I was happily eating meat one day and I realized that killing sentient things might be the kind of thing society takes in stride but is actually really bad, like slavery used to be. I should do something like a crisis of faith regarding whether or not it's okay.
The question at the heart of the matter is: How do we morally evaluate animal affairs? Here are my intuitions:
1. If you light a cat on fire, that's really bad. If you press a button to instantly vaporize an unsuspecting cow, that's morally neutral. If you step on a snail, no biggie.
2. If you keep a chicken in such a small cage that it can't turn around, that's bad. If you neuter a dog, that's better but still a little bad. If you make sure an un-neutered dog never gets to interact with a bitch (ensuring he can't have sex and puppies), that's morally neutral.
3. If an animal is already dead, the act of eating it is morally neutral. In fact I think the moral neutrality holds even for eating dead humans (although of course that activity will have a different context and there could be all kinds of other negative terms that go into the morality summation).
4. If you pet your dog, that's good because he likes it. (You like it too, which is another positive term in the morality summation. But the dog's enjoyment gets its own terminal value.)
5. If you wirehead an animal, it has the same moral value as any other orgasmium (orgasmium is the simplest configuration of matter which can be sentient and have the subjective experience of happiness, and whatever triggers the happiness sensation is constantly on at full blast). And I think orgasmium's existence is morally neutral.
So when an animal exists, goodness is some function of its happiness that increases while the happiness is within the animal's natural range, and subsequently drops to zero.
The point at which the animal's existence is morally neutral is around "somewhat happy". As you move left from that point, it monotonically decreases without bound. And even while you're within the animal's commonly experienced levels of pain, your trough in the graph is already deeper than the peak is high.
Here is a somewhat counterintuitive application of my tentative animal morality. Imagine there is an Animal Planet which is home to large populations of all the different animals from contemporary Earth in various ecosystems, but with no humans. It would be morally good to instantly vaporize Animal Planet, because putting all the suffering animals out of their misery will surely outweigh the cost of killing the few animals whose happiness is at the top of their natural range (and not higher).
Sunday, February 01, 2009
Sunday, October 19, 2008
Are Semicolons Pretentious?
I just realized it's impossible to use a semicolon when you're writing with a casual tone; it comes off as pretentious. Right?
Wednesday, April 30, 2008
Amazon's New Price Font
Maybe this is just me, but it seems like Amazon.com has managed a huge psychological breakthrough with the slightly altered fonts and styles on their product pages.
Somehow it seems like the larger, skinnier red price letters make it really easy to just click and buy, and even disappointing not to do so.
Somehow it seems like the larger, skinnier red price letters make it really easy to just click and buy, and even disappointing not to do so.
Wednesday, April 16, 2008
Overcoming Bias
Overcoming Bias is my favorite blog. I've always thought of myself as a highly rational person, but after spending about 40 hours reading the series on rationality by Eliezer Yudkowsky, I've realized that the methodology of rationality is a lot more subtle and fascinating than I thought.
It's fair to say this blog has changed my life more than anything else I've read in the last year. Here is one great quote among many:
It is a corruption of curiosity to prefer the question to its answer. Yet people seem to get a tremendous emotional kick out of not knowing something. Worse, they think that the mysteriousness of a mysterious phenomena indicates a special quality of the phenomenon itself, inferring that it is surely different-in-kind from phenomena labeled "understood". If we are ignorant about a phenomenon, that is a fact about our state of mind, not a fact about the phenomenon itself.
Sunday, March 30, 2008
On Undefinable Numbers
If you are asked to define the biggest number you can, and restricted to only using only 1000 characters of English text, then there are only finitely many things you can write, and only finitely many numbers you can define.
Out of all the numbers nameable with 1000 characters of English text, which is the biggest? Surely, it must be huge -- a lot bigger than "a googol to the power of (a googol to the power of a googol (...and so on, nested a googol times))".
Unfortunately, there is no such "biggest number", because there is no well-defined mapping from English phrases to numbers. In Who Can Name the Bigger Number, Scott Aaronson considers:
The problem with English isn’t that it’s unsuitable for a discussion of math. On the contrary, it’s too good at discussing math. The Berry paradox shows that if phrases in a language could all be unambiguously interpreted as numbers, then the language wouldn’t be able to refer to itself with anywhere near as much expressive power as English.
The Berry paradox only applies when one attempts to define big natural numbers using natural language. But there is also a second problem when you try to define a real number using any representation: the finite-length strings you use to represent real numbers aren't able to represent them all.
The problem is that the set of real numbers is uncountably infinite, while the set of finite-length strings is countably infinite. (If you don't know what that means, then read the Infinity lesson notes from X-treme Thinking). Thus, the set of real numbers that you can define is only a countable island in the uncountable ocean of reals.
OK now imagine you're given an infinitely long piece of paper with a real number printed on it. It starts like this: 0.821480865132823066470938446095...
As far as you look, the numbers seem completely random. You don't discern any pattern at all. Let's say you have an eternity to look at this number and try to understand it, but when you're done, you have to communicate which number this is to a mortal living in a finite universe. What do you do?
