Measurements are always established a priori. Go grab a meter stick and put it down somewhere. Then walk backwards and look at it again. Empirically, t looks smaller. But you rationally understand that it hasnt gotten smaller, you just changed your point of view, and that the measurement is consistent.
Same thing with the concept of mass. Mass is the amount of matter in the object, while weight is the gravitational energy in the object, these had to be discovered and differentiated a priori. It is difficult to measure mass, but it is easy to measure weight. Scientists rationally use the weight of an object to find the mass.
Same thing with numbers. No one test the number 5.5 it is rationally understood that it is equal to 4 + 1.5 or 2.75 + 2.75.
Physical sciences are rational because they depend on measurements, numbers, statistics, etc. which are established a priori.
Measurments require experience with the world, and hence are a posteriori judgments.
Again, this measurment requires experience with the world, and is hence no longer a priori. In both this example, and the above you are equivocating the word “a priori”, and “rational”. Plus, it requires experience to learn that weight can lead to a measurment of mass.
Indeed you are correct, if you’re interested; I would add that investigations in the physical sciences have even revealed some rather spooky consequences based on some of the physical laws that have been attained. For instance not only does the 2nd law of thermodynamics break the time reversibillity implicit in mechanics, with the aid of probabillity being explained by an increase in disorder(or a decrease in information if one is an information theorist); on a deeper level, without it there would be no way to establish time or causality in any meaningful way in the first place.
Also, in Quantum Mechanics the best information you can have a priori to measurements of a particle is simply knowledge of its wavefunction, from which one can obtain only knowledge of a probability distribution of a set a priori. Yet this is not the knowledge of any set in the ordinary frequentist interpretation of the term, yet only corresponds to the distribution of the same kind of particle passed through a slit for example repeatedly. No mechanical picture can be formed and such allusions are immediately destroyed as soon as one tries to observe the particle “on its way” before the regular point of measurement, creating a different distribution, changing the wavefunction by measurement.
In a sense Mises was right in 1949, we physicists have had to give up some of our “god like” delusions about the relation between us and our experiments! Empiricism like that of Einstein and Mach’s was seriously challenged in the heyday of Quantum Mechanics, things have not been the same since.