(Also incorrect is that people "don't want to cycle to work", I see literally hundreds of them every morning in SF, and it's growing every year.)
There are many arguments in favor of cycling infrastructure, but unfortunately I'm versed enough in anti-bike comment threads to recognize someone who won't be convinced :) But here goes...
* bikes do less damage to road than cars, and therefore the maintenance cost of bike lanes is lower than car lanes, which require frequent paving
* bikes are important as a "last mile" transportation option for public transit commuters. densely-placed bike share stations improve the feasibility of public transit for people whose homes or offices are 1-2 miles from a train station.
* bikes are far smaller than cars and so you can push more traffic through a given width of road than with cars. this is especially relevant if you ever get to a downtown stretch where one car turning right at a pedestrian crossing causes an entire green light to be wasted on the rest of the cars (basically every light along Market Street in SF). cars are just wide and clumsy when driving in dense downtowns.
* the government frequently subsidizes behaviors that are considered virtuous (or conversely, imposes "sin taxes"), and also subsidizes services that benefit lower-income citizens. both of those apply here. car parking in downtowns can run $40+ for a day. bike share and bus commutes are $2-5 and keep people active.
I could go on, but again, I don't think your mind will be changed much.