So I don't remember why I was trying to explain WOP (the integer one, not the set theory one) to Jake.
Ah! Yes I do. I was going to explain it so I could point out how people often try to apply it where it's not applicable. (IIRC, I thought of talking about this, decided it wasn't really relevant, then remembered Putnin's assertion that the rational numbers don't exist and started laughing, so I had to explain.)
Maybe I should go back a bit further - Jake is a first-year philosophy major who likes to start these philosophical discussions. And he asked me about Zeno's paradox, and I gave him the simple "the sum goes to one" explanation, and then said how I thought that personally I didn't think that by itself was a very satisfying explanation, and, with a lot of fumbling, tried to explain how I thought the basic problem was that Zeno simply doesn't understand how a continuum works, and was trying to apply things inappropriate to a continuum to one.
So the thing is apparently Jake has never taken calculus. Like, nothing beyond high school math. So this gets me to trying to explain to him countability and uncountability, because I had invoked this earlier - "the length of the individual points is 0, but that doesn't mean the length of the interval is 0 because there's uncountably many".
He doesn't really understand what I'm talking about, understandably; I probably should have gotten some paper. I don't actually get as far as proving that there are uncountable sets, because before that I break off to talk about what these "sizes" actually are - I say that aleph_0 is just a name, don't worry about what it actually *is*, it's just the name we give to the size of the integers - so this leads me to explaining about equivalence relations and equivalence classes, or as I like to explain it, "sledgehammer abstraction".
Well, turns out Jake doesn't much like sledgehammer abstraction. I try to explain it as best I can - "we want something that represents all things even, what better than the set of all things even itself?" ...but, I don't know. He's unsatisfied. What exactly does he expect? Something like Plato? There are all sorts of problems with that. Well, let's leave that alone for now.
So he starts saying about how he's not certain that the infinites I'm talking about is his idea of infinity. And I should tell him, you should probably get rid of any unified idea of infinity. First of all, what's even meant by "infinite" varies - there's no more unified idea of "infinity" in math than there is of "number" (I didn't go into that, though, I have to wonder what his reaction would have been) - well, actually, what they all seem to have in common is that they're greater than all whole numbers - but that basically, some things are finite and some things are infinite, and there's no one "infinity". Things are not inherently very similar just because they're infinite. We talk about negative numbers, not "negativity". If you want you can have a unified idea of infinity that is simply, all things infinite (good old sledgehammer abstraction!) but that won't really be very useful. He still seems to want some unified infinity... I don't know...
Now, everything up to this point is, I would say, just a non-math person having incorrect expectations of how math works. But what happened next was just utterly ridiculous.
This was where I somehow got to trying to explain WOP to him. So I say, the fundamental principle of the whole numbers is that every nonempty set of whole numbers has a least element. He doesn't see this as immediately intuitive, so I give some examples. The smallest whole number is 0; the smallest prime is 2. But it doesn't matter what set of whole numbers it is - as long as there is some whole number which can be written as the sum of 2 cubes in 2 different ways, there is necessarily some smallest, some first, whole number which can be written as the sum of 2 cubes in 2 different ways.
And he said, What about single-element sets? What? Of course those have a least element, they're only element. But it's not less than itself. It doesn't have to be; that's ridiculous! Least element means less than or equal to everything, or less than everything else; that's like saying 2 isn't the least prime because it's not less than itself! And I go on to say, this is not merely the mathematical definition of "least", but the common definition as well. If there were only one movie ever made, it would be the best movie ever made. No, he says, I don't think it would be. What the hell?! Yes, it would be the best, and also the worst. Assuming we had some method of comparison. That's the problem, he says, I don't know that we have a method of comparison. That wasn't the point of the example! Well then, let's take something unambiguous... chronological order. If there is never another American Civil War, then the one American Civil War would be the first and the last American Civil War. He disagrees. He apparently does not know what the word "first" means. Against every mathematical definition, dictionary definition, and common definition, he disagrees. I say, you apparently don't know English then! We're talking about the meaning of the word "first" here, not some inherent fact about the world. The word "first" means whatever people agree it means, and the agreement here is quite definitely on the side of inclusivity on this matter. And he says - talking about the word "first"! - that maybe that's not the case when the word has other associations, etc. Somewhere along the way Jim showed up to agree with me that this is, indeed, the common definition of the word "first".
Anyway, Jake apparently doesn't like the idea that it would be the first because it's a "tautology". Yes, I say, it's a tautology! That's why it's true! So Jake apparently doesn't think tautologies are true. Um, tautologies are true by definition. Formally speaking this isn't actually a tautology, but the idea's the same; it's trivially true. Jake apparently doesn't think tautologies are true because they're "meaningless". Yes, they convey no information, that doesn't make them not true! Let's see... and I'll have to avoid anything fuzzy because Jake apparently doesn't understand what is and isn't relevant in an example... I am at least 18 years old, or I am less than 18 years old! Definitely true! Either the earth is a cube, or it is not a cube! Definitely true! OK, I suppose if the statement in question were literally meaningless - which can't happen formally, but I suppose we're talking generally - then I suppose it would not be true, e.g. "Flarble is gorble or flarble is not gorble" you could say is not true, because it's meaningless - but it's certainly not false!
At this, Jake, after thinking for a bit, finally just says that he "tentatively concedes most if not all points", and goes to bed. Yeah.
And he plans to do philosophy? Does he expect to produce anything other than bullshit? I know they're supposed to "question the foundations", but this is just stupid! I feel sorry for his professors. Maybe he should do something that doesn't involve words.
