Today's Times features an oped which is sure to excite a swarm of comment. The author, a retired political scientist at one of the senior CUNY schools named Andrew Hacker, asks "Is algebra necessary", in the sense of "is the sine qua non status of algebra in our elementary-to-collegiate education tracking justified". His response is decidedly negative.
Now, normally I'd sarcastically begin my response with "There's only one thing wrong with this..." and then proceed to let the author know why he should leave and go feed penguins or something. Normally, this kind of attack begins from some deeply flawed assumptions, and it just gets worse from there. This piece, though, isn't that. So instead of sarcastically pointing out the one little thing the author got wrong, I think it's useful to mention the only thing the author got right: there is a deep and low-level problem with how we teach and learn school mathematics, and in particular with what role we expect mathematics instruction to play in a student's larger academic trajectory.
Sunday, July 29, 2012
Saturday, July 28, 2012
Monday, July 23, 2012
Saturday, July 21, 2012
Tourismus
Biked across the Golden Gate today. Fuckin gorgeous weather.
It was interesting watching the bikes of those I was biking near. Plenty, of course, were riding their own bikes, but a good percentage were tourists riding rentals, which of course were branded with the logo of the rental shop. The preferred method is to hang a bag on the front to hold the lock and simultaneously announce another satisfied customer for Blazing Saddles.
What was interesting was twofold: that so many people were in the market for a $30 day rental bike to cross the Gate (I patted myself on the back for paying half that due to negotiated rate via the hostel) and that, despite seeing several shops across the city offering this service, the vast majority had in fact rented from the aforementioned Mel Brooks-themed vendor.
Whoever opened up that shop can take a bath in the cash s/he made today.
Thursday, July 19, 2012
On modal logic and rights
There's been a lively discussion up on my Facebook wall for a while now. For those of you not in that loop, I've got an old friend, Joe, who holds and brandishes a clear and pure stripe of libertarian political and moral thinking, and will on occasion brandish it in comments on my wall. We used to do it in person, back when we both taught incorrigible youngsters how to disrespect their betters elders and I helped him figure out what the hell an integral domain is.
The current discussion centers on rights. The positions, roughly stated, stand as follows:
Everyone pretty well agrees that there is a sense, which I've been calling the statutory sense, in which rights are given/controlled/created/overseen by the state. In this sense, a statement such as "X has the right to T" or "X has the right to do A" means, definitionally, that the state does not claim to have the power to prevent X's having of T or doing of A; or often, that X must be provided with T through some mechanism the state sets up. (Ideally, what the state claims to not have the power to do is also a reality, but this sense of "right" is already quite an exercise in nominalisms, so I won't dig further in that direction.) A key fact here is that the state both can and frequently does deny rights-in-the-statutory-sense to people, and unabashedly so, even when those people vehemently claim that what is being denied to them is something they have a right to.
Clearly, what such a person means when they say that they, for example, have the right to carry a weapon at all times, or that they have the right to a free-tuition public education through the baccalaureate level, is then not this statutory sense, because neither of these named actions or goods is provided for by statute or recognized by the courts. (I don't want to get into the matter of branches and separation of powers here either.) Instead, I think it's pretty well agreed by everyone who's commented, that a claim like these is a moral claim. The differences start appearing when we start clarifying the detailed content of these claims.
I don't deny that I am at my most comfortable when making claims as formalizable as possible. (This may color my thinking, but I think it does leave me with fewer irreducible propositions.) So when I need to clarify a moral claim, my first instinct is to apply the tools of modal logic and use the standard deontic/modal \( \square \) and \( \diamond \) formulations. If \( \varphi \) is a factual proposition like "Heelfilcher drinks beer" or "the state prohibits X from voting", then \( \square \varphi \) is the statement that "It is obligatory that Heelfilcher drink beer", and \( \diamond \varphi \) is the statement "it is licit for the state to prohibit X from voting". (Textbooks frequently say "it is permissible" rather than "it is licit", but I don't like the imagery of someone giving "permission". Of course "licit" has its own problems, since etymologically it conjures up a legislator.) We have the following equivalences: \[ \square \varphi \equiv \neg \diamond \neg \varphi \qquad \diamond \varphi \equiv \neg \square \neg \varphi \]since it is morally obligatory to do an action iff it is illicit to omit that action, etc.
