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Tim Button's presentation of set theory motivates the subject by providing a history of 19th century mathematics where the notion of limit allowed definitions of the derivative and continuity. These led to proofs of non-intuitive results such as Bolzano and Weierstrass' discovery of a function that is continuous everywhere but differentiable nowhere or Peano's space-filling curve.

Button makes the following comment: (page 9)

For better or worse, these “pathological” geometric constructions were treated as a reason to doubt appeals to geometric intuition. They became something approaching propaganda for a new way of doing mathematics, which would culminate in set theory.

I can see this as a mathematical ax to grind against the use of geometric intuition, but Button makes a further comment: (page 10)

Part of the moral of the previous section is that the history of mathematics was largely written by the victors. They had axes to grind; philosophical and mathematical axes.

What I am interested in is what were the philosophical axes they had to grind?

One philosophical ax might have been doubting any argument, especially in metaphysics, because it was based on intuitive reasoning rather than the new logical formalism. However, there may have been other axes they had to grind? What might they have been?


Button, T. Set Theory: An Open Introduction. Retrieved on October 1, 2019 from the Open Logic Project at http://builds.openlogicproject.org/courses/set-theory/settheory-screen.pdf

  • It is ard to say, based on the above quotes... Maybe Button is alluding to the "traditional" (due to Aristotelian influence) rejection of the actual infinite. Maybe useful The Early Development of Set Theory : "Three historical misconceptions that are widespread in the literature should be noted at the outset: (1) It is not the case that actual infinity was universally rejected before Cantor. [...] 1/2 – Mauro ALLEGRANZA Oct 1 at 14:56
  • In 19th century German-speaking areas, there were some intellectual tendencies that promoted the acceptance of the actual infinite (e.g., a revival of Leibniz’s thought). In spite of Gauss’s warning that the infinite can only be a manner of speaking, some minor figures and three major ones (Bolzano, Riemann, Dedekind) preceded Cantor in fully accepting the actual infinite in mathematics." 2/2 – Mauro ALLEGRANZA Oct 1 at 14:56
  • Maybe relevant also Kant's Mathematical Antinomies. – Mauro ALLEGRANZA Oct 1 at 14:58
  • @MauroALLEGRANZA Accepting the actual infinite does sound like another ax they might have to grind. Thanks! – Frank Hubeny Oct 1 at 15:20
  • The other axes, perhaps see Antoine Cournot, determinism, beginning of calculus of probability. And Boussinesq (see JC Maxwell) informationphilosopher.com/solutions/scientists/maxwell. The Wikipedia on Cournot is bad, but see Encyclopedia articles. – Gordon Oct 1 at 20:23
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Interesting question. I think one part of the answer is that logicians and mathematicians such as Frege were concerned to remove 'psychologism' from their subjects. Psychologism in its various form is a set of views :

about the relationship between psychology and logic, but its traditional form holds that the laws of logic are grounded in psychological facts. So, the rules of logic yield the laws of thought by virtue of our psychological composition. This view is attributed to John Stuart Mill, among others, and was assailed by Frege.

According to Frege the plausibility of Millian psychologism trades on an ambiguity in the phrase 'law of thought'. The reading necessary for Mill's view entails that logical laws govern thinking in the same manner that phys- ical laws govern physical events, which means that logic is treated as a part of descriptive psychology. A logical rule of inference then is an abstraction from the psychological activity of drawing a demonstrative inference. But logic for Prege is a normative, substantive science, and no set of laws can be both normative and descriptive. So, Millian psy (Gregory Wheeler, 'Applied Logic without Psychologism', Studia Logica: An International Journal for Symbolic Logic, Vol. 88, No. 1, Psychologism in Logic? (Feb., 2008), pp. 137-156: 137.)

Frege believed that logic discloses how we should think, but that is a separate point.

  • Thanks for that. The quote seems cut off midway at end? – Rusi-packing-up Oct 3 at 12:55
  • Thank you. Mill's psychologism does seem like it would be a philosophical ax to grind motivating Frege. +1 – Frank Hubeny Oct 3 at 13:38
  • Does this gap between normative/deontic/prescriptive rules and descriptive/declarative laws have a more general name in philosophy? – Rusi-packing-up Oct 4 at 4:40
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I like Geoffrey Thomas's answer, and would add.

