Relativity & Objectivity

To the Editors:

In his extremely critical review of Robert Nozick’s Invariances: The Structure of the Objective World [NYR, June 27], Colin McGinn completely dismisses the central idea of Nozick’s book, the idea of objectivity as “invariance under transformations.” In support of this idea, Nozick quotes Paul Dirac, who wrote, “The important things in the world appear as the invariants…of… transformations,”¹ to which McGinn objects: “The quotation from Dirac does not in fact support Nozick’s proposal, since Dirac speaks of ‘importance’ and not ‘objectivity’ (important for what—calculation? the formulation of laws?).” But Dirac himself explained what sort of importance he had in mind. A few lines after the statement just quoted, he writes:

The growth of the use of transforma-tion theory, as applied first to relativity and later to the quantum theory, is the essence of the new method in theoreti-cal physics. Further progress lies in the direction of making our equations in-variant under wider and still wider transformations. This state of affairs is very satisfactory from a philosophical point of view, as implying an increasing recognition of the part played by the observer in himself introducing the regularities that appear in his observations, and a lack of arbitrariness in the ways of nature….

This obviously expresses an idea of ob-jectivity, and Nozick is completely justified in appealing to Dirac’s authority. Of course, the importance of invariances can be seen in different ways. To Max Born the notion of “invariance under transformations” is the essential component of our scientific notion of “reality”² ; to Eugene Wigner it is important with regard to the laws of nature³ ; but there can be no doubt that many others explicitly considered it as intimately con-nected with the notion of “objectivity.” Be-sides Dirac one should mention, first and foremost, the mathematician Hermann Weyl who was almost obsessed by this connec-tion. In his beautiful little book Symmetry he tersely says, “Objectivity means invari-ance with respect to the group of automor-phisms,”⁴ and the central source of this idea, and its most perspicuous domain of application, is the theory of relativity.⁵ When McGinn writes that “it appears quite wrong to construe relativity theory as as-serting that spatial and temporal intervals are not objective or are subjective, while the appropriate spatiotemporal intervals are deemed objective,” what McGinn de-scribes as “quite wrong” here is exactly what Weyl was devoted to and what Dirac regarded as “very satisfactory from a philo-sophical point of view.”

The idea of objectivity as invariance under transformations is nothing but a variant of Nozick’s third characteristic of objectivity: that an objective truth “holds independently of people’s beliefs, desires, hopes, and observations and measurements.” In the special theory of relativity, for example, what is transformed are the arbitrarily chosen inertial frames which serve as the bases of our observations and measurements; and invariance under these transformations then simply means independence of these bases—which obviously can be considered as a special case of Nozick’s third characteristic. And it is equally obvious that, and how, this case can be generalized in the ways displayed by Nozick. Furthermore, the idea of independence of arbitrary or contingent factors (of which inertial frames are only a special example) leads to another idea, the idea of “degrees of objectivity,” which too is dismissed by McGinn. But this idea actually suggests itself when one recognizes that there can be more or less factors, and factors of different sorts and weights, on which the truths of science may depend. Of course, this can be plausibly described as resulting in “more or less objectivity.” In view of the obviousness and naturalness of these ideas, McGinn’s vehement rejection of all of them is more than puzzling.

Felix Mühlhölzer
Professor of Philosophy
Universität Göttingen
Göttingen, Germany

Colin McGinn replies:

Felix Mühlhölzer is wrong to suggest that I completely dismiss Nozick’s idea of a con-nection between invariance and objectiv-ity; my point was that it entirely depends upon what the invariance involves. As I noted, the invariance characteristic of the Fahrenheit and centigrade scales is a case that exemplifies objectivity, since the two scales are a matter of human choice and convention (and hence not independent of states of mind). But such invariance only exemplifies objectivity in a case like this, not when the invariance is with respect to nonmental facts—for example, invariance of shape under change of position. In such a case the varying facts are themselves objective—as position surely is. Moreover, a point Mühlhölzer completely omits to mention, Nozick’s account runs into trou-ble when it comes to mental facts, since these can exhibit invariance too, despite being subjective. Nozick’s analysis of the notion of objectivity supplies neither a necessary nor a sufficient condition for a fact being objective—despite its working in certain limited contexts.

On relativity theory Mühlhölzer makes a familiar and unwarranted slide: from inertial frames to observations and measurements made within them. That spatial and temporal intervals are relative to an inertial frame does not imply that they are somehow dependent on our human observations; and of course they are not, since there were spatial and temporal intervals before there were any observing humans. The idea that relativity theory involves some sort of mind-dependence is just a bad philosophical interpretation of a good piece of physics (the same sort of misinterpretation is often made of quantum theory). Nozick, to his credit, does not make this mistake.

Lastly, Mühlhölzer thinks that the notion of degrees of objectivity makes sense and I do not. Of course, as I remarked in my review, views can be more or less objective (this is Nozick’s second notion of objectivity); the question is whether facts can be more or less objective. Since factual objectivity is basically a matter of mind-independence, it is easy to see that it cannot admit of degrees, since a fact is either mind-dependent or it is not. I am not sure what Mühlhölzer has in mind by speaking of the truths of science being dependent on various “factors”; I would have thought the only thing they depend upon is whether the world is as they say it is. Is there some kind of quasi-Kantian subjectivism at the back of his mind?