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.

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. — causative, Jul 15 '23 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). — Kristian Berry, Jul 15 '23 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" https://philarchive.org/archive/BURPOW — J Kusin, Jul 15 '23 at 16:41
@infinitezero And above that entire linegraph is the philosopher, who looks down at them all, acutely aware of the lack of the inherit importance in their infinitesimal endeavors! ;) — RockPaperLz- Mask it or Casket, Jul 17 '23 at 09:05
@causative - The problem with that argument though is that maths is completely missing its foundation. Thus, every formalization fails eventually. — ScottishTapWater, Jul 17 '23 at 10:37
@infinitezero is mathematics a science (as we define science)? — RonJohn, Jul 17 '23 at 16:33
My brain tells me math is equally as rigorous as philosophy. My brain tells me wrong stuff 99% of the time though. Do you feel lucky, punk? — Agent Smith, Jul 17 '23 at 17:16
@ScottishTapWater: Yes, but it's a matter of *rigor*, not of ultimate foundations. Consider [Metamath's set.mm database](https://us.metamath.org/mpeuni/mmtheorems.html); from a few dozen axioms, they prove thousands of theorems, and it takes only a few seconds for a computer to validate the database's correctness. What similar edifice do philosophers have? — Corbin, Jul 18 '23 at 18:48
Can you consider anything rigourous if it's built on non-existent foundations? I'd argue that you can't — ScottishTapWater, Jul 18 '23 at 20:32
There’re many similar questions on this site for the general relations including rigor between math and philosophy you could search and review, in summary it could be said it hides in plainsight… — Double Knot, Jul 21 '23 at 02:18

niels nielsen · Answer 1 · 2023-07-17T15:58:23.023

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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.

edited Jul 17 '23 at 15:58

answered Jul 15 '23 at 16:05

niels nielsen

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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. – Kristian Berry Jul 15 '23 at 19:10
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@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 – DreamConspiracy Jul 16 '23 at 00:02
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@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. – Kristian Berry Jul 16 '23 at 01:24
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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. – J Kusin Jul 17 '23 at 14:52
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@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. – Feathercrown Jul 18 '23 at 20:48

score 8 · Accepted Answer · answered Jul 15 '23 at 17:16

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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.

