I’m not sure that the example justifies a move to multisets. H-O-H has three atoms: one is an oxygen atom and the other two are hydrogen atoms. The right way to read H-O-H is that the first atom in the molecule is of type H, the second is of type O, and the third is of type H. The H, O letters are really types/predicates. When conceived this way the problem disappears: {H, O} is the set of non-null predicates of the molecule (a second-order set, if you will), rather than the set of atoms in the molecule. Conversely, the set {2, 1+1} has only one element, not two: the two names are different, but they name the same element.
Mr Frege
Karma: 34
They’re all interesting questions.
One point I’d make is that I tried to limit my discussion to ‘abstraction’ as it relates to concepts, as that seemed most pertinent to the quote you cited. The opposite of abstract is concrete. I’m not entirely sure that we should confound the abstract/concrete comparison with that of general/specific.
“pizza that is warm, has an exterior that is thin and crispy, an interior that is warm, chewy, and fresh, a thin layer of tomato sauce that is mildly sweet and acidic, and small dollups of fresh mozzarella cheese that is cool and soft”
Let’s say you’ve just described a ‘Napolitana’ pizza. Is “large Napolitana” specific in your sense? I can see a definitional regression emerging. For this reason, I suspect that specificity/generality might be best kept to comparisons relative to some ‘basis’ concepts, rather than made in some absolute sense.“Last weekend” has its own issues, being relative to the day it is uttered. Indeed, to account for any day of utterance, it might be a difficult concept to articulate precisely. As a ‘relative’ (or ‘relational’) concept, it might be general in that it ranges over many instances.
There is the further issue of vagueness. Many of the concepts you use (eg, warm, thin, etc.) are not precise. This issue is exacerbated with good, which is, in addition, inherently subjective.
Adding to @TsviBT.
“But I get a sense that “lattice” involves order in some way, and I am not seeing how order fits in to the question of how specific a concept is.”
Sounds to me like you’re on the right track. The claim made is that concepts can be ordered in terms of their abstractness. For example, the concept day would be taken to be more abstract than the concept sunny day in that day abstracts from the weather by admitting both sunny and cloudy days.
The order of concepts is ‘partial’ in that not all concepts can be compared by abstraction: for example, neither sunny nor day by themselves is more abstract than the other. So, unlike the familiar ‘total’ orderings that we see with, say, numbers, in which any two numbers can be compared/ordered by ‘less than’, the abstraction ordering on concepts is only ‘partial’ in that some pairs of concepts cannot be compared.
Minor point on the origin of the term NPC. As an old role-playing game nerd—of the tabletop variety—the origin of the term predates computer games. The term was introduced/used back in the days of the original tabletop role-playing games (eg, Dungeons and Dragons) of the 70s and 80s. In these games, players each play the role of a character in a story/adventure: a ‘player character’ or PC. Characters played by the game master (eg, ‘villains’), and not by any player, are called non-player characters (NPCs).
Of course, the first thing I did is put your post through it
This comment received a lot of backlash. Admittedly, my comment wasn’t as diplomatic as it might’ve been nor did I elaborate on my own reasoning. In my defence, I didn’t think the original article was making much of an effort to get at the truth (see other criticisms above). Rather, it is a (very) one-sided account advocating that we should not consider the other side of the story (ie, it is an attack on both-sidesism).
The attack on both-sidesism is consistent with findings referenced in the video below. Both sides are prone to such anti-democratic behaviour, but the findings also suggest that one side is “slightly more willing to sacrifice democracy (by supporting actions that benefit their own party at the expense of democracy)”. This might be a case in point.
Feel free to watch the whole thing, but the tldr part starts at 4:05.
https://www.youtube.com/watch?v=PVqjH6MaqRYThe events of Nov 6⁄7, 2024 might support the argument that the original argument was indeed self-defeating; ie, an argument against the argument against both-sidesism—effectively, an argument for both-sidesism.
Good arguments support some claim. Bad arguments provide little/no support. Some arguments are so bad that they support the opposite claim. This is one of those.
I think the issue might be that I’m interpreting circular reasoning as something stronger than you; ie, in the pernicious sense which explains why “The idea that circular reasoning is bad is widespread”.
I suspect that according to your interpretation all valid deductive reasoning is circular in some way, circularity thus being necessary for valid deductive reasoning. In this regard, circularity would be a desirable attribute.
In contrast, my interpretation is one in which in the process of affirming a belief, one presupposes something that would require to have already affirmed the same belief; what is sometimes called “begging the question”.
In this context, I don’t regard A |- A (circular in your sense, but not in mine) as problematic, as it just involves inferring something that has already been affirmed.
“Although I didn’t think the distinction was so important to focus on in this essay.” The distinction is important.
Do you agree that in logic, A |- A says something akin to “from a true A (a non-logical axiom), you may infer the truth of A”? If so, I don’t see any circularity there; anything on the left hand side of the |- is an ‘additional’ axiom of your theory, hence accepted as true. Circular reasoning, by my interpretation, is where you infer/assume as true what you’re in the process of trying to prove to be true. This is not the case with A |- A, in which the occurrence of A on the left of the |- means you are taking it as true to begin with. That’s why I assumed you meant A → A when you said “A implies A”.
The law (A → A) → A would allow you to infer A from A → A, which would, indeed, allow you to infer A without it having being established as true, doing so, rather, solely from the premise that if it were true, it would be true, thereby concluding that it is true. This is closer to the interpretation of circular reasoning I gave above.
“It also doesn’t fit the geometric intuition behind ‘circular’ well. leads back around to where it started”. I think this appeal to geometric intuition is misleading, which might explain the confusion. All A |- A it is saying is that if you already know/believe A, then you may infer it. In a formal system it may take several steps to derive (‘lead to’) A from A, but I don’t think that circular reasoning should be tied to proof theory.
“The first important thing to note is that circular reasoning is logically valid. A implies A. If circular arguments are to be critiqued, it must be by some other standard than logical validity.”
I challenge that premise: “A implies A” is not circular; this confuses a logical law with a valid deductive inference—reasoning, circular or otherwise, is about argument/inference. The logical law “A implies A” doesn’t say that ones believes A; no position on the belief/truth of A is being made. This is different from having the belief A, and inferring A, which is just saying that you can infer what you already believe. What would be circular would be the law “(A implies A) implies A”, as that would allow one to infer A from being able to infer A from A. That, however, isn’t a logical law.
In addition to the ‘war on math’, there’s also a ‘war within math (education)‘: discovery-based learning vs explicit/direct instruction, etc. Moreover, in relation to your related post on reading/literacy, I’d be interested in people’s take on these debates as they pertain to ‘the science of learning’: a methodology that seems to work for learning to read that is now being transplanted to learning mathematics/numeracy. I’m not convinced that there’s evidence for transfer: reading involves scanning, fixation, and saccades (all visual-perceptual skills—I exclude comprehension here), whereas mathematics is the opposite: abstract and relies on reasoning. Despite claims from advocates of ‘the science of learning’, I see little to suggest that we should adopt techniques that work for reading/literacy for mathematics/numeracy