In the case of mathematics, it turns out to be worthwhile to think about concepts in the decidedly unnatural mode of necessary and sufficient conditions.
Absolutely. But as you say, “there’s little reason to think that a common concept like ‘triangle’ is [stored in the brain in necessary and sufficient conditions] for most people.” The error happens when the philosopher thinks that defining “goodness” or “knowledge” as a set of necessary and sufficient conditions actually captures his pre-theoretic intuitive concept of “goodness” or “knowledge.” Mathematicians, I hope, are not making that mistake. They are working with a cleanly defined formal system, and have no illusions that their pre-theoretic intuitive concept of “infinity” exactly matches the term’s definition in their formal system.
The error happens when the philosopher thinks that defining “goodness” or “knowledge” as a set of necessary and sufficient conditions actually captures his pre-theoretic intuitive concept of “goodness” or “knowledge.” Mathematicians, I hope, are not making that mistake. They are working with a cleanly defined formal system, and have no illusions that their pre-theoretic intuitive concept of “infinity” exactly matches the term’s definition in their formal system.
(Emphasis added.)
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. One can use the formal concept in lots of the places where people were accustomed to using the pre-formal concept, with the bonus that the formal concept is far clearer, amenable to rigorous study, apparently free from contradiction, and so forth.
And many mathematicians would say that this NASC-ified concept may have meaning even outside of formal theories such as ZF. Most would consider it to be at least possibly meaningful to apply the NASC-ified definition to thought experiments about concreta, such as in Hilbert’s Hotel.
I suspect both philosophers and mathematicians succeed and fail on this issue in a wide range of degrees.
I’m not sure what you mean by “fail on this issue”. Are you saying that mathematicians who wonder about the physical realizability of a Hilbert-Hotel type scenario using the NASC definition of infinity are committing the error of which you, Lakoff, and Johnson accuse conceptual-analysis style philosophy? (You probably don’t mean that, but I’m giving you my best guess so that you can bounce off of it while clarifying your intended meaning.)
Sorry, I’m not familiar enough with the studies of infinity to say.
As thomblake said, you should still be able to clarify what you meant by “fail on this issue”. In particular, what is “this issue”, and in what sense do you suspect that mathematicians “fail” on it to some degree?
At any rate, my original example of a common mathematical concept was “triangle” (you brought up “infinity”). So perhaps you can make your point in terms of triangles instead of infinity.
I’ll try to clarify, but I’m very pressed for time. Let me try this first: Does this comment help clarify what I was trying to claim in my previous post?
I’ll try to clarify, but I’m very pressed for time. Let me try this first: Does this comment help clarify what I was trying to claim in my previous post?
Your linked comment doesn’t clarify for me what you meant by “mathematicians succeed and fail on this issue in a wide range of degrees.” That said, the comment does give a meaning to the sentence “Conceptual analysis makes assumptions about how the mind works” that I can agree with, though it seems weird to me to express that meaning with that sentence.
The “assumption” is that our categories coincide with “tidy” lists of properties. First, I would call this a “hope” rather than an “assumption”. Seeking a tidy definition is reasonable if there is a high-enough probability that such a definition exists, even if that probability is well below 50%. Second, it seems strange to me to characterize this as an issue of “how the mind works”. That’s kind of like saying that I can’t give a short-yet-complete description of the human stomach because of “how epigenesis works”.
This whole topic looks like a good candidate for saying “oops” about, though settling the details would take more work. (Specifically, does someone on LW understand your point and can re-state it?)
FWIW, I understood lukeprog’s comment to mean that both philosophers and mathematicians sometimes mistake the formal theoretical constructs they work with professionally for the related informal cognitive structures that existed prior to the development of those constructs (e.g., the formal definition of a triangle, of infinity, of knowledge, etc.), but that the degree of error involved in such a mistake depends on how closely the informal cognitive structure resembles the formal theoretical construct, and that he’s not familiar enough with the formal theoretical constructs of infinity to express an opinion about to what degree mathematicians who wonder about the physical realizability of a Hilbert-Hotel type scenario are making the same error.
That said, I’m not especially confident of that interpretation. And if I’m right, it hardly merits all this meta-discussion.