-Harry
Ah! Yes I do. I was going to explain it so I could point out how people often try to apply it where it's not applicable. (IIRC, I thought of talking about this, decided it wasn't really relevant, then remembered Putnin's assertion that the rational numbers don't exist and started laughing, so I had to explain.)
Maybe I should go back a bit further - Jake is a first-year philosophy major who likes to start these philosophical discussions. And he asked me about Zeno's paradox, and I gave him the simple "the sum goes to one" explanation, and then said how I thought that personally I didn't think that by itself was a very satisfying explanation, and, with a lot of fumbling, tried to explain how I thought the basic problem was that Zeno simply doesn't understand how a continuum works, and was trying to apply things inappropriate to a continuum to one.
So the thing is apparently Jake has never taken calculus. Like, nothing beyond high school math. So this gets me to trying to explain to him countability and uncountability, because I had invoked this earlier - "the length of the individual points is 0, but that doesn't mean the length of the interval is 0 because there's uncountably many".
He doesn't really understand what I'm talking about, understandably; I probably should have gotten some paper. I don't actually get as far as proving that there are uncountable sets, because before that I break off to talk about what these "sizes" actually are - I say that aleph_0 is just a name, don't worry about what it actually *is*, it's just the name we give to the size of the integers - so this leads me to explaining about equivalence relations and equivalence classes, or as I like to explain it, "sledgehammer abstraction".
Well, turns out Jake doesn't much like sledgehammer abstraction. I try to explain it as best I can - "we want something that represents all things even, what better than the set of all things even itself?" ...but, I don't know. He's unsatisfied. What exactly does he expect? Something like Plato? There are all sorts of problems with that. Well, let's leave that alone for now.
So he starts saying about how he's not certain that the infinites I'm talking about is his idea of infinity. And I should tell him, you should probably get rid of any unified idea of infinity. First of all, what's even meant by "infinite" varies - there's no more unified idea of "infinity" in math than there is of "number" (I didn't go into that, though, I have to wonder what his reaction would have been) - well, actually, what they all seem to have in common is that they're greater than all whole numbers - but that basically, some things are finite and some things are infinite, and there's no one "infinity". Things are not inherently very similar just because they're infinite. We talk about negative numbers, not "negativity". If you want you can have a unified idea of infinity that is simply, all things infinite (good old sledgehammer abstraction!) but that won't really be very useful. He still seems to want some unified infinity... I don't know...
Now, everything up to this point is, I would say, just a non-math person having incorrect expectations of how math works. But what happened next was just utterly ridiculous.
This was where I somehow got to trying to explain WOP to him. So I say, the fundamental principle of the whole numbers is that every nonempty set of whole numbers has a least element. He doesn't see this as immediately intuitive, so I give some examples. The smallest whole number is 0; the smallest prime is 2. But it doesn't matter what set of whole numbers it is - as long as there is some whole number which can be written as the sum of 2 cubes in 2 different ways, there is necessarily some smallest, some first, whole number which can be written as the sum of 2 cubes in 2 different ways.
And he said, What about single-element sets? What? Of course those have a least element, they're only element. But it's not less than itself. It doesn't have to be; that's ridiculous! Least element means less than or equal to everything, or less than everything else; that's like saying 2 isn't the least prime because it's not less than itself! And I go on to say, this is not merely the mathematical definition of "least", but the common definition as well. If there were only one movie ever made, it would be the best movie ever made. No, he says, I don't think it would be. What the hell?! Yes, it would be the best, and also the worst. Assuming we had some method of comparison. That's the problem, he says, I don't know that we have a method of comparison. That wasn't the point of the example! Well then, let's take something unambiguous... chronological order. If there is never another American Civil War, then the one American Civil War would be the first and the last American Civil War. He disagrees. He apparently does not know what the word "first" means. Against every mathematical definition, dictionary definition, and common definition, he disagrees. I say, you apparently don't know English then! We're talking about the meaning of the word "first" here, not some inherent fact about the world. The word "first" means whatever people agree it means, and the agreement here is quite definitely on the side of inclusivity on this matter. And he says - talking about the word "first"! - that maybe that's not the case when the word has other associations, etc. Somewhere along the way Jim showed up to agree with me that this is, indeed, the common definition of the word "first".
Anyway, Jake apparently doesn't like the idea that it would be the first because it's a "tautology". Yes, I say, it's a tautology! That's why it's true! So Jake apparently doesn't think tautologies are true. Um, tautologies are true by definition. Formally speaking this isn't actually a tautology, but the idea's the same; it's trivially true. Jake apparently doesn't think tautologies are true because they're "meaningless". Yes, they convey no information, that doesn't make them not true! Let's see... and I'll have to avoid anything fuzzy because Jake apparently doesn't understand what is and isn't relevant in an example... I am at least 18 years old, or I am less than 18 years old! Definitely true! Either the earth is a cube, or it is not a cube! Definitely true! OK, I suppose if the statement in question were literally meaningless - which can't happen formally, but I suppose we're talking generally - then I suppose it would not be true, e.g. "Flarble is gorble or flarble is not gorble" you could say is not true, because it's meaningless - but it's certainly not false!
At this, Jake, after thinking for a bit, finally just says that he "tentatively concedes most if not all points", and goes to bed. Yeah.
And he plans to do philosophy? Does he expect to produce anything other than bullshit? I know they're supposed to "question the foundations", but this is just stupid! I feel sorry for his professors. Maybe he should do something that doesn't involve words.
-Harry
