# Which field is more rigorous, mathematics or philosophy?

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.

• Yes, mathematics is by far more rigorous. "Rigor" in mathematics means that everything can be strictly formalized if necessary (even if this is not often done in practice). Philosophy is largely about appeals to intuition, which cannot be formalized and are to a degree subjective. Commented Jul 15, 2023 at 15:45
• The concept of rigor might itself be taken for a philosophical one; and we might distinguish between absolute and relative rigor (or I've seen, in the SEP, talk of "informal rigor," no less) or second-order rigor (rigorous definitions of rigor), etc. The intuitive sense of rigor as strict/mechanical, of "crisp edges," supports the picture of mathematics as more rigorous on average (though when it comes to the vagaries of set-theoretic multiverses and the ocean of category-theoretic enigmas, vagueness and ambiguity enter into this picture quite vividly). Commented Jul 15, 2023 at 15:46
• Math, unless you're Plato who put philosophy as more certain. Example, "dialectic will centre on explaining the hypotheses of mathematics in a way that mathematics does not, and cannot, do" philarchive.org/archive/BURPOW Commented Jul 15, 2023 at 16:41
• Obligatory xkcd Commented Jul 17, 2023 at 7:54
• @causative - The problem with that argument though is that maths is completely missing its foundation. Thus, every formalization fails eventually. Commented Jul 17, 2023 at 10:37

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.

• "mathematical rigor is easier to see." The reason for that is that mathematics does entail more rigor than philosophy, similar to how a stain on a white surface is more visible than a stain on a dark or motley surface (the question being which surface is more pure). "It's hard to see non-rigorous philosophy until one digs into the nitty-gritty of an analysis". It is not that hard. The lesser extent of rigor in philosophy oftentimes is palpable already in the ambiguous semantics of statements (thereby leading to hermeneutical difficulties), something that mathematical language precludes. Commented Jul 15, 2023 at 23:19
• @IñakiViggers: Sorry, but as I pointed out, you're conflating methodology and practice. Mathematics is easier to check than philosophy because it is a closed, formal system. That doesn't make it better, or more rigorous, or easier to practice than philosophy; maths can be quite challenging. It's just easier to check. Commented Jul 16, 2023 at 0:29
• "Mathematics is a closed, formal system" - no, it is not. Mathematics uses formal systems. Confusing the use of formal systems for what Mathematics is is like confusing the use of a specific Telescope for what Astronomy is.
– Yakk
Commented Jul 17, 2023 at 2:45
• @TedWrigley There are many many ways to formalize counting. Counting is the thing, the formalization is a tool. And formalization isn't the first step - you constantly want to check your fomalization to see if it does what you want it to do, and math regularly redoes formalization for the same set of concepts. I can give you a half dozen different formalizations of calculus off the top of my head, but they are all talking about mostly the same thing. And formilization isn't the same as abstraction!
– Yakk
Commented Jul 18, 2023 at 13:27
• @TedWrigley Mathematicians invent rules, axioms and rules of symbolic manipulation. That invention and application to the things they are working on is not itself governed by rules, axioms and principles of symbolic manipulation. Mathmaticians build formal systems, but the building of formal systems, which is an important part of mathematics, is not done in a formal system. The patterns mathematicians study are not determined (only) by the systems they build to study them. You are confusing the map for the territory.
– Yakk
Commented Jul 18, 2023 at 15:30

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.

• One of the more bizarre things I read about in the history of mathematics was how a lot of mathematicians thought that the concept of infinitesimals wasn't rigorous enough, which translated into thinking that calculus wasn't rigorous enough until they came up with the limit-based version and Cauchy sequences/Dedekind cuts. Then Robinson came along and "rigorized" infinitesimals. I'm not disagreeing with you per se but I thought it would be helpful to point out that mathematicians themselves seem to doubt their own rigorousness betimes. Commented Jul 15, 2023 at 19:10
• @KristianBerry I think the only thing that that shows is the limitations of phrasing like "infinitesimals [weren't] rigorous enough." Rigor is a property of a particular theorem/definition/argument , not of a vague concept like "infinitesimals." The original arguments that used a notion of infinitesimals to reason about calculus were indeed insufficiently rigorous. Later work was able to "fix" those arguments to make them rigorous. Those two facts are not incompatible Commented Jul 16, 2023 at 0:02
• @DreamConspiracy I should have talked about the social background factor I had in mind, which is that "non-mathematicians" are variously notorious for their bewilderment in the face of complex mathematical notation systems of this or that species; from "our" point of view, all mathematics is rigorous by default, so it is strange for "us" when we realize what the much higher standards to which mathematicians hold themselves are. Commented Jul 16, 2023 at 1:24
• there was 1500 years of debate about Euclid’s 5th postulate-could it be proved, could it be derived from other axioms, derived at all, etc. and thats one small example. Commented Jul 17, 2023 at 14:52
• @j Kusin Yes, but once the answer to a question like that is determined and verified to arise from base axioms then the issue is settled. Commented Jul 18, 2023 at 20:48

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.

• do you know what mathematics is?
– user66760
Commented Jul 18, 2023 at 12:34
• @doot_s: You can consult my publication list here: u.math.biu.ac.il/~katzmik/publications.html Commented Jul 18, 2023 at 12:35
• why are you doing this, then?
– user66760
Commented Jul 18, 2023 at 12:48
• @doot_s, You seem to be upset by my answer. Beyond that, I would need to have more details concerning your reservations in order to be in a position to respond. Commented Jul 18, 2023 at 12:50
• @Rushi, unfortunately I don't know much about CS (I wish I did) but what I heard is that the law of excluded middle cannot be implemented as part of a code, so it certainly seems reasonable that the kind of mathematics that's relevant for CS would be constructive mathematics. Commented Jul 24, 2023 at 9:27

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

• The author (Paseau) there writes: "The moral this suggests is that complete rigour is not even a defeasible epistemic ideal." Do you know what defeasible is? Commented Jul 18, 2023 at 12:29
• yes, i think so. i think he means to say that it is not even an epistemic ideal that can turn out to be wrong: it is just wrong @MikhailKatz
– user66760
Commented Jul 18, 2023 at 12:30