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Boolean Logic : George Boole's The Laws of Thought
Ian writes "The Globe and Mail has a piece about the man behind Boolean Logic - George Boole - The Isaac Newton of logic. 'It was 150 years ago that George Boole published his classic The Laws of Thought, in which he outlined concepts that form the underpinnings of the modern high-speed computer.'"
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I think Turing and von Neumann had far more to do with the underpinnings of modern computers than Boole.
Boole's great acheivement was his attempt to formalize logic algebraically at a time when logic was informal and far too meta for even mathematicians to consider formally. While this is great and all, it doesn't result in a general purpose computer.
However, Turing machines and von Neumann machines are in everyway a general purpose computer.
As wonderful as binary is, it falls utterly in capturing the fuzzy analog nature of life and the real world. Our recent debate on whether Sedna (or Pluto) is a planet is but one example of how the real world fails to fit into simple binary categories. Even at the subatomic level, the wave-particle duality gives lie to the fiction of discreteness.
I'm not saying that binary is not great for doing all manner of wonderfully powerful proofs, logic, and computation. I'm only saying that it is a mere approximation to the real world and can thus fail when the real world is does not dichotomize to fit into Boole's logic.
Boolean Logic illustrates both the tremendous power and weakness of mathematical systems. On the one hand the power of proof guarantees that man-made mathematical system with certain axioms will undeniably have certain properties. On the other hand, math gives one no guarantee that the real world obeys those axioms.
Two wrongs don't make a right, but three lefts do.
Ooh! Ooh! Ooh! 01101000 00111001 00110110 00110101 00110110 00110110 01101011 00101110 01101010 01110000 01100111
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Yes, I'm aware than an AC beat me to it, but he's at -1 right now, so I'm posting this because it's more visible.
Some complain that the intruduction of "null" into some systems (such as databases) ruins the simplicity of Boolean logic. It creates a "3-value logic" which can get messy to grok.
I generally agree. I think nulls are perhaps fine for numeric calculations in some cases, such as the average if there are zero records, but not Booleans and not strings. But sometimes it is hard to limit it to one but not the other. It is a controversial topic nevertheless. Chris Date has written some white papers on how to get rid of null.
Table-ized A.I.
Parent post is not completely wrong - I got the first part of the title right. :-) And I blame Dover
No electrons were harmed creating this post, though some may have been subjected to electrical and/or magnetic fields.
But, maybe the real world fits into complex binary categories. For example, suppose I ask you to pick a real number between 0 and 1
Excellent point. But again, I'm not sure that the real world actually obeys the laws of real numbers either. Again, wave-particle duality makes a mess of mathematically notions of pure discrete and pure continuous. Some theories of physics suggest the existence of a quantum mechanical foam at dimensions of about 10^-33 meters. Perhaps the physical world is neither continuous (in the infinite-digit real number sense) nor discrete (in the exactly N-bits binary sense) Perhaps continuous real numbers are a good approximation, but whether real numbers are real (or just a very convenient mathematical construct) is debatable
Similarly, a question asking the color of something (which has finitely many answers) could be reformulated as a sequence of yes/no questions. For example, if the color is in 24-bit format, start with: Is the first bit a 1? and so on.
An interesting example. Yet real-world colors aren't 24-bit, although they can be approximated with a 24-bit color measuring systems. Its a crude approximation, unfortunately. I don't even know of a 24-bit system that has the color gamut of human vision, let alone one that properly measures the hyperspectral reflectance, transflectance, absorption, & flourescence properties of real-world materials. Yes, if you assume a 24-bit approximation, then binary yes/no questions suffice. My point is that one is forced to make a big (sometime right, sometime wrong) assumption in reducing the physical world to any N-bit approximation.
After all, everything you do on a computer, from playing video games to chatting via Instant messaging, ultimately gets reduced to binary form.
So very true.
To me, the deeper issue is whether the real world obeys the mathematicaly axioms of an algebra, Boolean or otherwise. The real world is nonlinear and that throws a wrench in the axioms right there. I also wonder about the axiom of closure -- that interactions of physical quantities in physical systems have consequences outside the algebraic variables of the system.
Again, I'm sure that algebras and real numbers or N-bit numbers are excellent approximations as long as we don't forget that they are only approxmations.
Two wrongs don't make a right, but three lefts do.
Again, I'm sure that algebras and real numbers or N-bit numbers are excellent approximations as long as we don't forget that they are only approxmations.
... the red on white was a different color than the red on black! So, what is our reference point? Is a certain shade of red a fixed frequency of visible radiation? Or is it entirely perceptual?
