I mostly agree with this assessment. However, the key point is that such uses of numbers should be seen as metaphorical. The literal meaning of a metaphor is typically nonsensical, but it works by somehow hacking the human understanding of language to successfully convey a point with greater precision than the most precise literal statement would allow, at least in as many words. (There are other functions of metaphors too, of course, but this one is relevant here.) And just like it is fallacious to understand a metaphor literally, it is similarly fallacious to interpret these numerical metaphors as useful for mathematical purposes. When it comes to subjective probabilities, however, I often see what looks like confusion on this point.
It is wrong to use a subjective probability that you got from someone else for mathematical purposes directly, for reasons I expand on in my comment here. But I don’t think that makes them metaphorical, unless you’re using a definition of metaphor that’s very different than the one I am. And you can use a subjective probability which you generated yourself, or combined with your own subjective probability, in calculations. Doing so just comes with the same caveats as using a probability from a study whose sample was too small, or which had some other bad but not entirely fatal flaw.
I will write a reply to that earlier comment of yours a bit later today when I’ll have more time. (I didn’t forget about it, it’s just that I usually answer lengthy comments that deserve a greater time investment later than those where I can fire off replies rapidly during short breaks.)
But in addition to the theme of that comment, I think you’re missing my point about the possible metaphorical quality of numbers. Human verbal expressions have their literal information content, but one can often exploit the idiosyncrasies of the human language interpretation circuits to effectively convey information altogether different from the literal meaning of one’s words. This gives rise to various metaphors and other figures of speech, which humans use in their communication frequently and effectively. (The process is more complex than this simple picture, since frequently used metaphors can eventually come to be understood as literal expressions of their common metaphorical meaning, and this process is gradual. There are also other important considerations about metaphors, but this simple observation is enough to support my point.)
Now, I propose that certain practical uses of numbers in communication should be seen that way too. A literal meaning of a number is that something can ultimately be counted, measured, or calculated to arrive at that number. A metaphorical use of a number, however, doesn’t convey any such meaning, but merely expects to elicit similar intuitive impressions, which would be difficult or even impossible to communicate precisely using ordinary words. And just like a verbal metaphor is nonsensical except for the non-literal intuitive point it conveys, and its literal meaning should be discarded, at least some practical uses of numbers in human conversations serve only to communicate intuitive points, and the actual values are otherwise nonsensical and should not be used for any other purposes—and even if they perhaps are, their metaphorical value should be clearly seen apart from their literal mathematical value.
Therefore, regardless of our disagreement about subjective probabilities (of which more in my planned reply), this is a separate important point I wanted to make.
I mostly agree with this assessment. However, the key point is that such uses of numbers should be seen as metaphorical. The literal meaning of a metaphor is typically nonsensical, but it works by somehow hacking the human understanding of language to successfully convey a point with greater precision than the most precise literal statement would allow, at least in as many words. (There are other functions of metaphors too, of course, but this one is relevant here.) And just like it is fallacious to understand a metaphor literally, it is similarly fallacious to interpret these numerical metaphors as useful for mathematical purposes. When it comes to subjective probabilities, however, I often see what looks like confusion on this point.
It is wrong to use a subjective probability that you got from someone else for mathematical purposes directly, for reasons I expand on in my comment here. But I don’t think that makes them metaphorical, unless you’re using a definition of metaphor that’s very different than the one I am. And you can use a subjective probability which you generated yourself, or combined with your own subjective probability, in calculations. Doing so just comes with the same caveats as using a probability from a study whose sample was too small, or which had some other bad but not entirely fatal flaw.
I will write a reply to that earlier comment of yours a bit later today when I’ll have more time. (I didn’t forget about it, it’s just that I usually answer lengthy comments that deserve a greater time investment later than those where I can fire off replies rapidly during short breaks.)
But in addition to the theme of that comment, I think you’re missing my point about the possible metaphorical quality of numbers. Human verbal expressions have their literal information content, but one can often exploit the idiosyncrasies of the human language interpretation circuits to effectively convey information altogether different from the literal meaning of one’s words. This gives rise to various metaphors and other figures of speech, which humans use in their communication frequently and effectively. (The process is more complex than this simple picture, since frequently used metaphors can eventually come to be understood as literal expressions of their common metaphorical meaning, and this process is gradual. There are also other important considerations about metaphors, but this simple observation is enough to support my point.)
Now, I propose that certain practical uses of numbers in communication should be seen that way too. A literal meaning of a number is that something can ultimately be counted, measured, or calculated to arrive at that number. A metaphorical use of a number, however, doesn’t convey any such meaning, but merely expects to elicit similar intuitive impressions, which would be difficult or even impossible to communicate precisely using ordinary words. And just like a verbal metaphor is nonsensical except for the non-literal intuitive point it conveys, and its literal meaning should be discarded, at least some practical uses of numbers in human conversations serve only to communicate intuitive points, and the actual values are otherwise nonsensical and should not be used for any other purposes—and even if they perhaps are, their metaphorical value should be clearly seen apart from their literal mathematical value.
Therefore, regardless of our disagreement about subjective probabilities (of which more in my planned reply), this is a separate important point I wanted to make.