2,657 thoughts on “Elon Musk Thinks Evolution is Bullshit.

  1. BruceS,

    I’ve gone to a bunch of Suffolk Symposia over the last couple of years and have generally been impressed with how well the speakers give papers without reading them. I have a lot of trouble with that myself.

  2. BruceS: FWIW, Putnam’s late papers on perception say positive things about Noe, sensorimotor perception, and Gibson affordances, although their ideas are only part of the story according to Putnam.

    That’s interesting. Where are the late papers you have in mind?

    I agree that that’s only part of the story. There are at least two other important components: a theory of the subpersonal neurophysiology that’s part of how direct realism gets causally implemented, and a theory of how language and other Good Tricks (imitation, gesture, culture, symbolism) helps give us sapient critters a degree of cognitive grip on the world that is more than just directly perceiving affordances.

    (Also: I just finished a paper on direct realism and critical realism. If you’d like I’ll send it to you.)

  3. walto: Thanks for the Button link. I listened to about 25 minutes of it, but I think he goes too fast and gobbles up too much territory with each sentence.

    I agree. So thanks, BruceS. And I agree that his pace is a bit high. But I can go relisten if I want time to think over the major points.

    And, to connect that with my earlier controversial statement, Tim Button is an alien from outer space.

    He describes problems of reference. And, a note to keiths – I am not having difficulty understanding it. However, it has nothing whatsoever to do with the natural language that I use. That’s what makes it alien.

    Button repeatedly asks (in different contexts), “What fixes reference?” Well, nothing fixes reference. Natural language doesn’t work by reference. We use meaning, not reference.

    For sure, we can have debates, arguments, etc, where reference is important. In those cases, the participants in the debate “fix” (or, really, negotiate) reference for the duration of the debate. But that’s very ad hoc. And it doesn’t always work out (people sometimes admit that they are talking past one another).

  4. Kantian Naturalist: That’s interesting. Where are the late papers you have in mind?

    (Also: I just finished a paper on direct realism and critical realism. If you’d like I’ll send it to you.)

    I’ll send the Putnam to you: it’s currently only available in his latest book (it is based on a speech he gave).

    I had not heard of critical realism as a philosophy of perception so please do send the paper you mention.

  5. Kantian Naturalist,

    Literature is entertainment. As for its usefulness, there is no telling what musing will trigger a useful idea.

    Ideas are the alleles of social systems. They come and go, sometimes selected, sometimes drifting. They are not causes.

  6. Kantian Naturalist: Quine doesn’t think that physics needs intentionality because he thinks that physics can be done entirely in mathematics, and Quine thinks (rightly or wrongly) that mathematics is purely extensional (since it is reducible to set theory). We can eliminate intentions because we can eliminate intensions.

    That seems fine to me; I was concerned because I understood you to saythat Putnam held the same views.

  7. BruceS: Quine doesn’t think that physics needs intentionality because he thinks that physics can be done entirely in mathematics

    I don’t really understand what ‘done in’ means there. What would it mean for physics not to be ‘entirely doable in mathematics’?

  8. BruceS: That seems fine to me; I was concerned because I understood you to say that Putnam held the same views.

    Ah, sorry! I never meant to suggest that Putnam was an eliminativist about intentionality or regarded intentional discourse as second-hand! He clearly was very much opposed to Putnam’s tendencies in some regards.

    I think of “the good Putnam” as being what he gets from Wittgenstein (also James and Dewey) and “the bad Putnam” as being what he gets from Quine. He’s good when he recognizes that reason cannot be naturalized, because that means — as he is using those terms in that context! — that intentionality and normativity cannot be transposed into an extensional semantics. But he’s bad when he insists that we can use an extensional semantics for doing ontology, as his criticism of metaphysical realism requires.

  9. walto:
    BruceS KN: Quine doesn’t think that physics needs intentionality because he thinks that physics can be done entirely in mathematics

    Walto: I don’t really understand what ‘done in’ means there. What would it mean for physics not to be ‘entirely doable in mathematics’?

    It was KN that said that bit about Quine, physics, and math, not me.

  10. BruceS: It was KN that said that bit about Quine, physics, and math, not me.

    Right, I know. Sorry. You did say it seems fine to you though.

