Logic Quotes …

Logic will get you from A to B. Imagination will take you everywhere. Albert Einstein Contrariwise, if it was so, it might be; and if it were so, it would be; but as it isn’t, it ain’t. That’s logic. Lewis Carroll Logic! Good gracious! What rubbish! E. M. Forster Against logic there is no armor like ignorance. Laurence J. Peter Logic teaches rules for presentation, not thinking. Mason...

Zooming the Zone

We all know that the best way to get something done is to write down a list of what to do. Well, here for you need of course some idea of what it is you are doing and that usually implies a little of analysing or broadly spoken “cutting into pieces”. In doing so it is sometimes necessary to keep cutting the things again and again until they are small enough to be digested. Parallels to actual eating food? Yes, naturally we wouldn’t eat the whole salmon in one go – even if it smells delicious and we can’t wait to have it down our throats. We would separate it into smaller pieces and then start pleasing our taste buds … but what happens when our cutting results in pieces which have the same behavior or structure, lets say pattern as one of the previously found? Deadlock? Frustration? Maybe initially some excitement of recognizing something and some pleasure of presumptuous “Oh, I know that!”. but that will die away quickly when we realize that we actually didn’t understand anything. Point of depression is reached quickly. However, no need to be disappointed. Just change the zoom from in to out and look at it from the whole. Embrace the thing. Let it simmer in your brain and find its own way. Remember the fractals and Mandelbrot set pictures? How certain pattern popped up again and again and the recursiveness of the whole thing became more and more obvious to that extent that we actually do understand now what this is about. At least some of us. Almost not necessary as...

Hilbert’s program

Never forget that even the most solid buildings of thought are supported by sand only … In mathematics, Hilbert’s program, formulated by German mathematician David Hilbert, was a proposed solution to the foundational crisis of mathematics, when early attempts to clarify the foundations of mathematics were found to suffer from paradoxes and inconsistencies. As a solution, Hilbert proposed to ground all existing theories to a finite, complete set of axioms, and provide a proof that these axioms were consistent. Hilbert proposed that the consistency of more complicated systems, such as real analysis, could be proven in terms of simpler systems. Ultimately, the consistency of all of mathematics could be reduced to basic arithmetic. However, some argue that Gödel’s incompleteness theorems showed in 1931 that Hilbert’s program was unattainable. In his first theorem, Gödel showed that any consistent system with a computable set of axioms which is capable of expressing arithmetic can never be complete: it is possible to construct a statement that can be shown to be true, but that cannot be derived from the formal rules of the system. In his second theorem, he showed that such a system could not prove its own consistency, so it certainly cannot be used to prove the consistency of anything stronger with certainty. This refuted Hilbert’s assumption that a finitistic system could be used to prove the consistency of itself, and therefore anything else. >> Go to Source...
John Venn and the Diagrams

John Venn and the Diagrams

Venn diagrams were introduced in 1880 by John Venn (1834–1923) in a paper entitled On the Diagrammatic and Mechanical Representation of Propositions and Reasonings in the “Philosophical Magazine and Journal of Science”, about the different ways to represent propositions by diagrams.[1] The use of these types of diagrams in formal logic, according to Ruskey and M. Weston, is “not an easy history to trace, but it is certain that the diagrams that are popularly associated with Venn, in fact, originated much earlier. They are rightly associated with Venn, however, because he comprehensively surveyed and formalized their usage, and was the first to generalize them”.[2] Venn himself did not use the term “Venn diagram” and referred to his invention as “Eulerian Circles.”[1] For example, in the opening sentence of his 1880 article Venn writes, “Schemes of diagrammatic representation have been so familiarly introduced into logical treatises during the last century or so, that many readers, even those who have made no professional study of logic, may be supposed to be acquainted with the general nature and object of such devices. Of these schemes one only, viz. that commonly called ‘Eulerian circles,’ has met with any general acceptance…”[3] The first to use the term “Venn diagram” was Clarence Irving Lewis in 1918, in his book “A Survey of Symbolic Logic”.[2] Venn diagrams are very similar to Euler diagrams, which were invented by Leonhard Euler (1708–1783) in the 18th century.[note 1] M. E. Baron has noted that Leibniz (1646–1716) in the 17th century produced similar diagrams before Euler, but much of it was unpublished. She also observes even earlier Euler-like diagrams by...

Barber paradox

Suppose there is a town with just one barber, who is male. In this town, every man keeps himself clean-shaven, and he does so by doing exactly one of two things: shaving himself; or going to the barber. Another way to state this is that “The barber is a man in town who shaves all those, and only those, men in town who do not shave themselves.” From this, asking the question “Who shaves the barber?” results in a paradox because according to the statement above, he can either shave himself, or go to the barber (which happens to be himself). However, neither of these possibilities is valid: they both result in the barber shaving himself, but he cannot do this because he only shaves those men “who do not shave themselves”. >> Go to Source...