answered Jul 15 '23 at 17:16

Ted Wrigley

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"*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. – Iñaki Viggers Jul 15 '23 at 23:19
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@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. – Ted Wrigley Jul 16 '23 at 00:29
Your admission that mathematics is a *formal system* implies that it is more rigorous than a system that is less formal. Both terms *formal* and *rigor* indicate strictness, whence trying to draw a distinction between method and practice is pointless. – Iñaki Viggers Jul 16 '23 at 23:51
@IñakiViggers: Uh... no. Again, sorry, but methodology is long-recognized as distinct from practice (in the same way that the rules of baseball are distinct from an actual baseball game). That's why colleges have courses dedicated to nothing but methodology in each distinct field. Your comments boil down to an assertion that the rules of rules of baseball are 'better' than the rules of football, regardless of how well they are applied in individual games. That seems nonsensical to me. – Ted Wrigley Jul 17 '23 at 00:35
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"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 Jul 17 '23 at 02:45
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In support of Ted Wrigley's answer: why can't the rigor of mathematics and the rigor of philosophy be as incommensurable as scientific theories are sometimes said to be? Is the concept of clarity itself so clear? If you look at mathematics, as a social and historical community, from the inside-out, you can see that mathematicians themselves will accuse themselves and each other of a "lack of sufficient rigor" all the time. One might even compare the analytic/continental distinction in philosophy to, say, the Platonist/intuitionist distinction in mathematics. – Kristian Berry Jul 17 '23 at 14:07
@Yakk: If you care to point out which part of mathematics is not a formal symbolic system, I'd be interested to know. – Ted Wrigley Jul 17 '23 at 20:02
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@TedWrigley one sheep two sheep three sheep. There are many many formal systems you can use to describe counting. Counting is not the formal system. – Yakk Jul 17 '23 at 22:10
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(The same is true of piles of other mathematics, including calculus.) – Yakk Jul 17 '23 at 22:12
@Yakk: Counting is the basis of all mathematics, sure. But the very first step of even the most basic maths is abstraction: abandoning concrete objects like sheep and goats and rocks to make a formal system of numbers. The point of mathematics is that you don't have to know *what* you're counting for maths to work. If counting sheep were different from counting goats, mathematics could not possibly have been formalized. – Ted Wrigley Jul 18 '23 at 05:51
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@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 Jul 18 '23 at 13:27
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Like, {} is in N, succ(x) = {x,{x}}, for all x in N then succ(x) in N is one set-theoretic way to (start to) formalize counting. But it is far from the only way! It is a cute way, because it appears to only require naive set theory to get going. Of course, by the time you get deep enough into it, you realize that naive set theory doesn't describe what you think it describes. You still have counting as a concept, however, and when you realize your formalization failed to capture what you call counting, you can either compensate or rework your approach. – Yakk Jul 18 '23 at 13:29
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@Yakk: of course we can have different ways of constructing theories in maths, but all of the constructions have to work with the same rules, axioms, and principles of symbolic manipulation. That's what 'formal' means: "having a conventionally recognized form, structure, or set of rules". And maths are (as far as I can see) *always* abstract. Can you imagine teams of mathematicians developing set-theoretical propositions by rushing out to a pasture and herding sheep into groups? – Ted Wrigley Jul 18 '23 at 15:27
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@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 Jul 18 '23 at 15:30
"*methodology is long-recognized as distinct from practice*". My point is that this distinction is unavailing for denying that mathematics is more rigorous than philosophy. One source of such gap of rigor is the difference of tightness of each field's language, regardless of the practitioner's adherence thereto. The contradiction in which you end up when dissociating "rigor" and "formal system" illustrates the point. Other than that, your argument about methodology also would lead to the conclusion mathematics might not necessarily be more rigorous than the "field" of hairdressing. – Iñaki Viggers Jul 21 '23 at 14:15
@Yakk: Mathematicians didn't *invent* the rules of symbolic manipulation; they *discovered* the rules of symbolic manipulation by investigating and abstracting the principles of numeration, which is based in the object-oriented nature of human cognition. The first time some ancient shepherd carved notches in a stick to represent each of his sheep (the abstract n of 'number' from group) the rules for addition, subtraction and etc were set. We can play with those rules creatively, sure, but that's a different order of activity. – Ted Wrigley Jul 22 '23 at 15:39
@IñakiViggers: Hairdressing is art (or at least artisan). It's governed by norm, custom, and creativity, not reason. And I think you are failing to reflect on the distinction between logic and reason. Logic is a systematic process — an application of pure methodology — and thus subject to question of rigor. Reason is much more than that: it uses logic, but transcends it Mathematical logic is what one learns in classes; mathematical reason is how one uses those logical tools. Maths logic is more defined than phils logic, but maths reason is far more limited than phils reason. apples/ oranges – Ted Wrigley Jul 22 '23 at 15:51
@TedWrigley Except, I personally know of multiple different symbolic manipulation systems that can all consistently describe counting sheep, but aren't otherwise identical. The rules for addition, subtraction etc are *different* in these systems, despite the fact that they give consistent results for all "real" sheep counting situations. I'm not sure what you mean by *invent* vs *discover* here, but I doubt I agree with your definitions. Also, counting is usually not a group. – Yakk Jul 23 '23 at 23:54
@Yakk: I didn't mean 'group' in the maths sense; I meant 'group as in "group of sheep". The point was about the abstract equivalence between a group of sheep and a group of notches on a stick. And however many systems you can find to describe counting, they all have to rest on the cognitive fact than *n* distinct, indivisible objects are always *n* in number. If one has exactly 5 sheep, one doesn't have 6, or 3, or any other number. Someone who claims otherwise hasn't grasped the fundamental nature of counting. – Ted Wrigley Jul 24 '23 at 03:21
@TedWrigley Yes, and yet the formal systems that all seem to fully describe counting sheep are distinct, and claim different things are true. Yet every testable prediction they make, agrees with the test. One formal system will state "a group of sheep of some fixed size has property X", and the other says "no". Someone who claims differently hasn't grasped the fundamental nature of using formal systems of symbols to describe counting. Mathematicians would love it if it wasn't true, but this is why mathematics *fundamentally cannot only be about* manipulation of formal systems of symbols. – Yakk Jul 24 '23 at 13:40
This isn't sophistry -- this is highlighting a problem sufficiently powerful formal systems where we can determine if arguments are valid have. All of them will describe both counting *and things which aren't quite counting*. We cannot describe counting in a powerful enough way in a formal system of symbol manipulation with sufficient accuracy to only describe counting, we "miss" and also describe stuff that isn't quite counting. When trying to get around this, you either end up with a system where *you provably can't know if an argument is valid*, or one that is weaker than we want. – Yakk Jul 24 '23 at 13:46
And the point of this is that *symbol manipulation* is a tool of how mathematicians learn things about counting. But if mathematicians only did symbol manipulation, we'd be blind to "actual counting", the thing we are wrapping up in abstraction to learn about, which is also part of what mathematics is about. And the same holds for things far beyond "1, 2, 3, 4" - it holds for every abstract topic mathematicians poke at. The framework is not the (only) thing mathematicians study. The symbolic framework *is the tool* used to study stuff. – Yakk Jul 24 '23 at 13:50

Mikhail Katz · Answer 3 · 2023-07-18T15:15:17.403

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.

