Wednesday, September 24, 2008

Heis on Reed on Kant and Frege

Jeremy Heis has a useful review of Reed's recent book on Kant and Frege. Heis raises a number of issues for the way Reed structures his discussion, so the review is also helpful as a road map for where future discussion should go.

More on Traffic

The article cited in the last post makes a number of interesting claims about the "phase transition" from free flow traffic to a traffic jam. While such transitions had been reproduced in simulations, it was apparently only recently that they have been reproduced in experimental contexts. Exactly what this shows about traffic and phase transitions more generally is less than clear, but the video is worth watching a few times.

Saturday, September 20, 2008

Traffic and Shock Waves

As explained in elementary terms here traffic can be modeled using a density function ρ(x, t) and a flux function j(ρ(x, t)), i.e. we assume that the number of cars passing through a point at a given time is a function of the density of cars at that point. Making certain continuity assumptions, we can obtain a conservation law

ρ_t + j’(ρ)ρ_x = 0

where subscripts indicate partial differentiation and j’ indicates differentiation with respect to ρ. If we make j(ρ)=4ρ(2-ρ) and start with a discontinuous initial density distribution like

1 if x <= 1
1/2 if 1< x <=3
2/3 if x > 3

Then we can show how the discontinuity persists over time and changes.

Such persisting discontinuities are called shock waves and appear as lines across which the density changes discontinuously. For example, in this figure we have lines of constant density intersecting at x = 3. The philosophical question is "what are we to make of this representation of a given traffic system?" That is, what does the system have to be like for the representation of a shock wave to be correct? My suggestion is that we need only a thin strip around x = 3 where the density changes very quickly, i.e. so quickly that a driver crossing it would have to decelerate to zero speed. Then, on the other side of the strip, the driver experiences a dramatic drop off in density, and so can accelerate again. Still, there is something a bit strange in talking about shock waves in traffic cases where the number of objects involved is so small, as opposed to fluid cases where many more fluid particles interact across a shock wave. Here, then, I would suggest that we have a case where the mathematics works, but we are less than sure what it is representing in the world.

See this New Scientist article (and amusing video) for the claim that shock waves can be observed in actual traffic experiments (summarizing this 2008 article).

Sunday, September 14, 2008

Steven Weinberg on "Without God"

Physicist Weinberg offers some extended reflections ($) on science and religion before concluding that "there is a certain honor, or perhaps just a grim satisfaction, in facing up to our condition without despair and wishful thinking". Sensible advice, especially in the wake of David Foster Wallace's grim demise. (Carroll has posted a Wallace passage on Cantor.)

On a lighter note, Weinberg offers an amusing analogy with religion without religious belief:
To compare great things with small, people may go to college football games mostly because they enjoy the cheerleading and marching bands, but I doubt if they would keep going to the stadium on Saturday afternoons if the only things happening there were cheerleading and marching bands, without any actual football, so that the cheerleading and the band music were no longer about anything (75).

Zach is Back!

After an inexcusable period of light posting, Richard Zach is back on track with a host of updates on the world of logic, philosophy of mathematics and beyond. Especially useful is this summary of the new issue of the Review of Symbolic Logic.

Thursday, September 4, 2008

Rayo's "On Specifying Truth-Conditions"

This is a long, innovative and frustrating article (Phil. Review 117 (2008): 385-443), at least for someone like me who is not at all inclined to fictionalism or other non-standard approaches to mathematical language. But for those, like Rayo, who think that “expressions with identical syntactic and inferential roles can perform different semantic jobs” (415fn22) this paper may represent the current state of the art.

Rayo aims to defend the stability or coherence of a form of noncommittalism according to which the use of mathematical language does not commit one to the existence of any abstract mathematical objects. An important negative point that Rayo makes early on is that the noncommittalist is unable to accomplish this task by “translating each arithmetical sentence into a sentence that wears its ontological innocence on its sleeze” (385). Still, the positive program is to set out a method of specifying associated truth conditions for these sentences so that these conditions do not involve abstract objects, but only the world being a certain way. The key innovation, if I am not misunderstanding the paper, is to employ the full range of semantic machinery in explaining how the world has to be for a given sentence to be correct. Rayo carefully explores the issue of whether this is legitimate, and concludes that if one begins as a noncommittalist, then one will feel entitled to appeal to numbers and functions in one’s semantic theory. As a result, the noncommittalist can claim an internally coherent or stable package of views.

While much of the interest of the paper is in the details, the following frank admission by Rayo struck me as worth highlighting:
Whatever its plausibility as an explanation of how the standard arithmetical axioms might be rendered meaningful in such a way that their truth is knowable a priori, it should be clear that the arithmetical stipulation is not very plausible as an explanation of how the axioms were actually rendered meaningful or how it is that we actually acquire a priori knowledge of their truth (431-432).
A committalist seems well within her rights to wonder, then, what the point is of carving out a stable position like Rayo’s if it has no bearing on our actual mathematical knowledge or on the actual contribution that mathematics makes in applied contexts.