I am not talking about manifesting physical objects. I am saying that since math is about precise and absolute concepts, it should be possible to conceptually manifest any given part of a mathematical object. This can easily be done with numbers by the thought process of counting... However, when it cannot be done the object is obviously not well-defined.
No set of natural numbers necessarily exists either. Hence the objection of many mathematicians in the past to the notion of a completed infinity, a notion that naive set theory allows. In constructivist mathematics, countable sets are more something that can be generated in an ongoing process.
Most people's conception of set theory is wrong. You can't have completed infinities and you especially cannot have uncountable sets. Doing so is something that must be taken on faith, and mathematics cannot allow for faith.
Also, mathematics is not necessarily an abstraction on physical reality. Mathematics is simple about pure, concise, and absolute ideas. Hence the problems with ideas that cannot be "thought about absolutely" such as the set of real numbers. Sure you can think about "a set consisting of all sequences of infinite length of digits", but what does such a thing mean exactly? Can you think about, precisely, any given part of such an object?
No, of course you can't. The set is has the property that you can't give a means to list any given part of it... there are things such as Chaiten's Omega... which you assume exist, but you cannot manifest conceptually. So what justification do you have to assume that the set of all real numbers exists?
You are the one that started the ad hominem attacks with the "take some discrete math" comment, as if I was some short-sighted dumbass. How did you know enough to make such a comment to me? So you are a pot and I am a kettle. Get over it... sheesh!
I realize when someone points out that the emperor has no clothes, they are not very popular, but it has to be done.
Current constructivist alternatives are still short of being perfect formulations of mathematics... but this popular naive approach to math has got to go.
No my set contains such numbers as "pi" and "sqrt(2)". Look between those quotes, and what do you see? You see two strings of UNICODE text that describe numbers.
It is only nonsense because you, like religious fundamentalists, refuse to question what you were taught or what your current understanding of things is. You can use a library via interlibrary loan to get these books, though your library may have some of them:
Constructivism in Mathematics: An Introduction is the best book for an introduction to constructive mathematics as it gives an overview of the many different variations of constructive math: intuitionism, finitism, constructive recursive mathematics, etc...
Mathematical Logic is a good inexpensive text book which lightly addresses the aforementioned topics. However, it does not take a rigorous look at constructivist approaches.
L.E.J. Brouwer started the first comprehensive alternative formulation of mathematics on constructive principles. This involves throwing everything away, and starting from scratch. He was able to recover large parts of logic, set theory, and analysis. Not too shabby. His lectures at Cambridge were turned into a textbook, but I think that his PhD student's textbook on the topic is easier to find and more accesible.
Anyway, thats just some quick stuff I could find. None of this is obscure, new, or nonsense like you probably think. You are just close-minded. You have probably never seen non-classical logics, set theories, etc...
You have probably never asked yourself "what is mathematics" or "does it make sense to consider a completed infinity like the set of a reals?" Oh well, if you did and followed through with it you would be a far better mathematician... which everybody no matter their profession should strive for.
But then most people are religious with things like math and science. They refuse to always question... to play the devil's advocate.
And again, after Skolem discovered this, he realized that what we now know as ZFC is flawed, and he turned to more constructivist formulations of mathematics. Its not like the call for constructive mathematics is new, radical, or incoherent. It has and still requires people to realize that what they trusted or had faith in for so long is flawed.
I guess you have never studied mathematical foundations.
The "the set of all non-finite strings of decimal digits" is not well-defined because you cannot write down its constituent elements, let alone explain how I could write down any finite part of said object. So you basically describe a "thing" that you assume exists, but it can't actually be manifested Only its description exists, just as the description of "love" exists. Both do not describe mathematical things.
It is worse than that. All models of ZFC are countable. Hence the semantic inconsistency, i.e. ZFC lacks mathematical meaning. It allows us to prove the existence of objects (without demonstrating them) that do not actually exist.
Why don't you try and study some foundations of mathematics? I have studied "discrete math". The material that you were most likely taught pretends that ZFC set theory is the means by which mathematical objects are defined. However, ZFC is by no means perfect. It has incompleteness, uncomputability, and inconsistency issues.
Kronecker was close, as was Brouwer and other constructivists. Even today we are still not yet there as far as a perfect formulation of math goes... but we are closer than we were before, thanks to such aforementioned great thinkers. Just as there is more than one geometry, logic, set theory, programming language, etc... there is more than one formulation of mathematics. Maybe we will never get the perfect formulation, but we can at least strive for perfection.