Right. Let me take the way you said this and run with it. Here’s what I’m trying to say, which does indeed strike me as not worth all that much meta-discussion, and not requiring me to say “oops,” since it sounds uncontroversial to my ears:
Both philosophers and mathematicians sometimes mistake the formal theoretical constructs they work with professionally for the related informal cognitive structures that existed prior to the development of those constructs (e.g., the formal definition of a triangle, of infinity, of knowledge, etc.), but the degree of error involved in such a mistake depends on how closely the informal cognitive structure resembles the formal theoretical construct and what is specifically being claimed by the practitioner, and these factors vary from work to work, practitioner to practitioner.
Both philosophers and mathematicians sometimes mistake the formal theoretical constructs they work with professionally for the related informal cognitive structures that existed prior to the development of those constructs
Both (pre-theoretic) “informal cognitive structures” and “formal theoretical constructs” (clearly things of different kinds) are ways of working with (in many cases) the same ideas, with formal tools being an improvement over informal understanding in accessing the same thing. The tools are clearly different, and the answers they can get are clearly different, but their purpose could well be identical. To evaluate their relation to their purpose, we need to look “from the outside” at how the tools relate to the assumed purpose’s properties, and we might judge suitability of various tools for reflecting a given idea even where they clearly can’t precisely define it or even in principle compute some of its properties.
This actually seems to be an important point to get right in thinking about FAI/metaethics. Human decision problem (friendliness content), the thing FAI needs to capture more formally, is (in my current understanding) something human minds can’t explicitly represent, can’t use definition of in their operation. Instead, they perform something only superficially similar.
FWIW, I understood lukeprog’s comment to mean that both philosophers and mathematicians sometimes mistake the formal theoretical constructs they work with professionally for the related informal cognitive structures that existed prior to the development of those constructs (e.g., the formal definition of a triangle, of infinity, of knowledge, etc.), but that the degree of error involved in such a mistake depends on how closely the informal cognitive structure resembles the formal theoretical construct, and that he’s not familiar enough with the formal theoretical constructs of infinity to express an opinion about to what degree mathematicians who wonder about the physical realizability of a Hilbert-Hotel type scenario are making the same error.
If that is what Luke is talking about then he’s making a good point.
How do these things relate to each other in the context of Luke’s statement (and which are relevant): mathematician’s pre-theoretic idea of a triangle; formal definition of a triangle; mathematician’s understanding of the formal definition; the triangle itself; mathematician’s understanding of their understanding of the formal definition; mathematician’s understanding of their pre-theoretic idea of a triangle; mathematician’s understanding of a triangle?
It seems to me that a very useful way of looking at what’s going on is that both pre-theoretic understanding and understanding strengthened by having a formal definition are ways of understanding the idea itself, capturing it to different degrees (having it control mathematician’s thought), with a formal definition being a tool for training the same kind of pre-theoretic understanding, just as a calculator serves to get to reliable answers faster. There is no clear-cut distinction, at least to the extent we are interested in actual answers to a question and not in actual answers that a given faulty calculator provides when asked that question (in which case we are focusing on a different question entirely, most likely a wrong one).
How do these things relate to each other in the context of Luke’s statement (and which are relevant):
[A] mathematician’s pre-theoretic idea of a triangle; [B] formal definition of a triangle; [C] mathematician’s understanding of the formal definition; [D] the triangle itself; [E] mathematician’s understanding of their understanding of the formal definition; [F] mathematician’s understanding of their pre-theoretic idea of a triangle; [G] mathematician’s understanding of a triangle
In that context, my understanding of Luke’s statement becomes “Mathematicians sometimes mistake B for A, but the degree of error involved in such a mistake depends on how closely B resembles A.”
C-G have nothing to do directly with the statement made, but personally I suspect that the relationship between F and E has a lot to do with the likelihood of mistaking B and A. I also think D is either overly vague or outright meaningless, and consequently G (assuming it exists) is either vague, complex or confused insofar as “triangle” in G means D.
To say that a little differently: there are things in the world that humans would categorize as triangles in common speech, which might not conform to the necessary and sufficient conditions of a triangle used by mathematicians. For example, shapes made out of curved lines and shapes whose vertices are curves might not qualify as triangles in the mathematician’s sense while still being triangles in the layman’s sense.
I do not agree that focusing attention on the question of what exactly is going on when laymen make that categorization is a wrong question, but I agree that it’s not a question about geometry.
I both agree and disagree that the layman’s and the mathematician’s understanding of “triangle” are both tools for understanding a single thing, which you’ve dubbed “the idea itself”, depending on just what you mean by that phrase. In other words, I don’t think the phrase “the idea itself” has a precise enough meaning to be useful in this sort of conversation. (This is also why I say that D and G are overly vague.)