In the general case, this is impossible, because we know that most real numbers are undefinable. So do you just give up? But wait, the number you were given was actually Pi, except with the first 100 decimal digits taken out. You could have just told that to the mortal!
Every real number has infinitely many decimal digits after the decimal point. And in general, it takes an infinite amount of information to communicate which real number you're talking about. But it would be overkill for me to spend my life trying to say infinitely many 3's as in 0.333333... when I could just use a finite shorthand like "one third" or "zero point three three three and so on". Certain real numbers admit to being identified by finite pieces of information. These numbers include Pi, for example, as well as the number 0.56656565556... whose 2nd, 3rd, 5th, and all other prime-numbered digits after the decimal points are 6's, with the composite-numbered digits all being 5's, and way more elaborate constructions than this.
So what kinds of real numbers can't we define? What does an undefinable real number look like? It looks like a number that you can't say what it looks like. In other words, it looks completely and truly random, more random than it's logically possible for finite creatures to understand.
So not only are we unable to talk about "the biggest whole number nameable with 1,000 characters of English text", we also can't say anything interesting about which real numbers are definable. In other words, the vast majority of real numbers are undefinable, but we can't imagine which ones they are, and we wouldn't know them when we see them!
Does it even make sense for us finite humans to talk about the existence of "undefinable real numbers" and the supposedly "infinite amount of information" that they contain? Are we talking about anything at all? It seems like the "ocean of undefinable reals" is really a make-believe ocean, and the "island of definable reals" is really all that's there to talk about.
Out of all the numbers nameable with 1000 characters of English text, which is the biggest? Surely, it must be huge -- a lot bigger than "a googol to the power of (a googol to the power of a googol (...and so on, nested a googol times))".
Unfortunately, there is no such "biggest number", because there is no well-defined mapping from English phrases to numbers. In Who Can Name the Bigger Number, Scott Aaronson considers:
"One plus the biggest whole number nameable with 1,000 characters of English text."
This number takes at least 1,001 characters to name. Yet we’ve just named it with only 80 characters! Like a snake that swallows itself whole, our colossal number dissolves in a tumult of contradiction. What gives?
The paradox I’ve just described was first published by Bertrand Russell, who attributed it to a librarian named G. G. Berry. The Berry Paradox arises not from mathematics, but from the ambiguity inherent in the English language. There’s no surefire way to convert an English phrase into the number it names (or to decide whether it names a number at all).
The problem with English isn’t that it’s unsuitable for a discussion of math. On the contrary, it’s too good at discussing math. The Berry paradox shows that if phrases in a language could all be unambiguously interpreted as numbers, then the language wouldn’t be able to refer to itself with anywhere near as much expressive power as English.
The Berry paradox only applies when one attempts to define big natural numbers using natural language. But there is also a second problem when you try to define a real number using any representation: the finite-length strings you use to represent real numbers aren't able to represent them all.
The problem is that the set of real numbers is uncountably infinite, while the set of finite-length strings is countably infinite. (If you don't know what that means, then read the Infinity lesson notes from X-treme Thinking). Thus, the set of real numbers that you can define is only a countable island in the uncountable ocean of reals.
OK now imagine you're given an infinitely long piece of paper with a real number printed on it. It starts like this: 0.821480865132823066470938446095...
As far as you look, the numbers seem completely random. You don't discern any pattern at all. Let's say you have an eternity to look at this number and try to understand it, but when you're done, you have to communicate which number this is to a mortal living in a finite universe. What do you do?
In the general case, this is impossible, because we know that most real numbers are undefinable. So do you just give up? But wait, the number you were given was actually Pi, except with the first 100 decimal digits taken out. You could have just told that to the mortal!
Every real number has infinitely many decimal digits after the decimal point. And in general, it takes an infinite amount of information to communicate which real number you're talking about. But it would be overkill for me to spend my life trying to say infinitely many 3's as in 0.333333... when I could just use a finite shorthand like "one third" or "zero point three three three and so on". Certain real numbers admit to being identified by finite pieces of information. These numbers include Pi, for example, as well as the number 0.56656565556... whose 2nd, 3rd, 5th, and all other prime-numbered digits after the decimal points are 6's, with the composite-numbered digits all being 5's, and way more elaborate constructions than this.
So what kinds of real numbers can't we define? What does an undefinable real number look like? It looks like a number that you can't say what it looks like. In other words, it looks completely and truly random, more random than it's logically possible for finite creatures to understand.
So not only are we unable to talk about "the biggest whole number nameable with 1,000 characters of English text", we also can't say anything interesting about which real numbers are definable. In other words, the vast majority of real numbers are undefinable, but we can't imagine which ones they are, and we wouldn't know them when we see them!
Does it even make sense for us finite humans to talk about the existence of "undefinable real numbers" and the supposedly "infinite amount of information" that they contain? Are we talking about anything at all? It seems like the "ocean of undefinable reals" is really a make-believe ocean, and the "island of definable reals" is really all that's there to talk about.
Wednesday, February 20, 2008
My First Musical Composition
I want to be a good composer and pianist someday, instead of a bad composer and an intermediate pianist like I am now.
