A Quote by John Allen Paulos
The once-surprising existence of non-Euclidean models of Euclid's first four axioms can be seen as a sort of mathematical joke.
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The concept of congruence in Euclidean geometry is not exactly the same as that in non-Euclidean geometry. ..."Congruent" means in Euclidean geometry the same as "determining parallelism," a meaning which it does not have in non-Euclidean geometry.
...the differential element of non-Euclidean spaces is Euclidean. This fact, however, is analogous to the relations between a straight line and a curve, and cannot lead to an epistemological priority of Euclidean geometry, in contrast to the views of certain authors.
If the system exhibits a structure which can be represented by a mathematical equivalent, called a mathematical model, and if the objective can be also so quantified, then some computational method may be evolved for choosing the best schedule of actions among alternatives. Such use of mathematical models is termed mathematical programming.
Blaise Pascal used to mark with charcoal the walls of his playroom, seeking a means of making a circle perfectly round and a triangle whose sides and angle were all equal. He discovered these things for himself and then began to seek the relationship which existed between them. He did not know any mathematical terms and so he made up his own. Using these names he made axioms and finally developed perfect demonstrations, until he had come to the thirty-second proposition of Euclid.
When you're young and you first see the extent and depth of the world's hypocrisy, it's fun to go after it. But by the time you're sixty, it's so commonplace. What's the point in ridiculing people anymore? Their existence itself is a sort of sick joke.
The classical theorists resemble Euclidean geometers in a non-Euclidean world who, discovering that in experience straight lines apparently parallel often meet, rebuke the lines for not keeping straight as the only remedy for the unfortunate collisions which are occurring. Yet, in truth, there is no remedy except to throw over the axiom of parallels and to work out a non-Euclidean geometry.
The constructs of the mathematical mind are at the same time free and necessary. The individual mathematician feels free to define his notions and set up his axioms as he pleases. But the question is will he get his fellow mathematician interested in the constructs of his imagination. We cannot help the feeling that certain mathematical structures which have evolved through the combined efforts of the mathematical community bear the stamp of a necessity not affected by the accidents of their historical birth.
I claim that many patterns of Nature are so irregular and fragmented, that, compared with Euclid - a term used in this work to denote all of standard geometry - Nature exhibits not simply a higher degree but an altogether different level of complexity ... The existence of these patterns challenges us to study these forms that Euclid leaves aside as being "formless," to investigate the morphology of the "amorphous."
I do think that once a horror genre is commonly parodied in other movies it sort kills that genre or that specific take on that genre. Once it sort of becomes a joke in and of itself, so you have to push and find something new.
The only basis for even talking about global warming is the predictions spewed out by computer models. The only quote/unquote "evidence" of global warming is what models are predicting the climate and the weather will be in the next 50 to 100 years. Now, what those models spit out is only as good as the data that's put in, and it's an absolute joke. In terms of science, it's a total joke. There is no warming, global or otherwise!
The development of mathematics toward greater precision has led, as is well known, to the formalization of large tracts of it, so that one can prove any theorem using nothing but a few mechanical rules... One might therefore conjecture that these axioms and rules of inference are sufficient to decide any mathematical question that can at all be formally expressed in these systems. It will be shown below that this is not the case, that on the contrary there are in the two systems mentioned relatively simple problems in the theory of integers that cannot be decided on the basis of the axioms.
Modern economics is a set of formal models and equations purporting to fully determine human behaviour, at least in the economic realm. And there is no way that uncertainty can be compressed into determinate mathematical models.
Mathematics has two faces: it is the rigorous science of Euclid, but it is also something else. Mathematics presented in the Euclidean way appears as a systematic, deductive science; but mathematics in the making appears as an experimental, inductive science. Both aspects are as old as the science of mathematics itself.
You can analyse a joke and say it's funny because this guy thought this was going to happen, and that happened, and it's surprising. But not all surprising things are funny.
There are and can be only two ways of searching into and discovering truth. The one flies from the senses and particulars to the most general axioms, and from these principles, the truth of which it takes for settled and immovable, proceeds to judgment and to the discovery of middle axioms. And this way is now in fashion. The other derives axioms from the senses and particulars, rising by a gradual and unbroken ascent, so that it arrives at the most general axioms last of all. This is the true way, but as yet untried.
It's really surreal when I play shows, I'll have three or four people who are in the front row who are singing every word to my songs. The first time I experienced that I was like,"Are they mocking me? Is this a joke?" But it's not a joke. They actually identify with my music and that is something that I'm getting used to.
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