Kuhn 3: Incommensurability
Jan. 23rd, 2009 04:06 pm![[personal profile]](https://www.dreamwidth.org/img/silk/identity/user.png){kind=small}
koganbot"Incommensurability" is a metaphor. For the most part, what Kuhn calls "incommensurable" are words (though of course, when words don't match, the numbers you'd get from actually "measuring" what the words are supposed to designate would be suspect). Kuhn's metaphor compares words to geometry: The hypotenuse of an isosceles right triangle is incommensurable with its side in that there's no way you can measure one against another and get an integer. There's always some remainder. So, by analogy, certain crucial terms from Aristotle's physics are incommensurable with Newton's physics in that you can't find terms in the latter that can match up with those crucial terms from the former without there being a leftover, a remainder, a residue.
Here's a brief version (from the Preface to The Essential Tension, 1977) of what Kuhn said in more detail in "What Are Scientific Revolutions?":
Aristotle's subject was change-of-quality in general, including both the fall of a stone and the growth of a child to adulthood. In his physics, the subject that was to become mechanics was at best a still-not-quite-isolable special case. More consequential was my recognition that the permanent ingredients of Aristotle's universe, its ontologically primary and indestructible elements, were not material bodies but rather the qualities which, when imposed on some portion of omnipresent neutral matter, constituted an individual material body or substance. Position itself was, however, a quality in Aristotle's physics, and a body that changed its position therefore remained the same body only in the problematic sense that the child is the individual it becomes. In a universe where qualities were primary, motion was necessarily a change of state rather than a state.
So there's no way that Aristotle's concepts of position and motion can find a home in Newton's physics, where motion is a state, rather than a change in state, and position is not a quality, so change in position is not a quality.
I'd add to this the idea that a common measure between two things is often some third thing, e.g. you can compare the heat in Alaska and heat in Hawaii by seeing how high mercury rises in a tube in one place in comparison to the other. Metaphorically, I'll say that you can "measure" two competing theories by checking them against what we variously call the facts or the data or sense impressions or reality etc. So if my theory predicts that there will be a giraffe in the closet, and yours predicts there won't be a giraffe in the closet, we can figure out who's right by looking in the closet and seeing if there's a giraffe. And if your theory predicts that there's a square circle in the closet, we don't even have to look, since a square circle is impossible, and we know this by definition.** However, say that according to my theory, a body in motion stays in motion unless acted on by an outside force, and you say that that's as absurd as calling circles square, since you know that motion is asymmetrical change in quality that leads to a final end result, e.g., a stone falling towards its resting place in the center or a man returning from sickness to health.
Given those differences between my theory and yours, what can we look at to resolve them, or what definition can we consult? How can we tell just by looking at him that a man being restored to health is an example of motion, when you've already decided that it is and I've already decided that it isn't?
These are not rhetorical questions that are intended to lead you to answer "Nothing!" and "We can't!" Actually, I don't like the term "incommensurable" and wish that Kuhn had used some other, but that's for a later discussion. The question I'll leave you with is this: Kuhn has shown - decisively, to my mind - that there's no single transcendent measure that can decide certain issues, such as which concepts of motion and position to use. But does this mean that there is no way whatsoever to compare the differing concepts, no types of "measures" beyond those two grand transcendent measures: facts and definitions? After all, starting in the late middle ages, in a process that took several hundred years, one type of physics (the Aristotelian) evolved into and was displaced by another (the Newtonian), and people thought they had good reasons with every step on that road. Did they? Were they right? Or was their choice just a matter of personal taste? If so, why did Newton's ideas achieve such unanimity among physicists after Newton? And when Newton was finally overturned by quantum physics, why did every physicist after 1928 subscribe to quantum physics?* Compare to nonscientific disciplines, where there are competing schools and so forth and where the competition never seems to have a period of let-up.
*This isn't meant to imply that I have any idea if the disagreements now between string theorists and others amount to a break in unanimity in regard to basic terms. I'm guessing that the unanimity isn't broken, but I'm hardly the person to know.