  11. I might be deeply confused about a crucial issue here, but I’ll say it outright: if a language is to be about the world in which is used, then the users of that language must be able to keep track of and distinguish between good and bad inferences in the language and correct and incorrect applications of the language. This means that they need to distinguish between revising concepts to get a better referential grip on the same object and changing concepts so as to refer to different objects.

    Although there are ambiguous cases in the history of science (phlogiston, ether), the ambiguity is artificial, because philosophers post-Quine were so afraid of Meinong’s jungle with intensional entities and the weirdness of Fregean senses that they opted for overly austere desert landscapes instead.

    What they needed instead was the recognition that we must be able to distinguish between inference and reference, and therefore between concept and object, and therefore between intensions and extensions, and therefore between analytic and synthetic judgments. Put otherwise, the analytic/synthetic distinction is necessary for any conceptual framework that is to be used in the world. Morton White was right only that it ought not be an untenable dualism, and Quine’s criticisms of it merely presuppose the extensionalism that he prefers on dubious grounds.

    In mathematics, one is constructing a “pure” system of formal operations; it is not important to distinguish between concepts and objects. They fold into each other, so to speak. This is why a practicing mathematician can be a Platonist or an intuitionist or a fictionalist as you please. The ontology of mathematics is idle because there’s no principled basis for distinguishing between concepts and objects in an extensional semantics to begin with.

    But in science, and in everyday life, we are using frameworks and models to navigate the world and to improve our understanding of it. In those cases it is imperative to distinguish between inferences and references, between concepts and objects, and thus between intensions and extensions. Therefore no science — not even fundamental physics — can be done in a purely extensional semantics, whereas logic and mathematics can be and perhaps must be.

  12. Neil,

    It was intended to be about me (about my reaction to philosophy).

    Oh, please. It was a dig at philosophy and philosophers:

    All philosophical arguments are written by aliens from outer space, and addressed to other aliens from outer space.

    Or, at least, that’s the way that it often seems from this side of “The Two Cultures”.

    What sticks out like a sore thumb, is that philosophers use logic in a way that I would never think of using it, and in a way that I would not expect to get useful results. Putnam’s argument is just an extreme example of this.

  13. keiths:

    He’s [Neil is] dismissing what he doesn’t understand. I see you doing the same thing.

    petrushka:

    I don’t see that there is anything to understand.

    Right. You’re leaping from “I don’t understand this” to “there isn’t anything to understand”. It’s a bad inference.

  14. Kantian Naturalist: Put otherwise, the analytic/synthetic distinction is necessary for any conceptual framework that is to be used in the world.

    and

    Therefore no science — not even fundamental physics — can be done in a purely extensional semantics, whereas logic and mathematics can be and perhaps must be.

    Those both seem right to me.

  15. Neil Rickert: Kantian Naturalist: Put otherwise, the analytic/synthetic distinction is necessary for any conceptual framework that is to be used in the world.

    and

    Therefore no science — not even fundamental physics — can be done in a purely extensional semantics, whereas logic and mathematics can be and perhaps must be.

    Those both seem right to me.

    I have this strong vibe that both you and KN have taken peyote tonight.

  16. KN,

    Therefore no science — not even fundamental physics — can be done in a purely extensional semantics, whereas logic and mathematics can be and perhaps must be.

    Logic and mathematics neither can nor must be done in a purely extensional semantics.

    Consider the prime numbers. Good luck defining those extensionally!

    Even a set as straightforward as the integers can’t be defined extensionally, because it’s infinite.

  17. Kantian Naturalist,

    Pigose was an insult? I thought you would take it as an extreme compliment. I was praising your amazingly intelligent ability to be able to differentiate which truffles and cheetoes you stuff your face with.

    But I am curious as to why humans are eating themselves sick, if knowing what to eat is a form of intelligence. I guess not having your pignose has been quite helpful to me. Maybe if I was as smart as most humans I could be much fatter and diabetic, with corrosively clogged arteries. I sure am glad I am dumb.

    But then what does the lone wolf do….

  18. keiths: Even a set as straightforward as the integers can’t be defined extensionally, because it’s infinite.