What does this mean? Well, among other things, it means that when I (i.e. me, speaking for myself) want to clarify a moral claim, I start by identifying actions (to be represented by statements with no modal operators, i.e. no claims about their (il)licitness or obligatory character), and then try to say what I'd like to about their morality with as little modality as possible. In fact, this interpretation of the modal operators is problematic in that it's not clear (at least initially) what a modal nesting of depth greater than 1 would mean. (Here \[ \text{Heelfilcher drinks beer} \]has modal depth zero, \[ \square ( \text{Heelfilcher drinks beer} ) \]depth one, and \[ \square \left( \square ( \text{Heelfilcher drinks beer} ) \right) \]depth two. The statements of depth zero and one have clear interpretations, as I've indicated above, and in an abstract Kripke semantics so does the one of depth two; but damned if I can make it make much sense in an actual context of moral philosophizing.)
One of the nice features of this, for me, is that it is a theorem of pure logic that no set of statements of fact ("such-and-such-is-the-case" or "X performs action A") can ever imply a statement of modality, unless that modal statement is tautologically true. This is encapsulated in an old maxim, sometimes attributed to Hume, to the effect that simply telling me what the state of the world is has no bearing at all on how the world ought to be.
Or in English, if you want to argue that something is right or wrong, if you want to argue that something is obligatory or illicit or permissible, then you cannot simply argue from empirical premises about the world around us. If I want to convince you of a moral claim, the argument must begin at shared moral premises.
I haven't told you what I mean by "X has the right to do A", have I? Well take a guess. Have you guessed? OK. Here's my answer. "X has the right to do A" means, definitionally (to me), that it is illicit for X to be prevented from doing A by the power vested in agents by the society X lives in. At the bare minimum, this means the state; but societal coercion is exercised by more than just the institutions of the state.
And how do we go about deciding the truth of such a statement? How are we to determine whether an exercise of government power is wronging someone under that government's power?
To tell the truth, I don't have a complete theory there. That's ok, academic careers in moral philosophy are made by advancing more or less comprehensive theories of things like state power -- apparently the field is still staggering about holding their ears from the whelming overtones of the great bell rung by Rawls -- and I'd be happy to come up with a reasonably simple and not-too-outre thought-experiment that clarifies some of the core disagreements between someone like Joe and myself. (By "clarify", I mean "swings one or more party's general position, so that the positions are less in conflict".)
You see, Joe's position is that "X has the right to T" means, definitionally, that were X the only person in existence, then X would have T; "X has the right to do A" means definitionally, that were X the last person on earth then X would have the right to do A. The second is formally circular; I've asked Joe to expand on his ideas, but I don't feel comfortable paraphrasing them yet any further than this. The first, on the other hand, is meaningful. (And clearly wrong, I might add -- the last person on earth might not in fact possess any steel tools, but who would claim that s/he does not have the right to all the steel tools s/he could amass?) And I contend that it falls afoul of my observation about modal implication above: the hypothesis is supposed to be that a (counterfactual, but that's irrelevant) person possesses some thing. This is a modal-depth-zero claim, and if the statement "X has a right to T" is to have any moral meaning above that of an empirical claim, it can't be equivalent to such a statement.
I had meant for this post to be a parable illustrating that Joe's definitions don't accord with commonly held moral intuitions. (I said in the Facebook discussion that his definition is frankly bizarre; I'm sticking to that story, no matter how many Ayn Rand quotes he summons up in support). It appears that the introductory material has swallowed that post, so I'll try this again tomorrow.
