The World as a Mathematical Curve - The Misuse of Mathematics in the Humanities and Social Sciences

The World as a Mathematical Curve

The misuse of mathematics in the Humanities and Social Sciences

Translated from http://contradictio.de/missbrauch_mathematik.pdf. We split some paragraphs into smaller paragraphs to aid readability.

The pluralism of the bourgeois humanities and social sciences insists on the fundamental uncertainty of knowledge. Theories that claim to be more than one approach among many are excluded as dogmatism. Strangely, the same pluralism is increasingly enjoying a discipline like mathematics, which however insists very “dogmatically” that it formulates valid laws and nothing else. The interest is in mathematics as a tool for the social sciences. That this interest cannot be in a rational use of mathematics, e.g. as in physics, can taken from the relevant verdicts about the achievements of mathematics, as they make the rounds in the social sciences. They bear witness not so much to a knowledge of mathematics as to the will to attach to it invented idiosyncrasies that are intended to make it suitable for ideological utilisation.

Is Mathematics exact?

Clearly yes, is the first praise for this discipline, which ought to distinguish it from others. In contrast, it should be noted that the title “exact” is not a closer definition of science, but a pure pleonasm. For what else does science consist in than “exactly” deriving the determinations of an object that pertain to it and not something else?

In the

Equations

in mathematics, however, bourgeois scientists believe that they have found an indication of their nonsense. While the judgment “The bourgeois state is the political power of capitalist society” is at best a highly ambiguous hypothesis, the law of gravity \[s = 1/2 g \times t^2\] serves as a prime example of an exact judgment. And why is that? Because the latter is an equation? The law would lose nothing of its validity at all if it was expressed in the sentence that in free fall the path covered grows with the square of time. Finally: the fact that \(s\), \(g\), \(t\) are measured and quantified makes no difference in the “accuracy” of the judgement, but indicates a difference in the objects that are explained.

Even where the sphere of state and capital is concerned with quantifiable objects, their explanation does not lie in quantities and proportions thereof. Public debt is not that it amounts to 30 billion. The political purpose of the stately authority sets and changes the level of indebtedness, instead of its quantity and its change forming an irrevocable law to which the state would be subject like a law of nature. This peculiarity is in fact reserved to the law of gravity, which is the necessity of the given quantitative relationship of path and time squared.

So what does it consist of, the compliment “exact”? It disregards the content of the two verdicts as well as their truthfulness. The attribute “exact” for an equation neither disputes the theorem about the state, nor does the attribute prove the law of gravity to be true. The fact that it is given (greater) “accuracy” obviously means a quality that differs from that of content and truth. As a method of explaining the mathematical theorem receives the label “exact”. In this way, the form of mathematical-scientific laws is separated from their content and claimed as its reason. But: an equation and its content do not relate to each other like the means and end of knowledge. In other words: the form of the equation \(s = g/2 \times t^2\) applies to the law of gravity because it is this quantitative ratio of path and time. And not the other way round does it have this content because mathematical thinking would have opted for the form of the equation as an instrument of knowledge for free fall.

The desire for a correct method of reason is wrong in principle. A decision in favour of the “suitable” instrument of knowledge that is based on the object already presupposed what its application is to bring, the knowledge of the object. But when it was available, there was no need for a method to be used, because what the method was supposed to achieve would already have been achieved. Bourgeois thinkers nevertheless insist on the need for methods. The decision for a method and its content then lies before all verdicts about the object: it actually contains the prejudice, the interest according to which one would like to imagine a thing. And that is nothing more than a scientifically justified rejection of scientific knowledge. Thus, underhand, the results of mathematical science have become a way of thinking, a method. And their outstanding achievement is seen by bourgeois thinkers in the use of the

Symbols

which ought to guarantee that unambiguity which distinguishes the mathematical method as exact before all others. Now abbreviated notations per symbol are actually the order of the day in mathematics. However, it is a rumour that symbols here create the definiteness of terms. That \(f(x) = y\) clearly says nothing to all those who do not know the objects designated by the symbols. Knowledge of the designated things such as constantly differentiable functions and real numbers is a prerequisite for the symbols if they are to be understood at all as their symbols. This banality appears to bourgeois thinkers as a special achievement only because they have attributed a very fundamental deficiency to colloquial language: their words and terms are ambiguous.