    I might be quite badly confused.

    I thought that one can define a set as infinite if each subset of the set has the same cardinality as the set itself.

    Why is that not extensional?

  19. KN,

    To define a set extensionally, you explicitly list all of its members.

    To define it intensionally, you specify a defining property (or set of properties) that something must have in order to be included in the set.

  20. Neil,

    I’m not sure why you think fictionalism rescues you. The primes are defined intensionally by both Platonists and fictionalists.

  21. keiths:
    Neil,

    I’m not sure why you think fictionalism rescues you.The primes are defined intensionally by both Platonists and fictionalists.

    From SEP:

    Mathematics is typically extensional throughout—we happily write “\(1+4=2+3\)” even though the two terms involved may differ in meaning (more about this later).

    There seems to be some disagreement here regarding the meanings of “extensional” and “intensional”.

  22. Neil,

    First, you still haven’t explained why you think fictionalism helps you.

    Second, all the SEP author is saying is that “=”, like other operators, applies to extensions and not intensions. In other words, the left-hand intension “1+4” is replaced by its extension, 5, before the equality is evaluated. Likewise for the right-hand intension. 5 = 5, so the equality is true.

    The equality would be false if intensions were being compared instead of extensions, because “1+4” is a different intension from “2+3”.

  23. keiths:
    KN: Why is that not extensional?

    Keiths: To define a set extensionally, you explicitly list all of its members.

    To define it intensionally, you specify a defining property (or set of properties) thatsomething must have in order to be included in the set.

    I believe that extensioal and intensional can be applied to both terms and sentences/formulas.

    As Keith says, the extension of a term is the set of objects it applies to. So both “Sam Clemens” and “Mark Twain” have as their extension the person who wrote that book. Similarly, renates and cordates have in their extension (in the actual world) the same set of animals, since all animals with hearts have kidneys and conversely.

    Names/terms with the same extension are called co-extensive.

    When we talk of the intension of a term, we talk about its meaning. “Cordates” and “renates” have different meanings. As Keith says, the intension of a word refers to the way it picks out objects in order to define the set which is its extension.

    But we can also speak of an extensional context in a sentence or formula. An extensional context is one where co-referring terms can be substituted without changing the truth value of the sentence.* . For example
    Mark Twain wrote Huck Finn
    is extensional, since I can replace “Mark Twain” by “Sam Clemens” without changing truth value. Intensional contexts are not extensional. Psychological attitudes provide one example of intensional contexts. For example, suppose
    Joe believes Mark Twain wrote Huck Finn
    is true. If Joe does not know that Sam Clemens is Mark Twain, then the sentence becomes false when we make that substitution.

    Modal contexts are also intensional.
    Necessarily, 8 is greater than 7
    is true, but substituing the co-referring term “number of planets” for “8” yields a falsehold.

    Now since math involves neither psychological attitudes nor modality, I think it is correct to say math formulae are extensional.

    Is intentionality the same thing as intensionality? Intentionality involves psychological attitudes, so they overlap there. But intentionality does not involve modality, so they do differ at least there. In addition to that difference, some philosophers believe that the psychological attitude of knowledge by acquaintance through direct perception is not intensional.

    ————————–
    *(There is also a different type of extensional context relating to existential generalization that I will ignore).

  24. Kantian Naturalist:

    But he’s [Putnam’s] bad when he insists that we can use an extensional semantics for doing ontology, as his criticism of metaphysical realism requires.

    I would amend that to say he insists that extensional semantics can be used to criticize metaphysical realism as Putnam formulates that belief.

    I am going to retract my previous comment that Putnam does not require that the ideal theory can be formulated in an extensional, formal language, possibly first order logic. Listening more closely to Button in the section on Putnam “just more theory” gambit, and also skimming the relevant sections of Drew Khlentzos’ book, I see that this is in fact a controversial issue. Putnam thought he could use intensional semantics for his permutations arguments at least, and he thought he justified his claim to being able to formalize the ideal theory that an MR must believe is open to doubt.

    However, after providing a detailed review and analysis of the literature, Khlentzos concludes Putnam failed in that justification, and that is one reason why the MTA fails as an argument against MR according to Khlentzos.