**EDIT: Kuhn might well have objected to my using the phrase "by definition" here, since one reason for his use of the term "paradigm" is that he believed we learn how to use a term by using it (when a science is not undergoing a revolution the scientists model their subsequent uses of terms like, say, "force," "mass," etc. on how they were taught to use it in looking at examples or solving textbook problems or being given experiments to conduct, not by looking up the definitions of these terms, which are rarely adequate to tell you how to apply them). But I'm using the term "by definition" here idiomatically, where "by definition" and "by meaning" and "by how we use it" are synonymous, and none of those phrases are really explanatory.
Here's a brief version (from the Preface to The Essential Tension, 1977) of what Kuhn said in more detail in "What Are Scientific Revolutions?":
Aristotle's subject was change-of-quality in general, including both the fall of a stone and the growth of a child to adulthood. In his physics, the subject that was to become mechanics was at best a still-not-quite-isolable special case. More consequential was my recognition that the permanent ingredients of Aristotle's universe, its ontologically primary and indestructible elements, were not material bodies but rather the qualities which, when imposed on some portion of omnipresent neutral matter, constituted an individual material body or substance. Position itself was, however, a quality in Aristotle's physics, and a body that changed its position therefore remained the same body only in the problematic sense that the child is the individual it becomes. In a universe where qualities were primary, motion was necessarily a change of state rather than a state.
So there's no way that Aristotle's concepts of position and motion can find a home in Newton's physics, where motion is a state, rather than a change in state, and position is not a quality, so change in position is not a quality.
I'd add to this the idea that a common measure between two things is often some third thing, e.g. you can compare the heat in Alaska and heat in Hawaii by seeing how high mercury rises in a tube in one place in comparison to the other. Metaphorically, I'll say that you can "measure" two competing theories by checking them against what we variously call the facts or the data or sense impressions or reality etc. So if my theory predicts that there will be a giraffe in the closet, and yours predicts there won't be a giraffe in the closet, we can figure out who's right by looking in the closet and seeing if there's a giraffe. And if your theory predicts that there's a square circle in the closet, we don't even have to look, since a square circle is impossible, and we know this by definition.** However, say that according to my theory, a body in motion stays in motion unless acted on by an outside force, and you say that that's as absurd as calling circles square, since you know that motion is asymmetrical change in quality that leads to a final end result, e.g., a stone falling towards its resting place in the center or a man returning from sickness to health.
Given those differences between my theory and yours, what can we look at to resolve them, or what definition can we consult? How can we tell just by looking at him that a man being restored to health is an example of motion, when you've already decided that it is and I've already decided that it isn't?
These are not rhetorical questions that are intended to lead you to answer "Nothing!" and "We can't!" Actually, I don't like the term "incommensurable" and wish that Kuhn had used some other, but that's for a later discussion. The question I'll leave you with is this: Kuhn has shown - decisively, to my mind - that there's no single transcendent measure that can decide certain issues, such as which concepts of motion and position to use. But does this mean that there is no way whatsoever to compare the differing concepts, no types of "measures" beyond those two grand transcendent measures: facts and definitions? After all, starting in the late middle ages, in a process that took several hundred years, one type of physics (the Aristotelian) evolved into and was displaced by another (the Newtonian), and people thought they had good reasons with every step on that road. Did they? Were they right? Or was their choice just a matter of personal taste? If so, why did Newton's ideas achieve such unanimity among physicists after Newton? And when Newton was finally overturned by quantum physics, why did every physicist after 1928 subscribe to quantum physics?* Compare to nonscientific disciplines, where there are competing schools and so forth and where the competition never seems to have a period of let-up.
*This isn't meant to imply that I have any idea if the disagreements now between string theorists and others amount to a break in unanimity in regard to basic terms. I'm guessing that the unanimity isn't broken, but I'm hardly the person to know.
**EDIT: Kuhn might well have objected to my using the phrase "by definition" here, since one reason for his use of the term "paradigm" is that he believed we learn how to use a term by using it (when a science is not undergoing a revolution the scientists model their subsequent uses of terms like, say, "force," "mass," etc. on how they were taught to use it in looking at examples or solving textbook problems or being given experiments to conduct, not by looking up the definitions of these terms, which are rarely adequate to tell you how to apply them). But I'm using the term "by definition" here idiomatically, where "by definition" and "by meaning" and "by how we use it" are synonymous, and none of those phrases are really explanatory.