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.
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.
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.

@doot_s: You can consult my publication list here: https://u.math.biu.ac.il/~katzmik/publications.html — Mikhail Katz, Jul 18 '23 at 12:35
@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. — Mikhail Katz, Jul 18 '23 at 12:50
no i mean your comment on my answer. i just don't understand why you seem to say that i have misread the abstract, and then ignored my reply. i guess some people are like that, and that's all — , Jul 18 '23 at 12:51
@doot_s, I am not familiar with the word *defeasible* and since you are quoting this article, I thought that perhaps you do. I haven't looked into this closely but it seems to be a neologism by the author (Paseau). If so, then I would say that using a neologism in an abstract may cast doubt on the *rigor* of this particular scholar. — Mikhail Katz, Jul 18 '23 at 12:57
oh ok. well that's interesting i guess. sorry for any confusion from me, and good luck in your work :) it's definitely not a neologism, philosophical or otherwise (i've seen it before in epistemology). it's not a very fluent word for me either! — , Jul 18 '23 at 12:58
Interesting data, I had no idea about. I quoted it in a [comment](https://philosophy.stackexchange.com/a/100846/37256) where there's some further discussion if you'd care to join. I appreciate the Herbart-Riemann connection. But don't get the Hilbert reference. Hilbert was hardly a philosopher so not sure what you're saying there.. — Rushi, Jul 20 '23 at 04:30
@Rushi, the demarcation line between philosophers and mathematicians is not as clear-cut as it might seem. Keep in mind that there was definitely a philosophical battle going on at the time. One side was Hilbert and Klein at Goettingen, and the other side was the old-timers at Berlin. The struggle was over the nature of mathematics, and as such, was not merely a technical mathematical one. — Mikhail Katz, Jul 20 '23 at 06:46
I suppose you could view it that way. You could also view it as eminent mathematicians fighting over turf. From my pov Hilbert was a better mathematician than Brouwer, whereas Brouwer was the better philosopher. Certainly "No one will drive us out of Cantors paradise" doesn't look like high grade philosophy (to me!) — Rushi, Jul 20 '23 at 07:00
@Rushi: the battle of Goettingen versus Berlin is separate from the Hilbert-Brouwer thing which came later. We are talking about Hilbert's speech at the Paris congress, which took place before Brouwer's conversion to what is today called Intuitionism. You may be familiar with Brouwer's fixed point theorem, which dates from his pre-intuitionistic period (and is in fact constructively not provable). — Mikhail Katz, Jul 20 '23 at 07:03
@Rushi: what you say about fighting over turf does not necessarily contradict the point that the Goettingen-Berlin battle was a philosophical battle over the nature of mathematics, with Berliners claiming that mathematics is analysis exclusively, and Hilbert, Klein, and others seeking to widen its scope to include all sorts of fields that are considered central to mathematics today. — Mikhail Katz, Jul 20 '23 at 07:05
Well again I'm sure you understand this area better than I do. I just feel the enormous philosophical significance of the Göttingen-Berlin war is 20-20 post-facto hindsight. Hilbert →Brouwer→ Entcheidungs problem→Gödel → Turing → 20th century computer science took a good ½ century to flesh out. My version of the [history](http://blog.languager.org/2015/03/cs-history-0.html) — Rushi, Jul 20 '23 at 07:14
I looked up the narrative there. It is not very accurate historically. At any rate, the fight over set theory is an issue separate from the question whether, for example, Lie theory is a legitimate part of mathematics. The Berliners denied this and at Goettingen they of course accepted it (and Klein helped Lie find a job). The point I have been trying to make is that the issue whether mathematics should be broadened beyond analysis cannot be established from axioms, because it is not a mathematical issue. It is in part a philosophical issue. @Rushi — Mikhail Katz, Jul 20 '23 at 08:53
Tnx @MikhailKatz. Would be much obliged if you could point out the inaccuracies, especially the most egregious ones [I am neither a mathematician nor historian but a computer scientist]. I guess thats where I get the view (or prejudice) that comp sci is privileged in being the constructive parts of math — Rushi, Jul 24 '23 at 02:08
@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. — Mikhail Katz, Jul 24 '23 at 09:27

score 1 · Answer 4 · 2023-07-18T13:03:37.613

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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

edited Jul 18 '23 at 13:03

answered Jul 18 '23 at 12:02

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? – Mikhail Katz Jul 18 '23 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 – Jul 18 '23 at 12:30

Which field is more rigorous, mathematics or philosophy?

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