    On the other hand, Khlentzos does believe that an MTA argument can be used against a Fodor-stype representational theory of mind (RTM), that is that thinking is nothing more than computing using the syntactic properties of an internal language of thought. So if I understand him correctly, Khlentzos is offering MTA as justification for the anti-RTM position, and I understand you share this anti-RTM view. So maybe you can appreciate the MTA on that basis?!

  25. walto: Right, I know. Sorry. You did say it seems fine to you though.

    The relevant KN quote that I said was fine was “Quine doesn’t think that physics needs intentionality because he thinks that physics can be done entirely in mathematics”

    I suspect an issue is what “done” is doing to the meaning of this sentence.

    I do think Quine thought the science is ideally expressed in first order logic (plus sets), and since math could be replaced by first order logic plus sets, math would serve as an ideal way to express science theories.

    Now I think that the theories of physics can be written using math.

    I also believe that the results of experiments in physics can be predicted using only math and logic. You have to provide an interpretation for the terms, of course. And I’m also a bit concerned about the role of statistical inference in experiments in physics. But leave that aside.

    Is that enough to say physics is “done” using math? No, because we have to account for all the pragmatic principles scientists use to choose the best theories of those that fit the data.

    Is that the type of concern you were alluding to?

    (Of course, Quine was well aware of this issue.)

  26. Neil Rickert:

    Button repeatedly asks (in different contexts), “What fixes reference?”Well, nothing fixes reference.Natural language doesn’t work by reference.We use meaning, not reference.

    Putnam is assuming some standard philosophy of language that you may not accept.

    First, there is a need to distinguish between what the meaning of a singular term is and how such a term acquires its meaning.

    On meaning: Roughly speaking, singular terms refer to a specific object in the world. Is that reference the same as the meaning of the term? Some say yes (Millians). But some (eg Fregeans) say no, that names have a “sense” (somewhat like an intension) that is used to establish the reference.

    There is also the related but separate question of how a term acquires a meaning. Putnam assumes that MRs will have a naturalistic, empirical theory of how this happens (ie not “magic”). A popular choice among philosophers, both when he formulated the argument and today, is that reference is established by the “right” type of causal history. That is the specific view Putnam attributes to MRs in his arguments against them.

  27. Neil Rickert: He describes problems of reference. And, a note to keiths – I am not having difficulty understanding it. However, it has nothing whatsoever to do with the natural language that I use. That’s what makes it alien.

    Of course, Putnam was both a philosopher and a mathematician (in particular, he was a co-author on a paper which was key to solving one of Hilbert’s ten problems).

    As a philosopher, he is known for changing many of his positions over his career. Perhaps we can use your “alien” theory to provide a psychological explanation of those changes? Namely, the cognitive dissonance caused by the warring of the philosophical and mathematical modes of thinking led to him to constantly revisit his positions.

  28. KN/BruceS: “Quine doesn’t think that physics needs intentionality because he thinks that physics can be done entirely in mathematics”

    I still have no idea what is supposed to be expressed by that remark. Is it a structural realism claim? Let me put it this way: if physics needed intentionality, could it not be “done in mathematics”? What are you two trying to say?

  29. walto: I still have no idea what is supposed to be expressed by that remark.Is it a structural realism claim?Let me put it this way: if physics needed intentionality, could it not be “done in mathematics”?What are you two trying to say?

    I think the word “done” is confusing.

    I would say that Quine thought the theories of physics could be expressed in math and that met his criteria that ideally scientific theories were expressed using first order logic and sets ( in order to derive ontological commitments).

    Now Quine knew that pragmatic considerations were used to choose theories in physics. As I understand Putnam to insist, those considerations involve human cognitive values and intentionality. So is Quine saying physics does need intentionality after all, at least if Putnam is right about the pragmatic considerations?

    Is that your concern?

  30. BruceS: Putnam is assuming some standard philosophy of language that you may not accept.

    First, there is a need to distinguish between what the meaning of a singularterm is and how such a term acquires its meaning.

    There is also the related but separate question of how a term acquires a meaning.

    I left something out. There is a third consideration: how we learn the meanings of words.

