Re: [math-fun] sesqui-imponderables
On Sun, Jul 20, 2014 at 1:18 AM, Andy Latto <andy.latto@pobox.com> wrote: On Sat, Jul 19, 2014 at 12:49 PM, Adam P. Goucher <apgoucher@gmx.com> wrote: 1. ((a (chicken)) and (a (half))), not (a ((chicken) and (a (half)))). That doesn't explain why it isn't plural; compound subjects of the form "A and B" are normally plural. Frog and Toad are Friends. Andy So >1 isn't essential for plural, else "Two half-wits writes a grammar book, ..." --rwg
Committing my usual error of wading into a frivolous conversation with my Serious Hat on... Number in grammar is not a mathematical concept; it's a purely formal one. The "number" of a noun phrase, and hence the verb form you have to pick to agree with it, is determined by the formal structure of the phrase and of its component words, *not* by the meaning of the phrase. Now, different people can have form-agreement rules that differ in various details. For me, "A chicken and a half lays ..." feels wrong (even without the subjunctivizing "if"), so the phrase *is* plural for me. If the subject isn't plural for Gosper and Andy, it must be because they are (in some sense) parsing it differently, so that it has singular *structure*. My guess is (and I can't be certain, because I don't have their linguistic intuitions to guide me) that for them, the and-a-half construction converts the noun phrase from a "count noun" to a "mass noun"; mass nouns always parse as singular in American English. On Sun, Jul 20, 2014 at 5:24 AM, Bill Gosper <billgosper@gmail.com> wrote:
On Sun, Jul 20, 2014 at 1:18 AM, Andy Latto <andy.latto@pobox.com> wrote:
On Sat, Jul 19, 2014 at 12:49 PM, Adam P. Goucher <apgoucher@gmx.com> wrote:
1. ((a (chicken)) and (a (half))), not (a ((chicken) and (a (half)))).
That doesn't explain why it isn't plural; compound subjects of the form "A and B" are normally plural.
Frog and Toad are Friends.
Andy
So >1 isn't essential for plural, else "Two half-wits writes a grammar book, ..."
--rwg _______________________________________________ math-fun mailing list math-fun@mailman.xmission.com https://mailman.xmission.com/cgi-bin/mailman/listinfo/math-fun
Yes, you would say `a sesquichicken lays ...' not `a sesquichicken lay ...'. -- Adam P. Goucher
Sent: Sunday, July 20, 2014 at 1:41 PM From: "Allan Wechsler" <acwacw@gmail.com> To: math-fun <math-fun@mailman.xmission.com> Subject: Re: [math-fun] sesqui-imponderables
Committing my usual error of wading into a frivolous conversation with my Serious Hat on...
Number in grammar is not a mathematical concept; it's a purely formal one. The "number" of a noun phrase, and hence the verb form you have to pick to agree with it, is determined by the formal structure of the phrase and of its component words, *not* by the meaning of the phrase. Now, different people can have form-agreement rules that differ in various details. For me, "A chicken and a half lays ..." feels wrong (even without the subjunctivizing "if"), so the phrase *is* plural for me. If the subject isn't plural for Gosper and Andy, it must be because they are (in some sense) parsing it differently, so that it has singular *structure*. My guess is (and I can't be certain, because I don't have their linguistic intuitions to guide me) that for them, the and-a-half construction converts the noun phrase from a "count noun" to a "mass noun"; mass nouns always parse as singular in American English.
On Sun, Jul 20, 2014 at 5:24 AM, Bill Gosper <billgosper@gmail.com> wrote:
On Sun, Jul 20, 2014 at 1:18 AM, Andy Latto <andy.latto@pobox.com> wrote:
On Sat, Jul 19, 2014 at 12:49 PM, Adam P. Goucher <apgoucher@gmx.com> wrote:
1. ((a (chicken)) and (a (half))), not (a ((chicken) and (a (half)))).
That doesn't explain why it isn't plural; compound subjects of the form "A and B" are normally plural.
Frog and Toad are Friends.
Andy
So >1 isn't essential for plural, else "Two half-wits writes a grammar book, ..."
