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I'm a mathematician/statistician, and I've been recently reading about epistemology and philosophy of science in my particular field of study.

In statistics, there is a deep concern for the objective validity of statements in science, meaning by this evaluating the adequacy of a statement that has a clear mathematical formulation with a precise way to measure the results and test/formulate clear hypothesis. This, however, does not apply necessarily for mathematics, especially when dealing with the purest fields (like abstract algebra/geometry).

As I understand, Münchausen's Trilemma applies only to empirical sciences. When dealing with the called ``dogmatic argument'', the foundational statements or axioms cannot be proved and thus the fundamental problem remains, because how can one be really sure that these foundational statements are true?

But as I perceive, this does not apply to mathematics: we don't actually need axioms to be true, whatever you mean by "true". Axioms are just axioms. You either accept them and do the theory or propose other axioms and do a different theory. This couldn't apply either for other pure things such as theology, since there is a justification for theological axioms (revealed by God, for instance).

Is my reasoning right?

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    Your exceptionalism of pure math regarding Münchausen's Trilemma is understandable and perhaps commonly held among formalists (if not Platonists). However, WP seems contrary to such position at the start; Agrippan trilemma, is a thought experiment intended to demonstrate the theoretical impossibility of proving any truth, even in the fields of logic and mathematics, without appealing to accepted assumptions. If it is asked... proof may be provided. Yet that same question can be asked of the proof, and any subsequent proof... Feb 6 at 4:13
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    "True" is a red herring, it can be replaced by "acceptable", "justifiable" etc. And you do not escape the trilemma in meta-theory, something has to be assumed to even claim that X is derivable from axioms, whether they are true or not. That something (logic, rule following habits, or whatever) then still stands in need of justification, or has to be taken as foundational.
    – Conifold
    Feb 6 at 6:36
  • In principle yes: if the proof is not circular it has to start somwhere and thus we can say that it is dogmatic. Feb 6 at 8:48
  • But this is true for every human knowledge: we have to start somewhere. Feb 6 at 8:49
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    Another point of view that gets around the statement "Axioms are just axioms" is that mathematics is the art of proving conditional statements, i.e. statements of the form "[hypotheses] imply [conclusions]", and axioms are just another word for very commonly used hypotheses. However, such proofs are carried out in some system of logic, so this point of view does not get around the logic issues brought up in the answer of @user21820.
    – Lee Mosher
    Feb 7 at 14:07

8 Answers 8

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Münchausen's trilemma is often presented as a kind of skeptical argument, purporting to show that knowledge, or demonstration, or certain belief, or something related to these, is impossible. But it may be more helpful simply to think of it as a fork in the road. If you start with the naive idea that beliefs require justification, and that such a justification takes the form of a belief, then you are left with three possibilities. 1. Your chain of justifying beliefs regresses to infinity. 2. Your beliefs are justified in a circular fashion. 3. Your naive idea must be adjusted to allow for exceptions, i.e. a collection of foundational beliefs that do not require justification. You can of course also take the more drastic step of rejecting the demand for justification entirely, which some philosophers do.

The same structure arises in other contexts. If you start with the idea that all events have a cause, and a cause itself is an event, then you reach the same kind of trilemma. Either there is an infinite regression of causes, or a circular pattern of causes, or a privileged collection of one or more uncaused events that are the exception. The same thing happens with word meanings. If all words have a definition, and the definition is given in words, then you would either need an infinite regression of definitions, or the definitions will be circular, or there will be some words that are foundational and don't require a definition.

To point out the trilemma does not serve as an argument that the concept of cause is defective, or that words cannot have definitions, it just says there is a three-way fork in the road here and we must make a choice or reject the assumptions that got us onto this road. In the case of causes, all three options have been defended. In the case of definitions, the circular option seems most plausible: if you look up a word in a dictionary and then look up the definiens you will find yourself going round in circles. It does not make the dictionary useless.

In the case of epistemology, there are plenty of defenders of the second and third options. The second option is called coherentism, and the third foundationalism. Coherentists hold that there is no non-circular justification of beliefs, so the most we can hope for is a consistent web of beliefs that agrees with our observations and experience. Foundationalists hold that some beliefs are irrefutable and indubitable. For rationalists, these might be fundamental principles of reason; for empiricists, these might take the form of some kind of phenomenalism.