  31. BruceS: I think the word “done” is confusing.

    I would say that Quine thought the theories of physics could be expressed in math and that met his criteria that ideally scientific theories were expressed using first order logic and sets ( in order to derive ontological commitments).

    Now Quine knew that pragmatic considerations were used to choose theories in physics.As I understand Putnam to insist, those considerations involve human cognitive values and intentionality. So is Quine saying physics does need intentionality after all, at least if Putnam is right about the pragmatic considerations?

    Is that your concern?

    Not entirely. Again, I’d think everybody–including those who think intentions are not reducible–would expect their physics to utilize math. So the question is about the “entirely done.” I take it you are saying that the difference between the intentionalist and extentionalist philosophers here could be expressed by saying something like

    If we have every physical law and every current condition, then (leaving aside quantum effects) we can, using only math, predict every future event.

    Is that right?

    If so, you and KN have put it very strangely above, with the business about first order logic, ontological commitment, and who knows what all else. Some of the posts from last night looked like they were made from a very fun party.

  32. keiths,

    You are talking about being able to use your senses to know what food is safe, and KN seems to be talking about using the ability to communicate, after trial and error, to know what foods to eat.

    So I guess you all need to resolve your own internal conflicts before you worry too much about my understanding.

  33. BruceS: On meaning: Roughly speaking, singular terms refer to a specific object in the world. Is that reference the same as the meaning of the term? Some say yes (Millians). But some (eg Fregeans) say no, that names have a “sense” (somewhat like an intension) that is used to establish the reference.

    I do see names as distinct from nouns. But the distinction is not sharp, so I’d be more inclined to agree with the Fregeans there.

    However, I already have an issue with “refer to a specific object”. As I see it, a common sense object is not a logical object. A common sense object is really a narrow category. We find logical objects in places such as mathematics, but not in the real world.

  34. BruceS: Perhaps we can use your “alien” theory to provide a psychological explanation of those changes? Namely, the cognitive dissonance caused by the warring of the philosophical and mathematical modes of thinking led to him to constantly revisit his positions.

    That’s an interesting possibility.

  35. walto: Not entirely.Again, I’d think everybody–including those who think intentions are not reducible–would expect their physics to utilize math.So the question is about the “entirely done.”I take it you are saying that the difference between the intentionalist and extentionalist philosophers here could be expressed by saying something like

    If we have every physical law and every current condition, then (leaving aside quantum effects) we can, using only math, predict every future event.

    Is that right?

    No, I don’t recognize that as what I am trying to say,

    All I am saying is that Quine agreed physics theories could be expressed in math to start to meet his criteria for ideal expression of scientific theories.

    To get ontological commitments from that, I guess you’d need to somehow move to involving the existential operators, maybe through Ramsay sentences? I have no idea if the math formulas can be used in such formulations.

    Time for me to stop as (ETA: even I realize that) I am just hand-waving at this point.

    I will retract my support for the part of KN’s statement that relied on “done”, since I really don’t know what he meant, let alone what I might mean.

  36. Neil Rickert:

    However, I already have an issue with “refer to a specific object”.As I see it, a common sense object is not a logical object.A common sense object is really a narrow category.We find logical objects in places such as mathematics, but not in the real world.

    So right away I think your position differs from the MR position that Putnam was criticizing. So his arguments are not relevant to your thoughts (which BTW I sometime read as not that far from Putnam’s IR, except of course Putnam remained a scientific realist despite that IR).

  37. BruceS: The relevant KN quote that I said was fine was “Quine doesn’t think that physics needs intentionality because he thinks that physics can be done entirely in mathematics”

    I’m with walto, on not being clear on what that is supposed to mean.

    At first glance, it seems obviously wrong.

    I do think Quine thought the science is ideally expressed in first order logic (plus sets), and since math could be replaced by first order logic plus sets, math would serve as an ideal way to express science theories.

    That ignores the role of geometric thinking in both mathematics and physics.

    But, yes, Quine may well have thought that. It’s part of what has long bothered me about Quine.

  38. I think I’ve caused much confusion with my use of “done”, and for that I apologize.

    I think the confusion arises because it conflates the content and structure of the theories with the actual investigative practices of physicists. I ought to have said “expressed” rather than “done” — in short, that Quine thinks that physical theories can be expressed entirely in mathematics.