Here's my first composition, a 30-second piece that I wrote up in an hour on the computer using Finale SongWriter 2007:
Fantasy in C Minor
Here's my first composition, a 30-second piece that I wrote up in an hour on the computer using Finale SongWriter 2007:
Fantasy in C Minor
X-treme Thinking
This semester I teach a 1-unit class at UC Berkeley called X-treme Thinking. Here is the Course Website.
Saturday, January 05, 2008
What Is There in Mathematics?
“What is there?” is an important question in philosophy, as it applies to both the physical world, and the world of ideas. The branch of philosophy that studies this question is called ontology.
Mathematics is characterized by defining and studying various “mathematical objects”, such as sets, numbers, functions, graphs, sequences, polynomials, equations, Turing machines and complexity classes.
But two questions remain:
Even though we only study sets, the surprising thing is that we can still prove things about properties of numbers, functions, graphs, and all the other “mathematical objects” we wanted to study. Sets are so versatile that we can always make some construction out of them with properties that mirror those of a given mathematical object.
Thus, we don’t require a new ontological entity for each mathematical object, because we can simply redefine all our terms about the object and its properties so that they refer to certain sets and their properties.
Since sets are the only mathematical objects, you might still ask the one ontological question about mathematical objects left to pose in modern math: What is a set? But then, what is “what is”? Generally one answers an ontological question with the name of an entity, so any pure mathematical answer to this question must be circular.
So we leave the question of “what is a set” unanswered. Until further notice, you don’t know what sets are, you just define all your mathematical objects in terms of them.
This is all the philosophical underpinning you will need to start thinking about naïve set theory. We should just keep in mind what the source of our naiveté is: since we aren’t defining what a set is, we don’t say how to decide which definitions of sets are valid, and which are not.
But the hole we left in naïve set theory, that we didn’t say how to decide which definitions of sets are valid, allows us to construct a profound paradox (Bertrand Russel’s): Let S be the set of all sets which are not members of themselves. Is S a set? Well, we said “let S be a set”, so that should be enough – you don’t have any grounds to argue that it isn’t. Is S a member of itself? By the law of the excluded middle, you must believe that the answer is either yes or no. By the definition of S, you must also believe the opposite conclusion. But by the law of non-contradiction, you can’t do that. So the constraints of rational thinking make nonsense out of naïve set theory.
When we study axiomatic set theory, we define properties that a set must satisfy, instead of just letting intuition decide which definitions are valid sets and which aren’t. Axiomatizing set theory introduces a stunning array of counterintuitive results, but still enables us to avoid all known paradoxes.
What answer do we give to the question of “what is a set” in axiomatic set theory? Assuming that the axioms of set theory don’t contradict each other, Kurt Gödel’s completeness theorem tells us that axiomatic set theory has a model – meaning, in another meta-theory of sets which is a foundation for the study of axiom systems, there exists a meta-set whose elements satisfy our axioms' conditions of being sets. So we can say that those elements are the sets. But what kind of mathematical object is a model? It’s a set: not a set in our axiomatic set theory, but a set in the set meta-theory that underpins model theory which underpins the original axiomatic set theory.
Then what about meta-sets in the meta-theory of sets? All we can do is construct a model for one axiomatic set theory within another, and add arbitrarily many levels to the hierarchy of meta-sets inside meta-meta-sets.
If we look at any given level of meta-set in this hierarchy and ask what its definition is, there are two possibilities: either the set is an element of a model of an axiomatic set theory, or the question has no mathematical answer, because the set is the ontological foundation of the highest level model theory.
When we answered the question of what sets are in axiomatic set theory, we forced ourselves into a dead end by saying that they were elements of a model of set theory. But we can also give a second answer: sets are the symbols we write on paper as we mechanically apply inference rules to the set axioms (which are also symbols). So this is what we mathematicians do: define everything in terms of sets, define an axiomatic set theory which avoids all known paradoxes, and then field philosophical challenges by pretending to be blind mechanical theorem derivers. Then when the challenger goes away, we go back to abstracting and deriving meta-theorems.
If you want to talk about sets, and you don’t want to be stuck without a definition for sets at the highest level of the model hierarchy, then you have no choice but to take the notion of a set out of the scope of ontological study. You have to believe that every discussion about sets is shorthand for a discussion of symbols which purport to describe the sets. But at the highest level of axiomatic set theory, the one the mathematician writes in, the symbols can’t really be talking about anything.
It isn’t all that surprising that we initially reached an ontological dead end when we asked the question of what is a set. After all, every definition is made up of words, and there are only finitely many words. Thus, any chain of “what is” questions must end, or be answered with a circular definition.
And of course, we are still working our way down such a chain of questions. For what exactly are the “inference rules” and “strings of symbols” with which we confidently work? This is another discussion. But compared to the original question of what a set is, this is a discussion which seems quite alright for a mathematician to leave to a philosopher. A discussion of sets seems to strike much closer to the foundation of mathematics than a discussion of the mechanical execution of rules. Thus, we should be content to proceed with mathematics as usual, while leaving the philosophers to address a topic in the non-mathematical realm of ontological inquiry.
Mathematics is characterized by defining and studying various “mathematical objects”, such as sets, numbers, functions, graphs, sequences, polynomials, equations, Turing machines and complexity classes.