(Note: yes, I have been incredibly lazy while writing this post and haven't included links. I mean to clean that up some, but if you think one is in order somewhere, post a comment indicating where and I'll strongly consider including it.)
This post has been brought to you via the generous sponsorship of 21st Amendment Brewery's Back in Black IPA.
The current discussion centers on rights. The positions, roughly stated, stand as follows:
Everyone pretty well agrees that there is a sense, which I've been calling the statutory sense, in which rights are given/controlled/created/overseen by the state. In this sense, a statement such as "X has the right to T" or "X has the right to do A" means, definitionally, that the state does not claim to have the power to prevent X's having of T or doing of A; or often, that X must be provided with T through some mechanism the state sets up. (Ideally, what the state claims to not have the power to do is also a reality, but this sense of "right" is already quite an exercise in nominalisms, so I won't dig further in that direction.) A key fact here is that the state both can and frequently does deny rights-in-the-statutory-sense to people, and unabashedly so, even when those people vehemently claim that what is being denied to them is something they have a right to.
Clearly, what such a person means when they say that they, for example, have the right to carry a weapon at all times, or that they have the right to a free-tuition public education through the baccalaureate level, is then not this statutory sense, because neither of these named actions or goods is provided for by statute or recognized by the courts. (I don't want to get into the matter of branches and separation of powers here either.) Instead, I think it's pretty well agreed by everyone who's commented, that a claim like these is a moral claim. The differences start appearing when we start clarifying the detailed content of these claims.
I don't deny that I am at my most comfortable when making claims as formalizable as possible. (This may color my thinking, but I think it does leave me with fewer irreducible propositions.) So when I need to clarify a moral claim, my first instinct is to apply the tools of modal logic and use the standard deontic/modal \( \square \) and \( \diamond \) formulations. If \( \varphi \) is a factual proposition like "Heelfilcher drinks beer" or "the state prohibits X from voting", then \( \square \varphi \) is the statement that "It is obligatory that Heelfilcher drink beer", and \( \diamond \varphi \) is the statement "it is licit for the state to prohibit X from voting". (Textbooks frequently say "it is permissible" rather than "it is licit", but I don't like the imagery of someone giving "permission". Of course "licit" has its own problems, since etymologically it conjures up a legislator.) We have the following equivalences: \[ \square \varphi \equiv \neg \diamond \neg \varphi \qquad \diamond \varphi \equiv \neg \square \neg \varphi \]since it is morally obligatory to do an action iff it is illicit to omit that action, etc.
What does this mean? Well, among other things, it means that when I (i.e. me, speaking for myself) want to clarify a moral claim, I start by identifying actions (to be represented by statements with no modal operators, i.e. no claims about their (il)licitness or obligatory character), and then try to say what I'd like to about their morality with as little modality as possible. In fact, this interpretation of the modal operators is problematic in that it's not clear (at least initially) what a modal nesting of depth greater than 1 would mean. (Here \[ \text{Heelfilcher drinks beer} \]has modal depth zero, \[ \square ( \text{Heelfilcher drinks beer} ) \]depth one, and \[ \square \left( \square ( \text{Heelfilcher drinks beer} ) \right) \]depth two. The statements of depth zero and one have clear interpretations, as I've indicated above, and in an abstract Kripke semantics so does the one of depth two; but damned if I can make it make much sense in an actual context of moral philosophizing.)
One of the nice features of this, for me, is that it is a theorem of pure logic that no set of statements of fact ("such-and-such-is-the-case" or "X performs action A") can ever imply a statement of modality, unless that modal statement is tautologically true. This is encapsulated in an old maxim, sometimes attributed to Hume, to the effect that simply telling me what the state of the world is has no bearing at all on how the world ought to be.
Or in English, if you want to argue that something is right or wrong, if you want to argue that something is obligatory or illicit or permissible, then you cannot simply argue from empirical premises about the world around us. If I want to convince you of a moral claim, the argument must begin at shared moral premises.
