throwaway13qf85
Joined 12 years ago
Joined 12 years ago
I assume that what's relevant here will be his work on paraconsistent logics (which allow contradictions) but I think an equally interesting line of work is linear logics. It's of interest to computer scientists because of its relationship to linear type theory (in the same way that the Curry-Howard correspondence links type theory and classical logic) and because it has close relationships with quantum computing.
In particular, you are not allowed to delete or duplicate elements/propositions/types (corresponding to physical processes that cannot arbitrarily create or destroy particles, the "no deletion theorem" and "no cloning theorem" of quantum information theory), so functions of the type
duplicate :: a -> (a, a)
and delete :: a -> ()
are not allowed in a linear type system. Practically, this means that many operations can be optimized by the compiler to in-place mutations, because it is guaranteed that there is only ever one reference to a particular object.Absolutely! There are many poor areas in London (Whitechapel, Brixton or Dalston, for example) but the idea that you would have to avoid them is crazy. In fact there are great reasons to visit them - the best Indian or Pakistani food in the city is in Whitechapel, the music scene in Brixton is amazing, there are great clubs in Dalston if you don't mind the hipsters.
How has SF let things get so bad that there is a major area of the city that people are told to actively avoid because of all the homeless people?
https://en.wikibooks.org/wiki/Write_Yourself_a_Scheme_in_48_...
Hemingway responded by saying, "Poor Faulkner. Does he really think big emotions come from big words? He thinks I don't know the ten-dollar words. I know them all right. But there are older and simpler and better words, and those are the ones I use."
y = x ^ (x ^ x ^ x ^ ...)
= x ^ y
and hence x = y ^ (1/y)
= exp( 1/y * log(y) )
which is defined for y > 0.Graphing this using Google plot [0] or Wolfram Alpha [1], or just differentiating, reveals that it has a maximum at x = e, and that x takes the value sqrt(2) at both y = 2 and y = 4.
Therefore the inversion, which is the original equation
y = x ^ (x ^ x ^ x ^ ...)
is dual-valued, i.e. it is not a mathematical function. For it to be a function, you need to specify whether you are on the upper branch (so that y = 4) or the lower branch (so that y = 2). Deliberately obfuscating the difference between the two branches leads to the conclusion that 2 = 4.The square root function can lead to a similar confusion, since there are two solutions to the equation
y = x^2
and hence the "function" x = sqrt(y)
is not really a function unless we specify whether we are on the upper (x > 0) or lower (x < 0) branch. By convention we interpret sqrt(y) to be the upper branch, and write -sqrt(y) for the lower branch, but there's nothing that forces that choice.Obfuscating the difference between the two branches could lead one to conclude that 1 = -1, although the fallacy is more obvious in this case, since everyone is familiar with the fact that the square root function is dual-valued.
If you replay when a draw occurs, then of course you have a 1/2 chance of winning.
For example, you might always fold low pairs 22-66 initially (and perhaps conspicuously reveal that you folded them) but then start raising with them later. If your opponents were paying attention, then (a) they will think you have a good hand, and may be scared into folding right now, but (b) even if they don't, they won't suspect you of having a low pair, so that if e.g. a 2-6 comes up on the board and gives you three of a kind, you have an advantage because your opponents won't consider that you have this hand.
I'm amazed that GT let him teach that course. It's full of errors, omissions and bad habits. I vaguely followed along, and the most noticeable thing was how much the students complained about the low quality of the course.
The only result of someone implementing the advice in that course is that they'd lose a lot of money quite quickly.
More mathematically, you can think of the Hasse diagram as a lattice, with the meet and join operations being AND and OR. For example, we are used to thinking of the expression
as being true if p is true and q is true. But what if we have four values: {}, {T}, {F} and {T,F}? The Hasse diagram tells us how to interpret expression like this - p AND q is the meet (greatest lower bound) of p and q, and p OR q is the join (least upper bound) of p and q. So for example, The Hasse diagram in the article makes sure that expressions like this agree with our intuition in the cases where we have intuition for what the result should be, and give us a way to interpret expressions consistently when our intuitions fail us.We're used to seeing Hasse diagrams for subsets of a set S used to indicate inclusion, in which case you would have {} < {F}, but you can have a valid poset structure that's not based on inclusion.