--rwg _______________________________________________ math-fun mailing list math-fun@mailman.xmission.com https://mailman.xmission.com/cgi-bin/mailman/listinfo/math-fun
_______________________________________________ math-fun mailing list math-fun@mailman.xmission.com https://mailman.xmission.com/cgi-bin/mailman/listinfo/math-fun
Note that the rules for European English are different; European English has a class of nouns that are morphologically singular, but formally plural. We say "Congress has ..." and they say "Parliament have ...". These nouns are mostly "corporate" nouns with arguably plural semantics. American English doesn't have any such corporate plurals. On Sun, Jul 20, 2014 at 11:11 AM, Adam P. Goucher <apgoucher@gmx.com> wrote:
Yes, you would say `a sesquichicken lays ...' not `a sesquichicken lay ...'.
-- Adam P. Goucher
Sent: Sunday, July 20, 2014 at 1:41 PM From: "Allan Wechsler" <acwacw@gmail.com> To: math-fun <math-fun@mailman.xmission.com> Subject: Re: [math-fun] sesqui-imponderables
Committing my usual error of wading into a frivolous conversation with my Serious Hat on...
Number in grammar is not a mathematical concept; it's a purely formal one. The "number" of a noun phrase, and hence the verb form you have to pick to agree with it, is determined by the formal structure of the phrase and of its component words, *not* by the meaning of the phrase. Now, different people can have form-agreement rules that differ in various details. For me, "A chicken and a half lays ..." feels wrong (even without the subjunctivizing "if"), so the phrase *is* plural for me. If the subject isn't plural for Gosper and Andy, it must be because they are (in some sense) parsing it differently, so that it has singular *structure*. My guess is (and I can't be certain, because I don't have their linguistic intuitions to guide me) that for them, the and-a-half construction converts the noun phrase from a "count noun" to a "mass noun"; mass nouns always parse as singular in American English.
On Sun, Jul 20, 2014 at 5:24 AM, Bill Gosper <billgosper@gmail.com> wrote:
On Sun, Jul 20, 2014 at 1:18 AM, Andy Latto <andy.latto@pobox.com> wrote:
On Sat, Jul 19, 2014 at 12:49 PM, Adam P. Goucher <apgoucher@gmx.com> wrote:
1. ((a (chicken)) and (a (half))), not (a ((chicken) and (a (half)))).
That doesn't explain why it isn't plural; compound subjects of the form "A and B" are normally plural.
Frog and Toad are Friends.
Andy
So >1 isn't essential for plural, else "Two half-wits writes a grammar book, ..."
--rwg _______________________________________________ math-fun mailing list math-fun@mailman.xmission.com https://mailman.xmission.com/cgi-bin/mailman/listinfo/math-fun
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Mathematics are less confusing than English. Jim Propp On Sunday, July 20, 2014, Allan Wechsler <acwacw@gmail.com> wrote:
Note that the rules for European English are different; European English has a class of nouns that are morphologically singular, but formally plural. We say "Congress has ..." and they say "Parliament have ...". These nouns are mostly "corporate" nouns with arguably plural semantics. American English doesn't have any such corporate plurals.
On Sun, Jul 20, 2014 at 11:11 AM, Adam P. Goucher <apgoucher@gmx.com <javascript:;>> wrote:
Yes, you would say `a sesquichicken lays ...' not `a sesquichicken lay ...'.
-- Adam P. Goucher
Sent: Sunday, July 20, 2014 at 1:41 PM From: "Allan Wechsler" <acwacw@gmail.com <javascript:;>> To: math-fun <math-fun@mailman.xmission.com <javascript:;>> Subject: Re: [math-fun] sesqui-imponderables
Committing my usual error of wading into a frivolous conversation with my Serious Hat on...
Number in grammar is not a mathematical concept; it's a purely formal one. The "number" of a noun phrase, and hence the verb form you have to pick to agree with it, is determined by the formal structure of the phrase and of its component words, *not* by the meaning of the phrase. Now, different people can have form-agreement rules that differ in various details. For me, "A chicken and a half lays ..." feels wrong (even without the subjunctivizing "if"), so the phrase *is* plural for me. If the subject isn't plural for Gosper and Andy, it must be because they are (in some sense) parsing it differently, so that it has singular *structure*. My guess is (and I can't be certain, because I don't have their linguistic intuitions to guide me) that for them, the and-a-half construction converts the noun phrase from a "count noun" to a "mass noun"; mass nouns always parse as singular in American English.
On Sun, Jul 20, 2014 at 5:24 AM, Bill Gosper <billgosper@gmail.com <javascript:;>> wrote:
On Sun, Jul 20, 2014 at 1:18 AM, Andy Latto <andy.latto@pobox.com <javascript:;>> wrote:
On Sat, Jul 19, 2014 at 12:49 PM, Adam P. Goucher <apgoucher@gmx.com <javascript:;>> wrote:
1. ((a (chicken)) and (a (half))), not (a ((chicken) and (a (half)))).
