Every measurement is made with the purpose of increasing the knowledge of the person who performs it, and of anybody else who may be interested in it. This may be the members of a scientific community, a physician who has prescribed a certain analysis or a merchant who wants to buy a certain product. It is clear that the need to perform a measurement indicates that one is in a state of uncertainty with respect to something, e.g. a fundamental constant of physics or a theory of the Universe; the state of health of a patient; the chemical composition of a product. In all cases, the measurement has the purpose of modifying a given state of knowledge. One would be tempted to say `acquire', instead of `modify', the state of knowledge, thus indicating that the knowledge could be created from nothing with the act of the measurement. Instead, it is not difficult to realize that, in all cases, it is just an updating process, in the light of new facts and of some reason. Let us take the example of the measurement of the temperature in a room, using a digital thermometer -- just to avoid uncertainties in the reading -- and let us suppose that we get 21.7C. Although we may be uncertain on the tenths of a degree, there is no doubt that the measurement will have squeezed the interval of temperatures considered to be possible before the measurement: those compatible with the physiological feeling of `comfortable environment'. According to our knowledge of the thermometer used, or of thermometers in general, there will be values of temperature in a given interval around 21.7C which we believe more and values outside which we believe less.2.9
It is, however, also clear that if the thermometer had indicated, for the same physiological feeling, 17.3C, we might think that it was not well calibrated. There would be, however, no doubt that the instrument was not working properly if it had indicated 2.5C!
The three cases correspond to three different degrees of modification of the knowledge. In particular, in the last case the modification is null.2.10
The process of learning from empirical observations is called induction by philosophers. Most readers will be aware that in philosophy there exists the unsolved `problem of induction', raised by Hume. His criticism can be summarized by simply saying that induction is not justified, in the sense that observations do not lead necessarily (with the logical strength of a mathematical theorem) to certain conclusions. The probabilistic approach adopted here seems to be the only reasonable way out of such a criticism.