In that context, my understanding of Luke’s statement becomes “Mathematicians sometimes mistake B for A, but the degree of error involved in such a mistake depends on how closely B resembles A.”
By (A) I mean mathematician’s thoughts, and by (B) a syntactic definition. Since these things can’t possibly be confused, you must have read something different in my words… On the other hand, what these things talk about, that is whatever mathematician is thinking of, and whatever a formal definition defines, could be confused (given one possible reading, though not one I intended). Do you mean the latter?
In other words, I don’t think the phrase “the idea itself” has a precise enough meaning to be useful in this sort of conversation.
You can use a formal definition to access an idea, but separately from how the mathematician in question uses it to access the same idea. Using whatever means you have, you look both at how someone understands an idea (i.e. what thoughts occur, that have cognitive as well as mathematical explanations), and at the idea, and see how accurately it’s captured.
Ah, sorry: I understood by B what it seems you actually meant by C.
I should have said: “Mathematicians sometimes mistake C for A, but the degree of error involved in such a mistake depends on on how closely B resembles A.” (The relationship between C and B is also a potential source of mathematicians’ error, of course, but irrelevant to the mistake being described here.)
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Re: your last paragraph… I certainly agree that for any X that exists outside of our minds, if I and a mathematician both construct separate understandings in our minds of X, it’s in principle possible to compare those understandings to X to determine which of us is more accurate.
So, for example, I agree that if that mathematician and I both look at this image: http://a3.mzstatic.com/us/r1000/042/Purple/25/f7/a3/mzl.xofowuov.480x480-75.jpg and she concludes it isn’t a triangle (perhaps because the vertices are curved, perhaps because the bottom right vertex is missing, perhaps because the lines have nonzero thickness, perhaps for other reasons) while I conclude that it is, it follows that if a triangle (or an idea of a triangle, or a formal definition of a triangle) is a thing that exists outside of our minds then we can compare our different understandings to the actual triangle/idea/definition to determine which of us is more accurate.
And I agree that it further follows that, as you say, to the extent that we are interested in actual answers to a question and not in actual answers that a given faulty calculator provides when asked that question (in which case we are focusing on a different question entirely, most likely a wrong one), to that extent we care about the more accurate of the two understandings more than the less accurate of them.
That said, I am not certain how I would go about actually performing that comparison between my understanding and the actual formal definition of a triangle that exists outside of both my mind and the mathematician’s, and I’m not convinced it makes sense to talk about doing so.
Just to be clear, I’m not trying to go all postmodern on you here. I understand how I would use a bucket full of pebbles to keep track of a pen full of sheep and I’m perfectly happy to say that there’s a fact of the matter about the pebbles and the sheep and that it’s possible for me to be wrong about that and to determine the magnitude of my error through observation of the world, and how all of this depends on the fact that various things I can do with pebbles allow me to differentially predict what will happen when I count the sheep in the pen.
What I don’t understand what experience of the world I’d be differentially predicting when I say that this rather than that is the actual formal definition of a triangle.
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?
Absolutely. But as you say, “there’s little reason to think that a common concept like ‘triangle’ is [stored in the brain in necessary and sufficient conditions] for most people.” The error happens when the philosopher thinks that defining “goodness” or “knowledge” as a set of necessary and sufficient conditions actually captures his pre-theoretic intuitive concept of “goodness” or “knowledge.” Mathematicians, I hope, are not making that mistake. They are working with a cleanly defined formal system, and have no illusions that their pre-theoretic intuitive concept of “infinity” exactly matches the term’s definition in their formal system.
(Emphasis added.)
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. One can use the formal concept in lots of the places where people were accustomed to using the pre-formal concept, with the bonus that the formal concept is far clearer, amenable to rigorous study, apparently free from contradiction, and so forth.
And many mathematicians would say that this NASC-ified concept may have meaning even outside of formal theories such as ZF. Most would consider it to be at least possibly meaningful to apply the NASC-ified definition to thought experiments about concreta, such as in Hilbert’s Hotel.
I agree it’s a matter of degree. I suspect both philosophers and mathematicians succeed and fail on this issue in a wide range of degrees.