But two questions remain:
- How can a mathematician be sure that the object of conversation is in fact a mathematical object, and as such that the mathematician is justified in using the terminology and methods of dealing with mathematical objects?
- How does one ensure that one's definition of a mathematical object is unambiguous?
Even though we only study sets, the surprising thing is that we can still prove things about properties of numbers, functions, graphs, and all the other “mathematical objects” we wanted to study. Sets are so versatile that we can always make some construction out of them with properties that mirror those of a given mathematical object.
Thus, we don’t require a new ontological entity for each mathematical object, because we can simply redefine all our terms about the object and its properties so that they refer to certain sets and their properties.
Since sets are the only mathematical objects, you might still ask the one ontological question about mathematical objects left to pose in modern math: What is a set? But then, what is “what is”? Generally one answers an ontological question with the name of an entity, so any pure mathematical answer to this question must be circular.
So we leave the question of “what is a set” unanswered. Until further notice, you don’t know what sets are, you just define all your mathematical objects in terms of them.
This is all the philosophical underpinning you will need to start thinking about naïve set theory. We should just keep in mind what the source of our naiveté is: since we aren’t defining what a set is, we don’t say how to decide which definitions of sets are valid, and which are not.
But the hole we left in naïve set theory, that we didn’t say how to decide which definitions of sets are valid, allows us to construct a profound paradox (Bertrand Russel’s): Let S be the set of all sets which are not members of themselves. Is S a set? Well, we said “let S be a set”, so that should be enough – you don’t have any grounds to argue that it isn’t. Is S a member of itself? By the law of the excluded middle, you must believe that the answer is either yes or no. By the definition of S, you must also believe the opposite conclusion. But by the law of non-contradiction, you can’t do that. So the constraints of rational thinking make nonsense out of naïve set theory.
When we study axiomatic set theory, we define properties that a set must satisfy, instead of just letting intuition decide which definitions are valid sets and which aren’t. Axiomatizing set theory introduces a stunning array of counterintuitive results, but still enables us to avoid all known paradoxes.
What answer do we give to the question of “what is a set” in axiomatic set theory? Assuming that the axioms of set theory don’t contradict each other, Kurt Gödel’s completeness theorem tells us that axiomatic set theory has a model – meaning, in another meta-theory of sets which is a foundation for the study of axiom systems, there exists a meta-set whose elements satisfy our axioms' conditions of being sets. So we can say that those elements are the sets. But what kind of mathematical object is a model? It’s a set: not a set in our axiomatic set theory, but a set in the set meta-theory that underpins model theory which underpins the original axiomatic set theory.
Then what about meta-sets in the meta-theory of sets? All we can do is construct a model for one axiomatic set theory within another, and add arbitrarily many levels to the hierarchy of meta-sets inside meta-meta-sets.
If we look at any given level of meta-set in this hierarchy and ask what its definition is, there are two possibilities: either the set is an element of a model of an axiomatic set theory, or the question has no mathematical answer, because the set is the ontological foundation of the highest level model theory.
When we answered the question of what sets are in axiomatic set theory, we forced ourselves into a dead end by saying that they were elements of a model of set theory. But we can also give a second answer: sets are the symbols we write on paper as we mechanically apply inference rules to the set axioms (which are also symbols). So this is what we mathematicians do: define everything in terms of sets, define an axiomatic set theory which avoids all known paradoxes, and then field philosophical challenges by pretending to be blind mechanical theorem derivers. Then when the challenger goes away, we go back to abstracting and deriving meta-theorems.
If you want to talk about sets, and you don’t want to be stuck without a definition for sets at the highest level of the model hierarchy, then you have no choice but to take the notion of a set out of the scope of ontological study. You have to believe that every discussion about sets is shorthand for a discussion of symbols which purport to describe the sets. But at the highest level of axiomatic set theory, the one the mathematician writes in, the symbols can’t really be talking about anything.
It isn’t all that surprising that we initially reached an ontological dead end when we asked the question of what is a set. After all, every definition is made up of words, and there are only finitely many words. Thus, any chain of “what is” questions must end, or be answered with a circular definition.
And of course, we are still working our way down such a chain of questions. For what exactly are the “inference rules” and “strings of symbols” with which we confidently work? This is another discussion. But compared to the original question of what a set is, this is a discussion which seems quite alright for a mathematician to leave to a philosopher. A discussion of sets seems to strike much closer to the foundation of mathematics than a discussion of the mechanical execution of rules. Thus, we should be content to proceed with mathematics as usual, while leaving the philosophers to address a topic in the non-mathematical realm of ontological inquiry.
Saturday, November 10, 2007
Top 5 Most Poorly Executed Dead Baby Jokes
5. What's the difference between a pile of dead babies, and stuff I don't keep in my garage?
4. What's arguably more humane than one dead baby in ten dumpsters?
3. What's funnier than a live adult in a clown costume?
2. How many dead babies was I able to fit in a barrel last night after I killed them?
1. What's the funniest way to prevent human overpopulation while neither killing anyone over the age of 1, nor causing a decrease in births?
