Is mathematics as a concept value-laden?

Question

Joan Robinson in Economic Philosophy writes:

Perhaps Gunnar Myrdal is too sweeping when he says (speaking as an economist) that 'our very concepts are value-loaded' and 'cannot be defined except in terms of political valuation'

It is true that economic terminology is coloured. Bigger is close to better; equal to equitable; goods sound good; disequilibrium sounds uncomfortable; exploitation , wicked; and sub-normal profits, rather sad.

All the same, taking a particular economic system as a given, we can describe the technical features of its operation in an objective way. But it is not possible to describe a system without moral judgements creeping in.

Turning to mathematics - as a concept - is it too value-laden?

Its generally considered as the epitome of objectivity: one cannot argue with a number or indeed an axiom.

Answer 1

Mathematics as a concept may not be value-laden (it's debatable whether concepts or objects of any sort can or cannot have intrinsic value), but mathematics in practice certainly is. Ask any professional mathematician which of two proofs or axiomatic frameworks is more beautiful and the question will make perfect sense to them. Since beauty implies a value judgement, mathematicians must have values in regards to axioms and proofs. But axioms and proofs are all that mathematics is!

Put simply: if art is value-laden, so is mathematics.

Answer 2

I would argue that mathematics is indeed value-laden. One of the central ideas in statistics is that the individual case is not important, it's the aggregate of a large number of separate pieces of data that carry significance. Conversely, in chaos theory, a central concept is that an arbitrarily small change can have an arbitrarily large impact. I would call these value-laden perspectives. If you choose to analyze a given situation statistically, it implies a value judgement about what kind of things you'll chose to take as important, and what kinds of information you're willing to discard.

For a real world example, many schools are shifting to a data-driven approach to learning, based on statistical analysis of educational outcomes. However, one might argue that the really important things to pay attention in education to are the outliers --the successes or failures that lie outside the realm of statistical probability.

Even in the early days of mathematics, the very idea of evaluating things quantitatively and making comparative estimations of value based on number surely had a profound psychological and philosophical --and thus moral --influence. In more recent times, modern mathematics and logic have discarded the notion that "you cannot argue with an axiom" --many axioms are a matter of choice and preference, and choosing one over the other is surely value-laden, although the moral implications may be obscure.

Answer 3

Mathematics is to a large extent based on abstraction. Abstraction means to remove the irrelevant details in order to only keep the important aspects of something. This allows to apply those concepts to a wider class of problems.

But what are irrelevant details, and what are important aspects? I don't think you can rigorously define that. Rather it is a value judgement. And this value judgement is what determines which of the infinitely many possible mathematical structures are studied, and which are essentially ignored.

To give a concrete example, for millennia Euclid's parallel axiom was considered essential; it was not considered to be useful to consider axiomatic systems where it doesn't hold. Instead, they tried to find a proof that would derive it from the other polynomials (which would have changed it from an axiom to a theorem).

But then, people realized that a geometry violating that axiom indeed makes sense. The result was a rich new field of study, called non-Euclidean geometry. Now, already at Euclid's time, it would have been possible to do non-Euclidean geometry. After all, all you would have had to do is to change one axiom in a quite obvious way. But nobody did it in all those millennia, not because of a lack of knowledge, but because of a lack of interest. It was just not considered a valuable topic because the parallel axiom was considered essential.