No to “exactly matches”, but yes to “actually captures”, in the sense of “actually captures enough of”. A typical mathematical definition of “infinite” is “A set S is infinite if and only if there exists a bijection between S and a proper subset of S.” It’s not a coincidence that the pre-formal and formal concepts of “infinity” are both called “infinity”. The formal concept captures enough of the pre-formal concept to deserve the same name.
Actually, this is rather dubious. Lakoff and Nuñez’s work Where Mathematics Comes From includes an extensive case study of what they call the Basic Metaphor of Infinity, and they argue that transfinite numbers do not account for all uses of infinity. (And this is not even addressing the issue of potential vs. actual infinity, which is quite central to their analysis.)
Well, you didn’t. You stated that the definition “A set S is infinite if and only if there exists a bijection between S and a proper subset of S.” i.e. sets with transfinite cardinality accounts for the pre-formal concept of “infinity”, which it doesn’t. Lakoff and Núñez provide a cognitive, metaphor-based analysis which is much more comprehensive.
I haven’t read their book, but an analysis of the pre-theoretic concept of the infinitude of a set needn’t be taken as an analysis of the pre-theoretic concept of infinitude in general. “Unmarried man” doesn’t define “bachelor” in “bachelor of the arts,” but that doesn’t mean it doesn’t define it in ordinary contexts.
“Unmarried man” doesn’t define “bachelor” in “bachelor of the arts,” but that doesn’t mean it doesn’t define it in ordinary contexts.
Except that Lakoff and Núñez’s pre-theoretic analysis does account for transfinite sets. There is a single pre-theoretic concept of infinity which accounts for a variety of formal definitions. This is unlike the word “bachelor” which is an ordinary word with multiple meanings.
I’m having trouble seeing your point in the context of the rest of the discussion. Tyrrell claimed that the pre-theoretic notion of an infinite set—more charitably, perhaps, the notion of an infinite cardinality—is captured by Dedekind’s formal definition. Here, “capture” presumably means something like “behaves sufficiently similarly so as to preserve the most basic intuitive properties of.” Your response appears to be that there is a good metaphorical analysis of infinitude that accounts for this pre-theoretic usage as well as some others simultaneously. And by “accounts for X,” I take it you mean something like “acts as a cognitive equivalent, i.e., is the actual subject of mental computation when we think about X.” What is this supposed to show? Does anyone really maintain that human brains are actually processing terms like “bijection” when they think intuitively about infinity?
Actually, this is rather dubious. Lakoff and Nuñez’s work Where Mathematics Comes From includes an extensive case study of what they call the Basic Metaphor of Infinity, and they argue that transfinite numbers do not account for all uses of infinity. (And this is not even addressing the issue of potential vs. actual infinity, which is quite central to their analysis.)
I think that nearly everyone would agree with that.
Well, you didn’t. You stated that the definition “A set S is infinite if and only if there exists a bijection between S and a proper subset of S.” i.e. sets with transfinite cardinality accounts for the pre-formal concept of “infinity”, which it doesn’t. Lakoff and Núñez provide a cognitive, metaphor-based analysis which is much more comprehensive.
That would indeed be a strange claim, so it is fortunate that I did not make it.
I haven’t read their book, but an analysis of the pre-theoretic concept of the infinitude of a set needn’t be taken as an analysis of the pre-theoretic concept of infinitude in general. “Unmarried man” doesn’t define “bachelor” in “bachelor of the arts,” but that doesn’t mean it doesn’t define it in ordinary contexts.
Except that Lakoff and Núñez’s pre-theoretic analysis does account for transfinite sets. There is a single pre-theoretic concept of infinity which accounts for a variety of formal definitions. This is unlike the word “bachelor” which is an ordinary word with multiple meanings.
I’m having trouble seeing your point in the context of the rest of the discussion. Tyrrell claimed that the pre-theoretic notion of an infinite set—more charitably, perhaps, the notion of an infinite cardinality—is captured by Dedekind’s formal definition. Here, “capture” presumably means something like “behaves sufficiently similarly so as to preserve the most basic intuitive properties of.” Your response appears to be that there is a good metaphorical analysis of infinitude that accounts for this pre-theoretic usage as well as some others simultaneously. And by “accounts for X,” I take it you mean something like “acts as a cognitive equivalent, i.e., is the actual subject of mental computation when we think about X.” What is this supposed to show? Does anyone really maintain that human brains are actually processing terms like “bijection” when they think intuitively about infinity?