4. What's arguably more humane than one dead baby in ten dumpsters?
3. What's funnier than a live adult in a clown costume?
2. How many dead babies was I able to fit in a barrel last night after I killed them?
1. What's the funniest way to prevent human overpopulation while neither killing anyone over the age of 1, nor causing a decrease in births?
Wednesday, April 11, 2007
Tuesday, April 10, 2007
Wednesday, March 21, 2007
Computers Are Fast
What have you done in the last second? Because your computer could have done a billion things.
Do you ever stop to think about how much work a CPU running at 1Ghz is doing? A lifetime isn't enough for a human to consciously do a billion of anything (although the brain's equivalent clock speed is much higher for subconscious tasks like vision processing). The task of calculating Pi to a million digits was beyond the wildest dreams of a millennium of mathematicians. But we can give a computer the task, and it'll come back with the answer in one second!
One nanosecond is too small to ever comprehend, but that is the basic unit of time for a computer's operations. For a computer, one second must be an eternity.
Do you ever stop to think about how much work a CPU running at 1Ghz is doing? A lifetime isn't enough for a human to consciously do a billion of anything (although the brain's equivalent clock speed is much higher for subconscious tasks like vision processing). The task of calculating Pi to a million digits was beyond the wildest dreams of a millennium of mathematicians. But we can give a computer the task, and it'll come back with the answer in one second!
One nanosecond is too small to ever comprehend, but that is the basic unit of time for a computer's operations. For a computer, one second must be an eternity.
Thursday, March 01, 2007
Proving 1+1=2
This is my first philosophy of math post, and I got the ideas from reading Scott Aaronson.
I saw some wiseacre on an internet forum make a reference to this famous page in Russell and Whitehead's Principia Mathematica, and challenging people to prove that 1+1=2. So I thought,
What does it mean to prove that 1+1=2?
Generally when you prove something in formal math, you start by defining axioms, then use logical inference rules to derive the result. If you've learned Geometry, you are probably most familiar with Euclid's axioms for plane geometry.
[Interesting geometry fact: Do you remember learning that when two parallel lines are intersected by a transversal line, pairs of interior angles are congruent? There's no derivation of why that's true -- it's just accepted as an axiom of Euclidean geometry! But it's like the axiomatic black sheep -- much more complicated than Euclid's other four axioms. And it's also what makes the system Euclidean geometry, instead of a different geometry.]
Today, the most common axiom system mathematicians use as a foundation for proofs is called Zermelo-Fraenkel Set Theory. And in ZF set theory, 1+1=2 has a relatively simple proof (although I personally haven't seen it yet)! 1+1 is also easy to prove in a simpler axiom system called Peano Arithmetic.
In the last few centuries, there's been a huge quest by mathematicians to "justify" basic math statements like 1+1=2, in order to "place mathematics on a more secure foundation". That's partly what all this axiom system stuff is about.
Axiom systems are great; in fact, they are an area of concentration for me. But we can make axioms be whatever we want. If you get a kick out of seeing 1+1=3, you can just define a new axiom system where that's the case. You still have to avoid logical contradictions, though, so I hope you weren't too attached to the old semantic meaning of "+" and "=".
So, what is the ultimate secure foundation for 1+1=2? Well, I'm afraid 1+1=2 is part of the foundation itself. Yes, the axioms of set theory prove it, but that's not why anyone is actually inclined to believe that 1+1=2. On the contrary, mathematicians surely listened to their powerful intuition that 1+1=2 when they chose which axioms to accept as part of the "foundation of math".
So where does the 1+1=2 intuition come from? I think you can break it down a little more by seeing 1+1 as the definition of our intuitive notion of two-ness. That intuition, I'm guessing, is a connection we make when the fundamental perceptual notions of repetition or similarity are invoked. Perhaps you are holding a Jolly Rancher, and also a Sweet Tart. They're not identical, but they're both foods, and they're certainly both physical objects with pretty well-defined spatial boundaries. So two is afoot.
I saw some wiseacre on an internet forum make a reference to this famous page in Russell and Whitehead's Principia Mathematica, and challenging people to prove that 1+1=2. So I thought,
What does it mean to prove that 1+1=2?
Generally when you prove something in formal math, you start by defining axioms, then use logical inference rules to derive the result. If you've learned Geometry, you are probably most familiar with Euclid's axioms for plane geometry.
[Interesting geometry fact: Do you remember learning that when two parallel lines are intersected by a transversal line, pairs of interior angles are congruent? There's no derivation of why that's true -- it's just accepted as an axiom of Euclidean geometry! But it's like the axiomatic black sheep -- much more complicated than Euclid's other four axioms. And it's also what makes the system Euclidean geometry, instead of a different geometry.]
Today, the most common axiom system mathematicians use as a foundation for proofs is called Zermelo-Fraenkel Set Theory. And in ZF set theory, 1+1=2 has a relatively simple proof (although I personally haven't seen it yet)! 1+1 is also easy to prove in a simpler axiom system called Peano Arithmetic.
In the last few centuries, there's been a huge quest by mathematicians to "justify" basic math statements like 1+1=2, in order to "place mathematics on a more secure foundation". That's partly what all this axiom system stuff is about.