    The reason I object to Quine here is that a physical theory must be used— its terms must have referents which are observed or posited objects, processes, relations, events, etc. That’s not the case with a mathematical proof.

  39. Bruce,

    Now since math involves neither psychological attitudes nor modality, I think it is correct to say math formulae are extensional.

    Math does involve psychological attitudes and modality. For example, mathematical conjectures are considered likely to be true, though unproven.

    Equations are extensional in the sense that “=” signifies equality of extensions, not of intensions. But that hardly justifies KN’s (and Neil’s) contention:

    Therefore no science — not even fundamental physics — can be done in a purely extensional semantics, whereas logic and mathematics can be and perhaps must be.

    Mathematics would be hobbled without the use of intensional semantics. You’d never be able to refer to things like “the set of squares greater than 25”!

  40. keiths:

    Mathematics would be hobbled without the use of intensional semantics.You’d never be able to refer to things like “the set of squares greater than 25”!

    I’m perfectly willing to accept that as a provisional objection to Quine’s claim that mathematics is purely extensional.

  41. Kantian Naturalist: I’m perfectly willing to accept that as a provisional objection to Quine’s claim that mathematics is purely extensional.

    As in my long post on the matter above, I think there are different uses of “intensional” which are being confused. “Intension” and “extension” alone can be used to show the two ways we determine the set of objects a term refers to. But “intensional context” refers to sentences or formula*.

    Quine did a PhD correcting errors in Russell’s Principia relating to use and mention. Given that, I think I’ll trust Quine’s judgement on this issue. There is also the SEP link that Neil provided: that whole linked paragraph is worth reading.

    Here is a formal version of the definition of extensionality for formulas; it corresponds to the definition which I gave informally in previous long post.**

    Consider any two formulas A and B where B occurs as part of A. Further, consider a formula B* where B has same extension as A. Now replace all occurrences of B in A by B*, giving A*. Then a logical system is extensional iff A has same extension as A* (ie same truth value). Any logic which is not extensional is called intensional.

    Keith does point out that mathematicians doing math have psychological attitudes but I don’t think that affects the fact that mathematics is extensional in the above defined sense.

    (FWIW [not much], Keith’s point does however remind me of my failed attempts to explain the use of “done” in the quote from you on how Quine and physicists use math; see exchange with Walt.).

    ———————-
    * Of course, a single term can also be considered a subformula as per the definition I give later in the post, but why confuse things even more until people agree on the basic issue?

    ** From this Youtube video (starting at 6:00 for the definition).

  42. Bruce:

    Here is a formal version of the definition of extensionality for formulas; it corresponds to the definition which I gave informally in previous long post.**

    Consider any two formulas A and B where B occurs as part of A. Further, consider a formula B* where B has same extension as A. Now replace all occurrences of B in A by B*, giving A*. Then a logical system is extensional iff A has same extension as A* (ie same truth value). Any logic which is not extensional is called intensional.

    Bruce,

    That definition doesn’t make sense. Here’s the problem sentence:

    Further, consider a formula B* where B has same extension as A.

    That should read:

    Further, consider a formula B* where B* has same extension as B.

  43. Bruce,

    Keith does point out that mathematicians doing math have psychological attitudes but I don’t think that affects the fact that mathematics is extensional in the above defined sense.

    I was responding to your statement…

    Now since math involves neither psychological attitudes nor modality, I think it is correct to say math formulae are extensional.

    …and noting that mathematical conjectures involve both psychological attitudes and modality.

    So the reason math formulae can be considered extensional is not because math “involves neither psychological attitudes nor modality”; it’s because math formulae satisfy a particular definition of “extensional”. I described it this way to Neil:

    Second, all the SEP author is saying is that “=”, like other operators, applies to extensions and not intensions. In other words, the left-hand intension “1+4” is replaced by its extension, 5, before the equality is evaluated. Likewise for the right-hand intension. 5 = 5, so the equality is true.

    The equality would be false if intensions were being compared instead of extensions, because “1+4” is a different intension from “2+3”.

