come together when nuclear fission was discovered (in Nazi Germany!) and agreed to keep the discovery a secret, thereby sparing humanity such a burden. No. Given the development of nuclear physics up to 1938, development that physicists throughout the world pursued in all innocence of any intention of finding the engine of a new weapon of mass destruction—only one of them, the remarkable Hungarian physicist Leo Szilard, took that possibility seriously—the discovery of nuclear fission was inevitable. To stop it, you would have had to stop physics. If German scientists hadn’t made the discovery when they did, French, American, Russian, Italian, or Danish scientists would have done so, almost certainly within days or weeks. They were all working at the same cutting edge, trying to understand the strange results of a simple experiment bombarding uranium with neutrons. Here was no Faustian bargain, as movie directors and other naifs still find it intellectually challenging to imagine. Here was no evil machinery that the noble scientists might hide from the problems and the generals. To the contrary, there was a high insight into how the world works, an energetic reaction, older than the earth, that science had finally devised the instruments and arrangements to coart forth. “Make it seem inevitable,” Louis Pasteur used to advise his students when they prepared to write up their discoveries. But it was. To wish that it might have been ignored or suppressed is barbarous. “Knowledge,” Niels Bohr once noted, “is itself the basis for civilization.” You cannot have the one without the other; the one depends upon the other. Nor can you have only benevolent knowledge; the scientific method doesn’t filter for benevolence. Knowledge has consequences, not always intended, not always comfortable, but always welcome. The earth revolves around the sun, not the sun around the earth. “It is a profound and necessary truth,” Robert Oppenheimer would say, “that the deep things in science are not found because they are useful; they are found because it was possible to find them.”
...Bohr proposed once that the goal of science is not universal truth. Rather, he argued, the modest but relentless goal of science is “the gradual removal of prejudices.” The discovery that the earth revolves around the sun has gradually removed the prejudice that the earth is the center of the universe. The discovery of microbes is gradually removing the prejudice that disease is a punishment from God. The discovery of evolution is gradually removing the prejudice that Homo sapiens is a separate and special creation.
For any natural number $k$, let $a_k = 3^k$. The smallest natural number $m$ for which \[ (a_1)^1 \times (a_2)^2 \times \dots \times (a_{20})^{20} \;<\; a_{21} \times a_{22} \times \dots \times a_{20+m} \] is: