tag:blogger.com,1999:blog-8148573551417578681.post7651657070192751598..comments2021-01-15T10:36:20.149-08:00Comments on Dark Buzz: Rigorous math v physical heuristicsRogerhttp://www.blogger.com/profile/03474078324293158376noreply@blogger.comBlogger3125tag:blogger.com,1999:blog-8148573551417578681.post-34055371099810098792017-08-10T08:20:18.967-07:002017-08-10T08:20:18.967-07:00Traditional rigour is, in many cases, a kind of li...Traditional rigour is, in many cases, a kind of limiting behaviour with regards to heuristics. Consider a statement of the form 'there exists no positive integers a,b,c such that a^3+b^3=c^3'. Now there being no such integers with 'c < 10^10' will be enough for some (a very 'practical' heuristic). For others 'c < 10^10^10' will be plenty. What the 'traditional rigour' proof of FLT tells us is that we can increase our lower bound on c to get better and better 'heuristics' for as long as we like. As such, traditional rigour is the 'lazy mathematicians' way of doing away with the need for practical heuristics and the need to worry about their accuracy and reliability. But that 'tradtional rigour' comes at a cost in some cases: a 'good enough approximation' (e.g. pi to 100 decimal places in physics) is way easier to sort out than an 'arbitrarily good approximation' as happens when we have the computational p***ing contest to calculate pi to trillions of decimal places).<br /><br />One thing I do like about Zeilberger and others is that they at least take a serious interest in what mathematics is about, and what mathematicians do (as do quite a number of conventional mathematicians), as opposed to many who just 'play with symbols on paper for a living'.Anonymoushttps://www.blogger.com/profile/01383722741806122506noreply@blogger.comtag:blogger.com,1999:blog-8148573551417578681.post-19537204907635097732016-08-30T02:14:29.134-07:002016-08-30T02:14:29.134-07:00Hi Matthew!Hi Matthew!Anonymousnoreply@blogger.comtag:blogger.com,1999:blog-8148573551417578681.post-84882496800790289822014-12-30T18:07:30.626-08:002014-12-30T18:07:30.626-08:00He also has some examples of rejection from egghea...He also has some examples of rejection from egghead journals: http://www.math.rutgers.edu/~zeilberg/Opinion77.html<br /><br />Doron is quite right about the subject and also for being an ultrafinitist. Please note that he engages in a great deal of sarcasm. Someone like Galois would be a better example than Einstein.<br /><br />"Real" Analysis is a Degenerate Case of Discrete Analysis <br />http://www.math.rutgers.edu/~zeilberg/mamarim/mamarimPDF/real.pdf<br /><br />I find the argument that real numbers are sensible in the 21st century as being complete quackery. If you get rid of them, you keep solid math in another form and throw out all of the nonsense paradoxes. The point about proof is that the world has irreducible complexity. Is uncertainty on the order of 1*10^-100 good enough? What happens when proofs are so long that you can pile them as high as the Empire State building? Only small islands of mathematics are really accessible to human proof. The sooner we face that fact the better off we will be. Zeilberger is against the continuum, the potential and actual infinite and the reckless use of universal quantification. I am too!<br /><br />Norman Wildberger makes a similar case in his foundations series on YouTube:<br />https://www.youtube.com/playlist?list=PL5A714C94D40392AB<br /><br />I have understood that mathematicians are not the best logicians and seem to overuse the law of the excluded middle. There is no clear definition of Dedekind cuts, or Cauchy sequences:<br /><br />Dedekind cuts and computational difficulties with real numbers<br />https://www.youtube.com/watch?v=4DNlEq0ZrTo<br /><br />Real numbers as Cauchy sequences don't work! <br />https://www.youtube.com/watch?v=3cI7sFr707s<br /><br />On Cantor's important proofs (W. Mueckenheim)<br />http://arxiv.org/abs/math/0306200<br /><br />Chaitin on Real Numbers:<br />http://arxiv.org/abs/math/0411418<br /><br />(Note: he makes a slight error about halting being a stronger result when it is a weaker version of Godel's first incompleteness theorem because it requires soundness.)<br /><br />Logic of Actual Infinity and G. Cantor's Diagonal Proof of the Uncountability of the Continuum<br />http://projecteuclid.org/euclid.rml/1203431978<br />http://www.researchgate.net/publication/255490016_CANTOR%27S_DIAGONAL_ARGUMENT_A_NEW_ASPECT<br /><br />Solomon Feferman debunks the Cantor cranks:<br />http://www.amazon.com/In-Light-Logic-Computation-Philosophy/dp/0195080300<br /><br />"Cosmologists do not know if the universe is physically infinite in either space or time, or what it means if it is or isnâ€™t."<br /><br />Terry Tao does boring analysis. Another failed savant in an outdated field. How many angels are on the head of that pin again?<br /><br />Although the map is not the territory, I see how the gap between math and physics grows needlessly. People who defend this gap as sensible are just making an argument from narcissism. Get some logic or go home.Anonymousnoreply@blogger.com