Precautions required by the use of the mathematical method

The use of this method, both analytical and geometric, must always be

accompanied by extreme caution, and the more the reasoning tends to

become almost a mechanical operation – as happens when using algebraic

symbols – the greater become the probabilities of errors, which derive from

the uncertainty of the premises.

When we are reasoning according to usual logic, in passing from one

proposition to another we can examine it, and if we find it to be in contrast

with the concepts we hold true, we stop and decide whether we must modify

the concepts or reject the proposition. But the use of the mathematical

method – especially the algebraic, less so the geometric – prevents us from

doing so. The intermediate propositions escape our perception; we only

know the two extreme ones. We can state that one logically follows from the

other, but we do not know if along the way we have strayed too far from

reality.

Now, and this is an important point to make, all any science can do is to

approximate reality, without ever being able wholly to encompass it.

The phenomenon studied by science is always an ideal phenomenon, which

sometimes comes extremely close to the real phenomenon, but never entirely

Considerations, I, May 1892 7

coincides with it; hence the necessity to compare our deductions with

experience or with observation as often as possible, to make sure that we have

not strayed too much from the facts of nature.

This is all that is true in the common remark that theory and practice are

two different things. But to conclude that theory should be rejected is a

foolish thing, if not ignorance or bad faith. In this way every human science

would be discredited, and there is no need to recall all the sophisms at which

the ancients arrived through this path. The true conclusion is that it is necessary

to proceed cautiously and always to go back to experience and

observation.

* We shall express our concept in a better way by reasoning on a concrete

example. Let us consider the fall of bodies, which is precisely the example

chosen by WalrasXXXIII to show how mathematics is used in the study of

natural phenomena.

The problem of the fall of bodies to the surface of the earth looks extremely

simple, but not even that problem is completely solved. We have only studied

various abstract phenomena that more or less approximate the real one.

* The first and simplest is the case of a material point, or even, if one

wishes, of a sphere falling in a vacuum, assuming that the intensity of gravity

is constant for the whole duration of the fall, and that the part played in the

phenomenon by the rotation of the earth is irrelevant.

* The formulae recalled by Prof. Walras in the introduction to his book

Elements d’Economie Politique are in relation to this very case. The real phenomenon

of a platinum sphere falling to the surface of the earth is very close

to the abstract phenomenon.

* But the latter differs from the natural phenomenon in two ways. First

of all we must take into account that the body is not falling in a vacuum

but through air. The air causes the body to lose some of its weight, and in

the case of the oscillations of a pendulum, it has also been calculated how

such loss varies according to whether the body is at rest or swinging. All

these phenomena depend on the temperature of the air and on that of the

body. Furthermore, we have the resistance of the air; and we are stopped

here in our very first few steps by the difficulty of the topic. The rational

theory of the phenomenon is very imperfect, the empiric theory is worth

little more. If we take away the case of spheres and of a few more bodies

of very simple shapes, we do not know anything about the resistance of the

air.

* Then, even ignoring the air, we see that the study of this phenomenon

becomes progressively more difficult. Let us even not bother with fixing the

direction of the vertical, which gives rise to important studies, but gravity

varies according to latitude and to the distance of the body from the earth.

The main parts of these phenomena can be easily known through calculations,

but studying them in such a way as to exhaust the subject in all its

details involves considerable difficulties. One has also to take into account the

8 Considerations, I, May 1892

rotation of the earth. Finally, in theory, one should also consider the attraction

of celestial bodies. And then, if we are dealing not with a material point,

but with a solid body, the study becomes even more difficult. Luckily, in

practical terms many of these phenomena are absolutely negligible, but this

does not mean they do not exist, demonstrating that the real phenomenon is

different from the abstract phenomena we can study.