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no subject
Date: 2009-01-23 11:43 pm (UTC)a: "there's no way you can measure one [ie the hypotenuse] against another [ie one of the two same-length sides] and get an integer" -- what's meant here by "measure one side against another"?
b: "there's always some remainder" -- how? where? what? the act of measurement hasn't been clearly defined (see a) and besides, measurement doesn't "leave a remainder" (division leaves a remainder): what's meant by "remainder"
this is a one-time maths teacher (and philosophy of maths student) being picky obvously, but given that you're questing kuhn when he says that "you can't define a scientific revolution, you can only give examples," we should be super-exact i think which models precisely are being said to be incommensurable (and to need a revolution to move between them)
my guess would be that kuhn is restating the just-so story* collapse of the (pre-pythagorean?) model, which stated that any measurable length can be expressed, if not as an integer, then as a ratio of integers (these are known as rational numbers): but it's quite easy to prove that if the length of the short sides of a right isoceles can be expressed as a ratio of integers, then the ratio of either of the shorts sides (call them S) with the long side (call it H) is not expressible as a ration of integers -- viz that if S is a rational number S/H cannot be...
(in this interpretation S/H would be the result of "measuring one side against another", as per (a) above)
even so it's not clear what exactly the "remainder" is: there are actually an infinite number of rational approximations to the non-rational H, getting ever closer, so the "remainder" may not be a number at all, in thesense of any particular number -- it could be (to get a bit meta) the stubborn fact that H is not rational; or it could be all the numbers that aren't rational (there are an infinite number of these -- though this infinity is infinitely laerger that than the infinity in the first line of this paragraph!)
the above is technical and possibly hard to follow, esp. w/o a diagram -- but give or take it being late at night and me not stating as well as i could, ii think this is the geometrical phenomenon you're citing
the reason i'm being picky is that i don't know what it is in this example that you're calling a "residue" -- and not knowing that makes it hard to judge how the analogy works between this and the aristotle-newton example
(and the reason THIS is an issue is precisely because there's an overall question how possible -- or useful -- it is to generalise these various shifts into a broadly encompassing concept of "revolution")
*just-so story because it isn't i think known whether this was an actual historical conundrum leading to a revolutionary shift in perspective (about the nature of number) or a way of teaching number, as a drama, to demonstrate a fact about it
no subject
Date: 2009-01-23 11:52 pm (UTC)for example: you're saying incommensurability in geometry is a metaphor -- for kuhn -- for incommensurability in words; but i'm not sure if that's true -- isn't it possible he's arguing that incommensurability between models is actually incommensurability between geometries?
ok now i will go and actually read this piece properly, over the weekend!
no subject
Date: 2009-01-24 06:28 am (UTC)What Kuhn says about "incommensurability" as a metaphor isn't in that piece, but here is what Kuhn wrote in another one, not trying to explain the math, though I don't think it's crucial to get the math exact (which is to say that if what I wrote above is a misstatement, it's not one that matters). This is from the piece "Commensurability, Comparability, Communicability" which he wrote in 1982 and which is the second piece in The Road Since Structure:
Remember briefly where the term "incommensurability" came from. The hypotenuse of an isosceles right triangle is incommensurable with its side or the circumference of a circle with its radius in the sense that there is no unit of length contained without residue an integral number of times in each member of the pair. There is thus no common measure. But lack of a common measure does not make comparison impossible. On the contrary, incommensurable magnitudes can be compared to any required degree of approximation. Demonstrating this could be done and how to do it were among the splendid achievements of Greek mathematics. But that achievement was possible only because, from the start, most geometric techniques applied without change to both of the items between which comparison was sought.
Applied to the conceptual vocabulary deployed in and around a scientific theory, the term "incommensurability" functions metaphorically. The phrase "no common measure" becomes "no common language." The claim that two theories are incommensurable is then the claim that there is no language, neutral or otherwise, into which both theories, conceived as sets of sentences, can be translated without residue or loss. No more in its metaphorical than its literal form does incommensurability imply incomparability, and for much the same reason. Most of the terms common to the two theories function the same way in both; their meanings, whatever those may be, are preserved; their translation is simply homophonic. Only for a small subgroup of (usually interdefined) terms and sentences containing them do problems of translatability arise.