Axiom systems are great; in fact, they are an area of concentration for me. But we can make axioms be whatever we want. If you get a kick out of seeing 1+1=3, you can just define a new axiom system where that's the case. You still have to avoid logical contradictions, though, so I hope you weren't too attached to the old semantic meaning of "+" and "=".
So, what is the ultimate secure foundation for 1+1=2? Well, I'm afraid 1+1=2 is part of the foundation itself. Yes, the axioms of set theory prove it, but that's not why anyone is actually inclined to believe that 1+1=2. On the contrary, mathematicians surely listened to their powerful intuition that 1+1=2 when they chose which axioms to accept as part of the "foundation of math".
So where does the 1+1=2 intuition come from? I think you can break it down a little more by seeing 1+1 as the definition of our intuitive notion of two-ness. That intuition, I'm guessing, is a connection we make when the fundamental perceptual notions of repetition or similarity are invoked. Perhaps you are holding a Jolly Rancher, and also a Sweet Tart. They're not identical, but they're both foods, and they're certainly both physical objects with pretty well-defined spatial boundaries. So two is afoot.
Tuesday, February 13, 2007
I Am A Strange Loop
28 years after his classic book Godel, Escher, Bach (aka Introduction to Thinking About Awesomeness), author Douglas Hofstadter is releasing a sequel on March 5: I Am a Strange Loop.
If you've forgotten about the GEB topics over the decades since it came out, here is a quiz to refresh your memory:
If you've forgotten about the GEB topics over the decades since it came out, here is a quiz to refresh your memory:
- True or false: I am a strange loop.
- True or false: For every well-formed statement in formal set theory, there exists a proof that it is either true or false.
- True or false: The axioms of formal set theory do not contain a contradiction.
Will Intern For Money
Hey blog readers,
I am looking for a position in the South San Francisco Bay Area as a Summer programming intern. I am an undergraduate Computer Science and Engineering major, Math minor with experience and drive. Please let me know if you have anything to recommend.
Thanks,
--Liron
I am looking for a position in the South San Francisco Bay Area as a Summer programming intern. I am an undergraduate Computer Science and Engineering major, Math minor with experience and drive. Please let me know if you have anything to recommend.
Thanks,
--Liron
Tuesday, January 09, 2007
Secret Questions
Today I created a password for a website, and it asked me for a secret question in case I forget my password. So I typed in "What is the answer?" And then as the answer, I used a throwaway password because it was displayed on the screen in plaintext, and obviously not going to be encrypted. What an awkward and patronizing experience.
Friday, January 05, 2007
Identifying Yourself to Yourself
I just read a quick thought-provoking everything2.com post called creating a password to convince yourself you have traveled back in time. It got me thinking, if I were to travel back in time, how would I authenticate myself to myself as efficiently as possible? (It has to be quick, or else my present self will proceed to press a red button.)
I might start listing off passwords I use, but those could be hacked. I think a good approach would be to list the stuff I concern myself with and worry about. So I was imagining myself rapidly describing my innermost thoughts to myself, and I think I stumbled on a good exercise. Describing my innermost thoughts quickly and matter-of-factly in the second person gave me a bit of perspective.
I might start listing off passwords I use, but those could be hacked. I think a good approach would be to list the stuff I concern myself with and worry about. So I was imagining myself rapidly describing my innermost thoughts to myself, and I think I stumbled on a good exercise. Describing my innermost thoughts quickly and matter-of-factly in the second person gave me a bit of perspective.
Monday, January 01, 2007
All I've Got Against Moderate Religion
The religion that we see today comes in fundamentalist and moderate varieties. I consider the two to be wrong and meaningless, respectively, but that's a subject for another post.
As an atheist, what truly moves me to action is fundamentalism, which I believe is actively destroying society (e.g. preventing stem cell research because of idiotic absolutist classification of a clump of cells as a human in good standing). But the bone I have to pick with moderate religion is much smaller.
Moderate religion is usually pretty fine in practice, because our moral intuitions and enlightened time period generally override a rational interpretation of what's really written in the bible. However, I would still prefer that the moderates give it up already, and here are all the reasons I can think of why moderate religion is bad.
As an atheist, what truly moves me to action is fundamentalism, which I believe is actively destroying society (e.g. preventing stem cell research because of idiotic absolutist classification of a clump of cells as a human in good standing). But the bone I have to pick with moderate religion is much smaller.
Moderate religion is usually pretty fine in practice, because our moral intuitions and enlightened time period generally override a rational interpretation of what's really written in the bible. However, I would still prefer that the moderates give it up already, and here are all the reasons I can think of why moderate religion is bad.
- The prevalence of moderate religion masks the true egregiousness of fundamentalist religion. For example, a moderate Christian doesn't think that Genesis is literally true, but when incredibly radical "Intelligent Design" comes along, which robs modern science of so much integrity that it might as well be asserting the full literal truth of the Genesis story, the average moderate is tempted to give it equal time in the classroom. Richard Dawkins expands on this in The God Delusion.