Mathematics does not escape the trilemma. A formalist could say they are just playing a game of manipulating symbols, and that there is no justification for playing this game, but this extreme variety of formalism does not seem to do justice to the sheer usefulness of mathematics. Mathematical theories have interpretations under which their sentences are true or false. So, it makes sense to ask how we justify mathematical reasoning, and the same three options arise. The various different approaches to the philosophy of mathematics, viz, platonism, formalism, intuitionism, logicism, conventionalism, structuralism, constructivism, fictionalism, empiricism, etc., give different answers to this epistemological question.

Although mathematics makes use of logic, this does not resolve the problem, since it just invites us to ask in turn how we justify the logic. About once a year or so on this site we get someone asking whether it is possible to justify deductive logic, and while there is much that can be said, e.g. here and here, there is ultimately no non-circular justification.

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But as I perceive, this does not apply to mathematics: we don't actually need axioms to be true, whatever you mean by "true". Axioms are just axioms. You either accept them and do the theory or propose other axioms and do a different theory.

A pure formalist can try to claim this but it fails because one needs to assume at least some basic properties of finite binary strings (or equivalent) to even be able to meaningfully talk about even plain FOL (first-order logic), not to say more complicated axiomatic systems. I say more in this post.

So everyone (except those who disavow logical reasoning completely) must believe in the truth of some basic axioms about finite binary strings, such as TC (theory of concatenation) (given here). This is the bare minimum. But this opens the door to natural questions of whether other sentences over TC (with higher quantifier complexity) are also meaningful and whether they too have boolean truth values. In some sense, the answer to this question determines whether you are a strict finitist or not. If you doubt that anything beyond purely finitary things, then you may be stuck with just TC. But if you think that quantifying over all finite binary strings preserves boolean statements, then you get TC plus induction, which is equivalent to PA (Peano Arithmetic).

That's not all. There are some fundamental facts of logic, such as the semantic-completeness of FOL for countable theories and the syntactic incompleteness of formal systems that can interpret TC. The first one simply cannot be proven without some tiny bit of second-order arithmetic beyond PA. The second one can be argued to require second-order arithmetic in order to have genuine meaning, even though PA itself can prove a suitably encoded version. I say more about this here. Note that the logician Peter Smith also points out that anyone who accepts PA ought to also accept ACA.

ACA is actually very low in the general 'hierarchy' of foundational systems for modern mathematics. Yet it turns out that every known application of mathematics in the real world can be expressed as some sentence that is provable in ACA, and conversely every theorem of ACA appears to be true when appropriately interpreted about the real world (at least at human scales), so we actually have good empirical evidence for this part of mathematics, and such evidence is actually necessary to provide meaning and purpose to mathematics beyond just formal symbol-pushing!

In other words, it is untenable to say that mathematics is just as arbitrary as say the rules of chess. And so we do have to face the question of whether the foundational system we choose is in fact meaningful, which is why the above considerations are all important.

What about higher mathematics that deals with very abstract objects? Well, logicians who have been concerned with meaningfulness of foundations have generally agreed that predicativist concepts can reach roughly ATR0 (which is slightly beyond ACA but nowhere close to Z2). Since adding closure principles can reach roughly Π[1,1]-CA0, some might argue that Π[1,1]-CA0 is true too, but that's shaky ground. What is clear is that Z2 is impredicative, so one can legitimately doubt its meaningfulness even if one believes it is consistent.

On the other hand, the average mathematicians do not care about predicativism and usually say that they use ZFC. What most of them don't know is that ordinary mathematics very naturally stays within a very weak fragment of ZFC known as bounded ZFC, where roughly speaking you can have inbuilt function-symbols for pairing, power-set, union, and inbuilt constant ω, and can construct any set of the form { E : x∈S ∧ ... ∧ y∈T ∧ Q } where E is a term and Q is a formula with only bounded quantifiers. A bounded quantifier in set theory is of the form "∀x∈A" or "∃x∈A" where A is some term.

