We come to understand the subjects of science through observation, but observation of certain subjects is imperfect. For the astronomer, the motion of the planets is too far away; for the chemist, the activities of the atoms (and even their existence) is too remote and difficult to see; to the physicist, electricity, heat, magnetism, radiation etc. are all invisible- and often fleeting in nature (a flash of static, a lightening bolt, an explosion).
There are many ways of dealing with this problem, but the best and most common way is to understand the subject of the science mathematically. This does not change the subject of the science- the astronomer still strives to understand the motions of the planets as such; the chemist still tries to understand the ultimate material parts of things, etc. In the mathematized physical sciences, a certain subject matter is understood by means of being quantified and actually measured; as opposed to when we understand mathematics, when quantity is the subject matter. Simply put: astronomy studies orbits, math studies quantity.
The mathematizing of physical things allows us to understand more about the order of physical things- and sometimes it is the only order we understand about the things as such, or the foundation of all order*. Because every science is an ordered body of knowledge, it s therefore essential to mathematize certain scientific subjects.
But since mathematics cannot be identified with the subject matter as such, there will be some parts of a mathematized physical science that are not necessary parts of the scientific subject.What I mean is that since mathematized physical science deals with a mathematical and measured subject, there is in it things that are proper to mathematics and measurement, and things that are proper to the subject, just as a red dress has characteristics that are proper to red, and proper to a dress. Both characteristics can be said to belong to a certain thing, but they are not identical.
Compare, for example, the following quantifications: A boy’s body is three feet tall, and a boy’s body is divided in mean and extreme ratio at his navel. Or perhaps this: an orbit is one million miles around, and an orbit is an ellipse. The second thing in the comparisons (mean and extreme, an ellipse) exists separately from some artificially determined unit, whereas the first does not. If a planet’s orbit isn’t quite an ellipse, or a boy’s navel does not quite divide his body into mean and extreme ratio, we still say something like “this is what nature intended”. But if a boy is three feet tall, we don’t object that he should have been 90 centimeters. Not every quantity belongs to a thing in the same way. Some quantifications, in other words, are closer to the nature of the thing than others. Numbers are very close, for example, to the elements of the periodic table, (Hydrogen is 1, helium 2, lithium 3, etc.), as are the ratios that constitute the specific difference of tones (2:1 is an octave, 3:2 is a fifth)
A more subtle aspect of quantification is the “functional definition” which does not seek to tell us what something is, only how it is measured. Hence heat gets defined in chemistry by “temperature differences”, for where everything is the same heat, there can be no measurement of heat (heat is measured by a length, and the length is created by something hotter and something cooler). So too, energy is defined by actually moving something. People often get confused by the paradoxes that come from thinking that a functional definition is telling us what something is (for example, the chemist believes that if everything were the same heat, there would be no heat, and if nothing was actually moving, there would be no energy expended. These paradoxes proceed from confusing a thing with the measurement of a thing.)
A certain kind of functional definition is actually capable of appearing to transcend the distinction of contraries. The classical example was Galileo’s insistence that a circle was a polygon with infinite sides. But a better example is “zero velocity”which means the same thing as “rest”- is the opposite of motion (and a fortiori of velocity). Probability theory also defines “necessity” as being “a probability of one” i.e. 1/1 probability.
Mathematized physics will also lead to the identifying of two things when they are unified in quantity (equal). A well known example of this is electro magnetic force, which is founded on the discovery that electricity, light and magnetism all move at the same speed. The evidence will have to do.
The wild success of mathematized sciences has not been attended by thinkers who can accurately parse the distinctions that arise between measurement and reality, appearance and fact, etc. What’s new- philosophers have been dropping the ball at explaining nature for a while now.