    You provided a definition which (after correction) says essentially the same thing:

    Consider any two formulas A and B where B occurs as part of A. Further, consider a formula B* where B* has same extension as B. Now replace all occurrences of B in A by B*, giving A*. Then a logical system is extensional iff A has same extension as A* (ie same truth value). Any logic which is not extensional is called intensional.

    So, yes, formulae are extensional by that definition, but that hardly means that KN (and Neil) are correct to say that

    Therefore no science — not even fundamental physics — can be done in a purely extensional semantics, whereas logic and mathematics can be and perhaps must be.

    Without intensional semantics, mathematics would be restricted to trivialities like “5 = 5”. You couldn’t even say “1+4 = 2+3”!

  44. keiths: Consider any two formulas A and B where B occurs as part of A. Further, consider a formula B* where B* has same extension as B. Now replace all occurrences of B in A by B*, giving A*. Then a logical system is extensional iff A has same extension as A* (ie same truth value). Any logic which is not extensional is called intensional.

    So, yes, formulae are extensional by that definition, but that hardly means that KN (and Neil) are correct to say that

    Therefore no science — not even fundamental physics — can be done in a purely extensional semantics, whereas logic and mathematics can be and perhaps must be.

    I still don’t know what KN was getting at there. I’d think that so long as any “formula” is either necessarily true or necessarily false, then (if we take the extensions of formulae as their truth values as is done above), each will meet that definition of “extensional.”

    If we take the laws of physics as its formulae, if we again substitute formula “parts” with the same (now contingent) truth values as the “parts” we’re substituting for, then, because of quantum uncertainties, maybe we could get whole formulae with different truth values than we started with. I don’t know. But what have we learned from that question one way or the other? And what is it supposed to have to do with Quine?

  45. keiths:

    Thanks for correcting the typo in my definition of extensional applied to formulas..

    I was concerned with KNs preceding post which said:

    I’m perfectly willing to accept that as a provisional objection to Quine’s claim that mathematics is purely extensional.

    My point is that even though mathematicians may use either intensional or extensional definitions for terms, one can still say that mathematical formula are extensional according to the definition.

    I believe it is that lack of intensional contexts that Quine was looking for in the language for his ideal theory. So mathematics is extensional in the way Quine wants.

    I agree that the word “done” has been confusing (and so does KN in one of his posts)

  46. walto: I still don’t know what KN was getting at there.I’d think that so long as any “formula” is either necessarily true or necessarily false, then (if we take the extensions of formulae as their truth values as is done above), each will meet that definition of “extensional.”

    If we take the laws of physics as its formulae, if we again substitute formula “parts” with the same (now contingent) truth values as the “parts” we’re substituting for, then, because of quantum uncertainties, maybe we could get whole formulae with different truth values than we started with.I don’t know.But what have we learned from that question one way or the other?And what is it supposed to have to do with Quine?

    It is not that the subformula being substituted for have to have the same truth value, only the same extension. It is the preservation of the truth value of the overall formula that then matters. So for example could substitute “8+1” for “9” in “9<10”.

    Also, I don’t think necessarity should be involved since that will lead to intensionality. So I am unclear on why you use it.

    I am not sure what you mean by “contingent”. To talk about the truth value of a formula, we do need an interpretation including definition of predicates/operators and mapping of names to objects. Is that what you mean by “contingent”?

    I don’t see how quantum indeterminacy enters the picture. Perhaps you had in mind that the results of measuring could vary so maybe the truth value of a formula would depend on that? I think the relevant QM formula would involve assigning probabilities to measurements, not stating an specific outcome, so that type of situation would not apply.

  47. Bruce,

    I’m still trying to figure out exactly why (and where) you are disagreeing with me.

    I agree that formulae are extensional by the definition you provided; that was exactly my point to Neil here.

    That does not mean that logic and mathematics “can be and perhaps must be” done “in a purely extensional semantics”, as KN and Neil were saying. To the contrary, mathematics (including its formulae) relies on intension. This beautiful result…

    [latex]
    \int_{0}^{1}\frac {x^4(1-x)^4}{1+x^2} dx = \frac {22}{7} – \pi
    [/latex]

    …is chock full of intension.

    Could you describe where you think our disagreement lies?

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