I believe that approximations are the best we can do. I've been trained as a mathematician. But, I don't believe in the square root of 2 in any physical sense. Some may argue, well construct a square 1 unit on a side, then the diagonal is square root of two. I argue, is it possible to construct a physical square 1 unit to a side? Each side would have to have the same number of atoms in a regular array, otherwise, it's not a square. But even if you accomplish this magnificent feat, the atoms in the lattice are vibrating, so the length isn't constant in time. So, I don't even believe in 1 as a physically measurable number!
Point being, what is our reference point? Color perception has the same problem. I saw an excellent program on PBS (maybe NOVA? it's been several years) on color vision. One experiment was to project a red (that is a fixed frequency of light) circle onto a black background. Then they repeated the experiment with a white background. And I'll be damned
I say again, every measurement is an approximation. Ergo, choose N large enough that no one can practically tell the difference. Then the approximation becomes reality.
I believe that approximations are the best we can do.
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;)
So, I don't even believe in 1 as a physically measurable number!
Cool, I can tell that you and I are on a similar wavelength. Whether that wavelenth is representable as a real number or as a 24-bit color is another matter.
I say again, every measurement is an approximation. Ergo, choose N large enough that no one can practically tell the difference. Then the approximation becomes reality.
And I agree 100% that a large enough N creates a indistinguishably fine approximation with one important exception. If the physical system violates the axioms of the mathematical system used in measurement, then there will be physical states or dynamical behaviors that have no corresponding mathematical state or admissible mathematical transform. Thus, for example, there are physical and perceptual colors for which there are no 24-bit approximations (the gamut problem). Moreover, the inverse problem occurs too. A mathematical system can have states with no corresponding physical state (see the problem of illegal colors)
The extent that the physical and mathematical systems lack a bijective (1-to-1) mapping of both states and admissible transforms is the extent that mathematical reasoning has short-comings. Math is great. As an engineer who has studied math extensively, I can vouch for the power of math to construct axiomatic systems that represent novel physical systems. I can also vouch for the weakness of math in misconstructing those axiomatic analogs of physical systems.
Math is like a chainsaw -- very powerful at cutting into problems, but also very dangerous if one is not careful.
Two wrongs don't make a right, but three lefts do.
Well,Goerge Boole proposed the basic principles of Boolean Algebra in 1854 in his trearise "An investigation of the laws of thought on Which to Found the Mathematical Theories of Logic and Probablities". While admire Goerge Boole, and I certainly give hime credit for creating this branch of Algebra, it should be noted that Goerge Boole himself had nothing to do with computers or digital systems.In the middle of the 19th century, many mathematicians were working on something called "Principles of Logic". Their goal was to descibe the human thought, in pure mathematical format. They aimed to model the human logic, as a branch of science, and they wanted to formulate it and find the principles of human's way of thinking. If you have ever taken a Descrete Mathematic course, you certainly have seen nonsense statements that "If Today is Sunday" AND "if Betty is happy" THEN "The Sky is Red".
This was what those mathematicians were aiming for. Goerge Boole also proposed a set of principles, which at the time no one thought had any practical use. This branc of mathematics was a purely theoric one. Mathematicians mostly abondend this subject after it was proven by experience that the human thought can not be formulated in to some mathematical notations.
It wasn't untill in the 40s, when someone at the Bell Labs (forgot his name) suddenly found out that the Boolean Algebra can be used in digital systems, specifically in implementing digital circuits. Even the first computer built, the ENICA, used a decimal system, and didn't have anything to do with digital systems. It was only by an accident that it was found out that Boolean Algebra, which at the time was a completely useless and theoritic branch of math, found an application, and became a widely studied subject.
What I am trying to say, is that Goerge Boole himself, by no means had any interest in digital systems, in programming, in computers, or in anything even remotely related to electronics. While as I said, I we should all give him immense credit for his work on Boolean Algebra, it should be noted that many people, contributed much more to the computer and electrical science, than Goerge Boole. Charles Babage and Lady Ada were actually writing computer programs in the 19th century; their only problem was that they had no computer at that time! And certainly, the father of today's computer architecture, is von Neuman.
Give credit were credit is due, but over-crediting someone, like saying Goerge Boole invented the foundation of computers, is certainly not correct.
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I found it very interesting at first, but the final parts where really annoying. But the important thing is that after finishing the semester I started that even I've being a programer since 95 and having experience with turbo pascal, javascript, lambdaMOO, php, c, c++ and object pascal, I've just get better at programking because of boole!
Boole is one of those things that looks simple and useless at first, but that is not the truth.