In a long footnote in The Principles of Psychology, William James discussed the analytic/synthetic dichotomy.
He consigned it to a footnote because he was sick of the subject, thinking it a "waste of ink and paper." W.V.O. Quine would come to much the same conclusion, 60 years later, after much more paper and ink had been employed.
James provides us in the course of this footnote with a geometrical example of why the effort to maintain a consistently dichotomous relationship between these two sorts of statements is such a waste.
"No one will say that such analytic judgments as 'equidistant lines can nowhere meet' are pure tautologies. The predicate is a somewhat new way of conceiving as well as of naming the subject. There is something 'ampliative' in our greatest truism, our state of mind is richer after than before we have uttered them."
Now, if he had written "parallel lines can never meet" he would he stated neither a synthetic nor an analytic truth but … a falsehood, as we since Einstein's work have come to understand the
space of this cosmos. In standard definition, parallel lines are lines that each cross some third line at a 90 degree angle. If we draw three lines on a piece of paper that meet this qualification,
the two that are "parallel" will never cross each other because they'll stay the same distance away from each other at every point (they'll stay "equidistant" in James word). But notice, equidistance
doesn't define parallellism. The two 90 degree angles define it.
This is why, if you had a piece of paper the size of a continent, the experiment might no longer work. If the think of the equator as the base line, then the various meridians cross it at 90 degree angles,
yet they come closer to each other the further they get from the equator, and they intersect with each other, at the two poles. Likewise, if space itself is curved, then we can't be certain that
parallel lines however abstractly conceived will always fail to intersect.
The development of non-Euclidean geometry actually preceded Einstein's life and James' working life. Two great mathematicians, Gauss and Lobachevsky, were working out such systems in the early
1850s. Still, Gauss and Lobachevsky thought they were engaged in a thoroughly hypothetical endeavor, pure mathematics. They had no idea they might be describing real space. This shows that a pragmatic theory of truth or knowledge has to be fairly subtle to accommodate such situations. G & L weren't doing or trying to do anything "useful" in any very mundane sense of that word.
James didn't have any inkling of the Einsteinian revolution about to engulf our understanding of the physical world, and he had only a nodding acquaintance with the then-recent developments in geometry that paved the way for it. So we are left to speculate why he said "equidistant" there instead of "parallel." At any rate, it was a lucky word choice, because it has prevented people from saying "what an obsolete example" and writing it off.
Clearly, in anyone's geometry, if two lines really are two, and they really are equidistant, they never meet. If they are two than they aren't zero distance apart. So they are some distance apart, and by
definition will always stay that difference apart (however curvy that might require them to become, in a curvy space!), so they never meet.
I think James' conclusion is quite germane. Even with "two equidistant lines will never meet" we can see that the predicate adds something to our grasp of the subject, helps us to grasp it in a
different light than we otherwise might.
He consigned it to a footnote because he was sick of the subject, thinking it a "waste of ink and paper." W.V.O. Quine would come to much the same conclusion, 60 years later, after much more paper and ink had been employed.
James provides us in the course of this footnote with a geometrical example of why the effort to maintain a consistently dichotomous relationship between these two sorts of statements is such a waste.
"No one will say that such analytic judgments as 'equidistant lines can nowhere meet' are pure tautologies. The predicate is a somewhat new way of conceiving as well as of naming the subject. There is something 'ampliative' in our greatest truism, our state of mind is richer after than before we have uttered them."
Now, if he had written "parallel lines can never meet" he would he stated neither a synthetic nor an analytic truth but … a falsehood, as we since Einstein's work have come to understand the
space of this cosmos. In standard definition, parallel lines are lines that each cross some third line at a 90 degree angle. If we draw three lines on a piece of paper that meet this qualification,
the two that are "parallel" will never cross each other because they'll stay the same distance away from each other at every point (they'll stay "equidistant" in James word). But notice, equidistance
doesn't define parallellism. The two 90 degree angles define it.
This is why, if you had a piece of paper the size of a continent, the experiment might no longer work. If the think of the equator as the base line, then the various meridians cross it at 90 degree angles,
yet they come closer to each other the further they get from the equator, and they intersect with each other, at the two poles. Likewise, if space itself is curved, then we can't be certain that
parallel lines however abstractly conceived will always fail to intersect.
The development of non-Euclidean geometry actually preceded Einstein's life and James' working life. Two great mathematicians, Gauss and Lobachevsky, were working out such systems in the early
1850s. Still, Gauss and Lobachevsky thought they were engaged in a thoroughly hypothetical endeavor, pure mathematics. They had no idea they might be describing real space. This shows that a pragmatic theory of truth or knowledge has to be fairly subtle to accommodate such situations. G & L weren't doing or trying to do anything "useful" in any very mundane sense of that word.
James didn't have any inkling of the Einsteinian revolution about to engulf our understanding of the physical world, and he had only a nodding acquaintance with the then-recent developments in geometry that paved the way for it. So we are left to speculate why he said "equidistant" there instead of "parallel." At any rate, it was a lucky word choice, because it has prevented people from saying "what an obsolete example" and writing it off.
Clearly, in anyone's geometry, if two lines really are two, and they really are equidistant, they never meet. If they are two than they aren't zero distance apart. So they are some distance apart, and by
definition will always stay that difference apart (however curvy that might require them to become, in a curvy space!), so they never meet.
I think James' conclusion is quite germane. Even with "two equidistant lines will never meet" we can see that the predicate adds something to our grasp of the subject, helps us to grasp it in a
different light than we otherwise might.
Comments
Post a Comment