That doesn't explain why it isn't plural; compound subjects of the form "A and B" are normally plural.
Frog and Toad are Friends.
Andy
So >1 isn't essential for plural, else "Two half-wits writes a grammar book, ..."
--rwg _______________________________________________ math-fun mailing list math-fun@mailman.xmission.com <javascript:;> https://mailman.xmission.com/cgi-bin/mailman/listinfo/math-fun
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Yes, that's what I should've written! Jim On Sun, Jul 20, 2014 at 11:54 AM, Jon Ziegler <jonz@alumni.caltech.edu> wrote:
On Jul 20, 2014, at 8:50 AM, James Propp <jamespropp@gmail.com> wrote:
Mathematics are less confusing than English.
Surely that should be "fewer". (JZ) _______________________________________________ math-fun mailing list math-fun@mailman.xmission.com https://mailman.xmission.com/cgi-bin/mailman/listinfo/math-fun
Or better still: Maths are fewer than English. Jim On Sunday, July 20, 2014, James Propp <jamespropp@gmail.com> wrote:
Yes, that's what I should've written!
Jim
On Sun, Jul 20, 2014 at 11:54 AM, Jon Ziegler <jonz@alumni.caltech.edu <javascript:_e(%7B%7D,'cvml','jonz@alumni.caltech.edu');>> wrote:
On Jul 20, 2014, at 8:50 AM, James Propp <jamespropp@gmail.com <javascript:_e(%7B%7D,'cvml','jamespropp@gmail.com');>> wrote:
Mathematics are less confusing than English.
Surely that should be "fewer". (JZ) _______________________________________________ math-fun mailing list math-fun@mailman.xmission.com <javascript:_e(%7B%7D,'cvml','math-fun@mailman.xmission.com');> https://mailman.xmission.com/cgi-bin/mailman/listinfo/math-fun
Hello, Adam. Noticed that CP^4 recently discussed SO(4) in terms of quaternions, and the fact that its double cover is S^3 x S^3. Thought I'd mention an interesting fact about this (that had me confused for a while ca. 25 years ago): SO(4) is actually homeomorphic is SO(3) x Spin(3), i.e., to P^3 x S^3. (For any g in SO(4), map it to g(1) (=g) in S^3. The stabilizer of this point is clearly a copy of SO(3). So topologically, SO(4) is an SO(3) bundle over S^3. This must be a trivial bundle over each D^3 hemisphere, so is determined by the identification of these two trivial bundles along the S^2 equator. But pi_2 of any Lie group is the trivial group, and so the bundle over S^3 is trivial to begin with. But, as you may know, SO(4) is not isomorphic to SO(3) x Spin(3) as a Lie group. --Dan
Dan, Indeed, that's very interesting. Observe that Spin(4) = Spin(3) x Spin(3) has an obvious centre of order 4 (namely {(1,1), (1,-1), (-1,1), (-1,-1)}). Every subgroup of this is normal (that's how centres work), so we get five nice normal subgroups. Considering their quotients, we have: Subgroup: {(1,1)} Quotient: Spin(4) = Spin(3) x Spin(3) Subgroup: {(1,1), (-1,1)} Quotient: Spin(3) x SO(3) Subgroup: {(1,1), (1,-1)} Quotient: SO(3) x Spin(3) Subgroup: {(1,1), (-1,-1)} Quotient: SO(4) Subgroup: {(1,1), (1,-1), (-1,1), (-1,-1)} Quotient: PSO(4) = SO(3) x SO(3) Consequently, SO(4) enjoys the status of being the double cover of SO(3) x SO(3), and having Spin(3) x Spin(3) as its own double cover! It seems really weird for a group like SO(4) to be neatly sandwiched between two direct squares in this way. Sincerely, Adam P. Goucher
Sent: Tuesday, July 29, 2014 at 12:49 AM From: "Dan Asimov" <dasimov@earthlink.net> To: math-fun <math-fun@mailman.xmission.com> Subject: Re: [math-fun] sesqui-imponderables
Hello, Adam.
Noticed that CP^4 recently discussed SO(4) in terms of quaternions, and the fact that its double cover is S^3 x S^3.
