tag:blogger.com,1999:blog-8148573551417578681.post4766530380640817286..comments2021-01-15T10:36:20.149-08:00Comments on Dark Buzz: Definition of a functionRogerhttp://www.blogger.com/profile/03474078324293158376noreply@blogger.comBlogger3125tag:blogger.com,1999:blog-8148573551417578681.post-78438225367807631792015-07-17T16:52:08.612-07:002015-07-17T16:52:08.612-07:00Logicians have hard a time getting a concept of th...Logicians have hard a time getting a concept of the continuum because it's a flawed idea, just like set theory, given that it confuses intension and extension (you can't go backwards). Math will never be founded on it because it's a superfluous development that falsely claims to create a foundation for what was developed BEFORE! Real analysis was never put on a sound basis with set theory but ended in paradox and confusion. This is true of Dedekind cuts, Cauchy sequences, uncountable sets and the rest. People are simply ignorant about the history. <br /><br />Real fish, real numbers, real jobs <br />http://link.springer.com/article/10.1007%2FBF03024838<br /><br />Although Roger says otherwise (I tend to agree with most of what he writes), it isn't true that modern mathematics is based on ZFC (it certainly isn't creative within it). Truth is not Rortyian intersubjectivity! This is postmodern nonsense coming from "metamathematicians" that want people to buy into social constructivism and relativism. Philosophers like David Stove made the same charge of the crackpot Popper. Everyone knew, just like Lagrange, that calculus was just algebra with quacky proofs that used "infinity" to justify finite reasoning and derivations. What was unclear was what the math was ultimately talking about. What is a line, afterall? It can't be a continuum and that's why the discrete vs. continuous distinction is a false dichotomy. A continuum is like a round square, an impossible a priori. The argument has nothing to do with Platonism. There are even probability models where zero-probability events comprise infinitely many sample points!<br /><br />The main problem is that the constructivists believed in the self-contradictory completed infinity of the natural numbers and this was just as bogus as the completed infinities of the reals. Solomon Feferman at least put to death the transfinite theory of Cantor (and the diagonalizing of Gödel and Turing) building on Poincare's very obvious notion of impredicativity, which philosophers, like Russell (a womenizing flake), and mathematicians, like Bourbaki (pompous, postmodern French abstractionist) simplify embraced.<br /><br />"In his concluding chapters, Feferman uses tools from the special part of logic called proof theory to explain how the vast part--if not all--of scientifically applicable mathematics can be justified on the basis of purely arithmetical principles. At least to that extent, the question raised in two of the essays of the volume, Is Cantor Necessary?, is answered with a resounding no."<br />http://www.amazon.com/In-Light-Logic-Computation-Philosophy/dp/0195080300<br /><br />Gödel and Turing in 10 minutes seen to be trivial and meaningless<br />https://physics.stackexchange.com/questions/14939/does-g%C3%B6del-preclude-a-workable-toe/14944#14944<br /><br />It's about as arbitrary as the "law" of the excluded middle. The intutionists had points, as well as the constructivists and finitists, like Hilbert (the word "finite" is seriously misleading here). The only people that are serious about logic are the "strict finitists" or "ultrafinists".<br /><br />Jean Paul Van Bendegem has a great article on strict finitism and an entry on finite geometry in the Stanford Encyclopedia of Philosophy<br />http://www.univie.ac.at/constructivism/journal/7/2/141.vanbendegem<br />http://plato.stanford.edu/entries/geometry-finitism/<br /><br />Strict Finitism and the Logic of Mathematical Applications<br />http://www.amazon.com/Finitism-Mathematical-Applications-Synthese-Library/dp/9400713460<br /><br />The debate is over but we are waiting for unprofessional hacks in math departments to accept linear "approximations" in physics.<br />Anonymousnoreply@blogger.comtag:blogger.com,1999:blog-8148573551417578681.post-16931293930242230722015-06-29T23:18:45.857-07:002015-06-29T23:18:45.857-07:00Yes, an operator is just a function from one space...Yes, an operator is just a function from one space of functions to another.<br /><br />A distribution (like Dirac delta) is a more complex object. It can be defined as in the vector space to some function space, or in the completion of a function space with respect to a suitable metric. That is, the DIrac delta is defined by what it does in an integral, and not by the nature of an infinity value at some point.Rogerhttps://www.blogger.com/profile/03474078324293158376noreply@blogger.comtag:blogger.com,1999:blog-8148573551417578681.post-13362900107277880022015-06-29T23:08:05.429-07:002015-06-29T23:08:05.429-07:00Roger,
Could you please touch upon the difference...Roger,<br /><br />Could you please touch upon the difference between a function and an operator? I mean, I know that an operator acts on an input function to give you an output function (which possibly can be the same if it's an identity operator), but, my question here is: why is then an operator not regarded as a function, if the idea of a function indeed is so general (or am I over-reading something in the more general definition of a function that you mention here)? <br /><br />Similarly, what's the formal difference between a function and a distribution? Is there anything of consequence that this difference leads to in practice? As an engineer, I can understand that, perhaps, Dirac's delta cannot be treated as a function. It is defined via a limiting process, and in the limiting process, it not only "hits" infinity, but it also ends up being multi-valued, in a sense. But what precisely is the formal difference? And, why can't at least those simple PDFs, e.g. the Gaussian distribution, be directly treated as functions? Why are they still called distributions?<br /><br />Best,<br /><br />--Ajit<br />[E&OE]<br />Ajit R. Jadhavhttps://ajitjadhav.wordpress.comnoreply@blogger.com