1.You say that the points are brutely distinguishable and later you say that they are indistinguishable, which nevertheless you hold to be different properties.
The points are brutely distinguishable, but the sets aren’t.
2.Why are the sets indistinguishable? Although I don’t particularly understand what predicates you allow for brutely distinguishable entities, it seems possible to have X = set of all brutely distinguishable points (from some class) and Y = set of all brutely distinguishable points except one. It is, of course, not a definition of Y unless you point out which of the points is missing (which you presumably can’t), but even if you don’t have a definition of Y, Y may still exist and be distinguishable from X by the property that X contains all the points while Y doesn’t.
No predicates besides brute distinguishability govern it. Entities that are brutely distinguishable are different only by virtue of being different.
The sets that differ but for one element differ because their cardinality is different. This is how they differ from the infinite case.
3.If the argument were true, haven’t you just shown only that you can’t define an infinite set of brutely distinguishable entities, rather than that infinite sets can’t be defined at all?
If infinite sets and brutely distinguishable elements exist, infinite sets with brutely distinguishable elements should exist.
4.What is your opinion about the set of all natural numbers? Is it finite or can’t it be defined?
It is infinite, but it isn’t “actually realized.” (They don’t exist; we employ them as useful fictions.)
And how does the argument depend on infiniteness, after all? Assume there is a class of eleven brutely distinguishable points. Now, can you define a set containing seven of them? If you can’t, since there is no property to distinguish those seven from the remaining four, doesn’t it equally well prove that sets of cardinality seven don’t exist?
To make the cases parallel (which I hope doesn’t miss the point): take 7 brutely distinguishable points; 4 more pop into existence. The former and latter sets are distinguishable by their cardinality. When the sets are infinite, the cardinality is identical.
The frequentist definition of probability says that probability is the limit of relative frequencies, which is the limit of (the number of occurrences divided by the number of trials), which is not equal to (limit of the number of occurrences) divided by (limit of the number of trials).
This doesn’t seem relevant to actually realized infinities, since the limit becomes inclusive rather than exclusive (of infinity). The relative frequency of heads to tails with an actually existing infinity of tosses is undefined. (Or would you contend it is .5?)
And emphasis is usually marked by italics, not red.
Are my aesthetics off? I’ve decided that unbolded red is best for emphasizing text sentences, the reason being that it is more legible than bolding, centainly than italics. If you don’t use many pictures or diagrams, I think helps maintain interest to include some color whenever you can justify it.
Color seems increasingly used in textbooks. Perhaps its a status thing, as the research journals don’t use it. But blog writing should usually be more succinct than research-journal writing, and this makes typographical emphasis more valuable because there’s less opportunity to imply emphasis textually.
If infinite sets and brutely distinguishable elements exist, infinite sets with brutely distinguishable elements should exist.
Why? It doesn’t follow. (As a trivial case, imagine that there are only two brutely distinguishable things in the world.) (Assuming that by “infinite sets with brutely distinguishable elements” you mean “set with infinitely many b.d. elements”.)
Also, you say that sets are distinguishable whenever there is a predicate which applies to one and doesn’t apply to another. That is, X and Y are distinguishable iff for some P, P(X) and not P(Y). Right?
But then you argue as if the only allowed predicates were those about cardinality. To closely follow your example, let’s denote X = “the former set containing infinitely many b.d. points” and Y = “the latter set containing all those points plus the additional one which ‘popped into existence’”. Then we have a predicate P(Z) = “Z is a subset of X”, and P(X) holds while P(Y) doesn’t. What’s wrong here?
Are my aesthetics off? [...] Perhaps its a status thing, as the research journals don’t use it.
Your aesthetics are incompatible with most of the readers. You’ve got quite a lot of negative responses to your formatting, not a single positive response (correct me if I am wrong), yet you still persist and speculate about status reasons. Even if it were true, I’d suggest taking the readers’ preferences more seriously, if you want the readers take you more seriously.
To me, coloured text really doesn’t seem more legible than bold or italics. Moreover I like when a website has a unified colour scheme which your colours break. All violations of local arbitrary design norms are distracting; the posts aren’t art, therefore aesthetics shouldn’t trump practical considerations. But if you really that much insist on using colours for emphasis (but consider there may be colourblind people reading this), please at least use the same font and background colour as everybody else.