If we grant full power-sets to be meaningful, we get past Z2 since the impredicative quantification over the power-set of ℕ would not be an issue. But even then there does not seem to be any non-circular way to justify much more than bounded ZFC. To reach full ZFC requires believing in the meaningfulness of impredicative quantification over the entire (complete) set-theoretic universe, and that impredicativity also implies that we cannot use the iterative conception to justify full ZFC.

If you ask ordinary mathematicians whether they believe their theorems are true or just the final state of a symbol-pushing game, they will most likely tell you they believe their theorems are in some sense truths that they discovered. But as explained above, even that evidence from actual mathematical work only reaches bounded ZFC. Of course, modern logicians often do use the full impredicative power of ZFC, especially when studying ZFC itself, but not all of them believe that there is a 'true set-theoretic universe' (whatever that might mean).

So, no, you really cannot escape the question of meaning and truth no matter how low or high you look in the 'hierarchy' of foundations of mathematics!

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Mathematics, like everything we are able to think coherently, relies on logical reasoning, and logical reasoning relies on our logical intuitions, and most prominently among them the intuition that logical truths are, well, true, for example the modus ponens, the modus tollens, transposition, the hypothetical syllogism etc. Without these fundamental logical intuitions, and without the fact that all mathematicians have them, there would be no mathematics at all (although there would be no human being either). Thus, there is no fundamental difference between what people think of as empirical knowledge, and what is sometimes called "a priori sciences". The facts each science relies on may be sometimes different, but all sciences rely on facts, and our logical intuitions are facts.

Of course, it is proper to start from facts, including from the fact that logic is the only form of reasoning we know and all share but it is the case that mathematical reasoning can only be justified by the assumption that logic is the proper way to reason, an assumption which seems itself impossible to justify without resorting either to a circular argument, empirical evidence or the claim that we know that logical reasoning is valid and the only valid form of reasoning.

So, it is true that mathematicians don't actually need axioms to be true, but most people miss the fact that no theorem ever derive itself from axioms. Mathematics is done by human beings and require human beings to have a logical capability, something which in essence is no different from the need to have perception senses.

The notion that logic is an a priori science is a misunderstanding. Formal logic is an a priori science, and then only in the limited way that it does not absolutely require the logician to observe the material world because logicians have to be aware of their own logical intuitions, for example the intuition that the modus ponens is true. Thus, formal logic is an a priori science but logic is not because logic is a mental faculty and formal logic can only be done by subjectively experiencing logical intuitions. And this applies to mathematics as well.

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No, this is a popular misconception which arises from the study of axiomatic systems. The truth is more complex.

The primordial axiomatic system iscthat of Euclids. Its quite clear that here he begins from axioms that he considers indubitably to be true. These are what Descartes called "clear and distinct ideas".

The modern notion of an axiomatic system that has no relation to truth, but is merely self-consistent arose with Hilbert. This is now understood to be merely syntax or grammar and hence does not refer to meaning. And such a formal system must refer to some outside world if truth is to be established. This is what is studied in model theory which is the modern study of axiomatics systems qua axiomatic systems.

Thus no, a bare deductive system does not require truth, it merely requires consistency. However, it requires a model to establish truth. And for real axiomatic deductive systems it is always the case we look for and find models.

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  • I didn't think from historical perspective. In fact the Euclidean construction does assume that axioms in the Elements are "true". But it is much simpler to deal with mathematics in Hilbert's approach, as consistency does not have to do anything with truth. Feb 7 at 3:28
  • @YetAnotherUsr: Sure, but like I mentioned in the latter half of my answer, a bare deductive system isconly half the story and some relationship wirh a preexisting model iscrequired to establish its truth. A bare deductive system is not philosophically understood to incorporate deduction but is seen as grammar, and hence a form of syntax. Feb 7 at 13:52
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The presentation of the trilemma involves some presuppositions that undermine its cogency when stated explicitly. The 'obvious' error is that in presenting the trilemma, the presenter uses reasoning to 'show' why each horn of the trilemma is unsatisfactory. However, the deeper failure involved is in assuming that only one of the horns is satisfactory if any are at all.

We have gotten past this with things like Susan Haack's foundherentism or infinite coherentism, where we find two of the trilemma's horns merged. Personally, I see no moral obstacle (speaking of evidentialism as a sort of moral thesis, anyway) to all three positive solutions to the regress problem being admissible in their own way.