I’m not sure what you mean by “fail on this issue”. Are you saying that mathematicians who wonder about the physical realizability of a Hilbert-Hotel type scenario using the NASC definition of infinity are committing the error of which you, Lakoff, and Johnson accuse conceptual-analysis style philosophy? (You probably don’t mean that, but I’m giving you my best guess so that you can bounce off of it while clarifying your intended meaning.)
Sorry, I’m not familiar enough with the studies of infinity to say.
Even if that’s the case, it seems you should explain what you mean by “fail on this issue” without reference to “studies of infinity”.
As thomblake said, you should still be able to clarify what you meant by “fail on this issue”. In particular, what is “this issue”, and in what sense do you suspect that mathematicians “fail” on it to some degree?
At any rate, my original example of a common mathematical concept was “triangle” (you brought up “infinity”). So perhaps you can make your point in terms of triangles instead of infinity.
I’ll try to clarify, but I’m very pressed for time. Let me try this first: Does this comment help clarify what I was trying to claim in my previous post?
Your linked comment doesn’t clarify for me what you meant by “mathematicians succeed and fail on this issue in a wide range of degrees.” That said, the comment does give a meaning to the sentence “Conceptual analysis makes assumptions about how the mind works” that I can agree with, though it seems weird to me to express that meaning with that sentence.
The “assumption” is that our categories coincide with “tidy” lists of properties. First, I would call this a “hope” rather than an “assumption”. Seeking a tidy definition is reasonable if there is a high-enough probability that such a definition exists, even if that probability is well below 50%. Second, it seems strange to me to characterize this as an issue of “how the mind works”. That’s kind of like saying that I can’t give a short-yet-complete description of the human stomach because of “how epigenesis works”.
This whole topic looks like a good candidate for saying “oops” about, though settling the details would take more work. (Specifically, does someone on LW understand your point and can re-state it?)
Well, since you asked.
FWIW, I understood lukeprog’s comment to mean that both philosophers and mathematicians sometimes mistake the formal theoretical constructs they work with professionally for the related informal cognitive structures that existed prior to the development of those constructs (e.g., the formal definition of a triangle, of infinity, of knowledge, etc.), but that the degree of error involved in such a mistake depends on how closely the informal cognitive structure resembles the formal theoretical construct, and that he’s not familiar enough with the formal theoretical constructs of infinity to express an opinion about to what degree mathematicians who wonder about the physical realizability of a Hilbert-Hotel type scenario are making the same error.
That said, I’m not especially confident of that interpretation. And if I’m right, it hardly merits all this meta-discussion.
Right. Let me take the way you said this and run with it. Here’s what I’m trying to say, which does indeed strike me as not worth all that much meta-discussion, and not requiring me to say “oops,” since it sounds uncontroversial to my ears:
Both philosophers and mathematicians sometimes mistake the formal theoretical constructs they work with professionally for the related informal cognitive structures that existed prior to the development of those constructs (e.g., the formal definition of a triangle, of infinity, of knowledge, etc.), but the degree of error involved in such a mistake depends on how closely the informal cognitive structure resembles the formal theoretical construct and what is specifically being claimed by the practitioner, and these factors vary from work to work, practitioner to practitioner.
(See also this comment.)
Both (pre-theoretic) “informal cognitive structures” and “formal theoretical constructs” (clearly things of different kinds) are ways of working with (in many cases) the same ideas, with formal tools being an improvement over informal understanding in accessing the same thing. The tools are clearly different, and the answers they can get are clearly different, but their purpose could well be identical. To evaluate their relation to their purpose, we need to look “from the outside” at how the tools relate to the assumed purpose’s properties, and we might judge suitability of various tools for reflecting a given idea even where they clearly can’t precisely define it or even in principle compute some of its properties.
This actually seems to be an important point to get right in thinking about FAI/metaethics. Human decision problem (friendliness content), the thing FAI needs to capture more formally, is (in my current understanding) something human minds can’t explicitly represent, can’t use definition of in their operation. Instead, they perform something only superficially similar.
For the record, I agree with what this comment means to me when I read it. :)
If that is what Luke is talking about then he’s making a good point.
How do these things relate to each other in the context of Luke’s statement (and which are relevant): mathematician’s pre-theoretic idea of a triangle; formal definition of a triangle; mathematician’s understanding of the formal definition; the triangle itself; mathematician’s understanding of their understanding of the formal definition; mathematician’s understanding of their pre-theoretic idea of a triangle; mathematician’s understanding of a triangle?