“Language proves to be deficient when it comes to protecting thinking from mistakes… The same word is used to describe a term and an individual object falling under it … ‘The horse’ can be an individual, it can also describe the species …. Language is not dominated by logical laws in such a way that adherence to grammar guaranteed the formal correctness of movement of thoughts” — G. Frege: Über die wissenschaftliche Berechtigung einer Begriffsschrift, our translation

Frege uses his knowledge of the different meanings of the word horse to prove their indistinguishability as a shortcoming of the word. And, the fact that one can also reproduce nonsense in grammatically correct sentences proves not a shortcoming of language, but a mistake in the thought expressed. Frege confuses both because his ludicrous longing is for a structure of thought given by grammar that saves one from thinking and judging – and yet always guarantees “correctness”. This is a way to deny objectivity to thinking in principle: one should think correctly without thinking something; the truth of thoughts should stand separate from and before their content. This, their own absurdity, social scientists seriously want to consider as an achievement of mathematics. It is supposed to be home to an artificial language of symbols and formal logic that guarantees the “formal correctness of the movement of thoughts”, no matter what thoughts are moved there.

Inference

of the following kind the “power of mathematical thinking” is therefore revealed to a sociologist:

“The If-Then Paradigm: ’If John is Mary's husband, Mary is John's wife. Although this claim can be understood as the result of many observations in which a woman has always been her husband's wife, empirical observations are superfluous … If we know that John is Mary's husband, we are forced to conclude that Mary is John's wife because of the meaning of ‘husband' and wife.” — Rapoport: Mathematische Methoden in den Sozialwissenschaften, 15/16, our translation

The sentence is correct, but it is not a reasonable conclusion. Afterwards we do not know any more than before, because no new statement is concluded from the presumption. Conversely, the ending ends up with its - of all things! - premise John = husband means Mary = wife, because he never left her.

Such examples are not considered silly by admirers of mathematics. They represent the fallacies that they want to spread. The exactness of mathematics, which is to come from equations, symbols and formal logic, dissolves into the idea of a technique of “guiding” thoughts that is as indisputable as the thoughts empty. In this Popper sees quite unironically the achievement of each “mathematical theorem whose content may be taken to be nil” (Popper: Realism and the Aim of Science, Routledge 1996, p.139). The reinterpretation of mathematics into a method thus coincides with the disputation of its content, so that it now promptly becomes evident to every social scientist as an instrument that can be confidently applied to any content.

Is mathematics universal?

Clearly yes, say the friends of the guild from the other camp. “Mathematics is the lingua franca of all science, since it is without content in itself” (Rapoport, op. cit. 10, our translation) High school graduates should know better. Numbers and laws of arithmetic, equations and laws of their solutions (algebra), functions and their laws of continuity, differentiability (analysis) etc. /are/ the content of mathematics. The assertion of lack of content also becomes untenable with the following idea:

“The fall of a mature apple, the motion of the stars, the flight of bullets and today the orbits of satellites as well as the paths of space ships are all the subject of a single mathematical theory.” — Rapoport, 21, our translation.

If such disparate objects as apple, projectile and spaceship fall under one – btw. physical – law as “force = mass x acceleration”, it is because they actually have something in common. They are masses, and as such they are subject to the laws applicable to them. In these laws, the particular content of the mass, whether cellulose or steel, plays no role. One kilogram is just one kilogram, whether of an organic or inorganic nature (this is a difference of qualities, which falls into the chemistry). From this bourgeois thinkers would have drawn the following wrong conclusion: because apple and spaceship do not appear with regards to those moments, which do not play a role in the laws of mechanics, and thus even less as the sensually perceived object, in which all moments exist in unity, no content at all appears in the mechanical equations. So the equations are meaningless and therefore applicable to the most different contents. Against this, it should be noted that a sociologist should ask of the apple, instead of the rate at which it falls, whether is continuously differentiable. This suggestion would have to be revised quickly. He would not even notice that in function theory he deals with laws of a quality called function, which such a little fruit – in contrast to a mass! — does not have, so it doesn't fall under that theory.

The Misuse of Mathematics

in the humanities and social sciences consists of the following idealism:

“The unique success of the mathematical sciences is explained precisely by the combination (!) of these transcendent ‘reality’ of idealised concepts and the observable world.” — Rapoport, 16, our translation.