I haven't told you what I mean by "X has the right to do A", have I? Well take a guess. Have you guessed? OK. Here's my answer. "X has the right to do A" means, definitionally (to me), that it is illicit for X to be prevented from doing A by the power vested in agents by the society X lives in. At the bare minimum, this means the state; but societal coercion is exercised by more than just the institutions of the state.
And how do we go about deciding the truth of such a statement? How are we to determine whether an exercise of government power is wronging someone under that government's power?
To tell the truth, I don't have a complete theory there. That's ok, academic careers in moral philosophy are made by advancing more or less comprehensive theories of things like state power -- apparently the field is still staggering about holding their ears from the whelming overtones of the great bell rung by Rawls -- and I'd be happy to come up with a reasonably simple and not-too-outre thought-experiment that clarifies some of the core disagreements between someone like Joe and myself. (By "clarify", I mean "swings one or more party's general position, so that the positions are less in conflict".)
You see, Joe's position is that "X has the right to T" means, definitionally, that were X the only person in existence, then X would have T; "X has the right to do A" means definitionally, that were X the last person on earth then X would have the right to do A. The second is formally circular; I've asked Joe to expand on his ideas, but I don't feel comfortable paraphrasing them yet any further than this. The first, on the other hand, is meaningful. (And clearly wrong, I might add -- the last person on earth might not in fact possess any steel tools, but who would claim that s/he does not have the right to all the steel tools s/he could amass?) And I contend that it falls afoul of my observation about modal implication above: the hypothesis is supposed to be that a (counterfactual, but that's irrelevant) person possesses some thing. This is a modal-depth-zero claim, and if the statement "X has a right to T" is to have any moral meaning above that of an empirical claim, it can't be equivalent to such a statement.
I had meant for this post to be a parable illustrating that Joe's definitions don't accord with commonly held moral intuitions. (I said in the Facebook discussion that his definition is frankly bizarre; I'm sticking to that story, no matter how many Ayn Rand quotes he summons up in support). It appears that the introductory material has swallowed that post, so I'll try this again tomorrow.
(Note: yes, I have been incredibly lazy while writing this post and haven't included links. I mean to clean that up some, but if you think one is in order somewhere, post a comment indicating where and I'll strongly consider including it.)
This post has been brought to you via the generous sponsorship of 21st Amendment Brewery's Back in Black IPA.
Wednesday, July 18, 2012
San Francisco, Day 0
Uneventful flight to foggy San Francisco, unless you count extreme heat causing the insides of those metal sausages we fly around in to attempt to cook us on the tarmac. The flight crew literally requested that we keep our windowshades down until we reached you-can-now-use-your-MP3-player altitude to help keep the interior temperature down.
Also: felt strangely at home on the BART. Possibly because the cars are the same model that populate the DC Metro (actually, the Metro may have gotten some degree of facelift in the past few years, I can't exactly remember). Anyway, you gotta agree that the pay-as-you-exit system is much more equitable than a flat fare.
I'll try to do a little bit of blogging every day that I'm here, to get into some better habits of reflecting and verbalizing. And possibly engaging in political hectoring.
Also: felt strangely at home on the BART. Possibly because the cars are the same model that populate the DC Metro (actually, the Metro may have gotten some degree of facelift in the past few years, I can't exactly remember). Anyway, you gotta agree that the pay-as-you-exit system is much more equitable than a flat fare.
I'll try to do a little bit of blogging every day that I'm here, to get into some better habits of reflecting and verbalizing. And possibly engaging in political hectoring.
Monday, July 2, 2012
Shocking news
In today's round of news that is more surprising for actually seeing print than for its actual content,
- Support for Obamacare spiked after Thursday's SCOTUS ruling. Did the news coverage actually inform people what was in the bill? Do Americans have faith that the Court would have overturned Death Panels as unconstitutional? Or are most of us just authoritarians, ready to support whatever is the law of the land?
- Scully gets the scoop on Anderson Cooper. Centrists salute!
Subscribe to:
Posts (Atom)