As is done in ZFC, I can encode mathematical objects into the formalism, as is done for natural numbers via the null set and inclusion:
0,1,2,... is {}, {{}}, {{}, {{}}},...
Then I can also encode UNICODE strings via Godel numbering in a similar fasion. If you are getting to the point that ZFC won't let me derive a simple contradiction, then you are most likely correct (but we can't prove that is the case, yet another problem with such a formulation of math). Various axiomatic set theories don't necessarily decribe mathematical objects... they are just contrived languages that are engineered in a way so as to not allow the derivation of simple contradictions.
I can formulate other axiomatic systems, which are meaningless. For example, a system that has an axiom which states that every set is uncountably infinite... and no other axioms. In what way is such a thing meaningful? In what way is ZFC meaningful? ZFC was created so that mathematicians could keep doing what they had been doing, regardless of whether or not their previous work was correct.
Anyway, the point that I was trying to make with the example was Skolem's Paradox. Within ZFC you can prove the existence of uncountable sets, while semantically everything is countable. Hence the semantic contradiction, i.e. ZFC is meaningless in a mathematical sense. I do understand that the popular conception of mathematics is a dirty mixture of Platonism, Formalism, and axiomatic systems. Most people don't ask why such an approach or formulation of mathematics is the right way to go... they take it on faith or ignorance or some combination thereof.
Eventually society will see the light... mainstream mathematics will be based on constructive foundations as opposed to foundations which only exist because of legacy.
I realize that you think that I am some closed-minded wacko, but my post was nothing more than a overview of constructive mathematics. In fact, Albert Skolem, one of the creators of ZFC set theory realized the flaws with the classical axiomatic approach to mathematics when he discovered "Skolem's Paradox".
You are wrong that such formulations of mathematics have not been successful. Do a little research into Constructive Recursive Mathematics, for example. You probably haven't heard of it because it was a programme or school of mathematics roughly based on the Church-Turing Thesis as a definition of "math"... but it was developed by Russians... and well... the whole cold war thing kept good math outside of the "free world".
Finally, here is a little tid-bit for you. I define the set of real numbers R to be the set of all numbers than can be described with UNICODE text (a superset of ASCII text). Obviously such a set is countable, and obviously such a set contains the two real numbers you just described. However, using diagonalization, it can be proven that there exists a real number X that is not a member of the set R that I just described.
So your "successful" formulation of mathematics is inconsistent. X is in R, but it is not in R. Ha! Successful indeed!
It does not exist as a mathematical object, just as "love" does not exist as a mathematical object. You can write down words that describe "the set of all real numbers betwen zero and one" just as you can write down words to describe "love".
Exactly! If you want to call things like pi a "real number", then so be it. However, I would think of it as a sequence of digits, which can be written down to any sufficient expansion.
The concept that uncountable sets exist is just silly. The sets are simply not well defined. If you can't define something well enough for it to be calculated, then it is not mathematics. Just as I can describe "love" or "happiness", but I cannot give a formal definition of them... they are not math.
These supposed mathematical objects are claimed to exist because someone came up with a formal axiomatic system which assumes they exist. It is a self-fulfilling prophecy.
The problem is that such assumptions result in foundational or metamathematical problems. Formally you can prove the existence of uncountable sets, but semantically all sets are countable. So within the formal system you have one thing, while outside of the formal system you have another... its a sort of semantic inconsistency.
For example, in ZFC set theory you can easily prove that the set of all functions on natural numbers is uncountably infinite. However, the fact that ZFC is a formal system tells us that we can count every function on natural numbers that can be proven to exist in ZFC. This second part cannot be proven within the system, but it is immediate from the fact that finite strings have a one-to-one correspondence with the naturals. So if we assume that ZFC set theory is a formal language for describing the mathematical concept of sets, then we see that an inconsistency exists between the formalism and the mathematical concepts.
Many people, including mathematicians, only think it is necessary to avoid simple inconsistencies... while allowing semantic inconsistencies.
Others, including some of the pioneers of axiomatic set theory, realized that a more constructive foundation was required for mathematics. There are many variations of constructive mathematics. One such branch roughly states that something is mathematical if and only if it can be computed. So mathematical objects are algorithms. This is an interesting formulation of mathematics because all of math is complete, computable, and consistent.