- Moderate religion seems like an inconsistent worldview that straddles the boundary between proven rationality and comforting superstition. Scientists traditionally use the humble line of having nothing to say about philosophy. But the truth is that science had a lot to say about the subject. For example, every philosophy about the meaning of life that was written before Darwin must be completely re-examined -- the knowledge of our natural origin changes everything.
Likewise, modern biology tells us that our cells, organelles, DNA and proteins are made of the same passive atoms as any other matter in the universe. It seems that the modern, enlightened, scientific worldview is a completely materialistic one. There don't seem to be any gaps for supernatural intervention, even in what was once the most promising place -- our brains.
In light of the changes that science has had on our worldview, moderate religion seems to be nothing but a historical relic. Imagine that a baby were born into a completely secular, modern society, schooled in modern science, but largely ignorant of religion. Can you imagine a priest who finds him at age 20 and tries to convince him that there is a God? Even such a fixture of moderate religion as solitary prayer would seem absurd and outlandish to this rational character. In the world today, I believe moderate religion is using its moderation as an excuse to avoid some major burdens of proof. - My last argument is a very interesting one I read in a recent post by renowned philosopher Daniel Dennett, recovering from surgery after his heart almost exploded, entitled Thank Goodness! (highly recommended).
I am not joking when I say that I have had to forgive my friends who said that they were praying for me. I have resisted the temptation to respond "Thanks, I appreciate it, but did you also sacrifice a goat?" I feel about this the same way I would feel if one of them said "I just paid a voodoo doctor to cast a spell for your health." What a gullible waste of money that could have been spent on more important projects! Don't expect me to be grateful, or even indifferent. I do appreciate the affection and generosity of spirit that motivated you, but wish you had found a more reasonable way of expressing it.
Dennett's point is that while a little prayer by a religious moderate is not a big deal, it is as inappropriate to the situation as sacrificing an animal -- a completely unjustified waste of time. Furthermore, praying in the hope of actually being helpful can seem, to an intelligent sufferer, like a mockery of the things that truly are helpful and valuable (such as the practice of medicine with scientific rigor).
AIM Profile Dump 2
Note: All of this content is plagiarized. If you want to find the sources, just search for the strings in quotes :)
Some professors asked a monk to lecture to them on spiritual matters. The monk ascended a podium, struck it once with his stick, and descended. The academics were dumb-founded. The monk asked them, "Do you understand what I have told you?"
One professor said, "I do not understand."
The monk said, "Then I have concluded my lecture."
Another professor said, "We will not pay you for this lecture."
Two sages were standing on a bridge over a stream.
One said to the other, "I wish I were a fish. They are so happy."
The other replied, "How do you know whether fish are happy or not? You're not a fish."
The first said, "But you're not me, so how do you know whether or not I know how fish feel?"
The other thought for a moment and replied, "Because I was a fish in my previous life."
The first scowled at him. He said, "then you wouldn't mind if I threw you off this bridge, would you?"
At that moment, the first sage attained enlightenment.
He told the other sage what had happened.
"Yeah, I attained enlightenment too," the other sage said. He was lying.
There are two kinds of people in the world: those that can count.
"The truth is, I removed all my evidence of creating this world so that the smarter guys wouldn't believe in me. And what were the consequences? My Heaven only welcomes smart people. Those people will believe in me when they meet me in person! That's what I gave them the brains for, see? The last thing I need is a bunch of loony fanatics wandering around up there, getting their nose hairs all over the furniture."
--God
"I know when you are sleeping, I know when you're awake. I know if you've been bad or good... crap, there I go, confusing myself with Santa Claus again! Seriously, though, I do know." --God
Redundancy is the unnecessary use of either needless, tautological, pleonastic or superfluous text, by which one repeats, in duplication, the same, identical, aforesaid things over and over again, beyond what would be needed or required to explain, or make comprehensible, the intended or signified meaning of that which one wishes to convey. Customarily, it is usually common in redundancy to repeat, sometimes with different phrasing or words, the same idea or reasoning, thus restating one's thoughts, sometimes paraphrasing oneself and effectively saying the same thing twice, or double.
John J. Johnson Jr. II, the current and present president of the Society for Redundancy Society, has proposed that "Redundancy is an art, capable of being captured only by the minds of those with minds capable of capturing the art of redundancy."
All-Time Favorite Russian Reversals
In Firefox, you keep tabs on your browser. In Soviet Russia, browser keeps tabs on you!
Also: In Soviet Russia, fox fires you!
In Soviet Russia, ride pimps you!
In Soviet Russia, time kills you!
In Soviet Russia, day seizes you!
In Soviet Russia, remote controls you!
In Soviet Russia, Waldo finds you!
In Soviet Russia, joke overuses you!
If all the village idiots, in all the villages in the world, left their villages to form their own village, of village idiots, in that village, of village idiots, you would be the village idiot.
How many members of a (given demographic group) does it take to change a lightbulb?
N+1 (where N is a positive whole number) -- one to hold the lightbulb, and N to behave in a fashion generally associated with a negative stereotype of that group.
Some professors asked a monk to lecture to them on spiritual matters. The monk ascended a podium, struck it once with his stick, and descended. The academics were dumb-founded. The monk asked them, "Do you understand what I have told you?"
