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lukeprog comments on Living Metaphorically - Less Wrong
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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.
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).
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?