Formal axiomatic mathematics is flawed. In it only guarantees that you have a system for deriving strings in a formal language. It cannot guarantee that these strings have any mathematical meaning. Hence you can derive meaningless things such as a number that cannot be written down or computed to a sufficient decimal expansion.
Omega is not math, its just words. Math invovles precise, absolute concepts. Omega is nothing ore than a formal gesticulation.
What things, exactly, bug you about Mozilla's look-n-feel? Is it something that a skin could fix, i.e. something that is just "looks" and not "feel"?
What do you do when Microsoft changes its interfaces? They have done it before, they will do it again. They have done it with other apps too... even with their desktop! Are you going to cling to an old version of IE that is riddled with bugs?
I had been using only Windows for 7 years before I switched to 100% Linux. I knew that there would be a culture shock, so I slowly replaced my Windows apps with cross-platform apps like Mozilla and OpenOffice. This was back with Mozilla 1.0, and I remember being annoyed by slight differences in the feel. But now there are very few differences, "feel" wise, between the browsers. Also, the little differences there are, are very superficial.
So my point is that if you were a little more specific, then maybe things could be improved with Mozilla. However, there is little chance that anybody will want to create an IE clone. Skins are one thing, but a full-fledged clone is another.
So if you have usability suggestions for Mozilla, please speak loudly and make them. Otherwise people will just think that you are a koala bear that can only eat eucalyptus leaves.
Modern computers resulted from foundational mathematical questions of consistency, completeness, and calculability that were of great concern 100 years ago... It did not result because of the Turing test. Turing was one of many great mathematicians that played a part in the beginings of modern computer science. Other greats include:
1. Georg Cantor (1845-1918): created set theory and other mathematical things that led to paradoxes. Just as important, he created a certain proof technique known as diagonalization, which would prove to be an extremely important tool many years later.
2. L.E.J. Brouwer (1881-1966): created the programme of mathematical Intuitionism, a formulation of mathematics founded on constructivist ideals as a backlash against "unreliable" forms of mathematical reasoning used at the time such as the excluded middle, completed infinities, etc.
3. David Hilbert (1862-1943): started the programme of metamathematics as a backlash against Brouwer's Intuitionism. Hilbert wanted to keep concepts such as the exluded middle, completed infinities, etc... He laid out a series of problems to be solved in order to show that such concepts were safe to use mathematically.
4. Kurt Gödel (1906-1978): influential because he proved that one of Hilbert's goals was impossible - formal axiomatic mathematics could not be proven to be consistent. His proof used Cantor's diagonalization technique.
5. Alonzo Church (1903-1995): proved that certain supposedly mathematical things could not be calculated (using Cantor's diagonalization proof technique). He also created the first general purpose programming language, the lambda-calculus. At the time it was intended to be a formal mathematical language for describing calculation. However, it was a full fledged computational model. He also answered (via Church's Thesis) the foundational mathematical question of "What is computable"? Finally, he was a teacher of future greats: Turing and Kleene.
6. Alan Turing (1912-1954): Student of Church. He reaffirmed Church's thesis in more humanistic terms by defining a formal abstraction of a human carrying out mathematical calculations by hand with unlimited pencil and paper (i.e. a Turing machine). He reaffirmed Church's thesis by proving that a Turing machine could compute whatever the lambda-calculus could compute, and that the lambda-calculus could compute whatever Turing machines could compute.
So that is the high-level ultra-basic overview of how modern computer science started. It is far more complicated, and many more great mathematicians were involved, but those 6 people were the prominent ones. An interesting thing to note is that these guys most likely didn't set out to create computers as we know them today - proof that even great men can't grasp the effect they will have on the world. If you haven't heard of every one of these guys before, then I suggest this book "The Universal Computer", which is an easy read that shows how mathematicians ended up creating computer science.
Funny, I never realized that Debian required you to install packages by hand. Hell, that whole apt thing, with great tools such as apt-get, aptitude, synaptic... what the hell, lets just install crap by hand.
Slashdot Mirror
I am not talking about manifesting physical objects. I am saying that since math is about precise and absolute concepts, it should be possible to conceptually manifest any given part of a mathematical object. This can easily be done with numbers by the thought process of counting... However, when it cannot be done the object is obviously not well-defined.
No set of natural numbers necessarily exists either. Hence the objection of many mathematicians in the past to the notion of a completed infinity, a notion that naive set theory allows. In constructivist mathematics, countable sets are more something that can be generated in an ongoing process.
