Which field is more rigorous, mathematics or philosophy?

Question

I don't know if this question is best suited for this stack exchange, but I couldn't think of a better stack exchange. I want to know, which field of study is more rigorous, mathematics or philosophy? Personally, I believe it is mathematics, because philosophy has a lot of imprecise concepts, like causation, intension, conceivability, etc, which are not amenable to formalization. Mathematics, on the other hand, can be formalized in a computer and proofs of theorems can be checked mechanically. True, in practice, mathematicians use human-readable arguments rather than axiomatic derivations. But in principle, everything in mathematics can be formalized. However, I would be very interested to hear good arguments from both sides, and then I might change my mind.

Answer 1

Rigor is a methodological concept: it only applies to the system of analysis used within a particular investigation, as a measure of how thoroughly that investigation conformed to the intellectual standards of the field. The term isn't generally applied to an entire field, because trying to apply it to something like 'Mathematics' or 'Philosophy' would basically be an assertion that the specified field has no intellectual standards whatsoever, which is tendentious at best.

Mathematics is a closed, formal system. Investigations in Maths mean conforming to specific, pre-defined rules of symbolic transformation. Being non-rigorous in Maths means breaking one of those rules (intentionally or accidentally) for no good reason. Those violations can be sneaky and subtle, lost in a mass of other perfectly rigorous math, but once they are found they are usually D'oh moments in which it becomes obvious that rigor failed.

Philosophy is an open system which occasionally uses formal symbolic transformations (aka logic) but which extends well beyond it to reasoned argumentation. Rigor in philosophy is a matter of internal consistency and external acuity: seeing the way the world is and analyzing it systematically. It's hard to see non-rigorous philosophy until one digs into the nitty-gritty of an analysis, but that doesn't mean that philosophical investigations are less rigorous than mathematical ones, just that mathematical rigor is easier to see.

In both mathematics and philosophy the goal is the creative application of the system to produce productive results. Rigor is a secondary methodological check that keeps creative applications from imploding into nonsense. It's an important concept, but we should keep it in its lane.

Answer 2

Mathematics was and is intended to furnish precise numerical answers to precisely-posed questions involving numbers and logic. It is intended, by design, to be rigorous. Once the truth of a mathematical proposition has been established, it furnishes a foundation for further propositions and their tests, and the field expands- all on the basis of settled issues.

In contrast, there are no settled issues in philosophy; all its practitioners are free to open up and relitigate old questions based on their personal opinions, not by using numbers but by using human language. It is therefore necessarily less rigorous than mathematics.

Answer 3

The question whether mathematics or philosophy is more "rigorous" depends on what you mean by rigor. If your mention of rigor refers to the technical meaning often attached to it by modern mathematicians, namely the particular foundations for mathematical analysis as established around 1870 by Weierstrass and others, then the question is not entirely meaningful, since philosophy is not mainly concerned with the technical development of mathematical analysis. Thus to make sense of the question, one would need to refer to a more generic meaning of the term "rigor".

Once one starts analyzing the meaning of the term, one of the first things one discovers is that its meaning changed a number of times in the course of the development of mathematics. To give a quick example: in the 17th century, mathematician Paul Guldin published a book containing a number of quadrature results as well as a formulation of the (Pappus-)Guldin theorem on solids of revolution. Guldin attacked Cavalieri's principle (if two plane domains have the same height and the same cross-sectional length at every point along that height, then they have the same area) as unrigorous. Guldin's reason was that applying the principle involves an infinity of comparisons, and since actual infinity was impossible, Cavalieri's principle is incoherent.

When is a scholar's contribution (to math or philosophy) rigorous, then? A possible definition that avoids the pitfalls of being too technical (and therefore arbitrary), and also applies to both philosophy and mathematics, is a contribution that helps the scholar's contemporaries get out of an intellectual rut.

Thus, many mathematicians around 1870 felt that analysis was in a rut due to mistakes being committed through cavalier use of arguments exploiting infinitesimals. They thought they helped the field out by developing foundations for analysis that were infinitesimal-free.

Guldin felt that Cavalieri's arguments were unreliable, and therefore developed an approach to quadrature based on the exhaustion method, which he felt was more reliable and helped mathematics out of a rut (his co-religionist Tacquet even declared that if geometry is not to be destroyed by indivisibles, indivisibles themselves must be destroyed).

On the philosophy side, I would mention the philosopher Johann Friedrich Herbart. Herbart was a significant influence on the great Bernhard Riemann. This is discussed in the following article by Nowak:

Nowak, Gregory. Riemann's Habilitationsvortrag and the synthetic a priori status of geometry.The history of modern mathematics, Vol. I (Poughkeepsie, NY, 1989), 17-46. Academic Press, Inc., Boston, MA, 1989

Nowak goes on to make the following three points.

1. Herbart's constructive approach to space, already cited, mirrored the content of Riemann's reference to Gauss in that both discussed construction of spaces rather than construction in space.

2. Riemann followed Herbart in rejecting Kant's view of space as an a priori category of thought, instead seeing space as a concept which possessed properties and was capable of change and variation. Riemann copied some passages from Herbart on this subject, and the Fragmente philosophischen Inhalts included in his published works contain a passage in which Riemann cites Herbart as demonstrating the falsity of Kant's view.

3. Riemann took from Herbart the view that the construction of spatial objects was possible in intuition and independent of our perceptions in physical space. Riemann extended this idea to allow for the possibility that these spaces would not obey the axioms of Euclidean geometry. We know from Riemann's notes on Herbart that he read Herbart's Psychologie als Wissenschaft.

Thus, Herbart's philosophy helped Riemann escape from the rut of Kant's "absolute space", at a time when a vast majority of Riemann's contemporaries were still under its spell. Who knows whether Riemann would have been able to establish what is known today as Riemannian geometry without the liberating influence of Herbart's philosophy.

Another example I would mention is Hilbert. Around 1900, mathematics was still dominated by analysts in Berlin, and those analysts thought that mathematics = analysis, and that people like Lie and Felix Klein were charlatans (they said so explicitly). It is well known that Hilbert's list of 20 problems helped shape the course of 20th century mathematics. What is significant about Hilbert's list is that few of the problems are actually in analysis. In his speech at the Paris congress, Hilbert outlined a liberating philosophy that took mathematics out of the rut of Berlin's focus on analysis.

What I tried to illustrate is that philosophy can be considered in a significant sense as being more rigorous than mathematics, since only philosophy can take mathematics out of a rut when it gets stuck in one.

Answer 4

This article talks about "inferential rigour" in philosophy and maths, breaking down arguments into "atomised" "steps", and concludes that it is not an epistemic ideal for philosophy.

https://www.jstor.org/stable/43904709

So does it matter? That's not the same as saying that everyone (and their student) is right, it's impossible to bluff, and there's no standards of success, etc.

the quality of being extremely thorough and careful