And so in mathematics, I think we actually find (a) the solution to the trilemma as classically presented and (b) an application of this solution to mathematics itself. (a) comes from glossing the regress problem in a sort of 'algebraic' fashion and then holding that the positive solutions all 'exist' in the kind of way that we 'know that the imaginary unit exists because it is the solution to a given equation.' (Skepticism ends up being the 'empty solution,' which is roughly how the issue is framed in justification logic.) And then as for (b), the application follows quickly from a strongly graph-theoretic vantage, but more intriguingly (to me) follows if we conceive of forms of sets in terms of the regress solutions. I.e. there is a structural similarity between foundationalism and well-founded sets, between coherentism and looping sets (e.g. Quine atoms), and between infinitism and infinite descending ∈-chains; so why not have a set theory with all three forms of sets, the existence of each form justified as an interpolant of the respective justification type? This would perhaps be tantamount to a triple-extension set theory (holding fast to elementhood as the essence of extensionality), and if even double-extension set theory is not known to be consistent, a fortiori neither is its immediate 'successor' in conceptual space. But perhaps the question of absolute or relative consistency proofs does not arise for triple-extension set theory in the same way that it arises for other set theories (though note that double-extension set theory was introduced as a 'way to get around' the problematique of the Russell set, which here adverts to a function from/to well-founded justification structures, after all).

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Trying to solve the curse of statistics is less of a noble objective than it seems. After "I say everything is unique in a fundamentally digital universe" it has its place and is usable as such. The one big mistake in dealing with statistics is the attempt to draw objectively valid statements from it. More today than in any other era in human history, mankind suffers under the misinterpretation of statistic results. It is the appreciation of uncertainty that gets lost along the way while people fearfully search for a notion of truth.

So is there absolute truth in mathematics? No, there isn't. One plus one does not equal two, it never did and it never will, but that doesn't make mathematics useless.

Does Münchhausens trilemma apply on math? Yes of course it does. You can drag it a dozen parallel universes down the road and it will still apply. The whole thing depends on the appreciation of the existence of some fantastic concept called infinity, which most anyone religiously believes exists, in spite of the fact nobody has ever seen it or been there. The instant you drown that idea, all options prove flawlessly compatible.

Your concern is absolutely valid, but it can only be satisfied by making sure the result of your work seeds a soil of firm understanding of (the) truth behind uncertainty. The power of being able to determine the exact extend to which you don't know anything for sure. And if you can't teach the swimmers how to surf, or the turtles how to fly, then at least help them respect it correctly, so they don't think they're safe from misconception by praying to the god of condom.

I recently saw the leaders of my country on national television speak to the assembly of the peoples representatives, condemning millions of perfectly innocent citizens , blaming them for the current mayhem with misinterpreted statistics as their main argument. They got applauded. It's creepy.

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The a priori, by definition, cannot be proven by any system of logic that is built upon it. So what, then, gives us the clarity of our own propositions and conclusions?

The answer is the history of the universe. This is shared by all beings within it. About this, there is a lot to explain, yet it is all known and has to deal with the evolution of YHVH (providing one ring of reasoning of the universe) and our own. From this, the identity axiom (A=A) came from an agreement between these two and algebra. The two parallel lines of the equal sign can be seen as representing this agreement between two equal parties of reason.

What I've just revealed to you comes from the Messianic prophesy of the Jews and you won't find any reference for it.

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Your question touches the interesting point: How does mathematics differ from science?

The short answer: General results from mathematics can be proved, general results from science can be confirmed or refuted – but not proved.

Concerning the three horns of the Muenchhausen trilemma:

• A circular argument disqualifies any mathematical reasoning.

• The regressive path is the only way to prove mathematical theorems: One has to reduce the claim by logical argumentation to the axioms and definitions.

• The dogmatic argument may be used as a heuristic, and it may be interesting from a historical point of view. But using dogmatic statements as argument for mathematical reasoning would destroy the mathematical reasoning.

The deeper reason that mathematics has no problem with Muenchhausen’s trilemma: Mathematics makes no general claim about the exterior world. Therefore mathematics is free to create its own concepts and axioms, like creating the rules of a new game.

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