You’ve got quite a lot of negative responses to your formatting, not a single positive response (correct me if I am wrong), yet you still persist and speculate about status reasons.
I just found it curious: I’ve addressed typography issues in a blog posting, “Emphasis by Typography.”
I have to say I’m surprised by your tone; like you’re accusing me of some form of immorality for not being attentive to readers. This all strikes me as very curious. I read Hanson’s blog and so have gotten attuned to status issues. I’m not plotting a revolution over font choice; I’m only curious about why people find Verdana objectionable just because other postings use a different font.
If infinite sets and brutely distinguishable elements exist, infinite sets with brutely distinguishable elements should exist. Why? It doesn’t follow.
The argument concerns conceptual possibility, not empirical existence. If actually existing sets can consist of brutely distinguishable elements and of infinite elements, there’s nothing to stop it conceptually from being both.
You have located a place for a counter-argument: supplying the conceptual basis. But it seems unlikely that a conceptual argument would successfully undermine brutely distinguishable infinite elements without undermining brutely distinguishable elements in general.
Then we have a predicate P(Z) = “Z is a subset of X”, and P(X) holds while P(Y) doesn’t. What’s wrong here?
You can distinguish the cardinality of finite sets with brutely distinguishable points. That is, if a set contains 7 points, you can know there are seven different points, and that’s all you can know about them.
Sorry for the perceived tone, it wasn’t my intention to accuse you of anything immoral (although I think you aren’t being much attentive to readers, but that is hardly immoral). I was mainly trying to say that violating the local aesthetic code against the disagreement of everybody who cares to voice their opinion is instrumentally bad. Even if you think that your style makes communication easier, the disagreement of others is a strong piece of evidence that it doesn’t, at least with the LW audience.
As aesthetic preferences are usually difficult to explain or even describe, I probably cannot provide a deep reason why your style is unwelcome. Few particular things I find annoying:
I like stylistic unity, so when everything on the site is one style and one post is another style (yet all the surroundings are original), it is the one post which I perceive odd by default
too many fonts; you have a normal LW headline, then a link in some serifed font, then one paragraph in the standard LW sans-serif font, then the rest of the article in another sans-serif font (or perhaps the same but different size)
the section headlines are larger than the main headline (aaargh!)
blank line missing between sections 1 and 2, other sections are separated by blank lines
the grey background, which is only slightly different from the standard white background and (aargh again) surprisingly missing in the first paragraph and at some (but not all) blank lines; the most annoying thing is that it forms visual boxes which unite the body of a section with the headline of the following section
the red emphasis; on first reading I tend to interpret red not as emphasis, but rather as a text marked for further revision or deletion in a draft
Originally I thought that these “features” accidentally arose when you had copied the text from elsewhere. Now when you are defending the typography, I am curious whether you really have a reason for them all, especially no. 5.
Thank your for the astute response.
The points are brutely distinguishable, but the sets aren’t.
No predicates besides brute distinguishability govern it. Entities that are brutely distinguishable are different only by virtue of being different.
The sets that differ but for one element differ because their cardinality is different. This is how they differ from the infinite case.
If infinite sets and brutely distinguishable elements exist, infinite sets with brutely distinguishable elements should exist.
It is infinite, but it isn’t “actually realized.” (They don’t exist; we employ them as useful fictions.)
To make the cases parallel (which I hope doesn’t miss the point): take 7 brutely distinguishable points; 4 more pop into existence. The former and latter sets are distinguishable by their cardinality. When the sets are infinite, the cardinality is identical.
This doesn’t seem relevant to actually realized infinities, since the limit becomes inclusive rather than exclusive (of infinity). The relative frequency of heads to tails with an actually existing infinity of tosses is undefined. (Or would you contend it is .5?)
Are my aesthetics off? I’ve decided that unbolded red is best for emphasizing text sentences, the reason being that it is more legible than bolding, centainly than italics. If you don’t use many pictures or diagrams, I think helps maintain interest to include some color whenever you can justify it.
