Hilbert's Program Then and Now

Hilbert's program was an ambitious and wide-ranging project in the philosophy and foundations of mathematics. In order to "dispose of the foundational questions in mathematics once and for all, "Hilbert proposed a two-pronged approach in 1921: first, classical mathematics should be formalized in axiomatic systems; second, using only restricted, "finitary" means, one should give proofs of the consistency of these axiomatic systems. Although Godel's incompleteness theorems show that the program as originally conceived cannot be carried out, it had many partial successes, and generated important advances in logical theory and meta-theory, both at the time and since. The article discusses the historical background and development of Hilbert's program, its philosophical underpinnings and consequences, and its subsequent development and influences since the 1930s.Comment: 43 page

A Spectroscopic Study of the Environments of Gravitational Lens Galaxies

(Abridged) We present the first results from our spectroscopic survey of the environments of strong gravitational lenses. The lens galaxy belongs to a poor group of galaxies in six of the eight systems in our sample. We discover three new groups associated with the lens galaxies of BRI 0952-0115 (five members), MG 1654+1346 (seven members), and B2114+022 (five members). We more than double the number of members for another three previously known groups around the lenses MG 0751+2716 (13 total members), PG 1115+080 (13 total members), and B1422+231 (16 total members). We determine the kinematics of the six groups, including their mean velocities, velocity dispersions, and projected spatial centroids. The velocity dispersions of the groups range from 110 +170, -80 to 470 +100, -90 km/s. In at least three of the lenses -- MG0751, PG1115, and B1422 -- the group environment significantly affects the lens potential. These lenses happen to be the quadruply-imaged ones in our sample, which suggests a connection between image configuration and environment. The lens galaxy is the brightest member in fewer than half of the groups. Our survey also allows us to assess for the first time whether mass structures along the line of sight are important for lensing. We first show that, in principle, the lens potential may be affected by line-of-sight structures over a wide range of spatial and redshift offsets from the lens. We then quantify real line-of-sight effects using our survey and find that at least four of the eight lens fields have substantial interloping structures close in projection to the lens, and at least one of those structures (in the field of MG0751) significantly affects the lens potential.Comment: Accepted for publication in the Astrophysical Journal. Figure 6 posted as a JPEG image. Requires emulateapj.st

Towards a Neo-Copenhagen Interpretation of Quantum Mechanics

The Copenhagen interpretation is critically considered. A number of ambiguities, inconsistencies and confusions are discussed. It is argued that it is possible to purge the interpretation so as to obtain a consistent and reasonable way to interpret the mathematical formalism of quantum mechanics, which is in agreement with the way this theory is dealt with in experimental practice. In particular, the essential role attributed by the Copenhagen interpretation to measurement is acknowledged. For this reason it is proposed to refer to it as a neo-Copenhagen interpretation