Most people's conception of set theory is wrong. You can't have completed infinities and you especially cannot have uncountable sets. Doing so is something that must be taken on faith, and mathematics cannot allow for faith.
Also, mathematics is not necessarily an abstraction on physical reality. Mathematics is simple about pure, concise, and absolute ideas. Hence the problems with ideas that cannot be "thought about absolutely" such as the set of real numbers. Sure you can think about "a set consisting of all sequences of infinite length of digits", but what does such a thing mean exactly? Can you think about, precisely, any given part of such an object?
No, of course you can't. The set is has the property that you can't give a means to list any given part of it... there are things such as Chaiten's Omega... which you assume exist, but you cannot manifest conceptually. So what justification do you have to assume that the set of all real numbers exists?
You are the one that started the ad hominem attacks with the "take some discrete math" comment, as if I was some short-sighted dumbass. How did you know enough to make such a comment to me? So you are a pot and I am a kettle. Get over it... sheesh!
I realize when someone points out that the emperor has no clothes, they are not very popular, but it has to be done.
Current constructivist alternatives are still short of being perfect formulations of mathematics... but this popular naive approach to math has got to go.
If the truth hurts then stay ignorant. Don't blame the messenger.
No my set contains such numbers as "pi" and "sqrt(2)". Look between those quotes, and what do you see? You see two strings of UNICODE text that describe numbers.
It is only nonsense because you, like religious fundamentalists, refuse to question what you were taught or what your current understanding of things is. You can use a library via interlibrary loan to get these books, though your library may have some of them:
Constructivism in Mathematics: An Introduction is the best book for an introduction to constructive mathematics as it gives an overview of the many different variations of constructive math: intuitionism, finitism, constructive recursive mathematics, etc...
From Frege to Godel: A Source Book in Mathematical Logic, 1879-1931 is great source of the original great mathematicians' works who took part in mathematical foundations.
Mathematical Logic is a good inexpensive text book which lightly addresses the aforementioned topics. However, it does not take a rigorous look at constructivist approaches.
L.E.J. Brouwer started the first comprehensive alternative formulation of mathematics on constructive principles. This involves throwing everything away, and starting from scratch. He was able to recover large parts of logic, set theory, and analysis. Not too shabby. His lectures at Cambridge were turned into a textbook, but I think that his PhD student's textbook on the topic is easier to find and more accesible.
Anyway, thats just some quick stuff I could find. None of this is obscure, new, or nonsense like you probably think. You are just close-minded. You have probably never seen non-classical logics, set theories, etc...
You have probably never asked yourself "what is mathematics" or "does it make sense to consider a completed infinity like the set of a reals?" Oh well, if you did and followed through with it you would be a far better mathematician... which everybody no matter their profession should strive for.
But then most people are religious with things like math and science. They refuse to always question... to play the devil's advocate.
And many people are quite happy to 'believe' in various deities, but that doesn't make such concepts mathematical.
And again, after Skolem discovered this, he realized that what we now know as ZFC is flawed, and he turned to more constructivist formulations of mathematics. Its not like the call for constructive mathematics is new, radical, or incoherent. It has and still requires people to realize that what they trusted or had faith in for so long is flawed.
I guess you have never studied mathematical foundations.
The "the set of all non-finite strings of decimal digits" is not well-defined because you cannot write down its constituent elements, let alone explain how I could write down any finite part of said object. So you basically describe a "thing" that you assume exists, but it can't actually be manifested Only its description exists, just as the description of "love" exists. Both do not describe mathematical things.
It is worse than that. All models of ZFC are countable. Hence the semantic inconsistency, i.e. ZFC lacks mathematical meaning. It allows us to prove the existence of objects (without demonstrating them) that do not actually exist.
Why don't you try and study some foundations of mathematics? I have studied "discrete math". The material that you were most likely taught pretends that ZFC set theory is the means by which mathematical objects are defined. However, ZFC is by no means perfect. It has incompleteness, uncomputability, and inconsistency issues.
Kronecker was close, as was Brouwer and other constructivists. Even today we are still not yet there as far as a perfect formulation of math goes... but we are closer than we were before, thanks to such aforementioned great thinkers. Just as there is more than one geometry, logic, set theory, programming language, etc... there is more than one formulation of mathematics. Maybe we will never get the perfect formulation, but we can at least strive for perfection.