Color seems increasingly used in textbooks. Perhaps its a status thing, as the research journals don’t use it. But blog writing should usually be more succinct than research-journal writing, and this makes typographical emphasis more valuable because there’s less opportunity to imply emphasis textually.
Why? It doesn’t follow. (As a trivial case, imagine that there are only two brutely distinguishable things in the world.) (Assuming that by “infinite sets with brutely distinguishable elements” you mean “set with infinitely many b.d. elements”.)
Also, you say that sets are distinguishable whenever there is a predicate which applies to one and doesn’t apply to another. That is, X and Y are distinguishable iff for some P, P(X) and not P(Y). Right?
But then you argue as if the only allowed predicates were those about cardinality. To closely follow your example, let’s denote X = “the former set containing infinitely many b.d. points” and Y = “the latter set containing all those points plus the additional one which ‘popped into existence’”. Then we have a predicate P(Z) = “Z is a subset of X”, and P(X) holds while P(Y) doesn’t. What’s wrong here?
Your aesthetics are incompatible with most of the readers. You’ve got quite a lot of negative responses to your formatting, not a single positive response (correct me if I am wrong), yet you still persist and speculate about status reasons. Even if it were true, I’d suggest taking the readers’ preferences more seriously, if you want the readers take you more seriously.
To me, coloured text really doesn’t seem more legible than bold or italics. Moreover I like when a website has a unified colour scheme which your colours break. All violations of local arbitrary design norms are distracting; the posts aren’t art, therefore aesthetics shouldn’t trump practical considerations. But if you really that much insist on using colours for emphasis (but consider there may be colourblind people reading this), please at least use the same font and background colour as everybody else.
I just found it curious: I’ve addressed typography issues in a blog posting, “Emphasis by Typography.”
I have to say I’m surprised by your tone; like you’re accusing me of some form of immorality for not being attentive to readers. This all strikes me as very curious. I read Hanson’s blog and so have gotten attuned to status issues. I’m not plotting a revolution over font choice; I’m only curious about why people find Verdana objectionable just because other postings use a different font.
The argument concerns conceptual possibility, not empirical existence. If actually existing sets can consist of brutely distinguishable elements and of infinite elements, there’s nothing to stop it conceptually from being both.
You have located a place for a counter-argument: supplying the conceptual basis. But it seems unlikely that a conceptual argument would successfully undermine brutely distinguishable infinite elements without undermining brutely distinguishable elements in general.
You can distinguish the cardinality of finite sets with brutely distinguishable points. That is, if a set contains 7 points, you can know there are seven different points, and that’s all you can know about them.
Sorry for the perceived tone, it wasn’t my intention to accuse you of anything immoral (although I think you aren’t being much attentive to readers, but that is hardly immoral). I was mainly trying to say that violating the local aesthetic code against the disagreement of everybody who cares to voice their opinion is instrumentally bad. Even if you think that your style makes communication easier, the disagreement of others is a strong piece of evidence that it doesn’t, at least with the LW audience.
As aesthetic preferences are usually difficult to explain or even describe, I probably cannot provide a deep reason why your style is unwelcome. Few particular things I find annoying:
I like stylistic unity, so when everything on the site is one style and one post is another style (yet all the surroundings are original), it is the one post which I perceive odd by default
too many fonts; you have a normal LW headline, then a link in some serifed font, then one paragraph in the standard LW sans-serif font, then the rest of the article in another sans-serif font (or perhaps the same but different size)
the section headlines are larger than the main headline (aaargh!)
blank line missing between sections 1 and 2, other sections are separated by blank lines
the grey background, which is only slightly different from the standard white background and (aargh again) surprisingly missing in the first paragraph and at some (but not all) blank lines; the most annoying thing is that it forms visual boxes which unite the body of a section with the headline of the following section
the red emphasis; on first reading I tend to interpret red not as emphasis, but rather as a text marked for further revision or deletion in a draft
Originally I thought that these “features” accidentally arose when you had copied the text from elsewhere. Now when you are defending the typography, I am curious whether you really have a reason for them all, especially no. 5.