Since the belief in invariance of physical laws with respect to space and time translations is a pre-requisite to do science, and given that Noether's theorem establishes the equivalence between those types of invariance and the laws of conservation of momentum and energy, can one conclude that those laws were a priori necessary (synthetic a priori) ?


This is an ibteresting question; I suspect rather the invariance of physical laws with respect to translation in time & space is synthetic a priori.

First, the distinction that Kant made between analytic & synthetic propositions is this:

analytic proposition: a proposition whose predicate concept is contained in its subject concept

synthetic proposition: a proposition whose predicate concept is not contained in its subject concept

This distinction is mutually exclusive, a proposition must be one or the other. As the predicate 'invariance under translation' is not contained in the subject 'physical laws', we see that it must be a synthetic proposition.

Further, the distinction between a posterio & a priori is this:

a priori proposition: a proposition whose justification does not rely upon experience. Moreover, the proposition can be validated by experience, but is not grounded in experience. Therefore, it is logically necessary.

a posteriori proposition: a proposition whose justification does rely upon experience. The proposition is validated by, and grounded in, experience. Therefore, it is logically contingent.

Now, to translate the universe by one metre is not something that can be done in practise, it can only be done in the imagination; that it is a thought or gedankenexperiment in Einsteins nomenclature. Thus it cannot be an a posteriori proposition, thus it must be a priori proposition.

Thus, we have shown that invariance of physical law with respect to space, and similarly for time, is synthetic a priori.

Now, what about the conservation of momentum? This is easily seen to be synthetic, and because its conservation has been demonstrated by experiment, it is a posteriori.

But, Noethers theorem demonstrates a link between the synthetic a priori invariance of law under space translation and the synthetic a posteriori conservation of momentum. This explains the conceptual importance of Noethers theorem, for it establishes a link between a contingent law (a posteriori) with a neccessary one (a priori), and it is this that makes it a deep theorem; and it also underlines the importance of conservation laws - one could say with some justification that they are, despite appearances, logically neccessary.

It also raises a puzzling question; for how can a theorem establish a link between a contingent and a neccessary one? I suspect that the explanation in our contingently human experience - we directly experience translation but not momentum. One could suppose, in some other possible world our conditions for experience may be so structured that momentum is a direct perception but translation is not and is a derived concept; or that both are directly percieved.

  • You perfectly reformulated my question: how can a theorem establish a link between a contingent (synthetic a posteriori) truth (a conservation law), and a necessary (synthetic a priori) truth (invariance with respect to space and time) ? In other words : was it necessary that modern scientists "discovered" conservation laws, since they postulate invariance in time and space ? – Julien Nov 16 '13 at 13:28

No, the laws of conservation are not synthetic a priori. They are "idealizations" of observed empirical facts. Determining whether the "idealized" conservation law are "perfectly true" requires a bit more interaction between theory and experiments. I'm not even sure that the currently accepted belief is that they really are "perfectly true locally", or just true for macroscopic objects and sufficient long time intervals.

No assumptions about physical laws are a priori necessary to do science. For example, if you do chemistry, the laws governing chemical properties are more important for you than assumptions about the fundamental laws of physics.

  • It seems that something 'close' to laws of conservation are required. For example, chemistry probably couldn't be studied if molecules were popping into and out of existence too much of the time. Imagine if instead of ≥ ħ/2, HUP were ≥ some large number. Then the mind's ability to learn how the world works might be severely curtailed. – labreuer Nov 15 '13 at 23:42

There are several of holes in your statement:

First, The assumption of translation invariance is not pre-requisite of science. A great deal of ancient and medieval science was done w/out such assumption, which goes back only as far as Galileo.

Second, Noether's theorem applies not to physics but to a mathematical model of physics. It relates two mathematical concepts: geometric invariance under Lie group action (translations, rotations) and first integrals (called energy, momentum, angular momentum) of dynamical flows given by differential equations of classical mechanics. Fitness of these equations to the physical universe is empirical matter, which is of course synthetic aposteori.

Third, conservation laws of physics don't hold apriori; we conclude that they hold either by direct measurement (which is synthetic aposteori) or by a combination of fitting a mathematical model to the observed nature (also synthetic aposteori) and deriving the conserved quantities from analytic conclusions of the model (Noether's theorem). And the combination of synthetic aposteori and analytic apriori is still synthetic aposteori.

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