Mathematicians and rationalists more generally have always chased after certainty of reason. Plato was perhaps the first mathematical foundationalist who attempted to deal with infinite regress by creating a distinct, objective reality for abstraction with his theory of forms. For over two thousand years, Euclid's postulates and axiomatic method in Elements were seen as an exemplar in reasoning and imitated, since axiomatic reasoning relies on foundations. Rationalism with Descartes through works like Meditations sought to establish metaphysical certainty of rationalism. With mathematicians like Frege who came along and advocated a logicist position, ultimately the epistemological foundations of math were debated regarding the acceptance of non-Euclidean geometries which had to deal with how to be certain one has distinct, correct postulates. This was a major philosophical issue in the 19th century. While accepting that there were not necessarily right and wrong axioms, mathematicians also had to deal with Cantor's infinite paradise and the pursuit of set-theoretic axiomatic foundations of set theory which at the time were hotly contested. Men like Cantor bumped heads with Kronecker. These debates eventually lead to Hilbert's goal of certainty crushed by Kurt Gödel in the 20th century.

But the axe you are specifically referring to is a question regarding the role of intuition in mathematical reasoning. There is a school of thought called intuitionalism which sharply differs with that of formalism. A classic example of this is how geometric intuitions and analytical foundations are used in calculus. What is a curve? Since calculus, the derivative, which is a measure of slope of a line tangent to a curve at a point has generated a discussion about what constitutes a curve. Remnant of Zeno's paradoxes, the question of what a differential is invokes foundational questions in the same way that Cantor's transfinite numbers do. These maths work and were accepted, but the nature of their foundation either through intuitive concepts like 'parallel lines' or 'continuous curve' creates issues in reasoning axiomatically. Non-Euclidian geometries open up a bag of worms in terms of psychologism because if one can intuit contradictory, but functional postulates, how can one be certain about one's foundations at all? Characterizing mathematics as essentially a type of logic, would for mathematicians, kick the can down the road by asserting all mathematical foundations are certain, because they reduce to logical statements (leaving open the question of where the logical statements are ultimately grounded).

While non-Euclidian geometries open up the question, a reduction of math to logic would make things certain. This works to an extent. Consider the adoption of ZFC in the early twentieth-century in response to the ambiguities of intuition in naive set theory. ZFC postulates are commonly given in terms of quantitative and existential qualifiers. The null set axiom can be stated as "There exists some set such that for all elements pertinent to that set, there does not exist a relation such that any element exists in that set, and can be denoted by a special symbol called the empty-set symbol". (∅≡∃S∀e¬(e∈S)) Now, a mathematician can say, since the logic is accepted, the mathematical axiom is beyond doubt.

This is the axe. Some people see statements which rely on intuition as pathological (a metaphor from disease). And because they distrust intuition, they rally against formalisms built on it.

What I am interested in is what were the philosophical axes they had to grind?

As far as the philosophy of math can be extricated from philosophy in general (and I would argue it would be difficult to do well), I would suggest that there are several at play which are all interlinked:

intuitionism v. formalism (mathematical) which reduces to psychologism v. objective realism (19th century philosophical) which reduces to empiricism v. rationalism (18th century, philosophical) which reduces to what is the appropriate level of skepticism, the relationship between the mental and physical, and tackling ideas like the Agrippan trilemma which purports to show that rationality ultimately fails to provide truth or certainty.

Remember that most of the 19th century occurred before Wundt, James, and Freud established the legitimacy of psychology as a science. So psychologism v. objective realism is really linked to the credibility of what we now called psychology which hadn't been established. In the 21st century, cognitive science is pushing back on objective realism by arguing exactly how neurons create concepts, and establishing the mind-body duality as a category mistake. But back then, mind-body duality was widely accepted as a philosophical truth. To this day, many philosophers outside of the philosophy of mind are vigorously defensive of the notion of objectivity in contradistinction of intersubjectivity, and the debate continues.

  • Thank you. I will have to look into psychologism more. +1 – Frank Hubeny Oct 3 at 23:17

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