As is done in ZFC, I can encode mathematical objects into the formalism, as is done for natural numbers via the null set and inclusion:
0,1,2,...
is
{}, {{}}, {{}, {{}}},...
Then I can also encode UNICODE strings via Godel numbering in a similar fasion. If you are getting to the point that ZFC won't let me derive a simple contradiction, then you are most likely correct (but we can't prove that is the case, yet another problem with such a formulation of math). Various axiomatic set theories don't necessarily decribe mathematical objects... they are just contrived languages that are engineered in a way so as to not allow the derivation of simple contradictions.
I can formulate other axiomatic systems, which are meaningless. For example, a system that has an axiom which states that every set is uncountably infinite... and no other axioms. In what way is such a thing meaningful? In what way is ZFC meaningful? ZFC was created so that mathematicians could keep doing what they had been doing, regardless of whether or not their previous work was correct.
Anyway, the point that I was trying to make with the example was Skolem's Paradox. Within ZFC you can prove the existence of uncountable sets, while semantically everything is countable. Hence the semantic contradiction, i.e. ZFC is meaningless in a mathematical sense. I do understand that the popular conception of mathematics is a dirty mixture of Platonism, Formalism, and axiomatic systems. Most people don't ask why such an approach or formulation of mathematics is the right way to go... they take it on faith or ignorance or some combination thereof.
Eventually society will see the light... mainstream mathematics will be based on constructive foundations as opposed to foundations which only exist because of legacy.
Ok, and please define such a set. It is a contradictory object.
I realize that you think that I am some closed-minded wacko, but my post was nothing more than a overview of constructive mathematics. In fact, Albert Skolem, one of the creators of ZFC set theory realized the flaws with the classical axiomatic approach to mathematics when he discovered "Skolem's Paradox".
You are wrong that such formulations of mathematics have not been successful. Do a little research into Constructive Recursive Mathematics, for example. You probably haven't heard of it because it was a programme or school of mathematics roughly based on the Church-Turing Thesis as a definition of "math"... but it was developed by Russians... and well... the whole cold war thing kept good math outside of the "free world".
Finally, here is a little tid-bit for you. I define the set of real numbers R to be the set of all numbers than can be described with UNICODE text (a superset of ASCII text). Obviously such a set is countable, and obviously such a set contains the two real numbers you just described. However, using diagonalization, it can be proven that there exists a real number X that is not a member of the set R that I just described.
So your "successful" formulation of mathematics is inconsistent. X is in R, but it is not in R. Ha! Successful indeed!
It does not exist as a mathematical object, just as "love" does not exist as a mathematical object. You can write down words that describe "the set of all real numbers betwen zero and one" just as you can write down words to describe "love".
Exactly! If you want to call things like pi a "real number", then so be it. However, I would think of it as a sequence of digits, which can be written down to any sufficient expansion.
The concept that uncountable sets exist is just silly. The sets are simply not well defined. If you can't define something well enough for it to be calculated, then it is not mathematics. Just as I can describe "love" or "happiness", but I cannot give a formal definition of them... they are not math.
These supposed mathematical objects are claimed to exist because someone came up with a formal axiomatic system which assumes they exist. It is a self-fulfilling prophecy.
The problem is that such assumptions result in foundational or metamathematical problems. Formally you can prove the existence of uncountable sets, but semantically all sets are countable. So within the formal system you have one thing, while outside of the formal system you have another... its a sort of semantic inconsistency.
For example, in ZFC set theory you can easily prove that the set of all functions on natural numbers is uncountably infinite. However, the fact that ZFC is a formal system tells us that we can count every function on natural numbers that can be proven to exist in ZFC. This second part cannot be proven within the system, but it is immediate from the fact that finite strings have a one-to-one correspondence with the naturals. So if we assume that ZFC set theory is a formal language for describing the mathematical concept of sets, then we see that an inconsistency exists between the formalism and the mathematical concepts.
Many people, including mathematicians, only think it is necessary to avoid simple inconsistencies... while allowing semantic inconsistencies.
Others, including some of the pioneers of axiomatic set theory, realized that a more constructive foundation was required for mathematics. There are many variations of constructive mathematics. One such branch roughly states that something is mathematical if and only if it can be computed. So mathematical objects are algorithms. This is an interesting formulation of mathematics because all of math is complete, computable, and consistent.