...
It is not clear, however, that incommensurability can be restricted to a local region. In the present state of the theory of meaning, the distinction between terms that change meaning and those that preserve it is at best difficult to explicate or apply. Meanings are a historical product, and they inevitably change over time with changes in the demands on the terms that bear them. It is simply implausible that some terms should change meaning when transferred to a new theory without infecting the terms transferred with them...
So, my using the phrase "measure one against another" may have confused things (I honestly don't know if it is or isn't compatible with "no unit of length contained without residue an integral number of times in each member of the pair"), but the purpose of the metaphor is to produce the statement "there is no language into which both theories can be translated without residue or loss."
His phrase "no common language" is confusing, however, since one can explain and understand both Aristotle and Newton fully in English. And even if we say "well, Aristotle's concept of motion isn't part of modern English usage, even if it can be explained in modern English, so in effect we're adding it to English rather than translating it into current English," there are other incommensurable activities - think of how the term "fruit" is used in botany and how this differs from its use in cookery, or how "out" is used differently in tennis from how it's used in baseball, and even how the use of "out" differs within baseball (an infielder can throw a batter out at first, and the umpire can throw the player out of the game for bad sportsmanship [and I don't know if those baseball terms translate to cricket or not]) - where all uses of the terms are part of modern English. What Kuhn really means is something like "the language of Newtonian physics" and "the language of Aristotelian physics" - there's no common language of physics within which Aristotelian and Newtonian physics can make up a unified whole that functions in concord.
no subject
Date: 2009-01-24 06:29 am (UTC)no subject
Date: 2009-01-24 07:48 am (UTC)(2) the reason i'm being picky is that i don't know what it is in this example that you're calling a "residue" -- and not knowing that makes it hard to judge how the analogy works between this and the aristotle-newton example
I'd say (for our purposes in this convo, not overall) that we should draw a distinction between "metaphor" and "analogy," the latter needing to be tighter than the former - though of course there need not be a sharp or specifiable boundary between "metaphor" and "analogy," any more than there's a boundary between "warm" and "hot." And actually the distinction I'm drawing is between "metaphor" and "small-p paradigm." Kuhn's comparison between geometry and language is a metaphor that gives him a word ("incommensurable") and the related words "residue" and "loss." But I very much doubt that geometry here was a model that helped generate Kuhn's idea (my guess is that the concept of meanings not matching up was already in Kuhn's mind and he hit upon the example from geometry simply as what he hoped would be an effective way to express this concept). A crucial difference between "incommensurability" in geometry and "incommensurable paradigms" is that in geometry the hypotenuse and the sides are nonetheless part of the same triangle, despite their incommensurability, whereas Aristotelian motion and Newtonian motion aren't and can't be part of the same theory. (Question: to what extent does incommensurability result in incompatibility? Must it always?)
In any event, the analogy between geometry and word meanings is not a tight one, the examples from geometry not being models of what Kuhn means by "incommensurability" when applied to scientific theories (whereas the examples Kuhn gives in "What Are Scientific Revolutions?" of the difference between Aristotelian and Newtonian motion and the difference between Planck's resonators and his oscillators are good models for "incommensurability," even though he doesn't use the word in that essay); so examining the geometry in a deep way probably isn't necessary for understanding the difference between Aristotle's and Newton's concept of motion and why there's residue and loss there, which seems easy to understand even if we'd not known that the word "incommensurable" had its source in mathematics.
(3) (and the reason THIS is an issue is precisely because there's an overall question how possible -- or useful -- it is to generalise these various shifts into a broadly encompassing concept of "revolution")
Key words in thinking about this question would be "cumulative," "noncumulative," and "holistic."
no subject
Date: 2009-01-24 02:45 pm (UTC)no subject
Date: 2009-01-24 03:00 pm (UTC)(for an engineer, "usage" -- ie meaning actually building bridges that don't fall down -- trumps "usage" meaning "uses the same language as mathematicians; and thus totally sidesteps the issue of the residue of incommensurability that does indeed arise when looking at (and measuring) mathematically perfect right isoceles triangles... in effect, an engineer can approximate this residue to nothing, if he does it the right way, and not have his bridge fall down)