* The example we are now dealing with is also very good for allowing us

fully to appreciate the difference between the empirical method and the

experimental – or concrete deductive, as Mill calls it – method.XXXIV

* Theory tells us that a mass falling from a great height must deviate to the

east of the vertical. There is also a deviation to the south, but it is in the order

of the square of the speed of the rotation of the earth, and it is therefore too

small to be observed. The deviation to the east falls instead within the limits

of quantities we can actually observe. Many attempts were made to verify the

conclusions of the theory through experience. Abbot GuglielminiXXXV managed

to discover these deviations in 1790, by conducting experiments in the

Torre degli Asinelli, in Bologna. Other experiments were conducted by Dr

BenzenbergXXXVI in Hamburg and in a mine at Schlebush, and more still by

Prof. ReichXXXVII in the mines at Freiberg. All these experiments show a tendency

by falling bodies to deviate to the east, but none of them can be said to

agree entirely with the theory,11 so that yet again one should repeat Laplace’s

words about the objections that were once moved against Galileo:XXXIX ‘In

recording the influence of the rotation of the earth in the fall of bodies, we

find as many difficulties now as were found then in trying to demonstrate that

that influence was not significant’.

* However, no physicist has any doubt whatsoever about the results of the

theory. Is the experimental method being abandoned because of this? Most

certainly not. But, even without taking into account direct experiences such

as Foucault’sXL pendulum or the gyroscope, the movement of the earth is

proved by such a large number of observations, that we must accept the

consequences that derive from it, even when they perchance escape our direct

observation.

Similarly, we accept that theoretical Political Economy may set up theorems

that cannot be directly verified through observation, provided that these

are a necessary consequence of principles that elsewhere find broad and

effective demonstration from experience, which is therefore always guiding

us, either by directly leading us to the truth, or by indirectly letting us

know it.

We have seen that our premises are never entirely, but only approximately

true; we must add that conclusions are not always as close to reality as

premises, but may sometimes end up very far from it.

We believe it is possible to give examples of this proposition without making

use of the science of quantities; but here we are on the boundaries of its

domain, and therefore we cross them without hesitation.

Considerations, I, May 1892 9

y = φ (x)

is nothing but the conclusion of a reasoning, whose premises are some qualities and

the measurements of x and y. Now, it is generally true that to a slight variation of x

corresponds a slight variation of y, but it is also known that this is false in many cases.

Let us suppose, for instance, that by indicating with a the quantity of one kilogram,

one has found12

y = e

x−a ;

then, if x is equal to one kilogram and one milligram, it will be possible to conclude

that y is very large, since it will be equal to 2.71828 . . . to the power of one thousand.

Who would believe now, if they did not know any mathematics, that by changing the

premise in a minimal way, namely by supposing that x is equal to one kilogram minus

one milligram, the conclusion changes completely and y becomes very small? And yet

this is exactly how things are, and y is equal to one divided by 2.71878 . . . to the power

of one thousand.

In this case, mathematics also shows us the reason for the difference between the

conclusions, since it tells us that beside the absolute value of x, one must also bear in

mind the crucial circumstance whether that value is greater or smaller than a.

* Theoretical mechanics teaches us how to calculate the pressure on each

foot of a three-legged table. But if the legs are four, the problem becomes

indeterminate. The geometricians who first confronted the question found

this fact quite puzzling. How could indeterminateness ever exist in nature,

with regard to the weight supported by each of the four feet of a table? The

answer can now be found in any basic treatise of mechanics. The indeterminateness

ceases to exist when one stops considering rigid bodies, as theoretical

mechanics would like them to be, and starts considering elastic bodies

instead, as they are in nature.

Who could deny, now, that similar cases may arise, when one considers people

not as shrewd and perfect hedonists, as pure Political Economy would like

them to be, but with that mixture of hedonistic and altruistic qualities of

shrewdness and carelessness as we observe them in real life?

The theorem that the pressure on each of the four feet of a table is

indeterminate is not more or less close to reality, it is actually false. How,

then, can we ever make sure that the theorems of Pure Economics will not

lead us to similar mistakes, other than by sticking very closely to observation?

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