Formal axiomatic mathematics is flawed. In it only guarantees that you have a system for deriving strings in a formal language. It cannot guarantee that these strings have any mathematical meaning. Hence you can derive meaningless things such as a number that cannot be written down or computed to a sufficient decimal expansion.
Omega is not math, its just words. Math invovles precise, absolute concepts. Omega is nothing ore than a formal gesticulation.
We were talking about Mozilla, not Firefox. Firefox is still experimental.
Have you tried the real version of the virus? People claim that the test version gets caught by scanners, while the real version does not get caught.
What things, exactly, bug you about Mozilla's look-n-feel? Is it something that a skin could fix, i.e. something that is just "looks" and not "feel"?
What do you do when Microsoft changes its interfaces? They have done it before, they will do it again. They have done it with other apps too... even with their desktop! Are you going to cling to an old version of IE that is riddled with bugs?
I had been using only Windows for 7 years before I switched to 100% Linux. I knew that there would be a culture shock, so I slowly replaced my Windows apps with cross-platform apps like Mozilla and OpenOffice. This was back with Mozilla 1.0, and I remember being annoyed by slight differences in the feel. But now there are very few differences, "feel" wise, between the browsers. Also, the little differences there are, are very superficial.
So my point is that if you were a little more specific, then maybe things could be improved with Mozilla. However, there is little chance that anybody will want to create an IE clone. Skins are one thing, but a full-fledged clone is another.
So if you have usability suggestions for Mozilla, please speak loudly and make them. Otherwise people will just think that you are a koala bear that can only eat eucalyptus leaves.
Supposedly the virus can be obfuscated so as to circumnavigate scanners... it involves obfuscating Javascript code.
Modern computers resulted from foundational mathematical questions of consistency, completeness, and calculability that were of great concern 100 years ago... It did not result because of the Turing test. Turing was one of many great mathematicians that played a part in the beginings of modern computer science. Other greats include:
1. Georg Cantor (1845-1918): created set theory and other mathematical things that led to paradoxes. Just as important, he created a certain proof technique known as diagonalization, which would prove to be an extremely important tool many years later.
2. L.E.J. Brouwer (1881-1966): created the programme of mathematical Intuitionism, a formulation of mathematics founded on constructivist ideals as a backlash against "unreliable" forms of mathematical reasoning used at the time such as the excluded middle, completed infinities, etc.
3. David Hilbert (1862-1943): started the programme of metamathematics as a backlash against Brouwer's Intuitionism. Hilbert wanted to keep concepts such as the exluded middle, completed infinities, etc... He laid out a series of problems to be solved in order to show that such concepts were safe to use mathematically.
4. Kurt Gödel (1906-1978): influential because he proved that one of Hilbert's goals was impossible - formal axiomatic mathematics could not be proven to be consistent. His proof used Cantor's diagonalization technique.
5. Alonzo Church (1903-1995): proved that certain supposedly mathematical things could not be calculated (using Cantor's diagonalization proof technique). He also created the first general purpose programming language, the lambda-calculus. At the time it was intended to be a formal mathematical language for describing calculation. However, it was a full fledged computational model. He also answered (via Church's Thesis) the foundational mathematical question of "What is computable"? Finally, he was a teacher of future greats: Turing and Kleene.
6. Alan Turing (1912-1954): Student of Church. He reaffirmed Church's thesis in more humanistic terms by defining a formal abstraction of a human carrying out mathematical calculations by hand with unlimited pencil and paper (i.e. a Turing machine). He reaffirmed Church's thesis by proving that a Turing machine could compute whatever the lambda-calculus could compute, and that the lambda-calculus could compute whatever Turing machines could compute.
So that is the high-level ultra-basic overview of how modern computer science started. It is far more complicated, and many more great mathematicians were involved, but those 6 people were the prominent ones. An interesting thing to note is that these guys most likely didn't set out to create computers as we know them today - proof that even great men can't grasp the effect they will have on the world. If you haven't heard of every one of these guys before, then I suggest this book "The Universal Computer", which is an easy read that shows how mathematicians ended up creating computer science.
DOSBOX is great, but for Lucas Arts adventure games, you would be better off using ScummVM, which has also been ported to the XBOX, as far as I know.
Funny, I never realized that Debian required you to install packages by hand. Hell, that whole apt thing, with great tools such as apt-get, aptitude, synaptic... what the hell, lets just install crap by hand.
That was an extremely popular CPU back in the 70s and 80s. The poor man's RISC.