One professor said, "I do not understand."
The monk said, "Then I have concluded my lecture."
Another professor said, "We will not pay you for this lecture."
Two sages were standing on a bridge over a stream.
One said to the other, "I wish I were a fish. They are so happy."
The other replied, "How do you know whether fish are happy or not? You're not a fish."
The first said, "But you're not me, so how do you know whether or not I know how fish feel?"
The other thought for a moment and replied, "Because I was a fish in my previous life."
The first scowled at him. He said, "then you wouldn't mind if I threw you off this bridge, would you?"
At that moment, the first sage attained enlightenment.
He told the other sage what had happened.
"Yeah, I attained enlightenment too," the other sage said. He was lying.
There are two kinds of people in the world: those that can count.
"The truth is, I removed all my evidence of creating this world so that the smarter guys wouldn't believe in me. And what were the consequences? My Heaven only welcomes smart people. Those people will believe in me when they meet me in person! That's what I gave them the brains for, see? The last thing I need is a bunch of loony fanatics wandering around up there, getting their nose hairs all over the furniture."
--God
"I know when you are sleeping, I know when you're awake. I know if you've been bad or good... crap, there I go, confusing myself with Santa Claus again! Seriously, though, I do know." --God
Redundancy is the unnecessary use of either needless, tautological, pleonastic or superfluous text, by which one repeats, in duplication, the same, identical, aforesaid things over and over again, beyond what would be needed or required to explain, or make comprehensible, the intended or signified meaning of that which one wishes to convey. Customarily, it is usually common in redundancy to repeat, sometimes with different phrasing or words, the same idea or reasoning, thus restating one's thoughts, sometimes paraphrasing oneself and effectively saying the same thing twice, or double.
John J. Johnson Jr. II, the current and present president of the Society for Redundancy Society, has proposed that "Redundancy is an art, capable of being captured only by the minds of those with minds capable of capturing the art of redundancy."
All-Time Favorite Russian Reversals
In Firefox, you keep tabs on your browser. In Soviet Russia, browser keeps tabs on you!
Also: In Soviet Russia, fox fires you!
In Soviet Russia, ride pimps you!
In Soviet Russia, time kills you!
In Soviet Russia, day seizes you!
In Soviet Russia, remote controls you!
In Soviet Russia, Waldo finds you!
In Soviet Russia, joke overuses you!
If all the village idiots, in all the villages in the world, left their villages to form their own village, of village idiots, in that village, of village idiots, you would be the village idiot.
How many members of a (given demographic group) does it take to change a lightbulb?
N+1 (where N is a positive whole number) -- one to hold the lightbulb, and N to behave in a fashion generally associated with a negative stereotype of that group.
Sunday, November 12, 2006
Smart Voting Website
Last Tuesday, I voted in the midterm elections. I did it by filling out an absentee ballot in front of a computer at the library. I thought many of my votes were well informed, but frankly, my votes for the less prominent positions were not very well informed at all (I briefly consulted a few web pages), and for some votes I didn't feel informed enough to make any decision at all. This state of affairs is just unacceptable for this country in this age.
Today I got an idea from this post by Scott Adams. Scott says he doesn't vote because he doesn't know enough to make an informed decision. In the future, he thinks there will be a website where people can go to get informed to a sufficient degree, and when that time comes, he looks forward to voting.
I would like to develop a website with a very specific goal. For each election ballot item, there should be a single dynamic page that gives people the info they need to make a decision. It should allow people to quickly pinpoint the differences between the candidates that are relevant to their values. I don't know what that will look like yet, and I will appreciate any ideas other people suggest.
The site should be informative, smart, and unbiased the same way that Wikipedia is -- by fairly presenting input from lots of users. Maybe each voter can enter what their values are, and then the page will be modified to point out differences in stances that similar voters thought were important. One idea for ballot proposition issues is to have polls whose results are weighted by voters' scores on a short knowledge test (basic facts about the propositions).
By the way, I made my first web application last year -- a proof of concept for a new type of social networking site. A demo of RelationCraft is currently online. But I hope my politics site, or a future idea, will be a more practical candidate for wide release.
Today I got an idea from this post by Scott Adams. Scott says he doesn't vote because he doesn't know enough to make an informed decision. In the future, he thinks there will be a website where people can go to get informed to a sufficient degree, and when that time comes, he looks forward to voting.
I would like to develop a website with a very specific goal. For each election ballot item, there should be a single dynamic page that gives people the info they need to make a decision. It should allow people to quickly pinpoint the differences between the candidates that are relevant to their values. I don't know what that will look like yet, and I will appreciate any ideas other people suggest.
The site should be informative, smart, and unbiased the same way that Wikipedia is -- by fairly presenting input from lots of users. Maybe each voter can enter what their values are, and then the page will be modified to point out differences in stances that similar voters thought were important. One idea for ballot proposition issues is to have polls whose results are weighted by voters' scores on a short knowledge test (basic facts about the propositions).
By the way, I made my first web application last year -- a proof of concept for a new type of social networking site. A demo of RelationCraft is currently online. But I hope my politics site, or a future idea, will be a more practical candidate for wide release.
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