Thought I'd mention an interesting fact about this (that had me confused for a while ca. 25 years ago):
SO(4) is actually homeomorphic is SO(3) x Spin(3), i.e., to P^3 x S^3. (For any g in SO(4), map it to g(1) (=g) in S^3. The stabilizer of this point is clearly a copy of SO(3). So topologically, SO(4) is an SO(3) bundle over S^3. This must be a trivial bundle over each D^3 hemisphere, so is determined by the identification of these two trivial bundles along the S^2 equator. But pi_2 of any Lie group is the trivial group, and so the bundle over S^3 is trivial to begin with.
But, as you may know, SO(4) is not isomorphic to SO(3) x Spin(3) as a Lie group.
--Dan _______________________________________________ math-fun mailing list math-fun@mailman.xmission.com https://mailman.xmission.com/cgi-bin/mailman/listinfo/math-fun
Nice observations! In fact, I've seen numerous claims that the almost-squareness of SO(4) is the reason that 4-dimensional geometry is supposedly so strange. But as much as it's clear that 4-dimensional *topology* is extremely strange (e.g., the uncountably many inequivalent smooth structures on R^4 as compared with only 1 on every other R^n), I'm not that familiar with anomalies in 4D *geometry*. Other than the three regular polytopes not found in other dimensions, especially the 24-cell. --Dan On Jul 28, 2014, at 5:44 PM, Adam P. Goucher <apgoucher@gmx.com> wrote:
Indeed, that's very interesting.
Observe that Spin(4) = Spin(3) x Spin(3) has an obvious centre of order 4 (namely {(1,1), (1,-1), (-1,1), (-1,-1)}). Every subgroup of this is normal (that's how centres work), so we get five nice normal subgroups. Considering their quotients, we have:
Subgroup: {(1,1)} Quotient: Spin(4) = Spin(3) x Spin(3)
Subgroup: {(1,1), (-1,1)} Quotient: Spin(3) x SO(3)
Subgroup: {(1,1), (1,-1)} Quotient: SO(3) x Spin(3)
Subgroup: {(1,1), (-1,-1)} Quotient: SO(4)
Subgroup: {(1,1), (1,-1), (-1,1), (-1,-1)} Quotient: PSO(4) = SO(3) x SO(3)
Consequently, SO(4) enjoys the status of being the double cover of SO(3) x SO(3), and having Spin(3) x Spin(3) as its own double cover! It seems really weird for a group like SO(4) to be neatly sandwiched between two direct squares in this way.
I think you're exactly right about why it feels singular to me. "A chicken and a half lays an egg..." feels just like "Three and a half ounces of water weighs about a centigram" Thanks, Allan! Andy On Sun, Jul 20, 2014 at 8:41 AM, Allan Wechsler <acwacw@gmail.com> wrote:
Committing my usual error of wading into a frivolous conversation with my Serious Hat on...
Number in grammar is not a mathematical concept; it's a purely formal one. The "number" of a noun phrase, and hence the verb form you have to pick to agree with it, is determined by the formal structure of the phrase and of its component words, *not* by the meaning of the phrase. Now, different people can have form-agreement rules that differ in various details. For me, "A chicken and a half lays ..." feels wrong (even without the subjunctivizing "if"), so the phrase *is* plural for me. If the subject isn't plural for Gosper and Andy, it must be because they are (in some sense) parsing it differently, so that it has singular *structure*. My guess is (and I can't be certain, because I don't have their linguistic intuitions to guide me) that for them, the and-a-half construction converts the noun phrase from a "count noun" to a "mass noun"; mass nouns always parse as singular in American English.
On Sun, Jul 20, 2014 at 5:24 AM, Bill Gosper <billgosper@gmail.com> wrote:
On Sun, Jul 20, 2014 at 1:18 AM, Andy Latto <andy.latto@pobox.com> wrote:
On Sat, Jul 19, 2014 at 12:49 PM, Adam P. Goucher <apgoucher@gmx.com> wrote:
1. ((a (chicken)) and (a (half))), not (a ((chicken) and (a (half)))).
That doesn't explain why it isn't plural; compound subjects of the form "A and B" are normally plural.
Frog and Toad are Friends.
Andy
So >1 isn't essential for plural, else "Two half-wits writes a grammar book, ..."
--rwg _______________________________________________ math-fun mailing list math-fun@mailman.xmission.com https://mailman.xmission.com/cgi-bin/mailman/listinfo/math-fun
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-- Andy.Latto@pobox.com
participants (7)
-
Adam P. Goucher -
Allan Wechsler -
Andy Latto -
Bill Gosper -
Dan Asimov -
James Propp -
Jon Ziegler