The sociologist calls mathematics ‘transcendental reality’ because, according to him, it presents a hodgepodge of terms in which nothing is understood at all. So certainly not the “observable world”, which should nevertheless be understood with them. A “connection” is therefore also not inherent in mathematical concepts and reality. Because how should one notice about a term without content to what it refers! So what the sociologist calls connection is an act of pure arbitrariness. One must ascribe a mathematical lawfulness to a real object. Then you can look at it as such. Its qualities are therefore never determined. Conversely, the pure construction of laws is the starting point, as whose expression the “observable world” is then interpreted. Popper is very self-confident about this kind of metaphysics:

“By choosing explanations in the form of universal laws of nature (!), we offer a solution to precisely this last (Platonic) problem. For we conceive all individual things, and all singular facts, to be subject to these laws. The laws (…) therefore explain regularities or similarities of individual things or individual facts or events. And these laws are not (sic!) inherent in the singular things” — Popper, op. cit. p.137

This science thus discovers precisely the “regularities” which it has previously put into the things with its constructed “laws”, but which are not their laws at all. So the nonsense is inevitable.

“Assume we have a valid syllogism such as:

(a) All men are mortal (1)
(b) All Athenians are humans (2)
----------------------------------------
(c) All Athenians are mortal (3)

Now the rule of indirect reduction says:

(4) If a,b|c is a valid inference, then a,non-c|non-b is also a valid inference, too.

For example, owing to the validity of inference of (c) from the premises (a) and (b), we find that

(a) All men are mortal
(non-c) Some Athenians are non-mortal
------------------------------------------------
(non-b) Some Athenians are non-humans

must also be valid.” — Popper, What is Dialectic, in: Mind, Volume XLIX, Issue 194, 1 April 1940, Pages 403–426, https://doi.org/10.1093/mind/XLIX.194.403

“In the much cited syllogism:

All humans are mortal,

Now Gaius is a human,

Therefore Gaius is mortal,

the major premise is correct only because and to the extent that the conclusion is correct; were Gaius by chance not mortal, the major premise would not be correct. The proposition which was supposed to be the conclusion must be correct on its own, immediately, for otherwise the major premise would not include all singulars; before the major premise can be accepted as correct, the antecedent question is whether the conclusion may not be a counter-instance of it.” — Hegel, Science of Logic, Book II, Cambridge University Press 2010

Warning of misunderstandings

The reason for the mistakes made in the humanities and social sciences using mathematics lies in the idealism of these disciplines. Not in mathematics itself, as some critics think. Their conception of arithmetic

Mathematics: merely quantitative and abstract?

is no less wrong than the preference of the social department for mathematics, which we criticised. They, too, consider mathematics to be a method, but unlike the colleagues mentioned above, they consider it to be useless to dangerous.

The fact that the number is a “merely quantitative” determination is not a very intelligent criticism of the number. Here, of all things, its achievement is that it is accused of. It determines the unit according to its quantity, separate from the special content this unit may have when counting these or those items. That's exactly what it ought to do. The arithmetic operation \(2 + 2 = 4\) is by no means modified by the fact that the unit consists of apples one time and pears the other time. However, the number demands so much: because it is the number of a unit, the addition of different units makes no sense. 2 apples + 2 pears = compote at best.
This criticism is thus completely false where it refers to equations of dimensioned quantities as in physics. 2 meters never equals 2 seconds, no matter how much 2 is equal to 2. It depends on the quality of the quantities, which new quality is given by quantitative conditions. A distance of 120 km in relation to the time required for this of 2h does not simply make \(120 : 2 = 60\) but 120 km / 2h = 60 km/h speed.
That abstractions finally have their place in mathematics as in any science is correct. Criticising this is wrong. Abstractions have to be correct. Then they do their work for the explanation by recording the general of various peculiarities that determines them. The ‘state’ does not exist any more than the ’function’; both abstractions grasp the general that exists in the particular, in the English and German states, in this parable and the hyperbola. Hegel has already made fun of putting abstraction as an unreality against something particular and therefore demonising it:

“Such a procedure would strike us automatically as inappropriate and inept in the case of objects of everyday life, such as when someone were to ask for fruit and then rejected cherries, pears, and grapes simply because they are cherries, pears, and grapes, but not fruit.” — Hegel, Encyclopedia ofthe Philosophical Sciences in Basic Outline, Cambridge University Press 2010, p.42
The whole accusation “merely quantitatively” ultimately acquires its meager plausibility only by applying the number to such spheres in which it has no place. Marx said what was necessary.

“’can we halve reason or speak of the third of a truth?’”

asks a certain Karl Grün polemically. Marx’s justified counter question:

“can we speak of a green-coloured logarithm?” —Marx, German Ideology