It seems to me that a very useful way of looking at what’s going on is that both pre-theoretic understanding and understanding strengthened by having a formal definition are ways of understanding the idea itself, capturing it to different degrees (having it control mathematician’s thought), with a formal definition being a tool for training the same kind of pre-theoretic understanding, just as a calculator serves to get to reliable answers faster. There is no clear-cut distinction, at least to the extent we are interested in actual answers to a question and not in actual answers that a given faulty calculator provides when asked that question (in which case we are focusing on a different question entirely, most likely a wrong one).
In that context, my understanding of Luke’s statement becomes “Mathematicians sometimes mistake B for A, but the degree of error involved in such a mistake depends on how closely B resembles A.”
C-G have nothing to do directly with the statement made, but personally I suspect that the relationship between F and E has a lot to do with the likelihood of mistaking B and A. I also think D is either overly vague or outright meaningless, and consequently G (assuming it exists) is either vague, complex or confused insofar as “triangle” in G means D.
To say that a little differently: there are things in the world that humans would categorize as triangles in common speech, which might not conform to the necessary and sufficient conditions of a triangle used by mathematicians. For example, shapes made out of curved lines and shapes whose vertices are curves might not qualify as triangles in the mathematician’s sense while still being triangles in the layman’s sense.
I do not agree that focusing attention on the question of what exactly is going on when laymen make that categorization is a wrong question, but I agree that it’s not a question about geometry.
I both agree and disagree that the layman’s and the mathematician’s understanding of “triangle” are both tools for understanding a single thing, which you’ve dubbed “the idea itself”, depending on just what you mean by that phrase. In other words, I don’t think the phrase “the idea itself” has a precise enough meaning to be useful in this sort of conversation. (This is also why I say that D and G are overly vague.)
By (A) I mean mathematician’s thoughts, and by (B) a syntactic definition. Since these things can’t possibly be confused, you must have read something different in my words… On the other hand, what these things talk about, that is whatever mathematician is thinking of, and whatever a formal definition defines, could be confused (given one possible reading, though not one I intended). Do you mean the latter?
You can use a formal definition to access an idea, but separately from how the mathematician in question uses it to access the same idea. Using whatever means you have, you look both at how someone understands an idea (i.e. what thoughts occur, that have cognitive as well as mathematical explanations), and at the idea, and see how accurately it’s captured.
Ah, sorry: I understood by B what it seems you actually meant by C.
I should have said: “Mathematicians sometimes mistake C for A, but the degree of error involved in such a mistake depends on on how closely B resembles A.” (The relationship between C and B is also a potential source of mathematicians’ error, of course, but irrelevant to the mistake being described here.)
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Re: your last paragraph… I certainly agree that for any X that exists outside of our minds, if I and a mathematician both construct separate understandings in our minds of X, it’s in principle possible to compare those understandings to X to determine which of us is more accurate.
So, for example, I agree that if that mathematician and I both look at this image:
http://a3.mzstatic.com/us/r1000/042/Purple/25/f7/a3/mzl.xofowuov.480x480-75.jpg
and she concludes it isn’t a triangle (perhaps because the vertices are curved, perhaps because the bottom right vertex is missing, perhaps because the lines have nonzero thickness, perhaps for other reasons) while I conclude that it is, it follows that if a triangle (or an idea of a triangle, or a formal definition of a triangle) is a thing that exists outside of our minds then we can compare our different understandings to the actual triangle/idea/definition to determine which of us is more accurate.
And I agree that it further follows that, as you say, to the extent that we are interested in actual answers to a question and not in actual answers that a given faulty calculator provides when asked that question (in which case we are focusing on a different question entirely, most likely a wrong one), to that extent we care about the more accurate of the two understandings more than the less accurate of them.
That said, I am not certain how I would go about actually performing that comparison between my understanding and the actual formal definition of a triangle that exists outside of both my mind and the mathematician’s, and I’m not convinced it makes sense to talk about doing so.
Just to be clear, I’m not trying to go all postmodern on you here. I understand how I would use a bucket full of pebbles to keep track of a pen full of sheep and I’m perfectly happy to say that there’s a fact of the matter about the pebbles and the sheep and that it’s possible for me to be wrong about that and to determine the magnitude of my error through observation of the world, and how all of this depends on the fact that various things I can do with pebbles allow me to differentially predict what will happen when I count the sheep in the pen.
What I don’t understand what experience of the world I’d be differentially predicting when I say that this rather than that is the actual formal definition of a triangle.
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?